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Consider inviscid flows around any bounded simple polygon. For sufficiently small but nonzero Mach number, irrotational flows do not exist.

We consider pairs of self-similar 2d vortex sheets forming cusps, equivalently single sheets merging into slip condition walls, as in classical Mach reflection at wedges. We derive from the Birkhoff-Rott equation a reduced model yielding formulas for cusp exponents and other quantities as functions of similarity exponent and strain coefficient. Comparison to numerics shows that piecewise quadratic and higher approximation of vortex sheets agree with each other and with the model. In contrast piecewise linear schemes produce spurious results and violate conservation of mass, a problem that may have been undetected in prior work for other vortical flows where even point vortices were sufficient. We find that vortex cusps only exist if the similarity exponent is sufficiently large and if the circulation on the sheet is counterclockwise (for a sheet above the wall with cusp opening to the right), unless a sufficiently positive strain coefficient compensates. Whenever a cusp cannot exist a spiral-ends jet forms instead; we find many jets are so narrow that they appear as false cusps.

We prove existence of certain compressible subsonic inviscid flows with vortex sheets separating from a solid boundary. To leading order, any perturbation of the upstream boundary causes positive drag. We also prove that if a region of irrotational inviscid flow bounded by a vortex sheet and slip condition wall is enclosed in an angle less than $180^\circ$, then the velocity is zero.

We prove that there are no nonzero uniformly subsonic potential flows around bodies with three or more protruding corners, for piecewise analytic boundary and for equation of state a $\gamma$-law with $\gamma>1$. This generalizes an earlier result limited to the low-Mach limit for nondegenerate polygons. For incompressible flows we show the velocity cannot be globally bounded.

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.

Interaction of multiple shock waves generally produces a contact discontinuity whose circulation has previously been analyzed using “thermodynamic” arguments based on the Hugoniot relations across the shocks. We focus on “kinematic” techniques that avoid assumptions about the equation of state, using only jump relations for the conservation of mass and momentum but not energy. We give a new short proof for the nonexistence of pure (no contact) triple shocks, recovering a result of Serre. For Mach reflection with a zero-circulation but nonzero-density-jump contact, we show that the incident shock must be normal. Nonexistence without contacts generalizes to two or more incident shocks if we assume that all shocks are compressive. The sign of circulation across the contact has previously been controlled with entropy arguments, showing that the post-Mach-stem velocity is generally smaller. We give a kinematic proof assuming compressive shocks and another condition, such as backward incident shocks, or a weak form of the Lax condition. We also show that for 2 + 2 and higher interactions (multiple “upper” shocks with clockwise flow meeting multiple “lower” shocks with counterclockwise flow in a single point), the circulation sign can generally not be controlled. For γ-law pressure, we show that 2 + 2 interactions without contacts must be either symmetric or antisymmetric, with symmetry favored at low Mach numbers and low shock strengths. For full potential flow instead of the Euler equations, we surprisingly find, contrary to folklore and prior results for other models, that pure triple shocks without contacts are possible, even for γ-law pressure with 1 < γ < 3.

We consider self-similar solutions of the 2d incompressible Euler equations. We construct a class of solutions with vorticity forming algebraic spirals near the origin, in analogy to vortex sheets rolling up into algebraic spirals.

We prove non-existence of nontrivial uniformly subsonic inviscid irrotational flows around several classes of solid bodies with two protruding corners, in particular vertical and angled flat plates; horizontal plates are the only case where solutions exists. This fills the gap between classical results on bodies with a single protruding corner on one hand and recent work on bodies with three or more protruding corners. Thus even with zero viscosity and slip boundary conditions solids can generate vorticity, in the sense of having at least one rotational but no irrotational solutions. Our observation complements the commonly accepted explanation of vorticity generation based on Prandtl's theory of viscous boundary layers.

