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Low Magnetic Reynolds Number Hypersonic MHD Flow Using High Order WENO Schemes

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Low Magnetic Reynolds Number Hypersonic MHD Flow Using High Order WENO Schemes 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition AIAA 2009-459 5 - 8 January 2009, Orlando, Florida Low Magnetic Reynolds Number Hypersonic MHD Flow Using High Order WENO Schemes Jaejin Lee,∗ Manuel A. Huerta,y and Gecheng Zhaz University of Miami Coral Gables, Florida 33124 In this paper we incorporate a low magnetic Reynolds number MHD model to a high order CFD algorithm with WENO and a low diffusion scheme for 3D Navier-Stokes equa- tions. We present results for hypersonic laminar flows around a flat plate and around a blunt body, with different strengths of magnetic fields, to demonstrate the methodology. More simulations, including turbulence, are in progress and will be reported in the final paper. I. Introduction Hypersonic vehicles generate shocks that can heat the air sufficiently to partially ionize the air and create an electrically conducting plasma that can be studied using the equations of single fluid magnetohydrody- namics (MHD), also called magnetofluiddynamics (MFD). Introducing strong applied magnetic and electric fields into the flow could have beneficial effects such as reducing heat damage, providing a sort of MHD parachute, and generating electric power or thrust in the vehicle. Among the many authors that have done numerical simulations of hypersonic MHD flows we mention here Bityurin, Lineberry, Potebnia, et. al.,1 Bityurin and Bocharov,2 Bityurin, Zeigarnik and Kuranov,3 Fujino, Kondo and Ishikawa,4 MacCormack,5 and Hoffmann, Damevin, and Dietiker.6 To simulate hypersonic flow with MHD it is essential to have a flow solver that can capture shock waves and turbulence accurately. Recently a low diffusion E-CUSP (LDE) scheme with a fifth order WENO scheme was developed to resolve flow fields with shock discontinuities. The purpose of this paper is to incorporate the low magnetic Reynolds number MHD model to the CFD algorithm so that it can be used to simulate hypersonic flows with MHD effects. Realistic flows are turbulent, but in this paper we only show results for laminar low. In the final paper we will include turbulence. II. The MHD Model Single-fluid magnetohydrodynamics (MHD) describes an electrically conducting but electrically neutral fluid of density ρ, velocity u, pressure p, energy per unit mass e, viscosity tensor τ¯, and heat flux vector q. The electrical quantities are the magnetic field B, the electric field E, the current density J, and the electrical conductivity σ, which may be a scalar or a tensor, depending on the model. In principle σ should be calculated from an air chemistry model, such as in Ref. [5], but in this paper we assume that σ is given, and we do not discuss the air chemistry. The plasma is taken to obey Ohm's law J = σ(E + u × B): (1) The magnetic field is advanced in time using Faraday's law @B = −∇ × E: (2) @t ∗ Graduate Student, Department of Physics y Professor, Department of Physics, AIAA Member z Associate Professor, Department of Mechanical and Aerospace Engineering, Senior AIAA Member Copyright c 2008 by Baoyuan wang and Ge-Cheng Zha, Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. 1 of 17 Copyright © 2009 by the American Institute of Aeronautics andAmerican Astronautics, Institute Inc. All ofrights Aeronautics reserved. and Astronautics The Amp`ere-Maxwell equation, which could be used to advance E is @E µ = r × B − µ J: (3) 0 0 @t 0 However, MHD describes low frequency phenomena, in which the conduction current is much greater than the displacement current, that is, @E J : (4) 0 @t The displacement current is therefore neglected and we use the pre-Maxwell equation µ0J = r × B; (5) which gives J in terms of B. This means that E is not advanced in time, but is obtained in terms of B using Eqs. [1] and [5]. As a consequence of this, MHD cannot describe the usual type of high frequency electromagnetic waves. However, MHD does describe three interesting types of waves known as the Alfv´en wave and the fast and slow magneto-acoustic waves. Therefore E can be eliminated from Eq.