Construction of Low Dissipative High Order Well-Balanced Filter Schemes for Non-Equilibrium Flows

The goal of this paper is to generalize the well-balanced approach for non-equilibrium flow studied by Wang et al. [26] to a class of low dissipative high order shock-capturing filter schemes and to explore more advantages of well-balanced schemes in reacting flows. The class of filter schemes developed by Yee et al. [30], Sjoegreen & Yee [24] and Yee & Sjoegreen [35] consist of two steps, a full time step of spatially high order non-dissipative base scheme and an adaptive nonlinear filter containing shock-capturing dissipation. A good property of the filter scheme is that the base scheme and the filter are stand alone modules in designing. Therefore, the idea of designing a well-balanced filter scheme is straightforward, i.e., choosing a well-balanced base scheme with a well-balanced filter (both with high order). A typical class of these schemes shown in this paper is the high order central difference schemes/predictor-corrector (PC) schemes with a high order well-balanced WENO filter. The new filter scheme with the well-balanced property will gather the features of both filter methods and well-balanced properties: it can preserve certain steady state solutions exactly; it is able to capture small perturbations, e.g., turbulence fluctuations; it adaptively controls numerical dissipation. Thus it shows high accuracy, efficiency and stability in shock/turbulence interactions. Numerical examples containing 1D and 2D smooth problems, 1D stationary contact discontinuity problem and 1D turbulence/shock interactions are included to verify the improved accuracy, in addition to the well-balanced behavior

Variable high-order multiblock overlapping grid methods for mixed steady and unsteady multiscale viscous flows, part II: hypersonic nonequilibrium flows

The variable high-order multiblock overlapping (overset) grids method of Sjogreen & Yee (CiCP, Vol.5, 2008) for a perfect gas has been extended to nonequilibrium flows. This work makes use of the recently developed high-order well-balanced shock-capturing schemes and their filter counterparts (Wang et al., J. Comput. Phys., 2009, 2010) that exactly preserve certain non-trivial steady state solutions of the chemical nonequilibrium governing equations. Multiscale turbulence with strong shocks and flows containing both steady and unsteady components is best treated by mixing of numerical methods and switching on the appropriate scheme in the appropriate subdomains of the flow fields, even under the multiblock grid or adaptive grid refinement framework. While low dissipative sixth- or higher-order shock-capturing filter methods are appropriate for unsteady turbulence with shocklets, second- and third-order shock-capturing methods are more effective for strong steady or nearly steady shocks in terms of convergence. It is anticipated that our variable high-order overset grid framework capability with its highly modular design will allow an optimum synthesis of these new algorithms in such a way that the most appropriate spatial discretizations can be tailored for each particular region of the flow. In this paper some of the latest developments in single block high-order filter schemes for chemical nonequilibrium flows are applied to overset grid geometries. The numerical approach is validated on a number of test cases characterized by hypersonic conditions with strong shocks, including the reentry flow surrounding a 3D Apollo-like NASA Crew Exploration Vehicle that might contain mixed steady and unsteady components, depending on the flow conditions

Numerical Dissipation and Wrong Propagation Speed of Discontinuities for Stiff Source Terms

In compressible turbulent combustion/nonequilibrium flows, the constructions of numerical schemes for (a) stable and accurate simulation of turbulence with strong shocks, and (b) obtaining correct propagation speed of discontinuities for stiff reacting terms on coarse grids share one important ingredient - minimization of numerical dissipation while maintaining numerical stability. Here coarse grids means standard mesh density requirement for accurate simulation of typical non-reacting flows. This dual requirement to achieve both numerical stability and accuracy with zero or minimal use of numerical dissipation is most often conflicting for existing schemes that were designed for non-reacting flows. The goal of this paper is to relate numerical dissipations that are inherited in a selected set of high order shock-capturing schemes with the onset of wrong propagation speed of discontinuities for two representative stiff detonation wave problems

Spurious Behavior of Shock-Capturing Methods: Problems Containing Stiff Source Terms and Discontinuities

The goal of this paper is to relate numerical dissipations that are inherited in high order shock-capturing schemes with the onset of wrong propagation speed of discontinuities. For pointwise evaluation of the source term, previous studies indicated that the phenomenon of wrong propagation speed of discontinuities is connected with the smearing of the discontinuity caused by the discretization of the advection term. The smearing introduces a nonequilibrium state into the calculation. Thus as soon as a nonequilibrium value is introduced in this manner, the source term turns on and immediately restores equilibrium, while at the same time shifting the discontinuity to a cell boundary. The present study is to show that the degree of wrong propagation speed of discontinuities is highly dependent on the accuracy of the numerical method. The manner in which the smearing of discontinuities is contained by the numerical method and the overall amount of numerical dissipation being employed play major roles. Moreover, employing finite time steps and grid spacings that are below the standard Courant-Friedrich-Levy (CFL) limit on shockcapturing methods for compressible Euler and Navier-Stokes equations containing stiff reacting source terms and discontinuities reveals surprising counter-intuitive results. Unlike non-reacting flows, for stiff reactions with discontinuities, employing a time step and grid spacing that are below the CFL limit (based on the homogeneous part or non-reacting part of the governing equations) does not guarantee a correct solution of the chosen governing equations. Instead, depending on the numerical method, time step and grid spacing, the numerical simulation may lead to (a) the correct solution (within the truncation error of the scheme), (b) a divergent solution, (c) a wrong propagation speed of discontinuities solution or (d) other spurious solutions that are solutions of the discretized counterparts but are not solutions of the governing equations. The present investigation for three very different stiff system cases confirms some of the findings of Lafon & Yee (1996) and LeVeque & Yee (1990) for a model scalar PDE. The findings might shed some light on the reported difficulties in numerical combustion and problems with stiff nonlinear (homogeneous) source terms and discontinuities in general

