L.I.M.S.I. - U.P.R. C.N.R.S. 3251, B.P. 133, ORSAY CEDEX, FRANCE fax number:

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1 Large Eddy Simulations of a spatially developing incompressible 3D mixing layer using the v-ω formulation. C. TENAUD, S. PELLERIN, A. DULIEU and L. TA PHUOC L.I.M.S.I. - U.P.R. C.N.R.S. 3251, B.P. 133, ORSAY CEDEX, FRANCE tenaud@limsi.fr, pellerin@limsi.fr fax number: Abstract The LES of a 3D mixing layer spatially developing downstream of a flat plate has been conducted for a high Reynolds number (Re θ = 2835). To overcome the problem of the pressure condition on the free boundaries, use of the (v ω) formulation of the Navier-Stokes equations has been preferred over the primitive (v p) formulation. To deal with the difficult problem of the divergence free constraint on both the velocity and the vorticity field, a new and efficient numerical algorithm has been devised which turns out to be very attractive. The velocity components are computed as a solution of a Cauchy-Riemann problem using a fractional step method. As one of the main advantages of the vorticity-based formulation is the treatment of the free flow boundary conditions, special care has been devoted to these boundary conditions. An optimum approximation of the outflow boundary condition has been carried out which satisfies the conservation of mass, making the long time integration easier and more accurate. The numerical results are compared to a reference experiment [15, 16] for a rather high Reynolds number. Using inlet perturbations and the mixed scale model, the LES results agree very well with the reference experiments. The validation of the numerical procedure is reviewed on the mean and fluctuating quantities. Good prediction of the spatial evolution is demonstrated for the distribution of the vorticity thickness as well as for the Reynolds stress profiles and spatial correlations. In order to estimate the quality of the spatio-temporal development, a spectral analysis is also reported on the space and time spectra, pointing out a highly 3D arrangement with length scales and frequencies in rather good agreement with the ones generally admitted. Keywords: Incompressible Navier-Stokes equations, LES, mixed scale model, vorticity-velocity formulation, fractional step method, Turbulent Mixing Layer, High Reynolds flow. 1

2 Introduction The spatial development of an incompressible viscous flow at high Reynolds number, downstream of an obstacle is a classical problem in fluid mechanics. This problem is however of particular interest for many industrial applications; for instance, it is found either behind aerospace vehicles in external flows or downstream of an injection throat inside a combustion chamber. Indeed, the largescale coherent structures generated within the unsteady wake contain most of the turbulent kinetic energy and, interacting with the solid walls, they are mainly responsible for vibrations, noise generation, etc... Though, in general, the geometry of the obstacle is rather simple, the flow structure may be complex and many of the phenomena of fluid mechanics are present in the wake. Therefore, it seems very important to correctly describe the characteristics of these flow structures and to predict with sufficient accuracy their time evolution. At the present time, it is commonly admitted that Large Eddy Simulation (LES) is a powerful tool for performing fine scale analysis of turbulent flow dynamics at Reynolds numbers higher than could be reached by Direct Numerical Simulations (DNS). During the last several decades, as computer power has largely increased, this approach has seen a considerable gain of interest even for industrial configurations [42, 45, 47]. Most previous applications of LES of turbulent flow development have generally been performed using the popular pressure-velocity formulation of the Navier-Stokes equations. A review of LES for incompressible flows can be found in [47]. Many difficulties are encountered in the simulations of incompressible flows using the pressure-velocity formulation, especially those related to the pressure condition on the open boundaries. Regarding external flow LES, as many vortical structures pass through the free boundaries, the quality of the results within the computational domain is highly related to the quality of the boundary condition treatment. With the pressure-velocity formulation, the prescription of the open boundary conditions is very delicate unless special features and special care are incorporated. Beside the pressure-velocity formulation, important work has been conducted in the past on the velocity-vorticity (v ω) formulation of the Navier-Stokes equations (see below). Theoretical equivalence conditions between the (v ω) and the (v p) formulations have been introduced by several authors (see Daube et al. [12], for instance). Since the elliptic problem related to the pressure has been removed in the vorticity transport equation, it is more convenient when prescribing the open boundary conditions for external flow simulations [26] than the primitive formulation. In fact, the Galilean transformation and the simple construction of the open boundary conditions constitute some of the known advantages of the (v ω) formulation as pointed out by Speziale [51]. Moreover regarding LES, the (v ω) formulation allows direct access to the vorticity vector, which, of course, plays a crucial role in turbulence and thus this presents a formidable 2

3 advantage compared to the (v p) formulation. Despite these advantages compared to the (v p) formulation and the important algorithm developments in vorticity-based formulations, LES using the (v ω) equations are not very popular and the corresponding literature remains relatively scarce. Nevertheless, to our knowledge, work on subgrid-scale modeling has been conducted for LES in the velocity-vorticity formulation (see for instance [27, 57]) by using an a priori testing procedure from (v p) DNS data. In the concluding remarks of [57], it was then pointed out that the vorticity-velocity formulation might be a good candidate for real LES. Besides the studies on SGS-modelings, a few LES based on the velocity-vorticity formulation have also been conducted. However, up to now, these LES have been performed using particle-based Lagrangian formulations (see for instance, [36, 37]). Therefore, the goal sought after in this study is to develop a competitive LES method on an Eulerian-based (v ω) formulation and to evaluate the capability of this approach to real LES, especially to predict the spatial development of free turbulent flows at high Reynolds number. As the prescription of the boundary condition is one of the main advantages of the vorticity-based formulation, special attention will be devoted to the treatment of the conditions on the outlet boundaries. In the vorticity-velocity formulation, while the pressure has been removed from the vorticity transport equation, another elliptic problem arises from the definition of the curl. The larger number of equations added to the difficulty of ensuring the solenoidal property of the velocity and vorticity vectors are likely some of the reasons why the vorticity-based formulation is still not very popular for numerical simulations. However, numerous studies on algorithm developments have been undertaken to make this formulation attractive. An interesting review on the developments of the (v ω) formulation has been proposed by Gatski [22]. One of the pioneering works on the (v ω) formulation is due to Fasel [19], who solves the time-dependent problem using an iterative procedure. The velocity components are then calculated by means of Poisson equations [17, 19]. By using this numerical procedure, instability and transition phenomena for 3D boundary layers have been studied [20]. Nevertheless, the main difficulty encountered lies in the divergence free constraint enforcement. To overcome this main difficulty, an alternative procedure has been proposed [18, 49]. Knowing the vorticity vector by solving the vorticity transport equation, only two velocity components are calculated by a Poisson equation using an iterative algorithm. The remaining velocity component is then obtained by satisfying the divergence free constraint. Following these numerical approaches, several simulations of external flows around bodies have been performed [18, 21, 29, 49]. Nevertheless, these numerical approaches suffer from the large computational effort needed to solve a 2D or 3D vectorial Poisson equation even when powerful algorithms are applied [13]. To make the resolution of the (v ω) equations attractive, the computed velocity field is decomposed into an irrotational (solenoidal) part 3

