RECONSTRUCTION OF TURBULENT FLUCTUATIONS FOR HYBRID RANS/LES SIMULATIONS USING A SYNTHETIC-EDDY METHOD

RECONSTRUCTION OF TURBULENT FLUCTUATIONS FOR HYBRID RANS/LES SIMULATIONS USING A SYNTHETIC-EDDY METHOD N. Jarrin 1, A. Revell 1, R. Prosser 1 and D. Laurence 1,2 1 School of MACE, the University of Manchester, M601QD, UK 2 EDF R&D, 6 Quai Watier, Chatou, France dominique.laurence@manchester.ac.uk Abstract A coupling methodology between an upstream RANS simulation and a LES further downstream is presented. The focus of this work is on the RANSto-LES interface inside an attached turbulent boundary layer, where an unsteady LES content has to be explicitly generated from a steady RANS solution. The performance of the Synthetic-Eddy Method (SEM), which generates realistic synthetic eddies at the inflow of the LES, is investigated on a wide variety of turbulent flows, from simple channel and duct flows to the flow over an airfoil trailing edge. The SEM is compared to other existing methods of generation of synthetic turbulence, and is shown to reduce substantially the distance required to develop realistic turbulence. 1 Introduction In the aeronautical or automotive industry, engineers are interested in LES because it provides an unsteady turbulent flow field which allows to compute the aeroacoustic noise generated by the vehicle or the airfoil. In practise only a small specific region of interest such as the trailing edge of an airfoil or the rear view mirror of a car needs to be computed with LES. The specification of the upstream flow conditions for the embedded LES domain requires the simulation of the whole geometry, which can be achieved using RANS techniques at a relatively cheap computational cost. The challenge is then to generate a mature unsteady turbulent LES solution from a steady RANS solution within as short a distance as possible in order to achieve both a reduction of the total computational cost of the simulation by limiting the size of the embedded LES domain, and a better accuracy of the simulation by using a LES model in the region of interest. The present investigation thus focuses on the RANS-to-LES interface, for what is often referred to in the literature as zonal hybrid RANS-LES methods, where LES and RANS regions use separate domains. In this case unsteady turbulent velocity fluctuations must be explicitly reconstructed and prescribed at the inflow of the LES region from a steady upstream RANS solution. There exists an overwhelming variety of methods available to generate inflow boundary conditions for LES (see Keating et al. (2004) for a review). Although recycling methods as in Lund et al. (1998) produce very realistic inflow data, they increase the cost of the computation and lack generality to be employed in complex industrial applications. Synthetic turbulence generation methods provide an alternative, even though they yield a transition region downstream of the inlet where the synthetic fluctuations imposed at the inlet evolve towards real turbulence. In this paper, the Synthetic-Eddy Method of Jarrin et al. (2006) is used to generate fluctuations at the RANS-to-LES interface of several wall flows. All of the input parameters of the method are calculated using only statistical data that is available from the upstream RANS simulation. The SEM is compared to other existing methods of generation of synthetic turbulence. Cases simulated include simple channel and duct flows, and the more challenging case of the turbulent flow over an airfoil trailing edge. 2 Methodology The governing equations are the incompressible Navier-Stokes equations, filtered (in the LES region) or time averaged (in the RANS region). In both regions an eddy viscosity mode is used to close the equations. The SST model of Menter et al. (1993) is used in the RANS region, while the standard Smagorinsky model with C S = 0.065 and Van-Driest damping at the wall is used in the LES region. The RANS and LES equations are solved with the collocated finite volume code Code_Saturne (Archambeau et al., 2004). The LES and the RANS simulation are run on two different domains which overlap so that the RANS region can provide statistics to the inlet boundary faces of the LES domain. The statistics available from the upstream SST solution are the mean velocity U i, the Reynolds stress tensor R ij, and the dissipation rate per unit of kinetic energy. The performance of several methods of generation of inflow conditions for LES is investigated. The simplest method, referred to as the

