DEVELOPMENT AND NUMERICAL INVESTIGATION OF MAGNETO-FLUID-DYNAMICS FORMULATIONS

A Dissertation by

Ovais U. Khan

Master of Science, King Fahd University of Petroleum and Minerals, 2003

Bachelor of Engineering, NED University of Engineering and Technology, 2000

Submitted to the Department of Aerospace Engineering and the faculty of the Graduate School of Wichita State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy

December 2009 c Copyright by Ovais Khan 2009 ° All Rights Reserved DEVELOPMENT AND NUMERICAL INVESTIGATION OF MAGNETO-FLUID-DYNAMICS FORMULATIONS

The following faculty have examined the final copy of this dissertation for form and content, and recommend that it be accepted in partial fulfillmentoftherequirementforthedegree of Doctor of Philosophy with a major in Aerospace Engineering .

Klaus A. Hoffmann, Committee Chair

Leonard S. Miller, Committee Member

Roy Y. Myose, Committee Member

Hussein H. Hamdeh, Committee Member

Kamran Rokhsaz, Committee Member

Accepted for the College of Engineering

Zulma Toro-Ramos, Dean

Accepted for the Graduate School

J. David McDonald, Dean

iii DEDICATED TO

ALLAH subhanahu-wa-ta ala The most Merciful, The most Benevolent

iv ACKNOWLEDGEMENTS

Words cannot at all express my thankfulness to Almighty Allah, subhanahu-wa-ta ala,the most Merciful; the most Benevolent Who blessed me with the opportunity and courage to complete this task.

My heartfelt gratitude and special thanks to my thesis advisor learned Professor Klaus

A. Hoffmann. I am grateful to him for his consistent help, untiring guidance, constant encouragement and precious time that he has spent with me in completing this course of work. I do admire his exhorting style that has given me tremendous confidence and ability to do independent research. Working with him in a friendly and motivating environment was really a joyful and learning experience.

I must appreciate and thank Dr. J. F Dietiker for his extraordinary attention and thought- provoking contribution in my research. His assistance and encouragement can never be forgotten, working with him was really a good experience. I am also thankful to all the committee members, Dr. Myose, Dr. Miller, Dr. Hamdeh and Dr. Rokhsaz for their comments and suggestions.

I must appreciate and thank Professor Robert W. MacCormack from Stanford University for his constant guidance, constructive and positive criticism about my research. It was surely an honor and an exceptional learning experience to work with him.

v I am thankful to Dr. Datta Gaitonde and Dr. Jonathan Poggie from Air Force Research

Laboratory, Wright-Patterson Air Force Base, Ohio; for technical discussion and providing positive feed back about my research. I would also like to acknowledge the sponsorship of my research from the US Air Force Office of Scientific Research (AFOSR).

Sincere friendship is the spice of life. I owe thanks to my graduate fellow students from

Pakistan, India, France and the United States; most particularly, Arshed bhai, Hassan

Khurshid, Ikram bhai and Raheel bhai.

A special note of thanks to a person who contributed a lot in building up my academic skills and personality by precious advising, guidance, and encouragement.

Family support plays a vital role in the success of an individual. I must appreciate the efforts of my brothers and sisters who supported me a lot during my education. I am thankful to my entire family for its love, support and prayers throughout my life especially my dearest mother and father who devoted their lives in the endeavor of getting me a quality education. I am grateful to my parents for all the hardships in supporting their family and making bright future for their children.

May Allah help us in following Islam according to Quran and Sunnah ! (Aameen)

vi ABSTRACT

Magnetofluiddynamics(MFD)isthebranchoffluid dynamics that involves mutual interaction of electrically conducting non-magnetic fluids and magnetic fields. MFD offers promising advances in flow control and propulsion of future hypersonic vehicles. With the advent of computational fluid dynamics (CFD), the numerical study of inherently complicated fluid dynamics problems, such as flows at high velocities, high-temperature re-entry bodies, and mixed subsonic-supersonic flows, has become an interesting area of research. Further advancement in high-speed cluster machines and development of efficient algorithmshasmadeitpossibletoexploreMFDproblemsnumerically. In this work, development and validation of numerical algorithms for the simulation of MFD problems of supersonic and hypersonic flows have been conducted. Validity of low magnetic Reynolds number approximation has been checked with respect to the results obtained from full MFD equations. In addition to the two commonly used formulations for MFD, a third formulation based on the decomposition of a magnetic field for solving full MFD equations was explored. The governing equations were transformed to a generalized computational domain and discretized using a finite difference technique. A time-explicit multistage Runge-Kutta scheme augmented with total variation diminishing (TVD) limiters for time integration was implemented. The developed codes were validated with the existing closed form solution of the magnetic Rayleigh problem for both two- and three- dimensional cases. The results obtained from decomposed full MFD equations compare well with the results obtained by solving low magnetic Reynolds number approximation and classical full MFD equations for a wide range of magnetic Reynolds numbers. It is shown that the decomposed full MFD technique requires substantially less computation time compare to classical full MFD equations for the solution of flow fields with strong imposed magnetic fields. Finally, high-speed flows over a backward-facing step that is subject to an applied magnetic field were numerically simulated. The global domain of computation was

vii decomposed into upstream and downstream domains from the step location. The low magnetic Reynolds number approximation under a multiblock grid approach was used for modeling the backstep flow. Pressure distribution for the Navier-Stokes analysis was found to be in good agreement with the experimental data. Different types of magnetic field distributions were investigated. Both uniform and variable electrical conductivity distributions were considered. It was observed that an increase in the separation zone and displacement of oblique shock wave towards the exit section occurs subsequent to application of the magnetic field. A comparison of results obtained with uniform and variable electrical conductivities showed a reduction in magnetic interaction for variable electrical conductivity.

viii

LIST OF FIGURES

Figure Page

1.1Bowshockwaveinfrontofblunt-body...... 4 1.2 Comparison of flow field obtained without and with the application of mag- netic field...... 7

2.1 Hypersonic flowoverblunt-body...... 17 2.2 Illustration of supersonic flow fieldoverbackward-facingstep...... 39

3.1 Different types of boundary conditions for a typical external flow...... 75

5.1 Solution algorithm based on modifiedRunge-Kuttascheme...... 120 5.2Solutionalgorithmbasedonmultiblockapproach...... 131

6.1 Development of velocity profilesforMFDRayleighproblem...... 136 6.2 Comparison of numerical and analytical velocity distributions for different time intervals at Rm =2.5...... 139 6.3 Comparison of numerical and analytical induced magnetic field distributions for different time intervals at Rm =2.5...... 140 6.4 Comparison of velocities obtained by Full MFD and low magnetic Reynolds 3 number formulations at Rm =2.5 10− ...... 141 6.5 Comparison of induced magnetic fi×eld distributions obtained from Full MFD formulation for differentvaluesofmagneticReynoldsnumber...... 142 6.6 Percentage average error in velocities obtained by Full MFD and low mag- netic Reynolds number formulations for different values of magnetic Reynolds number...... 143 6.7 Wall clock time for full MFD and low magnetic Reynolds number formulations.144 6.8 Comparison of velocities obtained from exact solution and DFMFD formu- lation at differenttimeintervals...... 146 6.9 Comparison of induced magnetic fields obtained from exact solution and DFMFD formulation at differenttimeintervals...... 147 6.10 Comparison of velocities obtained by DFMFD, FMFD, and low magnetic 3 Reynolds number formulations at Rm =2.5 10− ...... 148 6.11 Comparison of induced magnetic fields obtained× from FMFD and DFMFD formulations for different time intervals at Rm =2.5...... 149 6.12 Wall clock time for FMFD, DFMFD, and low magnetic Reynolds number formulations...... 150 6.13 Comparison of wall clock times taken by FMFD and DFMFD formulations 3 for magnetic field strength of 1.422 10− TatRm =0.125...... 153 ×

xii 6.14 Comparison of wall clock times taken by FMFD and DFMFD formulations 2 for magnetic field strength of 1.3944 10− TatRm =0.125...... 154 6.15 Comparison of wall clock times taken× by FMFD and DFMFD approaches 3 2 for magnetic field strength of 1.422 10− TatRm =2.5 10− ...... 155 6.16 Comparison of wall clock times taken× by FMFD and DFMFD× approaches 2 2 for magnetic field strength of 1.3944 10− TatRm =2.5 10− ...... 156 6.17 Wall clock time for FMFD, DFMFD,× and low magnetic Reynolds× number formulations...... 158 6.18 Computational grid system for blunt body flow...... 160 6.19 Surface pressure for Navier-Stokes analysis...... 161 6.20 Pressure contours for Navier-Stokes computation...... 163 6.21 Imposed magnetic fieldalongy-direction...... 165 6.22 Comparison of pressure contours obtained from the Navier-Stokes, DFMFD and FMFD analyses...... 167 6.23 Surface pressure for DFMFD and FMFD formulations...... 168 6.24 Induced magnetic field streamlines’ distributions obatined from DFMFD and FMFD formulations...... 170 6.25 Velocity field for the Navier-Stokes and MFD analyses...... 171 6.26 Velocity field and total magnetic fieldvectors...... 172 6.27 Comparion of DFMFD, FMFD and low magnetic Reynolds number formu- lations...... 174 6.28 Comparison of FMFD, DFMFD and Low Rm formulations for Rm =1.79.. 175 6.29 Pressure contours obatined from DFMFD and Low Rm formulations for Rm =2.5...... 177 6.30 Comparison of surface pressures obtained from DFMFD and Low Rm for- mulations for Rm =2.5...... 178 6.31 Comparison of pressure contours obatined from DFMFD and Low Rm for- 2 mulations for Rm =2.5 10− ...... 180 × 6.32 Comparison of surface pressures obtained from DFMFD and Low Rm for- 2 mulations for Rm =2.5 10− ...... 181 6.33 Comparison of shock stand× off distances obtained from DFMFD and LowRm 2 formulations for Rm =2.5 10− ...... 183 6.34 Multiblock computational× grid for backward-facing step flow...... 186 6.35 Pressure contours obatined from the Navier-Stokes analysis...... 188 6.36 Comparison of pressure distribution with experimental data of [86]. . . . . 188 6.37 Uniform magnetic field distribution along with pressure contours obtained with different strengths of magnetic field...... 190 6.38 Pressure distributions along the horizontal surface for different values of uniform magnetic fieldstrengths...... 192 6.39 Magnetic field distributions generated by dipoles aligned with the x-axis and corresponding pressure contours obtained with different magnetic field strengths...... 194

xiii 6.40 Pressure distributions along the horizontal surface for different magnetic field strengthsofdipolesalignedwiththex-axis...... 195 6.41 Magnetic field distributions generated by dipoles pointed in the direction of y-axis and corresponding pressure contours obtained with different magnetic fieldstrengths...... 198 6.42 Pressure distributions along the horizontal surface for different magnetic field strengthsofdipolespointedinthedirectionofy-axis...... 199 6.43 Pressure contours obtained with variable electrical conductivity for different strengths of uniform magnetic fielddistribution...... 201 6.44 Pressure distributions along the horizontal surface for constant and variable electricalconductivitydistributions...... 202 6.45 Streamline patterns for the Navier-Stokes and MFD computations with dif- ferent magnetic fielddistributions...... 204

xiv LIST OF TABLES

Table Page

6.1 Operating conditions for MFD Rayleigh problem used for code validation. 137 6.2 Percentage reduction in wall clock time for two strengths of magnetic fields versusMagneticReynoldsnumber...... 157 6.3 Operating conditions for blunt-body hypersonic flowproblem...... 159 6.4 Geometry and operating conditions for supersonic backward-facing step flow. 186

xv NOMENCLATURE

m a Speed of sound s A, B, C Flux Jacobian matrices¡ ¢ in x, y, and z directions, respectively