4 and a rotational part [6, 23, 30, 59]. Furthest progress in the resolution of the unsteady vorticity-based Navier-Stokes equations has recently been reported by Bertagnolio and Daube [6] and Lardat et al. [30]. The method is based on an original projection algorithm (or fractional step method) by solving the three coupled parabolic transport equations for the vorticity components and only one scalar Poisson equation to compute the velocity components. This approach of a Helmholtz decomposition combined with projection onto a space of divergencefree vectors can be performed cheaply without time-consuming inversion of a Poisson equation. Therefore, in the present paper, this latter numerical method has been chosen to solve the LES by using the Eulerian-based (v ω) formulation (see 2). To validate the present LES approach for high Reynolds number flow configurations, the simulations are then performed on a mixing layer configuration studied experimentally in detail in earlier work [15, 16, 55]. An originality of our simulations is the computation of the spatial development of the mixing layer, leading to an easy comparison with the corresponding experiment. As opposed to the experimental approach, the calculations yield the whole velocity field and the pressure field in a deterministic way. Thus, the space correlations can be derived and the time evolution of the flow can be analyzed in detail. This space and time simulation can generate a database, particularly for the pressure, which is experimentally difficult to access. The basic equations are presented in section 1, associated with the LES approach and the subgrid scale model chosen, the mixed scale model, originally developed jointly at LIMSI and ONERA [47, 52]. This model smoothly damps the eddy viscosity in the regions where all the flow structures are well captured. Section 2 is devoted to the numerical method, including the fractional step algorithm which solves the div-curl problem. The numerical procedure associated with simulations of the spatially developing mixing layer is explained in section 3. The convergence of an initial state and the post-processing techniques are then detailed. Results of the main simulations are then described and discussed. Influences of the main contributing factors are studied: the subgrid scale model used considering the intrinsic numerical dissipation of the convective scheme, the upstream perturbation and also the grid and domain sizes. Instantaneous snapshots of the flow are also given. A detailed comparison between numerical and experimental values are provided. The main statistical values, expansion factor (vorticity thickness (δ ω )) and Reynolds stress are also shown and discussed. Experiment-LES confrontations allow an estimate of the quality of the method and its ability to both reproduce a given experiment and be extended to other external flow problems. 4

5 1 Basic equations and LES approach 1.1 Basic equations The present calculations concern free flow developments occurring at high Reynolds numbers close to those found in experiments. For such Reynolds numbers, the flow is fully turbulent and Large Eddy Simulations (LES) must be adopted. LES consists of a separation between large scale and small (subgrid) scale structures [47]. This separation is performed by using a filtering operation defined as follows: Φ = Φ + Φ (1) Φ = G ( x x, ) Φ(x )d 3 x (2) Ω G is the normalized filter function, depending on the vector (x x ) and the cutoff length scale. Φ denotes the resolved part of the quantity, corresponding to the large scale (greater than ) and Φ the subgrid (unresolved) part of the quantity, associated with the small scales. We suppose that the filter function commutes with the spatial and temporal derivatives. The basic equations in the present study are based on the (v ω) formulation of the Navier-Stokes equations. The vorticity equation is deduced from the momentum equation by applying the curl operator. Let (Ω) be an open computational domain with its boundary ( Ω). We note (n) the unit vector normal to this boundary. Applying the filter function to the vorticity-velocity formulation, the dimensionless filtered Navier-Stokes equations for an incompressible fluid can be written as follows: ω t (v ω) = Re 1 ω + τ v = ω v = 0 v n = b n v n = b n in Ω in Ω in Ω on Ω on Ω In these equations, (v) stands for the resolved (filtered) part of the velocity field, (ω) is its curl and Re is the Reynolds number of the flow. The prescribed velocity vector (b) on the boundary ( Ω) has to satisfy the compatibility condition b n ds = 0 (mass conservation). Ω For a p-multiply connected computational domain, the equivalence between the (v ω) and the pressure-velocity formulations is ensured by prescribing p supplementary conditions on p arbitrary and independent paths (γ q, q = 1,...,p) [12]: [ v t v ω + 1 ] Re ω dl = 0 (4) γ q 5 (3)

6 In the (v ω) formulation, the subgrid scale contributions are represented by the vector: τ = v ω v ω. We underline the fact that here the subgrid scale vector must be directly modeled and then related to the resolved quantities. 1.2 Subgrid scale models To account for the kinetic energy dissipation occurring at small-scale structures, the model of the subgrid scale vector (τ) assumes the vorticity transfer theory of Taylor [54]. The subgrid contribution (τ) is then related to the local filtered vorticity (ω) by means of an eddy viscosity (ν sg ): τ = ν sg ω. (5) The eddy viscosity model must mimic the dissipative exchanges between the small and the large scale structures. This eddy viscosity should result from the product of a length scale and a time scale representative of the small scale structures. Numerous LES studies used Smagorinsky [50] or vorticity [38] subgrid eddy viscosity models which both consider an energetic approach. We note that, using the (v ω) formulation, the vorticity model implementation is of course very useful. Though this kind of model is very popular thanks to its low computational cost and easy implementation, it is well known that, for many flows, it provides too high a kinetic energy level and consequently, too high a dissipation rate. Moreover, the vorticity model gives subgrid contributions even in regions where all the length scales are well resolved because the eddy viscosity (ν sg ) is related to the magnitude of the local filtered vorticity. Therefore, the vorticity model does not provide the right behavior, for instance, in the laminar regions as well as in the turbulent regions where the dissipative length scale is greater than the estimation of the filter cut-off length scale. As a consequence, transition delay or re-laminarization are predicted by the vorticity model [31]. In a previous study [44] concerning development in a turbulent mixing layer, we evaluated the poor behavior of the vorticity model. Besides these, more sophisticated models have been developed, based either on subgrid scales (TKE, [2]) or on both resolved and subgrid scales (mixed models, [24, 52]). As regards the velocity-vorticity formulation, vorticity-based models for subgrid scale modeling have also been developed by including a dynamic procedure [11]. Besides these sophisticated dynamic models [24, 11] which give accurate results with however a good deal of computational effort, the mixed scale model was then introduced to improve the behavior of the vorticity model in the specific regions mentioned above. This model was first introduced by Ta Phuoc [52] and 6

7 validated by Sagaut [47] for several flow problems. The mixed scale model is derived from a new class of models which supposed that the subgrid viscosity is a function of the transfer rate (ε) of the kinetic energy, the kinetic energy (E(k c )) at the cut-off and the cut-off wave number (k c ): ν sg = f (ε,e(k c ),k c ) Following a dimensional analysis and assuming a local spectral equilibrium, we may obtain a one parameter family model, written in the physical space as: ν sg = C ω α u (1 α) (1+α) Where C is the model constant which needs evaluation. u stands for a velocity scale representative of the subgrid scale velocity. Following Bardina et al. [2] about the TKE model, this velocity scale has been related to the subgrid kinetic energy by using a scale similarity assumption. Thanks to an analytical test filter (.) with a cut-off length scale larger than, the subgrid scale velocity is estimated by using the quantity at the highest resolved wave numbers: u = (v ṽ) In practice, the explicit test filter is expressed using a trapezoidal rule [35] and = 2.. The mixed scale model can also be re-written as the weighted product of the vorticity [38] and the TKE models [2]: ( ) ν sg = CS 2 2 α ( ω CB u ) (1 α) (6) C S and C B are respectively the constant of the vorticity and the TKE [2] models. In practice, for shear flows, common values of C s are: C s [0.1, 0.12] [14, 39, 43]. We note that, for certain values of α, we recover: α = 1: the vorticity model [38] α = 0: the TKE model of Bardina et al. [2] Regarding equation (6), the TKE model may be considered as a damping function of the vorticity model. The eddy viscosity is then damped smoothly and vanishes in the regions where all the scales are well resolved. In the literature [31, 47, 52], the weight is equally distributed among the two models. Hence, in the following, the presented results have been obtained by setting α = 1. In practice, a resulting 2 coefficient is introduced : C M = CS 2αC1 α B. Its current corresponding value is equal to

