Soledad Le Clainche
email: soledad.leclainche@upm.es

Xuerui Mao

José M Vega

O Tatin

Spatio-temporal Koopman Decomposition in offshore wind turbines

Keywords:

is article presents the spatio-temporal analysis of the flow structures in the wake of an offshore wind turbine using spatio-temporal Koopman decomposition (STKD). STKD is a new method that obtains and combines the main frequencies and wavenumbers (characterizing a specific spatial direction) contained in a flow, with the aim at representing the main flow structures as a whole of traveling waves. e results obtained show that the complex behaviour of the wake of the wind turbine is mainly driven by a single traveling wave moving downstream and a combination of small amplitude traveling waves, related to velocity fluctuations, that represent the flow rotation.

INTRODUCTION

e recent knowledge of environmental problems, such as the green-house effect or the hole in the ozone layer, yields the need to increase the use of renewable energies instead of fossil fuels, to produce several types of energy. Offshore wind energy is a type of renewable energy that uses wind power to generate electricity. Due to its good performance, this type of energy has become very popular during the last years. Offshore wind speeds availables are higher than the ones found in land, making this energy more efficient than others and consequently, making profitable the construction of wind farms offshore.

Due to its high complexity, the deep study and understanding of the complex behaviour that takes place in the wake of an offshore wind turbine is a research topic of high interest [ ]. Wake interactions in wind farms may lead to highly complex non-linear phenomena that need to be analyzed using sophisticated techniques, such as Spatio-Temporal Koopman Decomposition (STKD)[ ]. is technique express the evolution of spatio-temporal flow structures as a whole of traveling waves that moves with a certain phase velocity. In this way, it is possible to simplify very complex flow features, as the ones found in the wake of a wind turbine, and consequently, undestanding the flow behaviour. In this article STKD is applied to study the spatio-temporal flow structures of the wake of an offshore wind turbine.

e article is organized as follows. Section introduce briefly the STKD method. Section presents the model and the performance of the numerical simulations. Finally, Sec-tions and explain the main results and conclusions.

. SPATIO-TEMPORAL KOOPMAN DECOM-POSITION

Spatio-temporal Koopman decomposition (STKD) is a recently introduced method[ ] suitable to study spatio-temporal flow structures. e method combines either singular value decomposition (SVD) [ ] or higher order SVD[ ] with dyanamic mode decomposition (DMD) [ ] or higher order DMD [ ] to obtain the main frequencies ω n and wavenumbers κ m describing the flow. STKD combines these results to obtain spatiotemporal STKD modes q mn represented by traveling waves, whose phase velocity is defined as c = -ω n /κ m . erefore, it is possible to represent highly complex spatio-temporal structures as a whole of traveling waves, helping in this way to understand the reality.

STKD decomposes spatio-temporal data u(x, y, z, t) (where x, y, z and t are the streamwise, normal and spanwise components and t is the time) as a sum of spatio-temporal STKD modes in the following way:

u(x, y, z, t) ≃ M, N m,n=1
a mn q mn (y, z) e (ν m +iκ m )x+(δ n +iω n )t ( ) where ν m and δ n are the growth rates related to the spatial and temporal modes, respectively, a mn are the spatiotemporal amplitudes. Spatio-temporal diagrams ωκ, become determined by the value of the spatio-temporal amplitudes. ey relate the frequencies and wavenumbers, helping to understand the flow behaviour, since the main direction and slope of the traveling waves, defined as ∆t/∆x, is related to the traveling wave phase velocity as ∆t/∆ = -1/c.

STKD is applied to a set of spatio-temporal data u(x, y, z, t), equispaced in space ∆x and in time ∆t, and collected in a tensor as U(x i , y, z, t k ), for 1 ≤ i ≤ N x and 1 ≤ k ≤ K, with N x and K as the number or points in the spatial direction and number of temporal snapshots, respectively. e main algorithm considers two main steps, particularized for this problem they read as the following.

• As first step HOSVD is performed to the tensor U in order to clean spatial redudancies, usualy found in data coming from numerical simulations[ ], and noise, commonly found in experimental data [ ]. HOSVD performs singular value decomposition (SVD) in each one of the spatial directions and combines the results. Hence, the original data tensor is decomposed in spatial modes in x, in y and in z and temporal modes in t. e number or retained SVD mode is established by a tolerace ε, set by the user, determining the root mean square (RMS) error of the approximated solution.

• As second step HODMD is applied to the temporal and spatial SVD modes in t and x, previously obtained. HODMD is a technique that combines classical DMD with Takens' delay embedding theorem [ ], using d time-lagged snapshots, reducing not only data uncertainty, but also increasing their accuracy and covering a wider range of applications than DMD. us, depending on the case analyzed, using HODMD could be the best option [ ]. At this step, the main frequencies and wavenumbers are calculated. e number of retained modes is established by a second tolerace ε 1 , also set by the user. In addition, the parameter d for the HODMD analysis is also set by the user, a er some calibration, based on the robustness of the results obtained.

. MODEL DESCRIPTION, NUMERICAL SIM-ULATIONS AND COMPUTATIONAL DOMAIN e wake of an offshore turbine has been studied in detail at Reynolds number 1000, defined as Re= U ı D/ν, where U ı is the freestream velocity, D is the diameter of the wind turbine and ν is the kinematic viscosity. To this purpose, threedimentional numerical simulations have been performed to solve the incompressible non-linearized Navier-Stokes equations in a Cylindrical system. e x-r plane is decomposed into 4722 spectral elements, each of which is further discretized with nodal based Gauss-Laba o-Legendre basis functions with polynomial order 7. In the azimuthal direction, a Fourier decomposition is applied and Fourier modes are calculated.

e computational domain is a cylinder with dimensions L x = 63, R = 35, where the wind turbine is located at (x, r) = (0, 0). e wind turbine is modeled using an actuator disc model with unit radius. In this model, the thrust acting on the turbine is considered as uniform over a disk region and therefore a uniform external forcing is added to the momentum equations of the fluid flow in the disk region.

. FLOW STRUCTURES IN THE WAKE OF THE OFFSHORE WIND TURBINE

e wake of a wind turbine is mainly driven by the big bulk of fluid that travels downstream, following the wind direction, and that is rotating, due to the influence of the wind turbine axis rotation. ese two main movements, longitudinal and rotational, make that the description and evolution of velocity fluctuations in the wake of a wind turbine be complex and, consequently, unpredictable and unknown. On the one hand, a good way to simplify reality is studying the flow evolution as a simple combination of traveling waves. To this purpose, it is necessary first, to study the spatio-temporal behaviour of the flow. On the other hand, using the Reynolds decomposition, it is possible to decompose the unsteady, instantaneous, flow field u(x, t) as a combination of the time-averaged flow U(x) plus its fluctuations u ′ (x, t) as u(x, t) = U(x) + u ′ (x, t). e number of spatio-temporal modes retained is 4 (2 steady modes and a mode with frequency/wavenumber and its conjugate complex mode). As seen in the Figure, there is a traveling wave describing the ). On the one hand, the reconstructed flow field captures some of the main features of the original one, but it would be necessary to retain a larger number of modes to improve the solution. On the other hand, the method retains a large number of frequencies and wavenumbers, so it is difficult to identify the more relevants. e spatio-temporal diagram shows several frequencywavenumber combinations, in good agreement with the high complexity of this flow, that could be represented as a combination of multiple traveling waves. However, the simplest solution represents the flow as a combination of standing waves (horizontal lines with ω n = 2, 1, 0, -1, -2), modulated (the remaining horizontal lines). e method give accurate reconstruction of the flow field, however, the flow physics related to the wind turbine rotation might difficult to understand from the results obtained.

Secondly, STKD has been performed to the velocity fluctuations. e parameters used for the previous analyses are maintained, but, as in the previous case, the tolerances are increased to ε = ε 1 = 1 • 10 -1 for similar reasons as in the former case. e spatio-temporal diagram shows the 44 spatio-temporal modes retained (22 modes and their complex cojuated). Again, the main traveling wave is identified, and some other lower amplitude traveling waves with positive and negative slope. e negative slope traveling waves represent the flow field traveling in oposite direction.

is fact suggests that the rotational movement of the flow along the streamwise axis, may invoke unexpected movements of the very small and low intensity flow scales. e reconstructed flow field, with RMS error 4.99 • 10 -1 , matches be er the original solution than in the preivous case. Especially at points x/D > 13. Retaining a larger number of traveling waves (positive and negative slopes), the rotational movement of the flow, driving this region, is be er recovered. Nevertheless, the reconstruction error is almost the same as in the former case, meaning that these additional modes are very low amplitude ones.

= ε 1 = 1 • 10 -1 .
to the vertical and horizontal axis, in the origin. In the remaining modes, the streamwise component is not uniform, but presents groups or disorganized structures. e remaining components, preserve the symmetry with respect to the origin, but the structures are now rotated. Additionally, it is possible to distinguish the rotational movement within these modes, related to the flow structures shown in Figure . However, a more sophisticated STKD analysis should be performed in the azimuthal component in order to identify the direction of rotation of the identified flow structures. But this is far from the scope of this article. Lastly, it is remarkable that, in all the modes presented, the real part of the streamwise component shows a region with negative value at r/D ∼ 1, suggesting that the wind turbine foils may produce a very slight suction movement trough the velocity flctuations, probably related to the rotation of the flow field.

.