A, B, C Flux Jacobian matrices in ξ, η, and ζ directions, respectively

Bx ⎧ ⎫ B→ = Magnetic field vector (Tesla) ⎪ By ⎪ ⎪ ⎪ ⎨⎪ ⎬⎪ Bz ⎪ ⎪ ⎪ ⎪ m c ⎩⎪ ⎭⎪ Speed of light s CFL Courant-Friedrichs-Lewy¡ ¢ number cp Pressure coefficient

Ex ⎧ ⎫ E→ = ⎪ E ⎪ Electric field vector Volts ⎪ y ⎪ m ⎪ ⎪ ⎨ ⎬ ¡ ¢ Ez ⎪ ⎪ ⎪ ⎪ J et ⎩⎪ ⎭⎪ Total energy per unit mass Kg ³ ´ E = E Convective flux vector in the x direction 8 1 { } × E = E Convective flux vector in the ξ direction 8 1 × E =© Eª Viscous flux vector in x direction v v 8 1 { } ×

Ev = Ev Viscous flux vector in ξ direction 8 1 × F = ©F ª Convective flux vector in the y direction 8 1 { } × F = F Convective flux vector in the η direction 8 1 × F =© Fª Viscous flux vector in y direction v v 8 1 { } ×

Fv = F v Viscous flux vector in η direction 8 1 × © ª

xvi G = G Convective flux vector in the z direction 8 1 { } × G = G Convective flux vector in the ζ direction 8 1 × G =© Gª Viscous flux vector in z direction v v 8 1 { } ×

Gv = Gv Viscous flux vector in ζ direction 8 1 × g,h,i © ª TVD limiters in ξ, η and ζ directions, respectively

H = H Additional magnetic source term vector 8 1 { } × → → HM Intermediate magnetic vector, H = HM ( B) 5· = I Identity tensor

J Jacobian of transformation

→ Ampere J Current density vector m2 ¡W ¢ k Thermal conductivity mK L Characteristic length ¡(m) ¢

M Mach number

N p Pressure m2

Pr Prandtl number,¡ ¢ Pr = cpµ/k

Prm Magnetic Prandtl number, Prm = ν/νe = µeoσe/ρ

qx ⎧ ⎫ →q = Surface heat flux vector ⎪ qy ⎪ ⎪ ⎪ ⎨⎪ ⎬⎪ qz ⎪ ⎪ ⎪ ⎪ ⎩⎪ qJx⎭⎪ ⎧ ⎫ →q = Joulean heat flux vector J ⎪ qJy ⎪ ⎪ ⎪ ⎨⎪ ⎬⎪ qJz ⎪ ⎪ ⎪ ⎪ Q = ⎪Q ⎪ Unknown field vector (conservative variable vector) ⎩ 8 1 ⎭ { } × Q= Q Unknown field vector in computational space 8 1 × © ª

xvii Q´= Q´ Primitive variable vector 8 1 { } × 2 σeB L Q = ρU Magnetic interaction parameter rb = Body radius

B2 Rb Magnetic pressure number, Rb = 2 ρµeoU ρUL Re Reynolds number, Re = µ

Rm Magnetic Reynolds number, Rm = µeoσeUL

SMFD = SMFD 5 1 Electromagnetic source term in low magnetic { } × Reynolds number approach t Time (second)

T Temperature (Kelvin)

m U Magnitude of velocity vector s U,V,W Contravariant velocities in ξ,¡ η, and¢ ζ directions

u

→ ⎧ ⎫ U = ⎪ v ⎪ Velocity vector ⎪ ⎪ ⎨⎪ ⎬⎪ w ⎪ ⎪ ⎪ ⎪ va ⎩⎪ ⎭⎪ Alfvén wave velocity vf Fast magneto-acoustic wave velocity vs Slow magneto-acoustic wave velocity x, y, z Cartesian coordinates

X, Y, Z Right eigenvector matrices in x, y, and z directions

X,Y,Z Right eigenvector matrices in ξ, η, and ζ directions

1 1 1 X− , Y− , Z− Left eigenvector matrices in x, y, and z directions

1 1 1 X− ,Y− ,Z− Left eigenvector matrices in ξ, η, and ζ directions

xviii Greek Symbols

∆ Shock standoff distance (m)

δ Entropy correction parameter in TVD scheme

δ Weight term in Eqs. (5.32 to 5.34)

δij Kronecker delta

εe Electric permittivity of a medium

Farads 9 1 εeo Electric permittivity of vacuum , εeo =1/(36π.10 ) F m− m ·

γ Ratio of specificheat(cp/cv) ¡ ¢

ξ,η,ζ Generalized coordinates

λξ,λη,λζ Eigenvalues of A, B, and C, respectively

Kg µ Laminar dynamic viscosity of fluid ms ¡ ¢ µe Magnetic permeability of the medium

Henry 7 1 µ Magnetic permeability of vacuum , µ =4π.10− H m− eo m eo · m ¡ ¢ ν Kinematic viscosity s2 ¡ ¢ 1 νe Magnetic diffusivity, νe =(µeoσe)− ψ Entropy correction function

Φ, Θ, Λ Flux limiter function vector associated with A, B, and C, respectively

Kg ρ Density of fluid m3 ¡ ¢ Columbs ρe Electric charge density m3 ¡ ¢ Siemens σe Electrical conductivity of the medium m N ¡ ¢ τ Wall shear stress m2 = τ Viscous stress tensor¡ ¢

xix Indices i, j, k Einstien indicial notations, or indexes used in computational

domain for ξ, η, and ζ directions, respectively

IMAX,JMAX,

KMAX Maximum indexes in ξ, η, and ζ directions, respectively

Subscripts a Alfvén wave d Divergence wave e Electromagnetic quantity or entropy wave f Fast acoustic wave i Induced magnetic field o Imposed magnetic field ref Reference or free stream condition s Slow acoustic wave t Total magnetic field

Free stream condition ∞ v Diffusion quantity

Superscripts n Iteration level

Non-dimensional quantity ∗ Operators

→ Nabla vector ∇ Dot product · Cross product × Dyadic product ⊗

xx Abreviations and Acronyms

1-D One-dimensional

2-D Two-dimensional

3-D Three-dimensional

CFD Computational fluid dynamics

AUSM Advection upstream splitting method

CFL Courant-Friedrichs-Lewy number

DNS Direct numerical simulation

DFMFD Decomposed full magnetofluiddynamics equations

FMFD Full magnetofluiddynamics equations

LIIF Laser-induced fluorescence

LDV Laser Doppler velocimetry

MAD Magnetoaerodynamics

MFD Magnetofluiddynamics

MGD Magnetogasdynamics

MHD Magnetohydrodynamics

PIV Particle image velocimetry

RANS Reynolds Average Navier-Stokes

RK4 Four-stage Runge-Kutta scheme

TVD Total variation diminishing

TsAGI The Central Aerohydrodynamic Institute, Moscow

WT Wind tunnel

xxi Chapter 1

INTRODUCTION

Magnetohydrodynamics (MHD) or more generally Magnetofluiddynamics (MFD) is the branch of fluid dynamics that involves mutual interaction of electrically conducting non-magnetic fluids and magnetic fields. Examples of such fluids include hot ionized gases

(plasma), strong electrolytes, and liquid metals. The term MHD is not general because it implies that the subject pertains to applications in water or incompressible fluids. Other nomenclatures, such as magnetogasdynamics (MGD) or magnetoaerodynamics (MAD), have been utilized to embrace most ionized gaseous and compressible media. Theodore von Kármán [1] suggested a broad and all-inclusive term, magnetofluidmechanics, which appeals most to the generalist and is the basis of the term magnetofluiddynamics.

From a historical point of view, it appears that the firstattempttoinvestigatetheMFD problem was by Faraday, who designed an MHD converter in 1860. On the theoretical side, the first problem of MFD was treated by Hannes Alfvén in 1940.Infact,thefield of MHD was initiated by Hannes Alfvén, for which he received the Nobel Prize in 1970.

Formally, MFD deals with the combined effects of fluid and electromagnetic forces, and has been an area of research since the late 1930s. Several scientific areas have interest in magnetofluiddynamics. Geophysicists study the terrestrial magnetic field generated by fluid

1 motion in the earth’s core. The fluid core of the earth and other planets is hypothesized to be a huge MHD dynamo that maintains the terrestrial magnetic field due to the motion of the molten rock. Astrophysicists work on different kinds of magnetic fields present throughout the universe, e.g., the galactic magnetic field, which is supposed to influence the formation of stars from interstellar clouds, and solar magnetic fields, which generate sunspots and solar flares. Plasma physicists use the MHD technique for controlling the stability of plasmas confined by magnetic fields in thermonuclear fusion reactions.

The first engineering application of MFD is due to Hartmann and Lazarus [2, 3] from their well-known experiment of mercury flow in the presence of a homogenous magnetic

field, which has attracted considerable attention in propulsion. Subsequently, investigators began to explore the influence of magnetic fields in different areas of engineering. Well- known examples include the damping, modification, and even suppression of turbulence in a variety of flows. The material processing industry employes MHD, where Lorentz force provides a non-intrusive means of controlling the flow of metals. Likewise, magnetohydro- dynamics provides a unique means of controlling the casting and refining processes, so that

fine quality material with minimum cost can be produced. Metallurgists constantly use the

MHD technique to heat, pump, stir, and levitate liquid metals. In addition, the electric power generation system, and design of heat exchangers, pumps, and flow meters are a few examples where MHD principles have been utilized.

Following several successful applications related to the control of liquid-metal flows through the application of electromagnetic fields, many fluiddynamists and aerodynamists begantorealizethatmagneticfields could be wisely utilized for attaining an efficient control over a range of high-speed flows that occur in numerous engineering applications.

2 1.1 High-Speed Flows

High-speed flows encompass a broad range of Mach numbers that vary from subsonic to supersonic values. Significant physical and mathematical differences exist between subsonic and supersonic flow regimes. One of the important characteristics of supersonic flowsisthe formation of strong shock waves, which forewarn the uniform free stream of the existence of the body. Figure 1.1 shows the formation of a bow shock wave in front of a conical blunt object subjected to a supersonic flow regime. Flow properties, such as temperature, pressure, and density, increase abruptly behind the shock wave.