random method, generates uncorrelated random numbers for each component of the velocity at each point of the inlet mesh, and at each iteration. The random numbers are then transformed using the Cholesky decomposition of the Reynolds stress tensor to create one-point cross-correlations between the velocity components (see Appendix B of Lund et al. (1998)). The method of Batten et al. (2004) involves the summation of sines and cosines with random amplitudes and phases. In all simulations presented here we used 2,000 random modes. Further details of the method can be found in Batten et al. (2004). The main focus of this paper is on the applications of the SEM to RANS-LES coupling. In Jarrin et al. (2006), the input parameters of the SEM were derived from a precursor LES or from ad hoc formulae. In this paper, all of the input parameters of the method are calculated using only statistics available from the upstream SST simulation. The LES inflow plane on which we want to generate synthetic velocity fluctuations with the SEM is a finite set of points S = {x 1,x 2,,x s }. The first step is to create a box of eddies B surrounding S which is going to contain the synthetic eddies. Its minimum and maximum coordinates are defined by where is a characteristic length scale of the flow whose computation from RANS statistics will be detailed later. In order to ensure that the density of eddies inside of the box of eddies is constant, the number of eddies is set as N = max ( V B / 3 ), where V B is the volume of the box of eddies. The SEM decomposes a turbulent flow field in a finite sum of eddies. The velocity fluctuations generated by N eddies have the representation where the x k are the locations of the eddies, the k j are their respective intensities and a ij is the Cholesky decomposition of the Reynolds stress tensor. f (x x k ) is the velocity distribution of the eddy located at x k. We assume that the differences in the distributions between the eddies depend only on the length scale and define f by In all our simulation f is a simple tent function, and is a parameter that controls the size of the structures. It is taken as where = max( x, y, z), = C k is the rate of dissipation, and is the thickness of the boundary layer considered. The position x k and the intensity k j of each eddy are independent random variables. At the first iteration, x k is taken from a uniform distribution over the box of eddies B and k j = ±1, with equal probability to take one value or the other. The eddies are then convected through the box of eddies B with a constant velocity U c characteristic of the flow. In our case it is straight forward to compute U c as the averaged mean RANS velocity over the LES inflow plane. At each iteration, the new position of eddy k is given by where dt is the time step of the simulation. If an eddy k is convected out of the box through face F of B, then it is immediately regenerated randomly on the inlet face of B facing F with a new independent random intensity vector k j = ±1. The method generates a stochastic signal with prescribed mean velocity, Reynolds stresses, and length and time scale distributions. Although the SEM involves the summation of a large number of eddies for each grid point on the inflow, the CPU time required to generate the inflow data at each iteration did not exceed 1% of the total CPU time per iteration of the LES simulation. 3 Results 3.1 Spatially developing channel flow Hybrid RANS-LES simulations of the turbulent flow in a channel are performed at Re = 395. The RANS equations are solved for x/ < 0. The RANS grid is one-dimensional, and only uses one cell with periodic boundary conditions in the streamwise and spanwise directions. At x/ = 0 the LES domain, of dimensions 10 2, begins. The grid spacing in wall units satisfy the constraints x + 50, z + 15, y + = 2 at the wall and y = 0.1. The wall-normal grid resolution is the same in the RANS simulation and in the LES to avoid interpolation of the RANS data onto the LES grid. Several methods of generation of inflow conditions for LES are tested and the simulations performed are summarized now. A baseline simulation was performed as a comparison point for all other cases (run P1). Time series of instantaneous velocity planes were extracted from a periodic LES (performed on a shorter domain but with the same grid refinement and the same numerical options) and imposed at the inlet of the present LES domain. In all other simulations, methods of generation of synthetic turbulence are used. Three hybrid calculations were conducted, using the SEM (run S1), Batten s method (run B1) and the random method (run R1).