CONCLUSIONS

Flow structures in the wake of an offshore wind turbine have been studied in detail using a new method, STKD. is method combines either SVD or HOSVD with either DMD or HODMD to perform spatio-temporal analyses, obtaining as result the main frequencies and wavenumbers contained in a flow, and their combination. us, highly complex spatiotemporal structures, as the ones found in the wake of an offshore wind turbine, can be understood as a whole of traveling waves, whose phase velocity is determined by the combination of frequencies and wavenumbers. e main results show that the flow is mainly driven by a single traveling wave with phase velocity c = 2/3, moving in the wind direction. However, due to both, the rotational movement of the flow traveling downstream, and the complex movement of the velocity fluctuations, it is necessary to include additional flow strucures to describe the complete flow behaviour. ese flow features become represented by lower amplitude traveling waves with phase velocity very similar to the driven traveling wave, or even with negative phase velocity, representing the strongest rotation regions. ese results shed light on undertanding the complex behaviour found in the wake of a wind turbine, very useful to predict the effects produced in the interaction of the wake with another wind turbine.

  Figure le shows the spatio-temporal diagram of the streamwise velocity downstream the wind turbine at a representative point of the computational domain. e figure shows three types of flow behaviour. In the near field of the wind turbine (x/D < 5) the waves describing the flow are very low intensity and remain standing. In the intermediate field, 5 < x < 13, the figure suggests that there is a wave traveling along the streamwise direction, with slope ∆t/∆x ≃ 3/2. Finally, in the far field, x > 13, the flow behaviour changes, but the character of the traveling wave of the previous region still remains. Figure right shows the same spatio-temporal diagram, but, of the streamwise velocity fluctuations. As seen, there is a traveling wave driving the flow along the entire domain represented, with the same slope calculated before. Despite of the fact that it is still possible to distinguish the three regions previously mentioned, the near field and intermediate field are now joined to a single region.

Figure .

 . Figure . Spatio-temporal diagram along the streamwise direction of the streamwise component at r/D = 1 and θ = π/4. Le : unsteady flow field u x . Right: velocity fluctuations u ′ x . Figure shows the normal instantaneous velocity u y (le ), the mean flow (middle) and the velocity fluctuations (right) at two different locations of the computational domain (intermediate and far field) for 0 < r/D < 2. e instantaneous flow filed shows that it is possible to ditinghish the

Figure .

 . Figure . Radial velocity at x/D = 6 (top) and x/D = 18 (bo om). From le to righ: instantaneous field u y , time-averaged value U y and velocity fluctuations u ′ y .

  Figure shows the frequencies and wavenumbers as function of their amplitudes. STKD captures two temporal modes with ω = 0 related to the mean flow (mode with largest amplitude) and its correction, and a single mode with frequency ω 1 ≃ 2. Similar results are obtained for the spatial analysis, where two modes with κ = 0 and a mode with κ 1 ≃ 3 are found describing the flow. Figure le shows the spatiotemporal diagram that relates the wavenumbers and frequencies previously obtained.

Figure .

 . Figure . Computational domain for STKD analysis. Contours of streamwise velocity fluctuations along the streamwise component.

Figure .

 . Figure . STKD analysis of the instantaneous flow field with tolerances ε = ε 1 = 0.01. From le to right: frequencies and wavenumbers as function of their amplitudes.

Figure .

 . Figure . STKD analysis of the instantaneous flow field with tolerances ε = ε 1 = 0.01. Le : spatio-temporal diagram. Right: reconstruction of the original flow field.

  Figure shows the frequencies and wavenumbers as function of the spatio-temporal amplitudes obtained for this analysis. As seen, there is a dominant frequency and wavenumber, ω 1 ≃ 2 and κ 1 ≃ 3, respectively. ese results are in good agreement with the one obtained in the analysis of the instantaneous flow field. e spatio-temporal diagram of Figure top le shows the 14 spatio-temporal modes retained for this analysis (7 modes and their conjugate complex). e diagram shows that it is possible to describe the flow as a sum of traveling waves, with very similar positive slopes. e main traveling wave with phase velocity c = 2/3, previously obtained, is also identified in the diagram. e reconstruction of the flow field, whose RMS error is 5.02 • 10 -1 , is shown Figure top le . As in the analysis performed in the instantaneous flow field, it is possible to identify the structure of the main traveling wave, but the structure is now slightly more complex, due to the influence of the lower amplitude traveling waves (obtained from the diagrams of Figure ). With the aim at studying more in depth the evolution of the velocity fluctuations, the same analysis has been performed descreasing the tolerance to ε 1 = 5 • 10 -2 . Figure bo om shows the results obtained.

Figure .

 . Figure . Counterpart of Figure for the velocity fluctuations analysis using ε = ε 1 = 1 • 10 -1 .

Figure . Figure .

 .. Figure . Counterpart of Figure for the velocity fluctuations analysis using ε = ε 1 = 1 • 10 -1 (top) and ε = 1 • 10 -1 , ε 1 = 5 • 10 -2 (bo om).

ACKNOWLEDGMENTS

is work was partially supported by the Spanish Ministry of Economy and Competitiveness, under grant TRA --R.


Mi Wang
email: m.wang@leeds.ac.uk

Metrology of Tomography for Engineering

Keywords:

Tomographic imaging has unique features at "seeing" through the optical opaque medium and "building" up a volumetric view of multiphase dynamics in process pipelines or reactors in a nonintrusive manner. At other aspects, raw tomograms are reconstructed with a specific algorithm, which usually present distributions of a specific physical property of the process medium, e.g. the electrical conductivity of mixture in the use of electrical resistance tomography. However, these volumetric measurements are not conventionally understood, which may create a level of challenges in industrial application. Normally, specific theories or methods have to be employed to convert raw tomograms to meaningful engineering data. This paper reviews typical methods in tomographic data fusion and their implementations for process applications via a number of case studies carried out by the author and his team, providing a glance of view of the imaging metrology. The electrical resistance tomography is particularly expressed for measurement of various multiphase flows, including mixing processes, slurry, oil-in-water and gas-in-water two-phase flows, as well as 3-dimensional rendered flow regimes. It emphasises the important role of enginering data fusion in the metrology of tomography for engineering application.

INTRODUCTION

Tomography has become one of common methods for visualizing and characterizing process dynamics of multiphase flows in chemical, petroleum, nuclear engineering and many other areas over the past decades. Electrical tomographic imaging, comparing the conventional direct imaging methods such as Optical reflection, PIV and X-ray transmission, can "see" the multiphase flow dynamics through the optical opaque medium without the use of radioactive source. However, the physical properties of mediums sensed by tomography are normally not as engineering parameters as expected. An interpretation from tomographic data to engineering parameters is always required. However, the method of the interpretation might be different from one case to other, involving multiple sciences at physics, statistics, computing as well as process engineering knowledge, which raises the specific features of the tomographic metrology for solving engineering problems.

The aim of the paper is to emphasise the important role of data fusion in bridging between tomography and process engineering for advanced process measurement and control. Typical methods for interpretation of disperse phase or miscible mixing dynamics from tomography data are summarily expressed and a number of case studies carried out by the author and his team are reviewed in following sections, which address both aspects of science and engineering, typically based on electrical resistance tomography (ERT), in engineering measurement challenges.

Metrology of tomography

Tomography is recognized as an indirect imaging method comparing the conventional reflection, absorption and transmission imaging, which is based on measurement of electrical properties of materials by applying a low frequency (from DC up to few MHz) electric field or a magnetic field. A low frequency electromagnetic field can penetrate most process materials which are opaque to the light. The electrical techniques also avoid the hazards of ionising radiation generated from nuclear emission techniques, e.g. x-ray or -ray based techniques. They are inexpensive and relatively straightforward to implement with submillisecond temporal resolution. Tomographic images (tomograms) needs to be reconstructed from boundary measurements with a specific algorithm. The reconstructed images generally do not report the engineering parameters of the process and are suffered from a level of artificial errors from the inverse solution process. Due to the limited number of measurements and the propagation nature of low frequency electromagnetic waves, they provide images with a spatial resolution around 5% (the diameter of the object to the diameter of the vessel) and a homogeneity resolution better than 1% (e.g. the mean concentration of gas in water).

Further data process are normally required in order to derive meaningful engineering data from conductivity tomograms. Typically, it can be employed to derive the disperse phase concentration distribution or mixing homogeneity in multiphase flows or mixing processes respectively. It is also used to investigate the permeability of materials, foam structure, velocity distribution and further, the flowrate of disperse phase in multiphase flows. There are many industrials applications reported, demonstrating its extensive applicability but also the important role of engineering interpretation as a key in the metrology of tomography for engineering.

Principle of process tomography

Electrical tomography, including electrical capacitance tomography (ECT), electrical impedance tomography (EIT) and electromagnetic tomography (EMT), is based on the specific properties of materials principally sensed by each technique [START_REF] Wang | Industrial tomography: systems and application, 1 st Edition[END_REF]. Electrical resistance tomography (ERT) is a particular case of electrical impedance tomography when the real component of electrical impedance is the dominant property of materials in an EIT process application. ECT senses the permittivity distribution of dispersed materials in a fluid process with a non-conductive continuous phase. ERT is specified for a process that has a conductive continuous phase. EMT is mainly applied for high conductive fluids, which can induce measurable current under a magnetic field. In the case of EIT, the sensor is made from multiple electrodes arranged around the periphery of the internal wall of the process vessel or pipeline, in contact with the process medium but not intrusive to the medium. An alternating current is applied to some electrodes and voltages are measured from the remaining electrodes, according to a predefined sensing strategy. Then these voltage measurements are used to reconstruct the impedance distribution inside the vessel with a specific inverse algorithm. Based on the reciprocity and sensitivity theorem proved by Geselowiz [START_REF] Geselowitz | An application of electrocardiographic lead theory to impedance plethysmography[END_REF] and Lehr [START_REF] Lehr | A vector derivation useful in impedance plethysmographic field calculations[END_REF] and the assumption cited for sensitivity coefficient backprojection (SBP) approximation [START_REF] Wang | Inverse solutions for electrical impedance tomography based on conjugate gradients methods[END_REF], the relative change of boundary voltage measurement can be presented as Eq.1 with an assumption of (  ),

        w k k k j k w k k k j k j j s s V V 1 , 1 , ) ( ) (     (1)
where j is the measurement-projection location and k is the pixel number, sj,k denotes the sensitivity coefficient at pixel k under the measurement-projection j, P denotes the maximum number of measurements, w denotes the maximum number of pixels, k and k are the conductivity and conductivity change at pixel k, respectively, and Vj and Vj refer to the reference voltage and the voltage change at measurementprojection j.