As the free stream Mach number increases to higher supersonic speeds, the values of these parameters also increase. Subsequently, for a higher range of free-stream Mach num- bers, the shock wave moves nearer to the body, and the flow field between the shock and the surface becomes extremely hot–in fact, sufficiently hot to induce ionization or dissociation of the gas particles. This high temperature and chemically reacting flow is categorized as hypersonic flow, which is one of the most critical classes of high-speed flows occurring in aerospace engineering. Hypersonic flows take place at a relatively high supersonic Mach number: the limit at which chemical effects become significant. In addition, hypersonic

fluids carry a large amount of kinetic energy. Within the boundary layer, these flows are slowed down by the frictional effects, and as a result, a substantial part of the kinetic en- ergy is transformed into internal energy of the fluid–the well known viscous dissipation phenomenon; consequently, an increase in temperature occurs within the boundary layer.

3 Figure 1.1: Bow shock wave in front of blunt-body.

4 Subsequently, viscous effects become important due to proportionality of the dynamic viscosity coefficient with the temperature and cause thickening of the boundary layer. An extremely thick boundary layer produces a major displacement effect on the outer inviscid

flow. In turn, changes in the inviscid flow feed back to influence the boundary layer growth.

This key interaction between the boundary layer and the inviscid flow is usually known as viscous interaction and may have great impact on the surface pressure and, therefore, on lift, drag, and stability of the vehicle. In addition, skin friction and heat transfer rates to the surface within the boundary layer are also increased, due to viscous interaction and high temperature of the chemically reacting flows, and become crucial parameters for hypersonic

flow regime. In fact, aerodynamic heating results in high-heat transfer rates to the surface, and may cause vaporization of the surface layer or shielding in the case of hypersonic flight, and creates high temperature variations within the structural materials. Both viscous interaction and aerodynamic heating are imperative characteristics of hypersonic flows.

Therefore, it is important to control these parameters for achieving an economical and efficient flight. Among various available techniques for controlling high-speed flows, MFD control has gained considerable popularity in the last few decades and will be discussed in subsequent sections.

5 1.2 MFD Control of High-Speed Flows

Based on previous research efforts, it has been recognized that control through the application of magnetic/electric fields offersapotentialbreakthroughinbothhypersonic vehicle design and propulsion. Depending upon the flow conditions and certain Mach num- ber values, high temperatures in the post-shock region can cause ionization and dissociation of the fluid particles, which increases the electrical conductivity of the fluidinthepost- shock region. Thus, it becomes possible to control the flow by applying a suitable type of magnetic field distribution. This interaction of imposed magnetic field with ionized gas is expected to produce beneficial effects on drag, skin friction, and heat transfer for hy- personic vehicles. The ion motion caused by the applied and self-induced magnetic fields can create strong electrical currents in the shock layer and substantially change the energy balance in the flow. Consequently, the shock wave standoff increases, and flow gradients within the shock layers are reduced. Figure 1.2 shows the shock structure that develops around a blunt body subjected to hypersonic flow, with and without imposed magnetic

field distribution of a dipole placed at the body center. An overall enlargement of the bow shock wave structure is obvious for MFD analysis.

Magnetic and electric fields may also be utilized for enhancing combustion and improv- ing performance of a propulsion system for a hypersonic vehicle. By placing an MFD gen- erator ahead of the combustor, the electrical energy from the ionized flow can be extracted before entering the combustor. Consequently, air velocity is reduced without causing signif- icant loss in total pressure typically generated by conventional shock wave retardation; thus, full combustion can be achieved. Subsequently, the extracted energy is then returned back into the gas for further acceleration by placing an MFD accelerator after the combustion.

6

Figure 1.2: Comparison of flow field obtained without and with the application of mag- netic field.

7 A model of global range hypersonic aircraft AJAX [4,5] is a prospective concept of MFD propulsion application where coupled MFD generator and accelerator would be employed.

1.3 Historical Background of MFD Control

The concept of electromagnetic control of hypersonic flows originated during the mid

1950s. Most of the studies reported at that time were theoretical, semi-analytical, and empirical. A variety of system concepts were proposed to reduce local stagnation heat transfer, to generate asymmetric forces for altitude control, and to increase drag (desirable for re-entry) and overall heat reduction for the re-entry of a vehicle, thereby enabling the vehicle to pass more efficiently and safely through the atmosphere. Furthermore, results of early work on supersonic MFD aerodynamic flow control around blunt bodies showed an increase in shock standoff distance and a decrease in heat transfer and skin friction withtheincreaseofmagneticfield strength. However, research in this area tapered off by the early 1960s, because significant deficiencies associated with the magnetic system led investigators to realize that aerospace MFD control was not as efficient compared to other available techniques at that time [6]. For example, it was recognized that a very heavy and large magnet system was necessary to generate the magnetic field of desired strengths for controlling the flow around an air vehicle primarily due to low levels of air electrical con- ductivity. Another impediment was the cooling requirements of the electromagnet, which were too extreme for their realistic use. In brief, the practical and technical limitations on magnetic system design along with successful improvements in the design of ablative ther- mal protection technology abandoned pursuit of this tool in aerospace engineering during that era.

8 However, due to recent technological improvements in the design of superconducting magnets and with the development of artificial ionization techniques for enhancing the electrical conductivity of gas, MFD flow control has received reconsideration in aerospace community. Along with this, rapid advancements in high-speed clusters technology since the late 20th century and development of higher-order numerical schemes have made the available numerical techniques popular among fluid dynamists, so that computational fluid dynamics (CFD) has emerged as a new area of research, especially, for complex and in- herently difficult fluid dynamics problems, such as high velocity, high temperature re-entry bodies, and mixed subsonic-supersonic flows. Subsequently, CFD has been utilized to solve magnetofluiddynamic problems. Several computational investigations have been devoted to understanding the aspects of high-speed MFD flow fields with different types of applied electric and magnetic field distributions.

1.4 MFD Modeling

Substantial efforts, both experimentally and theoretically, have been dedicated to inves- tigate high-speed classical MFD flows for the design of hypersonic vehicles and propulsion systems. Notable interest has been paid to exploring the effects of applied electromagnetic

fields particularly on velocity and temperature profiles, shock standoff distance, and total drag.

Mathematically, MFD problems are modeled by a combination of Maxwell’s equations and Navier-Stokes equations, with added magnetic stress tensor. The combined set of these equations is known as complete MFD equations. Earlier, unavailability of fast-computing

9 machines forced scientists to solve these equations either analytically or semi-analytically for simple flows with extremely idealized models due to mathematical complexities asso- ciated with nonlinear governing MFD equations. Although, these investigations revealed qualitative aspects of the problems, it was found that there were significant limitations.

In fact, analytical solutions involve several simplifications and restrictive assumptions that limit their range of applicability to a relatively simple and limited number of problems.

On the other hand, experimental efforts produce reliable data with reasonable accuracy, but they are very expensive and limited to conduct. Furthermore, experiments are per- formed either on full-scale or small-scale models. Full-scale experiments are very pricey and sometimes very difficult to conduct, whereas small-scale experiments require extrapolation and may not reflect the features of the actual model.

With the advent of high-speed computers and the development of efficient higher-order algorithms, numerical techniques have been recognized as a cutting-edge tool for obtaining practical solutions to a myriad of fluid engineering problems. Fast computing, low cost and ability to simulate ideal and realistic conditions efficiently provide numerical methods an edge over other means of investigation. Numerical investigations are very popular to- day because it has become possible to explore complex MFD flows under less-restricting conditions on magnetic field, flow properties, geometry, and boundary conditions.

10 1.5 Present Study

Two approaches are used to model the effects of a magnetic fieldonanionizedflow: (1) the classical full MFD equations, including magnetic advection and diffusion effects, and

(2) the addition of Lorentz force and Joule heating effects in the Navier-Stokes equations as source terms—low magnetic Reynolds number approximation. Choosing between the two approaches depends upon the level of electrical conductivity of the gas. The most general case uses the classical full MFD set of equations that is composed of Maxwell’s equations along with fluid dynamics equations containing electromagnetic terms. The system de- scribes the evolution of eight scalar unknown–density, three components of velocity, three components of magnetic field, and energy. A special CFD solver based on an eight-wave structure is required to achieve the solution of this system of equations. On the other hand, when levels of electrical conductivity are very low, the second approach, known as the low magnetic Reynolds number approach, in which the magnetic effects are included via source terms with no provision for magnetic induction, is generally utilized. This method is much simpler and more efficient, even at substantially small values of electrical conductivity and strong magnetic fields. An ordinary CFD code can be utilized to solve this system of equa- tions, provided that the source terms are included. Further details of the two formulations are discussed in Chapter 3.

The current research activity developed and implemented efficient and accurate algo- rithms for the simulation of various MFD problems using both MFD modeling approaches.

A variety of computer codes having the features of conventional and multi-block strategies were developed and implemented to solve high-speed flow problems of a complex nature

11 under the influence of a magnetic field. Different types of magnetic field distributions were considered, and flow simulations were compared with the existing solution whenever available. The following major objectives for the present research work are outlined below:

1. An extensive literature review on MFD indicates that most investigators have used primarily two types of boundary conditions at the surface:

(a) An electrically insulating wall, which is equivalent to an applied magnetic field.

(b) A perfectly conducting wall, which is equivalent to specifying zero normal derivatives of all magnetic components.

These boundary conditions have shown great numerical difficulties in the computation of blunt body hypersonic flows at significantly low values of electrical conductivity when utilized with full MFD equations. The numerical schemes for full MFD equations have demonstrated significant instability at low values of magnetic Reynolds numbers. It is important to mention that magnetic induction causes many changes in magnetic field dis- tribution at the body surface. Therefore, it is necessary to specify the boundary condition of the magnetic field that represents the real physics of the problem. Moreover, a correct evaluation of magnetic components at the boundary will not only provide a correct solu- tion but also reduce the numerical difficulties at critical values of electrical conductivity associated with magnetic induction.

Thus, the first task of this present effort was the development of proper and realistic boundary conditions for a magnetic field over the body surface by utilizing Ohm’s law and the divergence free characteristic of a magnetic field for full MFD equations.

12 2. Full MFD equations are more difficult to solve and require a substantial amount of computational time and resources even for simple aerodynamic flows. The second choice, the low magnetic Reynolds number formulation, is far more efficient and has fewer numeri- cal difficulties related to magnetic effects than the first choice. However, some uncertainties exist relative to the authenticity and range of applicability of low magnetic Reynolds num- ber formulation for aerodynamic flows over simple geometries. It is important to mention that for certain aerodynamic configurations and operating conditions, the low magnetic

Reynolds number approach–the second objective of current research activity–has not been validated. Thus, validity of low magnetic Reynolds number approximation was evalu- ated for low and high values of magnetic Reynolds numbers (Rm) by conducting numerical experimentation with both the full MFD and the low magnetic Reynolds number formula- tions.

