Comparison of Fifth-Order WENO Scheme and Finite Volume WENO-Gas-Kinetic Scheme for Inviscid and Viscous Flow Simulation

Abstract

The development of high-order schemes has been mostly concentrated on the limiters and high-order reconstruction techniques. In this paper, the effect of the flux functions on the performance of high-order schemes will be studied. Based on the same WENO reconstruction, two schemes with different flux functions, i.e., the fifthorder WENO method and the WENO-Gas-kinetic scheme (WENO-GKS), will be compared. The fifth-order finite difference WENO-SW scheme is a characteristic variable reconstruction based method which uses the Steger-Warming flux splitting for inviscid terms, the sixth-order central difference for viscous terms, and three stages Runge-Kutta time stepping for the time integration. On the other hand, the finite volume WENO-GKS is a conservative variable reconstruction based method with the same WENO reconstruction. But, it evaluates a time dependent gas distribution function along a cell interface, and updates the flow variables inside each control volume by integrating the flux function along the boundary of the control volume in both space and time. In order to validate the robustness and accuracy of the schemes, both methods are tested under a wide range of flow conditions: vortex propagation, Mach 3 step problem, and the cavity flow at Reynolds number 3200. Our study shows that both WENO-SW and WENO-GKS yield quantitatively similar results and agree with each other very well provided a sufficient grid resolution is used. With the reduction of mesh points, the WENO-GKS behaves to have less numerical dissipation and present more accurate solutions than those from the WENO-SW in all test cases. For the Navier-Stokes equations, since the WENO-GKS couples inviscid and viscous terms in a single flux evaluation, and the WENO-SW uses an operator splitting technique, it appears that the WENO-SW is more sensitive to the WENO reconstruction and boundary treatment. In terms of efficiency, the finite volume WENO-GKS is about 4 times slower than the finite difference WENO-SW in two dimensional simulations. The current study clearly shows that besides high-order reconstruction, an accurate gas evolution model or flux function in a high-order scheme is also important in the capturing of * Corresponding author. Email addresses: luojun@ust.hk (J. Luo), maljxuan@ust.hk (L. Xuan), makxu@ust.hk (K. Xu) http://www.global-sci.com/ 599 c 2013 Global-Science Press 600 J. Luo, L. Xuan and K. Xu / Commun. Comput. Phys., 14 (2013), pp. 599-620physical solutions. In a physical flow, the transport, stress deformation, heat conduction, and viscous heating are all coupled in a single gas evolution process. Therefore, it is preferred to develop such a scheme with multi-dimensionality, and unified treatment of inviscid and dissipative terms. A high-order scheme does prefer a high-order gas evolution model. Even with the rapid advances of high-order reconstruction techniques, the first-order dynamics of the Riemann solution becomes the bottleneck for the further development of high-order schemes. In order to avoid the weakness of th...

Extending a Hybrid Finite-Volume-Element Method to Solve Laminar Diffusive Flame

“…To prevent numerical diffusivity, they implemented a conservative second-order-accurate TVD scheme. Luo et al [9] compared the robustness, performance, and accuracy of the fifth-order finite-difference WENO-Steger-Warming (WENO-SW) scheme with those of the FV WENO-gaskinetic (WENO-GK) scheme in solving 2-D flow problems. Tsoutsanis et al [10] implemented fifth-order WENO schemes to solve viscous flows on unstructured grids.…”

“…In Section 5, we suitably describe the physical-based influence scheme PIS, which is the essence of current FV formulations [2][3][4][5]. In addition to PIS, we also discuss some modern techniques, which focus on higher-order nonoscillatory or monotonicity-preserving schemes, e.g., ENO, WENO, and TVD [6][7][8][9][10].…”

An efficient high-order gas-kinetic scheme (I): Euler equations

“…A typical choice of reconstruction is given in Appendix A. After the replacement by f 0 = f 0 N S , the expansion yields [17] f (…”

“…The original HGKS possesses the high accuracy and strong robustness to solve the NS and Euler equations for compressible flows [29,13,17]. But the process to compute a number of high-order derivatives of particle distribution function holds back HGKS to higher-order extensions and their practical simulations.…”

“…HGKS is originally developed to solve the NS equations from the Taylor expansion and firstorder Chapman-Enskog expansion on the integral solution [29]. Through the modifications on τ to replace the physical dissipation by the numerical dissipation, HGKS is also applicable to the Euler equations [17]. Because the original HGKS provides the basis for higher-order extensions and simplifications, the state-of-art third-order HGKS [16,13] is reviewed firstly, followed by the existing simplifications [36] for a more efficient NS solver and an Euler solver [16,17] ultimately.…”