8 2 Numerical method The numerical resolution of the transport equation (3) of the vorticity vector is very classical and does not pose a numerical problem. The numerical algorithm is described in (2.4). Knowing the components of the vorticity, the velocity components must be solved by means of the div-curl problem occurring in the set of equations (3). The numerical resolution of this problem presents some difficulties since the first order discrete operators involved are ill-conditioned. One way to overcome these numerical problems is to apply the curl operator on the definition of the vorticity and to increase to the second order the operator applied on the velocity. This way, one needs to solve the following equation: 2 v = ω + ( v) (7) The vorticity transport equation associated with the above equation (7) results in six partial differential equations known as the classical vorticity-velocity formulation. Following these equations, the divergence free constraints on both the velocity and vorticity are satisfied at any time: v = 0. and ω = 0. The equivalence of the (v p) and (v ω) formulations has been mathematically proved in Daube et al. [12]. The method presented here is based on an original projection algorithm (or fractional step method) [6, 30]. Three coupled parabolic transport equations are solved for the vorticity components and only one scalar Poisson equation to compute the velocity components. 2.1 Solving the kinematic vorticity-velocity problem One way to reduce the number of equations to be solved is to consider the velocity components as the solution of a Cauchy-Riemann problem (kinematic problem). Once the vorticity field is known, following the resolution of the ω transport equation, the velocity must be determined by the set of equations: v = ω v = 0 v n = b n in Ω in Ω on Ω The method consists of the decomposition into two steps of the resolution of the Cauchy-Riemann problem. In the first step, one looks for an intermediate velocity field (v ) which satisfies the curl equation given by: (8) v = ω in Ω v n = b n on Ω (9) 8

9 If such a vector (v ) is known, the second step consists of the projection of this intermediate velocity vector (v ) onto the space of the divergence free vector fields. This is performed by using the gradient of a scalar function (Φ) as follows: v = v + Φ v n = v n in Ω on Ω (10) Applying the divergence free constraint on the velocity (equation 10), gives the set of equations to be solved: 2 Φ = v Φ n = 0 in Ω on Ω (11) If the intermediate velocity vector (v ) is easily found, the computation of the velocity components consists of the resolution of the scalar Poisson equation (11). 2.2 Boundary conditions Boundary conditions are necessary for the vorticity and velocity vectors in the (v ω) formulation. Because natural boundary conditions arise only for the velocity components (see equation (3)), the main problem occurring in the vorticityvelocity formulation is to find adequate boundary conditions for the vorticity vector. The boundary conditions to be prescribed on (ω) are dependent on the form used in the vorticity transport equation [12]. As we use the curl form of the vorticity transport equation (3), the boundary conditions are derived from the vorticity definition: ω Ω = v Ω The curl operator is then applied on the boundary condition showing that only tangential components of the vorticity are required: ω n = ( b) n on Ω (12) Since a staggered grid is employed, we need to prescribe the component of the velocity normal to the boundary surface and the components of the vorticity tangential to that surface, following: v n+1 n = b n (13) ω n+1 n = ( v n+1) n (14) The boundary conditions (13) couple the velocity and the vorticity variables which provide some numerical difficulties since the resolution of the vorticity and velocity are decoupled. This problem could be resolved by using the influence 9

10 matrix technique [13, 21]. However, this technique would become very expensive in LES. To overcome this difficulty, a temporal linear extrapolation is then employed and the boundary conditions on the vorticity read: ω n+1 n = ( 2.v n v n 1) n 2.3 Space and time discretizations A M.A.C. staggered grid (see figure 1) is used. This choice was motivated by the fact that the mathematical relations ( φ) = 0, ( v) = 0 and φ = 0 are algebraically satisfied in such a discretization. In order to keep the numerical diffusion to a low level which has been proven to be crucial in LES [25, 28, 41], the space discretization uses a finite difference method incorporating a third order QUICK scheme for the convective terms and a 2 nd order centered scheme for the diffusive terms. In addition, a 2 nd order centered scheme for the convective terms has also been employed to evaluate the hypothetical role played by the QUICK scheme on the results (see 4.1). The time integration of the vorticity transport equation is based on a semiimplicit 2 nd order backward Euler scheme. The diffusive term is solved implicitly. The time discretization of the vorticity transport equation (see equations (3) and (5)) then reads: [ ( ) ] 3ω n+1 4ω n + ω n 1 1 = (v ω) n+1 2 t Re + ν sg ω n+1 (15) where t denotes the time step and ω n the vorticity evaluated at time n. t. The convective term is obtained by using a 2 nd order Adams-Bashforth extrapolation: (v ω) n+1 = 2 (v ω) n (v ω) n 1 (16) 2.4 Numerical algorithm The vectorial Helmholtz equation (15) couples the components of the vorticity. This equation, associated with the boundary conditions can be written as: [( ) ] 1 σω n+1 + Re + ν sg ω n+1 = S in Ω (17) ω n+1 n = b n on Ω (18) with σ = 3 2. t and S = 4vn v n 1 2 t + 2 (v ω) n (v ω) n 1 (19) 10

11 The form of equation (17) implies that the new vorticity variable (ω n+1 ) is a solenoidal vector field since it is deduced from a curl operator. This vectorial Helmholtz problem is solved by a block Jacobi iterative algorithm with a SLOR resolution. The Cauchy-Riemann problem of the velocity field is solved using a projection (or fractional step) method: In the first step, knowing the new vorticity field (ω n+1 ), an intermediate velocity field (v ) is obtained which satisfies the equations: ( ) 1 σv = S Re + ν sg ω n+1 in Ω (20) v n = b n+1 n on Ω Where S is given by (19). The intermediate field (v ), solved by equation (20), satisfies the relation v = ω n+1. Moreover, the additional condition (4), for a multiply connected domain, is automatically satisfied in discrete form. This intermediate velocity (v ) is then projected onto the space of divergence free vector fields, using a gradient scalar function [6, 30]: σ ( v n+1 v ) = Φ in Ω and on Ω (21) The following relations are then automatically satisfied: v n+1 = v = ω n+1 Φ dl = 0. (22) 2 Φ n+1 = σ v in Ω Φ n = 0 on Ω (23) The Poisson equation (23) is solved, following the work of Schumann et al. [48], by using a FFT algorithm in the two directions (x, z) and a direct inversion for the remaining one (y). Summing the two steps in the velocity resolution we recover the discrete form of equation (3). The resulting scalar function (Φ n+1 ) should then be considered as the dynamic pressure through (see [6] and [31] for details): p n (v v)n+1 = Φ n+1 11

12 3 Numerical procedure for the mixing layer simulation 3.1 Mixing layers The spatial development of an unsteady incompressible viscous flow downstream of an obstacle is a classical problem in fluid mechanics. Mixing layer developments have been conducted through experiments or theoretical studies; numerical simulations are often limited to the case of a temporal development. Corcos and Sherman [9] proposed a theoretical approach of the flow using 2D deterministic models. They showed the evolution of the first instability associated with the rollup of the mixing interface. In addition, Corcos and Lin [10] tried to find the origin of the 3D motion responsible for the main roll deformation. One first and often referenced experiment is that of Bernal and Roshko [4]. This mixing layer results from the confluence of two upstream flows with different velocities and densities. For rather high Reynolds numbers, the authors obtained very clear visualizations of the spatial development of main vortices with spanwise axis, Kelvin-Helmholtz rolls. These vortices grow along the flow propagation direction. A secondary kind of structure was also observed, organized as streamwise vortices connecting two successive Kelvin-Helmholtz rolls. A more recent experiment [3] concerns a twostream mixing layer in a wind tunnel, at high Reynolds numbers and focuses on the secondary instability generating the typically 3D streamwise structures. This experiment is rather close to our reference experiment [15, 16], for an equivalent velocity ratio ( 0.6) with lower velocity magnitudes (15 and 9 m.s 1 in [3]; 42.8 and 25.2 m.s 1 in [15, 16]). More recently, Lebœuf and Mehta [33] analyzed these results in order to discuss the use of Taylor s hypothesis. They showed that this hypothesis should not be used to transform time onto a spatial domain for this 3D flow. The most common numerical approach of the plane mixing layer concerns studies about its temporal growth in a periodic box. Different studies have been interested in large-scale vortices and their pairing, associated with the generation of the streamwise vortices, called braids [40] or also in the existence of a self-similarity behavior on velocity and Reynolds stress tensor profiles [46]. Another investigation concerns a spectral DNS of a turbulent mixing layer in a 3D periodic domain [1], with a successful comparison to experimental data. More recently [56], one of the first LES studies was performed, still for a temporally growing turbulent mixing layer, in the case of a lower compressible flow. 3.2 Computational domain and grid generation The free flow studied is generated at the confluence of two boundary layers developing from each side of a flat plate (Figure 2). The computational domain starts 12