3. The solution of full MFD equations is a challenging task at low values of electrical conductivity and strong imposed magnetic field, even for simple two-dimensional geome- tries of general interest. Low levels of electrical conductivity result in exceedingly high coefficients of magnetic diffusion in the magnetic induction equation; as a consequence, the order of magnitude of the diffusion term becomes several times higher than the production term, even in the non-dimensional form of the induction equation. Eventually, strong stiff- ness occurs due to imbalance in diffusion and production terms, and the numerical scheme becomes highly unstable. Another major source of numerical instability is the strong mag- netic field strength. The strong magnetic fieldrequiredforgeneratingaLorentzforceof significant strength, with the expected low electrical conductivity of air, can cause a large

13 order of magnitude of the magnetic pressure as compared to the static pressure because magnetic pressure depends upon the square of magnetic field. Therefore, numerical diffi- culties arise due to inequality of production and diffusion terms of the magnetic induction equation and large differences in the order of magnitude of magnetic and static pressures.

In 2003, MacCormak [7] reported that the numerical significance of static pressure and a usually much smaller induced magnetic field can be lost by carrying the imposed mag- netic field within the derivative terms of full MFD governing equations. He introduced the concept of splitting the total magnetic field vector into imposed and induced components, and proposed that by utilizing the divergence and curl-free nature of the imposed magnetic

field, the products of the imposed magnetic field could be eliminated within the derivative terms. Based on this strategy, an alternative form of full MFD governing equations has been developed and implemented. Hence, the third objective of the present work was to imple- ment this alternate form of governing equations, “Decomposed Full MFD Equations” using the explicit modified Runge-Kutta scheme. First, formulation of the governing equations and development of the magnetic field boundary conditions were performed. Subsequently, eigenvalues, and eigenvectors associated with this formulation were evaluated. Finally, a computer algorithm based on a fourth-order modified Runge-Kutta scheme augmented with a total variation diminishing (TVD) limiter in the post-processing stage was developed for the decomposed full MFD equations. An explicit time marching numerical scheme was utilized to achieve the solution for different flow problems. Details of the “Decomposed

Full MFD” governing equations are discussed in Chapter 4.

14 4. Finally, the results obatined from numerical simulations were compared. The developed computer algorithms were validated with respect to the existing closed-form solutions and available normal shock data in the literature. Furthermore, time studies for evaluating the performance of classical full MFD equations, low magnetic Reynolds number approximation, and decomposed full MFD equations were conducted with weak and strong imposed magnetic fields. Subsequently, computations were performed with significantly low values of magnetic Reynolds numbers and with strong imposed magnetic

fields for demonstrating the capability of decomposed full MFD equations to efficiently model hypersonic blunt body flow.

5. In the final section of this research activity, supersonic flows over a backward- facing step were investigated under the influence of applied magnetic field. Within the last few decades, several experimental and theoretical investigations have been undertaken to determine the flow characteristics and associated shock/boundary layer interaction phe- nomenon for supersonic separated flows over a backward-facing step. However, little at- tention has been paid over the electromagnetic control of supersonic separated flows over a backward-facing step. Therefore, in this investigation the effects of applied magnetic field on supersonic separated flows over a backward-facing step have been explored. Low mag- netic Reynolds number approximation under multiblock grid approach is utilized. Various types of magnetic field distributions have been imposed and both uniform and variable electrical conductivity distributions have been considered.

15 Chapter 2

LITERATURE SURVEY

2.1 Introduction

After several decades of dormancy, the application of the electromagnetic field has received renewed reconsideration in the aerospace society as a tool for modifying heat transfer, drag, skin friction, and shock wave location for hypersonic flows over various configurations, e.g., blunt-body, compression-expansion corner, channels, and backward- facing step. Several analytical, semi-analytical, and experimental studies have been devoted to understanding high-speed flows under different types of magnetic field distributions for internal as well as external flows. The following literature review was preformed with respect to primarily hypersonic/supersonic blunt-body and backward-facing step flows.

2.2 Blunt-Body Flows

Objects in hypersonic flow are subjected to severe aerodynamic heating and pressure loads. Due to high heating, the geometry of aerospace objects in hypersonic flows have blunt shapes, e.g., leading edges of airfoils, wings, and noses of aircrafts and missiles. Figure 2.1 shows some features of a typical hypersonic flow over a blunt body. Flow between the shock and body becomes quite complicated: high temperatures can initiate chemical reactions

16 and ionization, and a strong curved-bow shock wave results in mixed supersonic-subsonic

flow in the downstream region. Therefore, flow control becomes a critical task for reducing the drag and heat transfer on the body.

Oblique shock wave

Body surface

M > 1 Uniform free stream

Mα > 1 M < 1 R

Stagnation line δ Stagnation point e in cl ni So

Figure 2.1: Hypersonic flow over blunt-body.

17 In 1958, following preliminary suggestions that electromagnetic waves could be utilized for controlling the high-speed flows, Bush [8] performed an analytical study of hypersonic

MHD blunt-body flow. A semi-analytical model was proposed to investigate the steady- state hypersonic flow near the stagnation region of an axisymmetric blunt body. By as- suming a spherical shock wave in front of the body, the solution had been initialized and marched toward the solid body. The low subsonic flow in the shock layer between the shock and body was treated as incompressible and nonviscous. Negligibly small electrical con- ductivity was assumed in the free-stream region before the shock wave; however, a uniform distribution of electrical conductivity had been assumed in the shock layer. A dipole at the center of the body was utilized for generating the magnetic field distribution. Without considering conduction heat transfer, fluid dynamic equations with magnetic effects were solved. With the introduction of suitable functions for the magnetic field and velocity, based on the similarity principal, the governing partial differential equations were trans- formed into ordinary differential equations to a point where they could be solve numerically.

Both cases of small and vanishing magnetic Reynolds numbers were considered for MHD analysis. An increase in shock standoff distance and a decrease in heat transfer rate to the body were predicted through magnetic field interaction.

Subsequently, other investigators developed their models on the basis of Bush’s theory.

For example, Wu [9] introduced viscosity effects in a model similar to Bush’s. Constant vis- cosity, density, and electrical conductivity distributions were considered. For MHD analysis, a decrease in the magnitude of the alteration to shock standoff distance was observed due to the effects of viscosity.

18 Lykoudis [10] developed an analytical solution for shock standoff distance similar to the

Bush analysis but assumed the Newtonian pressure relation. Hypersonic flows of an electri- cally conducting fluid around a sphere and a cylinder were considered with the assumption of constant radial magnetic field. It was reported that shock wave standoff distance not only depends on magnetic interaction but also the density ratio across the shock. Lyk- oudis [11] further utilized a mean value of the magnetic field, because a practical magnet would have a decreasing radial field; therefore, this fact was incorporated by using a mean value of the field.

PaiandKornowski[12]considered a variation of the magnetic field, specified at the

3 bow shock wave, followed by power of the radius (as for a dipole r− ); however, the results of pressure for different exponential values showed the same qualitative behavior as Bush’s prediction. Smith, Schwimmer and Wu [13, 14] considered the effects of viscosity for mag- netohydrodynamic blunt body flows in their investigation using a similar method as Bush.

Ericson and Maciulaitis [15] applied a similar analysis to that of Bush to the problem of magnetoaerodynamic lift.

In addition to blunt body configurations, other geometries were also considered for theoretical MHD research in late 1950s. For example, Rossow [16] obtained an approximate solution for incompressible laminar flow with constant electrical conductivity over a flat plate in the presence of a uniform magnetic field normal to the plate. Lykoudis [17] obtained a similarity solution for the MHD boundary layer over a wedge with constant electrical conductivity and negligible induced magnetic field. By considering variable thermodynamic and transport properties, including electrical conductivity of the gas, Bleviss [18] explored

19 the magnetic hypersonic Couette flow and obtained an exact solution that explained some important features of the boundary layer. Bush [19] analyzed a two-dimensional steady magnetic hypersonic flow over a flat plate and Couette flow. A qualitative comparison of the two flows that showed some similarities and dissimilarities was made. Chu and Lynn [20] and Mimura [21] investigated MHD flows over a wedge. In addition to the above-mentioned works, other theoretical MHD studies have been devoted to stagnation point flow and flows over a flat plate and blunt body in the late 1950sand1960s in [22] to [39].

Parallel to theoretical research, some experimental work was started in the late 1950s for exploring this complex and interesting class of flows.

Ziemer [40] was the first to conduct an experiment that investigated the effects of an applied magnetic fieldonthebowshockstandoff distance. A three-inch diameter elec- tromagnetic shock tube was utilized for the experiments. Initially, a discharge of electric current was used to ionize the air at one end of the tube, followed by a large capacitor dis- charge through the ionized air. A model of a 2.0 cm diameter cylinder with a hemispherical nose of 1.2 cm radius made of Pyrex glass was placed in the tube. A dipole-type magnetic

field was generated by placing a coaxial pulsed magnetic copper coil within the body nose; amagneticfield of strengths up to 4.0 T at the stagnation point was produced with the setup. The time period for steady flow was on order of 10 microseconds, and during this time the free stream Mach number was 2.72. However, the initial planar shock wave within the tube passed the model at about Mach 20.Theshockwavestandoff distance was scaled from the photographs of bow shock taken from a high-speed image convertor camera, which was utilized to capture the image of the incident shock past the blunt body and during the

20 steady equilibrium air flow. The shock wave standoff distance was found to increase with increasing strengths of the applied magnetic field.

Subsequently, Ziemer and Bush [41] compared theoretical and experimental magne- toaerodynamics results for spherically nosed cylinder with dipole-magnetic field strength of 3.5 T at the stagnation point. Reasonable agreement in the values of shock standoff distance, obtained through experimental and theoretical analyses, were shown.

In 1966, Seemann and Cambel [42] conducted tests on spherically capped cylinders in a continuous arc-heated chamber. An internal electromagnetic coil, composed of copper magnetwirewoundonacoreofmagnetingotiron,wasplacedwithinthebodynosefor generating a dipole-type magnetic field distribution. An increase in standoff distance and drag with the increase in interaction parameter was reported. They showed that agreement of the experimental data for shock standoff distance with the theories of Lykoudis and

Bush could not be achieved due to the existence of different density ratios. However, agreement with these theories was found by multiplying the magnetic interaction parameter by the shock density ratio. It was reported that shock standoff distance depends on both the interaction parameter and the density ratio. They suggested that the assumption of constant density in the shock layer could be accepted.

Nowak et al. [43] explored the same problem of Seemann and Cambel [42] for an ex- tended range of pressure, density, and interaction parameter. Again, the product of density ratio and the interaction parameter was utilized for comparison of experimental data to theoretical prediction. Results for shock standoff revealed that viscous and Hall effects were important. Similarly, the results for drag showed that viscous effects were important and that the Hall effect could not be neglected for large values of the interaction parameter.