Spurious behavior of shock-capturing methods by the fractional step approach: Problems containing stiff source terms and discontinuities

The goal of this paper is to relate numerical dissipations that are inherited in high order shock-capturing schemes with the onset of wrong propagation speed of discontinuities. For pointwise evaluation of the source term, previous studies indicated that the phenomenon of wrong propagation speed of discontinuities is connected with the smearing of the discontinuity caused by the discretization of the advection term. The present study focuses only on solving the reactive system by the fractional step method using the Strang splitting. Studies shows that the degree of wrong propagation speed of discontinuities is highly dependent on the accuracy of the numerical method. The manner in which the smearing of discontinuities is contained by the numerical method and the overall amount of numerical dissipation being employed play major roles. Depending on the numerical method, time step and grid spacing, the numerical simulation may lead to (a) the correct solution (within the truncation error of the scheme), (b) a divergent solution, (c) a wrong propagation speed of discontinuities solution or (d) other spurious solutions that are solutions of the discretized counterparts but are not solutions of the governing equations. The findings might shed some light on the reported difficulties in numerical combustion and problems with stiff nonlinear (homogeneous) source terms and discontinuities in general

Computational challenges for simulations related to the NASA electric arc shock tube (EAST) experiments

The goal of this study is to gain some physical insights and an understanding of the computational challenges for the simulations related to the hypersonic nonequilibrium multi-species and multi-reaction experiments on the NASA Electric Arc Shock Tube (EAST). While experimental measurement does not provide any information about the radial structure of this type of flow, accurate and reliable numerical simulations can provide more insight into the physical structure of the flow to aid the design of atmospheric entry spacecrafts. The paper focuses on the spurious numerics which take place in numerical simulations of the subject physics containing stiff source terms and discontinuities. This paper is based on the knowledge gained from Yee et al. on simple reacting test cases (Yee et al. 2013, [9]) as a guide to reveal the computational challenges involved for such an extreme flow type. The results of the 1D and 2D EAST viscous and inviscid simulations using a simplified physical model are presented. The computation reveals, for the first time, that the 2D viscous model which contains both shocks and shears exhibits Tollmien– Schlichting-like instability complex patterns at the boundary layer. In addition to exhibiting spurious numerical behavior of wrong propagation speed of discontinuities by typical shock-capturing methods, there is improved understanding on the cause of numerical difficulties by previous investigators. One example is that the relative distance between the shocks and shear/contact is different from one grid spacing to another for each considered high order shock-capturing scheme. The results presented can provide insight on the numerical instability observed by previous investigations and future algorithm development for this type of extreme flow

Variable high-order multiblock overlapping grid methods for mixed steady and unsteady multiscale viscous flows, part II: hypersonic nonequilibrium flows

Abstract not provide

Investigation of low-dissipation monotonicity-preserving scheme for direct numerical simulation of compressible turbulent flows

© 2014 Elsevier Ltd. The influence of numerical dissipation on direct numerical simulation (DNS) of decaying isotropic turbulence and turbulent channel flow is investigated respectively by using the seventh-order low-dissipation monotonicity-preserving (MP7-LD) scheme with different levels of bandwidth dissipation. It is found that for both benchmark test cases, small-scale turbulence fluctuations can be largely suppressed by high level of scheme dissipation, while the appearance of numerical errors in terms of high-frequency oscillations could destabilize the computation if the dissipation is reduced to a very low level. Numerical studies show that reducing the bandwidth dissipation to 70% of the conventional seventh-order upwind scheme can maximize the efficiency of the MP7-LD scheme in resolving small-scale turbulence fluctuations and, in the meantime preventing the accumulation of non-physical numerical errors. By using the optimized MP7-LD scheme, DNS of an impinging oblique shock-wave interacting with a spatially-developing turbulent boundary layer is conducted at an incoming free-stream Mach number of 2.25 and the shock angle of 33.2°. Simulation results of mean velocity profiles, wall surface pressure, skin friction and Reynolds stresses are validated against available experimental data and other DNS predictions in both the undisturbed equilibrium boundary layer region and the interaction zone, and good agreements are achieved. The turbulence kinetic energy transport equation is also analyzed and the balance of the equation is well preserved in the interaction region. This study demonstrates the capability of the optimized MP7-LD scheme for DNS of complex flow problems of wall-bounded turbulent flow interacting with shock-waves

On well-balanced schemes for non-equilibrium flow with stiff source terms

In the modeling of unsteady reactive problems, the interaction of turbulence with finiterate chemistry introduces a wide range of space and time scales, leading to additional numerical difficulties. A main difficulty stems from the fact that most numerical algorithms used in reacting flows were originally designed to solve non-reacting fluids. As a result, spatial stiffness due to reacting source terms and turbulence/chemistry interaction are major stumbling blocks to numerical algorithm development. One of the important numerical issues is the proper numerical treatment of a system of highly coupled stiff non-linear source terms, which will result in possible spurious steady state numerical solutions (see Lafon & Yee 1996). It was shown in LeVeque (1998) that a well-balanced scheme, which can preserve the steady state solution exactly, may solve this spurious numerical behavior. The goal of this work is to consider a simple 1-D model with one temperature and three species as studied by Gnoffo, Gupta & Shinn (1989) and to study the well-balanced property of various popular linear and non-linear numerical schemes in the literature. The different behaviors of those numerical schemes in preserving steady states and in resolving small perturbations of such states will be shown