13 at the trailing edge of this splitting plate. Thus, in the following x = 0 refers to the plate trailing-edge. The dimensions of the computational domain correspond exactly to those of a portion of the experimental domain [15, 16]. This domain spreads over 25 δ ω0 far downstream. Let us mention that δ ω0 = m was chosen as the reference length-scale. It corresponds to the experimental value of the vorticity thickness (see the definition (32)) recorded at 20 δ ω0, close to the end of the computational domain. The grid uses 501 points (n x ) equally spaced in the streamwise direction (x). In the inhomogeneous (vertical) direction (y), the domain lies from 3 δ ω0 to +3 δ ω0. The mesh is tightened around the centerline of the mixing layer, following a cosine distribution using 71 grid-points (n y ). The flow is supposed to be periodic in the spanwise direction (z); the domain lies over 5 δ ω0 using 55 points equally spaced (n z ). Following that point distribution, the maximum aspect ratios, recorded at the centerline of the mixing layer, are close to x / y = 4.24 in the (x, y) plane and z / y = 7.85 in the (y, z) plane. The number and the distribution of the grid points have been calculated so as to ensure that the grid filter width ( = ( x y z) 1/3 ), in the vicinity of the centerline of the mixing layer, is close to the Taylor micro-scale estimated by using the experimental results [15, 16]. 3.3 Boundary conditions At the inflow surface, meaning at the trailing edge of the splitting plate (x = 0), we must prescribe boundary conditions which mimic the boundary layer (time and space) evolution from each side of the splitting plate. To impose perturbations at the inflow boundary that are representative of real turbulence, auxiliary inflow simulations of a fully developed turbulent flat plate boundary layer should be performed at the same Reynolds number with the same grid resolution to provide instantaneous inflow data. Studies to generate adequate inflow conditions, for instance [8, 34], have been undertaken in the past. While these precursor computations gave very good results and reduced significantly the setting extent of the flow, they notably increase the CPU time, data management and storage capacity. The present work is not devoted to the study of the influence of the inlet boundary condition but just dedicated to the evaluation of the capability of the numerical method to reproduce the dynamical behavior of the mixing layer development. Therefore, the inflow conditions are based on a very rough hypothesis: we have preferred to prescribe the mean (in time) profiles of the velocity and to superimpose a broad-band spectral noise on these mean profiles. Non stationary behavior of the boundary conditions at the inlet is then performed by means of white noise perturbations through the use of a random number generator in order not to favor a special wave length or frequency. These boundary conditions are of course not representative of the spectral arrangement that could be recorded in a natural turbulent boundary layer. Several 13

14 perturbation magnitudes have been checked with constant amplitudes through the boundary layer thickness with rms values going from 0% to 7.5% of the external velocity (U high ). The influence of these perturbation magnitudes on the setting of the mixing layer has been studied (see (4.2) for the result discussion). For each side of the flat plate, the surface normal component of the mean velocity < v x > (<. > refers to the time average) is prescribed by using a turbulent Whitfield profile [58] which mimics very well the longitudinal velocity profile of a fully developed turbulent boundary layer. In order to fit as well as possible the experimental profiles, three profile parameters are used: the momentum thickness θ, the form parameter H and the Reynolds number Re θ based on the momentum thickness and the external velocity. For the two upstream turbulent flows, these values are the following: high velocity side: θ high = 10 3 m, H high = 1.35 and Re θhigh = low velocity side: θ low = m, H low = 1.37 and Re θlow = The very good agreement of the < v x > profile between the numerical boundary condition and the experiments at the trailing edge location, can be judged in Figure (3). The random perturbations are imposed on the spanwise component of the velocity (v z ) which is situated in a plane one half grid cell just upstream of the inlet surface. The perturbations are calculated to satisfy the mass conservation in the computational domain. The vertical component of the velocity (v y ), also defined in the plane one half grid cell just upstream of the inlet surface, is deduced from the equation of mass conservation, assuming that v x = 0 in the cell just x before the computational domain. Let us mention that the components (v y ) and (v z ), evaluated just upstream of the computational domain, do not constitute boundary conditions on the velocity. These two components are only generated to calculate the boundary conditions on the tangential components of the vorticity (ω y ) and (ω z ), by means of the spatial derivatives of (v x ), (v y ) and (v z ) profiles. Consequently, the tangential vorticity components are disturbed by means of the velocity components. At the upper and lower surfaces of the domain, (y = ±3 δ ω0 ), we prescribed a slip condition, assuming that the surfaces are located sufficiently away from the region where velocity gradient occurs: v y = 0 v x y = 0 ; v z y = 0 (24) For the spanwise boundary surfaces, (z = 0 and z = 5 δ ω0 ), periodic conditions are imposed on the independent variables. 14

15 At the outlet boundary, meaning at x = 25 δ ω0 downstream of the flat plate trailing-edge, the components of the vorticity (ω y ) and (ω z ), tangential to the outlet surface are calculated by extrapolation along the characteristic directions, assuming that the viscous diffusion and the stretching term can be neglected in the vorticity equation: Dω y = D ( vx Dt Dt z v ) z = 0 (25) x Dω z = D ( v x Dt Dt y + v ) y = 0 (26) x This transparent boundary condition approximation is similar to that adopted by Ta Phuoc and Bouard [53]. Regarding the evaluation of the surface normal component of the velocity (v x ), we assume transport without diffusion of the normal derivatives ( / x) of the tangential components of the velocity (v y ) and (v z ). Using equations (25) and (26), Bertagnolio and Lardat [5] show that the condition is equivalent to prescribing: D v x Dt z D v x Dt y = 0 (27) = 0 (28) The value of (v x ) might be calculated by integration of equation (27) or (28). However, the resulting value of (v x ) will depend on the integration path. Moreover, conservation of mass must be ensured by the compatibility relation: v n = 0 (29) Ω To overcome these problems, we have preferred to solve a 2D Laplacian operator applied to the surface normal component of the velocity: 2 v x y + 2 v x 2 z = ( ) vx + ( ) vx (30) 2 y y z z ( ) ( ) vx vx The estimations of and are then obtained by using characteristic z y extrapolation. Using this integration, the normal component of the velocity (v x ) is determined up to a constant. This constant of integration is prescribed by satisfying the mass conservation (29). As a slip boundary condition is applied on the upper and lower surfaces of the domain, the conservation of mass must be ensured by the relation: v n = v n (31) Γ 0 Γ 1 where Γ 0 and Γ 1 refers to the inlet and outlet surfaces, respectively. 15