21 Kranc et al. [45] performed a laboratory investigation of MHD supersonic flow around axisymmetric blunt bodies of hemispherical and flat front-end noses. A solenoidal magnet aligned with the flow axis was utilized for generating the magnetic field. They observed an increase in total drag with the application of magnetic field due to an increase in Lorentz drag; however, a decrease in pressure drag and viscous drag was predicted. Likewise, an increase in shock wave standoff was also reported for MHD analysis. It was stated that nonequilibrium effects had a tendency to keep electron temperature constant behind the shock, thus justifying the assumption of uniform electrical conductivity in this region. It was found that the experimental results were not in agreement with the Hall effect theory; thus, an emphasis was drawn in developing a theory that considers the Hall effect along with ion slip for strong strengths of magnetic fields.

The results obtained through analytical and experimental investigations were prelimi- nary and quantitative and demonstrated the possibility of magnetic control. However, defi- ciencies of the magnetic system combined with successful development of ablative thermal protection technology turned off enthusiasm for this technique in aerospace engineering by the mid-1960s. More than half a century since the MFD flow control was proposed, techno- logical improvements in the design of superconducting magnets and with the development of artificial ionization techniques for enhancing the electrical conductivity of gas, electro- magnetic control of flow has received renewed consideration in the aerospace community.

Especially, advancement in microprocessor technology and development of higher-order algorithms have made it possible to explore MHD through numerical techniques.

22 Computational fluid dynamics has proved to be an essential tool for modeling and sim- ulating MFD problems with less-restricting conditions of magnetic and fluid properties.

Therefore, using the CFD technique, Palmer [46] studied flow characteristics under the influence of a magnetic field over a Mars return aerobrake. An axisymmetric hemispherical blunt body with solenoidal magnet within the body was considered. Navier-Stokes equa- tions and the magnetic induction equation were solved separately, and a loosely coupled approach was utilized to communicate between them. A time-marching explicit algorithm was developed to solve the system of MFD equations. Shock wave standoff was increased, and convective heat transfer rate was decreased for MFD analysis. That is, results ob- tained with this numerical analysis showed same characteristic behavior to that of previous analytical or experimental investigations.

Coakley and Porter [47] analyzed the effects of a magnetic fieldoninviscidandideal gas flow over a hemispherical body under a low magnetic Reynolds number assumption. A dipole was placed at the body center for generating the magnetic field, and both scalar- and temperature-dependent electrical conductivity distributions were utilized. A time marching scheme based on the finite difference technique was used for solving the flow equations. An increase in shock standoff distance for uniform electrical conductivity and temperature- dependent electrical conductivity distribution was found to be similar, which has been ascribed to the fairly uniform temperature in the subsonic portion of the post-shock region.

The results of shock standoff distance were found in reasonable agreement with Porter and

Cambel’s [37, 38] analyses, based on Bush’s theory.

23 Shang et al. [48] used ideal MHD equations for infinitely conducting fluids. The finite volume numerical method for modeling the flow over a two-dimensional cylindrical nose blunt body was utilized. They predicted that application of a magnetic field not only causes an increase in shock standoff distance but also generates a secondary shock wave at the stagnation region.

Shang [49] and Shang et al. [50,51] investigated drag reduction by a plasma counterflow jet for hypersonic blunt-body flow. A high Reynolds number wind tunnel facility designed to achieve a nominal Mach number of 6.0 was utilized for the experiments. The plasma was injected from a hemispherical cylinder. It has been reported that the major fractions of drag reduction by plasma injection result from the interaction of favorable shock and counterflow jet and thermal energy deposition. Furthermore, the amplitude of unsteady shock wave movement associated with the counterflow jet and bow shock wave interaction is reduced.

Poggie and Gaitonde [52] conducted a preliminary study of two-dimensional, steady- state, non-ideal MHD equations with a non-uniform magnetic field distribution for flow over a cylinder. The flow field was assumed to be inviscid and thermally non-conducting, with constant electrical conductivity. It was found that the imposed magnetic field decelerates the flow in the shock layer. Subsequently, with the increase of magnetic field strength, a drop in surface static pressure at the stagnation region and increase in shock standoff distance were observed. They identified fluid electrical conductivity as one of the most sensitive parameters for controlling the magnetic field and fluid flow interaction. It was concluded that for achieving the same level of control, a higher value of fluid electrical conductivity results in a lower requirement of magnetic field strength, and vice versa.

24 Gaitonde and Poggie [53] used full MHD equations of finitely conducting fluid to simu- late inviscid flow over a two-dimensional cylindrical body with non-uniform magnetic field distribution. An increase in shock standoff distance, decrease in surface heat transfer, and decrease in surface static pressure near the stagnation region were observed with the appli- cation of a magnetic field. However, the decrease in static pressure was counter balanced by the increase in magnetic pressure, resulting in an overall insignificant variation in wave drag. They identified fluid electrical conductivity as one of the most critical parameters for enhancing the interaction between fluid and magnetic field. Furthermore, the influence of applied magnetic field was investigated for a three-dimensional cylindrical body. Reduction in heat-transfer rates was observed with the application of a magnetic field; this reduction was higher near the stagnation region. They reported that heat-transfer rates were higher when Joulean heating is retained in the analysis as compared to neglecting this term.

The low magnetic Reynolds number approach was utilized by Poggie and Gaitonde

[54,55] to model viscous and inviscid flows over a hemisphere. They showed that application of a magnetic field caused an increase in shock standoff distance for both viscous and inviscid

flows, and concluded that qualitative changes in the pressure field, obtained by applying a magnetic field, were independent of viscous effects. Also, the applied magnetic field moved the shock away from the body nose in both viscous and inviscid cases. A reduction in stagnation velocity gradient and velocity gradient near the wall region was observed with an increase in magnetic field strength. Furthermore, a reduction in wall heat flux was also observed after the application of a magnetic field. It was concluded that viscosity does not affect the overall shock structure for a high Reynolds number in simple blunt-body flows.

25 In a series of papers, Damevin et al. and Hoffmann et al. [56—58] investigated the effects of chemistry, magnetic field strength, and magnetic field distribution on high-speed two-dimensional (2-D) and three-dimensional (3-D) magnetogasdynamic flows over differ- ent shapes, including blunt body configurations. First, a hypersonic inviscid flow structure of ideal gas was simulated [56], subsequently viscous effects with different types of mag- netic field distributions were introduced [57], and finally, three-dimensional analysis was performed [58].

Damevin et al. [56] showed that under assumption of an infinitely conducting fluid, a uniform magnetic fieldorientedinthey-directionwasimposed.Itwaspredictedthat the presence of a magnetic field would cause the shock wave to move radially away from the body and generate secondary waves near the body surface that become stronger upon increasing magnetic field intensity. An increase in shock standoff distance and a decrease in surface pressure were observed with increasing magnetic field strength.

Hoffmann et al. investigated [57] full MHD equations without considering Joule heating for two-dimensional blunt-body configurations. They investigated unsteady, viscous blunt- body hypersonic flow for three different magnetic field distributions: uniform along the y- direction, dipolar, and radial. It was deduced that the secondary wave structure generated by applying a uniform magnetic field orthogonal to the free stream was the expansion waves, and their location and orientation significantly depended upon the type of magnetic

field distribution. For example, in the case of a uniform magnetic field, the secondary wave completely surrounds the body nose, but for dipolar and radial patterns, it does not remain in front of the body, instead moving slightly downstream of the stagnation point. Also,

26 a reduction in body surface pressure and increase in shock standoff distance were larger for a uniform magnetic field, compared to the other types of arrangements. It was been concluded that high temperatures in the region behind the bow shock can significantly affect the thermodynamic and electric properties of the air.

Damevin and Hoffmann [58] made an attempt to extend their work for three-dimensional blunt-body configurations. They modeled inviscid MHD flows over a hemisphere with the low magnetic Reynolds number approach, and over a cylindrical wedge with inviscid full

MHD equations of plasma having finite electrical conductivity. A dipole located at the center of the hemisphere was used to generate the magnetic field. An increase in the sub- sonic region ahead of the hemisphere and in shock standoff distance was observed with increasing magnetic field intensity. It was revealed that for chemically frozen flow, the increase in shock standoff distance was higher. In contrast, for flows in chemical equilib- rium, the increase in shock standoff distance was lower than the prediction of frozen flows.

For the cylindrical wedge, an inviscid and resistive flow was computed by employing full

MHD equations under a high magnetic Reynolds number assumption so that the effects of magnetic diffusion could be explored. After applying a magnetic field at the body surface aligned with the axis of the cylinder, a decrease in static pressure and temperature, and an increase in shock stand off distance were observed. It was shown that Joule heating has a dominant effect on body surface temperature and shock standoff distance. When

Joulean dissipation is retained and magnetic field is imposed, the body surface tempera- ture decreases and a temperature hill developes near the surface, which disappears upon neglecting Joulean dissipation. Shock standoff distance is also reduced when Joule heating

27 is omitted. In addition, the effects of magnetic diffusion by varying the magnetic Reynolds number via electrical conductivity of the flowing medium was also examined. It was found that the increase in the magnetic Reynolds number results in the reduction of temperature peak in the vicinity of the body surface and causes the shock to move closer to the body.

It was concluded that the magnetic Reynolds number has a significant effect on the flow structure. Later, Damevin and Hoffmann [59] explored the chemical effects at different altitudes for inviscid flow over a cylinder.

Recently, Khan et al. [60] investigated the aspects of different types of magnetic field distribution under the low magnetic Reynolds number approximation for a two-dimensional high-speed flow. It was found that the electromagnetic force causes flow compression in the post-shock region. They concluded that maximum standoff distance for a shock can be obtained with a vortex-type of magnetic field distribution.

Kato et al. [61], Tannehill et al. extended the parabolized Navier-Stokes (PNS) code of Miller et al. [62] for computing two-dimensional and axisymmetric MHD flow fields by using iterative parabolized Navier-Stokes algorithms. They introduced Maxwell’s equation of electromagnetodynamics to the PNS equations and solved the resulting set of MHD equationsbyalooselycoupled approach, where fluid and magnetic equations are decoupled from each other. Their results were in good agreement with the Navier-Stokes analysis of

Hoffmann et al. [57].

Borghietal.[63]studiedtheeffects of electrical configuration of a blunt-body on MHD interaction for a low magnetic Reynolds number by using numerical approach. Viscous, time-dependent fluid dynamics equations were discretized by means of finite volume for-

28 mulation, whereas electrodynamic equations were discretized by means of a finite element technique based on variational formulation. They found that in the absence of the Hall current, the applied magnetic field significantly influences the flow around the blunt-body and causes an increase in pressure and decrease in viscous stresses near the body surface. It was concluded that the introduction of the Hall current results in a weak MHD interaction.

In an accompaning paper, Borghi et al. [64] used their previously developed model [63] to possibly control the boundary layer phenomena in a hypersonic flow over an airfoil. A vortex-type magnetic field in the x-y plane was generated by passing an electrical current through conductors located inside the airfoil at different locations. It was shown that the magnetic field causes a substantial decrease in friction stresses and pressure distribution over the surface and in the vicinity of the stagnation point. Although, they proved the capability of magnetic interaction for controlling the hypersonic boundary layer flow, their computed values are questionable.

