Success Requires Action

This PhD was a long journey filled with difficulties overcame with support, motivational encouragements, and effort from beloved family and beloved fiancé.

I would like to share my gratitude to my director, co-director, and supervisors for this opportunity for obtaining my PhD degree in a respectful and a leading university in France (IMT Lille Douai) with the collaboration of my mother university in Lebanon (Lebanese International University LIU).

I would like to express my appreciation for my supervisor Dr. Charbel Habchi, that was my university instructor in LIU, then my Master thesis supervisor, finally my PhD Thesis supervisor. He believed in my work and passion for knowledge and that he always put his touches on enhancing my capabilities.

I would like to express my thankfulness for my supervisor Dr. Talib Dbouk, who always been beside me in every detail in the PhD journey. In addition, his dedicated time at every needed question who worked as much as me on this PhD with technical details, administrative tasks, traveling issues, and finally increasing my knowledge in many fields.

I would like to thank my supervisor Dr. Serge Russeil, who always give me the precise comments on my work that really affect the quality of the work, in addition explaining the procedure for doing experiments at IMT and of course his administrative work and following.

I would like to express my appreciation for my co-director Dr. Thierry Lemenand, who have been always a kind person with his words and for his scientific advice, knowledge sharing, and his time for discussion either in France or in his arrival to Lebanon. I would like to express my gratefulness to my director Prof. Daniel Bougeard, for his continues guidance and support in achieving PhD deliverable and the opportunity for selecting me to be a PhD student at IMT Lille Douai. Furthermore, thanks for all the staff that helps me with all the needed inquires in LIU and in IMT Lille Douai.

Lastly but not finally, a great respect and gratitude for my friends who were beside me in my PhD as Dr. Ahmad Al Takash, Dr. Ali Mohsen, Dr. Joseph Jabbour, Dr. Bineet Mehra, and Dr. Vignaesh Subramaniam.

Finally, I would express my sincere feelings and dedicate this work to my fiancé Ghena Madi, my parents Ali Karkaba, Eman Bou Salman and my supporting sister and brothers: Reem Karkaba, Hussain Karkaba, and Yehya Karkaba.

2.12 Different optimization methods used approaches [START_REF]Optimization Algorithms[END_REF]. . . . . . . . . . . . . . . . . . . Fluid and Thermal Engineering are the most interesting majors in the amount of the knowledge that can the engineer learn and discover. Thus, my choice for this major in mechanical engineering is based on the passion in lecturing and admiring to its physical science.

Previously, I've worked as a design engineer and dealt with several mechanical systems. In addition, teaching in different institutes as a mechanical and math instructor. Both enhance my field experience and teaching capabilities.

For the future, i will complete the research journey and am hopeful to continue work with my supervision team on other projects.

Formations

In the PhD study formations are planned to be attended by the doctoral school . In my contract i should take forty (40) credits that can be divided to be taken in the three years of the PhD study.

The formation that i have done are:

• EMR'2019 "Modeling and Control using Energetic Macroscopic Representation: Hybrid Electrical Vehicles and Other Applications" (20 credits): This was a summer school that has taken place in Lille (France) at University of Lille. It was a for days training. The first three days were divided into three main parts that are 11 hours of lectures on concepts and applications, eight hours of simulation training sessions, and four hours practical training. As for the final day (fourth day) i was focused on personal work and to present it to the international scientific committee.

• Deep Learning for 3D Humans (6 credits): The following points that are done in this course can be highlighted that are: Firstly the fundamentals of 3D data acquisition, and processing, point cloud and meshes; acquisition with depth cameras; from depth data to point clouds and skeleton; mesh triangulation, noise removal and smoothing. Secondly, classical methods for 3D shape and skeleton analysis. Thirdly deep learning for 3D data: basic concepts of deep learning; extending Convolutional Neural Network (CNN) to depth and 3D data. Finally generative methods for 3D data: classical models based on 3D Morphable Models (3DMM); Generative Adversarial Networks (GAN) methods for 3D data.

• Ethics & STICs: The course balances a pragmatic perspective, for conceptual developments aimed at opening up to the uncertainties and risks that are specific to every scientific domain.

This triggers my interest and guides me along my career, because the immense potential of Information Sciences and Technologies to transform our societies, come great responsibilities for the scientists who push them forward.

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• Advanced Concepts in Quality Improvement: This course covers foundations of modern methods of quality control and improvement that are used in the manufacturing and service industries.

It focuses on the statistical process control and related subjects, such as experimental design.

It enforces the integration of statistical concepts into quality assurance methods. Real business practice examples are used to illustrate the application of the studied concepts.

• Operational Research: Operations research (OR) has many applications in science, engineering, economics, and industry and thus the ability to solve OR problems is crucial for both researchers and practitioners. This course is designed to equip the student with the knowledge and techniques to formulate, analyze and solve mathematical models that represent real-world problems.

Being able to solve the real life problems and obtaining the right solution requires understanding and modeling the problem correctly and applying appropriate optimization tools and skills to solve the mathematical model.

• French language: I am learning French by taking online courses in order to communicate easily with my colleagues, And to ease my work in the future that i am planning to have in France if available.

General Introduction

Heat transfer enhancement for many engineering applications such as heat exchangers, electronic devices, solar heaters and other systems is crucial. Through inserting a vortex generator VG in heat exchangers, and secondary flow is added to the main flow to intensify the fluid exchange between the fluid domains leading to a heat transfer enhancement. Vortices can be generated in the longitudinal , and transversal direction in the flow depending on the shape and style of a VG. Thus, the choice of selecting a VG should be efficient due to its effect in the heat and mass transfer phenomena. In addition, various VG styles were studied over the past decades in different flow regimes (laminar and turbulent) and convection types (forced, mixed, and natural). In order to better quantify the best VG style to be used is each case study, optimization is introduced in order to obtain a novel design performed with a large space exploration designs.

This project includes a well detailed and organized report, by that the reader will have a full idea about the project. Thus, the report is organized as follows:

• Chapter 1 is a literature review that presents a background about different types of heat exchangers (HX) used in different industrial applications. In addition, an overview is added on heat transfer enhancement methods that are classified as passive or active techniques, and on heat transfer quantification methods that are used to examine heat enhancement that increase the overall performance of HX. Then, a review is presented about heat enhancement using different types of vortex generators installed in various types of heat exchangers with their results. A brief review is shown for optimization methods and building the strategies of optimization with their results for heat transfer applications.

• Chapter 2 deals with computational fluid dynamics methods and strategies that are used in the software cascaded computers. Governing equations are listed with all the numerical methods and strategies used to firstly define the geometry and physical properties of the design then how the dividing of the computational domain into fine cells (meshing) is shown along with the discretization methods of the governing equations over this mesh. Various mesh types that can be used are also presented, in addition to the solution methods with their accuracy methodologies.

Finally a brief background about turbulence modeling is presented with a detailed view of the used method in this project with its equations, in addition to a detailed section about the optimization algorithms.

• Chapter 3 concerns thermal enhancement using Vortex Generator (VG) using a multi-objective • Appendix A includes a brief review on the experimental methods with the done experimental results in this project.

• Appendix B includes a brief review on the finite volume method that deals with the discretization equations with the flux and sources terms integrated over surfaces and volumes.

Chapter 1

Literature Review Abstract

Heat exchanger (HX) is a main component in thermal systems that are used in large part of industrial and commercial applications. Due to its wide use, it grabs the attention of researchers in order to enhance their thermal performance and to be more efficient. Different methods of heat transfer enhancement can be used and they are classified into passive and active techniques, in addition to the variety of methods in heat quantification. Vortex generators is a passive method that enhance heat transfer with high percentages depending on their designs and configurations. Several optimization studies with various optimization algorithms are performed in order to quantify new sized or shaped designs which enhance the heat transfer. Heat is transferred through HX mainly by convection.

Convection can be classified into two main types: forced convection that the fluid is induced in order to pass through a HX, and free (natural) convection that is driven due to buoyancy forces. There is another type of convection known as mixed convection that is simply a mix between the forced and the free convection. Moreover, depending on the fluid velocity, two flow regimes are also studied in convection that are laminar and turbulent flow regimes.

Introduction

Heat exchangers (HX's) usage is wide in thermal and fluid domains, thus increasing its overall performance is a main goal for modern industry. Mainly the overall performance is increased using heat transfer enhancement. Heat transfer enhancement within HX is complex. Experimental measurements have a high cost both in time and budget; then using computational fluid dynamics simulation is a good option that can be utilized to study the complex phenomenon conducting heat transfer in HX [1][2].

In fact, there are different techniques to enhance heat that can be classified either passive or active; the active ones are by using vibration, electromagnetic field, and jet impingent and classified active because they require an external power. The passive ones are the most efficient and can be done using treated surfaces, rough surfaces, and vortex generators (VG's) [1,3]. In this study, different shapes of vortex generators (VG's) have been applied and studied to analyze the overall thermal performance in HX with various shapes of VG's by using Computational Fluid Dynamics (CFD) simulations.

Longitudinal Vortex Generators (LVG) have received attentions in the last decades due to their heat transfer enhancement properties by generating vortices that are a result of strong swirling secondary flows [3]. Thus studying different shapes of LVG is very interesting to obtain the best LVG design that deals with all the desired parameters to be maintained with. To do that, Star-CCM+ the CFD commercial software is used due to its feature to enforce optimization integrated modules.

Designing optimal shapes of vortex generators is an interest in the field of increasing the performance of heat exchangers in different ways. Thus to achieve that purpose, shape optimization of VG's is implemented that deals with a variation of geometric parameters for different convection types. The study is performed in steady and unsteady state conditions but depending whether the flow is laminar or turbulent.

Heat exchangers

Heat exchangers can be found in many applications such as heating, ventilation, air conditioning, and refrigeration (HVAC-R). Thus, due to its wide usage, increasing its performance has been a main objective for many researchers [2]. Heat transfer (HT) applications require different types of config- Compact heat exchanger is another type of HX's that is designed to realize a large heat transfer area per unit volume. The ratio of the heat transfer area to the volume of the heat exchanger is called density β. A heat exchanger with β > 700 m 2 m -3 is classified as a compact heat exchanger. This type is mostly used in the industry and can be found in car radiators, gas turbines, and stirling engine heat exchangers. Another advantage of compact heat exchanger is that it can achieve a high heat transfer rate between the fluids in a small volume configuration. These types are the most studied nowadays due to they reduced weight and volume. The large surface area in compact HX is obtained by attaching a spaced thin plate to the walls separating the two fluids. This type is mainly used in gas to gas and gas to liquid HX to increase the heat transfer coefficient (associated with gas flow with increased surface area). In compact HX's, the flow configuration is called a cross flow, where the two fluids move perpendicular to each other and they can be mixed or unmixed as shown in Figure 1.2.

urations
It is called mixed when the fluid is free to move in the transverse direction while it is unmixed if it is prevented to move in the transverse direction [2]. Regenerative HX's are commonly used in power plant applications, where there is a use of hot fluid. This hot fluid is stored in a thermal storage unit before its temperature is changed and becomes a cold fluid. The process is maintained by HT that occurs between hot and cold mediums which absorb heat. This arrangement is done by passing two fluids in separate parts that lead to this change of heat. But also in such designs the same fluid can be used to circulate in specific mediums. This type of HX take great importance in the technology development during the industrial revolution, also it is used in high pressure boilers and chemical reactors.

Water to air HX progress mainly is to transfer heat from water to air. They are implemented usually in outdoor boiler applications, such that the use of the air in the atmosphere in a certain thermal conditions is useful to transfer heat from water in the boilers. This type is commonly used in residential heating, air conditioning, dehumidification, hybrid systems, and all kinds of car radiators.

Chemical reactors HX use is a major trade in the field of HX's. The chemical reactions in the process industry are temperature dependent, thus removing heat generated by exothermic reaction is necessary to control the reaction temperature and to avoid thermal runway reactions or to suppress endothermic by-product reactions for instance [4]. Therefore the aim of thermal intensification is necessay to reduce the temperature; thus a use of HX reactors is obligatory. A comparison between different kinds of reactors is presented in Figure 1.6 that shows the major advantages of HX reactors in terms of thermal performance.

Heat transfer quantification in heat exchangers

Heat transfer quantification is obtained by computing parameters from the conservation equations in non-dimensional forms, that result in dimensionless numbers that are very useful for performing parametric studies of engineering problems. For incompressible viscous flow, the dimensionless parameters governing convection heat transfer are Grashof number Gr, Prandtl number P r, and Nusselt number N u [2,5,6]. Under different conditions (compressible flows, and porous flows) other types of dissipation terms may be included in the governing equations resulting in different non-dimensional groups. The non-dimensional forms used in this study are listed below with their definitions [7].

1. Reynolds number Re: is a measure of inertia forces to viscous momentum fluxes (equation 1.1).

If the viscous momentum fluxes and inertia forces are in the same direction, then the Reynolds number Re reveals the boundary layer characteristics of the flow. If the fluxes are defined such that the diffusion is in the cross stream direction, then Re conveys the flow regime (laminar, transitional, or turbulent).

Re = ρU D h µ (1.1)
where ρ is the density of the fluid, U is the flow velocity, D h is the hydraulic diameter of the studied configuration, and µ is dynamic viscosity of the fluid.

Grashof number

Gr = gβ∆T m L 3 ν 2 (1.2)
where g is the gravity, β = (1/T f ) is the thermal expansion coefficient where T f is the average fluid temperature, ∆T m = (T w -T ∞ ), T w is the wall temperature, T ∞ is the free stream temperature, L is the length of the plate, and ν is the kinematic viscosity.

3. The Prandtl number P r: is defined as the ratio of momentum diffusivity (kinematic viscosity ν)

to thermal diffusivity α (equation 1.3). The Prandtl number represents the ratio of hydrodynamic boundary layer to thermal boundary layer.

P r = µC p k = ν α (1.3)
4. Nusselt number N u: is the dimensionless form of the convection heat transfer coefficient h and provides a measure of the convection heat transfer at a solid surface. The Nusselt number does not arise as a dimensionless group when writing the conservation equations in non-dimensional forms; rather, it is widely used to report convection heat transfer data. The global Nusselt number is defined in equation 1. 4.

N u = hD h k (1.4)
where h is the heat transfer coefficient defined in equation 1.5 , and k is the thermal conductivity of the fluid.

h = q A(T w -T out ) (1.5) with q = ṁC p ∆T (1.6) and ∆T = (T w -T in ) -(T w -T out ) ln((T w -T in )/(T w -T out )) (1.7)
Writing Nusselt number in local format and substituting the value of h reveals to equation 1.8.

N u(x) = q x D h k(T w -T b (x)) (1.8)
where q x is the wall heat flux and T b (x) is the bulk temperature defined on a cross section area

A by:

T b (x) = A U (x)T (x, y, z)dA A U (x, y, z)dA (1.9)
where U (x) and T (x, y, z) are the velocity magnitude and temperature at a specific location (x, y, z) in the heat exchanger [7].

Thermal enhancement factor

Thermal enhancement factor (ζ) is the ratio of Nusselt number times the ratio of the friction factor.

This value is defined to evaluate the thermal efficiency at constant pumping power. ζ is defined as:

ζ = N u N u 0 f f 0 -1/3 (1.10)
where the subscript 0 refers to the results of empty channel. The thermal performance of the channel is increased once the value of ζ is greater than the unity. For more details about the derivation of the thermal enhancement factor formula refer to paper [?]. The friction factor f is computed in order to account the pressure drop in a channel to calculate the pumping power needed and is defined in equation 1.11 [8].

f = (P in -P out ) ρU 2 H L (1. 11
)
where P is the pressure at the inlet and outlet of the channel and H is the height of the channel.

Many studies were performed to show the convective heat enhancement by computing the thermal enhancement factor (ζ) known in some studies as η, TEF, or

N u N u 0 / f f 0 1/3 .
Habchi et al. [9] showed the variation of η while using a vortical structures that are generated by inclined baffles in a turbulent pipe flow while using three different configurations as shown in figure 1.7. It is clearly shown in figure 1.8 that the heat is enhanced since the value of η is greater than 1 in all configurations at different Re.

Guo et al. [10] showed also the effect of using different swirl flow structures to enhance heat by

computing the value of ζ = N u N u 0 / f f 0 1/3
as shown in figure 1.9. It is concluded that when using any type of swirling device the heat is enhanced due to the increase in thermal enhancement factor. As shown ζ remains increasing as Re increases until reaching a peak value and then it starts to decrease gradually. But also at high Re heat remains enhanced.

Oneissi et al [11] investigated a new design of VG called inclined projected winglet pair IPWP. where TEF for IPWP is around 1.3 higher than that of DWP, TEF around 1.26. Then it starts to decrease as increasing Re but always with a higher value of TEF for IPWP than DWP.

Khanjian et al. [12] also studied the effect of using VG in a flat plate channel on the heat transfer enhancement. Rectangular shape VG is used in the study and the angle of attack (α) is varied. To examine the results, the thermal enhancement factor η is computed and shown in figure 1.11. It is clearly shown that as the number of attack (α) increases at Re = 456, the value of the thermal enhancement factor (η) will increase until reaching an angle α = 25 • , then it starts decreasing till reaching α = 30 • .

Colburn factor

Another factor allowing to compare the energy efficiency of the HX is the Colburn factor j defined in equation 1.12. Many researches use the Colburn factor and friction factor versus the Reynolds number analogy to evaluate the enhancement performance of compact surfaces. Colburn factor quantifies the ratio of the thermal power transferred to the mechanical power consumed. For a given fluid, and a specific Reynolds number, a higher Colburn factor means a higher efficiency of heat transfer. It is considered as a frequent tool for presenting performance results for heat transfer surface geometries [8,13]. (1/3) (1.12)

j = N u ReP r
Lei et al. [14] studied the hydrodynamics and heat transfer characteristics of a novel heat exchanger with delta winglet vortex generators at different Re and by varying the angle of attack (α)

and the aspect ratio that means the ratio of the winglet parameters. The hydrodynamic result is studied by computing the Colburn factor, the results are shown in figure 1.12. It is clearly shown that as the angle of attack (α) increase the Colburn factor increase at a constant Reynolds number (Re).

In addition as (Re) increase from 600 to 2700 always the Colburn factor is higher for a higher angle of attack.

Field synergy principle

Field synergy principle is a concept to enhance convective heat transfer, simply due to the reduction of the angle between the velocity and temperature gradient. This can briefly be demonstrated starting from the energy equation [10]:

ρC p (U.∇T ) = k∇ 2 T (1.13)
where U and T are respectively the local velocity field and fluid temperature distribution in Cartesian coordinates. By integrating over the whole domain of interest using Gaussian integration and neglecting the axial conduction since the Péclet number is greater than 100 in most engineering applications, the energy equation becomes:

Figure 1.12: Colburn factor results for different angle of attack configurations at different Reynolds number [14].

ρC p V (U.∇T )dxdydz = q w (1.14)
But the total heat flux q w is proportional to the Nusselt number N u. And it can be in non dimensional form as:

N u x = Re x P r 1 0 (U.∇T )dy (1.15)
Thus it is clearly stated that any increase in the dot product of U and ∇T will increase the Nusselt number thus enhance heat transfer. The dot product can be expressed as:

U.∇T = ||U || × ||T || × cos(β) (1.16)
where β is the angle between the velocity vector and the temperature gradient (heat flow vector). Equation 1. 16 shows that in the convection domain there are two vector fields, U and ∇T , or three scalar fields, |U |, |T | and cos(β). Hence, the value of the integration or the strength of the convection heat transfer depends not only on the velocity, the temperature gradient, but also on their synergy.

Thus the principle of field synergy for the enhancement of convective heat transfer is satisfied. The better the synergy of velocity and temperature gradient/heat flow fields, the higher the convective heat transfer rate under the same other conditions. The synergy principle is satisfied when verifying the following conditions that all correspond to the same concept [10]:

• The angle β between velocity and temperature gradient should be as small as possible that means the velocity and the temperature gradient should be as parallel as possible.

• The local values of the three scalar fields should all be simultaneously large, that means, larger values of cos(β) should correspond to larger values of the velocity and the temperature gradient.

• The velocity and temperature profiles at each cross section should be as uniform as possible.

Better synergy among such three scalar fields will lead to a larger value of the Nusselt number.

In order to validate the field synergy, global value of β should be calculated since from equation 1.17, the angle computed is local due to the local velocity and temperature fields, Thus the following expression can be used to compute the desired value:

β = cos -1 - → u .∇T | - → u ||∇T | ∆V V (1.17)
Guo et al. [10] studied the heat transfer enhancement in a channel using different configurations of the plates. The first one is by using the discrete parallel plates and the other is the discrete staggered plates as shown in figure 1.13. The heat enhancement is analyzed and visualized by computing the intersection angle between the velocity and temperature fields β as shown in figure 1.14. It is observed that at the same value of Reynolds number, the heat enhancement for the stagger discrete plates is higher than that of parallel discrete plates depending on the lower value of intersection angle β. Same behavior is observed as the value of Re is increased. Lei et al. [14] studied the heat transfer enhancement as stated in the previous section at different aspect ratios. The results are obtained and shown in figure 1.16. It is clearly shown that a higher aspect ratio the value of the intersection angle is lower thus an enhanced heat transfer for a higher Reynolds number.

Habchi et al. [9] also, as stated in the previous section, observed the variation of the intersection angle between velocity vector field and temperature vector field. The results are shown in figure 1.17.

It can be seen from Figure 1.17-a and 1.17-b that the synergy is negative in the region of the common outflow in the aligned and alternating arrays while it is the opposite in the common outflow region of the reversed arrays as shown in Figure 1.17-c.

In this section, heat transfer quantification parameters for heat exchangers are presented. The main three parameters are the thermal enhancement factor, the Colburn factor, and the field synergy principle. Each quantification parameter is presented with a brief bibliography on its application. The most used parameter, and the one that will be used in this contribution is the thermal enhancement factor.

Heat transfer enhancement methods

Heat exchangers performance improvement grabs researchers to dedicate their studies, to achieve this goal. Due to the wide range of HX's usage in several field applications, they have different basic flow geometries that can attain various types of heat enhancement methods. Liquid and gas as working fluid can be used in HX's. Each is chosen according to the specific application obligations. Both of the fluids have limitation in their thermal performance. Thus, different heat transfer enhancement methods were introduced.

Heat transfer enhancement using passive and active techniques

Heat transfer enhancement is essential in designing any type of HX. Different methods of heat enhancement are employed and they can be classified as passive and active [16].

1. Passive techniques: These types of techniques are sufficient, since they don't need any external power. It employs special geometries and fluid additives for enhancement purposes. Listing some passive techniques with their figures is as follows [16]:

Note that all the figures presented are reproduced to be clearer. for enhancing condensation, where up to 60% improvements in heat transfer were recorded, at a constant pressure drop penalty [17]. (g) Vortex generators: One of the effective ways to increase the heat transfer coefficient is to decrease the thermal resistance of the sublayer adjacent to the wall immediately where the viscous effects of the sublayer are dominant. This can be achieved by increasing the turbulence of the fluid flow in the main stream so that the quasi-streamwise wall eddies will be closer to the wall and so more efficient. The thermal resistance can also be reduced and heat transfer can be augmented by using vortex generators (VG's) as shown in Figure 1.24 that allow disrupting the growth of the boundary layer and also make fluid swirling. This swirl effect causes exchange between the core and wall fluid, leading to the enhancement of heat transfer between the flowing fluid and the channel walls [8,14,9,11,12,15]. (h) Fluid additives: These additives can be classified either for gases or liquids. For the liquid fluid, additives can be solid particles or gas bubbles. While the additives for gas fluid are liquid droplets or solid particles [16].

2. Active techniques: These types of techniques are for special applications because they are not sufficient in energy saving since they require external power. Listing some active techniques are as follows [16]:

Note that all the figures presented are reproduced to be clearer.

(a) Surface vibration: This occurs at low or high frequencies that improve a single phase heat transfer. For example, a piezoelectric device may be used to vibrate a surface and impinge small droplets onto a heated surface to promote spray cooling as shown in Figure 1.25 [16]. (d) Injection: Injection is utilized by supplying gas through a porous heat transfer surface to a flow of liquid or by injecting the same liquid upstream of the heat transfer section. The injected gas intensifies single phase flow heat transfer and widely used in automotive cooling systems [22]. Figure 1.28a shows the injection of small bubbles inside the heat exchanger due to the air flow exit from tiny holes.

(e) Mechanical aids: They involves stirring the fluid by mechanical rotating machines that induce a secondary flow that enhance heat transfer but with an increase in pressure drop as shown in Figure 1.29 [16]. This category includes rotating tube and scraped surface heat exchangers. They are widely used for viscous liquids in the chemical process industry. Many applications benefit from such techniques such as transpiration cooling, controlling boundary layers over airplane wings surfaces, turbo machinery, lubrication of permeable Figure 1.30: Electrostatic field device [16].

bearing and many others [23]. Figure 1.31 shows an isometric view of a permeable tube that is used for suction applications.

Heat transfer enhancement using vortex generators

Heat exchangers can be classified as gas-to-gas or gas-to-liquid HX. In this study, the interest will be with the gas side to be analyzed because of its low heat transfer coefficient. This coefficient can be increased by enhancing the convective heat transfer part in the HX by using an induced vortices that increase the convection surface area upon applying "Vortex Generators (VG)" [3].

VG's have four basic configurations that are delta wings (DW), rectangular wings (RW), deltawinglet pair (DWP), and rectangular winglet pair (RWP). When the trailing edge is attached to the plate, the VG is a wing, when the chord length is attached to the plate, it is called a winglet as shown in Figure 1.32 [3]. New boundary layers will be generated around the surface of the VG, vortices will be generated as a result of the VG and flow will be less stable due to the reversed flows created because of the VG. LV are 3D swirling motion and persist over a long distance downstream the VG in the flow direction. Figure 1.33 shows various types of vortices with their direction.

The early studies of vortex generators treated Longitudinal Vortex Generator (LVG) that focuses on straight channel applications where the flow structure is simple and can be produced easily through experiments. The heat transfer enhancement in flat plate channels by using LVG's is an interest in present study. Dupont et al. [26] studied experimentally isothermal flow in compact heat exchangers channel with embossed type vortex generator. Gentry and Jacobi [27,28] carried out experiments of delta winglets VG (figure 1.34) in a channel showing an increase in the global heat transfer coefficient by 55%, but associated with an increase in pressure drop by 100% at low Reynolds number. Wang et al. [29] experiments showed an increase in the overall heat transfer by 10-45% due to LVG.

Moreover, different types of VG studies are also presented in the literature such as the helical and twisted VG [30,31], dimples or protrusions [32,33], cylindrical tubes [34], transverse vortex generators (TVG) [35], plane or curved surface of VG [36,37]. By that, variations in types of VG and for the operational conditions in the channel and the geometric configuration make the analysis of heat enhancement so disperse thus a difficult task.

Vortex generators can be implemented in different configurations, common flow down (CFD) and common flow up (CFU). The configuration is known as CFD when the transverse distance between leading edges is less than that of trailing edges and vice-versa for CFU (figures 1.35a, 1.35c) . Common flow down vortices create down-wash in between, and up-wash flow in the outside regions. Along the downstream direction, the vortices velocity vectors decrease while the distance between vortex cores increases which leads to the thinning of the thermal boundary layer (figures 1.35b, 1.35d) [25]. figure 1.41, on the heat transfer enhancement in a laminar channel flow. In this study, it was obvious that when increasing the roll angle value, not only the heat transfer will be enhanced but also the pressure drop increases. Thus, the enhancement factor η is used to compare the different designs.

After computing η with different designs on two Reynolds number this conclusion was obtained and shown in figure 1.42:

1. At Re = 456 the maximum enhancement factor of 1.2 is reached at β = 90%.

2. At Re = 911 the maximum enhancement factor of 1.32 is reached at β = 70%.

Zhang and He [3] studied hydrodynamics and heat transfer characteristics of a novel heat exchanger with two rows of tubes and delta winglet vortex generator. Firstly, effect of delta winglet on flow and heat transfer is studied. A vortex generator with common flow up arrangement is employed.

To examine the heat enhancement, exit temperature contours are plotted and shown in figure 1.43. It is observed that the temperature is higher in the case of vortex generator than the plain fin heat exchanger. Figure 1.44 shows the surface temperature distribution for plain fin case and for the induced VG's at Re = 1000. It can be seen that the temperature gradient in the region behind the tube for the case with VG is higher than in the case of plain fin.

Secondly, effect of configuration of tubes (inline and staggered) with and without VG on the performance of fin and tube heat exchanger is presented. Ratio of Colburn factor and drag coefficient (j/f ) for different tube arrangement is shown in figure 1.45. For both inline and staggered arrangements, the overall performances for the cases with delta winglets are better than that of the plain fins, which indicate that the enhancement of heat transfer is more significant than increase of the drag coefficient. It should be noted that as Reynolds number increases, increase of j/f due to delta winglet becomes less significant since increase of drag coefficient at higher Reynolds number is more significant than the case of low Reynolds number. Therefore, it can be concluded that the delta winglet is more effective on enhancing heat transfer at low Reynolds number.

