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. 

Monday, January 1, 2018

Are High-Order CFD Solvers Ready for Industrial LES?

The potential of high-order methods (order > 2nd) is higher accuracy at lower cost than low order methods (1st or 2nd order). This potential has been conclusively demonstrated for benchmark scale-resolving simulations (such as large eddy simulation, or LES) by multiple international workshops on high-order CFD methods.

For industrial LES, in addition to accuracy and efficiency, there are several other important factors to consider:

  • Ability to handle complex geometries, and ease of mesh generation
  • Robustness for a wide variety of flow problems
  • Scalability on supercomputers
For general-purpose industry applications, methods capable of handling unstructured meshes are preferred because of the ease in mesh generation, and load balancing on parallel architectures. DG and related methods such as SD and FR/CPR have received much attention because of their geometric flexibility and scalability. They have matured to become quite robust for a wide range of applications. 

Our own research effort has led to the development of a high-order solver based on the FR/CPR method called hpMusic. We recently performed a benchmark LES comparison between hpMusic and a leading commercial solver, on the same family of hybrid meshes at a transonic condition with a Reynolds number more than 1M. The 3rd order hpMusic simulation has 9.6M degrees of freedom (DOFs), and costs about 1/3 the CPU time of the 2nd order simulation, which has 28.7M DOFs, using the commercial solver. Furthermore, the 3rd order simulation is much more accurate as shown in Figure 1. It is estimated that hpMusic would be an order magnitude faster to achieve a similar accuracy. This study will be presented at AIAA's SciTech 2018 conference next week.

(a) hpMusic 3rd Order, 9.6M DOFs
(b) Commercial Solver, 2nd Order, 28.7M DOFs
Figure 1. Comparison of Q-criterion and Schlieren  

I certainly believe high-order solvers are ready for industrial LES. In fact, the commercial version of our high-order solver, hoMusic (pronounced hi-o-music), is announced by hoCFD LLC (disclaimer: I am the company founder). Give it a try for your problems, and you may be surprised. Academic and trial uses are completely free. Just visit to download the solver. A GUI has been developed to simplify problem setup. Your thoughts and comments are highly welcome.

Happy 2018!