Recently, Borghi et al. [65] discussed their solver for electrodynamics equations in MHD

flow regimes. A comparison between the direct iterative method and inexact Newton scheme in terms of efficiency, convergence, and robustness was made to numerically solve governing electrodynamics equations. They reported that convergence of the direct iterative scheme is efficient and reliable for magnetic Reynolds numbers less than half, whereas the

Newton method takes longer time and requires efficient preconditioning of the coefficient matrix.

A conservation form of ideal MFD equations for modeling supersonic flow of real gas in equilibrium over a blunt-body was used by MacCormack [66]. An Implicit modified

29 algorithm based on the finite volume approach of discretization was implemented. A dipole located at the body center was used for generating the imposed magnetic field. Bow shock wave displacement towards the upstream region and reversal of the magnetic field near the exit section at the body shoulder was observed for MHD computations.

MacCormack [67] investigated viscous MHD flow over the surface of a spherical nose cone with a dipole-type magnetic field placed at the center of the sphere. Supersonic flow of real gas in equilibrium and hypersonic flow of nonequilibrium gases with chemical and thermal effects were considered. Computations were performed for two types of boundary conditions for a magnetic field at the solid body. For the boundary condition of zero normal derivatives of all magnetic components at the body surface, a decrease in heat transfer rate, decrease in velocity gradient (skin friction), and increase in total drag with the increase of magnetic field strength for the isothermal wall boundary condition was noted. Reduction in surface pressure was, however, balanced by the increase of magnetic pressure. In the case of an applied magnetic field at the wall, less interaction between plasma and magnetic

field was achieved. For a hypersonic nonequilibrium flow, an insignificant reduction in drag and heat transfer was observed after the application of a magnetic field. It was concluded thatthecaseofanappliedmagneticfield at the wall resulted in less interaction between plasma and the magnetic field.

In an attempt to compare the two mathematical formulations for modeling MFD prob- lems, MacCormack [68] simulated high-speed viscous flow over an axisymmetric spherical cone and internal flow within an MFD accelerator. He raised an important issue about the validity of the low magnetic Reynolds number approach versus solving full MFD equa-

30 tions. A constant value of electrical conductivity over the entire domain was assumed, and a dipole at the center of the sphere was placed to generate the magnetic field of desired strength. With the application of a magnetic field, an increase in total drag around the stagnation region of the sphere for both techniques was observed. It was demonstrated that the surface pressures obtained from these two analyses were in very good agreement, except at the stagnation region. However, the low magnetic Reynolds number approach provided a decrease in heat transfer around the blunt section in contrast to an increase computed by the full eight-wave system of MFD equations. In the case of an MFD accelerator with imposed electric and magnetic fields, both approaches predicted similar thrust and axial velocity along the centerline in the accelerator section (downstream of the origin ). How- ever, velocity distribution was not in agreement for the two approaches in the combustor section. In fact, a shock wave was formed in the combustor section, traveling towards the entrance section of the MFD accelerator. The low magnetic Reynolds number approach could not predict this shock wave. Finally, there were some doubts over the validity of the low magnetic Reynolds number approximation, and further exploration of this research was emphasized.

Subsequently, it was reported by MacCormack [69] that a more realistic value of elec- trical conductivity in the flow domain can significantly affect the results and may provide the same solutions from a low magnetic Reynolds number approximation and full MFD set of equations. It was reported that large differences in results between the two approaches by MacCormack [68] mainly occurred because of the Lorentz force and Joule heating in the free stream ahead of the bow shock wave, which occurred due to setting up a uniform

31 value of electrical conductivity in the overall domain. It was mentioned that a low magnetic

Reynolds number approximation can perform very well if equilibrium ionization is assumed between the cylinder surface and the shock layer. Thus, by assuming ionization of gas particles occurrs only in the confined region between the shock wave and the blunt body surface, hypersonic flow over the sphere was simulated with both modeling techniques.

Low magnetic Reynolds number formulation provided similar results to that of full MFD equations. However, results of the two modeling approaches indicated some differences in the solution of internal flow through an accelerator.

More recently, MacCormack [70] considered Hall current and ion slip effects for mod- eling supersonic flow over a spherical blunt body with the axisymmetric assumption. A comparison between full MFD and low magnetic Reynolds number formulations was made.

An increase in shock standoff distance was observed with the increase of magnetic field strength for both approaches; however, some decrease in shock standoff values occurred for full MFD calculations, primarily due to the opposite sign of the induced magnetic field.

MacCormack explained that the opposite nature of the induced magnetic field has a ten- dency to nullify the applied magnetic filed strength, which reduces its effects on the flow structure. Computation performed with Hall effects and ion slip has shown less increase in the shock wave standoff as compared to the scalar electrical conductivity computation, which indicates a decrease in magnetic interaction between the gas and magnetic field under these two parameters. Furthermore, the previously mentioned issue about electrical con- ductivity distribution [69] was discussed for the low magnetic Reynolds number approach.

Itwaspointedoutthatanalysisperformedbyassumingionizationintheconfined region between body surface and shock wave may provide similar results when electrical conduc-

32 tivity distribution having Hall current and ion slip effects are selected for low magnetic

Reynolds number flow calculations. Moreover, the drag coefficient obtained from the two

MFD modeling approaches was found to be in good agreement; however, some differences in normal surface stress and heat transfer rates have been reported.

Bityurin et al. [71] discussed their MHD research activities, published earlier [72—77], for controlling hypersonic flows of ionized air with the application of magnetic fields. The potential of MHD flow and flight control over several geometries was demonstrated exper- imentally as well as numerically. Flows around a circular cylinder, wedge, cowel-wedge, blunt body, and re-entry airfoil were investigated. Experiments were performed at MHD wind tunnel (WT) facility of the Central Aerohydrodynamic Institute (TsAGI), Moscow, which is capable of accelerating the flow up to Mach number ˜10—15. Seeding was utilized in the MHD accelerator section for enhancing the ionization of gas. Experimental and computational MHD flow control of hypersonic flow over a circular cylinder was discussed.

The cylinder was positioned across the flow, and the azimuthal magnetic field distribu- tion was generated by flowing the current within the cylinder along its axis. Numerical computations were performed using a two-dimensional cylinder model with low magnetic

Reynolds number approximation. Two types of electrical conductivity distributions were employed. The first model assumed ionized atoms exist behind the shock only, whereas, the second model assumed a fully ionized medium exists in the entire flow regime.

The first model was referred to as an equilibrium ionization model, representing a condi- tion similar to a re-entry vehicle, and the second model was referred to as a frozen ionization model, representing a case for a ground test WT facility. Different pressure and temper-

33 ature distributions were observed for both electrical conductivity models. MHD analysis with the equilibrium ionization model showed a substantial increase in shock standoff dis- tance and significant enlargement of the bow shock wave; however, for frozen ionization, the bow shock structure slightly differed from its original shape.

For the equilibrium ionization model, an increase in shock standoff distance and a decrease in surface heat flux were observed with the increment of magnetic field intensity.

On the other hand, no significant effects of magnetic fieldontheflow field near the surface of cylinder were noticed for the frozen ionization model. The authors reported that the

flow structure using the frozen ionization model was similar to the WT test facility, despite large variations that occurred in the cylinder wake and upstream regions in the experimental investigation. It was concluded that magnetic field has a stronger effect on flow structure when the equilibrium ionization model was utilized, as compared to the frozen ionization model. Moreover, investigation about the Hall effect revealed that for the frozen model, the

Hall current was important and had a larger value, which ultimately resulted in reduction of overall efficiency of the MHD effect. Thus, interaction of a magnetic fieldinthevicinityof the cylinder was not substantial despite its maximum value. However, for the equilibrium model, the Hall effect was found to be of negligible value and could be omitted from the analysis.

Hypersonic forebody and inlets were considered in the second part of numerical and experimental MHD flow control investigations. A wedge was utilized to model forebody

flows, and a wedge with a cowl was used for simulating hypersonic inlets. Two scenarios of

MHD control for positioning the bow to cowl oblique shock structure have been discussed.

34 The MHD technique has been successfully utilized to reduce the spillage drag for flights lowerthandesignMachnumber,andtoforcethebowshockoutoftheinletforflights higher than the design Mach number. Reasonable agreement between experimental and numerical results was achieved.

For flow over a wedge, two models of electrical conductivity distributions were tested for numerical analysis, similar to the previous case involving the circular cylinder. The first model had ionized gas particles behind the oblique shock wave only, and the second model hadionizationeverywhereintheflow domain. Oblique shock wave deflection was observed for both electrical conductivity models. However, the shock deviation angle was large for electrical conductivity distribution of the first model, as compared to the second model.

Experimental results showed a brighter flow field and some increase in shock angle after the application of a magnetic field, which was similar to the numerical prediction of the second electrical conductivity model. An increase in shock wave angle and pressure levels was observed with the increase in magnetic field intensity for both electrical conductivity models; however, strong MHD interaction occurred in the first ionization model.

For simulating the inlet of an engine, a wedge with cowl plate was utilized. Formation of a bow shock in front of the cowl plate and intersection of several oblique shocks took place in the gap. Application of a magnetic field increased the pressure in the gap and resulted in generation of a secondary shock near the mid-section of the gap. Furthermore, the bow shock of the cowl plate was also affected after the application of a magnetic field.

Next, hypersonic flow over an axisymmetric blunt body at flight scale with the low magnetic Reynolds number approach was discussed. A dipole-type magnetic field near the

35 stagnation point of the body was generated by a coil. Non-equilibrium flow of gas was considered in the post-shock region. Movement of the bow shock away from the body and a decrease in surface heat flux were noticed with the increase of magnetic strength. It was concluded that MHD has the potential to protect a hypersonic body surface from severe thermal heating.

Finally, MHD control of a re-entry vehicle was demonstrated numerically. An airfoil having upstream and downstream cylindrical edges was utilized for modeling this problem.

Dipoles at each end of the airfoil body were placed for generating the magnetic field. An interesting, enlarged shape of the shock structure (parachute shape) was obtained after application of the magnetic field, which is termed the parachute effect. A reduction in the maximum value of heat load at the stagnation point was been achieved as a result of the parachute effect.

The effects of wall electrical conductivity on the control of aerodynamic heating with the

MHD technique were numerically studied by Fujino et al. [78]. Re-entry flight conditions of an axisymmetric blunt-shaped space vehicle OREX [79] with electrically conducting and insulating walls with a dipole at the body center were considered. A thermoechemical nonequilibrium MHD code including the Hall effect under a low magnetic Reynolds number approach was developed.

It was reported that significantly stronger Hall electric field occurred for the insulat- ing wall, compared to the conducting wall case, after the application of a magnetic field.

Consequently, a stronger electric current was achieved, resulting in larger shock standoff distance for the insulating wall. Similarly, a decrease in heat flux was found to be larger for

36 the insulating wall case, and no remarkable change was observed for the conducting wall boundary. It was concluded that magnetic flowcontrolcannotbeeffective for walls with the conducting wall boundary condition.

