Acoustic End Effects in Magnetohydrodynamic Submerged Vehicular Propulsors—Part 2: Solitary Waves

Abstract

In a magnetohydrodynamic (MHD) seawater propulsion system for a submerged vehicle, a region of high-uniform magnetic induction (magnetic flux density) inside the MHD propulsion channel is separated from the region of essentially zero magnetic induction outside the channel in the seaway by a region of nonuniform, fringing magnetic induction at each end of the channel. This paper treats the propagation of an a periodic fluid transient (solitary wave) which is produced by an arbitrary unit impulse in velocity or pressure at any cross section in the uniform-field region and which propagates through either fringing field region and into the zero-field region outside the channel. The time scale for the transients is sufficiently short that compressive effects are important, so that the fluid transients are acoustic waves. The channel is a straight, rectangular duct with electrically insulating walls and highly conducting walls perpendicular and parallel to the magnetic induction, respectively. The linearized acoustic equations are averaged over each cross section of the channel to obtain a pair of coupled equations governing the average pressure and average axial velocity as functions of the axial coordinate and time. Together these equations represent a simple wave equation with a retarding force which is proportional to the square of the local magnetic flux density. Results are presented for three values of the acoustic interaction parameter N, which is the characteristic ratio of the electromagnetic body force opposing motions across magnetic-induction lines to the inertial "force" in the fluid transients. An abrupt change in velocity produces a wave front which travels at the speed of sound. Without MHD effects, the entire change in velocity or pressure occurs suddenly as the wave front passes. With MHD effects, only part of each change occurs suddenly as the wave front passes, followed by a gradual evolution to reach the entire change. The split between the abrupt and gradual fractions of the entire change depends on N. In previous work, the authors treated the fundamentally different MHD acoustic problem of the transmission of periodic waves from the channel and the previous paper (Walker et al 1992) is Part 1 of the present work.