16 3.4 Initial state For every set of physical and numerical parameters given, the numerical solution is established after a transient time period. Using the inlet perturbations (see 3.3) applied at each time step, a statistically converged solution is first computed by using the mixed scale model (6) and obtained after a dimensionless time t = 234. Notice that the reference time is based on the reference length (δ ω0 ) ( 3.2) and the velocity difference (U high U low ). To save computational time, this 3D converged solution is then taken as the initial condition for most of the 3D computations on the reference grid. However, for a parametric study where some given parameters are modified (for instance the subgrid scale model or the upstream conditions), starting with this initial state, the calculations with the new parameters were run until the statistically converged state was reached. Once converged, we proceed to the post-processing treatment. 3.5 Post-processing treatment of the numerical data As the accuracy of the present numerical approach is judged by means of comparisons with experimental data, several post-processing treatments are necessary to analyze the LES results. These post-processing treatments can consist on snapshot visualizations, spectral analysis and mean quantity profiles to analyze the large-scale structure arrangement and to compare with the experimental data. The operator (noted < >) used to evaluate the mean quantities corresponds to the integrations in both time and spanwise (homogeneous) direction: < >= 1 dt dz, where L z is the spanwise integral length and T the L z T L z T integration period. Let us mention that the calculation of the components of the Reynolds stress tensor did take into account the subgrid scale contributions since they have been evaluated using: < v i v j >=< v i v j > < v i > < v j >, where v i refers to the temporal fluctuation of the velocity centered on the average value of the resolved velocity < v i >. To characterize the mixing layer thickness, one of the common length scales used is the vorticity thickness (δ ω ), defined as follows: ( ) < vx >max < v x > min δ ω (x) = ). (32) ( <v max x> y y The vorticity thickness (δ ω ) may follow a linear evolution as a function ( of the main flow direction x [7]. Based on δ ω, a self-similarity parameter η = y y ) 0 δ ω 16

17 can then be constructed. Note that y 0 is the theoretical center of the mixing layer, computed from < v x > profiles. In practice, y 0 refers to the location where < v x > reaches the value of the convective velocity (U conv = (U high + U low )/2). The comparison of the LES results with the experimental data has first been performed on the distribution of the vorticity thickness. To emphasize the validation of the LES results, the profiles of the mean velocity and Reynolds stress tensor components have been compared to the experiments within a portion of the computational domain. This specific region starts from 10 δ ω0 downstream of the trailing edge of the flat plate and lies over 5 δ ω0 downstream to minimize the influence of the outlet boundary condition, if it exists. This region coincides with a zone where the mixing layer is fully developed and where a self-similarity behavior is expected. Successive profiles can be plotted as function of η in order to emphasize a self-similarity behavior. In the present calculations, the average convergence is studied by using the vorticity thickness evolution (Figure 4), which is obtained from a velocity derivative, very sensitive to a good statistical convergence. A statistically converged state is supposed to be obtained after 9000 time steps. Different averaged profiles (mean velocity and Reynolds stress tensor components) are also compared at a given position x = 0.6 m, where the mixing layer is fully established. These profiles are plotted versus the normalized vertical direction ((y y 0 )/δ ω ). Figure (5) shows that the average convergence depends on the ratio between the intensity of the temporal fluctuation (φ ) and the mean value (< φ >) i.e. (φ / < φ >). It is then easier to obtain the average convergence for a variable with very small fluctuations such as the propagation velocity < v x > (1000 time steps) than for a very disturbed variable such as the spanwise velocity < v z > (at least 9000 time steps). In most of the numerical data presented, unless it is mentioned, the averaging will then be performed over a dimensionless time interval t = The averaging begins after the established solution (after the transient period) is reached. The reference dimensionless time t = 61.7 corresponds to 9000 time steps and also to about twenty large scale (roll structure) periods. 4 Parametric study on the influence of the numerical ingredients As the goal sought after in this study is to validate the presented numerical approach and then to check its ability to recover the dynamic behavior of a realistic high Reynolds number flow, the LES is performed on the same flow configuration as the experimental one [16], concerning a plane mixing layer spatially developing downstream of a flat plate. The velocity ratio is r = U low /U high = 0.59, where U high = 42.8 m.s 1 and U low = 25.2 m.s 1 are the magnitudes of the external 17

18 velocities of the boundary layers at the trailing edge of the flat plate. Up to the end of this article, the reference velocity and length-scale are respectively based on U = ( U high U low and δ ω0. Using these reference quantities, the Reynolds number Re = (U ) high U low ) δ ω0 is then Re = and corresponds to a ν fully developed turbulent flow. 4.1 Influence of the numerical scheme and subgrid scale model The interaction between the intrinsic dissipative numerical scheme and the diffusion of the subgrid scale (SGS) model is often pointed out in LES results. In order to both emphasize the essential role of the subgrid scale (SGS) model and evaluate the potential influence of the intrinsic dissipation of the numerical scheme, two numerical schemes have been employed for the discretization of the convective terms: a second order centered and a third order QUICK scheme. For each scheme, the simulations have been conducted with and without the SGS mixed scale model described in (1.2). The influences of the numerical scheme and the SGS model have then been evaluated by comparing the numerical results with the experimental data [16] taken as referenced results Second order centered scheme with and without SGS model When using the second order centered scheme without the SGS model, the resolution of the vorticity transport equation diverges even if the time-step is decreased to a value ten times smaller than the one used in the present LES calculations. No result can be produced by using such a numerical configuration. Therefore, the numerical procedure needs the use of a stabilization feature. In the literature, it is sometimes admitted that when a non dissipative numerical scheme is used for LES computations, the subgrid scale eddy viscosity supplies sufficient diffusion to act as both a stabilization process of the numerical procedure and a turbulent dissipation of the small scale structures. To check this proposition, the LES of the plane mixing layer development has been performed by using the 2 nd order centered scheme coupled with the mixed scale model ( 1.2). If the coefficient of the model (C M ) is prescribed at the commonly admitted value (i.e. C M = 0.04), the simulation diverges, even by using very low time-steps. To obtain a converged (stable) solution, the value of C M must be increased up to C M = The results are presented on the longitudinal evolution of the vorticity thickness (δ ω ) (Figure 6) and compared with the experimental data [16]. Though stabilization of the resolution is reached with C M = 0.08, too much diffusion is provided with the SGS model since the expansion rate of the δ ω (given 18

19 by the slope of the matching line of the curve) is too low and consequently, the numerical results do not fit the experimental data. We note that the greater the value of C M, the greater the subgrid scale diffusion and the less the expansion rate. Due to a too large amount of subgrid scale diffusion, the self-similarity behavior of the mixing layer is not recovered. Following these results, we claim that, by using the typical LES grid we designed to capture the Taylor micro scale estimated with the experimental data (see 3.2), the numerical procedure needs the use of an ad hoc stabilization feature for the treatment of the non-linear terms QUICK scheme with and without SGS model A third-order QUICK scheme has been employed for the stabilization of the convective term discretization in the vorticity transport equation. This numerical scheme exhibits a third-order dispersive truncation error. While this truncation error is of the same order of accuracy as the diffusive term introduced through the SGS model, it does not exhibit the same behavior and, consequently, theoretically does not interact with the diffusive property of the SGS model. To evaluate the influence of the stabilization feature on the results, the calculations have been performed on two grids: the reference grid (see 3.2) and a coarser grid in the vertical direction than the previous one. In fact, the coarse grid uses only 61 grid points from 5 δ ω0 to +5 δ ω0 in the vertical direction. Consequently, the maximum aspect ratios on the coarse grid are x / y = 1.78 and z / y = The results, obtained on the reference and the coarse grids, are validated for the longitudinal evolution of the vorticity thickness (Figure 7). On the coarse grid (Figure 7-a), if no SGS model is used, the QUICK scheme largely overestimates the expansion rate of δ ω because no extra diffusion is introduced for the dissipation of the small scale structures. The results are better predicted by using the mixed scale model coupled with the QUICK scheme. Indeed, the predicted expansion rate is the same as the one found in the experiments (Figure 7-a) though the δ ω values are slightly overestimated. On the fine grid (Figure 7-b), the differences between the results obtained with and without the use of a SGS model are less important than on the coarse grid. With the fine (reference) grid, the grid resolution is close to the Taylor micro-scale and the diffusion introduced through the subgrid eddy viscosity is less crucial than on the coarse grid. However, the QUICK scheme coupled with the mixed scale model gives results more in agreement with the experiments than by using no SGS model. In conclusion, we can claim that, on the fine reference grid we used, the stabilizing process induced with the QUICK scheme must be employed in addition to the SGS (mixed scale) model required for an efficient dissipation of the small scale structures. In the following, all the calculations presented have been performed with the QUICK scheme and the mixed scale model. 19

