Variational multi-scale finite element approximation of the compressible Navier-Stokes equations

Abstract

Purpose - The purpose of this paper is to apply the variational multi-scale framework to the finite element approximation of the compressible Navier-Stokes equations written in conservation form. Even though this formulation is relatively well known, some particular features that have been applied with great success in other flow problems are incorporated. \ud \ud Design/methodology/approach - The orthogonal subgrid scales, the non-linear tracking of these subscales, and their time evolution are applied. Moreover, a systematic way to design the matrix of algorithmic parameters from the perspective of a Fourier analysis is given, and the adjoint of the non-linear operator including the volumetric part of the convective term is defined. Because the subgrid stabilization method works in the streamline direction, an anisotropic shock capturing method that keeps the diffusion unaltered in the direction of the streamlines, but modifies the crosswind diffusion is implemented. The artificial shock capturing diffusivity is calculated by using the orthogonal projection onto the finite element space of the gradient of the solution, instead of the common residual definition. Temporal derivatives are integrated in an explicit fashion. \ud \ud Findings - Subsonic and supersonic numerical experiments show that including the orthogonal, dynamic, and the non-linear subscales improve the accuracy of the compressible formulation. The non-linearity introduced by the anisotropic shock capturing method has less effect in the convergence behavior to the steady state. \ud \ud Originality/value - A complete investigation of the stabilized formulation of the compressible problem is addressed.Peer ReviewedPostprint (author's final draft

Solution of low Mach number aeroacoustic flows using a Variational Multi-Scale finite element formulation of the compressible Navier–Stokes equations written in primitive variables

“…The usual compressible definition for the τ matrix (see [22] for a complete demonstration), includes the local sound velocity that arises from the linearized characteristic compressible flow problem. At the low Mach number limit the sound speed tends to infinity (c → ∞), and therefore, that stabilization matrix definition is not suitable.…”

“…From the numerical point of view, this transformation not only scales the problem, but also allows to implement iterative linear system solvers in the case of transient problems. We also incorporate some particular features of the VMS framework that we have applied in other flow problems (like in [21,22], for example). Since there are different ways to define the subscales, three different attributes are studied in this work.…”

Variational multiscale error estimators for the adaptive mesh refinement of compressible flow simulations

“…In this section, the finite element formulation of the compressible problem is described. The VMS formulation of the compressible problem that has been presented in [26,29] is recalled in the section. As it will be explained, the VMS framework is used for stabilizing the finite element approximation and allows us to use arbitrary interpolation spaces for the different variables of the problem.…”

“…This is the so called Orthogonal Sub-Grid Scales (OSGS) method, which defines the projection as the orthogonal projection onto the finite element space P = P ⊥ h = I − P h , being P h the L 2 −projection onto the finite element space. Apart from the construction of the spaces where the subscales belong, we call the subscales dynamic because the temporal derivative of subscales in (18) and (26) is taken into account, and non-linear, as the subscales are accounted for in all the non-linear terms of both the finite scale and subscale equations. This means that at all instances where U appears, it is replaced by U h + U .…”

“…In consequence, we refer to the subscales as to be orthogonal, dynamic and non-linear. Second, we model the effect of the subscales inside each element from the perspective of a Fourier analysis, as developed in [25,26], instead of using Green's functions in [1,6,11,12]. Third, we model the subscales at the boundaries of the elements, as first presented in [27], and calculate them as part of the error estimator.…”