S Benhamadouche
email: sofiane.benhamadouche@edf.fr

M Arenas

W J Malouf

Wall-Resolved

Keywords: CFD, wall-resolved LES, orifice plate, discharge coefficient, pressure drop coefficient

published or not. The documents may come

Introduction

Pressure differential devices remain the most common tools used to measure flow rates in industry. They work on the principle that variations in the flow pattern alter the pressure and velocity profiles. Pressure differential devices include diaphragms, nozzles, and Venturi tubes. They are favored since an international standard exists for regulating their use: ISO 5167. By respecting these standards, the flow rate measurements can be guaranteed within a quantifiable and commercially acceptable level of uncertainty. The orifice flow meter is a commonly used instrument for flow measurements in pipes. Given the simplicity and reliability of the tool, it is used across many industries, as it is the case at EDF, for single phase flow measurements. A key variable affecting the measurement is the straight pipe length upstream and downstream of the orifice plate. The standards take this constraint into account by indicating a minimum straight pipe length upstream and downstream of the orifice. This constraint is not always met in practice due to other factors including plant layout, piping arrangement and equipment positioning. Computational Fluid Dynamics (CFD) can play a major role in the calculation of discharge coefficients outside the scope of the ISO standards, as it would be very expensive, if not impossible, to perform an experiment for every configuration. The involved Reynolds numbers (Re) might be of the order of 10 5 to 10 6 , or even greater. It would then be unaffordable to perform parametric studies at such Reynolds numbers using fine techniques such as Large Eddy Simulation (LES). The only reasonable approach is the one based on Reynolds Averaged Navier-Stokes (RANS) equations.

The present study is the first step of a whole program which aims at using the RANS approach to compute different configurations. However, a RANS model sufficiently robust for this application must be selected first. For this purpose, a very fine LES is performed at Re=25000 on a simple configuration and validated against the ISO standard and available experimental data. This will provide significantly more data of high quality to choose the RANS approach for the industrial computations. The present validation of LES will also provide confidence on its use if needed, at a Reynolds number of the order of 10 4 , for another configuration for which no experimental data are available.

The literature shows that wall-modelled Reynolds Averaged Navier-Stokes (RANS) simulations can reasonably predict the discharge coefficient (see section 3 for the definition of this coefficient), albeit with a high dependency on discretization errors and on the turbulence model. [START_REF] Erdal | Numerical aspects of flow computation through orifices[END_REF], in a two-dimensional axisymmetric simulation of an orifice plate with a standard and modified k-ε model, show that the pressure drop across the orifice plate is highly dependent on the grid refinement around the orifice plate and the turbulence model used. [START_REF] Shah | Analysis of flow through an orifice meter: CFD simulation[END_REF] model the orifice flow meter with a standard k-ε model and show that the pressure recovery downstream of the orifice plate is not well predicted. More recently, [START_REF] Shaaban | Optimization of orifice meter's energy consumption[END_REF] shows that the realizable k-ε turbulence is within 1.4% of the ISO estimate of the discharge coefficient. Although RANS simulations could reasonably predict the discharge coefficient in certain simulations, first and second order statistics downstream of the orifice plate diverge significantly from experimental data. [START_REF] Erdal | Numerical aspects of flow computation through orifices[END_REF] demonstrate that the standard k-ε turbulence model does not accurately capture the physics of the flow especially around the region where the fluid accelerates across the pipe. A review of the literature has not found a simulation technique that can reasonably predict both the pressure drop and first and second order statistics downstream of the plate. Therefore, it is of interest to study the precision obtainable with a wall-resolved Large Eddy Simulation (LES) and compare the results to detailed experimental data.

Test-case

The present LES is based on the test case of [START_REF] Shan | Particle image velocimetry measurements of flow field behind a circular square-edged orifice in a round pipe[END_REF] who used a planar Particle Image Velocimetry (PIV) to measure the flow field downstream of a circular square-edged orifice plate in a round pipe. The computational domain of the test case is shown in Fig. 1. The relevant parameters are given in Table 1, and the pipe wall is assumed to be smooth. A dimensional analysis leads to three nondimensional numbers: The hydraulic Reynolds number Re and two aspect ratios, β the ratio between the orifice and the pipe inner diameters and α the ratio between the orifice plate thickness and the pipe inner diameter. The values of these three non-dimensional parameters are equal to 25000, 0.62 and 0.11, respectively. Note that in some references such as [START_REF] Idel'cik | Handbook of Hydraulic Resistance[END_REF], the ratio between the bulk surfaces β is used instead of β.

In all the profiles provided in the present paper, the distances in the stream-wise direction are downstream of the center of the orifice plate and the pipe centerline is at / = 1, where R is the radius of the main pipe. The velocity components and the Reynolds stresses are made non-dimensional using the maximum mean velocity umax. umax is equal to 2.36m/s and 1.85 m/s in the computations and in the experiment, respectively.

Discharge and Pressure Loss Coefficients

The discharge coefficient is defined by (1):

= ε 1 - (1)
where ε is the coefficient of compressibility and Δ the mean pressure drop between taps located 1D upstream of the orifice and 0.5D downstream (both measured from the upstream face of the contraction as required in ISO 5167-2, 2013). The relation that gives the bulk velocity and thus the flow rate is then given by (2):

= ε (2)
The coefficient of compressibility ε is equal to 1 as the velocity of the fluid tackled in the present article leads to Mach numbers significantly below 0.3. The discharge coefficient and its uncertainty can be calculated using the Reader-Harris/Gallagher equation (see ISO 5167-2, 2013). The coefficient is a function of the geometry, the Reynolds number, the placement of the pressure taps and upstream and downstream flow features (such as bends, tees or valves). ISO TR12767 (1998) covers some deviations from the scope of ISO 5167-1 ( 2013) and ISO 5167-2 (2013).

The pressure loss coefficient ! is given by ( 3):

! = " # $ % ( 3
)
where & is the mean pressure drop between taps located 2D upstream of the orifice and 6D downstream. The stream-wise variation in the wall pressure is deemed to be linear at these locations.

The pressure loss coefficient ! can be estimated using the discharge coefficient in accordance with ISO 5167-2 ( 2013) from ( 4):

! '() = * + ,-./ % 0 ,-./ % -11 (4)
The ISO 5167-2 (2013) estimates of the discharge and pressure drop coefficients are 0.628 ± 0.005 and 8.71 ± 0.07, respectively. Note that the value obtained for the pressure drop coefficient using [START_REF] Idel'cik | Handbook of Hydraulic Resistance[END_REF] correlations is equal to 8.61 for the present configuration.

Numerical approach 4.1. Large Eddy Simulation with Code_Saturne

Code_Saturne is a highly customizable open source (www.code-saturne.org) CFD package developed by EDF (see [START_REF] Archambeau | Code_Saturne: A finite volume method for the computation of turbulent incompressible flows: Industrial applications[END_REF]. It is based on a co-located finite volume discretization and deals with unstructured meshes with polyhedral cells of any type. It can deal with incompressible and compressible flows. Here, the flow is incompressible and the Velocity and pressure coupling is ensured via the SIMPLEC algorithm. The pressure Poisson equation is solved using an algebraic multigrid. The solver is parallelized (see [START_REF] Fournier | Optimizing Code_Saturne computations on Petascale systems[END_REF] using MPI and OpenMP. Further information can be found in [START_REF] Archambeau | Code_Saturne: A finite volume method for the computation of turbulent incompressible flows: Industrial applications[END_REF]. The LES capabilities of Code_Saturne have been validated on various academic and industrial cases including decaying isotropic turbulence, channel flows, side-by-side cylinders, tube bundles and gusts over a plate; see [START_REF] Benhamadouche | Large-Eddy Simulation with the unstructured collocated arrangement[END_REF], [START_REF] Afgan | Large eddy simulation of the flow around single and two side-by-side cylinders at subcritical Reynolds numbers[END_REF] and [START_REF] Afgan | Flow over a flat plate with uniform inlet and incident coherent gusts[END_REF] for example.