Lastly, effect of locations of LVG's on the overall flow and heat transfer is studied. The two It can be seen that the Nusselt numbers for both cases increase with increasing Reynolds number. In the ranges of Reynolds numbers studied, the heat transfer performance for the downstream case is better than that of the upstream case. At the end of this section, a variety of heat transfer enhancement methods are presented that are specified as active and passive technique. And after performing a literature review about these techniques. It was found that the passive techniques are the commonly used techniques due to its higher efficiency than that of active techniques which use electrical energy in order to be driven. In addition, a detailed literature is performed on the vortex generator type of the passive heat transfer enhancement methods that is selected. VG usage is widely analyzed through researchers due to its variety of insertion techniques into the channel and its different effects on the flow. The most recent publications concerning VG show that the geometric parameters have a very significant thermohydraulic influence on the creation of longitudinal vortices and on the intensification of the exchanges which results from it. Thus, a more systematic search for all the geometric parameters that have a real impact on performance would be relevant. Parametric optimization techniques with a large number of parameters are techniques particularly suited to finding the best VG configuration for an application and a given specification.

Optimization 1.5.1 Optimization Methods

Optimization definition is the act of obtaining the best result under given circumstances. Optimization is done by building different models from statistical or experimental results to obtain an accurate representation of the physical phenomenon results. Optimization is a mathematical problem that can be expressed as following [38]:

Σ =        x 1 x 2 . . . x n        (1.18) f j (X); j = 1, 2, 3, • • • , m g j (X) ≤ 0; j = 1, 2, 3, • • • , p l j (X) = 0; j = 1, 2, 3, • • • , q (1.19)
where X is a n-dimensional vector called the design vector that minimize, maximize, or zeroed an objective (cost) function f j (X) that can be subjected to defined linear and non-linear constraints, g j (X) l j (X) and are known as inequality and equality constraints respectively, where m, p, and q are positive real numbers. The objective function may include many local minima and local maxima that may be a problem in optimization problems that in some simple basic method the local minima can be considered as the optimum design and stop the optimization study. To start with a study, a constrained nodal domain defining the degrees of freedom clearly must be analyzed with a specified number of objective functions (cost functions). As the number of objective functions increase the optimization will be harder especially if the functions are conflicting due to maximizing one and minimizing another. Such functions will lead to a complexity in identifying a set of parameters that will be best for the objective functions. A well known approach known as Pareto Optimal Frontier (POF) or Pareto Front (PF) will be used. The approach is a trade off curve that contains all the solutions that are considered non-inferior, non-dominated efficient solution. This curve will assemble all the optimal solutions in the sense of Pareto in order to examine the best solution range. Figure 2.10 shows a schematic representation of the concept of the POF [38,39]. In order to obtain the POF curve a user should iterate on a number of the design parameters hoping to find a good designs. For this desire an automated numerical optimization algorithms are adapted to yield a better design in much less time. These algorithms will work on known tools of analysis that evaluate the solutions, they are classified as follows [38].

• Robustness: Robustness is the capability of reaching a global optimum point without being stuck in local extrema, or blocked for lack of useful data. This is the most important feature in measuring the efficiency of an optimization technique. The algorithm is considered robust, if the chance to reach the global optimum is higher or getting close to it [38].

• Accuracy: Accuracy is the ability of an algorithm to reach the actual extrema, either global or local, around its proximity. Usually, accuracy and robustness are conflicting attributes, so that robust algorithms are not accurate and vice-verse [38].

• Convergence rate: Convergence rate is a measure of the effort an algorithm has to carry on to reach its goal. Again, robust algorithms are usually slow. However, fast but not robust ones might not reach the goal at all.

One of the keys to have a successful optimization study is the effectiveness of the search algorithm based on the previous defined tools.

Different algorithms were investigated in the last decade. These algorithms will defer according to the approximation models used to fit cost functions. Below are the most used algorithms:

1. Genetic Algorithm (GA): Genetic Algorithm is a numerical based on Darwin's theory of survival of the fittest. It is the evolutionary search method that performs global exploration of the design space while searching for an optimal solution based on a genetic material method. Genetic material is subsequently exchanged between them (crossover) and altered within the offspring (mutation). Then followed by an evaluation of the new individual obtained. This process will create N individuals of a next generation. And this will be repeated for t generations and the best individual (design) of the last generation is considered optimum [40,41]. 3. Linear Programming (LP): Linear programming is an optimization method applicable for the solution of problems in which the objective function and the constraints appear as linear functions of the decision variables. The constraint equations in a linear programming problem may be in the form of equalities or inequalities. Linear programming is considered a revolutionary development that permits the user to make optimal decisions in complex situations. LP can be solved basically using simplex method, but this method requires a very large storage computers.

Thus an improvement methods for LP were proposed to enhance the work of LP. Those methods are the revised simplex method, duality theory, decomposition method, postoptimality analysis, and Karmarkar's method [39].

Non-Linear Programming (NLP):

The optimization problem that involves the objective function and/or constraints that are not stated as explicit functions of the design variables or which are too complicated to manipulate, cannot be solved using the classical analytical methods of LP.

In the numerical methods of optimization, an opposite procedure is followed in which the values of the objective function are first found at various combinations of the decision variables and conclusions are then drawn regarding the optimal solution. The quadratic and cubic interpolation methods involve polynomial approximations to the given function. The direct root methods are root finding methods that can be considered to be equivalent to quadratic interpolation. In most practical problems, the optimum solution is known to lie within restricted ranges of the design variables. In some cases this range is not known, and hence the search has to be made with no restrictions on the values of the variables. The sequential quadratic programming is one of the most recently developed and perhaps one of the best methods of optimization. The method has a theoretical basis that is related firstly to the solution of a set of nonlinear equations using Newton's method, and secondly the derivation of simultaneous nonlinear equations using Kuhn-Tucker conditions to the Lagrangian of the constrained optimization problem [39].

Response Surface Method (RSM):

Response surface methodology is a well known and developed optimization method that is a collection of mathematical and statistical techniques based on the fit of a polynomial equation to the experimental data, which must describe the behavior of a data set with the objective of making statistical previsions. It can be well applied when a response or a set of responses of interest are influenced by several variables. The objective is to simultaneously optimize the levels of these variables to attain the best system performance.

Before applying the RSM methodology, it is first necessary to choose an experimental design that will define which experiments should be carried out in the experimental region being studied.

To approximate a response function to experimental data that cannot be described by linear functions, experimental designs for quadratic response surfaces should be used, such as three level factorial, Box-Behnken, central composite, and Doehlert designs. One of the methods used to reduce the burden of computationally expensive simulations to optimize a design, numerical approximation models called surrogate models using the response surface method [42].

6. Simultaneous Hybrid Exploration that is Robust, Progressive and Adaptive (SHERPA): SHERPA is a hyper-parameter optimization method that includes a variety of powerful and interchangeable algorithms [43]. SHERPA algorithm is considered simultaneous hybrid exploration that is robust, progressive and adaptive. It uses multi-search methods simultaneously that take the advantage of the best attributes of each method, and reduce a method's participation in the search if/when it is determined to be ineffective [41]. Thus it allows to implement stopping rules of under-performing trails that have a lower performance based on the Median-Stopping-Rule [44]. The advantages of using SHERPA is that, it is adaptive with the studied field type; thus it can automatically employ the appropriate algorithms as it proceeds toward the optimal solution.

In addition, it makes the decisions about which method to use and tune them. Finally, due to its robustness and effectiveness in finding the optimal solution, the user can develop a huge study border with a large number of variables [41]].

Heat transfer enhancement optimization

In design, construction, and maintenance of any engineering system, engineers have to take many technological and managerial decisions at several stages. The ultimate goal of such decisions is either to minimize the effort required or to maximize the desired benefit. Since the effort required or the benefit desired in any practical situation can be expressed as a function of certain decision variables, optimization can be defined as the process of finding the conditions that give the maximum or minimum value of a function.

Convective heat transfer enhancement optimization studies grabs many researchers to do studies on this domain and it can be implemented in different industrial fields of heat exchangers applications.

Convective heat transfer in heat exchangers improvement can be achieved by generating vortices using VG's that may be inserted according to various types and shapes in the channels. Different shapes of VG's have been studied over the last decade, with many studies continuing in this purpose. Parametric and shape optimization techniques are applied in the literature optimizing for VG's. The variation of several geometric parameters were conducted in order to achieve optimum designs.

Kim and Choi [45] performed shape optimization of rectangular channel with dimples by applying a response surface method. Samad et al. [46] optimized the shape of channel dimples on a single surface using different surrogate models. They improved heat transfer and friction-loss by approximately 20%.

Khan et al. [47] studied a vortex generator optimization in a plate-fin channel. The study is done by varying Re and shape parameters like the angle of attach roll angle. The data obtained from numerical simulations are used by Bayesian-regularized artificial neural network (BRANN), and then used to run the multi-objective optimization algorithm (MOA), obtained using response surface methodology (RSM) [47]. In this study [48], a kriging surrogate model is developed to approximate the objective function that is the thermal resistance as function of design variables. Surface plotting of surrogate model were used to obtain the optimization model. At the end of this section, optimization was defined with the most known tools that are used to evaluate it. In addition, different optimization algorithms are presented. The Simultaneous Hybrid Exploration that is Robust, Progressive, and Adaptive (SHERPA) is selected due to its advantages in seeking for an optimum solution in a robust way, along with its advanced techniques used in seeking for optimum designs. And thus selection is based after performing a literature review on heat transfer enhancement by VG optimization in heat exchanger that shows the different techniques that are used by researchers in order to seek for an optimal solution. In addition, SHERPA method is used in this contribution due to its multi-search powerful technique in progressing for optimal designs under a large exploration designs and variety of choose the type of convection and flow regimes. In this contribution, the large number of variable parameters in the optimization process has an influence on the performance of the design. Thus in the continuity of the scientific bibliography, it seems relevant to propose an optimization using SHERPA with a large number of parameters in order to define the best performance of intensification by VG.

Passive intensification and hydraulic regime and type of convection

Convection is the heat transfer that occurs due to a fluid motion. There are two main types of convection, forced convection and free (natural convection). Another third type is called mixed convection that is simply a mix between forced and free convection. The type of convection is said to be forced, if the fluid motion is supplied by fan or pump. On the other hand, convection is said to be natural if fluid motion is naturally established due to a temperature difference that causes density variations which induce buoyancy forces to drive the flow [2].

Convection regimes can be fully studied in CFD software. In each situation, computational domain is modeled with different fluid and heat fields. For example, in the forced convection type the density can be modeled with an ideal gas model due to an external forced flow. While in the natural convection type, as mentioned previously, the flow is induced due to buoyancy forces, thus density variations should be taken into consideration with the gravity effect on the flow. There are different models to study this type of convection, for example if studying a case with a small variations in the field temperature, a Boussinesq approximation can be used to compute the density change in the field based on the flow temperature difference in the domain [50,51,52,53,54,55,56,[START_REF] Henniche | Enhanced Mixed Convection in a Vertical Channel by Addition of Staggered Inclined Baffles[END_REF]]. In such model, it is easier to write conservation equations by considering the simplifications and flow conditions. This procedure is done by defining the conservation equations in non dimensional form.

The non dimensional parameters that are used are the Grashof number Gr, and the Prandtl number P r [7].

Numerical solutions were established in order to examine if the type of the convection is considered forced, free (natural), or mixed. Metais and Eckert [START_REF] Metais | Forced, Mixed, and Free Convection Regimes[END_REF] defined the convection regimes in laminar, transitional, and turbulent fluid flow in a horizontal tubes. Figure 1.55 shows all the regimes with the fluid flow conditions. The x-axis in the chart is dimensional number that is a combination between Gr, P r, and the dimensionless ratio of the tube. As seen as the ratio of the number increases, the convection type is changing from a forced to free passing through the mixed regime. This value is directly affected by the dimensions of the channel, the fluid properties, and the thermal flow conditions. Thus different convection regimes can be established by various conditions that can be adopted in each study. On the y-axis Re is plotted in order to distinguish the flow condition, either laminar or turbulent flow.

Figure 1.55: Convection regimes in a horizontal tube [START_REF] Metais | Forced, Mixed, and Free Convection Regimes[END_REF].

Passive techniques are the mostly used methods for heat transfer enhancement [16]. Although, heat intensification by inserting a vortex generator is the most known one in the passive techniques.

Several studies are performed in the forced convection with a restricted values Reynolds number, thus a restricted flow regimes [12,15]. In turbulent flow regime, different turbulent models are used in performing simulations, especially the, SST k -, RNG k -, and the SST k -ω [11,[START_REF] Behfard | Numerical investigation for finding the appropriate design parameters of a fin-and-tube heat exchanger with delta winglet vortex generators[END_REF][START_REF] Zhao | Numerical investigation of heat transfer and erosion characteristics for H-type finned oval-tube with longitudinal vortex generators and dimples[END_REF] that will be settlet for the simulations in this contribution.

In addition, mixed convection under both flow regimes (laminar and turbulent) is a hard topic to be studied, where rare studies are performed for heat intensification in such domain due to its complexity. Rectangular channels type of heat exchangers that is usually found in solar collectors are only studied when exhibiting a forced convection type [START_REF] Chai | A Review of Air-side Heat Transfer Augmentation with Vortex Generators on Heat Transfer Surface[END_REF]. Thus, it would be an interesting study if the research is expanded on studying this type of channels over mixed convection type under laminar and turbulent flow.

In this contribution, the study is expanded to be performed over a wide range of the convection regions that are found in figure 1. 55. That means forced convection with laminar and turbulent flow is studied. Along with, the most interesting idea that is performing a heat transfer enhancement through VG's in the mixed convection regions also with laminar and turbulent flow regimes where there is a lack of data especially with the present configuration.

Conclusion

In this chapter, detailed overview is presented on different heat exchangers types. In addition to the heat quantification methods that can be applied inside a heat exchanger in order to study the heat enhancement inside it. The thermal enhancement factor quantification method is chosen to be studied in this contribution along with analyzing the thermal effects by computing the Nusselt number, and the flow effects through the friction factor. Moreover several enhancement methods are also presented that are classified as active or passive.

Then the light is shed on a well studied passive method that is vortex generators (VG's) that will be studied. After that, a well literature was performed for investigating all the types of VG's that can be used in heat enhancements that can be classified according to its shape, size, configuration and style. In another words, if the VG is rectangular or triangular or other than that, that can have a different size in each studied case. In addition if they are inserted with a common flow down or up configurations, and finally the VG style i.e. it is a winglet that it is punched from the HX surface or not.

Due to the different types of VG's that can be investigated, optimization methodologies were introduced in order to minimize the work and time for obtaining a new novel enhanced vortex generator.

SHERPA algorithm is selected to perform the optimization study under a large exploration designs of the geometric and shape parameters of the VG.

Finally, convection heat transfer types were presented that should be chosen precisely with the flow regime type whether laminar or turbulent in order to analyze the system correctly, by applying the suitable governing equations and discretization methods to the model studied that will be discussed in chapter 2. And the other chapters will show for each convection type and flow regimes a novel design is obtained and analyzed quantitatively and qualitatively with respect for other designs. The study objectives will obtaining a new novel design firstly in forced laminar convection, then forced turbulent convection that can be used in various applications like solar collectors. But since these applications undergoes different type of convection and the studies carried out in the bibliography are not satisfactory. This is why we will study different type of convection with an intensification by VG.

Firstly by numerically investigating mixed convection in an empty channel, then heat enhancement using VG with mixed convection under laminar and turbulent flow regimes.

Chapter 2

Numerical Methods and Strategies Abstract

Fluid problems found either in liquid state or in gaseous state are not an easy task to study numerically and experimentally. Depending on the fluid flow behavior in the studied domain the fluid can be classified to Newtonian or non-Newtonian, in addition if the fluid flow is considered in one, two or three dimensional with a single or multi phases. Moreover, the fluid flow can be also classified to be steady or not steady, viscous or in-viscid, laminar or turbulent and many other classifications.

Thus a computational domain should be chosen precisely with a specific methods and strategies that are used in the computers cascaded software. Then governing equations are selected and applied to the computational domain with all the numerical methods and strategies needed to firstly define the geometry and physical properties of the design. And then, dividing the domain into fine cells (meshing) in order to apply the discretization methods of the governing equations over this mesh. Different mesh types can be used for this purpose. Then the problem is solved with the suitable solution methods with their accuracy methodologies. If turbulence is introduced an added equation shall be taken into consideration thus, a brief background about turbulence models that are used through decades with a detailed information is introduced. Finally, due to the use of optimization in this project, a detailed overview about optimization algorithms is presented in order to better distinguish which methodologies can be used depending on the defined problem.

Numerical Modeling

Fluids, which denote liquids and gases are substances that do not permanently change under a large stress (force per unit area). Whereas a solid resists an applied shear or tangential stress by deforming, a fluid cannot and a shear stress applied to a fluid puts it to motion. Moreover, unlike solids which have well-defined shapes, fluids do not have a definite shape. While gases are fluids that completely fill their domains, liquids are fluids that form a free surface in the presence of a gravitational field. In analyzing fluid flow phenomena, attention is focused on what happens at the macroscopic rather than the microscopic scale [START_REF] Hauke | An Introduction to Fluid Mechanics and transport phenomena[END_REF]- [START_REF] Bird | Lightfoot EN. Transport phenomena[END_REF]. It is also assumed that the fluid is a continuum, so that its physical and flow properties are defined at every point in space. Within this assumption, fluid flow behavior can be categorized as either Newtonian or non-Newtonian. Newtonian fluids are characterized by a linear relationship between the shear stress and the shear rate, with the molecular viscosity, which is a measure of the ability of a fluid subjected to a stress to resist deformation, representing the slope of the linear function. On the other hand, for non-Newtonian fluids this relationship is nonlinear.

Similarly fluid flow can be classified into various classes, such as one-dimensional or multi-dimensional, single phase or multi-phase, steady or unsteady, real (viscous) or ideal (inviscid), compressible or incompressible, turbulent or laminar, and rotational or irrotational, among others. The purpose of these classifications is to simplify the process of analysis and modeling of fluid flow phenomena [7].

Governing equations

Conservation of mass (continuity equation)

The flow field in this study is a 3D fluid domain that includes heat transfer variations, thus the field is governed by three dimensional equations that can carry out the variations of temperature, pressure, and velocity in the system [START_REF] Bird | Lightfoot EN. Transport phenomena[END_REF][START_REF] Fay | Introduction to fluid mechanics[END_REF]. A 3D differential fluid element is considered in x, y, z components of mass m, density ρ, and velocity v. The mass conservation in an Eulerian coordinate system can be written as:

V ∂ρ ∂t + ∇.[ρv] dV = 0 (2.1)
For the integral in equation 2.1 to be true for any control volume V, the integrand (mathematical expression to be integrated) should be equal to zero, giving the flux form of the mass conservation or continuity equation as:

∂ρ ∂t + ∇.[ρv] = 0 (2.2)
In the absence of any significant absolute pressure or temperature changes, then the flow can be considered as incompressible; that means the pressure changes do not have any effect on the density.

The incompressible condition indicates that ρ remains constant which mathematically can be expressed as ∂ρ ∂t = 0. Thus the continuity equation of incompressible flow is given by:

∇.v = 0 (2.3)
Equation 2.3 states that the net flow across any control volume is zero, that means flow in is equal to flow out [7].

Conservation of linear momentum

The principle of conservation of linear momentum indicates that in the absence of any external force acting on a body, the body retains its total momentum. Since momentum is a vector quantity, its components in any direction are also be conserved [START_REF] Bird | Lightfoot EN. Transport phenomena[END_REF][START_REF] Fay | Introduction to fluid mechanics[END_REF].

For a 3D differential fluid element with volume V , Newton's second law of motion can be applied on this element. Taking into consideration both the body forces and the surface forces, the Newton's law in Lagrangian coordinates can be written as:

d(mv) dt M V = V f dV M V (2.4)
The conservative (or flux) term of the equation is obtained by applying the form of the Reynolds transport theorem where m = ρ/V and setting the integrand to zero can be written as:

∂ ∂t [ρv] + ∇.{ρvv} = f (2.5)
The full form of the momentum equation is obtained once the external surface and body forces acting on the control volume are specified. The force f is split into two parts: one denoted by f s representing the surface forces and the second by f b representing the body forces such that:

f = f s + f b (2.6)

Surface forces

For the arbitrary macroscopic volume element, the forces acting on its surface are due to pressure and viscous stresses which can be expressed in term of the total stress tensor Σ [7]. In general there are nine components of stress at any given point; one normal component and two shear components (parallel to the surface that receives the stress) in each coordinate plane. Thus in Cartesian coordinates the stress tensor is given by:

Σ =     Σ xx Σ xy Σ xz Σ yx Σ yy Σ yz Σ zx Σ zy Σ zz     (2.7)
where terms of the form Σ ii represent normal stresses and Σ ij shear stresses. In practice the stress tensor is split into two terms such that:

Σ = -pI + τ (2.8)
where I is the identity tensor of size (3 × 3), p is the pressure, and τ is the deviatoric or viscous stress tensor. Thus the surface force acting on a differential element can be written as:

f s = [∇.Σ] = -∇p + [∇.τ ] (2.9)

Body forces

Body forces, which are presented as forces per unit volume, may also arise due to a variety of effects. They can be classified as gravitational forces or system rotation [7].

Gravitational forces

The force representing the weight of the material volume per unit volume in the presence of a gravitational field is denoted by gravitational force and given by

f b = ρg (2.10)
where g is the gravitational acceleration vector.

Substituting the external force f in equation 2.4 by its equivalent expression, the general conservative form of the momentum equation is obtained as

∂ ∂t [ρv] + ∇.{ρvv} = -∇p + [∇.τ ] + f b (2.11)
To proceed further with the momentum equation, the type of fluid should be known in order to relate the stress tensor τ to the flow variables. For a Newtonian fluid, the stress tensor is a linear function of the strain rate and is given by:

τ = µ ∇v + (∇v) T + k(∇.v)I (2.12)
where µ is the molecular viscosity coefficient, k the bulk viscosity coefficient, the superscript T refers to the transpose of ∇v, and I is the unit or identity tensor of size (3 × 3).

The expanded form of the stress tensor in a three-dimensional Cartesian coordinate system can be written as:

Σ =        2µ ∂u ∂x + k∇.v
+ k∇.v        (2.13)
The divergence of the stress tensor is a vector that can be expressed as

[∇.τ ] = ∇. µ ∇v + (∇v) T + ∇(k∇.v) =        ∂ ∂x
∂ ∂t [ρv] + ∇.{ρvv} = -∇p + ∇. µ ∇v + (∇v) T + ∇(k∇.v) + f b (2.15)
For steady state flow, incompressible flow (the divergence of the velocity vector is zero), and if the viscosity is constant; the momentum equation is reduced to become:

ρ∇.{vv} = -∇p + µ∇ 2 v + f b (2.16)

Conservation of energy

The conservation of energy is governed by the first law of thermodynamics which states that energy can be neither created nor destroyed during a process; it can only change from one state (mechanical, kinetic, chemical, etc..) into another. Consequently, the sum of all forms of energy in an isolated system remains constant [START_REF] Bird | Lightfoot EN. Transport phenomena[END_REF][START_REF] Fay | Introduction to fluid mechanics[END_REF].

Considering a 3D differential element of volume V , of mass m, density ρ, and moving with a velocity v. Defining the total energy E of the material volume at time t as the sum of its internal and kinetic energies, then E can be written as:

E = m u + 1 2 v.v (2.17)
where u is the fluid specific internal energy (internal energy per unit mass). The first law of classical thermodynamics applied to the material volume states that the rate of change of the total energy of the material volume is equal to the rate of heat addition and work extraction through its boundaries.

Mathematically this is given by:

dE dt M V = Q -Ẇ (2.18)
The net rate of heat transferred to the material element Q is the sum of two components. The first component is the rate transferred across the surface of the element QS and the second generated/destroyed that means due to a chemical reaction within the material volume QV . Moreover, the net rate of work done by the material volume Ẇ is due to the rate of work done by the surface forces ẆS and the rate of work done by the body forces Ẇb [7]. Thus the first law can be written as:

dE dt M V = QV + QS -Ẇb -ẆS (2.19)
By definition, work is due to a force acting through a distance and power is the rate at which work is done. Therefore the rate of work done by body and surface forces can be represented by:

Ẇb = - V (f b .v)dV ẆS = - S (f S .v)dS = - V ∇.[(-pI + τ ).v]dV (2.20)
If q V represents the rate of heat source or sink within the material volume per unit volume and q S the rate of heat transfer per unit area across the surface area of the material element, then QV and QS can be written as:

QV = V q V dV QS = - S q S .ndS = - V ∇. q S dV (2.21)
Applying the Reynolds transport theorem and substituting the rate of work and heat terms by their equivalent expressions, and collecting terms together to obtain one volume integral. Then the integrand has to be zero if integrating over a control volume. The mathematical description of energy conservation or simply the energy equation written in terms of specific total energy can be written as:

∂ ∂t [ρe] + ∇.[ρve] = -∇. q S -∇.[pv] + ∇.[τ.v] + f b .v + q V (2.22)
where e is total energy per unit mass.

The energy equation may also be written in terms of specific internal energy, specific static enthalpy (or simply specific enthalpy), specific total enthalpy, and under special conditions in terms of temperature.

Conservation of energy in terms of temperature

To be able to write the energy equation with temperature as the main variable some constraints have to be imposed. Assuming h to be a function of p and T , the fluid is expected to be Newtonian.

Therefore the derivations to follow are applicable to Newtonian fluids only [7]. If h = h(p, T ) then d h can be written as:

d h = C p dT + V -T ∂ V ∂T p dp (2.23)
After mathematical operations and substituting the variables with their values and by simplifications the energy equation in terms of temperature can be obtained and written as:

∂ ∂t [ρC p T ] + ∇.[ρC p vT ] = ∇.[k∇T ] - ∂(lnρ) ∂(lnT ) p Dp Dt + λΨ + µΦ + q V (2.24)
where q S = -[k∇T ] with k is the thermal conductivity of the material. And Φ and Ψ are mathematical operations written for simplicity that deals with the term τ : ∇v.

The energy equation is rarely solved in its full form and depending on the physical situation several simplified versions can be developed. The dissipation term Φ has negligible values except for large velocity gradients at supersonic speeds. Moreover, for incompressible fluids the continuity equation implies that Ψ = 0 and because the density is constant it follows that (∂(lnρ)/∂(lnT )) = 0 .

Therefore the energy equation for incompressible fluid flow is simplified to:

∂ ∂t [ρC p T ] + ∇.[ρC p vT ] = ∇.[k∇T ] + q V + ρT DC p Dt (2.25)
In addition to the simplification done before, equation 2.25 is also applicable for a fluid flowing in a constant pressure system. For the case of a solid, the density is constant, the velocity is zero, and if changes in temperature are not large then the thermal conductivity may be considered constant, in which case the energy equation becomes

ρC p ∂T ∂t = [k∇ 2 T ] + q V (2.26)
Another arrangement of the energy equation is if the fluid studied is gas with an ideal gas density case. Then (∂(lnρ)/∂(lnT )) = -1 and the energy equation for the ideal gas case becomes:

C p ∂ ∂t (ρT ) + ∇.(ρvT ) = ∇.[k∇T ] + Dp Dt + λΨ + µΦ + q V (2.27)
And for further simplification, if viscosity is neglected thus the fluid is considered to be inviscid.

The to terms (λΨ) , and (µΦ) can be neglected [7].

General conservation equation

The governing equations describing the conservation of mass, momentum, and energy are written in terms of specific quantities. These quantities are expressed in a per unit mass form. For example, the momentum equation expressed the principle of conservation of linear momentum in terms of the momentum per unit mass (velocity). This general form will be used for the discretization process. A general form can be considered to define a variation of quantity over a control volume [7].

Let φ be the variable, then the variation of φ in a control volume over time can be expressed as a balance equation consisting of three forms:

T erm 1 = T erm 2 + T erm 3 where:

• Term 1 is the change of φ over time ∆t within the material volume (MV)

• Term 2 is the surface flux of φ over time ∆t across the control volume (CV)

• Term 3 is source/sink of φ over time ∆t within the control volume (CV)

For a fixed control volume, the change of φ over time within the material volume can be written using the Reynolds transport theorem as:

T erm 1 = V ∂ ∂t (ρφ) + ∇.(ρvφ) dV (2.28)
where ρ is the fluid density and V the volume of the control volume of surface area S. The term ρvφ represents the transport of φ by the flow field and is denoted by the convective flux and it can be written as:

J φ convection = ρvφ (2.29)
The second term represents variation of φ due to physical phenomena occurring across the control volume surface. For the physical phenomena of interest, the mechanism causing the influx/outflux of φ is due to diffusion, which is produced by molecular collision and is designated by J φ dif f usion . Denoting the diffusion coefficient of φ by Γ φ , the diffusion flux may be written as:

J φ dif f usion = -Γ φ ∇φ (2.30)
thus term 2 becomes:

T erm 2 = V ∇.(Γ φ ∇φ)dV (2.31)
Term 3 can be expressed with the generation/destruction of φ within a control volume per unit volume, which is called the source term. The generation/destruction can be denoted by Q φ . Then term 3 can be written as:

T erm 3 = V Q φ dV (2.32)
The final form of the general conservation equation, for the transport of a property φ is expressed as

∂ ∂t (ρφ) + ∇.(ρvφ) = ∇.(Γ φ ∇φ) + Q φ (2.33)
where they are divided as an unsteady term, convection term, diffusion term, and source term. It is clearly shown that by assigning the right values of φ, Γ φ , and Q φ in equation 2.33 that is the general form can lead to obtain the conservation equations defined before (example equation 2.25).