It is known that inverse solution of Eq.1 is not directly derivable due to the ill-condition of sensitivity matrix. Many indirect methods were developed to solve the linear equation in the past. Typically, they were reported as single step method, e.g. the back-project method [START_REF] Kotre | EIT image reconstruction using sensitivity weighted filtered back-projection[END_REF] and the Newton one-step reconstruction (NOSER) [START_REF] Cheney | NOSER: An algorithm for solving the inverse conductivity problem[END_REF].

However, it should be pointed out the solution for Eq. 1 if any exists, only satisfies the change of conductivity <<. The closest solution for large conductivity changes may be obtained from multi-step approach [START_REF] Yorkey | Comparing reconstruction algorithm for electrical impedance tomography[END_REF].

In the process, errors may also be introduced from a number of stages, e.g. (a) at the measurement stage due to limits on signal-to-noise ratio, (b) at the inverse solution stage due to the linear approximation (for a nonlinear electric field problem) and ill-condition of the inverse problem (therefore the sharp boundary of an object image not possible) and (c) at the visualisation stage due to limits of human visual capability (only about 30 shades of grey). These errors are certainly transferred to the engineering interpretation. Therefore, understanding of the tomography principles and error sources are necessary for development of the metrology for engineering. Due to the focuses of the paper, the error and uncertainty of given case studies are not extensively discussed, which may find from specific articles elsewhere.

Metrology for engineering

Since ERT can detect local changes in electrical conductivity, the technique is used to study unsteady mixing or fluid dynamics of flow mixtures, such as gas-liquid or solid-liquid mixtures, where the disperse phase of fluids or solids have different conductivities from the aqueous continuous phase. ERT may, therefore, be suitable for numerous aqueous-based processes. However, ERT only produces electrical conductivity distribution or maps. For most process applications, relevant engineering data have to be interpreted from conductivity-based information. Few of typical converting relationships are introduced here.

Volumetric fraction

A number of correlations were proposed in history to convert electrical conductivity of a system to volumetric fraction of a disperse phase [START_REF] Wang | Chapter 2 Electrical Impedance Tomography, Industrial tomography: systems and application[END_REF]. Among them, the Maxwell relationship [START_REF] Maxwell | A Treatise on Electricity and Magnetism[END_REF] is most widely used in ERT application for process engineering [START_REF] Sharifi | Electrical Resistance Tomography (ERT) applications to Chemical Engineering[END_REF], which is given in Eq2.

f s m s m f f s m m s f v C                    2 2 2 2 ( 2
)
where Cv is the local volumetric fraction of disperse phase, f, s and m denote the conductivities of aqueous continuous phase, disperse phase and mixture respectively.

If the disperse phase is nonconductive, e.g. gas, oil or sands, Eq.2 can be simplified as,

m f m f v C        2 2 2
(3)

Salt concentration

In research on miscible liquid mixing, the liquids under investigation can be labelled with sufficient conductivity contrast to enhance the imaging performance [START_REF] Wang | Electrical resistance tomographic sensing systems for industrial applications[END_REF]. High concentration saline solution can also be used as a tracer to track the dynamic trajectory of the mixing process. In an instance of single phase miscible liquid mixing, the conductivity of mixture, m, provides a linear relation to the concentration Cw of salt [START_REF] Mann | Resistance tomography imaging of stirred vessel mixing at plant scale[END_REF][START_REF] Holden | Macromixing using electrical resistance tomography[END_REF]:

f w m C      728636 . 1 (4)
) ( 5785 . 0 f mc w C      (5)
where the m and f are in unit of mS cm -1 ; the weight concentration Cw is in gl -1 , the range of Cw: 0≤Cw≤10.

Porosity

Based on Archie's law, the local porosity of materials may also be derived from conductivity value [START_REF] Qiu | Electrical resistance tomography for imaging sub-seabed sediment porosity: initial findings from laboratory-scale experiments using spherical glass beads[END_REF]. The conversion is given by Archie's law [START_REF] Archie | The electrical resistivity log as an aid in determining some reservoir characteristics[END_REF]:

k f m             (6)
where  is the porosity (%) and k the cementation index dependent on the shape and packing of the particles (k=1.5 for spherical glass beads).

Velocity distribution

The principle of cross correlation method has been widely used to derive the velocity component of moving objects from two sets of measurements or images taken with a known time interval, e.g. the velocity distribution derived in Particle image velocimetry (PIV). Unlike PIV to statistically derive velocity component from two photo images of many seeds' over an interrogation region with a known time interval, ERT derives disperse phase velocity (or called the structural velocity) is based on many tomograms obtained from a dual-plane ERT sensor with a known distance between the two sensing planes and frame rate [START_REF] Lucas | Measurement of the Solids Volume Fraction and Velocity Distributions in Solid-liquid Flows Using Dual-plane Electrical Resistance Tomography[END_REF][START_REF] Wu | Characterization of air-water two-phase vertical flow by using electrical resistance imaging[END_REF]. However, the concept of cross correlation method applied in PIV and tomography is the same.

The fundamental principle underpinning crosscorrelation flow measurement is the 'tagging' of signal's similarity. These concepts are best illustrated in the figure below. The basic method is to find t that can make the difference, , minimum. This can be achieved using the least squares criterion,

         dt t f t f 2 2 1 2 ) ( ) ( ( ) ( t t  (7)
A revised error function is given by Eq.8, which only remains the product of two function since other terms to be constant from the integration over a sufficient time. Hence, the error function is minimum, when the last term is maximum. This expression is commonly known as the cross correlation function denoted by R12(t).

      dt t f t f R ) ( ) ( ) ( 2 1 12 t t (8) ) ( ) ( ) ( 2 1 1 12 n m f m f n R l m     (9)
where l is the sample length, n is the offset number, f1 and f2 are the values at pixels positioned at the same location in two sequences of the up-flow and down-flow images respectively.

The disperse phase velocity can be simply derived as,

p s s sampling p s s N f d T N d d v      t (10)
where v is the velocity, ds is the distance between two sensors, Np is the number of offset frames to get the peak value of R12(Np) and fs is the sampling frequency of ERT.

The fractional velocity discrimination, , can be estimated with Eq.11 and also the necessary data collection speed (sec./dual-frame), , for a certain  can be expressed as Eq.12.

t   2  (11) t  2  ( 12
)
Above point-by-point cross correlation technique for flow velocity measurement is based on the assumption that flow trajectories are parallel to each other and perpendicular to the sensor plane. However in most cases this ignores the fact that the trajectories of particles of the dispersed phase exhibit a complex three-dimensional behaviour. The 'bestcorrelated pixels' method overcomes this problem by proposing that a signal from one pixel on plane X is somehow better correlated with a signal from a non-axially corresponding pixel on the plane Y [START_REF] Mosorov | The 'Best-Correlated Pixels' Method for Solid Mass Flow Measurements Using Electrical Capacitance Tomography[END_REF]. The pixels from the second plane are chosen from the axially corresponding pixel and its neighbours, as described by Eq.13.

         1 0 ] , [ ] , [ ] , [ ] , [ ] [ ] [ ] [ T k j m i n m n j m i n y m n x p k y k x p R B j i  ) , ( (13)
where T is the number of the images, for which the crosscorrelation is calculated; p=-(T-1), ...0,1, 2, 3... T-1; k=0,1,2,3...T-1; n, m are the coordinates of the pixel; x, y are the values of the pixels on plane X and plane Y and B is the group of neighbouring pixels on plane Y.

Visualization

Multiphase flow is of practical significance in oil and gas industries, which is extremely challenging to be visualised and characterised in industrial multiphase flows due to the opaque nature of most industrial multiphase flows and pipelines. At the other aspect, the low resolution of electrical tomography with the colour mapping in commonly use is not sufficient to visualise the distinctive interface between fluid phases. As a result, the visualisation by the systems conveys limited information regarding multiphase flow dynamics, e.g. bubble size and distribution. A novel approach, namely bubble mapping, was proposed to overcome the problem. In the approach, a new lookup table is built up by means of transferring mean concentration of an interrogation pixel into a number bubbles in a base size located randomly inside the pixel. Further, an enhanced isosurface algorithm was applied to isolate big bubbles based on the merging of neighbouring bubbles in the cell which have their mean concentration beyond a certain threshold value. With a proposed bubble mapping approach, a stack of crosssectional tomograms by electrical tomography is transformed and displayed as individual air bubbles with different size in respecting to the air concentration in a visualization pixel. With further increment of air concentration, large bubbles will be merged from a number of pixels with full air cavity and then all bubbles are computed to 3-dimensional bubbles with an enhanced isosurface algorithm [START_REF] Wang | Experimental tomographic methods for analysing flow dynamics of gasoil-water flows in horizontal pipeline[END_REF].

