Explicit discontinuous spectral element method with entropy generation based artificial viscosity for shocked viscous flows

Abstract

A spatio-temporal adaptive artificial viscosity (AV) based shock-capturing scheme is proposed for the solution of both inviscid and viscous compressible flows using a high-order parallel Discontinuous Spectral Element Method (DSEM). The artificial viscosity and artificial thermal conduction coefficients are proportional to the viscous and thermal entropy generating terms, respectively, in the viscous entropy conservation law. The magnitude of AV is limited based on the explicit stable CFL criterion, so that the stable artificial viscous time step size is greater than the convective stable time step size. To further ensure the stability of this explicit approach, an adaptive variable order exponential filter is applied, if necessary, in elements where the AV has been limited. In viscous flow computations a modified Jameson's sensor (Ducros et al., JCP 152 (2)) limits the AV to small values in viscous shear regions, so as to maintain a high-order resolution in smooth regions and an essentially non-oscillatory behavior near sharp gradients/shocks regions. We have performed a systematic and extensive validation of the algorithm with one-dimensional problems (inviscid moving shock and viscous shock-structure interaction), two-dimensional problems (inviscid steady and unsteady shocked flows and viscous shock-boundary layer interaction), and a three-dimensional supersonic turbulent flow over a ramped cavity. These examples demonstrate that the explicit DSEM scheme with adaptive artificial

On Shock Propagation through Double-Bend Ducts by Entropy-Generation-Based Artificial Viscosity Method

“…In this section, we briefly present the governing equations and numerical approach. Detail of the numerical procedure is reported in [ 34 ] and is not repeated here for brevity.…”

“…For this, a cost effective approach is the usage of AV compared to slope limiters. A short review of different AV approaches in conjunction with high-order methods is reported in [ 34 ]. The basic idea of an AV method is to explicitly add even-order dissipation term to stabilize the numerical scheme.…”

“…Nevertheless, dealing with Gibb’s oscillations for non-smooth fluid flow problems requires implantation of suitable slope limiters or artificial viscosity based approaches. In this regard, Chaudhuri et al [ 34 ] reported the stable shock capturing capabilities of explicit high-order DSEM in conjunction with entropy-generation-based artificial viscosity (AV) method. Although inviscid simulations quite accurately predict the shock wave diffraction wave patterns over different geometrical shapes, it is evident that viscous effects are important for resolving long-time behavior of shock-vortex evolution, shock-shear layer and shock-boundary layer interactions [ 35 , 36 , 37 , 38 , 39 , 40 , 41 ].…”

“…The s (shock-wave Mach numbers) are chosen similar to those presented in the experimental work of Igra et al [ 48 ]. A high-order artificial viscosity based DSEM [ 34 ] is used for this purpose. The flow analysis sheds light on numerical perspective about the robustness and accuracy of the scheme, as well as the shock diffraction and shock attenuation aspects in double-bend ducts.…”

Multi-element SIAC Filter for Shock Capturing Applied to High-Order Discontinuous Galerkin Spectral Element Methods

“…Over the decades, many counter mechanisms have been developed to control overshoots and stabilize high-order approximations in shocked regions. Altogether, these methods can be subdivided into three main categories, which are slope limiters [3,4,12,14,15,31], artificial dissipation techniques [7,16,17,18,28], and solution filters [2,7,27,29].…”