The temporal discretization for the LES is second order Crank-Nicolson in time with linearized convection. The spatial discretization is a pure second order central difference scheme. By default, there are no external sweeps on the pressure-velocity coupling. The effect of the external sweeps is shown to be negligible in the present case (see subsection 5.2). The sub-grid scale models used are the Dynamic Smagorinsky [START_REF] Germano | A dynamic subgrid-scale eddy viscosity model[END_REF][START_REF] Lilly | A proposed modification of the Germano subgrid-scale closure method[END_REF], as a base case, the standard Smagorinsky model (Smagorinsky, 1963) and no SGS model to test sensitivities. In the dynamic model, negative values of the Smagorinsky constant are not allowed and its maximum value is set de 0.065. In the standard Smagorinsky model, the constant is set to 0.065 and a Van Driest damping function is used. Further details about the implementation of LES in Code_Saturne can be found in [START_REF] Benhamadouche | Large-Eddy Simulation with the unstructured collocated arrangement[END_REF].

Computational domain, boundary conditions and mesh details

The inlet is located 186 upstream of the orifice and the outlet is located 126 downstream of the orifice (see Fig. 1). In all the profiles provided in this paper, the distances in the stream-wise direction are downstream of the center of the orifice plate and the pipe centerline is at / = 1, where R is the radius of the main pipe.

The inlet profile is simulated through a recycling method (see [START_REF] Lamballais | Direct numerical simulation of a turbulent flow in a rotating channel with a sudden expansion[END_REF] along 6D (as shown in Fig. 1). This is to be distinguished from a precursor computation. At the first time step, artificial eddies are created using the synthetic eddy method (see [START_REF] Jarrin | A synthetic-eddy-method for generating inflow conditions for large-eddy simulations[END_REF]. Then, the flow 6D downstream of the inlet at a given time step is reused as an input for the next time step, creating a fully developed turbulent flow. The outlet uses a standard condition based on an imposed pressure and zero Neumann boundary conditions for the other variables. As the mesh is fine enough at the wall, no-slip boundary conditions are utilized.

The computational mesh is created using ICEM CFD v14.0. It is structured in the stream-wise direction, and locally refined in the stream-wise direction near the orifice. It also must be noted that the mesh is perfectly aligned in the stream-wise direction. This is important for the recycling method to generate the fully developed turbulent flow. Moreover, the mesh is conformal throughout the domain. This is also crucial, as non-conformal interfaces are known to be unsuitable in LES since they may introduce significant numerical errors with the present numerical approach. The non-dimensional wall distance 7 8 is kept below 1 across almost the whole domain, ensuring an accurate wall resolved LES. The refinement in the two other directions is also below the requirements for a wall resolved LES and this is imposed by the criterion applied in the regions far from the wall (see below the statements about the estimated Taylor micro-scale).

The cell size far from the wall is determined using precursor RANS results to ensure that the solution is resolved beyond an estimate of the Taylor micro-scale, which is taken equal to 15ν9/ε (see [START_REF] Pope | Turbulent flows[END_REF], where 9 is the turbulent kinetic energy, : the kinematic viscosity and ε the turbulent dissipation rate. This roughly gives a maximum non-dimensional azimuthal length of 12, a maximum non-dimensional radial length of 10, and a maximum non-dimensional stream-wise length of 40 (the values are based on the friction velocity upstream of the orifice). The final mesh consists of approximately 53 million cells. Fig. 2 shows the type of the mesh which has been used perpendicularly to the stream-wise direction. Fig. 3 represents the non-dimensional distance to the wall obtained a posteriori around the contraction. Locally, 7 8 can be higher than 1 (up to 1.5) but the influence is expected to be minimal.

Temporal statistic and axisymmetry of temporal averages

The time step is constant, and it is initially chosen to ensure a CFL less than 1 throughout the solution domain. However, it is later increased (such that there are localized CFL values greater than one. Its initial and final values are equal to 0.5 10 -5 s and 10 -5 s, respectively. As the time scheme is second-order implicit, these localized CFL values greater than one are deemed acceptable since they do not cause any observed numerical instabilities.

The statistics start after 3 flow-passes in the computational domain downstream of the face used for recycling the inlet (length equal to 24D). This corresponds to 12 flow-passes in the recycled pipe (length equal to 6D). An instantaneous azimuthal velocity field in the recycled pipe is shown in Fig. 4 before starting collecting the statistics. The structures are obviously characteristic of a fully developed turbulent flow in a pipe. The non-dimensional wall distance 7 8 of the first cell shown in Fig. 3 exhibits the characteristic low speed streaks expected in the near-wall region (see [START_REF] Pope | Turbulent flows[END_REF]. An instantaneous stream-wise velocity field in the mid-plane near the orifice is also given in Fig. 5 to show that even after 1 flow-pass across the computational domain, the flow already looks fully developed. The first and second order statistics are collected in time during approximately 8 flow-passes across the computational domain downstream the face used for recycling (length equal to 24D), corresponding to 1.2 × 10 < iterations. The computations have been run on EDF Blue Gene/Q supercomputer using 4096 cores. The computational cost per timestep was around 2.2 s.

The axisymmetry of the mean velocity gradient and the Reynolds stress tensor has been verified a posteriori, at least close to the orifice plate and it the near wall region for the mean pressure. Fig. 6 shows the turbulent kinetic energy and the mean wall pressure coefficient ( = = > ? @ ⁄ ) profiles averaged in the azimuthal direction compared to local ones (not averaged in the azimuthal direction) after 6 flow-passes. The difference between the local profile and averaged profile is very small, and this is true for first and second order statistics. This is actually a very good indicator of the statistical convergence, at least in the considered regions.

Sensitivity Studies

Sensitivity studies of the LES are first performed evaluating the effect of the sub-grid scale model. Then, the effect of the number of external sweeps on the pressure-velocity coupling is examined. The sensitivity study is performed using time-averaged values.

Sensitivity to the sub-grid scale model

The most advanced sub-grid scale model used in the present study is the dynamic Smagorinsky one. Nonetheless, the standard Smagorinsky model and a simulation without a sub-grid scale model are compared. The profiles show no significant differences between the three different SGS models, as it can be seen in Fig. 7. Only the Reynolds stresses exhibit little differences (this is the reason for not showing the mean velocity component in Fig. 7). One can notice here that the dynamic Smagorinsky results are closer to the ones obtained by switching off the SGS model that to the ones obtained with a standard Smagorinsky. The very small differences between the three sub-grid scale models indicate that the modelling error is sufficiently small and that the mesh is fine enough to correctly predict first and second order statistics (one could call it Quasi-DNS). Besides the profiles, the pressure loss coefficient and the three recirculation reattachment point distances are given in Table 2. Figs. 8 and9 show the three reciculations in the present case and the definitions of L1, L2 and L3. The recirculation reattachment points are determined at the point at which the wall shear stress changes direction, and are given in non-dimensional distances from the downstream face of the orifice plate (the radius R of the main pipe is use to make the distances non-dimensional). The overall pressure loss is not very sensitive to the SGS model. Its variations stay within the uncertainty given by the ISO. The primary reattachment length is more dependent on the sub-grid scales model than the pressure loss coefficients, what was not clearly visible using the mean velocity component. The most significant difference arises between the dynamic Smagorinsky model and the constant Smagorinsky model, while the LES without a SGS model is in between. However, the close resemblance between all three models demonstrates that the LES is well resolved beyond the estimated Taylor micro-scale and wall-resolved, as the influence of the SGS model is low.