ENGINERING CASE STUDY

Phase volumetric or concentration distribution might be the first engineering characteristic data interpreted from electrical tomography. The electrical impedance in either conductivity or permittivity is converted to materials' concentration distribution, using relevant methods as stated in Section 1.2. Typical examples are expressed in the section.

Mixing process

The non-intrusive three-dimensional measurement of mixing inside a stirred vessel in three-dimension, using electrical resistance tomography, provided powerful opportunities for characterising and quantifying the process complexities [START_REF] Mann | Application of electrical resistance tomography to interrogate mixing processes at plant scale[END_REF]. One of the early works reported an application of ERT for three-dimensional (3D) imaging of the concentration of solids in a slurry mixer as a function of key process variables (particle size, impeller type, agitation speed) [START_REF] Mckee | Tomographic measurements of solidliquid mixing[END_REF][START_REF] Williams | Development of slurry mixing models using resistance tomography[END_REF]. It was demonstrated how ERT can provide a wealth of detailed data to allow model development. On-line two-dimensional (2D) imaging of miscible liquid/liquid mixing and gas-liquid mixing in a large scale baffled mixing vessel (2.3 m 3 ) fitted with eight planes of ERT sensor shown in Figure 1 was reported in 1996 [START_REF] Mann | Resistance tomography imaging of stirred vessel mixing at plant scale[END_REF][START_REF] Dickin | Electrical resistance tomography for process applications[END_REF]. Figure 1 also shows a typical set of resistivity contours interpolated from a stack of 8-plane 2D images reconstructed with the back-projection algorithm, rendered as a solid body isometric image [START_REF] Wang | Stirred vessel mixing in 3-D using electrical resistance tomography[END_REF]. Results were presented from times of 1, 2, 3 and 4 s following the surface addition of 10 dm 3 of concentrated brine (conductivity of 13.5 mS cm -1 ) into a background conductivity of 0.1 mS cm -1 at t = 0. The stirrer speed was 100 rpm generating an estimated internal flow of 0.686 m3 s -1 . The mixing time was estimated using a colorimeter probe and was approximately 14 s. In Figure 1, high conductivity is presented by red regions and low conductivity as blue regions (cut off by the isosurface). The salt concentration at the isosurface, as a mixing index derived from Eq.5, is 0.035 gl -1 (0.16 mS cm -1 ), which was adopted from the final conductivity of the liquids after mixing was completed. Further studies of a gas-liquid system (air-water) in the same mixing tank allowed tomographic gas distribution to be compared with established characteristic flow patterns. A multiisosurface 3D gas concentration distribution was produced from a stack of 8-plane 2D images using the Maxwell relationship (Eq.3) at a pseudo-stationary mixing, which presents two gas equal-concentration contours (Figure 2) with the values as indicated in the volume histogram shown at the bottom-right of the figure. In the experiment, gas was sparged from a pipe beneath a sixblade Rushton turbine at 3.5 litre sec -1 with agitation at 73 rpm. Figure 2 reveals the high gas concentration (in red) is located beneath the turbine as well as the region close to the wall, which are reasonably report the effect of high gas injection and agitation, respectively. In addition, a high gas concentration at the top of the figure is due to the air vertex produced by the agitation (the red colure region is hidden by the yellow isosurface).

Figure 1. Monitoring a dynamic miscible liquid mixing in a baffled 2.3 m 3 mixing tank using ERT (mixing index: 0.035 gl -1 (0.16 mS cm -1 ) at the isosurfaces of above images; conductivity of pulse brine: 13.5 mS cm -1 in volume 10 dm 3 , speed of Rushton stirrer: 100 rpm) [START_REF] Wang | Stirred vessel mixing in 3-D using electrical resistance tomography[END_REF].

Figure 2. Pseudo-stationary gas-liquid mixing [START_REF] Wang | Stirred vessel mixing in 3-D using electrical resistance tomography[END_REF] (the images of two blades and a shaft are for purpose of illustration, which are not from the tomographic imaging).

Slurry pipeline flow

Efficient slurry transportation is vital to many industries. It was proposed that helically formed pipes, which can be used to keep particulate solids in suspension, should be applied to enhance the distribution of solids in piped slurries. The use of an in-situ measurement method based on electrical impedance tomography was proposed to assist in understanding the effect of particle suspension and as well as the effect on the wear of pipes by solid particle impingement due to the application of such a swirlinducing pipe [START_REF] Wang | Visualisation of asymmetric solids distribution in horizontal swirling flows using electrical resistance tomography[END_REF]. The experiments were carried out on a 50 mm diameter hydraulic conveying pipe loop composed of transparent 1 m flanged sections in a vertical plane. The pumped medium was a mixture of water and spherical nonconductive beads (diameter 2 mm and specific gravity 1.45). ERT sensors were fitted at distances of 150 mm, 370 mm, 885 mm and 1150 mm downstream of the outlet of the swirling pipe (a meter-long with 900º of spiral). This corresponds to downstream distances in the length to the pipe diameter ratios (L/D) of 3, 7.4, 17.7 and 23. A series of experiments were carried out at mean in situ particle burden concentrations of 2.1%, 4.0%, 6.4% and 8.6% by volume (or 3%, 6%, 9% and 12% by weight) at flow velocities of 0.5, 1.0, 1.5, 2.0 and 2.5 m/s, which are interpreted from Eq.3.Through the application of an advanced impedance image reconstruction algorithm [START_REF] Wang | Inverse solutions for electrical impedance tomography based on conjugate gradients methods[END_REF] and other analysis software, the asymmetric solids concentration distribution in horizontal swirling flows can be quantified. Particle trajectories and concentration regimes are reported as a function of the water axial flow velocity and the downstream distance from the swirl-inducing pipe section. Figure 3 demonstrates the measurements obtained at the solid burden concentrations of 8.6%. In the absence of accurate predictive models for such complex flows, it is demonstrated that this method enables direct visualisation of the solids suspension as function of flow velocity in the pipes.

Two phase flow

This study presents the use of a high performance dualplane electrical impedance tomography system [START_REF] Wang | A high performance EIT system, accepted for publication[END_REF] to measure the vertical upward co-current oil-in-water pipe flows [START_REF] Li | Volume flow rate measurement in vertical oil-in-water pipe flow using electrical impedance tomography and a local probe[END_REF]. Experiments were carried on a flow loop with a transparent 2.5 m long, 80 mm inner diameter test section detailed in Figure 4, using kerosene and tap water. Measurement was carried out at imaging rate of 1000 dual frame per second. The oil phase concentration and velocity distribution were interpreted with Eq.3 and the crosscorrelation method introduced in previous section respectively. The flow conditions were predominantly of the dispersed type with the non-slip oil concentrations of 9.1%, 16.7% and 23.1% respectively. Typical oil concentration and velocity distributions were presented in Figure 5, which were compared with measurements obtained with a mechanical orientated local intrusive conductance probe, showing a good agreement [START_REF] Li | Volume flow rate measurement in vertical oil-in-water pipe flow using electrical impedance tomography and a local probe[END_REF].

Swirling flow

A series of air-in-water flow tests were carried out in a 0.08 m internal diameter vertical flow loop at the University of Huddersfield [START_REF] Wang | Visualisation of bubbly velocity distribution in a swirling flow using electrical resistance tomography[END_REF]. The working section was 2.5 m long. A dual-plane ERT sensor was installed in the upstream section of the flow loop 2 m from the base of the vertical working section. The distance between the two ERT sensing planes was 0.05 m. A 6-vane swirler with a 20 swirl angle was mounted 4D upstream of the ERT sensor (Figure 6). The volume fraction distribution for each flow condition was reconstructed from 5000 frames of data to give the 'steady state' local gas volume fraction distribution, averaged over a period of about 52.6 seconds. About 8000 dual-frames of data per flow condition were recorded and then used to obtain one velocity vector distribution. Results are shown in Figure 7 in regard the gas volume fraction and velocity distributions under three flow conditions. The results demonstrate the potential capability of electrical resistance tomography for measurement of 3-D flow vector distribution. It implies that with the assistance of a fast data collection rate, the flow vector distributions, not only in pipelines but also in other applications, can be extracted with good accuracy [START_REF] Wang | Visualisation of bubbly velocity distribution in a swirling flow using electrical resistance tomography[END_REF].

Periodical flow in OBR

Oscillatory baffled reactor (OBR) has been proven to be very efficient due to its enhanced mixing mechanism, good particle suspension and excellent mass and heat transfer comparing to conventional chemical reactors. OBR is also capable of operating certain batch processes as a continuous process and is mostly employed in the polymer industry. In this study, the focus was on how to derive the periodical velocity of an OBR, where the conventional crosscorrelation method based on the assumption of steady state flow condition was not applicable. By applying a period sample extraction from tomograms obtained with a sampling speed of 1000 dual-frames per second, a significantly improved temporal resolution allowing detailed analysis of the process and measurement of characteristic parameters was achieved [START_REF] Issa | Velocity measurements of multi-phase unsteady flow in oscillatory baffled reactor using electrical resistance tomography[END_REF]. It allowed to observe the effect of individual strokes of the OBR piston in the flow of the mixture, which established the foundation for getting instant velocity profiles. The test rig in 50cm diameter glassware pipe shown in Figure 8 was installed at the Online Instrumentation Laboratory, University of Leeds [START_REF] Vilar | On line analysis of structure of dispersions in an oscillatory baffled reactor using electrical impedance tomography[END_REF]. A stagnant water-oil emulsion was introduced by the piston and pulsed forward and reversely in the rig at 1.5 Hz oscillatory frequency, 20 mm amplitude. A dual-plane ERT sensor was installed in the test rig. Tomograms were reconstructed with the back-projection algorithm. The axial isosurface image of oil fraction distribution (Figure 9) and axial velocity profiles (Figure 10) provide essential information regarding the flow conditions in the OBR that can be used to improve the OBR's performance. It provides a promising method to acquire the periodical process characteristics such as velocity profiles in the oscillatory process. and (e) annular flow [START_REF] Wang | Experimental tomographic methods for analysing flow dynamics of gasoil-water flows in horizontal pipeline[END_REF].

