Saturday, December 15, 2018

Not All Numerical Methods are Born Equal for LES

Large eddy simulations (LES) are notoriously expensive for high Reynolds number problems because of the disparate length and time scales in the turbulent flow. Recent high-order CFD workshops have demonstrated the accuracy/efficiency advantage of high-order methods for LES.

The ideal numerical method for implicit LES (with no sub-grid scale models) should have very low dissipation AND dispersion errors over the resolvable range of wave numbers, but dissipative for non-resolvable high wave numbers. In this way, the simulation will resolve a wide turbulent spectrum, while damping out the non-resolvable small eddies to prevent energy pile-up, which can drive the simulation divergent.

We want to emphasize the equal importance of both numerical dissipation and dispersion, which can be generated from both the space and time discretizations. It is well-known that standard central finite difference (FD) schemes and energy-preserving schemes have no numerical dissipation in space. However, numerical dissipation can still be introduced by time integration, e.g., explicit Runge-Kutta schemes.     

We recently analysed and compared several 6th-order spatial schemes for LES: the standard central FD, the upwind-biased FD, the filtered compact difference (FCD), and the discontinuous Galerkin (DG) schemes, with the same time integration approach (an Runge-Kutta scheme) and the same time step.  The FCD schemes have an 8th order filter with two different filtering coefficients, 0.49 (weak) and 0.40 (strong). We first show the results for the linear wave equation with 36 degrees-of-freedom (DOFs) in Figure 1.  The initial condition is a Gaussian-profile and a periodic boundary condition was used. The profile traversed the domain 200 times to highlight the difference.

Figure 1. Comparison of the Gaussian profiles for the DG, FD, and CD schemes

Note that the DG scheme gave the best performance, followed closely by the two FCD schemes, then the upwind-biased FD scheme, and finally the central FD scheme. The large dispersion error from the central FD scheme caused it to miss the peak, and also generate large errors elsewhere.

Finally simulation results with the viscous Burgers' equation are shown in Figure 2, which compares the energy spectrum computed with various schemes against that of the direct numerical simulation (DNS). 

Figure 2. Comparison of the energy spectrum

Note again that the worst performance is delivered by the central FD scheme with a significant high-wave number energy pile-up. Although the FCD scheme with the weak filter resolved the widest spectrum, the pile-up at high-wave numbers may cause robustness issues. Therefore, the best performers are the DG scheme and the FCD scheme with the strong filter. It is obvious that the upwind-biased FD scheme out-performed the central FD scheme since it resolved the same range of wave numbers without the energy pile-up. 


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  2. Dear ZJ,
    Interesting article I enjoyed reading It!. It is certainly not unexpected when it comes to comparing the properties of these Discretization schemes (dissipation and dispersion errors) and their applicability for MILES. It is also a known fact that upwind schemes’ numerical dissipation (which is what drains the energy at the small scales, acting as a de-facto Sub-grid Scale model) is anisotropic. This is certainly not how physical dissipation acts at those small scales. Thus, I wonder if the conclusion (of using high order upwind schemes, like DG, for LES) would still stand if there would be an interest of actually providing an SGS model, and/or to better match the turbulence statistics, and especially high order moments of turbulence ... ? That is, I think, the numerical and physical (modeled) dissipation at those small unresolved scales must work together and be just enough to give us the right simulated turbulence spectrum..

    1. Dear Doru,
      Thanks for the thought-provoking message! I should have said “The ideal numerical method for implicit LES...”, in which no explicit SGS model is used. The blog has been revised accordingly. I agree with your point in that if there is a physics-based SGS model, high wave number dissipation may not be needed. The well-known Smagorinsky-type models do not correlate with physical SGS, and essentially provide numerical dissipation. That is perhaps why I am a little pessimistic about the prospect of developing physics-based SGS models. I totally agree with your last sentence!

  3. Interesting topic! Thanks for sharing the main ideas from the article here. Regarding the separate roles of numerical versus model dissipations on the outcomes of the LES, I would like to draw your attention to our recent work on the 3-level variational multiscale approaches for FR schemes. This formulation creates the ability of applying the SGS model to an arbitrary range of small resolved scales, thus covering all levels from ILES to classical LES. This would give one the ability to have a better control on the total amount of dissipation, as (lack of) evidence suggests that the high-order ILES might not provide sufficient dissipation for very coarse/high-Re industrial applications.
    A High-Order Variational Multiscale Approach to Turbulence for Compact Nodal Schemes.

  4. Hi Farshad,
    Such multi-scale approaches seem to have the right ingredients we are looking for, for industrial LES: the HO upwind scheme acting on the large resolved, energy containing, scales of the turbulence (upwind providing the necessary numerical stabilization for convection dominated regimes) , while the sub-grid modeling (SGS model) is based on the more isotropic, smaller (resolved, if good LES resolution) scales of turbulence. For large Re number (like most industrial LES in say Aeronautics) achieving that necessary level of resolution for the smaller scales is going to be really hard though ...