Sensitivity to pressure-velocity coupling

The number of sweeps (outer iterations) for the pressure-velocity coupling is increased with the dynamic Smagorinsky model. The standard cases only iterate once about the SIMPLEC pressure velocity coupling. In this test, the number of sweeps is increased to three, and it is averaged for approximately 4 flow-passes across the domain. It is found that the differences between the two simulations are negligible in the present case (thus figures are not provided). For example, the relative difference between the predicted pressure loss coefficients is approximately 0.5%, and the velocity and turbulent kinetic energy profiles do not change visually. It can thus be concluded that one sweep about the pressure-velocity coupling is sufficient in the present case. [START_REF] Shan | Particle image velocimetry measurements of flow field behind a circular square-edged orifice in a round pipe[END_REF]. Fig. 10 gives the centerline stream-wise velocity downstream of the orifice, normalized by the maximum average velocity. The tail end of the centerline velocity differs slightly, but the results of the LES and PIV overall seem to be in very good agreement. This is an indicator for the very good predictions of the discharge and pressure drop coefficients.

Moreover, the shapes of both the LES and PIV stream-wise and radial velocity profiles provide a close match, as seen in Figs. 11(a) and 11(b). The results differ in the regions of high gradients of velocity. With regards to the results in the regions of high gradients, the resolution of the PIV is larger than the grid size of the LES; thus probably smoothing the results. For example, near r/R = 0.6, as seen in Fig. 11(b), the LES predicts larger variations than the PIV for the radial velocity. At this point, the LES radial mesh length is 0.0018D, while the PIV pixel size is 0. are reasonably similar with the exception of areas with large gradients as shown. On the other hand, the normalized profiles of and show large differences even outside regions with large gradients. The values of experimental data are suspicious at the wall as they do not vanish. . This is not surprising as the PIV methodology is known to lose accuracy in the near wall region. The difference between the experimental and simulated and profiles is almost constant, and the qualitative behavior is the same. It would be very hazardous to speculate on the origin of the differences in the core region far from the area showing very high gradients.

The PIV predicts a reattachment length of the primary recirculation zone K = 3.64 downstream of the orifice using the forward flow probability (FFP) method (see [START_REF] Shan | Particle image velocimetry measurements of flow field behind a circular square-edged orifice in a round pipe[END_REF]. From the LES, the reattachment length is estimated at K = 3.92 downstream of the orifice. However, the difference between the data is due to the method with which the reattachment length is calculated. The LES estimate determines the reattachment length as the point at which the wall shear stress changes direction. Since the near wall region is resolved, the velocity distribution is linear in the radial direction (normal to the wall); therefore O PQRR = S # TUV J TUV where T V is the velocity at the centre of the first cell from the wall, and T V is the distance of that cell center to the wall. This is analogous to the mean stream-wise velocity zero-crossing estimate, where the reattachment zone is the point at which the velocity changes sign. The PIV estimate of the reattachment point uses a FFP at = 0.056 away from the wall which corresponds to twice the resolution of the technique adopted (see [START_REF] Shan | Particle image velocimetry measurements of flow field behind a circular square-edged orifice in a round pipe[END_REF]. The FFP method calculates a probability density function of the stream-wise velocity being positive. The reattachment length is then given by the point at which the probability of the velocity being positive is exactly 50%. Using the experimental data, it is possible to estimate the recirculation zone using a mean stream-wise velocity zero-crossing method. The first point located at = 0.028 away from the wall is used to estimate the velocity. This method predicts a primary reattachment length W = 3.62 downstream of the orifice, and the secondary reattachment length is found equal to W = 0.27 . A similar approach for the LES results in a primary reattachment length W = 3.60 and a secondary reattachment length W = 0.34 . Therefore, it is clear that the primary and secondary reattachment points calculated with the same methodology using PIV data and the LES are very similar.

The discharge coefficient is calculated by taking pressure taps 16 upstream of the orifice and 0.56 downstream (both measured from the upstream face as required in ISO 5167-2, 2013). The dynamic Smagorinsky model gives a discharge coefficient = 0.632. The ISO 5167-2 estimate for the discharge coefficient is 0.628 ± 0.005. The results are very close. The pressure loss across is measured between 2D upstream and 6D downstream, where the streamwise variation in the wall pressure is deemed to be linear. Using the dynamic Smagorinsky model, the pressure loss coefficient is ! = 8.64. Similarly, the estimates from ISO 5167-2 and from Idel'cik (1996) correlations are equal to 8.71 ± 0.07 and 8.61, respectively. The results are again very close. The results between the ISO standards and the LES are, for both the discharge coefficient and the pressure loss coefficient, in very close agreement -within the margin of error of the ISO estimate -which serves as further validation of the LES results.

Conclusions

The present study demonstrates that a very fine wall-resolved LES with a dynamic Smagorinsky SGS can accurately and precisely simulate a single phase flow through a square-edged orifice flow meter at Re of the order 10 . A sensitivity study shows that the effect of the SGS model and pressure-velocity coupling is small. The LES shows excellent agreement with the velocity from the experimental data. There is also good agreement observed between the respective profiles. However, the and profiles show an offset but a good qualitative agreement. Moreover, the recirculation lengths are well predicted by the LES. Lastly, the pressure loss and discharge coefficients are also shown to be in very good agreement with the predictions of ISO 5167-2 (2003). Therefore, the wall-resolved LES is shown to be accurate in simulating the flow across a squareedged orifice. The results from this simulation can be used to validate other simulation techniques such as RANS approaches or provide sufficient confidence to compute other configurations at moderate Reynolds numbers. Moreover, this study shows that the LES is a viable technique in predicting recirculation lengths, discharge coefficient, pressure loss coefficient, mean velocities and Reynolds stresses in an orifice flow meter. This has important repercussions for the assessment of heat and mass transfer characteristics of industrial orifice flow meters. This result also shows that the discharge coefficients and pressure loss coefficients should be predictable using the LES technique in non-standard configurations.

Fig. 1 .

 1 Fig. 1. Computational domain and boundary conditions.

Fig. 2 .

 2 Fig. 2. Mesh view in a plane orthogonal to the stream-wise direction.

Fig. 3 .

 3 Fig. 3. Non-dimensional wall distance 7 8 of the first cell center (dynamic Smagorinsky model).

Fig. 4 .

 4 Fig. 4. Azimuthal (out of plane) instantaneous velocity field in the recycled pipe after approximately 5 flow-passes along 6D (dynamic Smagorinsky model).

Fig. 5 .

 5 Fig. 5. Instantaneous stream-wise velocity after approximately 1 flow-pass across the entire domain (dynamic Smagorinsky model).

Fig. 6 .

 6 Fig. 6. Comparison between azimuthally and non-azimuthally averaged data (dynamic Smagorinsky model), □ time average, ─ azimuthal and time average, umax (LES) = 2.36 m/s.

Fig. 7 .

 7 Fig. 7. Comparison between different SGS models. □ dynamic Smagorinsky, ─ standard Smagorinsky, ---no SGS, umax (LES) = 2.36 m/s.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Visualization of the primary (L1) and secondary (L2) recirculations

Fig. 10 .

 10 Fig. 10. Stream-wise centerline velocity profile downstream of the orifice. □ experimental data, ─ LES, umax (exp) = 1.85 m/s, umax (LES) = 2.36 m/s. .

Fig. 11 .

 11 Fig. 11. Comparison between LES and PIV profiles. □ experimental data, ─ LES, umax (exp) = 1.85 m/s, umax (LES) = 2.36 m/s.

Table 1 .

 1 Test

	-case main parameters.
	Density [kg/m 3 ]		996.65
	Dynamic viscosity µ [kg/m/s]		8.54 10 -4
	Main pipe inner diameter D [m]		0.046
	Orifice diameter d [m]		0.0285
	Orifice plate thickness [m]		0.005
	Bulk velocity in the main pipe	[m/s]	0.466

Table 2 .