Visualisation

This study deals with a fully developed turbulent two phase flow with no phase changing, and presents the outcomes of tomographic imaging techniques to visualise gas-water flows in a horizontal pipeline.

Experiments were conducted in a 50 meter long test section consisting of 4 inch nominal diameter piping within the industrial-scale three phase flow facility at National Engineering Laboratory (TUV NEL), UK. Refined oil (HT9) of 830 kg m -3 density and 16.18 cP viscosity was used alongside Magnesium Sulphate saltwater substitute (MgS04) of 1049.1 kg m -3 density and 1.35 cP viscosity. Nitrogen gas was supplied externally from a storage tank with density of 12 kg m -3 and absolute viscosity of 0.0174 cP. The total liquid flowrates were varied between 0 and 140 m 3 hr -1 and the gas flowrates were varied between 0 and 530 m 3 hr - 1 . A total of 270 measurements were conducted, which produced a variety of flow regimes in the horizontal pipeline, including stratified flow, slug flow, plug flow, bubble flow, and annular flow [START_REF] Wang | Experimental tomographic methods for analysing flow dynamics of gasoil-water flows in horizontal pipeline[END_REF]. ). The left shows the photo from camera, the middle is the axial cross section of stacked raw tomograms with colour mapping scheme, and the right shows the image produced by the bubble mapping scheme. Compared to its counterpart by conventional colour mapping, the new approach is not only able to visualise the expected flow regimes but also reveal additional flow dynamic characteristics with high temporal resolution. It is clearly demonstrated that the bubble mapping can greatly enhance the reality of flow regime visualization comparing the raw tomograms.

DISCUSSION

Electrical tomography as one of process tomography technologies appears for about three decades. Many scientists and engineers have devoted great efforts to progress the science and engineering of the technology for industrial practice. However, the unconventional features of the technology made a long standing of consideration on whether it could deliver a new practical imaging metrology for engineering. With the advantages of the unique ultrahigh temporal resolution, good penetrating and safe in operation, it was somehow over expected on its spatial resolution and neglect the importance of data fusion, which may result in improperly bridged with industrial challenges. This paper reviews and emphasises the importance of engineering interpretation in bridging between tomography and process engineering for advanced process measurement and control, provided with typical data fusion methods and a number of case studies carried out by the author and his research team, which express the specific features of the metrology, hopefully, promote the process tomography for engineering applications.

Figure 1 .

 1 Figure 1. Cross-correlated two functions.

Figure 3 .

 3 Figure 3. Solids suspension in horizontal pipeline at a fixed particle volumetric concentration of 8.6% [25]. The white triangles on the top of tomograms indicate the top direction (at 12'clock) of the pipe line. The vertical axis denotes the flow velocity in unit of ms -1 and the horizontal axis denotes the downstream distance ration. The contours in topographic map indicate the equal-concentration values.

Figure 4 .

 4 Figure 4. Schematic of the test section [27].

Figure 5 .

 5 Figure 5. Oil concentration (the top image) and velocity (the bottom image) distributions from ERT (Qw=7.5 m 3 /h, Qo=0.75 m 3 /h) [27].

Figure 6 .Figure 7 .

 67 Figure 6. Dual-plane ERT Sensor and flow swirler [28].

Figure 8 .

 8 Figure 8. Basic diagram of the experimental setup [30].

Figure 9 .

 9 Figure 9. 3D isosurface representation of the periodical oil concentration distribution in an OBR section [29].

Figure 10 .Figure 11 .

 1011 Figure 10. The accuracy of forward (the top figure) and reversed (the bottom figure) axial velocity profiles is enhanced with the increment of the sampling periods [29]. (Sampled at T=(2n+1/2) and (2n+3/2), n: the sampling periods 1, 3, 9; Frames in a period: 666 frames; Frame length for cross-correlation: 80 frames (0.08s))

Figure 11

 11 Figure 11 demonstrates the results of the approach applied to gas-liquid horizontal flow, with the flow regimes of stratified flow (Figure 11.a.), bubbly flow (Figure 11.b.), plug flow (Figure 11.c.), slug flow (Figure 11.d.), and annular flow (Figure 11.e.). The left shows the photo from camera, the middle is the axial cross section of stacked raw tomograms with colour mapping scheme, and the right shows the image produced by the bubble mapping scheme. Compared to its counterpart by conventional colour mapping, the new approach is not only able to visualise the expected flow regimes but also reveal additional flow dynamic characteristics with high temporal resolution. It is clearly demonstrated that the bubble mapping can greatly enhance the reality of flow regime visualization comparing the raw tomograms.

ACKNOWLEDGMENT

This work is founded by the Engineering and Physical Sciences Research Council (EP/H023054/1, IAA (ID101204) and the European Metrology Research Programme (EMRP) project "Multiphase flow metrology in the Oil and Gas production" which is jointly funded by the European Commission and participating countries within Euramet.


Damiano Casalino
email: d.casalino@tudel.nl

Francesco Avallone

Ignacio Gonzalez-Martino

Daniele Ragni

S Y Mpo

O Tatin

Aeroacoustic study of a wavy stator leading edge in a realistic fan/OGV stage

Keywords:

e effect of sinusoidal serrations applied to the leading-edge of the vanes of a realistic fan stage is investigated using high-fidelity numerical simulations. e CFD solver PowerFLOW based on a hybrid la ice-Boltzmann/very-large-eddy-simulation model is used to compute the unsteady flow and radiated noise of the -in source diagnostic test fan rig of the NASA Glenn Research Center. A computational model validated for three different geometries of the outlet guide vanes with straight leading edge is used. A subset of validation results is reported to prove the capability of the solver to accurately predict the influence of the stator geometry on the far-field noise. Different sinusoidal leading edge serrations are investigated for a radial and a swept stator and the same rotor and operating conditions. e influence of the serrations on the acoustic far-field and noise power level is reported in relation to the statistical properties of the velocity fluctuations in the wake of the rotor. Some noise reductions are obtained when the undulation amplitude and wavelength are large enough compared to the integral scales of the impinging turbulence fluctuations.

INTRODUCTION

Current turbofan design trends towards large bypass ratio engines (up to 12-13), dictated by the necessity of reducing the fuel consumption without increasing the engine weight, result in axially compact engines[ ], with a smaller distance between the rotor and the Outlet Guide Vane (OGV). Rotor/stator interaction noise is the dominant source of tonal and broadband noise, which is expected to be enhanced by smaller interstage distances.

One of the passive noise reduction concepts for rotor/stator interaction noise that has received the a ention of the aeroacoustic community is the wavy leading edge. e effect of a serration is to reduce the correlation in the radial direction by decorrelating the source in adjacent undulation roots, to create a destructive interference along the serration edge through the generation of vanishing subcritical gust components, or simply to decorrelate the fluctuations induced at the root and the peak of the undulations. As discussed herea er, these effects have been experimentally observed and confirmed by Computational Aero-Acoustics (CAA) simulations for wing sections. is serves as motivation for the present numerical investigation carried out on a realistic fan stage configuration.

Inspired by whale flippers, the effects of leading edge serrations on the aerodynamic performances of wing sections have been studied since the Seventies. Soderman[ ] performed li and drag measurements in a closed wind tunnel of an aerofoil with small sharp serrations on the leading-edge. He observed an increase of the incidence of maximum li and a small drag penalty at low angle of a ack. e delayed stall was related to the generation of streamwise vortices at the leading edge that energize the boundary layer through momentum transfer and delay the leading edge flow separation. Further experiments confirmed the effects of serrations on the stall behaviour of wings [ , ] and whale flipper with tubercles [ , ].

Aeroacousticians have been also inspired by nature. In the pioneering work by Graham[ ], the silent flight of owl was related with the special adaptations of the feathers in several species. By comparing the wing and the feathers of silently flying owls with the ones of an owl that belongs to a species that does not fly silently, he identified three peculiarities that are responsible for the quiet flight: the leading edge comb, the trailing edge fringe and the downy upper surface of the feathers. He also proposed to use similar features for the reduction of aircra noise, together with lower wing loading and flight speed, as also inspired by owls. Using simple modelling assumptions, Lilley[ ] concluded that the feather adaptations of the owl lead to a major noise reduction above 2 kHz.