20 4.2 Influence of the magnitude of the inlet perturbations As we saw in (3.3), the tangential vorticity components, at the inlet boundary, are disturbed by means of velocity perturbations to mimic the turbulent behavior of the two boundary layers at the trailing edge of the flat plate. Because these boundary conditions, based on white noise perturbations superimposed on mean profiles, are not representative of the spectral arrangement that could be recorded in a natural turbulent boundary layer, the amplitudes of the inlet perturbations may have an influence on the setting of the mixing layer, mainly on the location of the transition between the flat plate wake and the mixing layer itself. Hence, the numerical results obtained from different LES associated with several upstream perturbations with rms values ranging from 0% to 7.5% of the external velocity (U high ), are compared to the reference experiments [16]. The influence of these inlet perturbations on the longitudinal evolution of the vorticity thickness is plotted in Figure (8). The simulation performed without any perturbation provides results that do not agree with the experiments at all. Two parts can be distinguished in the LES curve without perturbation. The first part, just downstream of the trailing edge, is attributed to the wake of the flat plate. Then, a transition occurs close to x = 20 δ ω0 and the δ ω distribution recovers a linear evolution in the second part which corresponds to the mixing layer zone. If a matching line fits the second part of the curve, its intersection with the x-axis gives the virtual origin of the mixing layer. In fact, without any perturbation, the mixing rate in the wake might not be strong enough to generate a transition sufficiently close to the trailing edge. Then, the virtual origin of the mixing layer is too far downstream of the trailing edge. However, note that the expansion rate, given by the slope of the matching line, seems to be in agreement with the experimental expansion rate. In order to obtain a mixing layer development closer to the trailing edge, non-zero upstream white noise must be employed [32, 44]. If white noise with a constant amplitude is employed at the inlet boundary, the vorticity thickness recovers rapidly a linear longitudinal evolution just downstream of the trailing edge (Figure 8). Whatever the perturbation amplitude studied (even at very high amplitudes, 7.5% of U high for instance), the same expansion rate is recorded in the LES results. These expansion rates are in good overall agreement with the experiments. The influence of the magnitude of the perturbations is then on the fictitious origin of the mixing layer. In fact, the more important the perturbations are on the spanwise component of the velocity (< v z >), the more perturbed the transverse components of the vorticity (< ω x > and < ω y >) are, thanks to the spatial derivatives in the (x) and (y) directions of < v z >. Increasing the perturbations on < ω x > and < ω y > favors an efficient mixing and thus introduces 3D effects more rapidly than perturbing the spanwise component of the vorticity. This might explain that the more important 20

21 the magnitude, the more upstream the fictive origin and the closer to the experiments the LES results are. Eventually, the best results that fit the experiments very well are obtained by using a perturbation amplitude of 7.5% of U high. Up to the end, all the presented results have been obtained by using this perturbation amplitude. 4.3 Comments on grid and domain size influences We already saw the influence of the grid resolution on the numerical results in (4.1.2). The results, obtained on both the reference grid and a coarse grid (see 4.1.2), are presented for the longitudinal evolution of the vorticity thickness (Figure 7). Results on both grids show a very good prediction of the expansion rate of the mixing layer. On the coarse grid, the values of δ ω are however overestimated compared to the experiments. The SGS model on that coarse grid does not supply sufficient turbulent dissipation to provide efficient mixing just downstream of the trailing edge. Consequently, the wake - mixing layer transition occurs too rapidly. The results obtained on the fine grid are in better agreement with the experiments than on the coarser grid since the fine referenced grid has been designed to ensure that the filter width is close to the Taylor micro-scale estimated by using the experimental data. Periodic boundary conditions associated with a fixed domain length contribute to the selection of preferred length-scales. As periodic boundary conditions have been employed in the spanwise direction, the influence of the spanwise extent of the computational domain should also be studied. Results obtained on a reference domain with L z = 5 δ ω0 are then compared with results obtained with a domain two times larger in the spanwise direction, namely L z = 10 δ ω0. In both meshes, the grid spacing ( z) is kept unchanged. No significant influence of the spanwise extent is recorded on the distribution of the vorticity thickness (Figure 9). Both results correspond to a linear δ ω evolution which agrees well with the experimental data. By considering the increase of the computational effort due to the largest domain we checked, in the following the validation of the numerical approach will be presented on the reference domain. 5 Detailed comparisons with the experimental measurements Following the results of the above mentioned parametric study, validations of the numerical approach have been carried out on the reference grid ( 3.2) by using the mixed scale model associated with the third-order QUICK scheme and through the use of inlet perturbations with a constant amplitude of 7.5% U high. 21

22 5.1 Instantaneous field and flow description As regards the instantaneous organization of the mixing layer, the isobaric surface of the dimensionless pressure field at a threshold p P 0 = is plotted for a dimensionless time t = 250 (Figure 10).The visualization easily shows the large scale arrangement in the whole of the computational domain. Just downstream of the trailing edge of the flat plate, the flow develops through a transition region that coincides with the wake of that plate. In that region, Kelvin-Helmholtz instabilities develop leading to vortex structure like rolls mainly aligned with the spanwise direction (z). Far downstream, due to secondary instability mechanisms, helical pairings take place and braid like-structures occur between these main rolls which give a typically highly 3D arrangement. Let us note that, while high amplitude perturbations have been employed to destabilize the inlet flow, the perturbations are only visible on the iso-surface of the pressure very close to the inlet of the domain. Regarding the development of the energetic large scale structures, these perturbations do not contaminate the pressure field in the external part of the flow in the whole of the computational domain. 5.2 Mean flow analysis The validation of the numerical approach is also performed on the mean flow quantities. As one could see earlier (see Figure 8), the vorticity thickness (δ ω ) and its longitudinal evolution (dδ ω /dx) are correctly predicted by the computation (LES: dδ ω /dx = ; Experiments: dδ ω /dx = ). A selfsimilarity behavior is recovered inside the comparison region 10 δ ω0 x 15 δ ω0. On Figure (11), several profiles of the dimensionless longitudinal component of the mean velocity ((< v x > U low )/(U high U low )) are plotted versus η at ten different locations equally spaced within the selected comparison region. The ten profiles collapse together very well, showing that a self-similarity behavior is recovered. Moreover, they also fit the experimental profiles well. On Figure (12), ten profiles of the Reynolds stress components are plotted in the transform plane. The self-similarity behavior can also be judged in these profiles, mainly on the longitudinal and cross correlations. The LES results recover a good agreement with the experimental data for all the components. In the outer regions of the mixing layer, the external turbulence level on the vertical component (< v y v seems very high (Figure 12-b). Though this artifact is only visible on < v y v y >) profiles, it is relative to the influence of the inlet perturbations. In fact, if lower amplitude inlet perturbations are used, no extra turbulence intensity is visible in the external parts of the mixing layer. To emphasize the good agreement of the LES results with the experimental data, the two-point space correlations (R ii (x 0,y 0,x,y )) recorded from the LES results are compared on figure (13) to experimental data [16]. From an experimental point of view, the two-point space y > 22

