Farzad Ismail

Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks

In this paper, an entropy-consistent flux is developed, continuing from the work of the previous paper. To achieve entropy consistency, a second and third-order differential terms are added to the entropy-conservative flux. This new flux function is tested on several one dimensional problems and compared with the original Roe flux. The new flux function exactly preserves the stationary contact discontinuity and does not capture the unphysical rarefaction shock. For steady shock problems, the new flux predicts a slightly more diffused profile whereas for unsteady cases, the captured shock is very similar to those produced by the Roe- flux. The shock stability is also studied in one dimension. Unlike the original Roe flux, the new flux is completely stable which will provide as a candidate to combat multidimensional shock instability, particularly the carbuncle phenomenon.

An Evaluation of Euler Fluxes for Hypersonic Flow Computations

Shock-capturing finite-volume schemes often give rise to anomalous results in hypersonic flow. We present here a wide-ranging survey of numerical experiments from eleven different flux functions in one- and two-dimensional contexts. Included is a recently-developed function that guarantees entropy stability.

On Carbuncles and Other Excrescences

We report observations of numerical experiments involving the carbuncle phenomenon. highlighting three distinct phases in its development. We stress that the second and third stages are logically consistent consequences of the rst stage, which can largely but not wholly be explained by a one-dimensional nonlinear stability analysis. However, we do not present a cure, and are careful not to promise one, although we do feel free to criticize others.

A Proposed Cure to the Carbuncle Phenomenon

A new finite volume methodology is introduced to combat the carbuncle. The method features a more accurate treatment of entropy in the flux formulation at the cost of a small computational overhead. This new flux function is tested on a hypersonic flow past a circular cylinder on both structured quadrilateral and unstructured triangular grids, producing encouraging results.

Study of an entropy-consistent Navier–Stokes flux

This paper presents a study to achieve discrete entropy consistency using artificial and physical diffusion mechanisms. The study begins with the one-dimensional viscous Burgers equation, specifically looking at the shock results of entropy-conserved fluxes combined with a few choices of artificial and physical viscous diffusions. The approach is then repeated for the Navier–Stokes equations. Overall, it is demonstrated that the artificial viscosity (or entropy) terms are still needed in addition to physical viscosity to achieve entropy consistency in shock predictions, although one of the artificial terms can be dropped for high viscosity or low Reynolds number flow.

Entropy Consistent Methods for the Navier---Stokes Equations

The concept of entropy conservation, stability, and consistency is applied to systems of hyperbolic equations to create new flux functions for the scalar and systems of conservation laws. Firstly, Burgers’ equation is modelled, followed by the Navier–Stokes equations. The new models are compared with the pre-existing entropy consistent fluxes at selected viscosity levels; it is found that the system flux requires additional entropy production at low viscosities, but not at higher viscosity values. Initial results herein demonstrate that the accuracy of the first order systems approach are comparable to the results produced by the original entropy-consistent Navier–Stokes flux.

A Grid-Insensitive LDA Method on Triangular Grids Solving the System of Euler Equations

The performance of the classic upwind-type residual distribution (RD) methods on skewed triangular grids are rigorously investigated in this paper. Based on an improved signals distribution, an improved second order RD method based on the LDA approach is proposed to faithfully replicate the flow physics on skewed triangular grids. It will be mathematically and numerically shown that the improved LDA method is found to have minimal accuracy variations when grids are skewed compared to classic RD and cell vertex finite volume methods on scalar equations and system of Euler equations.

Developments of entropy-stable residual distribution methods for conservation laws I

This paper presents preliminary developments of entropy-stable residual distribution methods for scalar problems. Controlling entropy generation is achieved by formulating an entropy conserved signals distribution coupled with an entropy-stable signals distribution. Numerical results of the entropy-stable residual distribution methods are accurate and comparable with the classic residual distribution methods for steady-state problems. High order accurate extensions for the new method on steady-state problems are also demonstrated. Moreover, the new method preserves second order accuracy on unsteady problems using an explicit time integration scheme. The idea of the multi-dimensional entropy-stable residual distribution method is generic enough to be extended to the system of hyperbolic equations, which will be presented in the sequel of this paper.

Non-unified Compact Residual-Distribution Methods for Scalar Advection–Diffusion Problems

This paper solves the advection–diffusion equation by treating both advection and diffusion residuals in a separate (non-unified) manner. An alternative residual distribution (RD) method combined with the Galerkin method is proposed to solve the advection–diffusion problem. This Flux-Difference RD method maintains a compact-stencil and the whole process of solving advection–diffusion does not require additional equations to be solved. A general mathematical analysis reveals that the new RD method is linearity preserving on arbitrary grids for the steady-state advection–diffusion equation. The numerical results show that the flux difference RD method preserves second-order accuracy on various unstructured grids including highly randomized anisotropic grids on both the linear and nonlinear scalar advection–diffusion cases.