Fluid dynamic turbulence is one of the most challenging computational physics problems because of the extremely wide range of time and space scales involved, the strong nonlinearity of the governing equations, and the many practical and important applications. While most linear fluid instabilities are well understood, the nonlinear interactions among them makes even the relatively simple limit of homogeneous isotropic turbulence difficult to treat physically, mathematically, and computationally. Turbulence is modeled computationally by a two-stage bootstrap process. The first stage, direct numerical simulation, attempts to resolve the relevant physical time and space scales but its application is limited to diffusive flows with a relatively small Reynolds number (Re). Using direct numerical simulation to provide a database, in turn, allows calibration of phenomenological turbulence models for engineering applications. Large eddy simulation incorporates a form of turbulence modeling applicable when the large-scale flows of interest are intrinsically time dependent, thus throwing common statistical models into question. A promising approach to large eddy simulation involves the use of high-resolution monotone computational fluid dynamics algorithms such as flux-corrected transport or the piecewise parabolic method which have intrinsic subgrid turbulence models coupled naturally to the resolved scales in the computed flow. The physical considerations underlying and evidence supporting this monotone integrated large eddy simulation approach are discussed.

New insights into large eddy simulation

High resolution schemes for hyperbolic conservation laws

An adaptive central-upwind weighted essentially non-oscillatory scheme

An efficient class of WENO schemes with adaptive order

A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws

This paper studies the weights stability and accuracy of the implicit fifth-order weighted essentially nonoscillatory finite difference scheme. It is observed that the weights of the Jiang-Shu weighted essentially nonoscillatory scheme oscillate even for smooth flows. An increased e value of 10 -2 is suggested for the weighted essentially nonoscillatory smoothness factors, which removes the weights oscillation and significantly improves the accuracy of the weights and solution convergence. With the improved e value, the weights achieve the optimum value with minimum numerical dissipation in smooth regions and maintain the sensitivity to capture nonoscillatory shock profiles for the transonic flows. The theoretical justification of this treatment is given in the paper. The wall surface boundary condition uses a half-point mesh so that the conservative differencing can be enforced. A third-order accurate finite difference scheme is given to treat wall boundary conditions. The implicit time-marching method with unfactored Gauss-Seidel line relaxation is used with the high-order schemes to achieve a high convergence rate. Several transonic cases are calculated to demonstrate the robustness, efficiency, and accuracy of the methodology.

Improvement of Stability and Accuracy for Weighted Essentially Nonoscillatory Scheme

An improved low diffusion E-CUSP upwind scheme

High order conservative differencing for viscous terms and the application to vortex-induced vibration flows

A new smoothness indicator for improving the weighted essentially non-oscillatory scheme

This paper has analyzed the weights of the 5th order WENO (weighted essentially non-oscillatory) scheme suggested by Jiang and Shu and a modifled smoothness estimator is suggested to improve the accuracy. Several numerical tests are presented to demonstrate the accuracy and robustness of the new scheme.

Yiqing Shen

Papers

Improvement of Stability and Accuracy for Weighted Essentially Nonoscillatory Scheme

An improved low diffusion E-CUSP upwind scheme

High order conservative differencing for viscous terms and the application to vortex-induced vibration flows

A new smoothness indicator for improving the weighted essentially non-oscillatory scheme

Improvement of the WENO scheme smoothness estimator