Elimination of spurious loss in Euler equation computations

Abstract

The accuracy of a standard implicit conservative Euler equation solver is greatly improved by incorporation of an entropy-conservation equation, used in lieu of the streamwise momentum equation in all regions of the flow except at shock waves. A slight modification is also required at stagnation points to maintain wellposedness. The overall method completely eliminates spurious entropy errors, and thus reduces the grid density required to achieve any particular level of accuracy. Results are presented for subsonic and transonic airfoil flows.Although the derivation and results are for 2-D Euler equations, the approach has no inherent limitations which preclude extension to 3-D or viscous flows. A brief discussion of these extensions is presented.

Dilation‐based shock capturing for high‐order methods

“…The artificial viscosity enters the governing equations through an artificial dissipation term that is added to the conservation law in the following manner: Here, u are the conserved variables in the Euler equations, F denotes the inviscid fluxes, and the terms on the right hand side represent the viscosity, which is based on a Laplacian‐like term applied to u AV ( u )={ ρ , ρ v , ρ H }. The difference between u and u AV on the last term is designed to ensure conservation of enthalpy across shocks for steady‐state simulations while still being a dissipative term in the transient case .…”

An Adaptive Shock-Capturing HDG Method for Compressible Flows

“…Note in the definition of G that we choose ∂ũ/∂x i over ∂u/∂x i in order to preserve the total enthalpy. 1,18 The artificial viscosity ε is determined as follows.…”

“…Since the exact solution can be determined for any spatial point, we take the domain Ω to be (−2, −1)×(1, 2). The boundary condition is imposed by setting the exact state on the boundary of the domain and using (18). We consider triangular meshes that are obtained by splitting a regular n × n Cartesian grid into 2n 2 triangles.…”