23 correlations are evaluated through the space-time correlations (R ii (τ,y,y )), by assuming a Taylor hypothesis [16] at a given x 0 location. On this figure (13), the reference point (x 0,y 0 ) is located at the center of the selected spatial domain and the positive values are plotted with solid lines and the negative ones with dashed lines. Besides the results obtained by the LES on the Reynolds stress profiles (Figure 12), a very good agreement with the experiments is also achieved by the LES on the spatial organization of mixing layer since the overall shape of the measured correlations has been very well described by the LES. We can then see that the longitudinal component (v x ) is characterized by fluctuations of opposite sign from part to part of the mixing layer, while the normal component (v y ) can be described by fluctuations in phase all over the inhomogeneous direction (y), alternating in sign in the (x) direction (Figure 13). Moreover, the shape of the spanwise component (v z ) indicates the presence of a strong organization of the streamwise vorticity component, as recovered on the pressure field (see Figure 10). 5.3 Spectral arrangement Besides the visualizations showing a strongly 3D large scale arrangement of the flow (Figure 10), the space and time large scale structure of the mixing layer can be deduced from a spectral analysis. Energy spectra in time and space are computed from the velocity fluctuations at two streamwise locations within the selected region and for three vertical locations, i.e. the center of the mixing layer (y y 0 = 0.) and the two borders of the layer (y y 0 = ±δ ω /2). The results of the spectral analysis in time are presented on the kinetic energy spectra of the vertical velocity fluctuations (v y ) (Figure 14), plotted versus the Strouhal number (St = fδ ω /U conv, where f is the frequency and U conv is the convective velocity, see 3.5). A main frequency, close to St = 0.3, is recorded everywhere in the selected region of the mixing layer. This value, characteristic of the time between two successive main rolls, agrees very well with the one generally admitted and recorded experimentally by numerous authors ([15, 16], for instance). After the maximum value, the decrease rate of kinetic energy fits with the well known -5/3 slope, at least on one decade. This result suggests that the numerical scheme coupled with the subgrid scale model does not introduce too much intrinsic diffusion and that the mesh seems well adapted (i.e. fine enough) to the solution. Let us recall that the mesh has been built to ensure that the grid filter width is close to the experimentally estimated Taylor micro scale. The results on the spanwise organization is presented on the spatial energy spectra of v y (Figure 15), plotted versus the dimensionless wave number (k z δ ω ). A first wave number (k z δ ω = 0.18) is encountered in this energy spectra, which is characteristic of the largest spanwise wave length found in the computational domain. This computed value is in agreement with the experiments since it is close to the one 23

24 recorded experimentally by Delville et al. [15, 16] (k z δ ω = 0.15). This wave number is representative of a very large scale turbulent phenomenon everywhere in the mixing layer. A second typical wave number found in the spatial energy spectra is representative of the 3D arrangement and characterizes the spacing between two successive braids. The wave length is not constant through the layer thickness. It is recorded close to k z δ ω = at the boundaries of the layer (y y 0 = ±δ ω /2) and close to k z δ ω = 0.7 in the centerline of the shear layer. Though these values do not agree very well with the experiments, they are not so far away from the experimental value recorded at k z δ ω = 0.5 [15, 16]. Nevertheless, in the outer part of the mixing layer, the ratio between the spanwise length scale and an estimation of the longitudinal one (deduced from the temporal spectrum, assuming a Taylor hypothesis) is close to Λ z /Λ x 2/3 which is in rather good agreement with the value generally admitted [4]. 6 Conclusions The LES of a 3D mixing layer spatially developing downstream of a flat plate has been conducted for a rather high Reynolds number (Re θ = 2835) equal to the experimental one. The subgrid scale contributions are modeled by means of the mixed scale model [47, 52]. To overcome the problem of the pressure condition on the free boundaries, the (v ω) formulation of the Navier-Stokes equations has been preferred to the primitive (v p) formulation. To deal with the difficult problem of the divergence free constraint on both the velocity and the vorticity field, an original numerical method has been proposed which turns out to be very attractive. The solution of the three coupled components of the vorticity transport equation has been first obtained by solving a Helmholtz problem. The velocity components are secondly calculated through the resolution of a Cauchy-Riemann problem by means of a fractional step algorithm. This resolution algorithm is especially competitive and makes the (v ω) formulation very attractive. As one of the advantages of the vorticity-based formulation is the treatment of the free flow boundary conditions, special care has been devoted to the boundary conditions particularly at the outlet. An optimum approximation of the outflow boundary condition has been carried out which satisfies the conservation of mass, making the long time integration easier and more accurate. The validation of the numerical approach has been performed through comparisons with the detailed experimental data from Delville et al. [15, 16]. A parametric study has first been conducted to check the sensitivity of the solution to the numerical ingredients we used. As the interaction between the numerical scheme truncation error and the subgrid scale model is often pointed out in LES results, we evaluated the potential interaction between these two numerical ingredients. Two numerical schemes have been applied for the convective term 24

25 discretizations: a second order centered and a third order QUICK scheme. The simulations using both schemes have been performed with and without the mixed scale model. Following the results obtained on a fine reference grid and a coarser grid, we can claim that the stabilizing process induced with the QUICK scheme is necessary and must be employed in addition to the SGS model required for an efficient dissipation of the small scale structures. The influence on the flow development of the boundary conditions has secondly been checked. On the one hand, the influence of the computational domain size has been reviewed and no domain dependence has been recorded. On the other hand, the influence of the inlet perturbations has been reported. Though the inlet boundary conditions are rather rough and not representative of the spectral arrangement that could be recorded in a natural turbulent boundary layer, the numerical results showed that whatever the white noise amplitude (ranging from 0 to 7.5% of U high ) was, the experimental expansion rate of the mixing layer was very well recovered. The noise amplitude however had an influence on the location of the fictitious origin of the mixing layer. The 7.5% U high uniform magnitude perturbation leads to the best results obtained since it mimicked very well the experimental data [15, 16]. Though there is a high level of perturbation magnitudes on the vorticity profiles at the inlet, it is interesting to notice that the pressure fluctuations are only concentrated within the vortical mixing layer regions and of course at the inlet surface. Hence, it seems more convenient to prescribe the perturbations on the vorticity components than on the velocity profiles thanks to the Dirichlet-type condition on the vorticity. Moreover, the vorticity is calculated by its own transport equation as opposed to the velocity which is calculated from the vorticity by a Poisson equation. This is one of the advantages recovered by the (v ω) formulation coupled with the numerical algorithm presented here. More detailed comparisons with the experimental data are then performed. The numerical simulations were conducted using the mixed scale model and an upstream perturbation with a constant magnitude of 7.5% U high. Very good agreement between the LES results and the experiments has been recorded on the mean velocity profiles, the mixing layer expansion factor and the profiles of the Reynolds stress components. The highly 3D arrangement of the large scale structures can be pointed out by looking at the snapshots. The spatial organization, deduced from the time and space spectral analysis, can favorably be compared to the experimental one [15, 16]. Following these good results, the present LES results can be considered an interesting data base for undertaking future investigations. Kinetic energy balances might be computed to analyze specific modelings adapted to such a flow development in a RANS (or URANS) approach. As regards flow control, a low order dynamical system might also be derived from a POD-base, to analyze the capability of such an approach to mimic mixing layer spatial development. 25