(2) and we rewrite it as @B J 1 = −∇ × − u × B = r × (u × B) + r2B; (6) @t σ µ0σ which is the equation that advances B in time, and can be written in conservation form as @B 1 + r · (u ⊗ B − B ⊗ u) = r2B: (7) @t µ0σ The fluid quantities ρ, u, and e are advanced in time using the usual conservation of mass, @ρ + r · (ρu) = 0; (8) @t conservation of momentum, @(ρu) + r · (ρu ⊗ u) = −∇p + r · τ¯ + J × B; (9) @t which now includes the magnetic force density J × B, and finally e is advanced with the equation of conser- vation of energy, @ 1 1 ρ(e + u2) + r · ρe + u2 + p)u = −∇ · q + r · (τ¯ · u) + E · J: (10) @t 2 2 This equation is rewritten using Eq. [1] as @ 1 1 J 2 ρ(e + u2) + r · ρe + u2 + p)u = −∇ · q + r · (τ¯ · u) + + (J × B) · u: (11) @t 2 2 σ Another form of Eq. [10] is obtained by using Eqs. [2] and [5], @ 1 B2 1 E × B ρ(e + u2) + + r · ρ(e + u2)u + pu + = −∇ · q + r · (τ¯ · u); (12) @t 2 2µ0 2 µ0 from which we may eliminate E using Eq. [1], and we get 2 @ B BB · u J × B (ρet) + r · ρetu + p + u − + = −∇ · q + r · (τ¯ · u); (13) @t 2µ0 µ0 µ0σ where et is the total energy density. 2 1 2 B et = e + u + : (14) 2 2µ0 In Eqs. [9], [11], and [13] J can be eliminated using Eqs. [1] or [5] as the case may be. The system is closed by giving appropriate expressions for e; τ;¯ q, and σ in terms of the other quantities. Typically a simple form for e is used, p e = ; where γ = 1:4 for air: (15) (γ − 1)ρ 2 of 17 American Institute of Aeronautics and Astronautics III. The Full MHD Model and the Low Magnetic Reynolds Number Approximation The full MHD model presented in Section II advances in time the eight-dimensional flux vector [ρ, ρu, ρv; ρw; ρet;Bx;By;Bz], where u; v; and w are the x; y, and z components of u, using Eqs. [8], [9], [13], and [7] in conservation form. The only electrical quantities that appear in the equations are B; σ, and µ0, although the presence of E is felt in the boundary conditions. Several authors have treated the full 8-dimensional problem. For example, a nozzle problem with a generator section to extract power from the incoming flow, and an accelerator section to produce thrust, using a tensor conductivity was treated in Ref. [5]. Ref. [6] also used the full 8-dimensional MHD model to treat flows around a flat plate and around a blunt body with zero electric field. The full MHD 8-dimensional problem is conceptually clear, because the magnetic and electric fields, B and E are calculated. In this case J is obtained from Eq. [5], and E is obtained from Eq. [1], although the boundary conditions have an important effect on what E must be. The numerical simulations are complicated, however. The vehicle may have an electrically insulating surface and carry a magnet that applies a strong magnetic field Ba to the flow if the intention is to reduce surface heating and increase the drag force in a sort of MHD parachute. The vehicle may also have conductors on its surface, which change the boundary conditions, to try to generate electrical power by extracting energy from the flow. The vehicle might also have a source of Emf, Ea, and apply a strong electric field Ea, such as in a nozzle to produce MHD thrust. The total magnetic and electric fields in the plasma, B = Ba + Bi and E = Ea + Ei, are the sum of the applied and induced fields. However, the ionized air produced in the hypersonic flow is a poor electrical conductor. Therefore the electrical current J induced in the plasma is small, and the magnetic Bi generated by the plasma current is much smaller than the applied field. This allows the use of a low magnetic Reynolds number approximation. When expressed in dimensionless variables, the electrical conductivity σ enters the problem via the magnetic Reynolds number Rm = µ0σUref Lref ; (16) where Uref and Lref are appropriate reference values of speed and length in the problem. When Rm is small it is possible to treat the problem in a low Rm approximation, which neglects the fields Bi and Ei induced by the plasma. In this model only the five fluid quantities in the five-dimensional flux vector [ρ, ρu, ρv; ρw; ρe] need to be advanced in time using Eqs.[8], [9], and [11], with MHD source terms proportional to J. Refs. [1] - [4] used the low Rm approximation to study interesting effects, such as the MHD parachute and MHD 7 power generation. MacCormack showed that the full MHD solution agrees with the low Rm solution for 8 flow in a thrust producing nozzle for Rm = 0:17. Damevin and Hoffmann compared the full MHD solution including air chemistry with the low Rm approximation in flows over a hemisphere and a cylinder. Khan, Hoffmann and Dietiker9 used the full MHD model to treat flows with zero electric field around a flat plate and a blunt body and tested for agreement between the solutions of the full MHD model and the low Rm approximation. They obtained good agreement for values of Rm < 0:125. To understand the validity of the low Rm approximation we can think in terms of a sort of singular expansion in the small parameter Rm that would allow one to calculate the O(Rm) quantities that are neglected in the low Rm approximation.
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