Geometric domain

Modeling of physical phenomena is the most important part in solving a CFD problem. A physical phenomenon will not be considered as understood unless the problem is modeled mathematically. Two parts of modeling are used, one in relationship with the geometry of the problem and the other in relationship with to the physical phenomena. For the two parts, some assumptions or simplifications can be employed. For example a three dimensional domain could be turned into a two dimensional domain, or symmetry can be taken into account to decrease the size of the computational domain [7].

The computational domain that is used in this contribution consists of a rectangular channel of height H, width W , and length L as shown in

Meshing

Discretization process

The discretization process consists of generally obtaining a numerical solution of a partial differential equation then finding the values of the dependent variable φ at specified points from which its distribution over the domain of interest can be constructed [7]. This process consists of several steps that can be listed as follows:

1. Modelization of the domain and defining the physical models or physical interest.

2. Discretization of the domain into grids that form the computational domain. This step is known as meshing that results in a set of non-overlapping volume elements (cells).

3. Numerical discretization of the equations that transforms the set of conservation partial differential equations governing the physical processes into an equivalent system of algebraic equations defined over each cell of the computational domain.

4. Obtaining the solution from the set of equations using an iterative solver to find best numerical solution.

These stages can be illustrated in chart (figure 2.2) that clearly shows how the steps of discretization process occur [7].

Figure 2.2: The discretization process.

Domain discretization

The geometric discretization of the physical domain results in a mesh where the conservation equations are solved over the computational domain. This results from subdividing the domain into discrete, and non-overlapping cells that fully cover computational domain. This can be accomplished using different types of techniques and many mesh types. These meshes are classified according to several characteristics: structure, orthogonality, blocks, cell shape, variable arrangement. In all cases the mesh is composed of discrete elements defined by a set of vertices and bounded by faces. In order to discretize the equations and to be implemented accurately on the computation domain, mesh typology information is needed about the mesh elements [7].

During discretization, the partial differential equations are integrated over each cell in the mesh resulting in a set of algebraic equations. These equations are linked from one cell two another by assigning the variable value to a cell and compute the next value according to the assigned values on the neighbor cell. Then all the algebraic equations are assembled together. These forms are called global assembly. The integration of the equations over each element is called a local assembly. Thus while the discretization of the equations is derived in terms of neighbor element, the assembly of the equations in the global matrix accounts for the local matrices on each element. The local assembly which is the form of topological information about elements, faces, and vertices are presented in terms of connectivity lists.

Element connectivity relates the local assembly matrix to the global matrix so that the equations formed for one element are consistent with the equations formed for the other elements in the computational domain.

Face connectivity is essential for solving, since it is more efficient to assemble flux terms by looping over faces. Information about faces for each shared element is stored during computation.

Vertex connectivity is useful for post processing and for gradient computation and it is not essential while solving the problem [7].

Equation discretization

The governing partial differential equations, are transformed into a set of algebraic equations once the problem is discretized. These algebraic equations are then assembled into a global matrix and vectors that can be expressed in the form:

A [T ] = b (2.34)
where T is the unknown variable and is defined on each element at the interiors and the boundaries over the computational domain. The interior values are generally computed and the boundary values for T are generally specified from the predefined boundary conditions [7].

Solution of the discretized equations using element field

To obtain the discrete values of the variable T , coefficients of algebraic equations must be computed.

These coefficients can be considered linear that means independent of T ; or non linear that is dependent on T . The techniques to solve these forms of equations are independent of the discretization method.

For the linear algebraic sets, the uniqueness of the solution is guaranteed. Therefore if the adopted solution method converges and results in a solution, it will be the desired solution. Otherwise it will not be converged and results in a solution. The method considered for solving algebraic equations is iterative method [7].

Iterative methods

Iterative methods follow a guess-and-correct procedure to gradually refine the estimated solution by repeatedly solving the discrete system of equations. Consider a simple Gauss-Seidel iterative method. The overall solution loop may be written as follows:

1. Initially guess the discrete values of T at all the node elements in the domain.

2. Then solve for each element node in turn and update the value of general form of T .

The neighboring values are required for the update of T . Thus they are assumed from either the predefined or the assumed values. By that, nodes which have already been visited will have an updated values of T and those that have not will have old values.

3. Moveover the domain until all the nodes are covered. Then one iteration is completed.

4. Convergence criterion is checked if it met or not. The used criteria is that the maximum change in node values of T must be less than a residuals with a very low values of error. If the criterion is met, stop calculation. Else, go back to step 2 and repeat the steps. If the case considered is unsteady, then a second method is used to solve the system. this method is by solving the governing equation at each time step with defined inner iteration number. Finally compare the node value at the end of each time step whether a low value of residual is obtained.

The Gauss-Seidel scheme can be implemented with a small storage. All that is required is storage for the discrete values of T at each node. The coefficients can be computed but not stored if desired, since the entire coefficient matrix for the domain is not required when updating the value of T . Also, the iterative scheme is practical for non-linear problems. Since the coefficients depend on T, they may be updated using predefined values of T as the iterations are solved. Many other schemes are used in CFD problems, each depends on the case studies. These can be classified according to the requirements of computation cost, that means the computational time and size, and order of accuracy [7].

Under-Relaxation of the iterative solution method

Consider the general conservative equation 2.33 where φ is the variable in the equation. And to obtain the solution of this variable, iterative method is used. Thus by that, the value of φ is dependent on preceding φ value. Therefore large variations in φ between iterations results in large changes in the coefficients, which may cause divergence of the solution. The divergence is due to the non-linearity introduced by the coefficients and the cross-diffusion term. To ensure the convergence and stabilize the iterative solution, under-relaxation technique can be used that slow down the changes in the value of φ between iterations.

The relaxation factor is defined as λ φ . The value of λ φ can vary between positive and negative number. It can be subdivided into two domains, the first domain is defined when 0 < λ φ < 1 and the equations will be under relaxed. The second domain is defined when λ φ > 1 and the equations will be over relaxed. In CFD applications, under relaxation is usually used. A value of λ φ close to 1 implies a little under relaxation, while a value close to 0 produces heavy under relaxation effects with very small changes in φ from iteration to iteration.

The optimum under relaxation factor is problem dependent and is not governed by any general rule. The factors affecting λ φ values include the type of problem defined, the size of the computational mesh, the grid spacing and its expansion rate, and the used iterative method. The values of λ φ are usually defined based on experience or from pre-done calculations. Moreover, it is not necessary to use the same under-relaxation value throughout the computational domain and values may vary from iteration to iteration [7].

Finite volume mesh

Physical domain discretization produces a computational mesh on which the governing equations are solved on the meshed elements. The computational mesh is formed from a finite volume mesh that with different element types. Each element has some characteristics in the context of structuring the grids. Thus choosing the type of element with its type of grids is essential in defining a CFD problem [START_REF] Thompson | Numerical grid generation[END_REF][START_REF] Cheng | Delaunay mesh generation[END_REF][START_REF] Thompson | Handbook of grid generation[END_REF].

Structured grids

Mesh formation is obtained when a system of grids is defined. There are two types of grids, a structured and a non structured design. For structured grid design, every interior cell in the domain is connected to the same number of neighboring cells. These neighboring cells can be identified using the indices i, j, and k respectively in the x, y, and z coordinate system. And those cells can be directly accessed by incrementing or decrementing the respective indices. This leads to a lower memory usage since a topological information is embedded in the mesh structure through the indexing system. This also leads to greater efficiency in coding, cache utilization, and vectorization. Structured grids are widely used in the development of the finite volume methods [START_REF] Chappell | The generation of hybrid grids incorporating prismatic regions for viscous flow calculations[END_REF].

Element types

The finite volume mesh needs information about the geometric entities used, such as the volume of elements, the area of faces, the centroids of elements and faces, and lastly the alignment of faces with the vectors joining the centroids of the owner and neighbor elements. Different elements types can be used in generating the mesh. Figure 2.3 shows different types of elements that can be used in any CFD problem when using the finite volume method approach and figure 2.4 shows the face shapes obtained when using these element types.

The first element in figure 2.3 is tetrahedron cells that have a three vertices as the cell is projected in any direction as shown in figure 2.4. The second element is a prism cell that is mainly used in defining the boundary layers. These cells have different projected surfaces and they can form both the triangular and the quadrilateral shape. While the hexahedron element shape is with a four vertices when projected to a 2D element in all directions that can form easily a structured grid. The polyhedron element is the general shape of the elements that have five vertices on each projected face.

Those elements are most widely used types in CFD domains that each is adopted according to its application [START_REF] Bern | Mesh generation. Handbook of computational geometry[END_REF][START_REF] Mavriplis | Mesh generation and adaptivity for complex geometries and flows[END_REF].

Mesh methodology

Various mesh methodologies can be applied to a computational domain with different mesh refinement techniques. In this contribution, trimmed cells are used to produce a uniform structure. As for the refinement techniques, usually volume control technique is used in order to refine the specified domain that undergoes a adhesive flows. But this technique increases the number of cells leading to an increase in the computational time. Thus due to this contribution goal that is using optimization for the seek of obtaining an optimal shape design. It is better to decrease the number of cells but keeping the Figure 2.6 shows a clearer view of the refinement techniques applied with the growth rate for one direction. For the VG, two refinement methods are used. First, a surface control with a smaller number of cells but with a wake refinement method. In this contribution, the wake refinement is applied parallel to the flow that is behind the vortex generator. Second, a curve control with a very small number of cells, with respect to the base size of the mesh, is applied on the curves of the VG. It is generated with a low growth rate that states that the growth from the small cell to the base cell is propagating slowly. This is to ensure that there is no sudden jump in the cell sizes that can be shown in the zoomed cut in figure 2.6.

Turbulence modeling

The flow can be considered laminar or turbulent depending on the relative importance of fluid friction and flow inertia, thus depending on Reynolds number. The flow is laminar until reaching a limit that a turbulent boundary layer starts to appear and then a separation in the flow occurs (figure 2.7). As known laminar flows are stable, turbulent flows are chaotic, diffusive causing rapid mixing, time-dependent, and involve three dimensional vorticity fluctuations with a broad range of time and length scales [START_REF] Tennekes | A first course in turbulence[END_REF]. But for steady state flow the time scales can be dropped from the Navier Stokes

Turbulence numerical methods

Researchers develop many theories and models to understand and analyze correctly turbulence. The most accepted theory of turbulence is based on the "energy cascade" concept developed by Kolmogorov [START_REF] Kolmogorov | Dissipation of energy in isotropic turbulence[END_REF]. This theory states that, turbulence is composed of eddies of different sizes with each one possessing a certain amount of energy that depends on its dimension. These eddies will break up transferring their energy to smaller eddies and lead up to a chain process.

Different numerical methods were established to study the eddies based on energy cascade concept.

They are as follows:

• Direct Numerical Simulation (DNS)

• Large Eddy Simulation (LES)

• Reynolds Averaged Navier-Stokes (RANS)

Direct Numerical Simulation (DNS) of the Navier-Stokes equation for turbulent flows is used when having a very small time step limited by a Courant number below 1 and a very fine mesh resulting in a large number of grid points proportional to Re 3 to resolve the entire spectrum of temporal and spatial turbulent scales involved. This method is limited and just used in simple studies due to its high computational cost [START_REF] Moser | Direct numerical simulation of turbulent channel flow up to Re = 590[END_REF]- [START_REF] Choi | Direct numerical simulation of turbulent flow over Riblets[END_REF].

To reduce the large computational cost associated with a direct solution of the Navier-Stokes equation, statistical analyses can be used to simplify the resolution of turbulent flows. This is done by Large Eddy Simulation (LES), that the large turbulent scales are directly simulated, and the small turbulent scales are modeled by using sub-grid scale methods. This can be accomplished by using a filter mathematical operations that determined the scales to analyze and which should be discarded.

The advantage is it directly resolves the larger eddies with the higher content of energy (anisotropic turbulence) while the smallest eddies are simply modeled [START_REF] Leonard | Energy cascade in large-eddy simulations of turbulent fluid flows[END_REF]- [START_REF] Nieuwstadt | Large eddy simulation of the convective boundary layer: a comparison of four computer codes[END_REF].

The most popular and used method for solving turbulence problem is the Reynolds Averaged Navier-Stokes (RANS). This approach decomposes the flow variables into a time mean value component and a fluctuating one that are next presented, substituting in the original equations, and time-averaging the obtained equations. Two techniques can be used to average the equations. The standard Reynolds averaging, which is used to derive the Reynolds-Averaged Navier-Stokes (RANS) equations [START_REF] Reynolds | On the dynamical theory of incompressible viscous fluids and the determination of the criterion[END_REF], and the mass-weighted or Favre averaging technique employed with turbulent compressible flows and leading to the Favre-Averaged Navier-Stokes (FANS) equations [START_REF] Favre | Equations des Gas Turbulents Compressibles[END_REF]. In this project RANS equations are used to solve turbulence with the methods used.

Turbulent eddies create fluctuations in velocity. Velocity is varying in time due to turbulent fluctuations. If the flow is considered steady and laminar then v = v for all time t, where the over-bar denotes a time average. For unsteady turbulent flow, velocity variations include both a mean and a turbulent component that can be written as:

v(t) = v + v (t) (2.35)
where v is the mean value of the velocity, and v (t) is the turbulent fluctuating. These variations are called Reynolds decomposition and they can be modeled to obtain a solution.

Boussinesq hypothesis

Turbulence can be defined using several models based on Boussinesq hypothesis. And they can be categorized into four main groups as follows:

• Algebraic (Zero-Equation) Models

• One-Equation Models

• Two-Equation Models

• Second-Order Closure Models Each flow condition have its applicable developed model. Summarizing the main advantages and strengths of these groups gives:

The zero-equation models use an algebraic equation to compute the turbulent eddy viscosity (µ t )

without the need to solve any differential equation [START_REF] Prandtl | Uber die ausgebildete Turbulenz[END_REF][START_REF] Cebeci | Analysis of turbulent boundary layers[END_REF]. The one equation models require solving only one transport differential equation to compute µ t [START_REF] Baldwin | A one-equation turbulence transport model for high reynolds number wall-bounded flows[END_REF]- [START_REF] Spalart | A one-equation turbulence model for aerodynamic flows[END_REF]. Two-equation turbulence models necessitate the solution of two transport equations for the calculation of µ t [START_REF] Jones | The prediction of laminarization with a two-equation model of turbulence[END_REF]- [START_REF] Menter | Review of the shear-stress transport turbulence model experience from an industrial perspective[END_REF]. The secondorder closure models are the most computationally expensive since separate transport equations are solved for the individual turbulent fluxes (six equations) [START_REF] Daky | Transport equations in turbulence[END_REF]- [START_REF] Wilcox | Multiscale model for turbulent flows[END_REF].

The popular and most used group is the two-equation models due to their usage in the industrial applications. This group requires the transport equations to be used. Taking a brief review about the history of the two-equation models shows that the k -model of Jones and Launder was among the earliest two-equation models and the most popular. While the k -ω model of Wilcox comes as a close second. Both models are modified and improved in order to extend their applicability [START_REF] Jones | The prediction of laminarization with a two-equation model of turbulence[END_REF]. Thus by these improvements and modifications, the two equation models become:

1. Standard k -model 2. k -ω model 3. Baseline (BSL) k -ω model 4. Shear Stress Transport (SST) k -ω model 2.3.3 k -ω model
Firstly, the k -model is based on the Boussinesq approximation that relates the Reynolds stress tensor τ R and the turbulent heat fluxes qR directly with two added relations: the turbulent kinetic energy k and its rate of dissipation per unit mass . The hypothesis of Boussinesq is to simplify the relations that can compute the Reynolds Averaged Navier-Stokes (RANS) equations that deal with the fluctuating and mean value components of velocity, pressure, and temperature [START_REF] Jones | The prediction of laminarization with a two-equation model of turbulence[END_REF].

k and relations can be defined as:

k = 1 2 v .v (2.36) = 1 2 µ ρ [∇v + (∇v ) T ] : [∇v + (∇v ) T ] (2.37)
The k -ω model of Wilcox is similar in structure to the k -model but with a modification of using a specific turbulence dissipation ω instead of [START_REF] Wilcox | Reassessment of the scale-determining equation for advanced turbulence models[END_REF][START_REF] Wilcox | Turbulence modeling for CFD[END_REF], defined as

ω = C µ k (2.38)
The advantages of replacing the -equation by the ω-equation are:

1. ω-equation is easier to integrate.

2. It can be integrated through the sub-layer without the need for additional damping functions.

3. It behaves better for flows with weak adverse pressure gradient.

In this model, the turbulent kinetic energy k and the specific turbulence dissipation ω are computed using the conservation equations written as

∂ ∂t (ρk) + ∇.(ρvk) = ∇.(µ ef f,k ∇k) + S k (2.39) ∂ ∂t (ρω) + ∇.(ρvω) = ∇.(µ ef f,ω ∇ω) + S ω (2.40)
where S k and S ω are two relations to provided later according to the specific model used.

k -ω model can be modified and till now studies are done behind this purpose. Shear Stress Transport (SST) k -ω model is the model that will be used in this project. Shear Stress Transport (SST) [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF]- [START_REF] Menter | Review of the shear-stress transport turbulence model experience from an industrial perspective[END_REF] model has an improved adverse pressure gradient performance compared to other models. The first improvement is due to satisfying Bradshaw's assumption. This assumption states that the principal shear stress and the turbulent kinetic energy in the boundary layer are linearly related. Thus, the turbulent eddy viscosity calculation is modified by bounding it using this formula

µ t = ρa 1 k max (a 1 ω, √ 2S t F 2 ) (2.41)
where a 1 = 0.31, S t is the magnitude of the strain rate, and F 2 is a blending function. To maintain the original formulation of the eddy-viscosity for free shear layers, the same blending function approach as for the baseline model is also adopted in the SST k -ω model. The S k and S ω relation equations are given by

S k = P k -β * ρkω (2.42) S ω = Cα ω k P k -Cβ ρω 2 + 2(1 -F 1 ) σ ω2 ρ ω ∇k.∇ω (2.43)
The second modification is related to the production of turbulence kinetic energy P k which is replaced by Pk given by Pk = min (P k , c 1 ) (2.44)

where is obtained from equation 2.37 and the blending function F 1 is calculated based on computation relation [7] with the coefficients computed using the following model constants that are also inserted directly in the CFD software:

C α1 = 0.5532 , C β1 = 0.075 , β * = 0.09, σ k1 = 2 , σ ω1 = 2 , c 1 = 10, C α2 = 0.4403 , C β2 = 0.0828 , σ k2 = 1 , σ ω2 = 1.
186 , and P r t = 0.9.

In addition, the turbulent thermal conductivity and effective turbulent viscosities for k and ω are computed as

k t = µ t P r t µ ef f,k = µ + µ t σk µ ef f,ω = µ + µ t σω (2.45)
Finally, the incompressible time-averaged continuity, momentum, energy, turbulence kinetic energy, turbulence dissipation rate, and specific dissipation rate equations can be written, respectively, as follows [7]:

∇.[ρv] = 0 (2.46) ∂ ∂t [ρv] + ∇.{ρvv} = ∇. (µ + µ t )∇v + Q v (2.47) ∂ ∂t (ρC p T ) + ∇.[ρC p vT ] = ∇. C p µ P r + µ t P rt ∇T + Q T (2.48) ∂ ∂t (ρk) + ∇.[ρvk] = ∇.[µ ef f,k ∇k] + Q k (2.49) ∂ ∂t (ρ ) + ∇.[ρv ] = ∇.[µ ef f, ∇ ] + Q (2.50) ∂ ∂t (ρω) + ∇.[ρvω] = ∇.[µ ef f,ω ∇ω] + Q ω (2.51)
These equations are of the same form of the general equation presented in section that include convection, diffusion and source terms. Their discretization follows the general procedures presented before.

Wall treatment

As the turbulent flow approaches a wall that fluctuates the components of velocity that creates a large gradients. In addition, away from the wall a very high turbulent is induced, while it decreases in the near wall layer to values of magnitude comparable to those of the viscous stresses. Therefore the near wall layer known as boundary layer needs to be resolved, thus a substantial number of grid points are required.

The wall functions approach is based on the universal flow profiles in the boundary layer along a wall, with the normalized distance to the wall y + defined as:

y + = yu τ ν (2.52)
where y is the normal distance to a wall, ν is the kinematic viscosity, and u τ is the friction velocity expressed in terms of the wall shear stress τ w

u τ = τ w ρ (2.53)
The inner region of the boundary layer can be split up into three sub-layers. In each of them the flow has different characteristics and can be modeled using different empirical approaches:

1. Viscous sublayer (0 < y + < 5): The fluid layer in contact with the wall is dominated by viscous effects and is almost laminar. The mean flow velocity only depends on the fluid density, viscosity, distance from the wall, and the wall shear stress.

2. Buffer layer (5 < y + < 30): The buffer layer is a transitional layer between the viscous sublayer and the log-law layer.

3. Log-law layer (30 < y + < 200): The turbulent log-law layer is dominated equally by viscous and turbulent effects.

In all CFD software upon using turbulence models, there is a wall treatment methods that deals with the sub-layers found in the mesh. In this contribution, k -ω model is selected with the option of all y + wall treatment that deals with all the flow changes in all layers.

Optimization

Optimization is the design and operation of a system or process to make it as good as possible in some defined sense. But due to the behavior of the optimization algorithms, different optimization techniques are introduced using different CFD software. Star-CCM+ is a widely used software in the heat transfer applications in addition to its ability of performing various optimization studies.

Optimization can be performed using different techniques that are as follows:

1. Design optimization: In an optimization study, more than one objective can be defined that the analysis must meet as best it can. Along with the objectives, you define input parameters.

During the analysis, a search algorithm chooses the parameter values for the designs so as to best meet the analysis objectives. There are two types of design optimization studies that are the weighted sum of all objectives technique and multiple objective trade-off study (Pareto front).

The weighted sum of all objectives technique is a type of Optimization study that allows the user to optimize based on a single objective or based on multiple objectives. If multiple objectives are defined, a linear weighting is used that combines all objectives into a single performance function. Running a weighted sum of all objectives analysis with multiple objectives returns a single best design as shown in figure 2.9. In this type of optimization study, many goals can be applied including minimizing pressure drop, maximizing outlet uniformity, minimizing component temperatures, or minimizing stresses. The SHERPA algorithm is used in the weighted sum of all objectives optimization technique. In this contribution, this technique is used with the SHERPA algorithm where the objective function used is the thermal enhancement factor.

And as defined before any increase of this value above 1, the design is considered enhanced and its value is based on a combination of the contradicting values parameters that are the Nusselt number and the friction factor that can be used in the Pareto front technique [START_REF]Optimization Algorithms[END_REF].

When using two competitive or contradicting objective functions, then the use of multiple objective trade-off study (Pareto front) is the best choice. In such cases, there is no single optimum design. Instead, the optimization returns a curve along which all designs are optimum in one objective for a given value in the other objective-known as the non-dominated design condition. The curve, known as Pareto front as shown in figure 2.10, expresses the optimum trade-off relationship between two competing objectives [START_REF]Optimization Algorithms[END_REF].

Design optimization study evaluates designs over a broad range of input parameters and parameter values. This study gives you the best design.

Design Of Experiments (DOE):

A DOE study is used for local exploration around a particular optimized design in order to identify the parameters that have the greatest impact on the performance. The user can assess the sensitivities of the input parameters and their interaction with each other. For a DOE, you supply the input parameter values at two levels, at three levels, or Accounting for variations leads to more reliable designs, which effectively pushes the optimal designs away from the constraint boundaries as shown in figure 2.11b [START_REF]Optimization Algorithms[END_REF]. In this Project, the optimization algorithm used is SHERPA technique via multi-objective functions. SHERPA is Simultaneous Hybrid Exploration that is Robust, Progressive and Adaptive.

SHERPA is a hyper-parameter optimization method that includes a variety of powerful and interchangeable algorithms [43]. The importance of efficiency and robustness in optimization algorithms are critical for optimization algorithm. And both are found in SHERPA [41].

Conclusion

In this chapter an overview is presented on the computational fluid dynamics methods and strategies that are used in the software cascaded computers. In addition, the Governing equations are listed with all the numerical methods and strategies used to firstly define the geometry and physical properties of the design then how computational domain is divided into fine cells (meshing), inaddition to the refinement strategies used, and how to apply the discretization methods of the governing equations over the selected mesh. Various mesh types that can be used are also presented, in addition to the solution methods with their accuracy methodologies. Finally a brief background about turbulence models that are used through decades with a detailed information about the k -ω turbulent model that is used in this project with its equations. Finally, a detailed overview about optimization methods used in Star-CCM+ is presented with the selected type of optimization method that is chosen to be Simultaneous Hybrid Exploration that is Robust, Progressive and Adaptive (SHERPA). In the following chapter heat transfer enhancement using a new novel VG design is performed under forced convection with laminar flow regime. empty channel at Reynolds number Re = 4600. Khanjian et al. [20] also studied the effect of using rectangular VG in a flat plate channel on the heat transfer enhancement which results in an increase in the heat transfer. The study was performed by varying the roll angle (β) in order to distinguish the best design angle. Li et al. [21,22] studied numerically and experimentally the effect of using longitudinal vortex generator in a parallel, fin-less HX which indicates that the use of double triangle VG increase the heat transfer coefficient by 20% with an doubled pressure drop. Tian et al. [23] studied the fluid flow and heat transfer in a flat plate channel with different types of longitudinal vortex generators with rectangular and triangular winglets with a common flow down and common flow up configurations (CFD, CFU); they show how those configurations affect the boundary layer thinining specialy for the CFD configuration. The conclusion was that, the triangular winglet with CFD configuration has a better overall performance. This conclusion was obtained by comparing the Nusselt number and the friction factor. Moreover, different types of VG studies dispersly exist in the literature such as the helical and twisted VG [24,25], dimples or protrusions [26,27], cylindrical tubes [28], transverse vortex generators [35], plane or curved surface of VG [30,31], longitudinal vorticity [32], and a longitudinal rectangular wings [33].

Several parametric and shape optimization techniques were applied for optimizing VG's. The variation of several geometric parameters were conducted in order to achieve optimal designs. For example, Kim and Choi [34] performed shape optimization of rectangular channel with dimples by applying a response surface method. Samad et al. [35] optimized the shape of channel dimples on a single surface using different surrogate models. They improved heat transfer and friction-loss by approximately 20%. Khan et al. [36] obtain a vortex generator design by applying a Bayesianregularized artificial neural network (BRANN) with using of a multi-objective optimization algorithm (MOA). Song et al [37] In recent years, many authors try to vary the forms of VG's which are initially quite simple and limited to rectangular triangular shapes or protrusions of basic forms. Several authors perform parametric form optimization by choosing a simple form (triangular, and rectangular) and by varying one or two parameters of the shape in general of the angles of the VG's in regard to the direction of the flow. The set of results of the bibliography on these variations of shape shows that the performances of the VG's are sensitive to these geometrical variations and that it is possible to increase substantially the thermo-hydraulic performances of the exchangers by using these optimized forms.

In this work, parametric form optimization is performed with a large number of parameters (seven parameters), the objective of this work is to verify if it is possible to increase more the performances obtained by increasing the degrees of freedom of the optimization operation. As the increase in the optimization work-space is an element that significantly increases the cost of the optimization operation, in this article advanced algorithms carried out to test the efficiency of optimizing with a large number of parameters. Hence in order to give insight of the physical phenomenon at the origin of the increase of thermo-hydraulic performance local analysis is performed on the flow structure with highlighting the main difference with traditional geometry know as delta and rectangular winglet pairs (DWP and RWP) respectively. Pairs of VG's were considered with a common flow down configuration.

Computational

Governing Equations

The flow field in the channel is considered steady, and three dimensional. The fluid is considered incompressible with constant fluid and thermal properties at a laminar flow regime. The governing equations are the conservation of mass (continuity) (equation 3.1), momentum (Navier Stokes) (equation 3.2), and energy equations (equation 3.3) are given as the following:

∇. - → u = 0 (3.1) ρ( - → u .∇) - → u = -∇p + µ∇ 2-→ u (3.2) ρCp∇.( - → u T ) = k f ∇.(∇T ) (3.3)
where -→ u is the velocity vector, Cp the specific heat at constant pressure, p the pressure, µ the fluid dynamic viscosity, T is the temperature, ρ the fluid density, k f is the fluid thermal conductivity and ∇ is the gradient operator. Steadiness was tested by computing the same simulation case using unsteady solvers and all the results was verified as steady.

Numerical Procedure

Finite volume approach was used to discretise the domain to solve the Navier Stokes and energy equations using the computational fluid dynamics CFD software Star-CCM+. This software has been widely used in studing heat transfer and mixing flow problems [20,38,39] . The system of equations are solved in a segregated manner for pressure and velocity. Momentum and continuity equations coupling were solved in a predictor-corrector approach. The convective terms in the governing equations for momentum and energy are discretized using a second-order upwind scheme. A second-order central scheme was applied for diffusion terms. The algebraic multi-grid (AMG) linear solver was applied to solve the velocity, pressure and temperature with Gauss-Seidel relaxation scheme.

