We present a model-reduced variational Eulerian integrator for incompressible fluids, which combines the efficiency gains of dimension reduction, the qualitative robustness of coarse spatial and temporal resolutions of geometric integrators, and the simplicity of sub-grid accurate boundary conditions on regular grids to deal with arbitrarily-shaped domains. At the core of our contributions is a functional map approach to fluid simulation for which scalar-and vector-valued eigenfunctions of the Laplacian operator can be easily used as reduced bases. Using a variational integrator in time to preserve liveliness and a simple, yet accurate embedding of the fluid domain onto a Cartesian grid, our model-reduced fluid simulator can achieve realistic animations in significantly less computational time than full-scale non-dissipative methods but without the numerical viscosity from which current reduced methods suffer. We also demonstrate the versatility of our approach by showing how it easily extends to magnetohydrodynamics and turbulence modeling in 2D, 3D and curved domains.

The properties of air at meteorological temperatures relevant to sound propagation and shock wave structure are reviewed. Of particular interest is the irreversible process of vibrational relaxation which describes the transfer of energy to or from the vibrational modes of the molecules and which dominates the absorption of audible sound in air. A detailed discussion of the structure and propagation of weak nonlinear waves in air shows that relaxation is again the dominant effect. As an alternative method to the usual approach of non-linear acoustics the exact gas-dynamic equations are solved numerically for a number of simple flow situations and exact results are obtained for the structure of steady waves. Estimates are obtained for the propagation distances required for the development of waveforms into steady profiles, and these distances are found under some circumstances to be greater than the dimensions of the Earth's atmosphere. The results are confirmed by laboratory experiments in CO2 and N2O and applied to waves in air with special reference to the sonic bangs of supersonic aircraft. Very recent results of applying the same alternative approach to periodic waves are reviewed briefly,

An approximate expression is given for the thickness of weak fully dispersed shock waves. Using available data on the thermodynamic properties of air, it is shown that shocks of the strength expected in sonic bangs are fully dispersed. Estimated relaxation times for dry and humid air lead to wide variations in possible thickness, varying from millimetres to metres.

The method of characteristics is used to calculate the flow ahead of an impulsively started piston moving at constant velocity. Particular attention is paid to the development of weak shock waves which are either fully or partly dispersed at very large distances from the piston. It is found that the global features of the flows may be represented in similarity form, and the graphs obtained allow extrapolation to very weak waves.

Vibrational relaxation of oxygen and nitrogen is shown to be important in determining the structure of weak shock waves in air. Of particular interest are waves with pressure jumps of 100Pa1 Pa ≡ 1 pascal = 1 N m−2. or less which are present in the atmosphere as sonic bangs. It is found that the structure of the waves depends on shock strength, ambient pressure, temperature and humidity.