All Computational Fluid Dynamics codes are dependent on the types of numerical schemes used to translate the continuous derivatives of the Navier-Stokes equations into a finite form appropriate for computational grids. Over time, a wide variety of methods have been developed, each balancing the competing needs of accuracy, stability, and computational speed. Generally speaking. low-order schemes use fewer data points (either in time or space), obtaining better speed but potentially lower accuracy than high-order schemes which use more data points. Other considerations include the choice of explicit or implicit time integration and the amount of artificial or numerical diffusion to include in the computation.
To this end, most Direct Numerical Simulation/Large Eddy Simulation (DNS/LES) codes rely on low-order, explicit numerical schemes that require regular corrections in order to avoid computational failure. This approach relies on the marginal stability to continually "pump" the small-scale turbulence motions against the smoothing effects of numerical diffusion, sacrificing accuracy and efficient time-integration. At UK CFD, we have taken the opposite approach by employing advanced, high-order, implicity numerical schemes that increase both the accuracy and the speed of the time integration and reduce the number of points needed for a grid-independent solution. This makes the code more amenable to actual engineering problems, but at the price of requiring schemes with minimal numerical diffusion since the small-scale motions are not artificially boosted by numerical instabilities. However, the current high-order schemes were developed for steady-state RANS applications, and therefore exhibit undesirable characteristics in complex, time-dependent simulations: upwind-type schemes produce significant numerical dissipation errors, while central-difference schemes are non-dissipative but produce non-physical oscillations in the region of flow discontinuities such as shocks.
To meet this challenge, the UK CFD group has been investigating numerics that combine the minimal diffusion of central methods and the non-oscillatory discontinuities of upwind methods. A promising new combined numerical approach is the ENO-Pade scheme. Pade schemes are a family of high-order, compact, implicit central-difference schemes that are non-dissipative and have been proven efficient and accruate in numerous flows, with the exception of discontinuities. On the other hand, ENO schemes maintain a high order of accuracy and non-oscillatory behavior through discontinuities, but exhibit large numerical diffusion in areas with steep gradients. The combination of these two approaches eliminates the non-physical oscillatory behavior of the Pade scheme across discontinuities (such as shocks) while improving the performance of the ENO technique with steeper-gradient regions. The ENO-Pade scheme has been tested against other high-order schemes in several basic flow cases, including the rotation of a cut-out cylinder and a shock-reflection compressible flow. So far, ENO-Pade has yielded excellent results with a minimum of numerical diffusion, and is currerntly being tested in LESTool.
Relevant Publications
Z. Wang & P. G. Huang, "An Essentially Non-Oscillatory High-Resolution Pade-Type (ENO-Pade) Scheme", AIAA-2000-0918,
Jan.10-13, 2000, Reno, NV. (8.4 MB pdf)