realistically reproduced by the SEM. The velocity fluctuations generated using the method of Batten et al. (2004) exhibit surprising features. In the nearwall region, the fluctuations seem to be decorrelated in space. The reason for this phenomenon is the decomposition into Fourier modes used in Batten s method. The frequencies and wavelengths of the cosine and sine functions are allowed to vary in the direction of non-homogeneity of the flow (in the present case the wall-normal direction). The velocities at two points separated even by an infinitesimal distance in the wall-normal direction will thus oscillate at different frequencies, and therefore be completely decorrelated from each other. In the direction of homogeneity of the flow however (the spanwise direction in the present case), this problem does not occur since the frequencies and wavelengths are constant. Thus although the method of Batten et al. (2004) might appear to be capable of generating nonhomogeneous turbulence by allowing the wavelengths to vary in space, it does so at the expense of destroying the spatial correlations in the non-homogeneous directions. Figure 2: Coefficient of friction for hybrid simulations of channel flow at Re = 395. Inflow conditions are generated using a precursor LES ( ), the SEM ( ), Batten's method ( _ ) and the random method(... ). Figure 1: Velocity vectors of LES inlet conditions for hybrid simulations of channel flow. From top to bottom: precursor LES, SEM, Batten's method and random method. Figure 1 shows instantaneous velocity fluctuations prescribed at the inlet of the LES domain. Although the SEM does not reproduce completely the complex structure of the near-wall turbulence observed in the periodic LES, the length scale and the magnitude of the fluctuations are The development of the prescribed fluctuations downstream of the inlet will now be studied. Figure 2 shows the downstream development of the coefficient of friction. The horizontal dashed line represents the value of the coefficient of friction in the periodic LES and will be used as a reference point for the present RANS-LES simulations. Run P1 has a coefficient of friction in very good agreement with the periodic LES over the whole domain. When the random method is used, the coefficient of friction drops continuously downstream of the inlet, which indicates that the flow laminarizes. The decay of the coefficient of friction is also quite important downstream of the inlet when the method of Batten et al. (2004) is used. However the coefficient of friction reaches a minimum after about 8, before slowly recovering towards its fully developed value about 25 downstream of the inlet. With the SEM, the

coefficient of friction decays downstream of the inlet to reach a minimum about 3 downstream of the inlet (where it has only lost 15% of its initial value), and recovers its fully developed value only after 10 downstream of the inlet. The performance of the SEM is now tested at two higher Reynolds number (Re =590 and Re =950), keeping a constant grid refinement in wall units. Figure 3 shows that the development of the coefficient of friction downstream of the inlet is the same for the three Reynolds numbers considered when expressed as a function of x u /. Analysis of other flow statistics not presented in this paper confirm that in the near-wall region, the length of the transition region with the SEM scales roughly as x+ ~ 3, 000. 3.2 Spatially developing duct flow The SEM is now compared to Batten s method and to the random method in the case of a turbulent flow through a square duct at Re = 600 (Huser and Biringen, 1993). The computational set-up is similar to the one used in the case of the channel flow. The SST domain is positioned upstream of the LES domain, and the upstream SST simulation uses periodic boundary conditions in the streamwise direction. As expected the SST solution does not exhibit any secondary motion. The ability of the SEM, the method of Batten et al. (2004), and the random method to yield, after a short development distance, a secondary motion in the LES region is investigated. convected from the central region to the walls along the corner bisectors, and the mean streamwise velocity distribution (see Figure 5 (a)) is in excellent agreement with the one from the reference LES. The simulation using the random method does not exhibit any secondary motion (see Figure 4 (c)), since all the fluctuations have already been dissipated. With Batten s method, two very weak