Numerical analysis of the nonlinear propagation of plane periodic waves in a relaxing gas

Abstract

The waves propagating from an oscillating plane piston into a vibrationally relaxing gas are calculated by an exact numerical method ignoring viscosity and heat conduction. Secondary effects due to the starting of the piston from rest and to acoustic streaming can be eliminated from the calculated flows, leaving a truly periodic progressive wave which can be analysed and compared with approximate solutions. It is found that for moderate amplitude waves nonlinearity is only important as a convective effect which produces higher harmonics, whereas dissipation is adequately described by linear theory.

Spherical nonlinear wave propagation in a vibrationally relaxing gas

“…One feature of the flows requires special attention, namely the presence of a mean pressure in the wavefield different from the undisturbed pressure p 0. This was not discussed by Southern & Johannesen (1980) as they only presented results for flows with relaxation in which the main contribution to the mean pressure is due to therm al effects. The plane case was in fact considered by Fubini Ghiron (1935) wh° obtained the correct result, although this has apparently escaped the notice of some later writers.…”

“…U nfortunately, no experimental d ata were obtained for comparison with their spherical model. Southern & Johannesen (1980) used the plane model based on the frozen speed of sound for comparisons with their numerical results for plane wave propagation in a vibrationally relaxing gas with a large value of the vibrational specific heat. Reasonable agreement was again achieved although dispersive effects were present in the numerical results, due to the large vibrational specific heat, and these are not included in the model.…”

“…T hat is, the neglect of the 'nonlinear atten u atio n ' terms is balanced by the incorrect attenuation rate applied to the nonlinear term in equation (10.5) based on the assumption th a t nonlinear growth is generated from the fundam ental only. Southern & Johannesen (1980) showed th a t the solution of the fundam ental and the second harmonic could be improved by including higher-order terms in the series expansion for the Bessel function. I t should be emphasized th a t this applies only to the fundam ental and the second harmonic since the assumption th a t harmonic growth depends only on the fundam ental is correct up to the second harmonic to the approximation made in equation (10.3).…”

“…The question whether, and if so when, shock waves appear has been discussed in an earlier paper (Johannesen & Scott 1978) with particular attention to atmospheric air. We shall use some of the results of this paper w ithout derivation just as we shall quote results from the paper by Southern & Johannesen (1980) on the corresponding plane problem. The reader is referred to the survey paper by Johannesen & Hodgson (1979) for a discussion of the basic principles for the treatm ent of unsteady flows with relaxation and of attenuation of sound in relaxing gases.…”

“…These appear to be particularly useful because they give details of the development of the wave profile whereas the approxim ate theories are essentially concerned with the changes in amplitude with distance. Southern & Johannesen (1980) presented numerical solutions of the exact equations for the one-dimensional flow due to a sinusoidally oscillating piston in a gas in which attenuation was due to vibrational relaxation only, with classical and ro tational relaxation attenuation ignored. The main conclusion was th a t attenuation is adequately described by linear theory and can be superimposed on the nonlinear isentropic flow.…”