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31 List of Figures 1 Elementary M.A.C. cell Spatially developed mixing layer test case Profiles of the longitudinal component of the mean velocity (< v x > /U high ) at the trailing edge location: experiments [16] and numerical inlet boundary condition [58] Influence of the convergence process on the longitudinal evolution of the vorticity thickness (δ ω ) in LES using the mixed scale model [52] Influence of the convergence process on the profiles at x = 0.6 m in LES using the mixed scale model [52]: mean velocity components (a) ((< v x > U low )/ U), (b) (< v y > / U) and (c) (< v z > / U), and (d) Reynolds stress tensor component (< v xv y > / U 2 ) (t. s.: time steps and U = U high U low ) Influence of the convective scheme on the evolution of the vorticity thickness (δ ω ): experimental results [16] and LES using the mixed scale model [52] and a centered scheme Role of the subgrid scale model on the evolution of the vorticity thickness (δ ω ): (a) coarse grid in the inhomogeneous direction (y) and (b) reference grid; experimental results [16] and LES with the mixed scale model [52] and without subgrid scale model Influence of the inlet perturbation on the longitudinal evolution of the vorticity thickness (δ ω ): experimental results [16] and LES results using the mixed scale model [52] for upstream white noise perturbations with a constant amplitude Influence of the spanwise domain dimension L z on the longitudinal evolution of the vorticity thickness (δ ω ): experimental results [16] and LES results using the mixed scale model [52] for the reference domain (L z = 5 δω 0 ) and a two times larger domain (L z = 10 δω 0 ) conserving an equivalent uniform discretization z Isobaric surface of the instantaneous calculated pressure field obtained at a dimensionless time t = 250 by using the mixed scale model and an inlet perturbation magnitude of 7.5% U high Profiles of ((< v x > U low )/( U)): experimental results [16] and LES results at ten (x) locations within the selected region using a constant white noise perturbation of 7.5% U high and the mixed scale model [52] ( U = U high U low )

32 12 Profiles of the Reynolds stress components: (a) < v x v x > / U 2, (b) < v y v y > / U 2, (c) < v x v y > / U 2 and (d) < v z v z > / U 2 ; experimental results [16] and 3D LES results at ten (x) locations within the comparison region, using a constant white noise of 7.5% U high and the mixed scale model [52] ( U = U high U low ) Comparison in a plane (z =cst; τ,y or x,y) of the experimentally measured space time correlations R ii (y,y ;τ) (bottom) with the two-point space correlations R ii (x,y,x,y ) obtained from L.E.S. (top): solid-line for the positive values, dashed-line for the negative ones Temporal energy spectra of the vertical component of the velocity fluctuations (v y) at two streamwise locations within the selected region and at three vertical locations Spatial energy spectra of the vertical component of the velocity fluctuations (v y) at two streamwise locations within the selected region and at three vertical locations

33 y v y ω z v z ω x (i,j,k) ω y v x x z Figure 1: Elementary M.A.C. cell. 33

34 upstream condition : mean velocity profiles + white noise perturbation U high flat plate δ ω y z U low x x=0 mm Figure 2: Spatially developed mixing layer test case <v x > / U high ( Y - Y 0 ) / δ ω0 Figure 3: Profiles of the longitudinal component of the mean velocity (< v x > /U high ) at the trailing edge location: experiments [16] and numerical inlet boundary condition [58]. 34

35 δ ω (m) time steps time steps 6000 time steps time steps time steps X / δ ω0 Figure 4: Influence of the convergence process on the longitudinal evolution of the vorticity thickness (δ ω ) in LES using the mixed scale model [52]. 35

36 (a) (b) (<v x > - U low ) / U t. s t. s t. s t. s t. s <v y > / U t. s t. s t. s t. s t. s ( Y - Y 0 ) / δ ω ( Y - Y 0 ) / δ ω (c) (d) <v z > / U t. s t. s t. s t. s t. s <v x v y > / U t. s t. s t. s t. s t. s ( Y - Y 0 ) / δ ω ( Y - Y 0 ) / δ ω Figure 5: Influence of the convergence process on the profiles at x = 0.6 m in LES using the mixed scale model [52]: mean velocity components (a) ((< v x > U low )/ U), (b) (< v y > / U) and (c) (< v z > / U), and (d) Reynolds stress tensor component (< v xv y > / U 2 ) (t. s.: time steps and U = U high U low ). 36

37 δ ω (m) experimental data centered scheme X / δ ω0 Figure 6: Influence of the convective scheme on the evolution of the vorticity thickness (δ ω ): experimental results [16] and LES using the mixed scale model [52] and a centered scheme. 37

38 (a) (b) δ ω (m) δ ω (m) experimental data mixed scale model no model X / δ ω0 experimental data mixed scale model no model X / δ ω0 Figure 7: Role of the subgrid scale model on the evolution of the vorticity thickness (δ ω ): (a) coarse grid in the inhomogeneous direction (y) and (b) reference grid; experimental results [16] and LES with the mixed scale model [52] and without subgrid scale model. 38

39 δ ω (m) experimental data white noise 7.5 % U high white noise 5 % U high white noise 2.5 % U high white noise 1 % U high no perturbation X / δ ω0 Figure 8: Influence of the inlet perturbation on the longitudinal evolution of the vorticity thickness (δ ω ): experimental results [16] and LES results using the mixed scale model [52] for upstream white noise perturbations with a constant amplitude. 39

40 δ ω (m) experimental data reference domain long spanwise domain X / δ ω0 Figure 9: Influence of the spanwise domain dimension L z on the longitudinal evolution of the vorticity thickness (δ ω ): experimental results [16] and LES results using the mixed scale model [52] for the reference domain (L z = 5 δω 0 ) and a two times larger domain (L z = 10 δω 0 ) conserving an equivalent uniform discretization z. 40

41 Y X Z Y X 5 Z Figure 10: Isobaric surface of the instantaneous calculated pressure field obtained at a dimensionless time t = 250 by using the mixed scale model and an inlet perturbation magnitude of 7.5% Uhigh. 41

42 1.000 ( <v x > - U low ) / U ( Y - Y 0 ) / δ ω Figure 11: Profiles of ((< v x > U low )/( U)): experimental results [16] and LES results at ten (x) locations within the selected region using a constant white noise perturbation of 7.5% U high and the mixed scale model [52] ( U = U high U low ). 42

43 (a) < v x v x > / U ( Y - Y 0 ) / δ ω (b) < v y v y > / U ( Y - Y 0 ) / δ ω (c) < v x v y > / U ( Y - Y 0 ) / δ ω (d) < v z v z > / U ( Y - Y 0 ) / δ ω Figure 12: Profiles of the Reynolds stress components: (a) < v x v x > / U 2, (b) < v y v y > / U 2, (c) < v x v y > / U 2 and (d) < v z v z > / U 2 ; experimental results [16] and 3D LES results at ten (x) locations within the comparison region, using a constant white noise of 7.5% U high and the mixed scale model [52] ( U = U high U low ). 43

44 R uu R vv R ww (y y 0 )/δ ω (y y 0 )/δ ω (y y 0 )/δ ω ω 0 0 ω 0 0 ω δ δ δ (x x 0 )/δ ω (x x 0 )/δ ω (x x 0 )/δ ω Figure 13: Comparison in a plane (z =cst; τ,y or x,y) of the experimentally measured space time correlations R ii (y,y ;τ) (bottom) with the two-point space correlations R ii (x,y,x,y ) obtained from L.E.S. (top): solid-line for the positive values, dashed-line for the negative ones. 44