streamwise corner vortices can be observed, but their weak intensity does not alter the mean streamwise velocity distribution in the correct manner as shown in Figure 5 (b). We saw that Batten s method destroys spatial velocity correlations in the direction of non-homogeneity of the flow. In the present case, the upstream k and profiles extracted from the SST solution and transmitted to Batten s method are nonhomogeneous in the two transverse directions. Consequently Batten s method does not generate any two-point velocity correlations in the inlet plane. The better results obtained than when using the random method can be explained by the better time correlation of the inflow data generated using Batten s method. Figure 3: Development of the coefficient of friction (normalized by the coefficient of fiction obtained in the periodic LES) as a function x u / at Re = 395 ( ), Re = 590 ( ), and Re = 950( ). The topology of the mean flow is studied in a cross-section at x / D = 15 (roughly x + = 9,000) downstream of the RANS-to-LES interface. The simulation using the SEM exhibits two mean streamwise counter-rotating vortices in the corner of the duct, as shown in Figure 4 (a). Their centre location and topology are in very good agreement with those from the reference periodic LES. Due to the action of the secondary motion, momentum is Figure 4: Transverse velocity vectors at x/d = 15 for hybrid simulations of square duct flow with (a) the SEM, (b) Batten s method, (c) the random method and (d) the reference periodic LES. 3.3 Turbulent flow over an airfoil trailing edge The airfoil considered is a two-dimensional flat strut with a circular leading edge and an asymmetric beveled trailing edge with a 25 o tip angle. The geometry of the airfoil and the flow conditions are described in details by Wang and Moin (2000).

2.4 and x/h = 2.1, respectively. When the SEM is used the mean velocity profiles are in much better agreement with the reference LES. The hybrid simulation using the SEM detaches at x/h = 0.75, slightly after the reference LES (x/h = 1.17). The early detachment observed with the random method and with Batten s method is caused by the lack of coherence of the prescribed inlet fluctuations. As a result the fluctuations are underestimated at the first station. Further downstream the presence of a large separation bubble produces larger levels of fluctuations in the recirculation region. Figure 5: Mean streamwise velocity distribution normalized by bulk velocity at x/d=15 for hybrid simulations with (a) the SEM, (b) Batten s method, (c) the random method and (d) the periodic LES. As shown in Figure 6, the RANS domain encloses the entire airfoil and only the rear part of the trailing edge and the near wake are simulated with LES (the non-equilibrium region of the flow). The RANS simulation is conducted on a C-grid domain using only 0.1M cells. The LES domain begins at x / h = -4 and x / h = -2 on the low- and high-pressure side of the airfoil, respectively. For the LES mesh, the grid spacing at the inlet on the upper surface is x+ = 41, z+ 26, and y+ ~ 2 at the wall. In total, the LES mesh has about 3.0 10 6 cells. At the inlet plane of the LES domain, data are extracted from the SST solution, interpolated onto the LES grid, and used for the generation of synthetic turbulence. Results on the embedded LES domain using inflow data generated with the SEM, Batten s method and the random method are compared with the finely resolved LES of Wang and Moin (2000). Figure 6: Sketch of the hybrid RANS-LES simulations of the airfoil trailing edge. Profiles of mean velocity magnitude (U 2 +V 2 ) 1/2 and streamwise velocity fluctuations u on the lowpressure side of the airfoil are shown in Figure 7. The hybrid simulations using the random method and Batten s method initially laminarize (as expected from previous computations), and consequently show very early separation at x/h = Figure 7: Profiles of (a) the mean velocity magnitude and (b) the rms streamwise velocity fluctuations normalized by the edge velocity as a function of vertical distance from the upper surface:, SEM; _, Batten et al. (2004);..., random method;, LES Wang and Moin (2000). The effect of the inflow data on the turbulent structures will now be described. Streamwise velocity fluctuations along the upper surface of the airfoil are shown in Figure 8. The simulation using Batten s method - although leading to early separation and weak magnitude fluctuations in the near wall region - still shows features similar to the simulation using the SEM: the weak near-wall streaks are elongated in the streamwise direction (due to the favorable pressure gradient experienced by the boundary layer), followed by a rapid transition towards a more turbulent state (after the removal of the pressure gradient), before finally separating from the wall. With the random method, no turbulent structures are present in the near-wall region of the boundary layer downstream of the inlet, which also leads to early separation. However

