The interaction of flow discontinuities with small disturbances in a compressible fluid

Abstract

The interaction of a random small disturbance field in a compressible fluid with shock waves and flame fronts is analyzed. The disturbance field, which may consist of fluctuations of pressure, entropy and vorticity, is found to be modified in passing through the shock or flame. In the case of the shock wave, it is found that all of the the three types of disturbances are generated in comparable strength in the downstream flow by the presence of any of the three in the upstream flow. Moderate fluctuations of either vorticity (turbulence) or entropy will produce intense noise fields in the downstream flow. If the shock is normal, the frequency of this noise field is much lower for very weak shocks than for strong shocks, given the same upstream velocity and disturbance wave length. If the weak shock is oblique to the flow, the frequency of the noise is increased. For the flame front, also, it is found that all three types of disturbances are generated in the downstream flow by the presence of one of them in the upstream flow. In this case, however, the normal propagation Mach number of the flame enters as a small parameter. It is found that the intensity of the downstream turbulence generated by sound waves inpinging on the upstream face of the flame is proportional to the reciprocal of this Mach number times the intensity of the upstream pressure fluctuation. Hence, rather strong turbulence may be generated downstream of a flame by comparatively weak sound upstream. The pressure amplitudes of the sound fields generated by entropy and vorticity fluctuations in the upstream flow are proportional, respectively, to the Mach number squared and cubed. For ordinary hydrocarbon flames, ten percent turbulent velocity fluctuations, or one percent entropy (temperature) fluctuations will cause audible sound to be emitted. The fequency is in the range of 20 to perhaps 100 cycles per second for an input disturbance wave length of one inch. Although the analysis is carried out for an isolated infinite discontinuity, it is felt that the results are applicable, at least qualitatively, to the complicated configuration of shock waves found in under or over-expanded nozzles, and to the flame configurations found in actual combustion processes