High-order non-compact and compact gas-kinetic schemes

A compact fourth-order gas-kinetic scheme for the Euler and Navier–Stokes equations

Implicit high-order gas kinetic scheme for turbulence simulation

A compact and efficient high-order gas-kinetic scheme

A compact high-order gas-kinetic scheme on unstructured mesh for acoustic and shock wave computations

High-order gas-kinetic scheme with parallel computation for direct numerical simulation of turbulent flows

ABSTRACTFor the one-stage third-order gas-kinetic scheme (GKS), successful applications have been achieved for the three-dimensional compressible flows [Pan, L., K. Xu, Q. Li, and J. Li. 2016. “An Efficient and Accurate Two-stage Fourth-order Gas-kinetic Scheme for the Navier-Stokes Equations.” Journal of Computational Physics 326: 197–221]. The high-order accuracy of the scheme is obtained by integrating a multidimensional time-accurate gas distribution function over the cell interface within a time step without using Gaussian quadrature points and Runge-Kutta time-stepping technique. However, to the further increase of the order of the scheme, such as the fourth-order one, the one step formulation becomes very complicated for the multidimensional flow. Recently, a two-stage fourth-order GKS with high efficiency has been constructed for two-dimensional inviscid and viscous flow computations ([Li, J., and Z. Du. 2016. “A Two-stage Fourth Order Time-accurate Discretization for Lax-Wendroff Type Flow Solvers I. Hyperbolic Conservation Laws.” SIAM Journal on Scientific Computing 38: 3046–3069]; Pan et al. 2016), and the scheme uses the time accurate flux function and its time derivatives. In this paper, a fourth-order GKS is developed for the three-dimensional flows under the two-stage framework. Based on the three-dimensional WENO reconstruction and flux evaluation at Gaussian quadrature points on a cell interface, the high-order accuracy in space is achieved first. Then, the two-stage time stepping method provides the high accuracy in time. In comparison with the formal third-order GKS [Pan, L., and K. Xu. 2015. “A Third-order Gas-kinetic Scheme for Three-dimensional Inviscid and Viscous Flow Computations.” Computers & Fluids 119: 250–260], the current fourth-order method not only improves the accuracy of the scheme, but also reduces the complexity of the gas-kinetic flux solver greatly. More importantly, the fourth-order GKS has the same robustness as the second-order shock capturing scheme [Xu, K. 2001. “A Gas-kinetic BGK Scheme for the Navier-Stokes Equations and its Connection with Artificial Dissipation and Godunov Method.” Journal of Computational Physics 171: 289–335]. Numerical results validate the outstanding reliability and applicability of the scheme for three-dimensional flows, such as the cases related to turbulent simulations.

Two-step multi-resolution reconstruction-based compact gas-kinetic scheme on tetrahedral mesh

A general framework for the development of high-order compact schemes has been proposed recently. The core steps of the schemes are composed of the following. 1). Based on a kinetic model equation, from a generalized initial distribution of flow variables construct a time-accurate evolution solution of gas distribution function at a cell interface and obtain the corresponding flux function; 2). Introduce the WENO-type weighting functions into the high-order time-derivative of the cell interface flux function in the multistage multi-derivative (MSMD) time stepping scheme to cope with the possible impingement of a shock wave on a cell interface within a time step, and update the cell-averaged conservative flow variables inside each control volume; 3). Model the time evolution of the gas distribution function on both sides of a cell interface separately, take moments of the inner cell interface gas distribution function to get flow variables, and update the cell-averaged gradients of flow variables inside each control volume; 4). Based on the cell-averaged flow variables and their gradients, develop compact initial data reconstruction to get initial condition of flow distributions at the beginning of next time step. A compact gas-kinetic scheme (GKS) up to sixth-order accuracy in space and fourth-order in time has been constructed on 2D unstructured mesh. In this paper, the compact GKS up to fourth-order accuracy on three-dimensional tetrahedral mesh will be further constructed with the focus on the WENO-type initial compact data reconstruction. Nonlinear weights are designed to achieve high-order accuracy for the smooth Navier-Stokes solution and keep super robustness in 3D computation with strong shock interactions. The fourth-order compact GKS uses a large time step with a CFL number 0.6 in the simulations from subsonic to hypersonic flow. A series of test cases are used to validate the scheme. The high-order compact GKS can be used in 3D applications with complex geometry.

A third-order gas-kinetic scheme for three-dimensional inviscid and viscous flow computations

Comparison of the performance of high-order schemes based on the gas-kinetic and HLLC fluxes

Two higher-order compact finite difference approaches, the Hermitian and the Lax-Wendroff are examined by applying them to the viscous Burgers' equation. The difference equations obtained by the two methods were integrated in time through the third-order Strong-Stability-Preserving Runge-Kutta scheme. Absolute errors are computed by using an exact solution. The results are also compared with a second order central difference scheme. The Hermitian approach is far easier to implement. On uniform grids the Lax-Wendroff scheme produces smaller errors during the initial stages, but both methods are equally good for larger durations of integration. The convergence rate of the Hermitian scheme is slightly higher than the Lax-Wendroff scheme although both are of fourth order. Both schemes are unstable beyond a certain step size in time and space. When numerical boundary conditions are imposed, second-order conditions produce the best results whereas linear extrapolation proved to be the worst. It was also observed that large domains were required to implement the numerical boundary conditions properly. There was no detrimental effect on the accuracy of the results obtained through either of the two schemes when the size of the domain was greatly increased. Both schemes showed remarkable improvement in accuracy when clustered grids were employed. However much smaller time steps are required for stable solutions.

A third order gas-kinetic scheme for unstructured grid

Parallel FFT algorithms for high-order approximations on three-dimensional compact stencils

In recent years there has been a great interest in compact (or Pade type) schemes [1], [2], [3]. On Cartesian uniform meshes these schemes offer a high order of accuracy with only a compact stencil. More importantly, the resolution of the shorter length scales of the solution of these schemes is better than for classical finite-difference methods [4], which makes these schemes especially attractive for applications such as DNS, LES and computational aeroacoustics [5], [6].

Many numerical and experimental researches have been done for the unsteady flow around a flapping airfoil. The main research topics are vortex shedding, dynamic stall phenomenon, and MAV''s lift and thrust generation for the flow over a flapping airfoil. Although many researches have been studied to analyze the flow for the variation of frequency and amplitude, much research is still needed. In this study, thrust generation, wake structures, and acoustic wave propagation characteristics were studied for a flapping airfoil. The governing equation was the Navier-Stokes equation with LES turbulence model, and the Optimized High-Order Compact (OHOC) scheme and 4th order Runge-Kutta method were used. The Mach number and Reynolds number were 0.3 and 1000, respectively. The analysis of mean lift and drag coefficients showed the thrust generation characteristics according to the phase angle and the reduced frequency. The wake vortex shedding and propagation characteristics of acoustic waves could be simulated using the high order method.