Assessment of an approach to generating inflow synthetic turbulence for large eddy simulations of complex turbulent flows

An approach to generating inflow synthetic turbulence recently developed by the authors has been applied to zonal Reynolds-Averaged Navier-Stokes (RANS) / Large Eddy Simulations (LES) of two complex turbulent flows: flow over a wall-mounted hump and hydrofoil trailing edge flow, and to a LES of a flow in a three-dimensional (3D) diffuser. The results show that the zonal RANS-LES approach with synthetic turbulence at the interface is in excellent agreement with experimental data for hump and trailing edge flows. For the diffuser flow, it is shown that results depend significantly on the RANS model used to provide averaged velocity and Reynolds stresses at the inlet.


INTRODUCTION
Large eddy simulation of spatially developing turbulent §ows requires speci¦cation of unsteady (with a ¤turbulent content¥) velocity ¦elds at inlet boundaries. For nearly self-similar §ows, such ¦elds can be created with the use of the socalled recycling techniques (see, e. g., [1]). However, for more complex §ows, applicability of the recycling methods, even improved ones (see, e. g., [2]), becomes questionable, and other approaches should be used. In a recent paper of the authors [3], a simple synthetic turbulence generator has been proposed and, based on the simulations of a set of canonical shear §ows (developed twodimensional (2D) channel §ow, §at plate boundary layer, free shear layer) shown to be superior over similar methods available in literature [46] thanks to a capability of creating turbulent structures rapidly transforming to real turbulence downstream of the inlet boundaries. An objective of the present study is a more extensive validation of the method in the framework of zonal RANSLES computations of complex turbulent §ows. These include an aerodynamic §ow with pressure induced separation and reattachment (the wall-mounted hump studied in the experiments [7] and used as a test case in many validation studies, e. g., [8]) and a hydrofoil trailing edge §ow with shallow separation investigated in the experiment [9] and used for validation of di¨erent hybrid RANSLES approaches in the EU project DESider [10].
One more validation test of the in §ow generation method has been done, namely, the §ow in a 3D di¨user studied in the experiments [11]. This §ow is di©cult to simulate by means of RANS turbulence models because of the presence of secondary §ows driven by normal Reynolds stresses anisotropy. Since synthetic turbulence is usually generated according to a RANS solution, this can signi¦cantly worsen the LES solution. The di¨user §ow using synthetic turbulence generated with kω shear stress transport (SST) [12] RANS and EARSMWJBSL (Explicit Algebraic Reynolds Stress Model WallinJohanson Baseline) [13] RANS ¦elds taken as the input has been simulated.
The rest of the paper is organized as follows: section 2 outlines the synthetic turbulence generation method, section 3 brie §y describes turbulence models and numerical methods used in the simulations, sections 46 present simulation results for the wall-mounted hump §ow, hydrofoil trailing edge §ow and 3D di¨user §ow, respectively, and, ¦nally, section 7 contains conclusions of the study.

SYNTHETIC TURBULENCE GENERATION METHOD
The method has been described in detail in a recent paper by the authors [3]. Below follows a brief outline of the method highlighting only its main features. The velocity ¦eld at the LES in §ow is de¦ned as a sum of steady RANS velocity ¦eld and synthetic ¦eld of velocity §uctuations multiplied by Cholesky decomposition of the Reynolds stress tensor: The velocity §uctuations ¦eld is prescribed in the form of weighted superposition of Fourier modes: √ q n σ n cos k n d n · r + ϕ n + s n t τ where wavenumbers k form geometric series, mode weights q are calculated using the local energy spectrum (see below), τ is the global time scale, σ, d, ϕ, and s are the random parameters: velocity direction of the mode, wave vector direction, phase, and time frequency (for details, see [3]).

TURBULENCE MODELING
The weights of the modes are de¦ned with the use of a modi¦ed von Karman spectrum: where f η and f cut damp the spectrum near wavenumber corresponding to the Kolmogorov length-scale and the maximum resolvable wavenumber on the grid, wavenumber k e corresponding to the size of the most energy-carrying eddies is de¦ned by the length scale l e . The length scale is de¦ned as follows: where C l = 3 is the empirical constant and l t is the length scale of the turbulence model used in RANS region (for kω model, ). The global time scale τ is de¦ned by the maximum value of the length scale and a macroscale of the velocity at the LES inlet: where C τ = 2 is the empirical constant. Such global de¦nition of the time scale coupled with the local scale of the most energy-carrying eddies (1) results in forming of physically realistic elongated in the streamwise direction eddies near the wall and nearly isotropic eddies away from the wall. The method has shown to produce quality in §ow turbulent content and ensure a rapid formation of realistic turbulent structures downstream of the in §ow for canonical turbulent shear §ows: plane channel §ow, boundary layer §ow, and mixing layer §ow [3]. It has been shown that for wall-bounded §ows, the synthetic turbulence needed relaxation region of about 2 boundary layer thickness lengths and did not worsen the wall friction signi¦cantly.