Recent works on the use of leading-edge serrations for the reduction of the aerofoil/turbulence interaction noise have focused on the effects of the ratio between the undulation amplitude h s and the turbulence length scale L t of the impinging flow, the effect of the serration angle θ s , and thus the undulation wavelength λ s . Recent experiments[ ] conducted on a flat plate and a NACA-( ) aerofoil of 0.15 m chord and 0.45 m span interacting with grid-induced turbulence in a free stream with velocity U in the range 20 to 60 m/s, determined the existence of an optimal undulation wavelength λ opt s ≃ 4L t , corresponding to a condition at which adjacent sources in the undulation roots are incoherently excited. ey finally argued that the noise power level ratio between serrated and straight edge aerofoil at the frequency f is inversely proportional to the Strouhal number St h = f h s /U, i.e. the ratio between h s and the streamwise wavelength U/ f , for St h 0.2 up to the value where serration self-noise starts to dominate. e goal of the present paper is to investigate the benefits associated with a wavy OGV leading edge for a realistic engine geometry and turbulence anisotropy, by performing non-linear compressible flow simulations of a fan stage configuration. An extensively validated numerical model of the -in Source Diagnostic Test (SDT) fan rig of the NASA Glenn Research Center [ , ] is used as a reference for a series of simulations performed by using different sinusoidally serrated OGVs. Two reference configurations are considered, a radial stator (baseline) and a swept stator (low-noise). All simulations are performed at approach conditions corresponding to 61.7% of the nominal rotational speed. e organization of the paper is as follows. Section provides information about the geometry of the problem and details about the wavy leading edge.

e flow solver and the computational model are identical to the ones used by Casalino et al[ ] and reported, for the sake of completeness, in Section , together with a subset of reference results for the straight OGVs in Section .

e integral scales of the turbulent fluctuations in the interstage are estimated using the approach presented in Section . Results for the wavy OGVs are reported and discussed in Section in relation to the radiated far-field noise and to the integral length scales in the wake of the rotor. Due to the limited number of serration considered, only provisional conclusions for one single operating conditions are given in the conclusive section.

. FAN STAGE CONFIGURATIONS AND SI-NUSOIDAL LEADING EDGE

e reference engine is the SDT fan rig of the NASA Glenn Research Center [ , ], made available by NASA in the framework of the AIAA Fan Broadband Noise (FBN) prediction workshop. e engine model and the wavy OGV are described in this section.

. Reference geometries All the other parts, with the exception of very small variations of the inner and outer casing profiles downstream the OGV, are the same for the three configurations. e nacelle is perfectly axisymmetric without struts. In order to reproduce the stinger used in the experiment, a cylindrical prolongation of the centerbody has been added to the model released by NASA. e rotor radius is 0.2786 m, the bypass exhaust radius is 0.2710 m and the lip intake radius is 0.2962 m. e rotor is constituted of 22 blades, and the casing/blade-tip gap is about 0.5 mm.

. Wavy OGV

e wave leading edge is originated using the following procedure:

• A structured mesh is initially created starting from an imported unstructured surface mesh of a stator blade by individuation of the leading-and trailing-edge lines, defined as a sequence of N s equally-spaced constant points along the span from a minimal to maximal radial distance, and by construction of the chord line for every radial section.

• For every radial strip j delimited by the grid cuts j and j +1, a fixed point x j F is individuated on the chord of length c j , such that x j F -x j LE = σc j , where x j LE is the local leading-edge point, and σ is a prescribed relative portion of the chord affected by the morphing procedure.

• For every radial strip, the coordinates of the blade points such that d = x jx j LE /c j ≤ σ are modified using the following formula:

x j w = x j F + x j -x j F [1+cos(2πr/λ s )] (1-d/σ) hs , ( )
where r is the radial coordinate of the point, hs = h s /(σ c j ) is the prescribed undulation amplitude factor, and h s is the amplitude.

Six different wavy stator designs are considered in the present study, all obtained using the chord ratio σ = 0.15 and the parameters reported in Table . Corresponding images of the stator blades are shown in Fig. . e choice of these parameters for designs #1 and #2, was quite arbitrary, whereas for the other ones was based on a prior estimation of the integral scales of the turbulent fluctuations in the interstage volume, as discussed herea er. . CFD/CAA computational approach PowerFLOW solves the Boltzmann equation for the distribution function f (x, t, v) on a hexahedral mesh automatically generated around bodies, which consist of one or more connected solid parts. e function f represents the probability to find, in the elementary volume dx around x and in the infinitesimal time interval (t, t + dt), a number of fluid particles with velocity in the interval (v, v+ dv). e Boltzmann equation is solved by discretizing the particle velocity space into a prescribed number of values, in magnitude and direction.

ese discrete velocity vectors are such that, in a prescribed time step, one particle is advected from one point of the mesh to 19 neighbouring points, including the point itself, which constitute the computational stencil of the so-called D Q scheme (three-dimensional 19 states model). It can be demonstrated that using 19 particle velocity states ensures sufficient la ice symmetry to recover the Navier-Stokes equations for an isentropic flow[ ]. For high subsonic Mach number simulations, e.g, flows with local Mach number greater than 0.5, as in the present case, the LB solver is coupled with the solution of the entropy equation through a Lax-Wendroff finite difference scheme on the Cartesian LB mesh[ ]. Once the distribution function is computed, the macroscopic flow quantities, density and linear momentum, are simply determined through discrete integration: ρ(x, t) = j f j (x, t) and ρu(x, t) = j f j (x, t) v j . All the other quantities are determined through thermodynamic relationships for an ideal gas.

Solving the la ice Boltzmann equation is equivalent to performing a Direct Numerical Simulation (DNS) of the Navier-Stokes equations in the limits of the dynamic range (Mach number) that can be accurately covered by the number of discrete particle velocity vectors, and in the limits of the la ice resolution required to capture the smallest scales of turbulence. For high Reynolds number flows, turbulence modeling is incorporated into the LB scheme by changing the relaxation time in the collision operator that is computed accord-ing to a Bhatnagar-Gross-Krook (BGK) approximation[ ].

e turbulent kinetic energy and the turbulent dissipation are obtained by solving a variant of the Re-Normalisation Group (RNG) k-ǫ model for the unresolved scales[ ]. is approach is referred to as LB Very Large Eddy Simulation (LB-VLES). Since it is prohibitive to resolve the wall boundary layer using a Cartesian mesh approach down to the viscous sub-layer in high Reynolds number applications, a wall function approach is used to model boundary layers on solid surfaces.

e LB equation is solved on a grid composed of cubic cells. A variable resolution by a factor two is allowed between adjacent regions. Consistently, the time step is varied by a factor two between two adjacent resolution regions. Solid surfaces are automatically facetized within each cell intersecting the wall geometry using planar surface elements. No-slip and slip wall boundary conditions on these elements are imposed using a particle bounce-back process and a specular reflection process, respectively[ ]. Extremely complex geometries can be treated automatically. .

Computational setup

Images of the computational setup are shown in Fig. . e rotor and the spinner are encompassed by a volume (purple) that defines the rotating mesh region. e centerbody is extended with a solid cylinder since, similarly to the experiments, no primary jet is included in the simulation. e green and yellow surfaces in the interstage volume in the top-le image denote regions explored by Hot-Wire (H-W) measurements and are referred to as station # and station # , respectively. e second one is at the same axial location as the baseline OGV leading edge, and measurements are therefore available only for the low-noise configuration. e FW-H integration surface consists of three parts: a spherical sector around the intake, a cylindrical connector and a conical surface in the exhaust region. e cone is opened at its downstream extremity in order to avoid contamination of the acoustic signals due to integration of jet shear-layer hydrodynamic fluctuations. e downstream extension of the cone is however sufficient to recover the bypass duct radiation over the angular range of interest. e present setup is identical to the one used by Casalino et al[ ], with the exceptions of the shape of the trip added on the suction side of the rotor blades to trigger the turbulent transition in the wake of the rotor. Instead of a trip with a rectangular section, a zig-zag trip of height linearly varying from 0.3 mm at 5% of the span (root) to 0.1 mm at 95% ot the span, 5 mm wavelength and 1 mm amplitude, located at 10% of the chord, has been used, and all the simulations have been repeated, revealing a slight improvement of the prediction of noise levels at high frequencies. For the sake of consistency, the reference results reported in Section are the new ones obtained using the zig-zag trip.

All simulations reported in this paper have been performed by using, as initial condition, the solution obtained by using a coarser mesh and achieving an acceptable statistical convergence. en, the simulations are performed by sampling the solution along 10 rotor revolutions a er a very short initial transient of three blade passages. e CPU cost is of the order of six thousand CPU hours per rotor revolution using 720 cores Intel Sandybridge 2.7 GHz.

All the narrow band far-field noise and sound power level spectra have been computed using the Welch's periodogram method, using a signal length of 2.738 • 10 -1 s, a bandwidth of 28.63 Hz, and 13 spectral averages with 43% overlap. e noise signal at each microphone has been obtained by appending four FW-H signals computed at four azimuthal locations (every 90 • ), and truncated at an integer number of blade passage periods.

. REFERENCE RESULTS

In this section a subset of the available results for the three OGV configurations with straight leading edge are reported and shortly discussed.

ese results have been presented by Exa during the th FBN Prediction Workshop held on June th in Denver, as a special session of the rd AIAA/CEAS Aeroacoustics Conference. Only results for the approach operating condition are reported in this work.

Fig . shows a comparison between H-W measurements and simulation results at station # for the baseline configuration. e three components of the phase-locked average velocity, and the corresponding standard-deviation (SDV) values are plo ed on an angular sector covering one rotor blade passage. e average field is fairly well predicted, the main discrepancy being an underestimation of about 10% of the axial velocity component. It is worth mentioning that, for all configurations, the mass flow rate is predicted within an error of 1%, whereas the total pressure ratio across the full stage is predicted within an error of 0.4%. e predicted SDV levels are also in fair agreement with the measurements, with two main negative marks: an overestimation of about 20% of the axial and azimuthal velocity fluctuation levels close to the outer wall, and an overestimation of about 10% of the radial velocity fluctuation levels in the half-span region. It is worth mentioning that the predicted SDV fields are also affected by a statistical convergence error, since averages have been performed over 10 rotor revolutions only.

