Volker Elling

The carbuncle phenomenon is incurable

Abstract Numerical approximations of multi-dimensional shock waves sometimes exhibit an instability called the carbuncle phenomenon. Techniques for suppressing carbuncles are trial-and-error and lack in reliability and generality, partly because theoretical knowledge about carbuncles is equally unsatisfactory. It is not known which numerical schemes are affected in which circumstances, what causes carbuncles to appear and whether carbuncles are purely numerical artifacts or rather features of a continuum equation or model. This article presents evidence towards the latter: we propose that carbuncles are a special class of entropy solutions which can be physically correct in some circumstances. Using “filaments”, we trigger a single carbuncle in a new and more reliable way, and compute the structure in detail in similarity coordinates. We argue that carbuncles can, in some circumstances, be valid vanishing viscosity limits. Trying to suppress them is making a physical assumption that may be false.

THE ELLIPTICITY PRINCIPLE FOR SELF-SIMILAR POTENTIAL FLOWS

We consider self-similar potential flow for compressible gas with polytropic pressure law. Self-similar solutions arise as large-time asymptotes of general solutions, and as exact solutions of many important special cases like Mach reflection, multidimensional Riemann problems, or flow around corners. Self-similar potential flow is a quasilinear second-order PDE of mixed type which is hyperbolic at infinity (if the velocity is globally bounded). The type in each point is determined by the local pseudo-Mach number L, with L 1) corresponding to elliptic (respectively, hyperbolic) regions. We prove an ellipticity principle: the interior of a parabolic-elliptic region of a sufficiently smooth solution must be elliptic; in fact L must be bounded above away from 1 by a domain-dependent function. In particular there are no open parabolic regions. We also discuss the case of slip boundary conditions at straight solid walls.

Counterexamples to the Sonic Criterion

We consider self-similar (pseudo-steady) shock reflection at an oblique wall. There are three parameters: wall corner angle, Mach number, angle of incident shock. Ever since Ernst Mach discovered the irregular reflection named after him, researchers have sought to predict precisely for which parameters the reflection is regular. Three conflicting proposals—the detachment, sonic and von Neumann criteria—have been studied extensively without a clear result. We demonstrate that the sonic criterion is not correct. We consider polytropic potential flow and prove that there is an open nonempty set of parameters that admit a global regular reflection with a reflected shock that is transonic. We also provide a clear physical reason: the flow type (sub- or supersonic) is not decisive; instead the reflected shock type (weak or strong) determines whether structural perturbations decay towards the reflection point.

Steady and Self-Similar Inviscid Flow

We consider solutions of the two-dimensional compressible (isentropic) Euler equations that are steady and self-similar. They arise naturally at interaction points in genuinely multidimensional flow. We characterize the possible solutions in the class of flows

Relative entropy and compressible potential flow

L^\infty

Non-existence of Irrotational Flow Around Solids with Protruding Corners

-close to a constant supersonic background. As a special case we prove that solutions of one-dimensional Riemann problems are unique in the class of small

Hölder continuity and differentiability on converging subsequences

L^\infty

Nonexistence of compressible irrotational inviscid flows along infinite protruding corners

functions. We also show that solutions of the backward-in-time Riemann problem are necessarily BV.

Steady and Self-Similar Solutions of Non-Strictly Hyperbolic Systems of Conservation Laws

Abstract Compressible (full) potential flow is expressed as an equivalent first-order system of conservation laws for density and velocity v. Energy E is shown to be the only nontrivial entropy for that system in multiple space dimensions, and it is strictly convex in ρ,v if and only if |v|

Supersonic Flow onto a Solid Wedge

We motivate and discuss several recent results on non-existence of irrotational inviscid flow around bounded solids that have two or more protruding corners, complementing classical results for the case of a single protruding corner. For a class of two-corner bodies including non-horizontal flat plates, compressible subsonic flows do not exist. Regarding three or more corners, bounded simple polygons do not admit compressible flows with arbitrarily small Mach number, and any incompressible flow has unbounded velocity at at least one corner. Finally, irrotational flow around smooth protruding corners with non-vanishing velocity at infinity does not exist. This can be considered vorticity generation by a slip-condition solid in absence of viscosity.