 2 Recirculation lengths and pressure loss coefficient for the three LES simulations.

	Dynamic	Standard	No SGS	ISO
	Smag.	Smag.	Model
	Pressure loss coefficient Primary reattachment [B C /D] Secondary reattachment [B E /D] Tertiary reattachment [B F /D]	8.64 3.92 0.42 0.025	8.79 4.25 0.37 0.020	8.71 4.11 0.40 0.023	8.71 ± 0.07 ---
	6. Comparisons with experimental data
	LES with the dynamic Smagorinsky model is compared with PIV data from

Acknowledgements

The authors would like to acknowledge Dr. Feng Shan for kindly sharing the experimental results for the present test case.


Amir Keshmiri
email: a.keshmiri@mmu.ac.uk

Karim Osman

Sofiane Benhamadouche

Nima Shokri

Assessment of Advanced RANS Models against Large Eddy Simulation and Experimental Data in the Investigation of Ribbed Passages with Passive Heat Transfer

Keywords: Eddy-Viscosity Models, Heat Transfer, Large Eddy Simulation, RANS, Nuclear Reactor Core, Rib-Roughness, Thermal-Hydraulics, Turbulence Modelling

Rough surfaces are widely used to enhance convective heat transfer by the promotion of higher turbulence levels. The present paper reports simulations of the flow and heat transfer in a 2D ribroughened passage using a number of advanced Reynolds-Averaged Navier-Stokes (RANS) turbulence models including Eddy-Viscosity Models (EVM) and a Reynolds Stress Model (RSM). Large Eddy Simulation (LES) is also conducted and results are compared against experimental measurements. In addition, the effects of rib thermal boundary condition on heat transfer are also investigated. In the present work, the blockage ratio of the transversely-mounted rectangular ribs is 10% and the rib pitchto-height ratio of 9 is selected. The Reynolds number, based on the channel bulk velocity and hydraulic diameter, is 30,000. The RANS-based turbulence models investigated here are the 'k-ω-SST', the 'v 2 -f', the 'φ-f' and the 'Elliptic Blending RSM'. All computations are undertaken using the commercial and industrial CFD codes, 'STAR-CD' and 'Code_Saturne', respectively. Of all the models, the LES predictions were found to be in the best agreement with the experimental data, while the k-ω-SST and EB-RSM returned the least accurate results.

Additional symbols are defined in the text.

INTRODUCTION

For many years, rough surfaces have been one of the most common methods for convective heat transfer augmentation through increasing turbulence levels. The penalty associated with such roughening is an increase in pressure loss. Despite numerous studies on heat transfer, turbulence and turbulent boundary layer in rough surface problems, the detailed physics of these flow problems are still a topic of research.

In the past few decades there have been several experimental studies of the rib-roughened surfaces including those of Webb et al. [START_REF] Webb | Heat transfer and friction in tubes with repeatedrib roughness[END_REF], Han et al. [START_REF] Han | An investigation of heat transfer and friction for rib-roughened surfaces[END_REF], Park et al. [START_REF] Park | Heat transfer performance comparisons of five different rectangular channels with parallel angled ribs[END_REF], Liou et al. [START_REF] Liou | LDV measurements of periodic fully developed main and secondary flows in a channel with rib-disturbed walls[END_REF], Okamoto et al. [START_REF] Okamoto | Turbulent shear flow and heat transfer over the repeated two-dimensional square ribs on ground plane[END_REF], Rau et al. [START_REF] Rau | The effect of periodic ribs on the local aerodynamic and heat transfer performance of a straight cooling channel[END_REF], Iacovides et al. [START_REF] Iacovides | LDA study of flow development through an orthogonally rotating U-Bend of strong curvature and ribroughened walls[END_REF][START_REF] Iacovides | The measurement of local wall heat transfer in stationary U-ducts of strong curvature, with smooth and rib roughened walls[END_REF], Krogstad et al. [START_REF] Krogstad | An experimental and numerical study of channel flow with rough walls[END_REF] and Coletti et al. [START_REF] Coletti | Flow field investigation in rotating rib-roughened channel by means of particle image velocimetry[END_REF], amongst others. In these experiments, the effects of various factors including rib pitch-to-height ratio (P/k), rib-to-channel height ratio (k/H), rib shape and Reynolds number were tested.

Detailed flow and heat transfer measurements were made by Rau et al. [START_REF] Rau | The effect of periodic ribs on the local aerodynamic and heat transfer performance of a straight cooling channel[END_REF] in a square channel with ribs presenting a blockage ratio of 10% (Figure 1). The results of the local measurements were discussed for three different P/k ratios (6, 9 and 12) in a one-side-ribbed channel ('1s'). Measurements for a twoside-ribbed ('2s') channel were reported for P/k = 9. Air was the working fluid, and the Reynolds number based on the bulk velocity and the hydraulic diameter was fixed at Re = 30,000. The data of Rau et al.

[6] will be used in the present work to validate the numerical simulations.

There have also been numerous attempts to numerically simulate rib-roughened channels. A number of those works used the Large Eddy Simulation (LES) approach including Ciafalo and Collins [START_REF] Ciafalo | Large eddy simulation of turbulent flow and heat transfer in plane and rib-roughened channels[END_REF],

Sewall et al. [START_REF] Sewall | Experimental validation of large eddy simulations of flow and heat transfer in a stationary ribbed duct[END_REF] and Ahn et al. [START_REF] Ahn | Large eddy simulation of flow and heat transfer in a rotating ribbed channel[END_REF], among others. However, the most widespread techniques adopted were based on solution of the Reynolds-Averaged Navier-Stokes (RANS) equations, where the choice of turbulence model plays a critical role in determining the accuracy of the simulations. Some of the numerical investigations of rib-roughened surfaces using RANS were carried out by Baughn and Yan [START_REF] Baughn | Local heat transfer measurements in square ducts with transverse ribs[END_REF], Baughn and Roby [START_REF] Baughn | Enhanced turbulent heat transfer in circular ducts with transverse ribs[END_REF], Iacovides and Raisee [START_REF] Iacovides | Recent progress in the computation of flow and heat transfer in internal cooling passages of turbine blade[END_REF][START_REF] Iacovides | Computation of flow and heat transfer in two-dimensional ribroughened passages, using low-Reynolds-number turbulence models[END_REF], Manceau et al. [START_REF] Manceau | Turbulent heat transfer predictions using the v^2-f model on unstructured meshes[END_REF], Ooi et al. [START_REF] Ooi | Reynolds averaged simulation of flow and heat transfer in ribbed ducts[END_REF], Bredberg et al. [START_REF] Bredberg | An improved k-w turbulence model applied to recirculating flows[END_REF], Raisee et al. [START_REF] Raisee | Application of a non-linear k-ε model in prediction of convective heat transfer through ribbed passages[END_REF], Ryu et al. [START_REF] Ryu | Analysis of turbulent flow in channels roughened by twodimensional ribs and three-dimensional blocks. Part I: Resistance[END_REF], Keshmiri and Gotts [START_REF] Keshmiri | Thermal-hydraulic analysis of four geometrical design parameters in rib-roughened channels[END_REF] and Keshmiri [START_REF] Keshmiri | Numerical sensitivity analysis of 3-and 2-dimensional rib-roughened channels[END_REF], amongst others.

Iacovides and Raisee [START_REF] Iacovides | Recent progress in the computation of flow and heat transfer in internal cooling passages of turbine blade[END_REF][START_REF] Iacovides | Computation of flow and heat transfer in two-dimensional ribroughened passages, using low-Reynolds-number turbulence models[END_REF] examined the capabilities of the low-Reynolds-number Launder and Sharma [START_REF] Launder | Application of the energy dissipation model of turbulence to the calculation of flow near a spinning disc[END_REF] model (LS) and second moment closures in predicting convective heat transfer in ribbed annular channels, pipes and plane channels. They showed that the most reliable results were obtained using the low-Reynolds-number second moment closures. The authors also obtained a more realistic variation of heat transfer levels in the separation region and by employing a differential form of the Yap length-scale correction term, originally introduced by Yap [START_REF] Yap | Turbulent heat and momentum transfer in recirculating and impinging flows[END_REF], in ε-equation.