Figs. and show comparisons between H-W measurements and simulation results at stations # and # , respectively, for the low-noise configuration. At station # the same trends observed for the baseline configuration can be reported. Indeed, at this distance from the OGV, the velocity field is not affected by the OGV geometry in a significant way. At station # we can also observe a fairly good agreement between measurements and predictions. In particular, the levels are quite well captured by the simulations, as well as the turbulent mixing and spreading of the rotor wake. Similarly to what observed at station # , an underestimation of about 10% of the average axial velocity component and an overestimation up to about 20% of the azimuthal velocity fluctuation levels.

e velocity maps at station # provide a certain confidence of the capability of the method to predict the correct velocity field upstream of the OGV. is is proven for the low-noise configuration, and reasonably expected for the baseline one, for which no measurements are available at station # . erefore, the main discrepancy between H-W measurements and simulations remains an underestimation of about 10% of the axial velocity and an overestimation of about 20% of the azimuthal velocity SDV. e impact of these errors on the predicted far-field noise can be evaluated by considering the formula obtained by Amiet[ ] for an isolated airfoil of chord c and span L at zero incidence in a turbulent stream. e Power Spectral Density (PSD) of the far-field noise in the airfoil midspan plane, at a distance r from the airfoil midpoint and an angle θ from the streamwise flow direction, reads:

S(r, θ, ω) = ρ ∞ U ∞ sin θ 2r 2 (kc) 2 L 2 |L| 2 l (ω) S v (ω) , ( )
where ρ ∞ and U ∞ are the free-stream density and velocity, k = ω/c ∞ is the acoustic wavenumber, ω is the radian frequency, c ∞ is the speed of sound, L is the airfoil response function, l is the spanwise correlation length, and S v is the PSD of the upwash velocity component. e Helmholtz number kc can be wri en as

U ∞ k x c/c ∞ ,
where k x = ω/U ∞ is the streamwise wavenumber of an impinging turbulent perturbation of frequency ω. Hence, the far-field noise PSD reads:

S(r, θ, ω) = ρ ∞ U 2 ∞ k x c sin θ 2rc ∞ 2 L 2 |L| 2 l (ω) S v (ω) . ( )
is expression reveals a U 4 ∞ power-law of the far-field noise levels, and a quadratic proportionality to the magnitude of the Fourier component v of the upwash velocity fluctuation. e relative error of S in logarithmic scale can be thus evaluated as:

δS dB = 10 log 1 + 4 δU ∞ U ∞ + 2 δv v . ( )
From this expression, an error of 10% of the free-stream velocity results in an error of 1.46 dB of the far-field noise levels. e same error is provided by an error of 20% of the upwash velocity component. It could be therefore argued that, in the present fan noise simulations, a certain fortunate compensation may occur between the underestimation of the average axial velocity and the overestimation of the azimuthal SDV velocity.

A more quantitative comparison between measurements and simulation results can be performed by considering the velocity spectra at the same interstage locations. Fig. shows, for the baseline configuration, spectra of the three velocity components plo ed as a function of the harmonic count at about 80% and 60% of the casing-relative radial location. e comparison is quite satisfactory, both in terms of tonal and broadband levels. Be er comparisons, in particular for the tonal components, are expected by covering a longer transient time.

Figs. and show comparisons between H-W measurements and simulation results at stations # and # , respectively, for the low-noise configuration. As for the baseline configuration, the agreement is quite satisfactory, at both stations, both in terms of broadband and tonal levels.

e analysis of the unsteady velocity field in the interstage volume have demonstrated that the accuracy of the numerical results is quite well preserved along the whole path of the rotor wake. is is a necessary but not sufficient condition to predict accurate absolute and relative rotor/stator interaction noise levels, and therefore capture the effects of OGV modifications on the radiated noise. As relevant to the present study, Fig . shows the comparisons between measurements and predictions in terms of deltas between the baseline configuration and the other two configurations. Two quantities are considered: the overall sound pressure level (OASPL) at microphone located on a linear sideline array, 2.25044 m away from the engine axis, with a constant angular spacing of 5 • from 30 • to 140 • , and the sound power level (PWL) computed by integration of the far-field acoustic intensity over a spherical surface portion corresponding to a circular array of 10 m radius, with a constant angular spacing of 5 both quantities is very good, being within the experimental uncertainty of ±1 dB.

From the analysis of the reference SDT results, we could argue that the employed computational methodology can provide reliable indications about possible noise reduction trends associated with the usage of wavy OGVs.

. INTEGRAL SCALES OF TURBULENCE IM-PINGING ON THE OGV

Similarly to a wing section in a turbulent flow, the acoustic effectiveness of an OGV leading edge serration is expected to depend on ratios between the integral scales of the turbulent fluctuations in the impinging rotor wake and the amplitude and wavelength of the undulation. As shown by Gea-Aguilera et al[ ] for cases of impinging anisotropy turbulence, the maximum noise reduction can be achieved when the integral scale of the chordwise velocity component along the chordwise direction is smaller than the root-to-peak serration distance (L x < 2h s ) and the integral scale of the spanwise velocity component in the spanwise direction is smaller than the undulation wavelength ( L z λ s /2).

e turbulence in the rotor wake is herein characterized in terms of its integral length scale (L i j ) along the streamwise (x), radial (r) and azimuthal (θ) directions.

ey are estimated by using the fluctuations of the velocity components a er removing the phase-locked average, and applying

L k i j (x) = ∞ 0 u ′ i x + se k u ′ j (x) u ′ i (x) u ′ j (x) ds, ( )
where e k is the unitary vector in the kth direction, • is the ensemble average, and s is the separation distance. In order to estimate the integral length scale of turbulent fluctuations in different frequency bands, the two-point coherence between different velocity components is computed and used to calculate the following integral scales:

L k i j (x, f ) = ∞ 0 γ i j f , x, x + se k ds, ( )
where

γ 2 i j f , x, x + se k = φ u i u j f , x, x + se k 2 φ u i u i ( f , x) φ u j u j f , x + se k , ( )
is the coherence square and φ u i u j is the cross-spectral density of the two velocity components u ′ i and u ′ j . Due to the nonhomogeneity of the turbulent flow in the wake of the rotor, usage of this expression can lead to wrong estimates of the integral scales. is risk can be reduced by comparing the length scales computed at different points, and by using the phase-locked average removed velocity to compute the crossspectrum in Eq. . is procedure should provide length scales that can be interpreted in relation with the canonical problem of an airfoil in a homogeneous turbulent field.

e correlations are computed between signals extracted along five lines from a volume transient solution file sampled at the frequency of 458 kHz and covering one blade passage, from the rotor to the stator.

ese lines are extracted along the radial and azimuthal directions about two reference points at the same radial location, about 80% of the passage, and two axial locations, the first one at at x = 0.18 m (5 mm upstream of the baseline OGV), and the second one at x = 0.15 m, the origin of the reference system being located in the midpoint of the rotor. e two pairs of radial and azimuthal extraction directions are denoted as (e r 1 , e θ 1 ) and (e r 2 , e θ 2 ), respectively. e solution is also extracted along the axial direction e x connecting the two points. e non-dimensional cross-correlation of the different velocity components is plo ed in Fig. , and the corresponding integral scales are reported in Table . Interestingly, the integral scales along the radial and azimuthal directions are slightly smaller at the location closer to the OGV, and this is probably due to a local distortion induced by the OGV. is observation is not necessarily in contrast with the measurements made by

Table . Integral scales of turbulent velocity fluctuations in the SDT engine interstage

Integral length Value ( mm)

L x uu 8.3 L r 1 v r v r 3.8 L θ 1 v θ v θ 5.0 L r 2 v r v r 5.7 L θ 2 v θ v θ

5.7

Podboy et al[ ] who evaluated the integral scales at the two axial locations, say the H-W stations # and # introduced in section , and reported integral scales that increase farther downstream of the rotor. Indeed these measurements were carried out for the swept stator for which both H-W stations are sufficiently far from the OGV leading edge to exclude any influence on the velocity cross correlation.

As expected from previous literature works[ ], the integral scale along the axial direction is larger than along the other directions. On the base of these results, only the baseline OGV designs # and # are expected to satisfy both the axial and radial criteria for noise reduction, although only marginally for designs # . It is indeed worth mentioning that, as shown by Maunus et al[ ], all the integral scales increase monotonically along the radial direction. erefore, since the correlations have been computed at about 80% of the interstage radial passage, it is reasonable to expect that, for the radial OGV design # , the axial integral satisfies the condition L x 2h s over a certain passage percentage. Whereas, the condition should be satisfied along the whole radial passage Finally, Fig.

shows the frequency-dependent integral scales. Consistently with the previous estimations, the axial integral scale is higher than in the other directions and below the threshold values of about 7 mm in both the axial and radial directions for the baseline OGV designs # . An interesting property of these results is that, a er a rapid drop below about 2-3 kHz, the integral scales decay very slowly with frequency. It is therefore expected that, in agreement with the experimental observation made by Chaitanya et al[ ], if the serration amplitude and wavelength are large enough compared to the overall integral scales, the noise reduction should have a broadband character, with a slowly increasing trend above a certain frequency threshold.

. EFFECTS OF SERRATIONS ree serration designs are initially considered for the Low-Noise OGV (Table ). According to the estimated integral scales of the interstage turbulence at a location close to the leading edge of the baseline OGV, non of them has the potential to reduce the noise. Successively, three serration designs are considered for the baseline OGV, and two of them (designs # and # ) are expected to satisfy the criteria for the integral scales and thus result in some significant noise reduction.

e results are reported in this section in their chronological order.

