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Large Eddy Simulation Requirements for the Flow over Periodic Hills

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Abstract

Large eddy simulations are carried out for flows in a channel with streamwise-periodic constrictions, a well-documented benchmark case to study turbulent flow separation from a curved surface. Resolution criteria such as wall units are restricted to attached flows and enhanced criteria, such as energy spectra or two-point correlations, are used to evaluate the effective scale separation in the present large eddy simulations. A detailed analysis of the separation above the hill crest and of the early shear layer development shows that the delicate flow details in this region may be hardly resolved on coarse grids already at Re = 10595, possibly leading to a non monotonic convergence with mesh refinement. The intricate coupling between numerical and modeling errors is studied by means of various discretization schemes and subgrid models. It is shown that numerical schemes maximizing the resolution capabilities are a key ingredient for obtaining high-quality solutions while using a reduced number of grid points. On this respect, the introduction of a sharp enough filter is an essential condition for separating accurately the resolved scales from the subfilter scales and for removing ill-resolved structures. The high-resolution approach is seen to provide solutions in very good overall agreement with the available experimental data for a range of Reynolds numbers (up to 37000) without need for significant grid refinement.

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Acknowledgments

This work was granted access to the HPC resources of IDRIS and CCRT under the allocation 1736 made by GENCI (Grand Equipement National de Calcul Intensif).

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    Xavier Gloerfelt & Paola Cinnella

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Appendix: Influence of the Spatial Discretization Resolvability on CPU Costs

Appendix: Influence of the Spatial Discretization Resolvability on CPU Costs

In this Appendix we report selected numerical results illustrating the role of the spatial discretization scheme for the convective terms for achieving a given LES quality/computational cost compromise. Specifically, results are presented for the 2D-hill flow at Re = 10595 using various FD schemes defined by Eq. 2. As in Section 6, we focus on the coarsest grid 64 × 32 × 32.

Two families of central finite-difference schemes are tested, namely the standard versions (second- to tenth-order scheme, denoted FDo2 to FDo10), and the optimized DRP11 scheme. Numerical stabilization and RT modeling are achieved by applying the baseline filter, i.e. the DRP11 filter, in all cases. Profiles of 〈u〉, 〈v〉 and 〈v′v′〉 at x/h = 3 are reported in Fig. 26. Standard finite-difference schemes of various orders provide closeby results, except for the 2nd-order scheme, which introduces significant phase errors on the considered grid. This suggests that, beyond 2nd-order of accuracy, the quality of the solution becomes weakly dependent of the consistent part of the spatial scheme. However, for the 11-point schemes, and more specifically the optimized DRP11, the results tend to be more accurate.

Fig. 26
figure 26

Influence of finite-difference schemes supplemented by the DRP11 filter at Re = 10595: Standard FDo2 , FDo4 , FDo6 , FDo8 , FDo10 and FD DRP11 . Profiles of 〈u〉 a; 〈v〉 b and \(\langle v^{\prime }v^{\prime }\rangle \)c at x/h = 3

Full size image

The preceding results are in line with theoretical studies based on Fourier analysis, showing that the accuracy limit, i.e. the number of grid points required to represent a given wavelength with a prescribed error level, decreases significantly when increasing the order of accuracy (for standard schemes) or by optimizing the coefficients (for DRP schemes). The accuracy limits required by the present family of schemes is reported in Table 2. The error criterion corresponds to the upper value considered in [34]. Of course, the price to pay is an increased computational complexity due to the larger and larger stencil in use, leading to a higher computational complexity. This criterion, also reported in Table 2, tends to saturate and even slightly increases for orders higher than 6. Note that the normalized resolvability (i.e. the product of the number of points per wavelength by the number of points in the stencil) remains always better for the DRP11 scheme. However, the computational complexity of the FD scheme is an over-simplistic cost estimate, not well suited to characterize the overall CPU cost associated with a given scheme. Indeed, the discretization of the convective terms is only one of the many ingredients involved in the flow solver. This is why, in Table 2, we report the effective computational costs of 2D-hill simulations based on various FD schemes and the DRP11 filter (normalized with respect to the CPU cost of the baseline DRP11 scheme), along with a cost criterion corresponding to the CPU cost per iteration and per point multiplied by the accuracy limit. The table shows that: i) the brute CPU cost is weakly dependent on the chosen FD scheme; ii) the cost criterion is much lower for the high-resolution schemes, due to the considerably reduction in terms of grid points.

Table 2 From top to bottom: accuracy limits (points per wavelength λ/Δx ) of the standard and optimized finite differences (FD) to achieve a dispersion error of 5 × 10− 4; accuracy limits multiplied by the number of points p = 2N + 1 of the stencil; CPU costs (Re = 10595; grid 64 × 33 × 32, RT model) relative to the baseline scheme FDDRP11/SFDRP11; cost criterion corresponding to the CPU cost, multiplied by the minimal number of points per wavelength to achieve the prescribed dispersion error
Full size table

The preceding results were obtained by applied the DRP11 selective filter. In order to discuss the computational cost associate for the filter, in Table 3 we report the accuracy limits (points per wavelength required to achieve a dissipation error threshold of 5 × 10− 4/χ) of the various selective filters, along with the effective CPU times per iteration and per grid point relative to the baseline scheme. In this case, all the simulations are based on the DRP11 FD scheme for the convective term. A cost criterion corresponding to CPU multiplied by the filter accuracy limit is also reported. The results were obtained for the typical filter amplitude χ = 0.2. The table shows that the brute CPU cost is even less sensitive to the chosen filter (applied only one time per time step) than to the FD scheme, while the dissipation accuracy limits vary even more sharply when increasing the filter resolution than the dispersion error limits. As a consequence, the normalized CPU times decreases greatly when increasing the filter resolution, achieving a minimum in the case of the DRP11 filter.

Table 3 From top to bottom: accuracy limits (points per wavelength λ/Δx ) of the standard and optimized selective filters (SF) (χ = 0.2) to achieve a dissipation error of 5 × 10− 4/χ; CPU costs (Re = 10595; grid 64 × 33 × 32, RT model) relative to the baseline scheme FDDRP11/SFDRP11; cost criterion corresponding to the CPU cost, multiplied by the minimal number of points per wavelength to achieve the prescribed dissipation error
Full size table

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Gloerfelt, X., Cinnella, P. Large Eddy Simulation Requirements for the Flow over Periodic Hills. Flow Turbulence Combust 103, 55–91 (2019). https://doi.org/10.1007/s10494-018-0005-5

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