All governing equations were solved iteratively until the solution is converged with 10 -12 residuals values considered as a convergence criteria. In addition, balancing of mass and energy have been checked and checking that there is no difference in temperature contour and velocity streamlines.

Optimization using a Large Space Exploration Design Technique

In this contribution, variation of important number of design parameters with a large range of variation were defined to seek optimal VG's. Around 10 4 simulation cases using a sweep optimization method were performed firstly in order to obtain the optimal design. This huge number of simulation is not performed at the same time. The study was decomposed into three groups each has changed according to the results obtained form the previous one according to the observed heat transfer enhancement via parameters changes. These groups are listed in Table 3.1 with the parameters and different interval ranges being used. This optimization method will path over all the defined VG designs without applying an optimization algorithm to detect just the good designs. The design study is done under the forced convection and laminar flow regime (Re = 1020). Typical objective functions that can be applied in this study are as follows:

1. Friction factor f :

f = (P in -P out ) ρu 2 H L (3.4) 2. Nusselt number N u: N u = hD h k (3.5)
where

h = q A∆T (3.6) q = ṁ C p (T out -T in ) (3.7) ∆T = (T w -T in ) -(T w -T out ) ln[(T w -T in )/(T w -T out )] (3.8)
3. Thermal enhancement factor TEF (ζ):

ζ = N u N u 0 f f 0 -1/3 (3.9)
where h is the heat transfer coefficient and D h = 2H is hydraulic diameter. The subscript 0 in the Nusselt number and friction factor stands for the empty channel geometry results. Nusselt number and friction factor are usually conflicting: if the Nusselt number increases then the heat transfer increases but it may cause an increase in the pressure drop. Thus it is not only efficient only to examine these numbers. For that, the thermal enhancement factor was introduced in order to compare better all the VG's designs. For any value of ζ > 1, heat transfer enhancement occurs.

Mesh Sensitivity

Trimmed cells are generated in the computational domain of the channel. Near the walls, a prism layer meshing strategy is considered in Star-CCM+ in order to cover all the boundary layers. Along with, refinement method is introduced to refine the positions with the most velocity and temperature variation gradients in the channel that are mainly downstream the VG, via surface and curve control.

A surface control is imposed at the inlet and outlet to increase the number of cells with a low growth rate between the refined (smaller) cells and the base cells in the channel. Figure 3.4 shows clearly the refinement techniques applied with the growth rate. For the VG, two refinement methods are used. First, a surface control with a smaller number of cells but with a wake refinement method. In this project, the wake refinement is applied parallel to the flow that is behind the vortex generator.

Second, a curve control with a very small number of cells, with respect to the base size of the mesh, is applied on the curves of the VG. It is generated with a low growth rate that states that the growth from the small cell to the base cell is propagating slowly. This is to ensure that there is no sudden jump in the cell sizes that can be shown in the zoomed cut in figure 3.4.

A mesh sensitivity test is applied using the grid-convergence index (GCI) and the chosen mesh is with 4 × 10 5 cells with 0.2% GCI value [40]. The convergence of Nusselt number and friction factor values with respect to the number of grids is an another method applied for mesh sensitivity; and the results obtained are shown in figure 3.5, it shows that the same mesh can be selected. This mesh test is applied on the case having the maximum adverse pressure gradient that can affect the convergence of the flow. This case has the following dimensions 60 • , 0 • , 100 • , and 100 • for the attack α, roll β, and two base γ 1 , γ 2 angles respectively and 20 mm, 20, and 15 mm for the height h, length l, and shape length SL respectively.

In the optimization studies when the VG shape, roll and attack angles are changed resulting in a new case, the same mesh strategy is applied with the same base size thus a new mesh is generated in each case.

Computational Fluid Dynamics

Simulation Validation

In order to validate the computational model choice, the numerical results obtained from Stephan correlation [20] were compared to present results. Stephan's correlation case is with a smooth channel (developing flow). This is achieved by computing the local Nusselt number N u x as function of x * as the following:

x * = x ReP rD h (3.10)
The local Nusselt number N u x computed from Star-CCM++ is by using equation ??, and N u x from the Stephan correlation (equation 3.14). where

N u

x = h x × D h k (3.11)
h x = q x T w -T x (3.12) T x = u T dA u dA (3.13)
and N u x = 7.55 + 0.024x * -1.14 (0.0179P r 0.17 x * -0.64 -0.14) (1 + 0.0358P r 0.17 x * -0.64 ) 2 (3.14)

where q x boundary heat flux, T x temperature at each position, T w wall temperature, h x local heat transfer coefficient, N u x local Nusselt number, k thermal conductivity, and P r Prandtl number. Wu and Tao [41,42] In order to enhance the behavior of the Pareto frontier, a second optimization method is intro-duced known as SHERPA. It is a hyper-parameter optimization method that include a variety of powerful and interchangeable algorithms [44]. SHERPA algorithm is considered simultaneous hybrid exploration that is robust, progressive and adaptive. It uses multi search methods simultaneously that takes the advantage of the best attributes of each method, and reduces a method's participation in the search if/when it is determined to be ineffective [45]. Thus it allows to implement stopping rules of under-performing trails that have a lower performance based on the Median-Stopping-Rule [46]. The advantages of using SHERPA is that, it is adaptive with the studied field type; it can automatically employ the appropriate algorithms as it proceeds toward the optimal solution. Finally, due to its robustness and effectiveness in finding the optimal solution, the user can develop a huge study border with a large number of variables. A schematic diagram that shows the difference between traditional optimization approach with the SHERPA optimization approach is shown in figure 3.8. In Star-CCM+ the optimization models used in SHERPA cannot be accessed to figure its internal algorithm.

Pareto Frontier

(a) Traditional optimization approach (b) SHERPA method approach.

Figure 3.8: Optimization methods approach differences [47].

In SHERPA optimization, huge number of simulation cases are defined by varying 7 parameters with a small incremental values with respect to the sweep method. The design parameters with their variations are listed in Table 3.3 with the parameters and different interval ranges being used.

The new Pareto frontier curve obtained is shown in figure 3.9 illustrating 1516 designs corresponding to the SHERPA algorithm. To distinguish the optimal design, the maximum value of (ζ) in figure 3.10 is considered, that is denoted by a star ( * ) symbol. By comparing the previous results to these ones, it is clearly shown the difference between the two Pareto frontier curves that stands for an enhancement in the performance in the optimization topology due to the increase the slope which means a better performance of the algorithm . The interesting behavior about the SHERPA method is how it directly concentrates its designs beside the Pareto frontier curve, enabling to better determine the optimal design.

Furthermore, an important parameter that should be taken into consideration is the computation time of the whole simulation. In the SHERPA optimization method, a wide range of the design parameters is investigated by taking a smaller parameter intervals for the predefined parameters.

Each VG design was run on 64 processors and required approximately t = 6 min as a computational time for each simulation. As for the computational time the SHERPA method obtain the optimal design after computing just 1516 simulations. Using SHERPA method, the overall computational time of the Pareto frontier designs was reduced by a factor of 6.18 (compared to the Pareto obtained by the sweep method). The SHERPA Pareto frontier equation is found to be as following:

N u = a 0 + a 1 f + a 2 f 2 + a 3 f 3 (3.15)
where a 0 = -17.795 , a 1 = 698.123, a 2 = -4, 420.115, and a 3 = 9, 568.609.

The dimensions of the optimal design ( * ) symbol of figures 3.9 and 3.10 are listed in Table 3.4.

The unit from the height h to distance between VG's s is in (mm).

The objective function values of the optimal design are summarized in Table 3. to the empty channel geometry (smooth channel).

With respect to the several designs obtained from the literature, the heat transfer for the present optimal design (OD) is enhanced by 14% with respect to the highest ζ value (RWP [23]). Table 3.6

shows a comparison between the present results and these obtained from the literature where DVG travels downstream the VG. Moreover, the effect of vortex C is noticed but with a small mixing effect due to its small strength. Common flow down vortices create down-wash in between, and up-wash flow in the outside regions. Along the downstream direction, the vortices effects decrease while the distance between vortex cores increases which leads to the thinning of the thermal boundary layer.

Another interesting result was observed, is that the mixing in the flow that increases the heat transfer is occurring in all the channel region.

Another method to analyze the development of the vortices is to track each vortex propagation from upstream to downstream parts (X direction) in the channel on (Y -Z) plane locations. This analysis is performed by tracing the location of each vortex to obtain its y and z coordinates at different x/H locations that are as follows:

x/H = 1.5, 1.75, 2.25, 2.5, 2.75, 3, 3.5, 4, 5, 6, 7, 8, and 9. second number for the final x/H section that the vortex is traced in it. The vortices are named as primary and secondary vortices that are the two main vortices, corner vortices that are at the right corner side of the section, and induced vortices that are located beside a main vortex which in general have a small effect on the flow comparing with the primary or secondary vortex.

It is obvious that by tracing the generated vortices in the present optimal design, it covers all the (Y -Z) plane that lead to a whole mixing of the flow, while in the DWP and RWP the vortices do not cover the whole (Y -Z) plane area, it is much smaller. In the DWP design, the generated vortices flow and propagate in a small region in the channel and the primary vortex owes the main effect of mixing the flow due to its generation just downstream of the VG at x/H = 2.75, while the other vortices are generated far away from the VG (at the middle of the channel x/H = 5, 6, and 8) and will have a weak effect on the flow. For corner and induced (A) vortices, they appear for just two x/H sections and then vanish and those have a very weak effect on the flow. Also a formation of secondary and induced (B) vortices beside the wall that have a weak effect on the flow. In the RWP design, the vortex propagation occurs in a bigger region zone with respect to the DWP for the primary vortex, and an induced vortex appears approximately at the leading edge of the RWP (at x/H = 2.25) and then vanishes at x/H = 2.75 which have a small effect on swirling the flow. Corner vortex appears before the trailing edge (at x/H = 2.75) of RWP, and remains to the end of the downstream region being concentrated at the lower surface of the channel. In addition, a generation of secondary vortex at the upper part of the wall side that have a weak effect on the flow. In the present optimal VG design, the most interesting observation is that the vortices cover all the cross section part in the channel that allows to better mix the flow. A difference between this design compared to previous classic designs, is that there are two main vortices, primary and secondary vortices, that have a strong effect on the flow. Those vortices start at the leading edge of the VG x/H = 1.5 and remain until the end of the channel at x/H = 9. An induced vortex in figure 3.13 appears at x/H = 2 and then vanishes at x/H = 5 and then appears at x/H = 7 and have an added weak effect on mixing the flow.

Moreover, a corner vortex in the present design have a good effect on the lower part corner of the channel just after the trailing edge of the VG and remain until the end of the channel. As it is shown above, the shape of the vortex generator and its deviation angles strongly affect the formation and the behavior of the vortices. Comparing the three shapes, it is hard to predict the behavior of the vortex propagation, but only for the main vortices it can be concluded that they are always generated in the left middle part of the channel at the beginning of the leading edge of the VG. The main geometric parameters that affect the formation of the vortices are the height of the VG's and the angle of attack that increase the frontal area that disturb the flow in addition to the roll angles with the base angles that expand the area of the VG towards the leading and trailing edge.

The vortex propagation study is done between three designs to show the different local behaviors of the vortex. An additional analysis is applied to study the variation of the velocity streamlines and temperature distribution between the present optimal design, DWP, and RWP designs at x/H = 4 location chosen due to the behavior just downstream the VG. Figure 3.14 shows the temperature contour and the velocity streamline to show the effect of the vortex on temperature at x/H = 4 location. It's obvious that where the vortex is generated the temperature is affected due to the mixing that results in enhancing heat transfer. For the DWP and RWP designs, a main vortex is found to be at the middle of the section, and it mixes the flow in that region, but for the present optimal design two strong vortices affect the left and right parts that increase the mixing of the flow thus an enhanced heat transfer.

Local Performance Analysis Study

In addition to the global and the flow structure analysis, a local performance analysis is applied to the present optimal design compared to RWP, DWP, and empty channel.

For that reason the local bulk temperature T b was computed as the flow passes through the channel. It represents the mass-flow averaged temperature at each transversal cross section. Figure with respect to the highest temperature found for RWP. The homogeneity of scalar is also crucial especially for exothermal or endothermal chemical reactions, therefore the use of CoV to quantify this homogeneity is applied [33]. The coefficient of variation CoV of temperature is defined in equation 3.16 as the following:

CoV (T ) = σ (T ) θ T (3.16)
where σ (T ) is the temperature standard deviation and θ T is the normalized temperature at each x/H section and is defined as following:

θ T = T w -T b T w -T in (3.17)
Figure 3.17 shows that the coefficient of variation as function of x/H with temperature homogeneity that affect the heat transfer. Comparing all the results of the designs, the present design has the lowest CoV along the channel which indicates the best temperature homogeneity that means better mixing efficiency in this VG design.

Increasing the surface area of the VG lead to increase the opposing area or the block area of the flow in the channel. This increases the value of the friction factor that decreases the value of thermal enhancement factor and also increases the pumping power. Thus managing a value that is acceptable in the range of friction loss is an important goal. Some industries tend to increase the heat transfer despite of increasing the pumping power. The local friction factor f x defined in equation 3.18 was also used to study the pressure drop inside the channel.

f x = P x -P in ρ u 2 H L (3.18)
where P x is the local pressure value at different different cross sections. Figure 3.18 shows the variation of the friction factor f x as function of x/H. For the present study, f x is found to be larger than the other designs. The increase of f x can be explained by the fact of larger surface area of the present VG design. But, looking to the total effect of the present optimum design with respect to the RWP shape, the friction factor value is increased by a factor of 1.2 while the area increased by factor of 2.4;

while comparing the RWP with the DWP the friction factor is increase by a factor of 1.2 also while the area increase by a factor of 2. Thus comparing these results in the present design the surface area is increased but with lower pressure drop value with respect to the classic designs. This is due the effect of the rolling angle β.

Vorticity Flux

In order to better quantify the intensity of the secondary flow (i.e. to examine the effect of the longitudinal vortices), the value of absolute spanwise-averaged streamwise vorticity flux j can be computed [48] as the following:

j = S |ω x | dy dz S dy dz (3.19)
where ω x is the streamwise vorticity and S is the cross sectional area of the channel. A dimensionless value of the absolute cross-averaged streamwise vorticity flux called hereafter vortices density, can be obtained from the following expression:

Ω = j S u H (3.20)
where u is the mean flow velocity. to increase rapidly when reaching the leading edge of the VG to reach a maximum at the trailing edge of the VG, and then downstream the VG it starts decrease slowly. It is much higher with respect to the others throughout the whole channel.

Heat Transfer Performance

Another technique of analysis is to look to the heat performance parameters. LaHaye et al. [49] developed two dimensionless groups. These two parameters are the heat transfer performance factor J and the pumping power factor F as defined by equations 3.21 and 3.22 respectively. These dimensionless numbers were used in order to overcome the problems associated with colburn and friction factors curves when plotted versus Re that tend to vary over a wide range in magnitude along with the slope (for more details [50]).

J = N u Re P r -1/3 (3.21) F = f Re 3 (3.22)
where P r is the Prandtl number.

Conclusion

In the present contribution, large space exploration with multiobjective functions have been conducted

to find an optimal VG design inside a rectangular parallel plate channel under forced convection and laminar flow regime. A new enhanced optimal VG design is proposed as a result of optimization through SHERPA algorithm with (seven design parameters). The overall computational time with this algorithm was reduced by a factor of 6.18 compared to sweep method. In addition to better behavior of the Pareto frontier curve. The optimal design found and was compared to previous VG's from the literature.

Thermal enhancement factor was increased by 35% with respect to an empty channel and about 14% with respect to other designs from the literature. This order of magnitude of intensification is large with respect to the literature. This shows that when the space exploration design is enlarged then, the degrees of freedom increase and this allows finding new innovative design with better performance. A flow topology and temperature distribution analysis were conducted to examine the physical phenomenon behind the heat enhancement. They showed clearly that the proposed new design induces more vortices in the flow due to its new geometric shape that increases the mixing of the flow with the homogeneity of the temperature, thus increasing the overall heat transfer enhancement.

A new analysis was done by tracing the vortex propagation in the channel for the present optimal VG design compared classical designs known as RWP and DWP. This analysis showed that the obtained VG design induces stronger vortices along the channel.

Finally, local analysis was also performed looking into different parameters like the bulk temperature, Nusselt number, coefficient of variation, pressure loss (friction factor), and a normalized vorticity strength. This analysis confirmed our understanding for the reasons why this proposed design is better.

This study was performed as mentioned under forced laminar convection. In further developments different convection regions under different flow regimes will be studied in order to find the optimal design in each case and if they will share the best optimum design as per this contribution.

new optimized VG's and for other classical VG shapes in order to exhibit the physical phenomenon involved in the heat transfer enhancement.

Introduction

Increasing heat exchangers (HX's) performance through heat transfer enhancement and pressure drop reduction leads to energy savings in numerous wide applications worldwide. There are different techniques for heat transfer enhancement classified as passive, active, and dynamic; the active ones e.g. Fiebig et al. [9] studied experimentally the effect of delta and rectangular winglets on a flat plate by observing the fluid flow pattern along with a local performance study on heat and fluid flow coefficients. Gentry and Jacobi [10,11] carried out experiments on delta winglets VG in a channel, showing an increase in the global heat transfer coefficient by 55%, but associated with an increase in pressure drop up to 100% at low Reynolds numbers. Wang et al. [12] experiments illuminated an increase in the overall heat transfer by 10-45% due to longitudinal vortex generator at Reynolds numbers between 3000 and 20000.

Habchi et al. [13] showed the variation of the thermal enhancement factor (ζ) using vortical structures that are generated by inclined baffles in a turbulent pipe flow. Heat transfer intenstification in chemical reactors HX was also studied to monitor exothermal chemical reactions through mixing by the use of offset strip fins [14] and VG's [15]. Guo et al. [16] showed also the effect of using different swirl flow structures to enhance heat transfer by computing the value of ζ. Khanjian et al. [17] examined the effect of using rectangular VG in a flat plate channel on the heat transfer enhancement.

Their study was performed by varying the roll angle (β) in order to find the best β design. Li et al. [18,19] studied numerically and experimentally the effect of using longitudinal VG in a parallel, finless HX. They showed that the use of double triangle VG increases the heat transfer coefficient by 20% at a doubled pressure drop. Tian et al. [20] studied the fluid flow and heat transfer in a flat plate channel with different types of Longitudunal VG's. They investigated common flow down (CFD) and common flow up (CFU) configurations that affected the boundary layer thinning. By comparing the Nusselt number and the friction factor, they found that the CFD configuration has a better overall performance. Moreover, many VG studies dispersly exist in the literature such as the helical and twisted VG [21,22], dimples or protrusions [23,24], cylindrical tubes [25], transverse vortex generators [26], plane or curved surface of VG [27,28], longitudinal vorticity [29], and a longitudinal rectangular wings [30].

Behfard and Sohankar [31] studied the flow structure dynamics for a flow with Delta Winglet Pair (DWP) in a finned circular heat exchanger using a steady SST k -turbulence model. They showed an increase in the convective heat transfer ratio and thermal performance factor by about 59% and 43% , respectively. Zhao et al [32] studied the Rectangular Winglet Pair (RWP) in an oval tube heat exchanger employing RNG k -turbulence model. Oneissi et al. [33] investigated a new design of VG called inclined projected winglet pair which enhanced ζ by 30% with respect to the empty channel at

Reynolds number Re = 4600 (using the SST k -ω model).

Several parametric and shape optimization techniques were applied for optimizing VG's. For example, Kim and Choi [34] performed shape optimization of rectangular channel with dimples by applying a surface response method. Samad et al. [35] optimized the shape of channel dimples on a single surface using different surrogate models. They improved heat transfer and friction-loss by approximately 20%. Khan and Li [36] obtain an optimized vortex generator design by applying a Bayesian-regularized artificial neural network (BRANN) coupled to a multi-objective optimization algorithm (MOA). Song et al [37] studied heat sink design optimization based on surrogate-based algorithm technique. They developed a Kriging model to minimize the objective function (thermal resistance) as function of the design variables. Karkaba et al. [38] performed a large space exploration study with multiobjective functions and seven design parameters were considered. A novel enhanced VG was found with thermal enhancement factor 35% relative to an empty channel and 14% better than literature results.

In recent years, many authors vary the forms of VG's which are initially quite simple and limited to rectangular, or triangular shapes or even protrusions of basic forms. Several authors perform parametric form optimization by choosing a simple form (triangular, or rectangular) and by varying one or two parameters of the shape in general of the angles of the VG's in regard to the direction of the flow. The set of results of the bibliography on these variations of shape shows that the performances of the VG's are sensitive to these geometrical variations and that it is possible to increase substantially the thermo-hydraulic performances of the exchangers by using these optimized forms.

In this contribution, the objective is to shed the light on the optimal design of VG's and its possible dependence on the type of the flow regime. For this reason, VG shape optimization is performed at a turbulent flow regime with a large number of parameters (seven parameters) as performed in [38].

The knowledge of the variation of the optimal shape of the vortex generator as a function of the flow regime is a crucial element in the sizing of the exchangers in order to optimize energy efficiency. Local analysis is performed on the flow structure highlighting the main difference with a traditional geometry known as delta and rectangular winglet pairs (DWP and RWP) respectively for flow at different Re values. Pairs of VG's were considered with a common flow down (CFD) configuration as it is the most studied configuration due to its favorable physical behavior. The following parameters are also modified: the length of the VG (l), its height (h), and its two base angles (γ 1 and γ 2 ) as shown in figure 4.2-a. The angle of attack α may vary in the (x, y) plane, and the roll angle β in the (y, z) plane.

Computational Fluid Dynamics

Governing Equations, and Boundary Conditions

The flow field in the channel is unsteady and three dimensional. The fluid is considered as incompressible with constant fluid and thermal properties. The values of Reynolds numbers considered in the present study are Re 1 = 4600, Re 2 = 9920, and Re 3 = 30050. It was shown in previous studies [39,40] on similar flow physics that the heat transfer enhancement is a macromixing process caused by the counter-rotating longitudinal vortex pairs, and that the wall hairpins vortices do not have a significant effect on the heat transfer enhancement. where v is the velocity vector, Cp the specific heat at constant pressure, µ the fluid dynamic viscosity, µ t is the turbulent viscosity, T is the temperature, ρ the fluid density, ∇ is the gradient operator, and As for the turbulence properties, the turbulent intensity factor is T u = 5% usually taken as turbulence characteristic of the increasing airflow in such designs with a turbulent length scale: T uL = 0.07D h = 2.8mm , with turbulent Prandtl number: P r t = 0.9 , and normal defined turbulence factor constants.

P

Numerical Procedure

In this contribution numerical procedure used are as [38]. While, for the unsteadiness and turbulence the following numerical models are used. Firstly, the convective terms in the governing equations defined for momentum, energy, and k -ω energy equations are discretized using a second-order upwind scheme. Secondly, an option with all y + wall treatment for turbulence model is chosen with the projection model to ensure the viscous sublayer is meshed due to the adverse gradients beside the walls. Thirdly, all governing equations are solved with implicit unsteady solver where the maximum physical time t computed from the formula t = 10 L/u in (entrance length) with a time step t s = t/100 and 15 inner iterations. Finally, for the convergence criterion, all the governing equations are solved iteratively until the residuals stabilized at a very small value << 1.

Mesh Sensitivity

Trimmed cells are generated in the computational domain of the channel with all the methods for mesh refinement as stated in [38].

A mesh sensitivity test is applied using the grid-convergence index (GCI) and the chosen mesh contains 1.2 × 10 6 cells with less than 2% GCI value [41]. The convergence of Nusselt number and friction factor values with respect to the number of grids is an another method applied for mesh sensitivity; and the results obtained are shown in figure 4. 4. It shows that the same mesh can be selected. The y + values for all of the design cases has a value smaller than 2 beside the VG and below than 0.5 beside the walls that are considered typical. This mesh test is applied on the case having the maximum adverse pressure gradient that can affect the convergence of the flow in addition to the unsteadiness of the design variables due to turbulence at Re = 9920. After testing more than 5 cases the following case is chosen with those dimensions 45 • , 0 • , 90 • , and 90 • for the angle of attack (α), roll angle (β), and two base angles (γ 1 , γ 2 ) respectively and 10 mm, 20 mm, and 10 mm for the height (h), length (l), and shape length (SL) respectively.

In the optimization studies when the VG shape, roll and attack angles are changed resulting in a new case, the same mesh strategy is applied with the same cell base size thus a new mesh is generated in each case.

Results and Discussion

Simulation Validation

In order to validate the computational model choice, the numerical results obtained from Stephan correlation were favorably compared to the present cases, for more details see [38].

Wu and Tao [46,47] studied experimentally and numerically the laminar convection heat transfer in a channel with longitudinal vortex generator. The experimental and numerical results obtained by Wu and Tao [46,47] are compared to the present results as shown in Figure 4.5. The minimum and maximum relative difference in the heat transfer coefficient h (for the two numerical results) are 0.4% and 2.3%, respectivley. The minimum and maximum error, between the experimental results of Wu and Tao [46,47] and the present numerical results are 5.3% and 8.7% respectively. This is taking into consideration that there is a 5% uncertainty in the experimental values as reported by Moffat [48].

There is an agreement between the results, which proves that the models and methods are validated.

Optimization using a Large Space Exploration Design Technique

In this contribution, variation of important number of design parameters (seven parameters) with a large range of variation are defined to seek optimal VG's for different Re with the SHERPA method.

SHERPA is a hyper-parameter optimization method that includes a variety of powerful and interchangeable algorithms [49]. SHERPA algorithm is considered simultaneous hybrid exploration that is robust, progressive and adaptive. For more details about SHERPA optimization method refer to [38].

The set of parameters used in all Re's cases are listed in Table 4.1 with the interval ranges being used. The objective functions used for optimization are the friction factor f , Nusselt number N u, and thermal enhancement factor TEF (ζ) defined in [38].

The results show that the designs with the highest values of thermal enhancement factor (ζ) have a border values for the defined parameters; thus it is possible to obtain a better design at a larger design space of parameters. For example, for the optimization study done at Re = 30050, the best design obtained has the following dimensions 25 • , 60 • , 130 • , and 150 • for the attack angle α, roll angle β, and two base angles γ 1 , γ 2 respectively, and 14 mm, 30 mm, and 19 mm for the height h, length l, and shape length SL, respectively. By comparing those dimensions with the parameters defined in Table 4.1, it is shown that the roll angle β, and the two base angles γ 1 , γ 2 are at the maximum defined range, and for the height h it is at the minimum defined range. This issue is observed in all the studies, thus the parameter ranges are defined again and the optimum design is observed in each Re case. If it includes a border value parameter, a design set is defined and optimization executed until the best design is obtained iteratively. Tables 4 4.2) after running 3360 designs and overall time of 1984 hours. The OD for Re 2 = 9920 is found from Case 2 (Table 4.3) after running 3092 designs and overall time of 1400 hours. And the OD for Re 3 = 30050 is from Case 4 (Table 4.4) after running 3755 designs and overall time of 1557 hours.

The Pareto frontiers are found to fit well with polynomial law of fourth order (equation 4.4) and for each Re the equation constants are given in table 4.5:

N u = a 0 + a 1 f + a 2 f 2 + a 3 f 3 + a 4 f 4 (4.4)
The dimensions and shapes of the optimal designs ( * symbol of The results obtained for the optimization studies at different Re are compared to four results: classical vortex generators known as DWP and RWP and empty channel computed in this present study and the IPWP design, i.e. inclined projected winglet pair vortex generators publised by Oneissi et al [33]. RWP and DWP designs are well used in several literature studies [3] and have the following

geometric parameters α = 45 • , β = 0 • , γ 1 = 90 • , γ 2 = 90 • , h = 0.5H, l = H,

Qualitative Local Performance Analysis

In order to better analyze the heat transfer enhancement mechanisms involved when using vortex generators, induced vortices are investigated. For that reason, flow topology and temperature distribution should be studied with the occurrence of the flow changes. In Figure 4.9 at Re 1 = 4600, vortex A is induced just at the upstream tip of the VG, and starts to propagate downstream. In addition to vortex A, a second vortex B is also induced at the lower side of the VG beside z/W = 0. It propagates downstream the VG and shifts gradually from y = 0 to approximately beside y = H/2. Those vortices have a main effect in mixing the flow at a high intensity leading to a better heat transfer. As for the temperature contours, it is clearly noticed that at VG upstream tip, the thermal boundary layer at the upper and lower walls change once the flow reaches the VG. In other words, the vortices disturb strongly the boundary layer development on the walls. Thus heat transfer is enhanced due to the mixing effect but taking into consideration the weak effect of thinning the thermal boundary layer. In addition to the main vortices, it is observed that small vortices are induced inside the channel at different positions (corners of the channel), between the main vortices, symbolized by on Figure 4.9. Those vortices travel downstream the VG towards the outlet of the channel, and have a noticeable effect on mixing. These are known common flow down vortices that create a down-wash in between, and up-wash flow mainly beside the walls of the channel.

Along the downstream direction, the vortices effects decrease while the distance between vortex cores increases. Another interesting result was observed, is that the mixing in the flow that increases the heat transfer is occurring in all the channel parts due to the strong turbulence occurring inside the channel. For Figures 4.10, and 4.11 with higher Re it is clearly observed the effect of the VG's shape on inducing vortices. That results in mixing the flow and increasing the heat transfer but with a weaker effect compared with lower Re (Figure 4.9) due to VG shape and its lower turbulence level. Another method to analyze the development of the vortices is to track their formation and effect on the flow at different successive locations. This method is known as Lambda 2 (λ 2 ) criterion.