Fig.

shows deltas of the far-field OASPL and noise PWL for the three serrated low-noise OGVs. Far-field noise reductions up to 0.5 dB can be observed, with slightly be er performances for designs # and # . is confirms that the undulation amplitude for design # is too small compared to the integral scales of the impinging fluctuations, both in the axial and radial directions. However, all the achieved noise reductions are too small and likely in the order of the statistical convergence error of the simulations. e PWL deltas exhibit a large spectral variance around 1 dB and it is hard to identify a clear noise reduction trend, as well as a clear difference between the three designs. Both the axial and radial integral scales are indeed too large compared to the root-to-peak distance and semi-wavelength of the serrations, thus resulting in no significant noise reduction. e only visible effect in the PWL spectrum is a 1 dB reduction of the BPF2 peak.

Fig.

shows deltas of the far-field OASPL and noise PWL for the serrated baseline OGV. Far-field noise reductions between 0.2 dB and 0.75 dB have been predicted for design # , between 0.5 dB and 1.25 dB for design # , and between 0.75 and 1.5 for design # . e PWL spectra exhibit a clear noise reduction trend for the three designs starting from about 2-3 kHz. In agreement with the integral scale geometric criteria,

CONCLUSIONS

Different designs of serrated OGV leading edges have been simulated using a validated CFD/CAA solver based on the LB-VLES method. e NASA SDT engine geometry has been considered as a reference. Serrations have been applied both to a radial and a swept stator. In the explored design space, which is indeed limited to six different serration designs, no significant noise reduction has been observed, with the exception of two designs for the radial OGV, for which the serration root-to-peak distance and spanwise wavelength were sufficiently large compared to the integral scales of the impinging turbulent fluctuations. More precisely, average noise reductions along the sideline microphone array of about 1 dB and 0.2 dB for the radial and swept OGVs, respectively, have been obtained. e second value is below the numerical and experimental accuracy, thus revealing the usefulness of the leading-edge serration to reduce the noise of the lownoise OGV. Although the explored design space is rather small, the present results seem to confirm previous literature observations, both in terms of scale thresholds, say root-topeak distance larger than the axial integral scale and the half-wavelength smaller equal to or slightly larger than the radial integral scale, and the 0.2 Strouhal number threshold. Additional designs will be explored in the future in order to achieve more definitive conclusions about the interest of a wavy leading edge for rotor/stator interaction noise reduction. e analysis will be also extended to different engine configurations, like a modern ultra-high-bypass-ratio engine, and higher power operating conditions. whale (megaptera novaeangliae) flippers.

  Several numerical studies have been conducted by using Euler equations with ingested turbulence modelled as a convected harmonic upwash gust[ ], a Kraichnan-Fourier superposition of upwash gust components matching the spectral energy content of homogeneous isotropic turbulence[ ], and a stochastic three-dimensional divergence-free velocity field also matching the spectral energy content of homogeneous isotropic turbulence[ ]. ese studies showed that the noise reduction increases when the ratio of the leading edge undulation amplitude to the gust wavelength is increased. Moreover, Clair et al[ ], in the framework of the European project FLOCON, separated the effects of subcritical and supercritical gust components along the serration edge, as analytically investigated by Roger et al[ ], thus recognising the role of the serration angle and wavelength on the radiated noise. More recently, Gea-Aguilera et al[ ] solved linearized Euler equations past a NACA aerofoil with stochastic anisotropic turbulence of prescribed levels and integral scales in the streamwise (L x ) and spanwise (L z ) directions. ey observed that the maximum decorrelation between noise sources along the leading edge and consequent maximum noise reduction are achieved when L x < 2h s (root-to-peak decorrelation) and L z λ s /2 (root-to-root decorrelation).

Figure .

 . Figure . SDT engine configurations: baseline (top-le ), low-noise (top-right), low-count (bo om-le ), nacelle (bo om-right).

Fig. shows

  Fig. shows three variants of the OGV configurations: a 54-vane baseline radial OGV, a 26-vane low-noise swept OGV and a 26-vane low-count radial OGV.All the other parts, with the exception of very small variations of the inner and outer casing profiles downstream the OGV, are the same for the three configurations. e nacelle is perfectly axisymmetric without struts. In order to reproduce the stinger used in the experiment, a cylindrical prolongation of the centerbody has been added to the model released by NASA. e rotor radius is 0.2786 m, the bypass exhaust radius is 0.2710 m and the lip intake radius is 0.2962 m. e rotor is constituted of 22 blades, and the casing/blade-tip gap is about 0.5 mm.

Figure .

 . Figure . Wavy OGVs: design # , to # and # (top), and design # , # and # (bo om)

  e far-field noise is computed through an integral extrapolation based on a solid FW-H acoustic analogy formulation. A forward-time solution[ ] of the FW-H equation based on Farassat's formulation A[ ] is used. e free-stream convective effects are taken into account directly in the integral formulation[ ].

Figure .

 . Figure . Computational setup (low-noise OGV) and detail view of the zig-zag trip and FW-H surface.

Figure .

 . Figure . Phase-locked average (le block) and SDV velocity (right block) at station # for the baseline OGV [ m/ s]. Axial, radial and azimuthal velocity in the top, middle and bo om rows, respectively. Measurements on the le column, simulations on the right.

Figure .

 . Figure . Phase-locked average (le block) and SDV velocity (right block) at station # for the low-noise OGV [ m/ s]. Axial, radial and azimuthal velocity in the top, middle and bo om rows, respectively. Measurements on the le column, simulations on the right.

Figure .

 . Figure . Phase-locked average (le block) and SDV velocity (right block) at station # for the low-noise OGV [ m/ s]. Axial, radial and azimuthal velocity in the top, middle and bo om rows, respectively. Measurements on the le column, simulations on the right.

Figure .

 . Figure . Velocity spectra at station # for the baseline OGV. Axial, radial and azimuthal velocity on the top, center and bo om, respectively (in m/s). Comparison between H-W measurements (symbols) and PowerFLOW results at r = 0.223292 m (r/R = 80.07%) on the le and r = 0.167640 m (r/R = 60.11%) on the right.

Figure .

 . Figure . Velocity spectra at station # for the low-noise OGV. Axial, radial and azimuthal velocity on the top, center and bo om, respectively (in m/s). Comparison between H-W measurements (symbols) and PowerFLOW results at r = 0.223292 m (r/R = 80.07%) on the le and r = 0.167640 m (r/R = 60.11%) on the right.

Figure .

 . Figure . Velocity spectra at station # for the low-noise OGV. Axial, radial and azimuthal velocity on the top, center and bo om, respectively (in m/s). Comparison between H-W measurements (symbols) and PowerFLOW results at r = 0.223292 m (r/R = 80.07%) on the le and r = 0.167640 m (r/R = 60.11%) on the right.

Figure .

 . Figure .Non-dimensional cross-correlation of the velocity fluctuations as in Eq. .

Figure .

 . Figure .Frequency-dependent integral scales as in Eq. .

Figure .

 . Figure . Effect of different leading-edge serrations for the low-noise configuration (designs # -# ). Relative far-field OASPL along the sideline array (top) and PWL (bo om).

Fig. shows absolute

  Fig. shows absolute far-field OASPL and noise PWL for the reference Low-Noise OGV and the serrated edge design # .e only visible effect in the PWL spectrum is a 1 dB reduction of the BPF2 peak.Fig.shows deltas of the far-field OASPL and noise PWL for the serrated baseline OGV. Far-field noise reductions between 0.2 dB and 0.75 dB have been predicted for design # , between 0.5 dB and 1.25 dB for design # , and between 0.75 and 1.5 for design # . e PWL spectra exhibit a clear noise reduction trend for the three designs starting from about 2-3 kHz. In agreement with the integral scale geometric criteria,

Figure .

 . Figure . Effect of different leading edge serrations for the baseline configuration (designs # -# ). Relative far-field OASPL along the sideline array (top) and PWL (bo om).

Table .

 .

	Wavy OGV design parameters
	Design # OGV Conf. λ s ( mm) hs	h s ( mm)
	Low-Noise		0.15	1.88
	Low-Noise	.	0.075 0.94
	Low-Noise		0.3	3.76
	Baseline		0.3	1.80
	Baseline		0.6	3.60
	Baseline		0.8	4.80
	. NUMERICAL MODEL
	e La ice-Boltzmann (LB) CFD/CAA solver PowerFLOW
	. a is used to compute the transient flow inside and around
	the engine. A Ffowcs-Williams and Hawkings (FW-H) ap-
	proach is then used to extrapolate the near field solution
	sampled on a permeable surface to the far-field. Exa fan-
	noise best practice setup is used for all simulations.

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[

ACKNOWLEDGMENTS

e authors wish to acknowledge Dr. Edmane Envia from NASA Glenn Research Center for providing the geometry and experimental data of the NASA -in fan source diagnostic test.

design # is more efficient than designs # and # . Very interestingly, the far-field OASPL exhibits the same noise reduction directivity pa erns. Finally, Fig.

shows absolute far-field OASPL and noise PWL for the reference Baseline OGV and the serrated edge design # . In this case, the PWL shows some broadband noise reduction starting from a frequency of about 3 kHz, and a net reduction trend above 6 kHz. e la er value corresponds to a Strouhal number f h s /U of about 0.2, the average axial velocity U being about 110 m/s at a distance of 5 mm from the leading edge. Interestingly, Chaitanya et al[ ], determined a threshold Strouhal number of 0.2 also for the case of an aerofoil in isotropic turbulence. Further simulations are required in order to confirm the generality of this result.