For a relatively low range of supersonic Mach numbers and using argon as the flow medium, Otsu et al. [80] predicted temperature along the stagnation line of the flow regime.

A cylindrical body under the assumption of axisymmetric geometry was modeled with the low magnetic Reynolds number approach. A dipole at the body center was placed for generating the magnetic field, and temperature dependence of electrical conductivity was considered. An implicit finite volume method with AUSUM-DV scheme of high resolu- tion for shock capturing was utilized. It was stated that the Hall effect and ion slip were neglected due to the excessive requirement of computational time. Their numerical predic- tions for a temperature profile along the stagnation line were found to be in agreement with the experimental data predicted by Matsuda et al. [81]. An increase in shock wave standoff, withtheincreaseinmagneticfield strength and electrical conductivity, was demonstrated.

Furthermore, the shock wave standoff was validated with the experimental results of Mat- suda et al. [81].

In addition, electromagnetic fields were successfully utilized for controlling the bound- ary layer seperation. For example, the possibility of delaying the boundary layer seperation has been demonstrated both experimentally and numerically [82—85].

37 2.3 Backward-Facing Step Flows

Subsequent to overviewing the MFD research for hypersonic/supersonic blunt-body

flows, the available literature has been reviewed for supersonic flows over a backward- facing step. Backward-facing steps are often utilized to model seperated flows that occur in various applications. At present, the research in this area has sufficiently matured to be able to claim that flow separation, flow reattachment, and zones of recirculation have a profound impact on drag and heat-transfer distributions occurring between the wall and the flow. However, both the dynamics and thermodynamics of this subject are challenging, due to the complex nature of the flow. Nonetheless, the multiplicity of aerospace vehicles and structures exist, about which the external and internal flows often separate; some applications of great interest are supersonic inlets, engine combustors, and flow field behind blunt-nose projectiles and ballistics.

The basic features of a supersonic flow over a backward-facing step are depicted in Fig- ure 2.2. The complex nature of the flow field is evident in the vicinity of the backward-facing step, which involves the formation of an expansion fan at the corner, flow recirculation, a free shear layer, and an oblique shock wave. The literature available for this class of supersonic flows indicates that few, if any, attempts have been made to explore the effects of an applied magnetic field over backward-facing steps flows–the main thrust behind this case study. Much research has been devoted to understanding supersonic separated flows over backward-facing steps, experimentally as well as theoretically.

38

Expansion fan

Supersonic Flow

Reattachment Shock Boundary Layer Dividing Streamline

F ree Sh ear La ye r Free Streamlines

Step Height Height Step Recirculation Region

Figure 2.2: Illustration of supersonic flow field over backward-facing step.

39 Despite acceptable analytical predictions of total base pressure and drag, these meth- ods cannot predict detailed information about the flow field. Experimental techniques have always remained popular for providing adequate information about these types of flows.

Earlier experimental studies employed conventional measurement techniques for determin- ing pressure, temperature, drag, and heat transfer rates over backward-facing steps [86].

However, since the late 1980s, classes of non-intrusive optical measurement techniques, e.g., laser-induced fluorescence (LIIF) [87—89], laser Doppler velocimetry (LDV) [90—92], and particle image velocimetry (PIV) [93] have gained popularity for measuring supersonic separated flow properties for free and confined environments.

Parallel to experimental efforts, several numerical investigations have been performed to develop a comprehensive theory of supersonic separated flows. Kronzon et al. [94] used

finite difference approximation for two-dimensional steady viscous supersonic flows over a backward-facing step. Calculations were performed for a high Reynolds number within the laminar flow range. The appearance of an expansion fan at the corner and a reattachment shock downstream of the step were obvious through the results. They showed that the separation point occurred on the rear wall of the step below the corner. The calculated heat transfer rates and base pressure were in good agreement with experimental data.

Loth et al. [95] modeled supersonic flows over an axisymmetric backward-facing step for a range of Mach numbers. Their algorithm used adaptive unstructured grids and a shock-capturing scheme based on a finite element technique for modeling Euler equations of compressible flows. They observed that for an axisymmetric step, the corner expansion wave turns in a downward direction away from the corner, resulting in a relatively weak

40 recompression shock, which does not occur in two-dimensional cases. It has been reported that relatively high levels of base pressure would occur in the recompression region after the oblique shock wave due to axisymmetric effects.

In an attempt to validate some CFD codes for modeling hypersonic flows, Ebrahimi

[96—98] performed a series of tasks. Several engineering applications including supersonic

flows over a rearward-facing step were considered to determine the validity of different computer codes for predicting the expansion fan region, compression effects of shock, and relatively uniform pressure between the shock and shear layers. Results were compared with the experimental data of Smith [86]. Two supersonic Mach numbers (M = 2.5 and M = 3.5) were considered for all investigations with different turbulence models. Results indicated some overprediction in surface pressure at the separation region; however, analyses were in good agreement in the flow-reattachment region.

Ebrahimi [96] showed that the computer code TUFF, based on the finite volume tech- nique, was reviewed. Turbulence effects were included through κ ε and algebraic Baldwin- − Lomax turbulence models. Results obtained for backward-facing step flows were compared with the experimental data, which showed some overprediction in the surface pressure at the base region. It was suggested that the overprediction may be due to an experimen- tally unknown location of the transition point and inaccuracies of the turbulence models.

However, the code prediction of the reattachment length was in good agreement with the measured data. In the second investigation [97], the code named GASP, based on the finite volume method, was investigated. Similar conclusions as that of the previous effort [96] were reached, except overpredictions in the base pressure were similar for both M = 2.5 and

41 M=3.5 cases in contrast to the previous effort [96]. Finally, simulations were conducted by utilizing the CFD code GIFS, based on the implicit finite volume algorithm [98]. The predicted results and errors in surface pressure were found similar to that of the previous effort [97].

Ingelese and Acharya [99] proposed an improved κ ω model for modeling the turbu- − lence effects of supersonic compressible flows in a rectangular backward-facing step channel.

They stated that their improved κ ω model could predict values of skin friction, base − pressure, and reattachment length well and in very good agreement with the experimental data, as compared to Wilcxo’s revised κ ω model. They concluded that the κ ε model − − of Jones and Launder has deficiencies in modeling these effects.

Forsythe et al. [100, 101] used unstructured and a structured, implicit, finite-volume solver for modeling turbulence effects over base flows. A detached-eddy simulation for mod- eling supersonic flows around an axisymmetric base was utilized. The Spalart-Allmaras, shear stress transport, and Wilcox’s κ ω models were implemented. − Manna and Chakraborty [102] investigated supersonic flow over a free and confined backward-facing step. They performed a three-dimensional analysis using a commercial

CFD solver based on the finite volume approach. The results of both geometries showed evidence of expansion at the base corner, a circulation bubble, a reattached shear layer, and a recompressed oblique shock. However, unconfined geometry provided a free shock wave as compared to the confined geometry, where shock wave reflection was observed from the upper wall. Their predicted values of pressure, temperature, and velocities were in fair agreement with the experimental data. However, for a confined environment, the pressure

42 distribution near the wall region did not agree with experimental values. They stated that this discrepancy may be due to the ineptness of the turbulence model. It was concluded that despite a complex reflected shock structure created in confinement, recirculating flow remained the same as that in the free-environment case.

In an attempt to introduce different error indicators for adaptive remeshing algorithms for hybrid grids, Yang [103—105] performed a series of numerical investigations. A remesh- ing strategy of unstructured grid for modeling supersonic backward-facing step flows was introduced [103]. An extended locally implicit scheme with a dissipation model was devel- oped to solve unsteady Euler equations. Different grids strategies were utilized to achieve a close match with the experimental pressure distributions beyond the rearward-facing step.

Subsequent to performing the initial simulation on a non-adaptive mesh, an adaptive mesh strategy based on the error indicator was used for further simulations. For an adaptive grid, the number of triangles and quadrilaterals were controlled by a modified error indicator that contains unified magnitudes of density gradient and gradient of vorticity. Several hy- brid meshes were investigated to capture the physics of the problem and to obtain a better resolution of the flow field.Itwasreportedthatadensemeshatthecornerandatthe regions where strong convective fluxes occurred is necessary to capture the circulation zone and to enhance the accuracy of the results. An adaptive mixed triangular-quadrilateral grid with an increased number of cells at the zones of intense flow variations–corner vortex, expansion fan, and oblique shock wave–is recommended to perform the analysis within a reasonable amount of time. Subsequently, viscous terms were added to the Euler model, and turbulence effects were introduced in the adaptive remeshing algorithm [104]. The modi-

43 fied error indicator incorporated a unified magnitude of substantial derivatives of pressure and vorticity for controlling the grid spacing. Unsteady mass-averaged Navier-Stokes equa- tions with a low Reynolds number κ ε turbulence model were utilized with the locally − implicit scheme. Similar conclusions as that of the previous work [103] were reported.

Subsequently, Yang [105] developed two error indicators for achieving adaptive grids in the regions of severe flow variations. Supersonic turbulent flow over a rearward-facing step with time dependent mass-averaged Navier-Stokes equations was considered. Initially, analyses wereperformedonanunstructuredmeshcomposedofrelativelycoarsequadrilateraland triangular elements. Regardless of a poor resolution of the flow field, a reasonable so- lution was obtained. Finally, adaptive, refined, unstructured meshes were generated for subsequent numerical experiments, resulting in a high-resolution solution. Despite some differences observed in the downstream pressure near the upper-wall region of the flow, the calculated pressure distributions for refined mesh agree well with the experimental data.

Recently, Hermann et al. [106] presented a flow simulation methodology (FSM) for investigating the time-dependent behavior of complex compressible turbulent flows for subsonic and supersonic flow regimes. Subsonic flow over a backward-facing step and two-dimensional bluff bodies were considered, whereas an axisymmetric wake flow (ax- isymmetric backward-facing step flow) was modeled under a supersonic flow regime. A contribution function was developed to allow a consistent transition between Reynolds Av- erage Navier-Stokes (RANS) calculations and direct numerical simulations (DNS) within the same computation, depending on the local flow behavior and physical resolution. Time dependent simulations were carried out and time-averaged results were presented with

44 turbulence effects. Good agreement of FSM with DNS was achieved for a supersonic ax- isymmetric backward-facing step flow.

45 Chapter 3

GOVERNING EQUATIONS

The magnetofluiddynamic theory basically contains four partial differential equations– three generalized fluid dynamics equations with added magnetic effects, plus the Maxwell’s equations–for modeling any magnetic-plasma interaction phenomenon. However, there exists an approximation known as the low magnetic Reynolds number approach, which assumes a negligible induced magnetic field. In this chapter the basic assumptions made for deriving these equations will be reviewed. Both modeling approaches of MFD will be discussed and presented in the forms suitable for numerical methods. Subsequently, the development of boundary conditions will be presented.