Jeong and Hussain [53] proposed that a vortex corresponds to a region where two eigenvalues of the symmetric part of the square of the velocity gradient are negative. a stronger turbulent flow disturbance. The vortex grows a lot while it moves downstream the channel mainly at the location of the large main vortex A, with a strong effect compared to other designs.

While in the second case at Re 2 = 9920, the effect of the vortex core decreases due to the weaker effect of turbulence with the higher Reynolds number value that decrease slightly the heat transfer rate by increasing the friction factor. This phenomenon can be seen in the third case at Re 3 = 30050 where less vortical structure induced is compared to the other cases. It is concentrated around the main vortex due to the difference in VG designs that is according to β parameter value, that makes the VG more inclined with respect to the lower surface of the channel.

Quantitative Local Performance Analysis

In addition to the qualitative analysis, a quantitative local performance analysis is applied to the obtained optimal design compared to empty channel, and classical vortex generators known as DWP and RWP. RWP and DWP designs are well used in several literature studies [3] and have the following In Figure 4.15 at Re 3 = 30050, it is noticed that at the end of the channel there is no improvement with respect to the RWP design. Both the optimal design and RWP have approximately the same value of temperature. Thus, there is a need to seek and understand why ζ is enhanced in such design while the temperature remains the same. For that reason, local the Nusselt number, defined in equation (4.5), is computed and presented in figures 4.16, 4.17, and 4.18 as function of x/H locations.

dimensions α = 45 • , β = 0 • , γ 1 = 90 • , γ 2 = 90 • , h = 0.5H, l = H,
N u x = h x D h k (4.5)
where

h x = q x T w -T x (4.6)
and

T x = u T dA u dA (4.7)
where q x is the boundary heat flux, T x the temperature at each position, T w the wall temperature, h x the local heat transfer coefficient, N u x local Nusselt number, k the thermal conductivity.

In Figure 4.16 at Re 1 = 4600, it is well noticed that the value of the local Nusselt number for the optimum design is much higher than all the other designs and the empty channel. For all designs it jumps at the leading edge of the VG. For the optimum design, a higher Nusselt number is observed at

x/H = 1 to x/H = 2.75. In addition, N u levels remain higher than other designs all over the channel and this is due to the strong effects of the vortices induced in such design. At Re 2 = 9920, similar conclusions can be drawn from figure 4.17.

In figure 4.18 at Re 3 = 30050, the local Nusselt number has a higher values compared to other designs just at x/H = 1 to x/H = 2, and then at x/H = 2.5 it remains the same as DWP and an obvious higher value is observed for the RWP case. In this case the heat transfer performance cannot be considered enhanced with respect to other designs, thus another parameters shall play the role in improving the overall heat transfer.

The local friction factor f x (defined in equation 4. a value that is acceptable in the range of friction loss is an important goal. The friction factor is given by:

f x = P x -P in ρ u 2 H L (4.8)
where P x is the local pressure value at different cross sections, P in is the inlet pressure, and u is the uniform inlet velocity.

Figure 4.19 shows the variation of f x as function of x/H at Re 1 = 4600. For the present study, f x is found to be larger than the other designs. As the new design of VG exhibit larger surface area, it is interesting to analyze the increase in f x with the increase of surface area compared to ordinary designs. But, looking to the present optimum design with respect to the RWP shape, the friction factor value is increased by a factor of 1.7 while the area is increased by a factor of 3.3. This area increase affects strongly the formation of vortices and thus increasing the heat transfer. Comparing the RWP with the DWP, the friction factor is increased by a factor of 1.1 while the area is increased by a factor of 2. Thus, the present design surface area is increased but with lower pressure drop value with respect to the classic designs. This can be explained by a contribution effect from the rolling angle β. In Figure 4.20 at Re 2 = 9920, it is clearly shown that the friction factor in the optimum design is lower than that of RWP for x/H between 2.5 and 5, and higher than that of DWP. By taking the relation of increasing the area, the area increased by a factor of 2.7 w.r.t. RWP and 5.4 w.r.t. DWP while the friction factor increases by a factor 1.1 and 1.7, respectively.

The optimum design at Re 3 = 30050 has the lowest friction factor, as shown in Figure 4.21, w.r.t.

to RWP and DWP (decrease by a factor of 0.5 and 0.8, respectively). It is slightly higher than the

Vorticity Flux

To better quantify the intensity of the secondary flow (i.e. to examine the effect of the longitudinal vortices), the value of absolute spanwise-averaged streamwise vorticity flux j is computed [54] as the following:

j = S |ω x | dy dz S dy dz (4.9)
where ω x is the streamwise vorticity, and S is the cross sectional area of the channel. A dimensionless value of the absolute cross-averaged streamwise vorticity flux called hereafter vortices density, can be obtained from the following expression:

Ω = j S u H (4.10)
where u is the mean flow velocity. In Figure 4.24 at Re 3 = 30050, the vorticity flux Ω is approximately the same compared to RWP, but higher compared to DWP. At the leading edge of the obtained optimal design, the vorticity flux is higher than that of the RWP due to the geometric difference in the position of the starting of the leading edge of the VG. It remains higher till reaching x/H = 3.5 where Ω, for the optimal design, keep decreasing. For RWP, it fluctuates between x/H = 3 and x/H = 7.5 until it decreases toward the end of the channel. It is noticed that Ω is higher compared to DWP, but is approximately the for the optimal design and others, respectively).

Heat transfer performance analysis

LaHaye et al. [55] developed two dimensionless groups: the heat transfer performance factor J, and the pumping power factor F (equations 4.11 and 4.12, respectively). These dimensionless numbers were used in order to overcome the mislead associated with Colburn and friction factors curves when plotted versus Re that tend to vary over a wide range in magnitude along with the slope (for more details see [56]). J = N u Re P r -1/3 (4.11)

F = f Re 3 (4.12)
where P r is the Prandtl number.

Conclusion

In the present contribution, large space exploration with multiobjective functions has been conducted

to find an optimal vortex generator (VG) design inside a rectangular parallel plate channel under forced convection and at different Reynolds numbers (Re 1 = 4600, Re 2 = 9920, and Re 3 = 30050).

A new enhanced optimal VG design is proposed with seven design parameters. The optimal designs found at different Reynolds number are compared to previous VG's from the open literature.

Results show that thermal enhancement factor is increased by 34.7%, 23.9%, and 10.9% relative to an empty channel and 3.5%, 4.2%, and 14.4% better than literature results for different Re (Re 1 = 4600, Re 2 = 9920, and Re 3 = 30050) respectively. This order of magnitude of intensification is large with respect to the literature. This shows that when the space exploration design is enlarged then, the degrees of freedom increase and this allows finding new innovative design with better performance. Flow topology and temperature distributions analysis are conducted to examine the physical phenomenon involved in the heat enhancement. It is found that the proposed new designs induce more vortices in the flow due to their new geometric shapes that increase the mixing of the flow with the homogeneity of the temperature, thus increasing the overall heat transfer enhancement.

Further detailed analysis is done by tracing the vortex propagation in the channel for the present optimal VG design compared to classical designs known as RWP and DWP. This analysis showed that 123 the obtained VG design induces stronger vortices along the whole channel.

Finally, local analysis is also performed looking into different parameters like the bulk temperature, Nusselt number, pressure loss (friction factor), and a normalized vorticity strength. This analysis confirmed our understanding for the reasons why this proposed design is better and highlighting the main effect of the friction factor values in cases with high Reynolds number.

This study is performed as mentioned under forced turbulent convection. In further developments, different convection regions under different flow regimes will be studied in order to find the optimal design in each case and if they will share the best optimum design as per this contribution.

Introduction

Mixed convection heat transfer enhancement in plate channel flow is an important topic due to its various industrial applications such as electronic cooling, and compact heat exchangers (HX) used in solar systems [1,2,3]. Many scientists have been studying heat transfer enhancement in channels by adding a variety of vortex generators (VG's) shapes and types [4] such as rigid VGs [5,6], flexible VGs [7,8]; and turbulators such as fins, ribs, baffles, and other devices placed in inline or staggered arrangements [9]. These methods enhance mixing and heat transfer throughout the channel.

Different shapes of ribs have been investigated in the literature. They generate a secondary flow and a counter-rotating vortices that enhance the heat transfer. The basic shapes of ribs are: inclined, transverse, V-shape, and arc-shaped ribs [9]. They can be inserted in various configurations as downstream or upstream, discrete or staggered which can form many geometrical forms such as circular, rectangular, pentagonal, trapezoidal, etc. Han et al. [10] studied transverse-cornered, transverse and inclined ribs. They investigated the rib-roughened surfaces and determined the effects of rib shape, angle of attack and pitch to height ratio on friction factor and heat-transfer rate in a parallel plate geometry. They found that ribs at 45 • angle of attack have a superior heat transfer performance at a given friction power when compared to 90 • angle of attack and to a sand-grain roughness.

Henniche and Korichi [11] investigated numerically the heat transfer enhancement in a vertical channel with inclined baffles under the mixed convection regime. They found that heat transfer rate increases up to 1.4 times in the steady-state, while it increases up to 3.5 times in the unsteady-state. This is thanks to the inserted baffle height at different baffle tilt angles compared to a smooth channel.

Habchi et al. [12] showed the variation of the thermal enhancement factor (ζ) while using vortical structures generated by inclined baffles in a turbulent pipe flow, using three different configurations.

Skullong and Promvonge [13] studied experimentally the delta-winglet (DW) vortex generator inserted in two transverse pitches, with three angle of attacks, and at different relative heights in a solar air heater channels. The effect of the shapes on heat transfer and friction are investigated for Reynolds numbers between 5000 and 24000. They showed that when using DW with 30• at b/H = 0.5 yields to the highest heat transfer rate and friction factor. Oneissi et al. [14] investigated a new design of VG called inclined projected winglet pair which enhances ζ by 30% with respect to the empty channel at Reynolds number Re = 4600. Khanjian et al. [15] studied the effect of using rectangular VG in a flat plate channel on the heat transfer enhancement which induce an increase in the heat transfer. This study was performed by varying the roll angle (β) in order to find the best design.

Mixed convection cases are rarely studied in the literature. Kennedy and Zebib [16] studied the mixed convection between horizontal parallel plates with a flash mounted local heat source. Bratten and Patankar [17] studied numerically the laminar mixed convection in shrouded arrays of heated rectangular blocks mounted on the horizontal shrouds, spaced periodically in the spanwise direction.

Kim et al. [18] conducted numerical investigations on the mixed convection from multiple-layered boards with cross-streamwise periodic boundary conditions in a horizontally and vertically oriented channels.

Wu and Perng [19] studied heat transfer enhancement of mixed convection for heated blocks using vortex shedding generated by an oblique plate in a horizontal channel. Maughan and Incropera [20] investigated the use of VGs and ribs for heat transfer enhancement in a mixed convection flow in a uniformly heated horizontal channel. They found that using VGs have the same effect of buoyancyinduced flow at low Rayleigh numbers (Ra) and enhanced heat transfer at the top surface. Although, at higher Ra, perforated ribs were found more effective than VGs in heat transfer enhancement.

In the present contribution, the objective is to investigate the laminar and turbulent mixed convection for heat transfer enhancement by the use of VGs. The main focus is to increase the heat transfer while decreasing the pressure drop that is found due to the reversibility induced in empty channels in mixed convection cases. Thus, reversibility in the flow occurring in an empty channel at different Richardson number is investigated. The effect of inserting various number of VG sets is quantified inside the channel on the heat transfer rate and friction factor. To do so, local analysis is performed on the flow structure induced by the new VG's compared to a traditional geometry known as delta and rectangular winglet pairs (DWP and RWP) respectively. Symmetry is imposed at the upper surface of the channel (y = H). The applied boundary

Computational Fluid Dynamics

Governing equations

The flow field in the channel is unsteady and three dimensional. The working fluid is air and is assumed to be incompressible. All thermo-physical properties of the fluid have been kept constant, except the The solution for the flow and thermal fields obtained from the numerical simulations are postprocessed by computing the values of friction factor and Nusselt number as follows:

f = (P in -P out ) ρu 2 in H L (5.5)
and

N u = hD h k (5. 6
)
where P is the pressure, and h is the heat transfer coefficient defined as:

h = q A∆T (5.7)
where

q = ṁ C p (T out -T in ) (5.8) and ∆T = (T w -T in ) -(T w -T out ) ln[(T w -T in )/(T w -T out )] (5.9)
where q is the amount of heat exchanged between outlet and inlet, and ṁ = ρAu in is the mass flow rate.

Empty Channel Simulation

In this section, the behavior of the flow inside a vertical heated channel is investigated and verified based on previous studies for heated channels. The main goal is to distinguish when mixed convection

Mixed convection analysis for laminar flow regime

Heat transfer enhancement using vortex generator

In this contribution, heat transfer enhancement is the main objective where VG is added to the vertical channel with different shapes in order to find the best possible design in terms of overall performance enhancement. The chosen conditions are Ri = 2.7 and Re = 340 for which there is a flow reversal starting from the middle of the channel until its end.

VG optimization studies were performed in order to obtain an optimum design for a defined flow regime and type of convection. Karkaba et al. [21] performed optimization studies for laminar and turbulent flow regimes at different Reynolds number. It was found that the optimum design shape varies according to the case studied and the flow regime. For example, under turbulent flow regime at high Reynolds numbers, the optimum VG shape was found at a higher value of roll angle (β) that decreases the friction factor compared to the same VG design but with lower β angle. Thus for this seek, the VG dimensions in laminar mixed convection should be chosen precisely for the case conditions. Thus based on several simulation cases a VG design is proposed that enhances the heat transfer in the channel along with slightly breaking of the flow reversal beside the outlet.

The dimensions of the proposed VG design are listed in Table 5.1 and its shape is shown in Figure 5.10 In order to distinguish that the proposed design is a enhanced one, thermal enhancement factor (ζ) defined in equation 5.13 is introduced. For any value of ζ > 1, heat transfer is considered to be enhanced.

ζ = N u N u 0 f f 0 -1/3 (5.13)
where the subscript 0 in the Nusselt number and friction factor stands for the empty channel geometry results.

The Nusselt number (N u), friction factor (f ), and thermal enhancement factor (ζ) are computed for the new design and compared to empty channel, and classical vortex generators known as (DWP and RWP). The results obtained are summarized in Table 5. ) that means 0.3% enhancement. But, as observed the value of the N u for the present design is slightly higher than that of the empty channel. And that can be explained that in the empty channel, the heat is enhanced due to the presence of the transverse vortices at the end of the channel, but that increases the pressure drop in the channel and this is obvious by a higher value of the friction factor. While in the present design, the heat is enhanced due to the formation of streamwise vortices along the flow, but these vortices doesn't affect highly the fluid pressure like the transverse vortices. And the presence of the streamwise vortices in the present design breakdown the transverse vortex at the end of the empty channel which will be discussed in the next sections. In order to observe the heat transfer enhancement in the channel, figure 5.12 shows the temperature contours at y/H = 0.5. It can be clearly observed that for the empty channel and all the three VG designs the temperature is increased at the outlet of the channel due to the flow reversal. But it is obviously shown that for the present proposed design, the temperature is increased at the leading edge of the VG. That is due to the formation of the streamwise vortices in a higher rate in such design with respect to the RWP and DWP that has a lower effect rate on the temperature mixing. Thus it is better to investigate the streamwise vortices induced by the VG and its enhancing effect on the heat transfer.

Velocity streamlines and temperature contours at different x/H locations are post-processed for the new proposed design to better analyze and understand the heat transfer mechanism by means of cross-section based analysis. x/H at Re = 340. It can be seen that the temperature is approximately the same for all the cases.

But, the curve for the present proposed design is slightly deviated from the other curves at the leading edge of the VG and remains deviated until becoming coherent again at x/H = 13 till the end. This deviation can be explained by the formation of the flow reversal that affects the temperature and it increases its slope.

The main parameter used to verify the heat transfer enhancement is the Nusselt number defined in equation 5.10. The increase in the value of the friction factor decreases the value of thermal enhancement factor and also increases the pumping power. The local friction factor f x is defined in equation 5.14.

f x = P x -P in ρ x u 2 x H L (5.14)
Figure 5.17 shows the variation of the friction factor f x as function of x/H. The value of the friction factor for the present proposed design is higher in the location where the streamwise vortices affect the pressure drop in the system. While at the location of the flow reversal the friction factor

Effect of the VG numbers on heat transfer enhancement

In this section, the effect of inserting several VGs in the channel on heat transfer enhancement is investigated. As shown previously, the presence of the reversibility in the empty channel, enhances The study is performed with the same present proposed design and also compared to the empty channel. This study includes different number of VG's inserted in the channel starting with a single VG as in the previous section, then doubling the number to be two VGs, then doubling them to be four VG's and finally adding two more VGs to be six VGs. location is downstream a VG as shown in Figure 5.19, that means x/H = 3 is downstream the first VG, and x/H = 5 is downstream the second VG when using two VGs and so on. It is worth-noting the effect of the VG is shown after the trailing edge of the VG and has no effect in the backward direction. This can be observed for example when using six VGs where the same velocity streamline at x/H = 5 is for the two others sets (two and four VGs). These induced vortices will have a different effect on mixing the flow and thus increasing the overall heat transfer.

In order to quantify the strength of the streamwise vortices on mixing the flow, the temperature contours at x/H = 3, 5, 7, 9, 11, and 13 are presented in Figure 5.22. It is observed that by adding more VGs inside the channel, the streamwise vortices have higher residential time values, as the flow passes through the channel. Moreover as inserting more VGs, the temperature is increased and thus the heat transfer is enhanced. In the empty channel and using a single VG, the velocity keep increasing until reaching a value around 0.33 m/s (and a lower value when using double VGs around 0.29 m/s). This can be explained by the presence of the large transverse vortex in the empty channel when using a single VG, and the presence of a transverse vortex when using double VGs.

Mixed convection analysis for turbulent flow regime

In this section, we investigate the performance of the obtained designs under a turbulent mixed convection flow regime. The idea is to re-evaluate the performance of the previous designs, obtained under laminar mixed convection, of previous section when performing under a turbulent mixed convection flow regime. For this reason, turbulent mixed convection study is performed at two Reynolds numbers For the numerical procedure used in this case, the same one is used as in Karkaba et al. [21]. The convective terms in the governing equations of momentum, energy, and k -ω equations are discretized using a second-order upwind scheme. Wall treatment is applied using y + models and mesh refinement near the walls. The governing equations are solved with implicit unsteady solver where the maximum physical time t is estimated as t = 10 L/u in (entrance length) with a time step t s = t/100 and 15 inner iterations. For the convergence criterion, all the governing equations are solved iteratively until the residuals stabilized at a very small value << 1.

As for the turbulence properties, a turbulent intensity of 5% is applied with a turbulent length scale: T uL = 0.07D h = 2.8mm , and turbulent Prandtl number: P r t = 0.9. It is noticed in the temperature contours for both designs, upon the formation of a streamwise vortex, the temperature starts to increase leading to a better heat transfer enhancement.

The

The local bulk temperature T b was computed as the flow passes through the channel for the two cases. It represents the mass-flow averaged temperature at each transversal cross section. Figure 5.32 shows the variation of f x as function of x/H at Re 1 = 1240. Using the present design, f x is found to be higher than that obtained by using the other designs. But considering the overall performance ζ, the present design outer-performs all the other designs.

In Figure 5.33, the variation of f x as function of x/H at Re 2 = 3810 is presented. It can be observed by inserting the proposed vortex generator, the friction factor increases, but both RWP and DWP designs exhibits the lower f x . For the present optimum design, the maximum pressure drop is occurring. This is expected and can be explained by the large surface area of the present VG with respect to the VGs.

Conclusion

In the present chapter, the focus is on the heat transfer enhancement using VG's under the mixed

General Conclusion and Perspectives

The present PhD thesis is carried out at the Department of Energétique Industrielle of Institut Mines-Télécom Lille Douai (France) in collaboration with the Mechanical Department of Lebanese International University. This project is titled as:

"Large design space exploration for designing optimal vortex generators for heat transfer enhancement"

In this doctoral thesis, a detailed overview about different heat exchangers types is presented in chapter 1. In addition to the heat quantification methods that can be applied inside a heat exchanger in order to analyze the heat enhancement inside it. The thermal enhancement factor (ζ) quantification method is chosen to be considered in the present contribution along with analyzing the thermal effects by computing the Nusselt number (N u), and the flow effects through the friction factor (f ). Moreover several enhancement techniques are also presented that can be classified as active or passive. Then the light is shed on the passive method will be considered in this study that is vortex generators (VGs).

After that, a well literature was performed for investigating all the types of VG's that can be used in heat enhancements that are classified according to its shape, size, configuration and style. In addition if the VGs are inserted with a common flow down or up configurations, and finally the VG style i.e.

it is a winglet that it is punched from the HX surface or not. Due to the different types of VGs that can be investigated, optimization methodologies were introduced in order to minimize the work and time for obtaining a new novel enhanced vortex generator. Simultaneous Hybrid Exploration that is Robust, Progressive and Adaptive (SHERPA) algorithm is selected to perform the optimization study under a large exploration designs of the geometric and shape parameters of the VG. Then, convection heat transfer types were presented that should be chosen precisely with the flow regime type whether laminar or turbulent in order to analyze the system correctly, by applying the suitable governing equations and discretization methods to the model studied that is discussed in chapter 2.

The study objectives is obtaining a new novel design firstly in forced laminar convection, then forced turbulent convection that can be used in various applications like solar collectors. But since these applications undergoes different type of convection and the studies carried out in the bibliography are not satisfactory. That is why, different type of convection with a heat intensification by VG is studied.

Firstly by investigating mixed convection in an empty channel, then heat enhancement using VG with mixed convection under laminar and turbulent flow regimes.

In chapter 2, an overview is presented on the computational fluid dynamics methods and strategies that are used in the software cascaded computers. In addition, the Governing equations are listed with all the numerical methods and strategies used to firstly define the geometry and physical properties of the design then how computational domain is divided into fine cells (meshing), inaddition to the refinement strategies used, and how to apply the discretization methods of the governing equations over the selected mesh. Various mesh types that can be used are also presented, in addition to the solution methods with their accuracy methodologies. Finally a brief background about turbulence models that are used through decades with a detailed information about the k -ω turbulent model that is selected to be used in this project with its equations. Finally, a detailed overview about optimization methods used in Star-CCM+ is presented with the selected type of optimization method that is chosen to be SHERPA.

In chapter 3, the first objective is achieved; through a large space exploration design with multiobjective functions that have been conducted to find an optimal VG design inside a rectangular parallel plate channel under forced convection and laminar flow regime. A new enhanced optimal VG design is proposed as a result of optimization through SHERPA algorithm with (seven design parameters). An optimal design is found and compared to previous VG's from the literature. Thermal enhancement factor was increased by 35% with respect to an empty channel and about 14% with respect to other designs from the literature. This order of magnitude of intensification is large with respect to the literature. This shows that when the space exploration design is enlarged then, the degrees of freedom increase and this allows finding new innovative design with better performance. A flow topology and temperature distribution analysis are conducted to examine the physical phenomenon behind the heat enhancement. They showed clearly that the proposed new design induces more vortices in the flow due to its new geometric shape that increases the mixing of the flow with the homogeneity of the temperature, thus increasing the overall heat transfer enhancement. In addition, a new analysis technique is done by tracing the vortex propagation in the channel for the present optimal VG design compared classical designs known as RWP and DWP. This analysis showed that the obtained VG design induces stronger vortices along the channel. Finally, local analysis is also performed looking into different parameters like the bulk temperature, Nusselt number, coefficient of variation, pressure loss (friction factor), and a normalized vorticity strength. This analysis confirmed our understanding for the reasons why this proposed design is better.

The first object is performed under forced laminar convection. The second objective is to spread the optimization study over turbulent forced convection flow regime. Thus, also in chapter 4, a large space exploration with multiobjective functions has been conducted to find an optimal vortex generator (VG) design inside a rectangular parallel plate channel under forced convection and at different Reynolds numbers (Re 1 = 4600, Re 2 = 9920, and Re 3 = 30050). A new enhanced optimal VG design is proposed with seven design parameters. Results show that thermal enhancement factor is increased by 34.7%, 23.9%, and 10.9% relative to an empty channel and 3.5%, 4.2%, and 14.4% better than literature results for different Re (Re 1 = 4600, Re 2 = 9920, and Re 3 = 30050) respectively. Flow topology and temperature distributions analysis are conducted to examine the physical phenomenon involved in the heat enhancement. It is found that the proposed new designs induce more vortices in the flow due to their new geometric shapes that increase the mixing of the flow with the homogeneity of the temperature, thus increasing the overall heat transfer enhancement. Finally, local analysis is also performed looking into different parameters like the bulk temperature, Nusselt number, pressure loss (friction factor), and a normalized vorticity strength. This analysis confirmed our understanding for the reasons why this proposed design is better and highlighting the main effect of the friction factor values in cases with high Reynolds number.

In chapter 5, the final objective that is heat transfer enhancement using VGs under mixed convection is achieved.

Perspective

In the present doctoral study, heat transfer enhancement in heat exchangers is achieved by generating vortices that enhance the mixing in the flow by inserting vortex generators (VGs). The heat transfer enhancement is performed in forced and mixed convection types with laminar and turbulent flow regimes. Shape optimization is introduced in order to seek for an optimal VG design with a large space exploration design via multi-objective functions with seven design parameters.

As a perspective of this doctoral thesis, an additional objectives can be achieved around the studied subjects. Firstly, heat transfer enhancement in natural (free) convection by VG can be performed.

Thus, optimization study can be applied in such study in order to figure out the best VG design that stands for the highest heat transfer enhancement. Then, experiments can be performed for all the cases studied using particle image velocimetry (PIV) experimental method. In order to validate and defend the obtained results at IMT Lille Douai labs. Finally, an extend in the optimization studies can also be performed in order to cover different convection types and flow regimes. And that can be achieved by variation of the flow velocity, changing the fluid type, and changing the shape and dimensions of the channel.

It was a pleasure working with such supervising team that increases my knowledge in many domain not just the heat transfer domain. Hoping in the future we can do more research on this topic or even different topics.

A.2.1 Laser Surface Velocimeter

Laser surface velocimeter (LSV) method is designed to measure precisely the velocity and length of billets, slabs, plates, tubes and profiles for online control and inspection of hot and cold processes.

This method is based on a new measurement technique with respect to the old ones since it is accurate, required low-maintenance and uses the next-generation laser technology. These features will provide benefits for the use of such method in the industry since it doesn't need compromising important features such as accuracy, velocity direction (+/-) or zero speed measurement and also eliminates common problems that plague traditional contact-wheel measurement methods such as slippage, chatter, changing wheel diameter and frequent maintenance. The benefits of using such method is that it is simple to install, setup and integrate into the mill process and is an essential step to reducing scrap, increasing up time and improving material throughput. The LSV Series system combines a sensor head, a controller and software into a rugged industrial package designed specifically to measure velocity and length under the strenuous demands of mill environments. As in most designs LSV uses the laser doppler principle for setup the physics in this method. This principle states moving object is frequency shifted proportional to the speed and direction of the object relative to the observer (detector). Laser surface velocimeter measure this frequency shift, then the instantaneous surface velocity can be calculated [3].

LSV can be used in different applications but it is mainly designed for typical applications that can listed as follows [3]:

• Length/speed measurements in hot and cold rolling mills.

• Online control of continuous casters.

• Cut-to-length control of billets/slabs, plates, tubes, profiles...

• Length inspection for quality assurance.

• Measurements for automatic gauge control (AGC), mass flow and elongation control.

A.2.2 Laser Doppler Velocimeter

Detection of the Doppler frequency shift that occurs when light is scattered by a moving surface is the basis of LDV [4]. This frequency shift is directly proportional to the surface velocity and so its detection enables convenient and non-contact measurement of vibration velocity. Detection is not entirely straightforward as the laser has a frequency typically 6 or 7 orders of magnitude higher than the Doppler shifts, which are typically in the low MHz range. Scattered light from the target has to be mixed interferometrically with a mutually coherent reference beam to produce a beat in the collected light intensity at the difference in frequency between the target and reference beams, i.e. down in the MHz range where demodulation is possible electronically. The successes of LDV in providing measurement solutions in challenging applications is the innovation of this instrument with respect to the alternative vibration measurement techniques [5].

LDV is considered as an effective alternative to traditional contacting vibration transducers in diverse applications. Some of these applications are listed as follows [5]:

• Structural health monitoring.

• Micro-ElectroMechanical Systems (MEMS).

• Rotating machinery

• Hearing.

• Acoustics.

A.2.4 Particle Tracking Velocimetry

The methodology and application of particle tracking velocimetry (PTV) for surface flow velocity measurements are mainly to study the effect by large vortices and turbulence, making use of the fact that PTV allows to take reliable measurements in flows exposed to strong deformations typical in turbulent shear layers. PTV is a special method of Particle Image Velocimetry (PIV) that is an image velocimetry technique, that is based on a principle methodology that is tracing particles are added over the time delay between the recording of the frames. The difference between PTV and PIV is the art of the evaluation of the recording images. PIV provides a a velocity vectors for a set of particles as an integration area but this will lead to several fulfillment conditions that are the size and the number of particles that will be provided in the next section. While the PTV recognize and identify separately each particle that will provide a displacement vector for each particle and it is efficient at PIV represents a quantitative extension of the qualitative flow visualization techniques that have been practiced for several decades [9].