in this case, the separation is laminar and leads to the formation of large scale two-dimensional Kelvin-Helmotz vortices in the subsequent shear layer. Finally Figure 9 shows the frequency spectrum of the v fluctuations at x/h=4 downstream of the trailing edge. A strong peak around f h/u 0 = 0.6 can be observed with the random method and Batten s method, indicating the presence in the flow of Kelvin-Helmotz vortices shedding in the wake of the airfoil. On the contrary the frequency spectrum in the case of the SEM does not exhibit any peak, in agreement with observations of instantaneous fluctuations in the recirculation region which did not exhibit any clear vortex shedding. This is the physical behaviour of the flow observed in the reference LES of Wang and Moin (2000). (after separation) of quasi two-dimensional structures characteristic of transitional flows. Figure 9: Frequency spectrum of the v fluctuations in the near wake at x/h = 4 and y/h = 0.5:, SEM; _, Batten et al. (2004);..., random method. Figure 8: Streamwise fluctuations in a plane parallel to the wall at y+ = 1 for hybrid simulations of the trailing edge flow. From top to bottom: SEM, method of Batten et al. (2004) and random method. 3 Conclusions The SEM was used to generate inlet conditions for a LES using only information available from an upstream SST simulation. This hybrid RANS-LES coupling strategy was first tested in the case of channel and duct flows. The development length of the eddies in the near wall region was shown to be approximately 3,000 wall units. This offers significant promise for the application of the method to high Reynolds number flows of engineering interest. With Batten s method, the use of Fourier harmonics with spatially varying wavelengths leads to a destruction of the spatial correlations of the signal in the direction of nonhomogeneity of the flow. Finally hybrid simulations of the flow over an airfoil trailing edge were performed. With the SEM, realistic turbulence is generated upstream of the separation, and thus flow predictions downstream of the separation are accurate. The LES domain used did not allow either the random method or Batten s method to generate a realistic boundary layer upstream of the region of interest. This lead to an early separation and hence a higher production of turbulent kinetic energy; leading to the growth The use of small embedded LES domains (without significant loss of accuracy compared to the reference data) in the hybrid simulations using the SEM led to substantial savings in terms of number of cells used. The reduction in terms of CPU time achieved with the present LES domain is over 40% when compared to the domain used in the reference LES of Wang and Moin (2000), and over 80% when compared to a full domain LES enclosing the entire airfoil. References Archambeau, F., Mehitoua, N. and Sakiz M., (2004): Code_Saturne: a finite volume code for the computation of turbulent incompressible flows. Int. J. of Finite Volumes, Vol. 1, No. 1. Batten, P., Goldberg, U. and Chakravarthy, S. (2004): Interfacing Statistical Turbulence Closures with LES. AIAA Journal, Vol. 42, 3, pp. 485-492. Huser, A. and Biringen, S. (1993): Direct numerical simulation of turbulent flow in a square duct. Journal of,fluid Mechanics, 257:65 95. Jarrin, N., Benhamadouche, S., Laurence, D. and Prosser, R. (2006): A synthetic-eddy method for generating inflow conditions for LES, Int. J. of Heat and Fluid Flow, Vol. 27, pp. 585-593. Keating, A., Piomelli, U., Balaras, E. and Kaltenbach H.J. (2004): A priori and a posteriori tests of inflow conditions for large-eddy simulation, Physics of Fluids, Vol. 16, 12, pp. 4696-4712. Lund, T.S., Wu X. and Squires D. (1998): Generation of turbulent inflow data for spatiallydeveloping boundary layer simulations. Journal of Computational Physics, Vol. 140, pp. 233-258. Wang, M. and Moin, P. (2000). Computation of trailing-edge flow and noise using large-eddy simulation. AIAA Journal, 38:2201 220.