Manceau et al. [START_REF] Manceau | Turbulent heat transfer predictions using the v^2-f model on unstructured meshes[END_REF] applied the v 2 -f model to a number of test cases including a 2D periodic ribbedchannel. They concluded that the v 2 -f model is a good compromise between simplicity and accuracy for simulating separated flows. They also showed that the v 2 -f model is an accurate turbulence model especially in estimating the near-wall turbulence anisotropy which is essential for reproducing the correct levels of heat transfer.

Ooi et al. [START_REF] Ooi | Reynolds averaged simulation of flow and heat transfer in ribbed ducts[END_REF] carried out simulations of the flow and heat transfer in 3-dimensional rib-roughened ducts using the v 2 -f and Spalart-Allmaras (S-A) turbulence models; they compared their results with the experimental data of Rau et al. [START_REF] Rau | The effect of periodic ribs on the local aerodynamic and heat transfer performance of a straight cooling channel[END_REF] and the k-ε simulations of Chen and Patel [START_REF] Chen | Near-wall turbulence models for complex flows including separation[END_REF]. Configurations with various geometrical parameters including pitch, rib height, and cavity depth were considered. It was shown that, while the k-ε model severely underestimates heat transfer levels, the S-A model gave results that were closer to the experimental data, but nonetheless the computed values of Nu were still far from the measured values. The authors reported that heat transfer results generated by the v 2 -f model were closest to the experimental values of Rau et al. [START_REF] Rau | The effect of periodic ribs on the local aerodynamic and heat transfer performance of a straight cooling channel[END_REF]. However, none of the above models could capture the secondary flow structure which consequently led to incorrect predictions of Nu on the heated side wall.

Bredberg et al. [START_REF] Bredberg | An improved k-w turbulence model applied to recirculating flows[END_REF] modified the k-ω formulation in an attempt to improve its performance in recirculating flows. Comparison was made against the experimental data of Rau et al. [START_REF] Rau | The effect of periodic ribs on the local aerodynamic and heat transfer performance of a straight cooling channel[END_REF] and three other EVMs including the k-ε model of Abe, Kondoh and Nagano [START_REF] Abe | A new turbulence model for predicting fluid flow and heat transfer in separating and reattaching flows -I. Flow field calculations[END_REF], the k-ω model of Wilcox [START_REF] Wilcox | Comparison of two-equation turbulence models for boundary layers with pressure gradient[END_REF], and the v 2 -f model of Lien and Kalitzin [START_REF] Lien | Computations of transonic flow with the v 2 -f turbulence model[END_REF]. For the Nusselt number distribution, it was found that both their proposed model and the v 2 -f model gave reasonable predictions, while the Wilcox k-ω model underpredicted the Nusselt number. Once again, they confirmed that a close connection exists between heat transfer and the predicted turbulence levels.

Recently, Ryu et al. [START_REF] Ryu | Analysis of turbulent flow in channels roughened by twodimensional ribs and three-dimensional blocks. Part I: Resistance[END_REF] carried out a series of simulations using the k-ω model of Wilcox [START_REF] Wilcox | Turbulence Modeling for CFD[END_REF] to obtain the resistance coefficient and the velocity profile for a turbulent flow in channels with 2D ribs and 3D blocks. Various rib configurations were tested including ribs with square, triangular, semicircular, and wavy cross-sections over a range of rib pitch and Reynolds numbers. It was found that the k-ω model can successfully capture essential features of the flow.

More Recently, Keshmiri and Gotts [START_REF] Keshmiri | Thermal-hydraulic analysis of four geometrical design parameters in rib-roughened channels[END_REF] tested the effects of various geometrical factors on heat transfer and friction coefficient using two low-Reynolds number turbulence models including the k-ω-SST [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF] and a variant of Durbin's v 2 -f formulations [START_REF] Iaccarino | Predictions of a turbulent separated flow using commercial CFD codes[END_REF]. They concluded that the v 2 -f model generally returns more accurate results than the k-ω-SST turbulence closure. Moreover, Keshmiri [START_REF] Keshmiri | Numerical sensitivity analysis of 3-and 2-dimensional rib-roughened channels[END_REF] used the v 2 -f and k-ε models to carry out various numerical sensitivity analyses including the accuracy of a 2D channel approximation of a 3D configuration.

Large Eddy Simulation technique, although more expensive, particularly at high hydraulic Reynolds numbers, seems to globally give very satisfactory results in the literature. One of the pioneering works in the present configuration to the author's knowledge is by Ciafalo and Collins [START_REF] Ciafalo | Large eddy simulation of turbulent flow and heat transfer in plane and rib-roughened channels[END_REF]. They carried out LES based on the Smagorinsky model which showed the increase of heat transfer upstream the rib.

Sewall et al. [START_REF] Sewall | Experimental validation of large eddy simulations of flow and heat transfer in a stationary ribbed duct[END_REF] and Ahn et al. [START_REF] Ahn | Large eddy simulation of flow and heat transfer in a rotating ribbed channel[END_REF] studied a configuration close to the present one but with ribs on lower and upper walls of the channel. They concluded that LES is an appropriate tool for the present configuration.

In the present study, the performance of four advanced RANS models and a Large Eddy Simulation are evaluated for a rib-roughened channel flow using a commercial and an industrial CFD codes, 'STAR-CD' and 'Code_Saturne', respectively. Details of the configuration used in this work are given in Section 4.

COMPUTATIONAL CODES

Code_Saturne

Code_Saturne [START_REF] Archambeau | A finite volume method for the computation of turbulent incompressible flows -Industrial applications[END_REF] is the Electricité de France (EDF) in-house CFD tool for incompressible flows.

It is based on an unstructured collocated finite-volume approach for cells of any shape with Rhie and Chow interpolation and a SIMPLEC algorithm for pressure correction. While using LES, a fully centred scheme is applied for the velocity components and the temperature. For this latter, a slope test which switches to a 1 st order upwind scheme is utilized to limit the overshoots. The time scheme is second order based on Crank-Nicolson/Adams Bashforth scheme (the diffusion is totally implicit whereas the convection is semi-implicit). While using RANS or URANS, a centred scheme with a slope test is used for the velocity components and the temperature and a first-order upwind scheme for the turbulent quantities. The time scheme is first order Euler in this case.

STAR-CD

The second code to be used here is 'STAR-CD' [START_REF] Cd-Adapco | STAR-CD Methodology[END_REF], a commercial unstructured CFD package. In common with Code_Saturne, the program solves the governing equations using a collocated finite volume approach. Similar to Code_Saturne, STAR-CD is a co-localized cell centred incompressible Navier-Stokes solver. In the present STAR-CD computations, the momentum and turbulence transport equations are discretized using second-order central differencing and first-order upwind differencing schemes, respectively. The energy equation is discretized using the 'Monotone Advection and Reconstruction Scheme' (MARS) [START_REF] Cd-Adapco | STAR-CD Methodology[END_REF][START_REF] Hirsch | Numerical Computation of Internal and External Flows[END_REF]. The SIMPLE algorithm is adopted for pressure-velocity correction. The simulations are run at steady-state conditions.

It is worth noting that STAR-CD and Code_Saturne have recently been tested by Keshmiri et al. [START_REF] Keshmiri | Benchmarking of Three Different CFD Codes in Simulating Natural, Forced and Mixed Convection Flows[END_REF] in a benchmarking exercise to assess their performance in simulating two distinct heat transfer problems, both of which had direct relevance to nuclear reactor flows. Despite some differences in the numerical procedures used by each code, in all cross-code comparison tests, good agreement was obtained for velocity and temperature profiles between both codes.

