# Development of an Ideal Magnetohydrodynamics Flowsolver for High Speed Flow Control

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## Abstract

This paper presents the baseline development of an ideal magnetohydrodynamics (MHD) solver towards enhancing the knowledge base on the numerical and flow physics complexities associated with MHD flows. The ideal MHD governing equations consisting of the coupled fluid flow equations and the Maxwell’s equations of electrodynamics are implemented in the three dimensional finite volume flowsolver, CERANS. Upwind flux functions such as AUSM-PW+, KFVS and the local Lax-Friedrichs schemes were used for solving the discretized form of governing equations. The solenoidal constraint which requires that the magnetic field to be divergence free all through the flow field evolution is ensured using the artificial compressibility analogy method or the Powell’s source term method. The code had been validated for standard MHD test cases involving complex flowfields such as the MHD shock tube, blast, vortex, cloud-shock interaction and cylinder shock interaction problems. The flow control effect of MHD interaction had been demonstrated for supersonic flow past a wedge and the results are compared with analytical results obtained by solving the MHD Rankine Hugoniot relations. Further, MHD flow control for high speed flows had been demonstrated for the hypersonic blunt body problem. Through rigorous testing and validation, it is observed that the CERANS-MHD code is able to mimic the complex flows due to MHD interactions and the comparison of results are found to be in good agreement with similar literature.

## Keywords

Ideal MHD CERANS Artificial compressibility analogy (ACA) Powell source Flow control## Notations

- A
Area of surface

- B
_{j} jth component of magnetic field intensity

- C
Characteristic speed

- e
Specific energy

- F
_{j} jth component of flux vector

- M
Mach number

- \({\hat{\text{n}}}_{\text{j}}\)
jth component of surface outward normal

- p, \({\text{p}}_{\text{t}}\)
Static and total pressure

- t
Time in seconds

- \({\text{S}}\)
Source term vector

- T
Fluid temperature

- \({\text{U}}\)
Conserved variable vector

- U
_{n} Contra-variant velocity

- u
Velocity vector

- V
Volume of cell element

- V
_{ref} Reference velocity for ACA method

- u, v, w
Velocity components along x, y, z directions

- x, y, z
Cartesian coordinate directions

- L, R
Left, right states

## Greek symbols

- γ
Ratio of specific heats

- ϕ
Divergence free variable for ACA method

- ρ
Density of fluid

- λ
Eigenvalue

- μ
_{o} Magnetic permeability of free space (4π × 10

^{−7}N/A^{2})

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