Stability of Acoustic Wave in Two-Phase Dilute Flow with Mass Transfer

Abstract

The behavior of an acoustic wave propagating in a two-phase dilute ow is analytically and numerically investigated. The focus is on the effects of a mass transfer modeled by the so-called rapid-mixing model. An analytical solution is carried out that shows a possible unstable ow regime, which means that the magnitude of a pressure wave may be ampli ed under particular conditions. The neutral stability condition is mainly driven by a mass transfer number, which links the heat of phase change and the equilibrium temperature. Even the mass transfer is a simpli ed one and far from the actual combustion of metal particles, when the analysis is applied to aluminum particles in solid rocket motor environment, unstable ow behavior is seen at low frequencies. One-dimensional simulations of the propagation of an acoustic wave are performed, and the results recovered the theoretical ones. A simulation in a two-dimensional motor leads to an oscillatory ow, which is sustained, and the amplitude of the pressure oscillation reaches an asymptotic value. This result, obtained by solving the nonlinear coupled two-phase ow equations shows that the mass transfer might be a driven mechanism for instabilities in solid rocket motor two-phase ows. Nomenclaturea = sound speed, p .°RT), m/s C m = loading mass ratio D p = particle diameter, m I N N d = identity tensor L = heat of change phase, J/(kg ¢ m 3 ) Nu = Nusselt number R = gas constant, J/(kg ¢ K) St = acoustic Stokes number, !¿ d°= gas ratio of speci c heatş = thermal conductivity, W/(m ¢ K) ¹ = dynamic viscosity, kg/(m ¢ s) ½ ¤= material density of a particle, kg/m 3 ¿ t ; ¿ d = thermal and dynamic relaxation time, s ! = wave pulsation, 1/s P v = rate of mass transfer, kg/(m 3 ¢ s) Subscripts g = gas phase p = particulate phase