TURBULENCE MODELS

In this section, brief descriptions of the turbulence models used here are given. Interested readers are referred to Keshmiri et al. [START_REF] Keshmiri | Turbulence Models and Large Eddy Simulations Applied to Ascending Mixed Convection Flows[END_REF] for more detailed information on the turbulence models and the codes used in the present paper.

k-ω-SST Model (STAR-CD)

Advantages of both the k-ε and k-ω models are combined in the Shear Stress Transport (SST) model of Menter [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF]. Through a blending function, this model effectively uses the Low Reynolds Number • The coefficients were modified to improve the overall performance of the model. Note that the functions and coefficients of the k-ω-SST model may be found in several textbooks and are not included here for the sake of brevity. However, it is worth noting that the k-ω-SST models implemented in STAR-CD uses the original version of k-ω-SST model proposed by Menter [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF] (except the value of σk1 in STAR-CD which is set to 0.85 instead of 0.5).

v 2 -f Model (STAR-CD)

Another model to be considered here is the f v -2 model (or simply 'v 2 -f') which was originally proposed by Durbin [START_REF] Durbin | Near-wall turbulence closure modeling without damping functions[END_REF] and was designed to handle near-wall and non-local effects in turbulent boundary layers. In this model, the conventional ad-hoc damping functions are replaced with a third transport equation for 2 v , where v is the wall-normal component of the fluctuating velocity. The revised turbulent viscosity equation then becomes:

s t T k k v C 2 µ ν = (1)
where Ts is the integral time scale and is defined as

              = 5 . 0 , max ε ε v C k T kT s (2)
where CkT is a constant. In addition, an elliptic equation for f22 (the redistribution term in the 2 v equation) is included to account for near-wall and non-local effects.

Note that, the original version being unstable because of the boundary conditions, there are currently several v 2 -f formulations in the literature, however, the version implemented in STAR-CD is that given by Iaccarino [START_REF] Iaccarino | Predictions of a turbulent separated flow using commercial CFD codes[END_REF]. Therefore, the interested reader is referred to Iaccarino [START_REF] Iaccarino | Predictions of a turbulent separated flow using commercial CFD codes[END_REF] and STAR-CD user manual [START_REF] Cd-Adapco | STAR-CD Methodology[END_REF] for further details on the present formulation.

φ -f Model (Code_Saturne)

Laurence et al. [START_REF] Laurence | A robust formulation of the v 2 -f model[END_REF] developed a robust formulation of the v 2 -f model which was based on the original v 2 -f model proposed by Durbin [39]. This formulation has been shown to be more stable than the original one and has been implemented in Code_Saturne in order to be used in industrial studies. The idea behind this model is to solve an equation for the ratio of v 2 and k (named φ) which modifies the wall asymptotic behaviour and thus increases the stability of the system. For the temperature equation, the Simple Gradient Diffusion Hypothesis (SGDH) is used with a turbulent Prandtl number set to σt = 1 (note that in all the present STAR-CD computations, the turbulent Prandtl number is set to a constant value, σt = 0.9).

Elliptic Blending -Reynolds Stress Model (Code_Saturne)

As noted earlier by Iacovides and Raisee [START_REF] Iacovides | Recent progress in the computation of flow and heat transfer in internal cooling passages of turbine blade[END_REF][START_REF] Iacovides | Computation of flow and heat transfer in two-dimensional ribroughened passages, using low-Reynolds-number turbulence models[END_REF], the use of a low Reynolds second order closure performed quite well on the present test case. Manceau and Hanjalić [START_REF] Manceau | Elliptic blending model: A new near-wall Reynolds-stress turbulence closure[END_REF] developed a new version of the second moment closure turbulence model of Durbin [START_REF] Durbin | A Reynolds-stress model for near-wall turbulence[END_REF]. They noticed that the six relaxation variables of Durbin's original model essentially accounts for geometrical effect to modify the pressurestrain process as the wall is approached. Hence, Manceau and Hanjalić proposed to define this transition by a single non-dimensional variable, solution of an elliptic equation (so called 'Elliptic Blending'). The blending factor is used to switch between high Reynolds and near-wall models for the scrambling term and the dissipation rate. The high Reynolds model is based on the SSG model [START_REF] Speziale | Modelling the pressure-strain correlation of turbulence: an invariant dynamical system approach[END_REF]. The turbulent heat fluxes are solved using a transport equation which also uses the concept of elliptic blending (EB-DFM for Elliptic Bending -Differential Flux Model, see Dehoux [START_REF] Dehoux | Modélisation statistique des écoulements turbulents en convection forcée, mixte et naturelle[END_REF]).

Large Eddy Simulation (Code_Saturne)

The sub-grid scale tensor is modelled using the dynamic Smagorinsky model based on Germano et al. [START_REF] Germano | A dynamic subgridscale eddy viscosity model[END_REF] identity and Lilly [START_REF] Lilly | A proposed modification of the Germano subgrid-scale closure method[END_REF] minimization.

For the present LES procedure, an implicit (grid) filter is assumed, whereas an explicit filter is applied only to compute the dynamic constant. The width of the explicit filter is based only on the immediate neighbours of every cell (cells sharing a common face). The widely used approach of averaging the constant in the homogeneous (span-wise) direction in order to smooth it and to add stability is not utilized in the present work. The constant is capped between values of 0 and its value for the channel flow simulations. The code has been validated on several academic (decaying isotropic turbulence and channel flow) and industrial (T-junctions, tube bundles, etc.) test cases -see Benhamadouche [START_REF] Benhamadouche | Large eddy simulation with the unstructured collocated arrangement[END_REF]. The temperature equation uses a turbulent Prandtl number set to σt = 0.5.

GEOMETRY AND GRID

Geometry

Based on the experimental data of Rau et al. [START_REF] Rau | The effect of periodic ribs on the local aerodynamic and heat transfer performance of a straight cooling channel[END_REF], the rib height to channel hydraulic diameter ratio in the present simulations is k/De = 0.05, and the blockage ratio (k/H) is 10%. The rib pitch-to-height ratio of P/k = 9 is selected for the present analysis. Ribs are also assumed to have square cross-sections (i.e., k = b). The Reynolds number based on the bulk velocity and the hydraulic diameter is fixed at Re = 30,000 and the Prandtl number is set to 0.71.

The geometry considered here consists of a 2-dimensional channel, the lower wall of which is roughened by square ribs of height k. In STAR-CD simulations, the computational domain is of length 2P, i.e., it includes 2 ribs, while Code_Saturne uses a configuration with only 1 rib. Streamwise periodicity is assumed for all the computations. The domain is of height H and the thermal boundary condition at both the lower and upper walls is set as uniform wall heat flux for STAR-CD computations and with an insulated rib for Code_Saturne computations. The effects of these thermal boundary conditions on the heat transfer results are investigated in Section 5.

It is worth noting that using a 2D configuration in the present work is justified based on the findings of Keshmiri [START_REF] Keshmiri | Numerical sensitivity analysis of 3-and 2-dimensional rib-roughened channels[END_REF] who showed that the flow over the centre-line of a 3D rib-roughened channel (such as the one used in [START_REF] Rau | The effect of periodic ribs on the local aerodynamic and heat transfer performance of a straight cooling channel[END_REF]) can be represented by a 2D configuration with a relatively good accuracy.

STAR-CD Computations

The mesh used for STAR-CD computations is a structured Cartesian grid with approximately 160,000 computational cells. Since low-Reynolds-number turbulence models are employed, the grids were generated so as to be very fine near the wall (the wall-adjacent cell typically extends only to y + ≤ 0.5).

Code_Saturne Computations

RANS:

The mesh used for Code_Saturne RANS computations is a structured Cartesian grid with approximately 54,000 computational cells. Several tests have been performed to ensure that the solution is mesh-independent. The y + distribution is below 1 everywhere along the boundary.