We consider shock reflection which has a well-known local non-uniqueness: the reflected shock can be either of two choices, called weak and strong. We consider cases where existence of a global solution with weak reflected shock has been proven, for compressible potential flow. If there was a global strong-shock solution as well, then potential flow would be ill-posed. However, we prove non-existence of strong-shock analogues in a natural class of candidates.

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 $L^\infty$-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 $L^\infty$ functions. We also show that solutions of the backward-in-time Riemann problem are necessarily BV.

We consider a particular instance of reflection of shock waves in self-similar compressible flow. We prove that local self-similar regular reflection (RR) cannot always be extended into a global flow. Therefore the detachment criterion is not universally correct. More precisely, consider the following "angle condition": the tangent of the strong-type reflected shock meets the opposite wall at a sharp or right downstream side angle. In cases where the condition is violated and the weak-type reflected shock is transonic, we show that global RR does not exist. Combined with earlier work we have shown that none of the classical criteria for RR-MR transition is universally correct. A new criterion is proposed. Moreover, we have shown that strong-type RR is unstable, in the sense that global RR cannot persist under perturbations to one side. This yields a definite answer to the weak-strong problem because earlier work shows stability of weak RR in the same sense., Comment: Formatting corrected

A well-known theorem of Lax and Wendroff states that if the sequence of approximate solutions to a system of hyperbolic conservation laws generated by a conservative consistent numerical scheme converges boundedly a.e. as the mesh parameter goes to zero, then the limit is a weak solution of the system. Moreover, if the scheme satisfies a discrete entropy inequality as well, the limit is an entropy solution. The original theorem applies to uniform Cartesian grids; this article presents a generalization for quasi-uniform grids (with Lipschitz-boundary cells) uniformly continuous inhomogeneous numerical fluxes and nonlinear inhomogeneous sources. The added generality allows a discussion of novel applications like local time stepping, grids with moving vertices and conservative remapping. A counterexample demonstrates that the theorem is not valid for arbitrary non-quasi-uniform grids.

Compressible (full) potential flow is expressed as an equivalent first-order system of conservation laws for density $\rho$ 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 $\rho,v$ if and only if $|v

We consider solutions of two-dimensional $m \times m$ systems hyperbolic conservation laws that are constant in time and along rays starting at the origin. The solutions are assumed to be small $L^\infty$ perturbations of a constant state and entropy admissible, and the system is assumed to be non-strictly hyperbolic with eigenvalues of constant multiplicity. We show that such a solution, initially assumed bounded, must be a special function of bounded variation, and we determine the possible configuration of waves. As a corollary, we extend some regularity and uniqueness results for some one-dimensional Riemann problems.

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, it has been an open problem to predict precisely for what 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 \emph{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.

A particular case of initial data for the two-dimensional Euler equations is studied numerically. The results show that the Godunov method does not always converge to the physical solution, at least not on feasible grids. Moreover, they suggest that entropy solutions (in the weak entropy inequality sense) are not well posed.

High performance applications and the underlying hardware platforms are becoming increasingly dynamic; runtime changes in the behavior of both are likely to result in inappropriate mappings of tasks to parallel machines during application execution. This fact is prompting new research on mapping and scheduling the dataflow graphs that represent parallel applications. In contrast to recent research which focuses on critical paths in dataflow graphs, this paper presents new mapping methods that compute near-min-cut partitions of the dataflow graph. Our methods deliver mappings that are an order of magnitude more efficient than those of DSC, a state-of-the-art critical-path algorithm, for sample high performance applications.

We study the classical problem of self-similar reflection of shocks at a ramp, modeled by potential flow with γ-law pressure. Depending on corner angle θ and upstream Mach number MI , either regular (RR) or Mach reflections occur. There are several conflicting transition criteria predicting the corner angle at which the type of reflection changes. We show that in some cases, in particular MI =1 and γ =5 /3, an exact RR solution exists for all θ specified by the sonic criterion. Thus all weaker criteria are false.