Magneto-Fluid Dynamics Calculations for Aerodynamics

Improved capabilities for simulating flows in magnetic fields are now available.

Governing differential equations, and algorithms to solve the equations numerically, have been developed to enable computational simulation of weakly ionized aerodynamic flows in the presence of electromagnetic fields. The equations and algorithms are intended mainly for application to airflows about, and within the engines of, contemplated hypersonic vehicles. There, strong imposed magnetic fields and perhaps imposed electric fields might be used, variously, for reducing transfer of heat to solid surfaces, for extracting electric energy from flows, or for accelerating or decelerating flows to enhance combustion of fuel or otherwise increase energy efficiency.

In order to perform the desired simulations, it is necessary to account for effects that are not present in pure aerodynamics, including not only magnetic forces on flows but also magnetic induction and diffusion. In principle, it is necessary to solve the full set of governing equations of magneto-fluid dynamics (MFD), which include the Navier-Stokes equations of flow, Maxwell's equations for the electromagnetic field, the Lorentz-force equation, and a generalized form of Ohm’s law. In the case of a chemically reacting flow (e.g., an ionizing and/or combusting flow), it is also necessary to include governing equations of thermodynamics and conservation of chemical species. The full governing equations are typically accompanied by equations expressing simplifying assumptions — for example, the assumption that the fluid remains electric-charge neutral. However, simplifying assumptions can introduce significant errors, and even in the presence of simplifying assumptions, the task of solving the equations numerically can involve severe difficulty.

The present governing equations, denoted the reduced MFD equations, are mathematically equivalent to the full equations of MFD but are modified from traditional forms into forms more amenable to numerical solution. One salient feature of the modified formulation is that the imposed magnetic field is treated somewhat separately from the magnetic field induced by the flow, in such a manner as to reduce the number of arithmetic operations involving the imposed magnetic field. The algorithms for solving the reduced MFD equations include numerical procedures for casting the governing equations in forms that enhance numerical precision, and numerical procedures for application of boundary conditions, scalar and tensor formulations for generalized electrical conductivities accounting for Hall-current and ion-slip effects, calculation of equilibrium properties for an open-ended set component gas species, and suppression of spurious numerical divergence of the magnetic field.

This work was done by Robert W. MacCormack of Stanford University for the Air Force Office of Scientific Research.
AFRL-0080

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