LES:

The present LES is well-resolved and uses a mesh containing 33 Million cells (160 cells in the span-wise direction). The maximum Courant number does not exceed 1 and the averaging is performed in time and space (30 flow passes based on the bulk velocity are used for time averaging). In Figure 2, it can be seen that the results of the v 2 -f model and LES are closest to the experimental data, while the k-ω-SST and EB-RSM under-predict the rate of heat transfer especially in the recirculation area and therefore return the poorest predictions i.e., under-predict the magnitude of the wall heat flux. In the figure, it is also evident that different models predict different shapes for the heat transfer distribution; while the φ-f and v 2 -f formulations tend to predict a uniform profiles, the LES predicts a rather skewed distribution towards the first rib. The k-ω-SST and EB-RSM, however, both predict a distribution which is rather skewed towards the second rib.

RESULTS AND DISCUSSION

Note that despite their contradicting predictions for the overall shape, LES and EB-RSM predict a sudden rise in heat transfer rate near the second rib which is consistent with the observations of earlier LES results reported in Ciafalo and Collins [START_REF] Ciafalo | Large eddy simulation of turbulent flow and heat transfer in plane and rib-roughened channels[END_REF], Sewall et al. [START_REF] Sewall | Experimental validation of large eddy simulations of flow and heat transfer in a stationary ribbed duct[END_REF] and Ahn et al. [START_REF] Ahn | Large eddy simulation of flow and heat transfer in a rotating ribbed channel[END_REF]. This abrupt rise in heat transfer corresponds to the secondary vortex (recirculating bubble) formed in the vicinity of the downstream rib (this point is discussed further below in connection with Figure 5).

In addition to the above turbulence models, the non-linear EVM of Suga [START_REF] Craft | Development and application of a cubic eddy-viscosity model of turbulence[END_REF] was also tested for the present geometry (not included in the results), where severe numerical instability were encountered. Similar numerical instabilities have also been reported by [START_REF] Keshmiri | Assessment of a Common Non-Linear Eddy-Viscosity Turbulence Model in Capturing Laminarization in Mixed Convection Flows[END_REF] for mixed convection flows.

It is also worth mentioning that due to measurement limitations, the experimental data of Rau et al. [START_REF] Rau | The effect of periodic ribs on the local aerodynamic and heat transfer performance of a straight cooling channel[END_REF] do not extend all the way to the faces of the ribs and therefore the accuracy of the simulations in these regions cannot be assessed by direct comparison. However, in the experiment data of Cho et al. [START_REF] Cho | Local heat/mass transfer measurements in a rectangular duct with discrete ribs[END_REF] and Liou and Hwang [START_REF] Liou | Effect of ridge shapes on turbulent heat transfer and friction in a rectangular channel[END_REF] one can find such behaviour with a peak of the Nusselt number occurring just upstream the ribs. v 2 -f models generally over-predict the magnitude of the velocity, while the LES, φ-f and EB-RSM tend to predict smaller velocity magnitude. At x/k = 4, Figure 7 shows that all the models generally under-predict the magnitude of the velocity. However, further away from the wall (y/H > 0.2), the predictions of the k-ω-SST and v 2 -f models are in better agreement with the data.

The size of the separation bubble predicted by each model is illustrated more clearly in the inset of Figure 7. It is shown that all the models yield longer separation bubble compared to the data. The results shown in the inset of Figure 7 are also consistent with the magnitude of the streamwise velocity in Figure 4 at x/k = 4. It is worth mentioning that the experimental data shown in Figure 7 have been carefully extracted from a velocity vector plot reported in Rau et al. [START_REF] Rau | The effect of periodic ribs on the local aerodynamic and heat transfer performance of a straight cooling channel[END_REF] and therefore the exact height at which U/Ub = 0 was not possible to identify, hence not shown in this figure .   Attention is turned next to the inter-rib pressure distribution. Impingement on the upstream face of the rib leads to high static pressure, while there is a low pressure zone downstream of the rib. These effects can be seen in Figure 8, where the pressure coefficient distributions between the two ribs are plotted against x/k. Note that in this figure, Cp in all cases is offset to the experimental value at x/k = 4.5 (the centre of the inter-rib cavity). In Figure 8, the performance of the present numerical results can be assessed by splitting the inter-rib cavity into recirculation (0.5 ≤ x/k ≤ 4.5) and recovery (4.5 ≤ x/k ≤ 8.5) regions. Clearly most of the models (except the φ-f model) fail to return a correct level of pressure in the recirculation region which is directly linked to the predicted size of the recirculation bubble (Figure 3) e.g., the longer the separation bubble, the lower the pressure magnitude. The predictions of the models tend to improve within the recovery region but still only the LES, EB-RSM and v 2 -f are in good agreement with the data.

A more detailed examination of the results shown in Figure 8 reveals that the LES and EB-RSM predict a sharp increase in the value of Cp near the second rib, while the eddy-viscosity-based turbulence models (i.e., k-ω-SST, v 2 -f and φ-f) return quasi-linear distributions. Similar findings also emerged from the heat transfer results shown in Figure 2. These discrepancies clearly show the advantages of the LES and EB-RSM in simulating the counter-rotating recirculating flow ('secondary strain') near the second rib in comparison to linear eddy-viscosity models.

Furthermore, in the inset to Figure 8, the predicted pressure coefficient distributions are re-plotted but now Cp is offset to the experimental value at x/k = 0.5. The models return a wide range of pressure differences between x/k = 0.5 and x/k = 8.5, with the v 2 -f and EB-RSM predicting the maximum and minimum, respectively. The pressure difference predicted by the LES is clearly in very good agreement with the data.

Figure 9 shows the contours of the normalized turbulent kinetic energy for the k-ω-SST and v 2 -f models, which respectively represent one of the least-and most-accurate turbulence models tested in the present study. In general, both models predict that the maximum level of turbulent kinetic energy occurs around upstream and above the rib, at separation onset. A dramatic difference can be seen between the predictions of the models, especially around the ribs and the inter-rib cavity. The v 2 -f model predicts a stretched region with the highest turbulent kinetic energy levels which extends over the top of the rib, a shape which is consistent with the PIV measurements of Casarsa and Arts [START_REF] Casarsa | Aerodynamic performance investigation of a rib roughened cooling channel flow with high blockage ratio[END_REF]. It is worth noting that the turbulent kinetic energy levels shown in Figure 9 can be related to the level of pressure drop predicted by each model; energy coming from the mean flow is an average pressure drop times (constant) mass flow rate which feeds into the turbulent kinetic energy and in turn is transferred into heat via the dissipation of turbulent kinetic energy.

-Thermal Boundary Conditions of the Ribs

In the present study, the application of two different thermal boundary conditions imposed on the ribs are investigated; in the first case, the thermal boundary condition on both the ribs and over the rib interval is one of the same uniform wall heat flux (Figure 10a), while in the second case, the ribs are insulated (adiabatic walls), as indicated in Figure 10 (b). This test provides some assessment of the uncertainties present in the experimental conditions. These uncertainties are usually associated with the measurement techniques, the rib and channel wall material properties and the presence of unsteady effects [START_REF] Iaccarino | Conjugate heat transfer predictions in twodimensional ribbed passages[END_REF].

The v 2 -f formulation is used here to carry out the simulations to compare the effects of these two thermal boundary conditions. It was found that the thermal field was slightly affected especially in regions near the ribs. Nu/Nu0 distributions for both cases are shown in Figure 11. While both cases have similar heat transfer patterns, the case with insulated ribs produce slightly higher levels of heat transfer.

It can also be seen that for the case with heated ribs, the heat transfer levels immediately downstream and upstream of the ribs are much lower than those of the insulated one. Similar results were also reported by Iaccarino et al. [START_REF] Iaccarino | Conjugate heat transfer predictions in twodimensional ribbed passages[END_REF].

From the above findings, it can be concluded that the thermal boundary conditions on the ribs is relatively an insignificant parameter in the case studied here and affect heat transfer levels only in regions very close to the ribs, while naturally for a forced convection flow the dynamic field is not affected by the thermal boundary condition.

