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We show that shock polars for an ideal non-polytropic gas (thermally but not calorically perfect) have a unique velocity angle maximum, the critical shock, assuming a convex equation of state (positive fundamental derivative) and other standard conditions. We also show that the critical shock is always transonic. These properties are explained by a brief informal mass flux argument, which is then extended into a precise calculation. In the process we show that temperature, pressure, energy, enthalpy, normal mass flux and entropy are increasing along the forward Hugoniot curves, and hence along the polar from vanishing to normal shock; speed is decreasing along the entire polar, mass flux and, importantly, Mach number are decreasing on subsonic parts of the polar. If the equation of state is not convex, counterexamples can be given with multiple critical shocks, permitting more than two shocks that attain the same velocity angle, in particular, more than one shock of weak type, which would cause theoretical problems and practical risks of misprediction. For dissociating diatomic gases, numerical experiments suggest that positive fundamental derivative and uniqueness of critical shocks hold at all realistic pressures, although both fail at very low purely theoretical pressures.

Building on work of Prandtl and Alexander, we study logarithmic vortex spiral solutions of the two-dimensional incompressible Euler equations. We consider multi-branched spirals that are not symmetric, including mixtures of sheets and continuum vorticity. We find that non-trivial solutions allow only sheets, that there is a large variety of such solutions, but that only the Alexander spirals with three or more symmetric branches appear to yield convergent Biot–Savart integral.

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 construct a class of self-similar 2d incompressible Euler solutions that have initial vorticity of mixed sign. The boundaries between regions of positive and negative vorticity form algebraic spirals, similar to the Kaden spiral and as opposed to Prandtl’s logarithmic vortex spirals. Also unlike the Prandtl case, spirals are not initially present.

We construct a class of self-similar 2d incompressible Euler solutions that have initial vorticity of mixed sign. The regions of positive and negative vorticity form algebraic spirals.

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 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.