The principle of a PIV system can be summarized as follows: Small seed particles are introduced to the flow in order to visualize the flow behavior. These particles must be illuminated into the flow at least twice within a short time interval. The flow is illuminated by a strong light source (laser), and the particle positions are recorded by capturing the light scattered from tracer particles through a recording medium with single frame or on a sequence of frames. The displacement of the particle images between the light pulses has to be determined through evaluation of the PIV recordings. Then, the recorded images by the PIV technique must be analyzed in order to extract the data from these images, thus a post processing methods is required. In order to extract the displacement information from a PIV recording an interrogation scheme must be used. Initially, the interrogation was performed manually on the recorded images but since it needs to much time to analyze the high number of recorded images computerized methods were introduced to enhance the time and accuracy in evaluating the images.

The evaluation process can be expressed through divided the PIV recording in small subareas called "interrogation areas". The local displacement vector for the images of the tracer particles of the first and second illumination is determined for each interrogation area by means of statistical methods (auto-and cross-correlation). It is assumed that all particles within one interrogation area have moved homogeneously between the two illuminations with a specific dimension ∆x. The projection of the vector of the local flow velocity into the plane of the light sheet (two-component velocity vector) v is calculated taking into account the time delay ∆t between the two illuminations and the magnification at imaging. Then the process is repeated for all the interrogation areas of the captured PIV images to obtain the required results of velocity. Nowadays, with the modern charge coupled device (CCD) cameras (1000×1000 sensor elements and more) it is possible to capture more than 100 PIV recordings per minute. High-speed recording on complementary metal-oxide semiconductor (CMOS) sensors even allows for acquisition in the kHz range [9]. The evaluation of one digital PIV recording with several thousand instantaneous velocity vectors (depending on the size of the recording, the interrogation area and processing algorithm) is of the order of a second with standard computers. If data is required at even faster rates for online monitoring of the flow, dedicated software algorithms which perform evaluations of reduced precision within fractions of a second are commercially available. Then velocity vectors can be plotted on the recording planes and thus the local analyses can be performed [9]. PIV measurements include some errors due to several sources which are mentioned below [9]:

• Random error in the recorded images due to the noise.

• Bias error arising from the process of computing the signal peak location to sub-pixel accuracy.

• Gradient error resulting from rotation and deformation of the flow within an interrogation spot leading to loss of correlation.

• Tracking error resulting from the inability of a particle to follow the flow without slip.

• Acceleration error caused by approximating the local Eulerian velocity from the Lagrangian motion of tracer particles.

Certain errors can be minimized by careful selection of experimental conditions and operating conditions.

The main devices used in the PIV experiments with their specification can be summarized as follows [10]:

• Laser: A pulsed laser is used from with a continues wave lasers up to 5 W or even more, in addition to a good match of repetition rate with all used PIV Cameras, and the laser have a rugged and design and it is a user friendly.

• Cameras: In PIV experiments different types of cameras can be used like CCD, sCMOS, and CMOS. They come high sensitivity and low noise specifications in addition to a sensors that differ form type to type with a range of 1 MPixel to 16 MPixel or even more. The data are directly transferred to a disk or a RAM.

• Synchronization devices: In recent PIV systems the synchronization devices are with an advanced analog specifications with an independent triggers at nanosecond resolution, sampling up to 2

GHz or even more, and an advanced waveform analysis.

There is a different types of PIV systems that have a special features that include enhanced analysis methods like using a unique lagrangian approach for velocity calculations, based on feature detection and tracking. Another one is that the features can be scaled down to a single particles like PTV. As for the data visualization the PIV system provides a new integrated modules with advanced capabilities to create a 2D and a 3D graphics and animations for the measured data. Thus, no need to use another graphical programs [10].

PIV experiments can be implemented in a different area of research. A few applications are presented to show how this technique has spread out with respect to other techniques. Those selected applications are the boundary layer, supersonic, and transonic flows. In addition to the PIV in shock tubes and shock tunnels. Also different PIV in naval hydrodynamics like surface-ship flow, propulsor hydrodynamics, underwater ship flows, two phase bubble flows. Finally with a helicopter aerodynamics, multiphase flow, and base-flow-plume interaction [9].

A.3 Experimental geometry design and velocity measurement

In order to validate numerical results, experiments should be performed and compared. By comparison, it means the same geometrical design is selected for both the experimental and numerical studies. The experimental bench used is in the Notre Dame University (NDU) in Lebanon. The channel found in the bench is a parallel plate channel. To perform the numerical studies, a channel is considered with the same dimensions as the channel found in NDU. As the geometric design of the channel should be the same, also same boundary conditions used in the experimental studies should be used in numerical simulations. Thus, initial and boundary conditions are essential to be the same in both studies. To do so, inlet flow conditions for velocity are measured experimentally to be imported as a boundary condition in the numerical study. where vectors are indicated by bold letters, the dot product operator is (.), the source term is represented by Q φ , surface vector is indicated by S, velocity vector is indicated by v, the conserved quantity is φ, and the surface integral is represented as ∂Vc over a volume V c .

B.1.2 Flux term integration over element faces

The total flux J φ is defined as the sum of the convection J φ,C and diffusion fluxes J φ,D , thus it can be written as: Keywords: générateurs de vorticité, intensification du transfert thermique, échangeurs thermiques, convection mixte et forcée, régime laminaire et turbulent, optimisation

J φ = J φ

3. 1 - 2 h 1 H

 121 Rectangular channel dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Vortex generator dimensions and geometric parameters definitions. . . . . . . . . . . . 3.3 Boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Mesh cut including the VG to show the refinement methods. . . . . . . . . . . . . . . 3.5 Mesh sensitivity test to show the convergence of N u and f on different number of grids. 3.6 Local Nusselt number N u x as function of x * . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Nusselt number (reversed scale) as function of friction factor. . . . . . . . . . . . . . . 3.8 Optimization methods approach differences [47]. . . . . . . . . . . . . . . . . . . . . . 3.9 Nusselt number (reversed scale) as function of friction factor (SHERPA Method). . . . 3.10 Thermal enhancement factor (ζ) versus Nusselt number (SHERPA Method). . . . . . 3.11 Optimum design shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Velocity streamlines and temperature distribution at different x/H locations (optimum design). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Vortex propagation comparison between the present study, DWP, and RWP at different x/H (1.5 → 9) locations as indicated beside each vortex (an animation is available in the online version). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 Temperature contour and velocity streamlines at x/H = 4 location for DWP, RWP, and optimum design (without imposing symmetry). . . . . . . . . . . . . . . . . . . . . 3.15 Local bulk temperature T b as function of x/H locations [Dashed black line and pointdashed blue line are the leading edge of the VG geometry for the optimal design and others respectively]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.16 Local Nusselt number N u x as function of x/H locations [Dashed black line and pointdashed blue line are the leading edge of the VG geometry for the optimal design and others respectively]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17 Local coefficient of variation of normalized temperature CoV (T ) as function of x/H locations [Dashed black line and point-dashed blue line are the leading edge of the VG geometry for the optimal design and others respectively]. . . . . . . . . . . . . . . . . . 3.18 Local pressure coefficient f x as function of x/H locations [Dashed black line and pointdashed blue line are the leading edge of the VG geometry for the optimal design and others respectively]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.19 Longitudinal evolution of the cross section-averaged vorticity flux as function x/H locations [Dashed black line and point-dashed blue line are the leading edge of the VG geometry for the optimal design and others respectively]. . . . . . . . . . . . . . . . . . 3.20 Comparison of pumping factor power F with the heat transfer performance J at different Re (Re 1 = 360, Re 2 = 760, and Re 3 = 1020). . . . . . . . . . . . . . . . . . . . . . . . 4.1 Rectangular channel dimensions and axis system [38]. . . . . . . . . . . . . . . . . . . 4.2 Vortex generator dimensions and geometric parameters definitions [38]. . . . . . . . . . 4.3 Boundary conditions [38]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Mesh sensitivity test to show the convergence of N u and f on different number of grids. ix 4.5 Heat transfer coefficient h as function of Reynolds number v for the present numerical study compared to the numerical results and experimental results obtained by Wu and Tao [46, 47]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Nusselt number as function of friction factor for Re 1 , Re 2 , and Re 3 . . . . . . . . . . . . 4.7 Optimum design shape for the three Re studies. . . . . . . . . . . . . . . . . . . . . . . 4.8 Present and literature results for different Re. . . . . . . . . . . . . . . . . . . . . . . . 4.9 Velocity streamlines and temperature distribution at different x/H locations (optimum design [Re 1 = 4600]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Velocity streamlines and temperature distribution at different x/H locations (optimum design [Re 2 = 9920]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Velocity streamlines and temperature distribution at different x/H locations (optimum design [Re 3 = 30050]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Vortex core detected by λ 2 criterion for the optimum designs at Re = 4600, 9920, and 30050 respectively, at different x/H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 Local bulk temperature T b as function of x/H at Re 1 = 4600 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 Local bulk temperature T b as function of x/H at Re 2 = 9920 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.15 Local bulk temperature T b as function of x/H at Re 3 = 30050 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16 Local Nusselt number N u x as function of x/H at Re 1 = 4600 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17 Local Nusselt number N u x as function of x/H at Re 2 = 9920 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.18 Local Nusselt number N u x as function of x/H at Re 3 = 30050 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.19 Local friction factor f x as function of x/H at Re 1 = 4600 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.20 Local friction factor f x as function of x/H at Re 2 = 9920 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.21 Local friction factor f x as function of x/H at Re 3 = 30050 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 4.22 Longitudinal evolution of the cross section-averaged vorticity flux as function x/H at Re 1 = 4600 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively). . . . . . . . . . . . . . . 4.23 Longitudinal evolution of the cross section-averaged vorticity flux as function x/H at Re 2 = 9920 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively)). . . . . . . . . . . . . . 4.24 Longitudinal evolution of the cross section-averaged vorticity flux as function x/H at Re 3 = 30050 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively). . . . . . . . . . . . 4.25 Comparison of the heat transfer performance J versus pumping factor power F at different Re (Re 1 = 4600, Re 2 = 9920, and Re 3 = 30050). . . . . . . . . . . . . . . . . 5.1 Rectangular channel dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 (a) Top view of the channel with VG inserted, (b) side view of the channel with the VG, and (c) VG dimension parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Mesh sensitivity analysis showing the converging values of N u and P out as function of the increasing total number of cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 The streamwise variation of the local bulk temperature of the present study compared to the numerical results obtained by Desrayaud and Lauriat [23] . . . . . . . . . . . . . 5.6 The streamwise variation of the center-line axial velocity of the present study compared to the numerical results obtained by Desrayaud and Lauriat [23] . . . . . . . . . . . . . 5.7 Velocity streamlines at different Richardson number (Ri = 93.7, 23.8, 6.1, 2.7, and 1.0) at the location y/H = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Local Nusselt number N u x variation at different Richardson numbers (Ri = 93.7, 23.8, 6.1, 2.7, and 1.0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Local pressure P x variation at different Richardson numbers (Ri = 23.8, 6.1, 2.7, 1.0 and 0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Present proposed design shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Velocity streamlines at y/H = 0.5 location for the empty channel, DWP, RWP, and present design at Re = 340. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Temperature contours at y/H = 0.5 location for the empty channel, DWP, RWP, and present design at Re = 340. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Velocity streamlines and temperature distribution at different x/H locations (present proposed design) at Re = 340. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14 Local bulk temperature T b as function of x/H locations at Re = 340. . . . . . . . . . . 5.15 Local Nusselt number N u x as function of x/H locations at Re = 340. . . . . . . . . . 5.16 Local pressure P x as function of x/H locations at Re = 340. . . . . . . . . . . . . . . . 5.17 Local friction factor f x as function of x/H locations at Re = 340. . . . . . . . . . . . . 5.18 Local velocity u x as function of x/H locations at Re = 340. . . . . . . . . . . . . . . . 5.19 Vortex generator insertion dimensions according to VG numbers. . . . . . . . . . . . . 5.20 Temperature contours at y/H = 0.5 location for various VG numbers at Re = 340. . . xi 5.21 Velocity streamlines at different x/H locations for various VG numbers at Re = 340. . 5.22 Temperature distribution at different x/H locations for various VG numbers at Re = 340.150 5.23 Local bulk temperature number T b as function of x/H locations at Re = 340. . . . . . 5.24 Local Nusselt number N u x as function of x/H locations at Re = 340. . . . . . . . . . 5.25 Local velocity u x as function of x/H location at Re = 340. . . . . . . . . . . . . . . . 5.26 Velocity streamlines and temperature distribution at different x/H for the present proposed design at Re 1 = 1240. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.27 Velocity streamlines and temperature distribution at different x/H for the present proposed design at Re 2 = 3810. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.28 Local bulk temperature T b as function of x/H locations at Re 1 = 1240. . . . . . . . . 5.29 Local bulk temperature T b as function of x/H locations at Re 2 = 3810. . . . . . . . . 5.30 Local Nusselt number N u x as function of x/H locations at Re 1 = 1240. . . . . . . . . 5.31 Local Nusselt number N u x as function of x/H at Re 2 = 3810. . . . . . . . . . . . . . . 5.32 Local friction factor f x as function of x/H at Re 1 = 1240. . . . . . . . . . . . . . . . . 5.33 Local friction factor f x as function of x/H at Re 2 = 3810. . . . . . . . . . . . . . . . . A.1 Schematic diagram for a well studied research. . . . . . . . . . . . . . . . . . . . . . . A.2 Schematic diagram of mill application of the LSV series [3]. . . . . . . . . . . . . . . . A.3 Layout of image tracking LDV [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 LDA parts and working principle [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Schematic example for an experimental setup of PTV [8]. . . . . . . . . . . . . . . . . A.6 Example of an typical experimental setup of PIV [10]. . . . . . . . . . . . . . . . . . . A.7 Channel dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.8 Velocity profile at location x = 16cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.9 Velocity profile at location x = 45cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.10 Velocity profile at location x = 70cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Conservation in a discrete element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Different number of surface integration points and their positioning. . . . . . . . . . . B.3 Different number of volume integration points and their positioning. . . . . . . . . . . xii List of Tables 3.1 Design parameters ranges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Relative difference between the heat transfer coefficient h for the present study numerical result and Wu and Tao [41, 42] numerical and experimental results. . . . . . . . . 3.3 SHERPA design parameters ranges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Optimum design parameters values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Global results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Present and literature results comparison. . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Design parameters ranges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Design parameters ranges for Re 1 = 4600. . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Design parameters ranges for Re 2 = 9920. . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Design parameters ranges for Re 3 = 30050. . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Fitting equation (4) constants for the Pareto Frontier at different Re. . . . . . . . . . 4.6 Optimum design parameters values for the three Re studies. . . . . . . . . . . . . . . . 4.7 Global results for the three Re studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Present proposed design parameters values. . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Thermal enhancement factor (ζ), Nusselt number (N u), and friction factor (f ) comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Nusselt number comparison for multiple VGs in the channel. . . . . . . . . . . . . . . 5.4 Design parameters values for Re 1 = 1240, and Re 2 = 3810. . . . . . . . . . . . . . . . 5.5 Thermal enhancement factor (ζ), Nusselt number (N u), and friction factor (f ) comparison at Re 1 = 1240. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Thermal enhancement factor (ζ), Nusselt number (N u), and friction factor (f ) comparison at Re 2 = 3810. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Height of the VG, and heat transfer coefficient mm, and W.m -2 .k -Height of the channel mm j streamwise vorticity flux, Colburn factor -J Heat transfer performance factor k thermal conductivity W.m -1 .k -1 k f fluid thermal conductivity, and turbulent kinetic energy W.m -1 .k -1 , and j l Length of the VG mm L Length of the channel mm m mass kg ṁ mass flow rate kg.s -1

•

  large space exploration design and Sherpa techniques. Simulations are performed over the VG shape in a rectangular parallel plate channels under forced convection laminar flow regime. The results obtained for the best design is illustrated. Then, local analysis of the flow structure and heat transfer processes are conducted for the new optimized VG and for other classical VG shapes. Chapter 4 is to shed the light on the designs of VG's and its possible dependence on the turbulent flow velocities. For this reason, VG shape optimization is performed at a turbulent flow regime with a large number of parameters (seven parameters) and different Reynolds number. New xx optimum designs with an increase in the thermal enhancement factor are achieved for different Reynolds number. Also, local analysis is performed on the flow structure highlighting the main difference with traditional geometries. • Chapter 5 concerns the relation between Richardson number Ri and the flow reversal and to investigate mixed convection in vertical empty channels . In addition of introducing new design for heat transfer in laminar mixed convection under with Richardson number Ri = 6.1 and Reynolds number Re = 340. Then, novel designs are simulated under turbulent mixed convection and compared with known literature shapes and empty channel. Then, different number of VGs are investigated to quantify the enhancement in heat intensification. A qualitative and quantitative analysis are performed for the each case studied in this chapter.

  of heat transfer equipments. The attempts to match requirements with specific constraints result in using several types of HX designs. The simplest type of heat exchangers consists of two concentric tubes called double pipe heat exchanger. It can be utilized in wo arrangements, in parallel flow and/or in counter flow as shown in Figure 1.1 and these names are dependent on the direction of the fluid flow inside the tubes. Note that all the figures presented are reproduced and modified to be clearer.

Figure 1 . 1 :

 11 Figure 1.1: Double pipe heat exchanger [2].

Figure 1 . 2 :

 12 Figure 1.2: Compact heat exchanger [2].

Figure 1 . 3 :

 13 Figure 1.3: Shell and tube heat exchanger [2].

Figure 1 . 4 :

 14 Figure 1.4: Plate heat exchanger [2].

Figure 1 . 5 :

 15 Figure 1.5: Spiral heat exchanger [2].

Figure 1 . 6 :

 16 Figure 1.6: Heat exchange capability for different reactors [4].

  Gr: represents the ratio of buoyant to viscous forces. It plays in natural convection the same role played by the Reynolds number in forced convection. It is defined in equation 1.2.

Figure 1 . 7 :

 17 Figure 1.7: Different inclined baffles configurations [9].

Figure 1 . 8 :

 18 Figure 1.8: Thermal enhancement factor (η) values at different Reynolds numbers for the three configurations [9].

Figure 1 . 9 :

 19 Figure 1.9: (a) and (b) are different swirl flow structures, (c) ζ values of the two structures at different Reynolds numbers [10].

Figure 1 . 10 :

 110 Figure 1.10: Thermal enhancement factor (TEF) values for DWP and IPWP configurations of vortex generator at different Reynolds number [11].

Figure 1 . 11 :

 111 Figure 1.11: Thermal enhancement factor (η) values for different angle of attack (α) at Re = 456 [12].

Figure 1 .

 1 Figure 1.13: (a) Discrete parallel plate configuration (b) Discrete staggered plate configuration [10].

Figure 1 . 14 :

 114 Figure 1.14: Intersection angle β results for the two configurations at different Reynolds number [10].

Figure 1 . 15 :

 115 Figure 1.15: Intersection angle results for various configurations at different Reynolds number [15].

Figure 1 . 16 :

 116 Figure 1.16: Intersection angle results for various aspect ratios at different Reynolds number [14].

Figure 1 . 17 :

 117 Figure 1.17: Distribution of the field synergy in the tube cross section for (a) aligned, (b) alternating, and (c) reversed arrays at Re = 15000 [9].

  (a) Coated surfaces: They involve metallic or nonmetallic coating of the surfaces. It promotes condensation on the evaporator fins; which reduces the wet air pressure drop and increase the heat transfer (figure 1.18) [16].

Figure 1 . 18 :

 118 Figure 1.18: Coated surface cross section [16].

Figure 1 . 19 :Figure 1 . 20 :

 119120 Figure 1.19: Rough surfaces [16].

Figure 1 . 21 :

 121 Figure 1.21: Different shapes of insert devices [16].

Figure 1 . 22 :

 122 Figure 1.22: Twisted tape and helical inserts [17].

Figure 1 . 23 :

 123 Figure 1.23: Helical coiled tube heat exchanger [18].

Figure 1 . 24 :

 124 Figure 1.24: Vortex generators [8].

Figure 1 . 25 :

 125 Figure 1.25: Piezoelectric device for surface vibration [16].

Figure 1 . 26 :

 126 Figure 1.26: Cut view for a jet flow characteristics [19].

Figure 1 . 27 :

 127 Figure 1.27: Fluid vibration characteristics [20].

Figure 1 . 28 :

 128 Figure 1.28: Air injection [22].

Figure 1 . 29 :

 129 Figure 1.29: Mechanical stirring machine [16].

Figure 1 . 31 :

 131 Figure 1.31: Suction tube pipe [23].

Figure 1 . 32 :

 132 Figure 1.32: Schematic diagram of known shapes of LVG's [3].

Figure 1 . 33 :

 133 Figure 1.33: Various vortices types [24].

Figure 1 . 34 :

 134 Figure 1.34: Delta winglet VG's in flat plate channel [3].

Figure 1 . 35 :

 135 Figure 1.35: Vortex generators configurations [25].

Figure 1 . 36 :

 136 Figure 1.36: Average Nusselt number vs Reynolds number [15].

Figure 1 . 37 :

 137 Figure 1.37: Drag coefficient vs Reynolds number [15].

  number and the friction factor to analyze clearly and perfectly the propagation and development of the flow.

Figure 1 . 38 :

 138 Figure 1.38: Inclined projected winglet pair (IPWP) shape [11].

Figure 1 . 39 :

 139 Figure 1.39: Normalized local Nusselt number for the IPWP and DWP configurations [11].

Figure 1 . 40 :

 140 Figure 1.40: Normalized local friction coefficient for the IPWP and DWP configurations [11].

Figure 1 . 41 :

 141 Figure 1.41: Rectangular winglet pair roll angle (β) [11].

Figure 1 . 42 :

 142 Figure 1.42: Thermal enhancement factor (η) for Re = 456 and Re = 911 [12].

Figure 1 . 43 :

 143 Figure 1.43: Exit temperature distribution [3].

Figure 1 . 44 :

 144 Figure 1.44: Fin surface temperature distribution [3].

Figure 1 .

 1 Figure 1.46 shows the average Nusselt number versus Reynolds number for the two different structures. It can be seen that the Nusselt numbers for both cases increase with increasing Reynolds

Figure 1 .

 1 Figure 1.47 shows the average intersection angle between velocity and temperature gradients versus Reynolds number for the two different structures. It can be seen that the placing the LVGs downstream can effectively improve the heat transfer in the wake zone where heat transfer is poorest and decrease the intersection angle (θ). Consequently, heat transfer in the entire heat exchanger is enhanced.

Figure 1 . 45 :

 145 Figure 1.45: Overall heat transfer and drag for different arrangements [3].

Figure 1 . 46 :

 146 Figure 1.46: Nusselt number vs Reynolds number for different placements of LVG's [3].

Figure 1 . 47 :

 147 Figure 1.47: Average intersection angle (θ) vs Reynolds number for different placements of LVG's [3].

Figure 1 .

 1 48 is a simple example for searching to find the optimum design based on a minimization of the objective. The objective function shows two minima. The points really known through a CFD-based evaluation are shown using symbols on top of the (in fact unknown) objective function represented by the solid line in the background. A gradient-based algorithm starting at Point 1 would probably find the right optimum (symbols * )while the same algorithm starting from Point 2 (symbols +) would most probably get stuck in the local minimum. Thus, depending on the goal of optimization (maximizing or minimizing, locally or globally) particular optimization method must be chosen to improve the searching algorithms to reach the optimal design[38].

Figure 1 . 48 :

 148 Figure 1.48: Schematic representation of a simple optimization problem involving a single parameter and a single objective [38].

Figure 1 . 49 :

 149 Figure 1.49: Schematic representation of the concept of Pareto Optimal Frontier [38].

2 .

 2 Simulated Annealing (SA): Simulated Annealing method algorithm can be summarized as follows. Start with initial design vector and a high value of the design objective. Generate a new design point randomly in the vicinity of the current design point and find the difference in function values. If the value of the chosen design is smaller the proposed high value with a specified difference accept the design and search for another at the same procedure, if not, reject it and search for another. Thus the simulated annealing is a single point algorithm that is capable of finding global optima often by passing through local ones[40, 41].

Figure 1 .

 1 Figure 1.50 shows the data obtained by BRANN prediction with two different scaled objectives (optimization objectives), that then are used to obtain the Pareto front curve using MOA model to obtain the optimum design as shown in figure 1.51 . Song et al. [48] study a heat sink design optimization based on surrogate-based analysis model.

Figure 1 .

 1 52 shows the variation of the objective function, thermal resistance as function of the design parameters that are the height H, the width W, the base height b, and the gap size G. These surface analysis are used to obtain the optimum design as shown in figure1.53 when varying also Reynolds number.Samad et al.[49] use a weighted-average surrogate model to optimize the shape of a dimpled channel for enhance heat transfer. Figure 1.54 shows the optimum values with different weighting factors, in which 1.54-a is optimum design variables and 1.54-b is optimum objective function and percentage improvement.

Figure 1 . 50 :

 150 Figure 1.50: Prediction with training data for (a) N u and (b) f [47].

Figure 1 . 51 :

 151 Figure 1.51: Pareto front of two objectives, N u and f with CFD data [47].

Figure 1 . 52 :

 152 Figure 1.52: Multiple plots of thermal resistance with respect to design variables [48].

Figure 1 . 53 :

 153 Figure 1.53: Influence of the Reynolds number on the thermal resistance [48].

Figure 1 . 54 :

 154 Figure 1.54: Optimum values with different weighting factors: (a) optimum design variables, and (b) optimum objective function and percent improvement [49].

Figure 2 . 1 .

 21 Different shapes of vortex generators are inserted in the rectangular channel and attached on the lower surface of the channel as shown in figure 2.1.

Figure 2 . 1 :

 21 Figure 2.1: Rectangular channel dimensions.

2. 1 . 4

 14 Physical boundary conditionsBoundary conditions of the computational domain should be defined as a constraints for a specific flow problem and determines the type of the simulation performed. There are several types of boundary conditions. But, the two most used ones for general scalars are the Dirichlet and the Neumann boundary conditions. Dirichlet boundary conditions are a value specified conditions, while Dirichlet boundary conditions are a flux specified. Based on the problem studied considering the physical and computational domain, several boundary conditions can be specified as follows: 1. Inlet: Different types of inlet conditions can be used that are as follows: (a) Velocity flow inlet: The inlet face velocity vector is specified directly. This type of boundary condition is used in this contribution. (b) Mass flow inlet: A mass flow inlet boundary represents an inlet for which the mass flow rate is known. (c) Stagnation inlet: The stagnation inlet boundary is an inlet condition that is well-posed for compressible flows, although it is equally valid for incompressible flows. 2. Outlet: Two main types are usually used as boundary conditions for the outlet that are shown below. Both types are used in this contribution based on the case studied. (a) Outflow: The outflow boundary represents an outflow condition where no back-flow occurs. (b) Pressure Outlet: The pressure outlet boundary is an outflow condition which imposes the working pressure. 3. Walls: A wall boundary represents an impermeable surface that confines fluid or solid regions. You can use the wall boundary conditions to model different types of walls. An adiabatic and constant temperature walls are used in this contribution. 4. Symmetric Planes: A symmetry plane boundary represents an imaginary plane of symmetry in the simulation. You can use the symmetry plane boundary condition to reduce the extent of the computational domain in locations where the geometry and the flow are symmetric. Also this type of boundary condition is used in this contribution. 5. Free Stream: The free-stream boundary allows you to model free-stream compressible flow conditions at a far-field boundary. And it can be used for both inlet and outlet conditions.

Figure 2 . 3 :

 23 Figure 2.3: Three dimensional element types [7].

Figure 2 . 4 :

 24 Figure 2.4: Two dimensional element types projected from three dimensional elements [7].

Figure 2 . 5 :

 25 Figure 2.5: 3D mesh view using the refinement methods.

Figure 2 . 6 :

 26 Figure 2.6: Mesh cut including the VG to show the refinement methods.

Figure 2 . 7 :

 27 Figure 2.7: Development of the flow from laminar to turbulent passing by transition with their boundary layers over a flat plate [73].

  Figure 2.8 illustrates the mean and turbulent variations of velocity.

Figure 2 . 8 :

 28 Figure 2.8: Velocity variations in the turbulent region [7].

Figure 2 . 9 :

 29 Figure 2.9: Weighted sum of all objectives technique example [109].

Figure 2 . 10 :

 210 Figure 2.10: Pareto front optimization curve example [109].

  (a) Variation in input parameters values give large number of infeasible design around deterministic optimum [109]. (b) Pushing optimum away from constraint boundary reduces number of infeasible designs [109].

Figure 2 . 11 :

 211 Figure 2.11: Stochastic optimization study performance example [109].

Figure 2 .

 2 12 shows the flow chart of the used methods in Star-CCM+ software.

  (a) Sweep method approach (b) SHERPA method approach.

