Modern infinitesimals and the entropy jump across an inviscid shock wave

This article applies nonstandard analysis to study the generalized solutions of entropy and energy across one-dimensional shock waves in a compressible, inviscid, perfect gas. Nonstandard analysis is an area of modern mathematics that studies number systems that contain both infinitely large and infinitely small numbers. For an inviscid shock wave, it is assumed that the shock thickness occurs on an infinitesimal interval and that the jump functions for the field variables are smoothly defined on this interval. A weak converse to the existence of the entropy peak is derived and discussed. Generalized solutions of the Euler equations for entropy and energy are then derived for both theoretical and realistic normalized velocity profiles.

Spectral energy cascade in thermoacoustic shock waves

We have investigated thermoacoustically amplified quasi-planar nonlinear waves driven to the limit of shock-wave formation in a variable-area looped resonator geometrically optimized to maximize the growth rate of the quasi-travelling-wave second harmonic. Optimal conditions result in velocity leading pressure by approximately $40^{\circ }$ in the thermoacoustic core and not in pure travelling-wave phasing. High-order unstructured fully compressible Navier–Stokes simulations reveal three regimes: (i) modal growth, governed by linear thermoacoustics; (ii) hierarchical spectral broadening, resulting in a nonlinear inertial energy cascade, (iii) shock-wave-dominated limit cycle, where energy production is balanced by dissipation occurring at the captured shock-thickness scale. The acoustic energy budgets in regime (i) have been analytically derived, yielding an expression of the Rayleigh index in closed form and elucidating the effect of geometry and hot-to-cold temperature ratio on growth rates. A time-domain nonlinear dynamical model is formulated for regime (ii), highlighting the role of second-order interactions between pressure and heat-release fluctuations, causing asymmetry in the thermoacoustic energy production cycle and growth rate saturation. Moreover, energy cascade is inviscid due to steepening in regime (ii), with the $k$th harmonic growing at $k/2$-times the modal growth rate of the thermoacoustically sustained second harmonic. The frequency energy spectrum in regime (iii) is shown to scale with a $-5/2$ power law in the inertial range, rolling off at the captured shock-thickness scale in the dissipation range. We have thus shown the existence of equilibrium thermoacoustic energy cascade analogous to hydrodynamic turbulence.

A new model of roll waves: comparison with Brock’s experiments

We derive a mathematical model of shear flows of shallow water down an inclined plane. The non-dissipative part of the model is obtained by averaging the incompressible Euler equations over the fluid depth. The averaged equations are simplified in the case of weakly sheared flows. They are reminiscent of the compressible non-isentropic Euler equations where the flow enstrophy plays the role of entropy. Two types of enstrophies are distinguished: a small-scale enstrophy generated near the wall, and a large-scale enstrophy corresponding to the flow in the roller region near the free surface. The dissipation is then added in accordance with basic physical principles. The model is hyperbolic, the corresponding ‘sound velocity’ depends on the flow enstrophies. Periodic stationary solutions to this model describing roll waves were obtained. The solutions are in good agreement with the experimental profiles of roll waves measured in Brock’s experiments. In particular, the height of the vertical front of the waves, the shock thickness and the wave amplitude are well captured by the model.

A variational approach to the shock structure problem

A variational approach to the shock structure problem is proposed. The set of governing equations, consisted of n first-order ordinary differential equations accompanied with 2n boundary conditions at ??, is put into variational form by means of least-squares method. The corresponding variational principle is adjusted for application of Ritz method. This direct method is used for construction of approximate analytical solutions to the shock structure problem and derivation of the estimates for the shock thickness. General procedure is applied to the study of Burgers' equation and equations of gas dynamics.