CONCLUSIONS

Numerical simulations of the flow and heat transfer in a 2-dimensional rib-roughened passage have been performed using k-ω-SST, v 2 -f, φ-f, Elliptic Blending Reynolds Stress Model (EB-RSM) and Large Eddy Simulation (LES). Comparison was made against the experimental data of Rau et al. [START_REF] Rau | The effect of periodic ribs on the local aerodynamic and heat transfer performance of a straight cooling channel[END_REF]. All simulations were undertaken using the commercial and industrial CFD codes, 'STAR-CD' and 'Code_Saturne', respectively. Of all the models, the LES predictions were found to be in the best agreement with the dynamic and thermal field data of Rau et al., while the k-ω-SST and EB-RSM returned the least accurate results due to their poor wall heat flux predictions. The v 2 -f and φ-f models also returned acceptable predictions, especially the v 2 -f model for the heat transfer.

It was also shown that the effects of some of the flow features such as the counter-rotating separation bubble near the second rib cannot be successfully simulated by linear eddy-viscosity models and were captured only by the LES and EB-RSM. Furthermore, computations using the v 2 -f model showed that the thermal boundary condition imposed on the ribs has insignificant effects on the Nusselt number prediction. However, the local values of the Nusselt number in the regions close to the ribs are affected by the type of thermal boundary condition through convection.

Finally, it is also worth noting that one of the principal applications of the present work is to the ribroughened fuel pins of the UK fleet of Advanced Gas-cooled Reactors (AGRs). Although the results presented here are for a simple 2D-plane passage, this study would form a valuable precursor to the simulation of the more complex reactor geometries [see 55, for example].

Figures Captions

2 xω

 2 Mean and fluctuating velocity components in Cartesian coordinatesUτFriction velocity, (|τw|/ρ) 1/Streamwise coordinate y coordinate or distance to the wall y + Dimensionless distance from the wall, yUτ/ν Dissipation rate per unit of kinetic energy, ε /(Cμ k)

(

  LRN) formulation of the k-ω model in the boundary layer and a version of the k-ε model (usually the 'standard k-ε model') in the free shear layer. This is based on the observations that the k-ε model is much less sensitive to the free-stream value of ε than the k-ω model is to ω. Apart from this unique feature, the main differences between the standard k-ω model and the SST one are the following:• The SST model includes a damped cross-diffusion derivative term, as well as a blending function, in the ω-transport equation.• The definition of the turbulent viscosity in the SST was modified to improve the prediction of the turbulent shear stress.

Figure 2

 2 Figure 2 compares heat transfer results of the k-ω-SST, v 2 -f , φ-f and EB-RSM against the LES and the experimental data of Rau et al. [6]. Following Rau et al., all Nusselt number distributions for the ribbed duct calculations are normalized by the value associated with a smooth channel (the Dittus-Boelter equation):

Figure 1 .

 1 Figure 1. Schematic diagram of a rib-roughened surface.

Figure 2 .

 2 Figure 2. Heat transfer distributions.

Figure 3 .

 3 Figure 3. Comparison of streamlines and reattachment lengths for various models.

Figure 4 .

 4 Figure 4. Streamwise velocity distributions at y/k = 0.1.

Figure 5 .

 5 Figure 5. Wall-normal velocity distributions at y/k = 1.

Figure 6 .

 6 Figure 6. Streamwise velocity profiles on the rib-top (x/k = 0).

Figure 7 .

 7 Figure 7. Streamwise velocity profiles at x/k = 4 plane. Legends are the same as in Figure 6.

Figure 8 .

 8 Figure 8. Pressure coefficient distribution.

Figure 9 .

 9 Figure 9. Contour plots of the turbulent kinetic energy for P/k = 9; 1s for k-ω-SST and v 2 -f models.

Figure 10 .Figure 11 .FiguresFigure 1 .

 10111 Figure 10. The computational domains used with different rib thermal boundary conditions: (a) Heated ribs (b) Insulated ribs

Figure 5 .

 5 Figure 5. Wall-normal velocity distributions at y/k = 1.

Figure 7 .

 7 Figure 7. Streamwise velocity profiles at x/k = 4 plane. Legends are the same as in Figure 6.

Figure 9 .Figure 10 .Figure 11 .

 91011 Figure 9. Contour plots of the turbulent kinetic energy for P/k = 9; 1s for k-ω-SST and v 2 -f models.

ACKNOWLEDGEMENTS

The authors are pleased to acknowledge the contribution of their colleagues in the School of Mechanical, Aerospace and Civil Engineering (MACE) at the University of Manchester, especially Dr. M.A. Cotton, Professor D. Laurence and Dr. I. Afgan. The authors are also grateful to EDF for the use of its 'BlueGene P' supercomputer for the LES simulations.

Streamlines obtained by all four RANS turbulence models as well as LES are shown in Figure 3. In this figure the main flow features are clearly visible; at the face of the first rib, a curved free shear layer is formed which reattaches within the inter-rib cavity. This is followed by the development of a new boundary layer which is accelerated by shear forces and separates once more before reaching the second rib. In the current configuration, the inter-rib cavity is mainly dominated by a large recirculation bubble which contains reversed flow in the near-wall region due to an adverse pressure gradient. [START_REF] Coletti | Flow field investigation in rotating rib-roughened channel by means of particle image velocimetry[END_REF] In Figure 3 it can be seen that the k-ω-SST model and the LES/φ-f models return the largest and smallest separation bubbles, respectively. The former is probably due to a turbulent viscosity limiter that exists in the k-ω-SST model which limits the shear stress when the production of k exceeds its dissipation rate (by about an order of magnitude, for example). This limiter tends to eliminate the unrealistic buildup of eddy viscosity in the stagnation regions [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF][START_REF]Recent experience with different turbulence models applied to the calculation of flow over aircraft components[END_REF].

Another interesting observation from Figure 3 is that the models with the poorest heat transfer predictions shown in Figure 2 (i.e., k-ω-SST and EB-RSM) return the longest recirculation bubble. This over-prediction of the reattachment length results in an under-estimation of flow renewal in the cavity which could explain the poor predictions of the k-ω-SST and EB-RSM in Figure 2. Furthermore, in Figure 3, it is seen that the φ-f and v 2 -f formulations return the smallest counter-rotating separation bubble near the second rib compared to the other models. Consequently, both models fail to capture the sudden increase of heat transfer which occurs in LES and EB-RSM within 8 ≤ x/k ≤ 8.5 -see Figure 2.

Figure 4 shows the normalized streamwise velocity distributions at one-tenth of the rib height. The profile predicted by the LES is the closest to the experimental data, while k-ω-SST has the worst prediction. In addition, all RANS and LES models and the experimental data clearly show a reattachment point (U/Ub = 0), except for the k-ω-SST model which indicates that the flow remains reversed at this elevation. The k-ω-SST model therefore fails to predict flow renewal in the inter-rib cavity which is partly responsible for the under-prediction of heat transfer rate seen in Figure 2. The reattachment lengths predicted by the LES and the φ-f model are the closest to that found by Rau et al. [START_REF] Rau | The effect of periodic ribs on the local aerodynamic and heat transfer performance of a straight cooling channel[END_REF].

Results for the wall-normal velocity at rib height are presented in Figure 5. It can be seen that the results of the LES, v 2 -f and φ-f models are in reasonably good agreement with the experimental data, with the LES being the closest. However, the EB-RSM and the k-ω-SST model generally under-predict the magnitude of the wall-directed velocity. The poor performance of the EB-RSM and the k-ω-SST model in Figure 5 could also be anticipated from the streamlines shown in Figure 3, where these models returned the largest separation bubbles.

Normalized streamwise velocity profiles at two wall-normal planes, namely over the rib-top (x/k = 0) and in the separation region (x/k = 4) are shown in Figure 6 and Figure 7, respectively. In Figure 6, it can be seen that all the models are in reasonable agreement with the measurements. The k-ω-SST and