Figure 2 . 12 :

 212 Figure 2.12: Different optimization methods used approaches [109].

  the study concerning the optimization of vortex generators in laminar forced convection flow regime for heat transfer enhancement. The aim of this study is to achieve the best VG design for the heat transfer enhancement in a rectangular channel under laminar flow regime with Reynolds number equal to 1020. In recent years, several authors perform parametric optimization by choosing a simple form (triangular, and rectangular) and by varying one or two parameters of the shape in general; and varying the angles of the VGs in regard to the direction of the flow. In this contribution, optimization studies is performed using a large space exploration design (seven parameters) via multi objective functions. The optimization studies are carried out in commercial CFD code STARCCM+ based on finite volume method. Based on the optimization studies performed, an optimum design is obtained with an increase in the thermal enhancement factor by 35%, and 14% relative to an empty channel and literature result respectively. This work has been published as a research article in the journal "Chemical Engineering and Processing -Process Intensification" and can be found by: H.Karkaba, T.Dbouk, C.Habchi, S.Russeil, T.Lemenand, D.Bougeard. Multi objective optimization of vortex generators for heat transfer enhancement using large design space exploration. Chemical Engineering and Processing -Process Intensification, Volume 154, August 2020, 107982.

  studied a heat sink design optimization based on surrogate-based algorithm with a use of developed kriging model to analyze the objective function (thermal resistance) as function of design variables.

3. 2 . 1

 21 Geometry, Computational Domain, and Boundary ConditionsThe computational domain consists of a rectangular channel of height H, width W = 4H, and length L = 10H as shown in Figure3.1.

Figure 3 . 1 :

 31 Figure 3.1: Rectangular channel dimensions.

Figure 3 . 2 :

 32 Figure 3.2: Vortex generator dimensions and geometric parameters definitions.

Figure 3 . 3 :

 33 Figure 3.3: Boundary conditions.

Figure 3 . 4 :

 34 Figure 3.4: Mesh cut including the VG to show the refinement methods.

Figure 3 . 5 :

 35 Figure 3.5: Mesh sensitivity test to show the convergence of N u and f on different number of grids.

Figure 3 .

 3 Figure 3.6 shows the variation of N u x as function of x * . The curves for N u x versus x * are very close to each other with a slight difference at the final region (x * > 0.006). The average error is about 2%, which validates the present CFD results.

Figure 3 . 6 :

 36 Figure 3.6: Local Nusselt number N u x as function of x * .

A

  Pareto frontier curve obtained is shown in figure 3.7 illustrating all the 10 4 designs corresponding to the sweep method. It shows the variation of the Nusselt number as function of the friction factor using the optimization using a large space exploration design technique. It can be clearly observed that the Pareto frontier exists where the user has the choice to select an optimal design depending on the two objective functions, N u and f . The best design can be distinguished with the design having the maximum value of ζ defined before. In the sweep method the best design obtained has a value of ζ = 1.33.

Figure 3 . 7 :

 37 Figure 3.7: Nusselt number (reversed scale) as function of friction factor.

Figure 3 . 9 :

 39 Figure 3.9: Nusselt number (reversed scale) as function of friction factor (SHERPA Method).

  Figure 3.10: Thermal enhancement factor (ζ) versus Nusselt number (SHERPA Method).

Figure 3 . 11 :

 311 Figure 3.11: Optimum design shape.

3. 3 . 3 Flow

 33 Structure Analysis and Temperature Distribution for the Optimal Design In order to better analyze heat transfer enhancement mechanisms using vortex generators, induced vortices must be investigated. The flow topology and temperature distribution as the flow passes through the channel should be studied with the occurrence of the flow changes.

Figure 3 .

 3 Figure3.12 shows the velocity streamlines at different successive locations in the flow represented as x/H cross sections. The heat is enhanced by VG which generate vortices, disturb and mix the flow. As shown in the figure, the vortices starts to appear just upstream the VG at the beginning of the VG edges. It's clearly shown that the present shape of VG induce large main vortex. Vortex A is induced upstream of the VG at the upper side of the VG, and starts to propagates along the channel downstream the VG. In addition to vortex A, a second vortex B is also induced at the lower side of the VG beside z/W = 0 and propagates downstream the VG. Those vortices have a main effect in mixing the flow and a high intensity leading to a better heat transfer. The obtained design due to its large shape induces more vortices than classical VG shapes. At the corner of the channel, it is observed that an additional vortex D is induced. This vortex also travels downstream of the VG towards the outlet of the channel. Due to the strength of those vortices, the mixing is considered strong. A small vortex D appears upstream the VG's at the lower part of the channel and beside z/W = 0 and then appears at the final x/H cross sections.

Figure 3 .

 3 Figure 3.12 shows the temperature distribution at different successive locations (x/H sections) at and downstream the VG. The effect of the vortices on the temperature distribution is obvious. Upstream the VG, the thermal boundary layer at the upper and the lower walls was clearly noticed and once the flow reaches the VG, the boundary layer thickness starts to change, in other words the vortices disturb strongly the boundary layer development. The heat transfer is enhanced due to the mixing effect due to the vortices A and B is examined at the leading edge of the VG and

Figure 3 . 12 :

 312 Figure 3.12: Velocity streamlines and temperature distribution at different x/H locations (optimum design).

Figure 3 .

 3 13 shows the vortex propagation along the channel for the present optimum design (OD) and classical vortex generators (DWP and RWP). RWP and DWP designs are well used in several literature studies[3] and have the following dimensions α = 45• , β = 0 • , γ 1 = 90 • , γ 2 = 90 • , h = 0.5H, l = H,d = 0.5H, and s = 2H shown in figure 3.2. The numbers written near each vortex locations are the x/H cross sections. These numbers stands for the first x/H traced section, and the

Figure 3 . 13 :

 313 Figure 3.13: Vortex propagation comparison between the present study, DWP, and RWP at different x/H (1.5 → 9) locations as indicated beside each vortex (an animation is available in the online version).

3. 15 Figure 3 . 14 :

 15314 Figure 3.14: Temperature contour and velocity streamlines at x/H = 4 location for DWP, RWP, and optimum design (without imposing symmetry).

Figure 3 . 15 :

 315 Figure 3.15: Local bulk temperature T b as function of x/H locations [Dashed black line and pointdashed blue line are the leading edge of the VG geometry for the optimal design and others respectively].

Figure 3 . 16 :

 316 Figure 3.16: Local Nusselt number N u x as function of x/H locations [Dashed black line and pointdashed blue line are the leading edge of the VG geometry for the optimal design and others respectively].

Figure 3 . 17 :

 317 Figure 3.17: Local coefficient of variation of normalized temperature CoV (T ) as function of x/H locations [Dashed black line and point-dashed blue line are the leading edge of the VG geometry for the optimal design and others respectively].

Figure 3 . 18 :

 318 Figure 3.18: Local pressure coefficient f x as function of x/H locations [Dashed black line and pointdashed blue line are the leading edge of the VG geometry for the optimal design and others respectively].

Figure 3 .

 3 Figure 3.19 shows the longitudinal evolution of the dimensionless vorticity flux. A high value of Ω is observed for the present design compared to the other designs. Ω of the present VG design starts

Figure 3 . 19 :

 319 Figure 3.19: Longitudinal evolution of the cross section-averaged vorticity flux as function x/H locations [Dashed black line and point-dashed blue line are the leading edge of the VG geometry for the optimal design and others respectively].

Figure 3 .

 3 Figure 3.20 shows the variation of the pumping power factor compared to different classical and empty channel at different Re values. The comparison between the present VG design and the other designs have a lower pumping power factor F at same heat transfer performance J and vice-verse.

Figure 3 . 20 :

 320 Figure 3.20: Comparison of pumping factor power F with the heat transfer performance J at different Re (Re 1 = 360, Re 2 = 760, and Re 3 = 1020).

  vibration, electromagnetic field, jet impingent techniques require an external power. The passive ones (no external power) which are the most robust, use surface deformation techniques e.g. vortex generators (VG's) [1, 2, 3, 4, 5, 6]. The dynamic techniques employ dynamic and elastic surfaces such as flexible VG's [7, 8]. Optimal designs of VG's shapes suiting the flow regime (laminar, turbulent, and transitional) is a challenging topic for research and development. Chai et al. [3] conducted a review on heat transfer enhancement by means of different shapes of HX's including VG's under different flow regimes.

4. 2 . 1

 21 Geometry, computational domain The computational domain consists of a rectangular channel of height H, width W = 4H, and length L = 10H as shown in Figure 4.1. Various shapes of the VG such as triangular, rectangular, and trapezoidal can be controlled by varying a parameter called shape length (SL) from [SL = 0 to SL = h] as shown in figure 4.2-a.

Figure 4 . 1 :

 41 Figure 4.1: Rectangular channel dimensions and axis system [38].

Figure 4 . 2 :

 42 Figure 4.2: Vortex generator dimensions and geometric parameters definitions [38].

  r t is the turbulent Prandtl number. The Shear-Stress Transport (SST) k -ω model developed by Menter [42] is employed in this study for all the turbulent flow cases. This model computes two additional partial differential equations: a modified version of the turbulence kinetic energy equation k and a transport equation for the specific dissipation rate ω. The shear stress transport uses the k -ω formulation in the inner parts of the boundary layer and the k -formulation in the freestream. The Shear Stress Transport (SST) model has an improved adverse pressure gradient performance compared to other models [43]. The equations for the turbulence are found in the appendix. The applied boundary conditions are a uniform velocity profile, with a constant inlet temperature T in imposed at the inlet. The upper and the lower surfaces of the channel, and the VG surfaces are each with an imposed constant temperature T w . The side walls of the channel are assigned a symmetric boundary condition. At the outlet, a Neumann boundary condition is applied with a zero pressure value (Figure 4.3). Air is used as a working fluid with the following properties: Prandtl number P r = 0.708, density ρ = 1.16 kg m 3 , thermal conductivity k f = 0.02787 W m.K , dynamic viscosity µ = 1.916 × 10 -5 P a.s, specific heat Cp = 1008 J kg.K .

Figure 4 . 3 :

 43 Figure 4.3: Boundary conditions [38].

Figure 4 . 4 :

 44 Figure 4.4: Mesh sensitivity test to show the convergence of N u and f on different number of grids.

Figure 4 . 5 :

 45 Figure 4.5: Heat transfer coefficient h as function of Reynolds number v for the present numerical study compared to the numerical results and experimental results obtained by Wu and Tao [46, 47].

4. 3 . 3

 33 Pareto Frontier The obtained Pareto frontier results are shown in Figure 4.6 which shows the variation of the Nusselt number as function of the friction factor. It can be clearly observed that the Pareto frontier exists in which the user has the choice to select an optimal design depending on the two objective functions, N u and f . For each flow regime depending on the value of Re, the design with the maximum value of ζ is considered the optimum design. The optimum designs obtained have the values of ζ = 1.347, 1.239, and 1.109 respectively denoted by ( * ) symbol in each figure. The optimum design (OD) obtained for Re 1 = 4600 is from Case 2 (Table

Figure 4 . 6 )

 46 are listed in Table 4.6 and shown in Figure 4.7).

Figure 4 . 6 :

 46 Figure 4.6: Nusselt number as function of friction factor for Re 1 , Re 2 , and Re 3 .

Figure 4 . 7 :

 47 Figure 4.7: Optimum design shape for the three Re studies.

  d = 0.5H, and s = 2H shown in Figure 4.2.Figure 4.8 shows a comparison between the present results and those obtained from the literature. The heat transfer for the present optimal design (OD) is enhanced by 3.5%, 4.2%, and 14.4% with respect to the highest ζ value obtained previously by Oneissi et al. [33]. These results clearly show the advantage of large exploration design that significantly enhanced VG performance at different flow regimes. In order to physically better understand the optimization results and the reasons of intensification mechanism, local investigation are now performed.

Figure 4 . 8 :

 48 Figure 4.8: Present and literature results for different Re.

Figures 4. 9

 9 Figures 4.9, 4.10, and 4.11 show the velocity streamlines with temperature distribution at different

, 4 .

 4 Figures 4.9, 4.10, and 4.11 show the velocity streamlines with temperature distribution at different successive locations in the flow represented as x/H cross sections. The heat is enhanced by VG which generate vortices, disturb and mix the flow. As shown in the figures, the vortices start to appear just at the upstream tip of the VG. It is clearly shown that the present shapes of VG's at different Reynolds numbers, induce large vortices that affect the flow due to mixing and thus improve the heat transfer.

Figure 4 . 9 :

 49 Figure 4.9: Velocity streamlines and temperature distribution at different x/H locations (optimum design [Re 1 = 4600]).

Figure 4 .

 4 Figure 4.12 shows the vortex cores detected by λ 2 criterion for the optimum designs. It is shown that upstream the VG at x/H = 2, where it is the formation beginning of the vortices, a very small red point appears inside the channel approximately at z = W/4 in the three cases. But, by comparing the three designs at x/H = 4, 6, and 8; the design at Re 1 = 4600 highlights the larger λ 2 , that means

Figure 4 . 10 :

 410 Figure 4.10: Velocity streamlines and temperature distribution at different x/H locations (optimum design [Re 2 = 9920]).

Figure 4 . 11 :

 411 Figure 4.11: Velocity streamlines and temperature distribution at different x/H locations (optimum design [Re 3 = 30050]).

Figure 4 . 12 :

 412 Figure 4.12: Vortex core detected by λ 2 criterion for the optimum designs at Re = 4600, 9920, and 30050 respectively, at different x/H.

  d = 0.5H, and s = 2H shown in

Figure 4 . 2 .

 42 Figure 4.2.For that reason, the local bulk temperature T b was computed as the flow passes through the channel for the three cases. It represents the mass-flow averaged temperature at each transversal cross section. Figure4.13 shows the variation of the local bulk temperature for four designs: optimum

Figure 4 .

 4 13 shows the variation of the local bulk temperature for four designs: optimum design, RWP, DWP, and empty channel as function of x/H at Re 1 = 4600. The dashed black vertical line is the leading edge of the optimum VG, and the dashed blue vertical line is the leading edge of the RWP and DWP VG's in all the figures. It is clearly shown that, immediately downstream the leading edge, the temperature increases due to the presence of the VG. But with the present optimum design, the temperature increases with a larger slope downstream the VG, compared to the other designs and empty channel. It is also noticed that the temperature in the present design increases gradually and do not exhibit a decrease downstream the VG as seen in the RWP design. It is shown that in the present design at Re 1 = 4600, the temperature at the channel end is 4.4 • C higher with respect to the highest temperature found for RWP.

Figure 4 . 13 :

 413 Figure 4.13: Local bulk temperature T b as function of x/H at Re 1 = 4600 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively).

Figure 4 . 14 :

 414 Figure 4.14: Local bulk temperature T b as function of x/H at Re 2 = 9920 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively).

Figure 4 . 15 :

 415 Figure 4.15: Local bulk temperature T b as function of x/H at Re 3 = 30050 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively).

  8) was used to analyze the pressure drop inside the channel. It plays an important role in the pumping power throughout the channel. Thus managing

Figure 4 . 16 :

 416 Figure 4.16: Local Nusselt number N u x as function of x/H at Re 1 = 4600 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively).

Figure 4 . 17 :

 417 Figure 4.17: Local Nusselt number N u x as function of x/H at Re 2 = 9920 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively).

Figure 4 . 18 :

 418 Figure 4.18: Local Nusselt number N u x as function of x/H at Re 3 = 30050 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively).

Figure 4 . 19 :

 419 Figure 4.19: Local friction factor f x as function of x/H at Re 1 = 4600 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively).

Figure 4 . 20 :

 420 Figure 4.20: Local friction factor f x as function of x/H at Re 2 = 9920 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively).

Figure 4 . 21 :

 421 Figure 4.21: Local friction factor f x as function of x/H at Re 3 = 30050 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively).

Figures 4. 22

 22 Figures 4.22, 4.23, and 4.24 show the longitudinal evolution of the dimensionless vorticity flux

  Figures 4.22, 4.23, and 4.24 show the longitudinal evolution of the dimensionless vorticity flux at different Re (Re 1 = 4600, Re 2 = 9920, and Re 3 = 30050). In figure 4.22, a high value of Ω is observed for the obtained optimum design compared to the other designs. Ω of the obtained VG starts to increase rapidly when moving from the leading edge to reach the trailing edge of the VG. It then decreases slightly downstream the VG and then increases to a fluctuating value between 2.75 and 3 until reaching the end of the channel. Ω of the optimal design has larger values compared to other designs all along x/H.

Figure 4 . 22 :

 422 Figure 4.22: Longitudinal evolution of the cross section-averaged vorticity flux as function x/H at Re 1 = 4600 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively).

Figure 4 . 23 :

 423 Figure 4.23: Longitudinal evolution of the cross section-averaged vorticity flux as function x/H at Re 2 = 9920 (Dash-point black line and point-point blue line are the leading edge of the VG geometry for the optimal design and others, respectively)).

Figure 4 . 24 :

 424 Figure 4.24: Longitudinal evolution of the cross section-averaged vorticity flux as function x/H at Re 3 = 30050 (Dash-point black line and point-point blue line are the leading edge of the VG geometry

Figure 4 .

 4 Figure 4.25 shows the variation of the heat transfer performance J versus the pumping power factor F of the optimum designs obtained at different Re (Re 1 = 4600, Re 2 = 9920, and Re 3 = 30050) compared to RWP, DWP, and empty channel. The comparison between the present VG designs and the other designs have a higher pumping power factor F at same J and vice-verse.

Figure 4 . 25 :

 425 Figure 4.25: Comparison of the heat transfer performance J versus pumping factor power F at different Re (Re 1 = 4600, Re 2 = 9920, and Re 3 = 30050).

5. 2 . 1

 21 Geometry, computational domain, boundary conditions The computational domain consists of a rectangular channel of height H = 20mm, width W = 4H, and length L = 20H as shown in Figure 5.1.

Figure 5 . 1 :

 51 Figure 5.1: Rectangular channel dimensions.

Figure 5 . 2 :

 52 Figure 5.2: (a) Top view of the channel with VG inserted, (b) side view of the channel with the VG, and (c) VG dimension parameters.

Figure 5 . 3 :

 53 Figure 5.3: Boundary conditions.

3 h ν 2 ( 5 . 4 ) where β 0 = 1 2 .

 325412 density and the kinematic viscosity to account for buoyancy effects. The governing equations are the conservation of mass (continuity) (equation 5.1), momentum (Navier Stokes) (equation 5.2), and total energy (equation 5.3) given as the following: + ∇.{ρvv} = -∇p + µ∇ 2-→ u + 4Gr Re 2 ∇T (5.2) ∂ ∂t (ρC p T ) + ∇.[ρC p vT ] = ∇. C p µRe P e T (5.3) where v is the velocity vector, Cp the specific heat at constant pressure, µ the fluid dynamic viscosity, T the temperature, ρ the fluid density, ∇ the gradient operator, Re = u in D h ν the Reynolds number, D h = 2H the hydraulic diameter, ν the kinematic viscosity, P e = ReP r the Peclet number, P r = ν α 0 the Prandtl number, α 0 = k ρCp the thermal diffusivity, k the fluid thermal conductivity, and Gr the Grashof number that is defined in (equation 5.4) Gr = gβ 0 (T w -T in )D Tavg is the thermal expansion coefficient, and T avg = Tw+T in The Richardson number Ri = Gr Re 2 represents the relative importance of natural convection (buoyancy forces) to forced convection (inertial forces).

Figure 5 . 4 :

 54 Figure 5.4: Mesh sensitivity analysis showing the converging values of N u and P out as function of the increasing total number of cells.

Figure 5 . 5 :

 55 Figure 5.5: The streamwise variation of the local bulk temperature of the present study compared to the numerical results obtained by Desrayaud and Lauriat [23] .

Figure 5 . 6 :

 56 Figure 5.6: The streamwise variation of the center-line axial velocity of the present study compared to the numerical results obtained by Desrayaud and Lauriat [23] .

  is occurring based on the flow properties measured by the variation of the Richardson number as follows Ri = 93.7, 23.8, 6.1, 2.7, and 1.0 . Along with investigating reversed flow and the magnitude of velocity distortion. This is clearly shown in Figure 5.7 which shows the velocity streamlines at the location y/H = 0.5 (middle of the channel in the y-axis direction) with variable Richardson numbers (Ri = 93.7, 23.8, 6.1, 2.7, and 1.0) which stands for various values of Reynolds numbers (Re = 75, 160, 340, 560, and 870), respectively. When Ri = 93.7 the reverse flow is near the inlet with a large effect on the flow; the value of Ri decreases the reverse flow which moves toward the exit, until the flow reversal totally disappears at Ri = 1.0, which states that natural convection vanishes and forced convection manifests. The local Nusselt number is analyzed in the streamwise direction for different Ri values. Figure 5.8 shows the variation of the local Nusselt number N u x at different x/H sections at (Ri = 93.7, 23.8, 6.1, 2.7, and 1.0). It shows that when Ri = 93.7, 23.8 and 6.1 that means the natural convection is present, the Nusselt number increases at the position where the reverse flow is occurring and the flow is slowing down, until it reaches the same behavior of the forced convection heat transfer at Ri = 1.0 reaching a value around N u = 7.5. This is a well known value in such types of channels and found in different literature studies (see Desrayaud and Lauriat [23] for more details). The behavior of the pressure distribution in such cases should be also studied to ensure the relation between the natural, mixed and forced convection and the linearity of the local pressure variation in the streamwise direction. For this reason, Figure 5.9 shows the variation of the local pressure at different x/H sections at different Richardson number values (Ri = 23.8, 6.1, 2.7, 1.0 and 0.5). An added value of Ri = 0.5 at Re = 1240 is taken into account to ensure that as decreasing the value

Figure 5 . 7 :

 57 Figure 5.7: Velocity streamlines at different Richardson number (Ri = 93.7, 23.8, 6.1, 2.7, and 1.0) at the location y/H = 0.5.

Figure 5 . 8 :

 58 Figure 5.8: Local Nusselt number N u x variation at different Richardson numbers (Ri = 93.7, 23.8, 6.1, 2.7, and 1.0).

Figure 5 . 9 :

 59 Figure 5.9: Local pressure P x variation at different Richardson numbers (Ri = 23.8, 6.1, 2.7, 1.0 and 0.5).

Figure 5 . 10 :

 510 Figure 5.10: Present proposed design shape.

2 .

 2 RWP and DWP designs are well used in several literature studies[4] and have the following dimensions α = 45• , β = 0 • , γ 1 = 90 • , γ 2 = 90 • , h = 0.5H, l = H,d = 0.25H, and s = 2H shown in figure 5.2. It is clearly shown that the choice of the present design is based on having the highest value of the thermal enhancement factor with a slight increase (ζ = 1.003

Figure 5 .

 5 Figure 5.11 shows the velocity streamlines at the location y/H = 0.5 that is the middle of the channel in the y-axis direction. This plane is selected in order to show the transverse vortices induced in the flow under laminar mixed convection regime. The heat transfer is enhanced by the presence of the flow reversal which disturbs and mixes the flow with an increase in the pressure drop in the channel. As shown in figure5.11, the largest transverse vortex is found in the empty channel, and it starts to breakdown as the VG is inserted into the channel. It is clearly observed that for the RWP and the present design shapes, the transverse vortex near the outlet is broken due to the flow distribution induced by the VG in the streamwise direction along with a formation of a small transverse vortex just near the VG.

Figure 5 .Figure 5 . 11 :

 5511 Figure 5.11: Velocity streamlines at y/H = 0.5 location for the empty channel, DWP, RWP, and present design at Re = 340.

Figure 5 .Figure 5 . 12 :

 5512 Figure 5.12: Temperature contours at y/H = 0.5 location for the empty channel, DWP, RWP, and present design at Re = 340.

Figure 5 . 13 :

 513 Figure 5.13: Velocity streamlines and temperature distribution at different x/H locations (present proposed design) at Re = 340.

Figure 5 . 14 :

 514 Figure 5.14: Local bulk temperature T b as function of x/H locations at Re = 340.

Figure 5 . 15 :

 515 Figure 5.15: Local Nusselt number N u x as function of x/H locations at Re = 340.

Figure 5 . 16 :

 516 Figure 5.16: Local pressure P x as function of x/H locations at Re = 340.

Figure 5 .

 5 Figure 5.18 shows the local velocity at each x/H locations for the empty channel and the three VG designs. It can be seen that close to the VG position, due to the induced vortices, the velocity increases in that region. And then it keeps increasing gradually until reaching the end. The empty channel have the highest slope in the increase of the velocity while the opposite is shown for the proposed VG. At the end of the channel both approximately reach the same speed near the exit.

Figure 5 . 17 :

 517 Figure 5.17: Local friction factor f x as function of x/H locations at Re = 340.

Figure 5 . 18 :

 518 Figure 5.18: Local velocity u x as function of x/H locations at Re = 340.

Figure 5 . 19 :

 519 Figure 5.19: Vortex generator insertion dimensions according to VG numbers.

Figure 5 .

 5 20 shows the temperature contours at y/H = 0.5. It can be clearly observed that adding more VGs increases more the heat transfer enhancement due to the formation of more streamwise vortices in the flow.The velocity streamlines at x/H = 3, 5, 7, 9, 11, and 13 are shown in figure5.21. It can be seen by using a single VG that the flow passes through the channel and the streamwise vortex starts to vanish at x = 13. But through inserting a second VG, the streamwise vortex strength is increased. Each x/H

Figure 5 .

 5 Figure 5.23 shows the variation of the bulk temperature as function of x/H. It is well noticed that the value of the bulk temperature increases with the number of VG.

Figure 5 .

 5 Figure 5.24 shows the variation of the local Nusselt number as function of x/H. It is well noticed that the value of the local Nusselt number increases as the number of VG increases due to the VG effect on the flow. The latter induces streamwise vortices that enhance the mixing of the flow leading to an increase in the value of the Nusselt number.

Figure 5 . 20 :

 520 Figure 5.20: Temperature contours at y/H = 0.5 location for various VG numbers at Re = 340.

Figure 5 .

 5 Figure 5.25 shows the local velocity at different x/H locations for all the designs. It can be noticed that when increasing the number of VG's the local velocity difference decreases between the inlet and the outlet. Velocity increases downstream the VG, and remains at a moderate value around 0.2 m/s. In the empty channel and using a single VG, the velocity keep increasing until reaching a

Re 1 =

 1 1240, and Re 2 = 3810. Local analysis is re-conducted on the flow structure highlighting the main difference compared to a traditional geometry known as delta and rectangular winglet pairs

Figure 5 . 21 :

 521 Figure 5.21: Velocity streamlines at different x/H locations for various VG numbers at Re = 340.

Figure 5 . 22 :

 522 Figure 5.22: Temperature distribution at different x/H locations for various VG numbers at Re = 340.

Figure 5 . 24 :

 524 Figure 5.24: Local Nusselt number N u x as function of x/H locations at Re = 340.

Figure 5 . 25 :

 525 Figure 5.25: Local velocity u x as function of x/H location at Re = 340.

  Nusselt number (N u), friction factor (f ), and thermal enhancement factor (ζ) are computed for the present proposed designs at Re 1 = 1240, and Re 2 = 3810 and compared to empty channel, and classical vortex generators known as (DWP and RWP) . The results obtained are summarized in tables 5.5, and 5.6 respectively. The heat transfer was enhanced by 13.5%, and 42.8% based on the thermal enhancement factor (ζ = 1.135, and 1.428) with respect to the empty channel geometry (smooth channel) for different Re (Re 1 = 1240, and Re 3 = 3810) respectively.

Figure 5 . 28 shows

 528 the variation of T b for the four designs: proposed design, RWP, DWP, and empty channel as function of x/H at Re 1 = 1240. It is observed that upon inserting a VG, the heat transfer enhancement is increased by increasing the temperature throughout the channel. Although the highest temperature increase was achieved by using the present proposed design.

Figure 5 .

 5 Figure 5.29 shows the variation of the local bulk temperature for the four designs at Re 2 = 3810. The same conclusion can be drawn concerning the effect of inserting a VG on the heat transfer enhancement. But by observing the curves of the RWP and DWP, the effect of such designs at Re 2 = 3810 are approximately the same and that can be noticed from close values of the global Nusselt number (N u), and friction factor (f ) found in table 5.6. Although the present proposed design has the highest value of temperature increase. The local Nusselt number N u x is plotted as function of x/H in figure 5.30 at Re 1 = 1240. It is well noticed that the value of the local Nusselt number for the present design is much higher than all the other designs including the empty channel. For all the VG designs, N u x starts to increase slightly at the leading edge of the VG. The present design produces the highest heat transfer enhancement. At Re 2 = 3810, as shown in figure 5.31, a higher Nusselt number is observed at x/H = 2 and this increase remains until the end of the channel due to the strong effects of the vortices induced by the present design.

Figure 5 . 26 :

 526 Figure 5.26: Velocity streamlines and temperature distribution at different x/H for the present proposed design at Re 1 = 1240.

Figure 5 . 27 :

 527 Figure 5.27: Velocity streamlines and temperature distribution at different x/H for the present proposed design at Re 2 = 3810.

