Hanxin Zhang

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.

A new class of central compact schemes with spectral-like resolution I: Linear schemes☆

Abstract In this paper, we design a new class of central compact schemes based on the cell-centered compact schemes of Lele [S.K. Lele, Compact finite difference schemes with spectral-like resolution, Journal of Computational Physics 103 (1992) 16–42]. These schemes equate a weighted sum of the nodal derivatives of a smooth function to a weighted sum of the function on both the grid points (cell boundaries) and the cell-centers. In our approach, instead of using a compact interpolation to compute the values on cell-centers, the physical values on these half grid points are stored as independent variables and updated using the same scheme as the physical values on the grid points. This approach increases the memory requirement but not the computational costs. Through systematic Fourier analysis and numerical tests, we observe that the schemes have excellent properties of high order, high resolution and low dissipation. It is an ideal class of schemes for the simulation of multi-scale problems such as aeroacoustics and turbulence.

A new class of central compact schemes with spectral-like resolution II

In this paper, we develop a class of nonlinear compact schemes based on our previous linear central compact schemes with spectral-like resolution (X. Liu et al., 2013 20). In our approach, we compute the flux derivatives on the cell-nodes by the physical fluxes on the cell nodes and numerical fluxes on the cell centers. To acquire the numerical fluxes on the cell centers, we perform a weighted hybrid interpolation of an upwind interpolation and a central interpolation. Through systematic analysis and numerical tests, we show that our nonlinear compact scheme has high order, high resolution and low dissipation, and has the same ability to capture strong discontinuities as regular weighted essentially non-oscillatory (WENO) schemes. It is a good choice for the simulation of multiscale problems with shock waves.

Topological structure of shock induced vortex breakdown

Using a combination of critical point theory of ordinary differential equations and numerical simulation for the three-dimensional unsteady Navier–Stokes equations, we study possible flow structures of the vortical flow, especially the unsteady vortex breakdown in the interaction between a normal shock wave and a longitudinal vortex. The topological structure contains two parts. One is the sectional streamline pattern in the cross-section perpendicular to the vortex axis. The other is the sectional streamline pattern in the symmetrical plane. In the cross-section perpendicular to the vortex axis, the sectional streamlines have spiral or centre patterns depending on a function λ(x, t )=1 /ρ(∂ρ/∂t +∂ρu/∂x), where x is the coordinate corresponding to the vortex axis. If λ > 0, the sectional streamlines spiral inwards in the near region of the centre. If λ < 0, the sectional streamlines spiral outwards in the same region. If λ =0 , the sectional streamlines form a nonlinear centre. If λ changes its sign along the vortex axis, one or more limit cycles appear in the sectional streamlines in the cross-section perpendicular to the vortex axis. Numerical simulation for two typical cases of shock induced vortex breakdown (Erlebacher, Hussaini & Shu, J. Fluid Mech., vol. 337, 1997, p. 129) is performed. The onset and time evolution of the vortex breakdown are studied. It is found that there are more limit cycles for the sectional streamlines in the cross-section perpendicular to the vortex axis. In addition, we find that there are quadru-helix structures in the tail of the vortex breakdown.

Classification and sound generation of two-dimensional interaction of two Taylor vortices

Two-dimensional interaction between two Taylor vortices is simulated systematically through solving the two-dimensional, unsteady compressible Navier-Stokes equations using a fifth order weighted essentially nonoscillatory finite difference scheme. The main purpose of this study is to reveal the mechanism of sound generation in two-dimensional interaction of two Taylor vortices. Based on an extensive parameter study on the evolution of the vorticity field, we classify the interaction of two Taylor vortices into four types. The first type is the interaction of two counter-rotating vortices with similar strengths. The second type is the interaction of two co-rotating vortices without merging. The third type is the merging of two co-rotating vortices. The fourth type is the interaction of two vortices with a large difference in their strengths or scales. The mechanism of sound generation is analyzed.

High Order and High Resolution Numerical Schemes for Computational Aeroacoustics and Their Applications

A class of high order central compact schemes with spectral-like resolution are designed for the computational aeroacousitcs (CAA). The schemes have the features of high order, high resolution, low dissipation and the ability to capture strong shock wave in flow field, which are perfect methods for computational aeroacoustics. Typical problems are solved through direct numerical simulation, including the merging process of two co-rotating Gaussian vortices, the interaction between an oblique shock wave and a shear layer and cavity flow. The mechanisms of noise generation are studied.