TURBULENCE MODELS AND NUMERICAL METHODS
For RANS simulations, kω SST model [12] has been used for all the §ows and EARSMWJBSL [13] model for the 3D di¨user §ows. For LES and hybrid RANSLES simulations, the Improved Delayed Detached Eddy Simulation (IDDES) [14] has been used. This model is solution-dependent and functions as wall-modeling LES model if the turbulent content is present in the solution and as a RANS model in attached boundary layer without resolved turbulent §uctuations. For all the simulations, NTS ¦nite-volume multiblock structured code [15] with overlapping grids capability has been used. The ability to use overlapping grids is crucial to simultaneous combined RANSLES simulation using synthetic turbulence at the RANSLES interface. The NTS code uses the method of Rogers and Kwak [16] for incompressible §ows. Convective §uxes are computed with the use of the 4th order central-di¨erencing scheme for LES and 3rd order upwind scheme for RANS. For di¨usive §uxes, the code uses 2nd order central di¨erencing scheme. Time integration is done using implicit 3-step 2nd order scheme with subiterations.

WALL-MOUNTED HUMP FLOW
The §ow over a 2D wall-mounted hump has been studied in the experiments [7] and widely used as a validation test for turbulence modeling approaches [8]. Scheme of the §ow is shown in Fig. 1.
The Reynolds number based on maximum inlet velocity and chord length is 936,000. Upper wall is slippery and slightly adjusted to account for partial blockage e¨ect as recommended in [8]. The computational domain extends from x/c = −2.14 to 4.0. Velocity and turbulence variables pro¦les at the inlet plane have been obtained in a separate RANS calculation of zero-pressure gradient boundary layer at the Reynolds number based on momentum thickness equal to Re θ = 7500. The computational grid in xy plane has dimensions 375×111 and is nearly isotropic in the separation zone with -x/c ≈ -y/c ≈ 5 · 10 −3 . For hybrid and zonal RANSLES simulations, the grid has 101 points with equal spacing -z/c = 4 · 10 −3 in z direction amounting to spanwise width of L z /c = 0.4. Periodic boundary conditions have been used in z direction. Three types of simulation have been done for this §ow: 2D RANS using kω SST model, hybrid RANSLES using IDDES method in the whole domain, and zonal RANSLES using synthetic turbulence at the interface. For zonal RANSLES simulation, the LES inlet plane was at near top of the hump. The RANS outlet was located somewhat farther downstream (20 grid points) to avoid contamination of RANS solution with resolved turbulent §uctuations (Fig. 2). Synthetic velocity ¦eld was prescribed at LES inlet, while at RANS outlet, the velocity and pressure were taken directly from the LES domain.
Some results of the simulations are shown in Figs. 3 and 4. Isosurfaces of λ 2 criterion showing resolved turbulent §uctuations are presented in Fig. 3. It can be seen from this ¦gure that when IDDES is used in the whole domain, the separated boundary layer contains only unphysical large almost 2D vortices in the vicinity of separation point. This is typical for hybrid RANSLES methods when the boundary layer does not contain resolved turbulent content before separation point. Zonal RANSIDDES simulation is free from this drawback. Such a di¨erence in structure of resolved turbulent §uctuations ¦elds manifests itself also in di¨erent prediction of wall friction in the separation zone shown in Fig. 4. Zonal RANSIDDES simulation provides correct level of wall friction in the whole separation zone while for IDDES in the whole domain, it is noticeably overpredicted at 0.8 < x/c < 1.0. The RANS simulation using kω SST model severely overpredicts the length of the separation zone.