Figure 5 . 28 :

 528 Figure 5.28: Local bulk temperature T b as function of x/H locations at Re 1 = 1240.

Figure 5 . 29 :

 529 Figure 5.29: Local bulk temperature T b as function of x/H locations at Re 2 = 3810.

Figure 5 . 30 :

 530 Figure 5.30: Local Nusselt number N u x as function of x/H locations at Re 1 = 1240.

Figure 5 . 31 :

 531 Figure 5.31: Local Nusselt number N u x as function of x/H at Re 2 = 3810.

Figure 5 . 32 :

 532 Figure 5.32: Local friction factor f x as function of x/H at Re 1 = 1240.

Figure 5 . 33 :

 533 Figure 5.33: Local friction factor f x as function of x/H at Re 2 = 3810.

  convection regime. Firstly, mixed convection in empty channel is investigated at different Richardson numbers Ri (Ri = 93.7, 23.8, 6.1, 2.7, and 1.0). It is shown that as the value of Richardson number increases, natural convection dominates and vice-versa. A validation study was conducted and compared to a reference case from the literature. Then new designs are introduced for heat transfer under laminar mixed convection at Ri = 6.1 and Re = 340; and under turbulent mixed convection at Re 1 = 1240, and Re 2 = 3810. The proposed new designs of VGs induce an increase in the thermal enhancement factor by 0.3%, 13.5%, and 42.8%, relative to an empty channel respectively. Finally, different numbers of VGs are investigated to quantify the enhancement in heat intensification. Qualitative and quantitative analysis are applied for the present proposed designs compared to other designs and empty channel. The effect of vortex generated is traced along the channel by viewing the velocity streamlines and temperature contours at different

  Firstly, mixed convection is numerically investigated at different Richardson numbersRi (Ri = 93.7, 23.8, 6.1, 2.7, and 1.0). It is shown that as the value of Richardson number increases, natural convection dominates and vice-versa. Then, a validation study is done and compared to a reference case from the literature. Then new designs are introduced for heat transfer under laminar mixed convection at Ri = 6.1 and Re = 340; and under turbulent mixed convection at Re = 1240, and Re = 3810. The proposed new designs of VGs induce an increase in the thermal enhancement factor by 0.3%, 13.5%, and 42.8%, relative to an empty channel respectively. Finally, different numbers of VGs are investigated to quantify the enhancement in heat intensification. Qualitative and quantitative analysis are applied for the present proposed designs with other designs and empty channel. The effect of vortex generated is traced along the channel by viewing the velocity streamlines and temperature contours at different x/H locations in addition to a local analysis performed to highlight the effect of the designs.

Figure A. 1 :

 1 Figure A.1: Schematic diagram for a well studied research.

Figure A. 2

 2 Figure A.2 shows a schematic diagram of mill application of the LSV series that is the mass flow regulation in a rolling mill with a C-frame and an integrated Laser Surface Velocimeter. The mass

Figure A. 2 :

 2 Figure A.2: Schematic diagram of mill application of the LSV series [3].

Figure A. 3

 3 Figure A.3 shows the layout of an example that is the tracking LDV for rotating machines. A camera is aligned with the optical axis of the LDV system and its image is observed through the same mirrors used for the LDV scanning head. The instantaneous position of the target point relative to that of the laser beam is used as the error for the feedback signal, overcoming limitations of the open-loop LDV system including the need for geometrical information and the encoder.

Figure A. 3 :

 3 Figure A.3: Layout of image tracking LDV [5].

  to the flow, assumed to move with the same local flow velocity of the fluid used. And the flow with the illuminated tracer particles is recorded with cameras. Local flow velocities are reconstructed by determining the displacements of the tracer particles between two subsequent images (called frames)

Figure A. 4 :

 4 Figure A.4: LDA parts and working principle [6].

Figure A. 5

 5 Figure A.5 shows an schematic example for an experimental setup pf PTV, showing the orientation of the camera and the illuminating projector. Note the intensity-gradient direction is in the depth direction with respect to the camera.

Figure A. 5 :

 5 Figure A.5: Schematic example for an experimental setup of PTV [8].

Figure A. 6

 6 Figure A.6 sketches briefly a typical setup for PIV system recording process. Small tracer particles are added to the flow; then a light plane sheet within the flow is illuminated twice by the laser with a the time delay ∆t between pulses. It is assumed that the tracer particles move with local flow velocity between the two illuminations. The light scattered by the tracer particles is recorded "via a high quality lens" with a use of either single frame or on two separate frames on special cross-correlation digital cameras. After development the photo-graphical PIV recording is digitized by means of a scanner. The output of the digital sensor is transferred to the memory of a computer directly.

Figure A. 6 :

 6 Figure A.6: Example of an typical experimental setup of PIV [10].

Figure A. 7

 7 Figure A.7 shows the considered rectangular channel of height H = 13.5cm, width W = 30cm, and length L = 1meter.

Figure A. 7 :

 7 Figure A.7: Channel dimensions.

Figure A. 8

 8 Figure A.10: Velocity profile at location x = 70cm

Figure B. 1

 1 Figure B.1 shows the characteristics of a discrete element. C in the middle of the element represents element (cell) name. The large term F indicates neighboring elements names, where the small term f indicates the face of the element that is common between the base element and the neighbor one. Convection and diffusion directions are shown in a discrete element, in addition to the source/sink term with the transient term that is dropped in this case study [7].

Figure B. 1 :

 1 Figure B.1: Conservation in a discrete element.

  ,C + J φ,D = (ρvφ) -(Γ φ ∇φ) (B.4) Replacing the surface integral over element C by a summation of the flux terms over the faces of this element, the surface integral of the total fluxes becomes:Large design space exploration for designing optimal vortex generators for heat transfer enhancement Abstract: Convective heat transfer enhancement is essential in numerous applications involving components such multi-functional heat exchangers/reactors. This intensification can be achieved by generating streamwise vortices and coherent flow structures using Vortex Generators (VGs). Shape optimization of VGs have been carried out over the last decades to find enhanced designs. Meanwhile, the design variables were limited to two or three such as angle of attack, roll angle and shape factor. In the present study, optimization using a large space exploration design via multiobjective function are conducted by means of numerical simulations to find optimal VG designs. Simulations are performed over the VG shape in rectangular parallel plate channel flows under forced and mixed convection in laminar and turbulent flow regimes. Firstly numerical results are validated vs experimental or numerical data from the open literature. Then, seven design parameters are considered with different incremental values which are: Attack Angle, Roll Angle, Base Angles, Height, Length, and Shape Length. Based on the optimization studies performed, new enhanced VGs are found with an increase in the thermal enhancement factor relative to an empty channel and literature results in each flow regime. Local analysis of the flow structure and heat transfer processes are conducted for the new optimized VG and for other classical VG shapes in order to better understand phenomenological mechanisms involved.Keywords: vortex generators, heat transfer enhancement, heat exchangers, mixed and forced convection, laminar and turbulent, shape optimization Optimisation à grand nombre de paramètres pour l'intensification des transferts par générateurs de vorticité Resume: L'intensification des transferts de chaleur est essentielle pour de nombreuses applications comportant des composants tels que les échangeurs thermiques ou échangeurs-réacteurs multifonctionnels. Cette intensification peut être obtenue par la génération dans l'écoulement de structures cohérentes telles que les tourbillons longitudinaux à l'aide de générateurs de vorticité (VGs). Des optimisations de forme ont été conduites ces dernières décennies pour identifier des géométries optimales. Cependant, le nombre de variables de dimensionnement reste souvent limité à deux voire trois paramètres : l'angle d'attaque, l'angle de roulis et le facteur de forme. Dans la présente étude, des optimisations sont conduites par voie de simulation numérique en explorant un large espace de dimensionnement avec des fonctions multi-objectives pour trouver des géométries optimisées de générateurs de vorticité. Les simulations sont menées pour différentes formes de VG dans des écoulements de canal entre plaques planes parallèles pour des régimes d'écoulement laminaires ou turbulents en convection mixte ou forcée. Systématiquement les modèles numériques mis en oeuvre sont tout d'abord validés par des données expérimentales ou numériques disponibles dans la littérature. Puis sept variables géométriques sont considérées pour les études d'optimisation : angle d'attaque, angle de roulis, deux angles de base, hauteur, longueur et un paramètre de forme. Sur la base des études d'optimisation réalisées, des nouvelles géométries de promoteurs de vorticité ont été identifiées comme ayant un facteur de performance supérieur à celui de la géométrie de canal lisse de référence et aux résultats trouvés dans la littérature, pour chaque régime d'écoulement. Des analyses locales de la topologie de l'écoulement et des transferts sont conduites pour les différentes géométries de générateurs de vorticité afin de comprendre les mécanismes mis en jeu.

  PhD student at IMT Lille Douai, and a mechanical engineer instructor at Lebanese International University specialized in heat transfer and fluid mechanics domain. IMT Lille Douai is a leading university and research unit in north of France, and, Lebanese International University is one of the most growing universities in Lebanon and the Middle East.

	u -→ u V Personal Information mean flow velocity General Overview velocity vector Volume	m.s -1 m.s -1 m 3
	W I am a	Width of the channel	mm
	Ẇ	Rate of work	W
	x, y, z Cartesian coordinates	-
	x *	Dimensionless Cartesian coordinate	-
	y +	normalized distance to wall	m
	Greek Symbols
	α	Attach angle, and thermal diffusivity	• , and m 2 .s -1
	β	Roll angle	•
		rate of dissipation	-
	γ 1,2 Base angles	•
	Γ	Diffusion flux	-
	ζ	Thermal enhancement factor	-
	ρ	Density	kg.m -3
	µ	Dynamic viscosity	P a.s
	ν	kinematic viscosity	m 2 .s -1
	∇	Mathematical operator stands for partial derivative	-
	Ψ	Mathematical operator	P a.s
	Φ	Dissipation term	-
	φ	Variable term	-
	σ	Standard deviation	-
	Σ	Stress tensor	-
	τ	Shear stress tensor	-
	θ	Normalized temperature	-
	ω	Streamwise vorticity, and specific turbulence dissipation rate	-
	Ω	Vorticity flux	-
	N u Subscript Nusselt number	-
	P b	Pressure Bulk, and body	P a
	P e ef f effective Peclet	-
	P r in	Prandtl number Inlet parameters	-
	q in	Boundary heat flux Integration points	W.m -2
	Q out Outlet parameters Rate of heat transfer	W
	Re s	Reynolds number surface	-
	s w	distance form channel inlet to VG leading edge Wall	mm
	S x	Surface area Variable Cartesian coordinate	m 2
	SL 0	Shape length of the VG Empty channel parameters	mm
	T	Temperature	k
		xv

Table 3 .

 3 1: Design parameters ranges.

	Design		Group		Increment
	parameter	A	B	C	A, B, C
	α ( • )	15 : 60	15 : 60	15 : 60	15, 15, 15
	β ( • )	0 : 60	0 : 60	0 : 60	20, 20, 20
	γ 1 ( • )	70 : 100 110 : 140 110 : 140 10, 10, 10
	γ 2 ( • )	70 : 100 110 : 140 110 : 140 10, 10, 10
	h(mm)	5 : 15	5 : 15	16 : 19	5, 5, 1
	SL(mm)	0 : 15	0 : 15	16 : 19	5, 5, 1

Table 3 .

 3 studied experimentally and numerically the laminar convection heat transfer in a channel with longitudinal vortex generator. The experimental and numerical results obtained from Wu and Tao are compared to present results at low Re as shown in table 3.2. The relative difference between the heat transfer coefficient h for the two numerical results is 2.2%. While the 2: Relative difference between the heat transfer coefficient h for the present study numerical result and Wu and Tao[41, 42] numerical and experimental results.

	difference error between the experimental results and present numerical results is 6.8%, taking into
	consideration that there is a 5% uncertainty in the experimental values as reported by Moffat [43].
	Wu and Tao [41, 42]	Numerical Experimental
	Relative % error with present results	2.2	6.8

Table 3 .

 3 3: SHERPA design parameters ranges.

	Design parameter Design interval Increment value
	α ( • )	20 : 60	5
	β ( • )	0 : 60	5
	γ 1 ( • )	100 : 140	5
	γ 2 ( • )	100 : 140	5
	h (mm)	14 : 19	1
	SL (mm)	14 : 19	1
	l (mm)	15 : 25	5

Table 3 . 4

 34

	α β	γ 1	γ 2	h	SL	l	d	s
	•	•	•	•	mm mm mm mm mm
	45 55 110 125 19	19	20	10	40

: Optimum design parameters values.

Table 3 .

 3

			5: Global results.
	ζ	N u	N u/N u 0	f	f /f 0
	1.351 18.481	1.628	0.113 1.747
	and RVG stands for delta and rectangular vortex generators respectively, and DWP and RWP for
	delta and rectangular winglet pairs respectively.
	Table 3.6: Present and literature results comparison.
	Author Khan et al. [36] Tian et al. [23] Present
		2018	2009	design
	Case	DVG	RVG	DWP RWP	OD
	N u N u 0	1.151	1.309	1.200	1.370	1.628
	f f 0	1.155	1.384	1.161	1.448	1.747
	ζ	1.097	1.119	1.142	1.211	1.351

Table 4 .

 4 1: Design parameters ranges.

	Design parameter Design interval Increment value
	α ( • )	20 : 60	5
	β ( • )	20 : 60	5
	γ 1 ( • )	90 : 130	5
	γ 2 ( • )	120 : 150	5
	h (mm)	14 : 19	1
	SL (mm)	14 : 19	1
	l (mm)	20 : 35	5

Table 4 .

 4 .2, 4.3, and 4.4 show the design parameters ranges defined in each case for different Re (Re 1 = 4600, Re 2 = 9920, and Re 3 = 30050) respectively with the same increment value used in table 4.1. 2: Design parameters ranges for Re 1 = 4600.

	Design	Design interval
	parameter	Case 1	Case 2	Case 3
	α ( • )	20 : 60	20 : 50	20 : 45
	β ( • )	20 : 60	40 : 70	40 : 60
	γ 1 ( • )	90 : 130	70 : 110	80 : 130
	γ 2 ( • )	120 : 150 130 : 170 110 : 160
	h (mm)	14 : 19	14 : 19	17 : 19
	SL (mm)	14 : 19	15 : 19	15 : 19
	l (mm)	20 : 35	20 : 35	25 : 35

Table 4 .

 4 3: Design parameters ranges for Re 2 = 9920.

	Design	Design interval
	parameter	Case 1	Case 2	Case 3
	α ( • )	20 : 60	30 : 40	35 : 45
	β ( • )	20 : 60	40 : 65	50 : 65
	γ 1 ( • )	90 : 130	80 : 120 110 : 130
	γ 2 ( • )	120 : 150 140 : 170 135 : 160
	h (mm)	14 : 19	15 : 19	17 : 19
	SL (mm)	14 : 19	17 : 19	17 : 19
	l (mm)	20 : 35	20 : 30	15 : 30

Table 4 .

 4 4: Design parameters ranges for Re 3 = 30050.

	Design		Design interval
	parameter	Case 1	Case 2	Case 3	Case 4
	α ( • )	20 : 60	20 : 40	20 : 40	20 : 40
	β ( • )	20 : 60	50 : 80	60 : 80	60 : 80
	γ 1 ( • )	90 : 130 110 : 150 90 : 125 110 : 140
	γ 2 ( • )	120 : 150 140 : 170 120 : 150 130 : 155
	h (mm)	14 : 19	10 : 17	11 : 16	14 : 19
	SL (mm)	14 : 19	17 : 19	17 : 19	17 : 19
	l (mm)	20 : 35	25 : 35	20 : 30	15 : 25

Table 4 . 5 :

 45 Fitting equation (4) constants for the Pareto Frontier at different Re.

	Reynolds			Equation constants
	Number	a 0	a 1	a 2	a 3	a 4
	4600	-4.278 158.4 -20, 247 121, 635 -273, 196
	9920	8.485 2, 347 -37, 503 288, 968 -835, 024
	30050	63.58 2, 590 -29, 950 129, 000	0

The objective function values of the optimal designs are summarized in Table

4

.7 for the three studies. The heat transfer was enhanced by 34.7%, 23.9%, and 10.9% based on the thermal enhance-

Table 4 .

 4 6: Optimum design parameters values for the three Re studies.

	Re	α β	γ 1	γ 2	h	SL	l
	-	•	•	•	•	mm mm mm
	4600 35 50 75 150 19	19	30
	9920 40 55 120 145 19	19	20
	30050 30 70 130 140 17	19	20

Table 4 .

 4 7: Global results for the three Re studies.

	Re	ζ	N u	N u/N u 0	f	f /f 0
	4600 1.347 41.094	1.799	0.063 2.378
	9920 1.239 61.089	1.677	0.044 2.480
	30050 1.109 98.328	1.254	0.016 1.445

Table 5 .

 5 1: Present proposed design parameters values.

	α β	γ 1	γ 2	h	SL	l	d	s

Table 5 .

 5

	2: Thermal enhancement factor (ζ), Nusselt number (N u), and friction factor (f ) comparison.
	Case Present design RWP DWP Empty channel
	ζ	1.003	0.957	0.993	1
	N u	9.611	9.308	9.542	9.602
	f	0.0258	0.0269 0.0259	0.0260
	For better analysis, flow topology and temperature distribution are investigated as the flow passes
	through the channel.

Table 5 .

 5 3: Nusselt number comparison for multiple VGs in the channel.

	Case Empty channel 1 VG 2 VGs 4 VGs 6 VGs
	ζ	1	1.003	0.980	0.974	1.071
	N u	9.602	9.611	9.653	9.859	10.82
	f	0.0260	0.0258 0.028 0.0303 0.0302

Heat transfer enhancement is quantified in the channel using multiple VG.

Table 5 .

 5 4: Design parameters values for Re 1 = 1240, and Re 2 = 3810.

	Reynolds α β	γ 1	γ 2	h	SL	l	d	s
	Number	•	•	•	•	mm mm mm mm mm
	1240	45 55 110 125 19	19	20	5	40
	3810	35 50 75 150 19	19	30	5	40

Table 5 .

 5

	5: Thermal enhancement factor (ζ), Nusselt number (N u), and friction factor (f ) comparison
	at Re 1 = 1240.
	Case Present design RWP DWP Empty channel
	ζ	1.135	1.103 1.031	1
	N u	11.41	10.79 9.989	9.543
	f	0.080	0.073 0.071	0.068

In order to physically better understand the results obtained by the proposed designs, local investigations are performed. Figures 5.26, and 5.27 show the velocity streamlines with the temperature distribution at different

Table 5 .

 5 6: Thermal enhancement factor (ζ), Nusselt number (N u), and friction factor (f ) comparison at Re 2 = 3810. Re 1 = 1240, and Re 2 = 3810 . The heat transfer is usually enhanced by a VG which generates vortices, disturb and mixes the flow. As shown in figures 5.26 and 5.27,streamwise vortices appear at the the VG for the two designs at x/H = 3 and start to propagate through the channel. Both designs induce a large primary vortex located approximately in the middle of the channel that has the major effect in mixing the flow. For both designs, induced vortices appear at the lower corners of the channel (z = 0, and z = W/2). These vortices increase the mixing leading to an enhanced heat transfer. It is noticed in both designs left and right vortices appear at x/H = 3 but then start to disappear at x/H = 9, and x/H = 7 for Re 1 = 1240, and Re 2 = 3810 respectively.

	Case Present design RWP DWP Empty channel
	ζ	1.428	1.236 1.217	1
	N u	22.56	17.45 17.19	13.03
	f	0.042	0.030 0.030	0.024
	successive x/H locations for

Acknowledgements

Abbreviations

Abstract

In many applications, convective heat transfer enhancement is readily achieved by generating streamwise vortices and coherent flow structures using Vortex Generators (VG's) such as in multifunctional heat exchangers/reactors. Shape optimization of the VG's have been carried out over the last decades to find enhanced designs. Meanwhile, the design variables were limited to two or three such as angle

of attack, roll angle and shape factor. In the present study, optimization using a large space exploration design via multiobjective function are conducted to find optimal VG designs. Simulations are performed over the VG shape in a rectangular parallel plate channels under forced convection laminar flow regime. Firstly the numerical results are validated versus experimental and numerical data from the open literature. Then, seven design parameters are considered with different incremental values which are: Attack angle (α), Roll angle (β), Base angles (γ 1 and γ 2 ), Height (h), Length (l),

and Shape Length (SL). Based on the new optimization study, a novel enhanced VG is found with thermal enhancement factor 35% relative to an empty channel and 14% better than literature results.

Local analysis of the flow structure and heat transfer processes are conducted for the new optimized VG and for other classical VG shapes.

Introduction

Heat exchangers (HX's) usage is wide in thermal and fluid domains, thus increasing HX overall performance is a main goal for different industrial application. Mainly the overall performance is increased using heat transfer enhancement. In fact, there are different techniques to enhance heat transfer that can be classified passive, active, and dynamic; the active ones are by using vibration, electromagnetic field, jet impingent and are qualified as active because they require an external power. The passive ones are the most efficient and can be done using treated surfaces, rough surfaces, and vortex generators (VG's) which will be this project topic [1,2,3,9,10,4,5,6]. The dynamic ones are also efficient due to using flexible VG's [7,8]. Heat exchangers can be designed to undergoes laminar and turbulent flows [9,10]. The advantages in designing with laminar flows is due to the thermodynamic and hydrodynamic irreversiblities and hence increasing HX system efficiency such as in compact multifunctional HX, tabular exchanger reactor and other reactor shapes [11].

Dupont et al. [12] studied experimentally isothermal flow in compact heat exchangers channel with embossed type vortex generator. Gentry and Jacobi [13,14] carried out experiments of delta winglets VG in a channel showing an increase in the global heat transfer coefficient by 55%, but associated with an increase in pressure drop by 100% at low Reynolds number. Wang et al. [15] experiments showed an increase in the overall heat transfer by 10-45% due to longitudinal vortex generator examined by Reynolds number variation from 3000 to 20000. Habchi et al. [16] showed the variation of the thermal enhancement factor (ζ) while using vortical structures that are generated by inclined baffles in a turbulent pipe flow, using 3 different configurations. Chemical reactors HX thermal intenstification was also studied to monitor exothermal chemical reactions through mixing by the use of offset strip fins [17] and VG's [11]. Guo et al. [18] showed also the effect of using different swirl flow structures to enhance heat by computing the value of ζ.

Abstract

Convective heat transfer enhancement is essential in multi-functional heat exchangers/reactors by the use of Vortex Generators (VG's) through generating streamwise vortices and coherent flow structures.

Shape optimizations of VG's have been carried out over the last decades to find enhanced designs.

Meanwhile, the design variables were limited to two or three geometric parameters such as angle of attack, roll angle and shape factor. In the present study, optimizations using a large space exploration design via multiobjective function are conducted to find optimal VG designs. Simulations are per- A local performance analysis is applied to compute the variation along the channel. The local Nusselt number N u x is given by

where

and

where q x is boundary heat flux density, T x is the bulk temperature at each position, and h x is the local heat transfer coefficient.

Numerical Procedure

The Finite Volume Method (FVM) is chosen for the discretization of the system of equations which are then solved in a segregated manner for pressure and velocity. Momentum and continuity equations coupling are solved in a predictor-corrector approach. The convective terms in the governing equations for momentum and energy are discretized using a second-order upwind scheme. A second-order central scheme is applied for diffusion terms. The algebraic multi-grid (AMG) linear solver is applied to solve the velocity, pressure and temperature with Gauss-Seidel relaxation scheme. All governing equations are solved with implicit unsteady solver where the maximum physical time t is computed from the formula t = 10 L/u in entrance length with a time step t s = t/100 and 15 inner iterations. Finally, for the convergence criterion, all the governing equations are solved iteratively until the residuals stabilized at a very small value << 1. Along with, checking the mass and energy balances.

Mesh Sensitivity

Trimmed cells are generated in the computational domain of the channel following the appraoch proposed by Karkaba et al. (See [21] for further details).

A mesh sensitivity test is applied using the grid-convergence index (GCI) and the chosen mesh is 0.56 × 10 6 cells (base cell size 0.0035 m) that corresponds to 2% GCI value [22]. The convergence of N u and P out have been also considered in the mesh sensitivity analysis as shown in Figure 5.4. This mesh test is applied on the case having the maximum adverse pressure gradient that can affect the convergence of the flow. After testing more than 5 cases the following case is chosen with dimensions 45 • , 55 respectively and 19 mm, 20 mm, and 19 mm for the height (h), length (l), and shape length (SL) respectively.

Results and discussion

Validation simulation

Desrayaud and Lauriat [23] investigated the phenomenon of the flow reversal in laminar mixed convection of air in a vertical parallel plate channel of finite length. In order to validate the computational

x/H locations in addition to a local analysis performed to highlight the effect of the designs.

After this study, mixed convection in empty channel is investigated. In addition new enhanced designs of VG's are proposed in laminar and turbulent mixed convection with different Reynolds numbers. Moreover the effect of increasing the VG numbers inside the channel for the laminar mixed convection case is investigated.

Appendix A

Experimental Study

A.1 Introduction

Experiments are essential in order to prove and validate theoretical or numerical results. Experiments in fluid mechanics can demonstrate the flow topology and its effects in numerical simulation studies.

Nowadays, researchers are focusing on the numerical studies due to its capability to study a wide range of designs with a low cost in time and money wise. But the numerical results cannot be validated Experiments can be implemented on all the research domain. Mechanical engineering domain specially heat transfer in heat exchangers grabs the attention of scientist due to its wide range of applications. Heat exchangers can be found in all applications concerning heating, ventilation, air conditioning, and refrigeration. Velocity, pressure, and temperature are the main parameters that are studied in heat transfer problems in addition to the fluid and solid properties. An engineer will think that applying an experiment to verifying some result obtained from a numerical study is an easy task.

However it is very hard due to several factors that are as follows: Firstly, if more than one parameter is to be studied thus different measurement devices for each parameter must be used. Secondly, the type of application and the hardness of implementing the measurement devices in the testing bench.

Thirdly, the experience in the use of instruments and understanding the relative merits and limitations at different experimental situations. Finally, adapting the calibration methods and the different errors subjected during the experiment [1,2]. In this contribution, velocity is the parameter that will be studied throughout the experiment.

A.2 Velocity measurement devices

Velocity measurement can be done using different devices depending on the applications and the aim of study. These application can be classified depending on the test bench used. For example: wind tunnels are used to study the effect of the fluid passing over a solid objects, another is just inserting a test object in a vibrating insulated medium in order to measure the vibrating velocity of an object.

Below some velocity measurement devices are described to show their features, application field, and operating conditions. velocity distribution for the three Reynolds numbers at different locations (x = 45cm, and x = 70cm

) in order to exhibit the velocity behavior inside the channel. In order to transform the set of partial differential equations into a system of linear algebraic equations, finite volume method is used. The discretization process used in this method is distinctive and it consists of two steps [1,2]. In the first step, the partial differential equations are integrated and transformed into algebraic equations over volume or surface element, using integration quadrature of a specified order of accuracy. This step results in a set of semi-discretized equations. In the second step, interpolation profiles are used to approximate the variation of the variables within the element.

Using some approximations in both steps will result in affections in the level of accuracy and robustness of the resulting numerical solution [3]. Thus, it is important to define some limitations and boundaries when applying these approximations.

B.1.1 Semi-Discretized equation

Firstly, the governing equations are integrated over finite cells that fully cover the computation domain when using the finite volume discretization process. Then the Gauss theorem is applied to transform the volume integrals of the convection and diffusion terms into surface integrals. Then, these integrals are transformed into discrete ones, and lastly numerically integrated through the use of integration points (ip). A recall of the conservation equation for a general scalar variable φ can be expressed as: Then, by evaluating the surface fluxes at the faces of the element f rather than integrated within it. And using a Gaussian quadrature the integral at the face of the element C can be written as:

where ip refers to the integration points and ip(f ) the number of integration points along surface f .

A number of options are available with their accuracy depending on the number of integration points used and the weighing function ω ip .

With ip(f ) denoting the number of integration points along face f , the general discretized equation for the convection and diffusion terms become:

Different number of integration points can be used on each face f that relates the values calculated from element to another. This will lead to a high accuracy in the solution but with an increase in the memory storage due to an increase number of terms of the integration. Different number of integration points are shown in figure B.2 that indicates also the positioning of these points [7].

B.1.3 Source term volume integration

For the source term in the conservative equation Q φ , volume integration is used due to formation of the source over the whole element. Adopting a Gaussian quadrature integration, the volume integral of the source term is computed as:

Volume integration like surface integration can be solved on different number of integration points (ip) and the weighing function ω ip [7]. This will affect the accuracy of the numerical solution obtained.

B.1.4 Implicit versus explicit numerical methods

Finite volume discretization method results in a set of algebraic equations that are defined according to the number of integration points and type of linearization due to the weighing functions. These equations leads to an unknown values of the variables at the center at each element. These unknown can be organized to be solved in different ways that classify the computational approach adopted.

Explicit and implicit solvers schemes are two types of the computational approaches.

An explicit numerical method is the method where the dependent variables are computed directly using already known values. In this case any discretization operator can be directly evaluated based on the actual variable values.

An implicit numerical method is the method where the dependent variables are treated as unknowns and assembled to form a coupled set of equations which are then solved using iterative solution algorithm.

Computational fluid dynamics problems, that are mainly characterized by conservative equations, are defined by nonlinearity condition. And for solving such cases, implicit approach is preferred over the explicit method. Then as noticed, when the proper discretization and linearization functions are defined, solving the obtained system of algebraic equations is applicable [7].