Numerical Solutions of Transient MHD Phenomena

Abstract

ANUMERICAL model generating simultaneous solutions to the coulpled gasdynamic-electromagnetic equations of motion is developed. The transient relationship existing between the hydrodynamic flow properties and electromagnetic field quantities is explored. The parabolic gasdynamic equations of motion are discretized by a set of firstorder accurate, explicit finite difference expressions and are solved in an Eulerian reference frame. Employing MHD approximations to Maxwell's relations, a linear second-order elliptic partial differential equation describing the current stream function \l/ or the electric potential is developed and is solved by the method of successive overrelaxation. The time-dependent hydrodynamic and electromagnetic properties of a conducting fluid are evaluated by including the Lorentz force (jxB) and the electrical energy (j-E) in the equations of momentum and energy, respectively. Simultaneously, or \j/ is computed as a function of the updated flow velocity, temperature, and pressure, demonstrating that stable simultaneous solutions with severe pressure gradients are obtainable with this technique. Contents An electrically conducting fluid flowing in the presence of a transverse magnetic field may either be accelerated or decelerated, depending on the orientation of the pondermotive or Lorentz force yx£. If the flow is accelerated, as in an accelerator, energy (j-E) must be added to the fluid; whereas if the flow is decelerated, energy may be extracted from the fluid, as in an MHD generator. The governing equations describing the above phenomena consist of the fluid equations of motion: mass, momentum, energy, and equation of state, and the electromagnetic equations: Maxwell's relations and Ohm's law. The development of a viable numerical procedure operating in the flow regime of MHD generators is hampered not only by typical numerical obstacles inherent in both gasdynamics and electromagnetic methods, but also by the unique difficulties arising from the marriage of these procedures. Because of these difficulties, various schemes13 have been developed to solve the electromagnetic and gasdynamic equations for certain defined flowfields and configurations. In the present formulation, stable simultaneous solutions of the fluidic and electromagnetic equations are obtained. The governing fluidic equations are written as