HYDROFOIL TRAILING EDGE FLOW
The scheme of the trailing edge §ow is shown in Fig. 5. The Reynolds number based on hydrofoil thickness h and freestream velocity U ∞ is equal to Re h = 10 5 according to the experiments [9].
Layout of the zonal RANSLES simulation is shown in Fig. 5. The LES zone covers only the trailing edge and near wake, the rest is simulated by RANS using  In the LES zone near the trailing edge, the grid is close to isotropic with spacing -x/h ≈ -y/h = 0.02. In z direction, the grid has 101 points evenly spaced by -z/h = 0.01, so that spanwise width is L z = h.
Freestream conditions u = U ∞ , v = 0 have been used at the inlet boundary which is located at x/h = −50. Constant pressure boundary conditions have been used at the outlet boundary at x/h = 20. In the z direction, periodic boundary conditions have been used.
Some results of the zonal RANSLES simulation are shown in Figs. 6 and 7. Figure 6 shows instant ¦elds of vorticity magnitude in the xy plane demonstrating resolved turbulent content in the LES zone. Comparison of streamwise velocity pro¦les at selected locations (see Fig. 7) shows excellent agreement both with resolved LES using recycling methods [17] and experimental data [9]. The pro¦les of root mean squared (rms) streamwise velocity §uctuations also show good agreement with resolved LES simulation using turbulence recycling.

THREE-DIMENSIONAL DIFFUSER FLOW
Separated §ow in a 3D rectangular di¨user has been studied in the experiments [11]. It was shown that the separation zone is strongly sensitive to geometric characteristics of the di¨user. This §ow presents a challenge for RANS modeling approaches, RANS simulations of this §ow have generally produced nonsatisfactory results [18]. The LES and hybrid RANSLES studies were more successful in predicting §ow behavior for this case [19]. However, when using synthetic turbulence, one usually obtains velocity and Reynolds stresses used to generate the synthetic velocity ¦eld from the RANS solution. Thus, unphysical velocity and stresses ¦elds at the in §ow can signi¦cantly worsen the results of the LES solution in the whole domain. To estimate the e¨ect of inlet averaged velocity and Reynolds stresses on the LES solution, the simulations of the di¨user §ow have been done with synthetic turbulence generated using RANS solution produced by kω SST model and EARSMWJBSL model. Also, a LES run using recycling in §ow generation method has been done.
The schematic of the di¨user is shown in Fig. 8. The Reynolds number based on bulk velocity U b in the inlet channel and height of the inlet channel H is equal to Re = 10 4 . Flow in the inlet channel is assumed to be developed. Five simulation runs have been done for the §ow: RANS simulations using kω SST  At the outlet boundary, constant pressure conditions have been used in RANS simulations. For LES simulation, a sponge zone with length L/H = 10 has been used where the velocity and pressure ¦elds were smoothly blended with RANS solution using cubic blending function. This was done to damp strong pressure waves re §ecting from the outlet variable in unsteady simulation.
The comparison of the simulation results with experimental data [11] is shown in Figs. 9 and 10. The LES simulation using turbulence recycling produced results in excellent agreement with experimental data both for pressure distribution on lower wall (see Fig. 9) and averaged velocity ¦elds (see Fig. 10). Thus, it is shown that LES using this grid and model produces good results for this §ow.
The RANS results with kω SST model were in complete disagreement with experimental data while EARSMWJBSL model predicted the pressure on the lower wall much better but still with di¨ering signi¦cantly from the experimental results.
The results of LES runs with synthetic turbulence at the inlet boundary depend signi¦cantly on the averaged velocity and Reynolds stresses ¦elds used to generate synthetic turbulent §uctuations signi¦cantly worsened the predictions of pressure and velocity ¦elds.

CONCLUDING REMARKS
A recently developed method to generate synthetic turbulent velocity §uctuations has been applied to zonal RANSLES simulations of complex turbulent §ows including pressure-driven separation with downstream reattachment of the boundary layer and to a simulation of a complex 3D turbulent §ow with secondary corner §ows. It has been shown that zonal RANSLES approach to simulation of turbulent §ows provides the results in excellent agreements with experimental data for shallow the separation §ows. Arti¦cial turbulent content at the RANSLES interface greatly improves prediction of the mean §ow in the separation zone (compared to one of the most advanced existing hybrid RANS LES methods ¡ IDDES) without signi¦cant degradation of the solution near the RANSLES interface. For a turbulent §ow in a 3D di¨user, it has been shown that the results depend strongly on the RANS solution used to create synthetic turbulent content at the in §ow. When the in §ow synthetic turbulence was created using the ¦elds of velocity and turbulence variables produced with Reynolds-stress model EARSMWJBSL [13] taken as the input, the results of the LES simulation agreed well with experimental data and with LES using recycling method. Noticeably worse results were obtained when using ¦elds produced with linear eddy-viscosity model.