On hybrid residual distribution–Galerkin discretizations for steady and time dependent viscous laminar flows

Abstract

This paper is concerned with the extension of second order residual distribution (RD) schemes to time dependent viscous flows. We provide a critical analysis of the use of a hybrid RD-Galerkin approach for both steady and time dependent problems. In particular, as in Ricchiuto (2008) and Villedieu etal. (2011), we study the coupling of a Residual Distribution (RD) discretization of the advection operator with a Galerkin approximation for the second order derivatives, with a Peclet dependent modulation of the upwinding introduced by the RD scheme. The final objective is to be able to achieve uniform second order of accuracy with respect to variations of the mesh size or, equivalently, of the local Peclet/Reynolds number. Starting from the scalar formulation given in the second order case in Ricchiuto etal. (2008), we perform an accuracy and stability analysis to extend the approach to time-dependent problems, and provide thorough numerical validation of the theoretical results. The schemes, formally extended to the system of laminar Navier-Stokes equations, are also compared to a finite volume scheme with least-squares linear reconstruction on the solution of standard test problems. While the scalar tests show the potential of this approach in providing uniform accuracy w.r.t. the range of values of the Peclet/Reynolds number, the results for the laminar Navier-Stokes equations show that in practice, the use of Peclet number based corrections does not change dramatically the quality of the solutions obtained. This justifies the quest for new formulations. This paper is concerned with the extension of second order residual distribution (RD) schemes to time dependent viscous flows. We provide a critical analysis of the use of a hybrid RD-Galerkin approach for both steady and time dependent problems. In particular, as in (Ricchiuto et al. J.Comp.Appl.Math. 215, 2008, Villedieu et al. J.Comput.Phys, 230, 2011, we study the coupling of a Residual Distribution (RD) discretization of the advection operator with a Galerkin approximation for the second order derivatives, with a Peclet dependent modulation of the upwinding introduced by the RD scheme. The final objective is to be able to achieve uniform second order of accuracy with respect to variations of the mesh size or, equivalently, of the local Peclet/Reynolds number. Starting from the scalar formulation given in the second order case in (Ricchiuto et al. J.Comp.Appl.Math. 215, 2008), we perform an accuracy and stability analysis to extend the approach to time-dependent problems, and provide thorough numerical validation of the theoretical results. The schemes, formally extended to the system of laminar NavierStokes equations, are also compared to a finite volume scheme with least-squares linear reconstruction on the solution of standard test problems. While the scalar tests show the potential of this approach in providing uniform accuracy w.r.t. the range of values of the Peclet/Reynolds number, the results for the laminar Navier-Stokes equations show t...

An enthalpy‐preserving shock‐capturing term for residual distribution schemes

“…In all cases considered, the shock-capturing term is capable of recovering the correct upstream total-enthalpy level. Solutions obtained with the LDA + LxF,H scheme present less wiggles and better postshock temperature fields and converge faster toward the steady state than those obtained with the classical Bx scheme of Dobeš et al 42 The favorable properties of the new shock-capturing term are preserved as well when applied to viscous simulations; both wall pressure and skin-friction predictions obtained with the LDA + LxF,H scheme improve those obtained with the classical Bx scheme.…”

“…In all cases considered, the shock‐capturing term is capable of recovering the correct upstream total‐enthalpy level. Solutions obtained with the L D A + δ L x F , H scheme present less wiggles and better postshock temperature fields and converge faster toward the steady state than those obtained with the classical B x scheme of Dobeš et al…”

“…Because the distribution matrices are uniformly bounded, the L D A scheme is linearity preserving, ie, it is second‐order accurate at steady state, (that is, it preserves linear solutions). It is widely used for the simulation of smooth flow fields …”

“…The resulting scheme is therefore multidimensional upwind, positive, and linearity preserving. Many variants can be constructed depending on the choice of . Here, we will use the shock detector function of Dobeš and Deconinck; this combination is known in the literature as the B x scheme.…”