Shuhai Zhang

Development of nonlinear weighted compact schemes with increasingly higher order accuracy

In this paper, we design a class of high order accurate nonlinear weighted compact schemes that are higher order extensions of the nonlinear weighted compact schemes proposed by Deng and Zhang X. Deng, H. Zhang, Developing high-order weighted compact nonlinear schemes, J. Comput. Phys. 165 (2000) 22-44] and the weighted essentially non-oscillatory schemes of Jiang and Shu G.-S. Jiang, C.-W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996) 202-228] and Balsara and Shu D.S. Balsara, C.-W. Shu, Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys. 160 (2000) 405-452]. These nonlinear weighted compact schemes are proposed based on the cell-centered compact scheme of Lele S.K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys. 103 (1992) 16-42]. Instead of performing the nonlinear interpolation on the conservative variables as in Deng and Zhang (2000), we propose to directly interpolate the flux on its stencil. Using the Lax-Friedrichs flux splitting and characteristic-wise projection, the resulted interpolation formulae are similar to those of the regular WENO schemes. Hence, the detailed analysis and even many pieces of the code can be directly copied from those of the regular WENO schemes. Through systematic test and comparison with the regular WENO schemes, we observe that the nonlinear weighted compact schemes have the same ability to capture strong discontinuities, while the resolution of short waves is improved and numerical dissipation is reduced.

A New Smoothness Indicator for the WENO Schemes and Its Effect on the Convergence to Steady State Solutions

The convergence to steady state solutions of the Euler equations for the fifth-order weighted essentially non-oscillatory (WENO) finite difference scheme with the Lax–Friedrichs flux splitting [7, (1996) J. Comput. Phys. 126, 202–228.] is studied through systematic numerical tests. Numerical evidence indicates that this type of WENO scheme suffers from slight post-shock oscillations. Even though these oscillations are small in magnitude and do not affect the “essentially non-oscillatory” property of WENO schemes, they are indeed responsible for the numerical residue to hang at the truncation error level of the scheme instead of settling down to machine zero. We propose a new smoothness indicator for the WENO schemes in steady state calculations, which performs better near the steady shock region than the original smoothness indicator in [7, (1996) J. Comput. Phys. 126, 202–228.]. With our new smoothness indicator, the slight post-shock oscillations are either removed or significantly reduced and convergence is improved significantly. Numerical experiments show that the residue for the WENO scheme with this new smoothness indicator can converge to machine zero for one and two dimensional (2D) steady problems with strong shock waves when there are no shocks passing through the domain boundaries.

Multistage interaction of a shock wave and a strong vortex

The interaction between a shock wave and a strong vortex is simulated systematically through solving the two-dimensional, unsteady compressible Navier–Stokes equations using a fifth-order weighted essentially nonoscillatory finite difference scheme. Our main purpose in this study is to characterize the flow structure and the generation of sound waves of the shock–strong vortex interaction. The simulations show that the interaction of a shock wave and a strong vortex has a multistage feature. It contains the interaction of the shock wave and the initial vortex, of the reflected shock wave and the deformed vortex and of the shocklets and the deformed vortex. The shocklets are generated by the secondary interaction. Due to the complex reflected shock structure, there exist interactions between the reflected shock waves and the sound waves. Many pressure waves are embedded in the second and third sound waves.

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.

The mechanism of sound generation in the interaction between a shock wave and two counter-rotating vortices

The interaction between a shock wave and two counter-rotating 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 the interaction between a shock wave and two counter-rotating vortices. It is found that there are two regimes of sound generation in this interaction. The first regime corresponds to the shock interaction with two isolated vortices, in which the sound wave generated by the interaction between the shock wave and two counter-rotating vortices equals to the linear combination of the sound waves generated by the interactions between the same shock wave and each vortex. The second regime corresponds to the shock interaction with a coupled vortex pair, in which the sound wave comes from two processes. One is the vortex coupling, and the second is the interaction between the shock wave and the coupled vortex pair.

Interaction of an oblique shock wave with a pair of parallel vortices: Shock dynamics and mechanism of sound generation

The interaction between an oblique shock wave and a pair of parallel 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 characterize the flow structure and the mechanism of sound generation in the interaction between an oblique shock wave and a pair of vortices. We study two typical shock waves of Mach number Ms=1.2 and Ms=1.05, which correspond to two typical shock structures of Mach reflection and regular reflection, respectively, in the problem of shock-vortex interaction. The effects of the strength of the vortices and the geometry parameters are investigated. In addition, we have also considered both cases of passing and colliding vortex pairs. The interaction is classified into four types for the passing case and seven types for the colliding case according to different patterns of the shock structure. Our simulati...

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.