Construction of low dissipative high-order well-balanced filter schemes for non-equilibrium flows

Abstract

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], Sjögreen & Yee [24] and Yee & Sjögreen [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.

Hydrodynamic Instabilities in Well-Balanced Finite Volume Schemes for Frictional Shallow Water Equations. The Kinematic Wave Case

“…7. Because these numerical instabilities appear in the simulation of both the background and of the perturbed flow, the origin of the problem might be the stiff source term which fixes the value of Fr p (7). Therefore, this benchmark may serve to check more sophisticated and accurate techniques valid for out-equilibrium shallow-water flows [4,5] with stiff source terms [6] and, in particular, to check the performance of low dissipative high-order well-balanced filter schemes for non-equilibrium flows [7] with variable cut-off wavenumber [13].…”

“…Furthermore, (16) is not only an exact solution to (1)- (2) for Fr p = 2 but also an asymptotic solution for Fr p > 2 when φ 1-typically φ (9) should be O(10 −4 ) to avoid non-normal effects-and λ 0 > λ ∞ (φ, V 0 , θ) (10) at t = 0 [3]. Notice that, according to (16), the normalized perturbation (h, u) T decays and its wavelength increases linearly with time at a rate independent of the initial wavelength λ 0 and of the plane-parallel Froude number Fr p (7). Conversely, for λ 0 < λ ∞ and values of Fr p (7) greater than Fr cr (8), the normalized perturbation (h, u) T will grow ultimately developing hydraulic jumps.…”

“…Therefore, the adoption of well-balanced numerical methods (which are able to resolve quasi-steady problems) become an interesting option. In the case of shallow water flows with moving water, different well-balanced methods have been developed [4,5], and specific formulations have been presented for hyperbolic system of balance laws in the presence of stiff source terms [6] as well as multiscale phenomena [7]. However, in the present work we shall restrict our analysis to well-balanced schemes in the sense described in [8,9].…”

Variable High-Order Multiblock Overlapping Grid Methods for Mixed Steady and Unsteady Multiscale Viscous Flows, Part II: Hypersonic Nonequilibrium Flows

“…Recently, these filter schemes were proved to be well-balanced schemes 6 in the sense that these schemes exactly preserve certain non-trivial steady-state solutions of the chemical nonequilibrium governing equations. With this added property the filter schemes can better minimize spurious numerics in reacting flows containing both steady shocks and unsteady turbulence with shocklet components than standard non-wellbalanced shock-capturing schemes.…”

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

“…[1][2][3][4][5][6] Importantly, the code implements many innovative low dissipative algorithms that adaptively use numerical dissipation from shock-capturing schemes as postprocessing filters on non-dissipative high-order centered schemes. [1][2][3][4][5][6] These filter schemes were especially designed for improved accuracy over standard high-order shock-capturing schemes in capturing turbulence with strong shocks and density variations. For multi-dimensional curvilinear grids, the metrics are evaluated at the same high-order as the spatial base scheme with high-order freestream preservation.…”