Guohua Tu

Geometric conservation law and applications to high-order finite difference schemes with stationary grids

The geometric conservation law (GCL) includes the volume conservation law (VCL) and the surface conservation law (SCL). Though the VCL is widely discussed for time-depending grids, in the cases of stationary grids the SCL also works as a very important role for high-order accurate numerical simulations. The SCL is usually not satisfied on discretized grid meshes because of discretization errors, and the violation of the SCL can lead to numerical instabilities especially when high-order schemes are applied. In order to fulfill the SCL in high-order finite difference schemes, a conservative metric method (CMM) is presented. This method is achieved by computing grid metric derivatives through a conservative form with the same scheme applied for fluxes. The CMM is proven to be a sufficient condition for the SCL, and can ensure the SCL for interior schemes as well as boundary and near boundary schemes. Though the first-level difference operators ?3 have no effects on the SCL, no extra errors can be introduced as ?3=?2. The generally used high-order finite difference schemes are categorized as central schemes (CS) and upwind schemes (UPW) based on the difference operator ?1 which are used to solve the governing equations. The CMM can be applied to CS and is difficult to be satisfied by UPW. Thus, it is critical to select the difference operator ?1 to reduce the SCL-related errors. Numerical tests based on WCNS-E-5 show that the SCL plays a very important role in ensuring free-stream conservation, suppressing numerical oscillations, and enhancing the robustness of the high-order scheme in complex grids.

Extending Weighted Compact Nonlinear Schemes to Complex Grids with Characteristic-Based Interface Conditions

There are still some challenges, such as grid quality, numerical stability, and boundary schemes, in the practical application of high-order finite difference schemes for complex configurations. This study presents some improved strategies that indicate potential engineering applications of high-order schemes. The formally fifth-order weighted compact nonlinear scheme developed by the authors is implemented on point-matched multiblock structured grids, which are generated over complex configurations to ensure the grid quality of each component block. The information transmission between neighboring blocks is carried out by new characteristic-based interface conditions that directly exchange the spatial derivatives on each side of an interface by means of a characteristic-based projection to keep the high-order accuracy and high resolution of a spatial difference scheme. The high-order scheme combined with the interface conditions is shown to be asymptotically stable. The engineering-oriented applications of the high-order strategy are demonstrated by solving several two- and three-dimensional problems with complex grid systems.

Evaluation of Euler fluxes by a high-order CFD scheme: shock instability

The construction of Euler fluxes is an important step in shock-capturing/upwind schemes. It is well known that unsuitable fluxes are responsible for many shock anomalies, such as the carbuncle phenomenon. Three kinds of flux vector splittings (FVSs) as well as three kinds of flux difference splittings (FDSs) are evaluated for the shock instability by a fifth-order weighted compact nonlinear scheme. The three FVSs are Steger–Warming splitting, van Leer splitting and kinetic flux vector splitting (KFVS). The three FDSs are Roes splitting, advection upstream splitting method (AUSM) type splitting and Harten–Lax–van Leer (HLL) type splitting. Numerical results indicate that FVSs and high dissipative FDSs undergo a relative lower risk on the shock instability than that of low dissipative FDSs. However, none of the fluxes evaluated in the present study can entirely avoid the shock instability. Generally, the shock instability may be caused by any of the following factors: low dissipation, high Mach number, unsuitable grid distribution, large grid aspect ratio, and the relative shock-internal flow state (or position) between upstream and downstream shock waves. It comes out that the most important factor is the relative shock-internal state. If the shock-internal state is closer to the downstream state, the computation is at higher susceptibility to the shock instability. Wall-normal grid distribution has a greater influence on the shock instability than wall-azimuthal grid distribution because wall-normal grids directly impact on the shock-internal position. High shock intensity poses a high risk on the shock instability, but its influence is not as much as the shock-internal state. Large grid aspect ratio is also a source of the shock instability. Some results of a second-order scheme and a first-order scheme are also given. The comparison between the high-order scheme and the two low-order schemes indicates that high-order schemes are at a higher risk of the shock instability. Adding an entropy fix is very helpful in suppressing the shock instability for the two low-order schemes. When the high-order scheme is used, the entropy fix still works well for Roes flux, but its effect on the Steger–Warming flux is trivial and not much clear.

Applications of High-Order Weighted Compact Nonlinear Scheme for Complex Transonic Flows

The results of the four drag prediction workshops (DPW) of AIAA indicate that the most current low-order CFD methods trend to given poor prediction for pitching moment of transonic aircraft configurations. The high-order weighted compact nonlinear scheme (WCNS) and algorithms for viscous terms as well as grid metrics are packaged together to establish a high-order code for complex geometries. The numerical results of the RAE2822 airfoil and the DLR-F6 wing-body configuration indicate that the accuracy for shock position and pitching moment may be improved by high-order methods. The American Institute of Aeronautics and Astronautics (AIAA) Applied Aerodynamics Committee have sponsored four drag prediction workshops (DPW) with the aim of assessing the state-of-the-art computational methods as practical aerodynamic tools for aerodynamic force and moment prediction of transonic aircraft configurations (http://aaac.larc.nasa.gov/tsab/cfdlarc/aiaa-dpw). Transonic flow occurs when there is mixed sub- and supersonic local flow in the same flowfield (typically with freestream Mach numbers from Ma = 0.6 or 0.7 to 1.2). The transonic flow regime provides the most efficient aircraft cruise performance, hence, most large commercial aircraft cruise in this regime [20]. However, transonic flow fields tend to be sensitive to small perturbations in flow conditions or to slight changes in geometrical characteristics, either of which can cause large variations in the flow field [20]. This complicates both computations and wind tunnel testing. As noted by Hafez [21], transonic aerodynamics is a rich field full of challenging problems for CFD, for example, the discretization process introduces errors, without controlling these errors, the results can be misleading. The four DPW show that current CFD codes may produce successful aerodynamic prediction for the transonic configurations. However, the results are somewhat scattered, especially for shock location, separation bubble, and drag as well as pitching moment. As the numerical methods of the four DPW are mainly low-order ones (less than 3th-order), the present invested the performance of high-order WCNS for complex transonic flows.