Stabilized finite‐element computation of compressible flow with linear and quadratic interpolation functions

Abstract

SUMMARYThe unsteady compressible flow equations are solved using a stabilized finite‐element formulation with C0 elements. In 2D, the performance of three‐noded linear and six‐noded quadratic triangular elements is compared. In 3D, the relative performance is evaluated for 6‐noded linear and 18‐noded quadratic wedge elements. Results are compared for the solutions to Euler, laminar, and turbulent flows at different Mach numbers for several flow problems. The finite‐element meshes considered for comparison have same location of nodes for the linear and quadratic interpolations. For the turbulent flow, the Spalart–Allmaras model is used for closure. It is found that the quadratic elements yield better performance than the linear elements. This is attributed to accurate representation of the stabilization terms that involve second‐order derivatives in the formulation. When these terms are dropped from the formulation with quadratic interpolation, the numerical results are similar to those obtained with linear interpolation. The absence of these terms result in added numerical diffusion that seems to give the effect of a relatively reduced Reynolds number. For the same location of nodes, the computations with the linear triangular and wedge elements are approximately 20% and 100% faster than those with quadratic triangular and wedge elements, respectively. However, if the same quadrature rule for numerical integration is used for both interpolations, the computations with quadratic elements are approximately 20% and 45% faster in 2D and 3D, respectively. Copyright © 2014 John Wiley & Sons, Ltd.

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

“…Results are presented for the refined mesh driven by the subscales-based error estimation measured with the scaled L 2 −norm. This flow corresponds accurately to the referenced result in [34], so that, the approximation achieved by the subscales-driven AMR of the compressible flow solver is satisfactory.…”

“…at the center of the element and/or at the nodes). Instead, the subscales at the element boundaries (34) are defined by the inter-element jump operator, and therefore, the contribution of neighbor elements must be accounted for. This type of calculation is inconvenient for parallel implementations of the AMR strategy, in which neighboring elements may be located on a different partition of the computational domain.…”

“…Let us first explain how the boundary subscales are developed. Expanding (34) and writing it for each equation of the compressible problem, results in…”

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

“…This type of shock capturing operator was introduced by Hughes and Mallet (1986) and later by Le Beau and Tezduyar (1991) into the SUPG compressible flow formulation. Mittal and Tezduyar (1998), Mittal (1998), Hughes et al (2010), and Kotteda and Mittal (2014) continued developing shock capturing stabilized formulations. In contrast with previous formulations (e.g.…”

“…Adequate physical solutions are found using both the residual-based and the orthogonal projection-based shock capturing methods. Overshoots are smoothed according to the benchmarked solutions (Mittal and Tezduyar, 1998;Rispoli and Saavedra, 2006;Tezduyar and Senga, 2006;Kotteda and Multi-scale finite element approximation Mittal, 2014). Figure 8 shows the steady state contours for density and velocity magnitude solved with the anisotropic orthogonal projection-based shock capturing method and the mesh composed by P 2 elements.…”