3.1 Governing MFD Equations

Generalized Fluid Dynamics Equations

The following assumptions have been made for deriving the generalized fluid dynamics equations of the MFD problem.

The fluid is assumed to be continuous, with no voids or empty spaces in the • domain at the macroscopic level. This allows the development of the fluid model based on the continuum mechanics approach.

46 The fluid is assumed to be in a chemically frozen state and follows Newtonian • theory.

There is a negligible body force other than the electromagnetic force. • No internal heat gereration will occur. • Under these assumtptions, for an electrically conducting fluid in the presence of electric and magnetic fields, the generalized fluid dynamics equations having electromagnetic effects can be expressed as

Continuity Equation: ∂ρ + → (ρU→)=0 (3.1) ∂t ∇ ·

Momentum Equation:

→ ∂(ρU) = = + → ρU→ U→+pI = → τ + ρ E→ + →J B→ (3.2) ∂t ∇ · ⊗ ∇ · e × h i

Energy Equation:

2 ∂(ρet) = J → → → → → → → → → → + (ρet + p) U = (U τ ) q+ ρeE + J B U+ (3.3) ∂t ∇ · ∇ · · − ∇ · × · σe h i ³ ´ where 1 p B2 ρe = ρU 2 + + (3.4) t 2 γ 1 2µ − eo

→ The electric force, ρeE, is due to the presence of electric charges in the conducting medium, and the magnetic force, →J B→, is due to interaction of the current and magnetic field. × Both of these forces are present in the momentum equation, and their corresponding work

47 is depicted in the energy equation. However, the term J2 represents dissipation of heat σe through kinetic energy due to magnetic load. This term is known as Joulean heating.

Maxwell Equation of Magnetic Induction

Addition of electromagnetic effects in fluid dynamics equations is not sufficient to model an MFD problem, because in an electrically conducting medium, electromagnetic waves may advect with a finite velocity and diffuse at a finite rate. Thus, advection and diffusion of electromagnetic waves need to be considered for a realistic MFD modeling approach. This canbeachievedwithamagneticdiffusion equation, based on the well-known Maxwell’s equation and generalized Ohm’s law. For an isotropic medium with no electric and magnetic polarization, Maxwell’s equations and generalized Ohm’s law are described as follows:

Ampere-Maxwell equation:

∂E→ 1 → B→ = ∇× →J (3.5) ∂t εeo à µeo − ! where Faraday’s law is ∂B→ = → E→ (3.6) ∂t −∇ ×

Gauss’s law for magnetism is

→ B→ = 0 (3.7) ∇·

Gauss’s law for electricity is ρ → E→ = e (3.8) ∇· εeo

48 Conservation of charge is ∂ρ → →J = e (3.9) ∇· ∂t where generalized Ohm’s law is

→ → → → J =σe(E + U B) (3.10) ×

MFD Assumption

In MFD, charge density does not produce significant effects and is beneficial to con- sider a neutrally conducting medium. This results in a negligible charge density that is,

ρe =0, or, more specifically, a negligible change in charge density with respect to time.

Consequently, the electric field E→ also becomes time invariant. Mathematically,

∂ρ ∂E→ e 0 and 0 ∂t → ∂t →

This assumption leads to significant simplifications in the Ampere-Maxwell equation and conservation of charge density equation, which are given by

→ B→ →J = ∇× and → →J = 0 µeo ∇·

Incorporating the above simplifications, the reduced form of Maxwell’s equation can be ob- tained. Finally, the combination of Faraday’s law, Ohm’s law, and Ampere’s law eliminates the electric field and provides an expression known as the “Maxwell equation of magnetic induction,” which models the electromagnetic effects by magnetic field only.

49 That is, → ∂(B) → → → → → 2→ + U B B U = νe B (3.11) ∂t ∇ · ⊗ − ⊗ ∇ ³ ´ 1 where νe =(µeoσe)− is the magnetic diffusivity. Equation 3.11 is also known as the magnetic transport equation because if U→ is known, then B→ can be evaluated by specifying initial and boundary conditions.

In addition to the above critical MFD assumption, the following assumptions have been made for a neutrally conducting medium:

The electromagnetic medium is assumed to be isotropic and homogeneous, thus • electric permittivity, magnetic permeability, and electrical conductivity are scalar parame- ters.

No electric and magnetic polarizations have been considered, that is, µ = µ • eo e and εeo = εe.

Hall effects and ion-slip have been neglected. • The governing equations are presented in their non-relativistic form, that is, all • velocities are small compared to the speed of light U<

Subsequently, the assumption of neutral plasma further simplifies the generalized fluid dynamics equations, and with some manipulation, the three fluid dynamics equations along with the magnetic induction equation can be represented in the following form:

Continuity equation: ∂ρ + → (ρU→)=0 (3.12) ∂t ∇ ·

50 Momentum equation:

→ 2 → → ∂(ρU) B = B B = + → ρU→ U→+ p + I ⊗ = → τ (3.13) ∂t ∇ · ⊗ 2µ − µ ∇ · " ½ eo ¾ eo #

Maxwell equation of magnetic induction:

→ ∂(B) → → → 2→ (U B)=νe B (3.14) ∂t − ∇ × × ∇

Energy equation:

2 → → → 2 ∂(ρet) → B → B → → → → = → ( B) + ρet + p + U (U B) = (U τ ) →q+νe ∇ × (3.15) ∂t ∇· 2µ −µ · ∇· · −∇· µ "½ eo ¾ eo # eo where 1 p B2 ρe = ρU 2 + + (3.16) t 2 γ 1 2µ − eo

It is important to note that an electric field does not appear in the MFD equations, and the effects of the electromagnetic field are represented by a magnetic field only. However, an electric field is implicitly present in these equations, and for some applications, it is prescribed as an initial or boundary condition. Further details of evaluating the electric

field have been discussed by Gaitonde and Poggie [107]. The term B2 , usually known as 2µeo magnetic pressure, is not a physical quantity but enters the momentum equation through the Lorentz force and may generate mechanical stresses. The above-mentioned MFD partial differential equations are generally known as full MFD equations and can be rewritten more compactly, as shown in the following section.

51 3.1.1 Full MFD Partial Differential Equations

Full MFD equations involve a coupled magnetic field induction equation and fluid dy- namics equations that also contain magnetic terms, such as magnetic pressure and advec- tion in momentum and energy equations. The unsteady full MFD equations for neutrally conducting plasma can be expressed in vector form as

ρ ρU→ = ⎡ → ⎤ ⎡ → → 1 2 1 → → ⎤ ∂ ρU ρU U+ p + 2µ B I µ B B ⎢ ⎥ + → ⎢ ⊗ eo − eo ⊗ ⎥ ∂t ⎢ ⎥ ∇ · ⎢ n o ⎥ ⎢ B→ ⎥ ⎢ U→ B→ B→ U→ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⊗ − ⊗ ⎥ ⎢ ⎥ ⎢ 1 2 → 1 → → → ⎥ ⎢ ρet ⎥ ⎢ ρet + p + 2µ B U µ B(U B) ⎥ ⎢ ⎥ ⎢ eo − eo · ⎥ ⎣ ⎦ ⎣ n o ⎦ 0

⎡ = ⎤ → τ = ⎢ ∇ · ⎥ (3.17) ⎢ → → → ⎥ ⎢ (νe B) ⎥ ⎢ −∇ × ∇ × ⎥ ⎢ = ⎥ ⎢ → → → → νe → → 2 ⎥ ⎢ (U τ ) q+ µ ( B) ⎥ ⎢ ∇ · · − ∇ · eo ∇ × ⎥ ⎣ ⎦ where 1 p B2 ρe = ρU 2 + + (3.18) t 2 γ 1 2µ − eo

In fact, the above system of equations are developed based on some critical assumptions with respect to electromagnetic effects; therefore, they are not the full MFD equations.

However, in order to retain consistency with the available literature, they are termed here as full MFD equations. If the time variation of a magnetic field by self induction is signifi- cant and the induced magnetic field is relatively high, then the full MFD set of equations

52 will correctly model the MHD phenomenon. This is essentially true because induced cur- rents →J will significantly affect the surrounding magnetic field; thus, the magnetic induction equation needs to be solved. For example, for a typical problem where electrical conduc- tivity has a high value, the solution of full MFD equations can be easily achieved.

However, at low values of electrical conductivity (which commonly occurs in most aerodynamic applications) where strong magnetic fields are required, finding the solutions to classical full MFD equations is not an easy task (even for flowsoversimplegeometries) due to severe stability requirements associated with the strong stiffness of the magnetic induction equation. Indeed, a low level of electrical conductivity results in an exceedingly high coefficient of magnetic diffusivity in the Maxwell transport equation, whereas a strong magnetic field increases the order of magnitude of the magnetic pressure several times larger than the static pressure. Because of these two factors, the system of full MFD equations becomes very stiff to solve numerically and shows great numerical instabilities. Another major source of numerical difficulty is the condition of the divergence-free magnetic field associated with the additional term H in full MFD equations, which has to be satisfied at each time level. In fact, it may be difficult for the magnetic field to remain divergence free at all time levels, which would certainly decrease the rate of convergence of the numerical scheme. Finally, considering all these factors results in a difficult and cumbersome task in the solution procedure of full MFD equations. Therefore, a second choice, known as low magnetic Reynolds number approximation, is envisioned to solve MFD aerodynamic problems, which avoids the above-mentioned numerical difficulties by assuming that the induced magnetic field is negligible as compared to the applied magnetic field. Before

53 reviewing the second choice, a brief discussion of a non-dimensional parameter known as a magnetic Reynolds number is presented, which is usually utilized as a tool for the selection between the two available MFD modeling approaches.

Physical Significance of Magnetic Reynolds Number

A dimensionless parameter known as a magnetic Reynolds number represents the rel- ative magnitude of the induced magnetic field components. A magnetic Reynolds number is defined as the ratio of advection to diffusion of the magnetic field. Mathematically,

advection of B Uref Lref Rm = = (3.19) diffusion of B νe or

Rm = µeoσeUref Lref (3.20)

where Uref and Lref are the reference speed and length, respectively, νe is the magnetic dif- fusivity, µeo is the magnetic permeability of vacuum, and σe is the electrical conductivity of the medium. The magnetic Reynolds number also represents the non-dimensional electrical conductivity of the medium. At low magnetic Reynolds number values (Rm << 1.0) , any self-induced magnetic field can immediately diffuse away; consequently, the relative inten- sity of the induced magnetic field becomes smaller than the applied magnetic field. Thus, the induced magnetic field can be neglected, and a great simplification in evaluating the electromagnetic effectscanbeachievedwiththeadditionofasourceterminfluid dynamics equations. The details of this approach are provided in the next section.

54 3.1.2 Low Magnetic Reynolds Number Formulation

Generally, the electrical conductivity of air is sufficiently low, which results in signifi- cantly small magnetic Reynolds numbers. A low magnetic Reynolds number value ensures a negligible induced magnetic field associated with the induced current in comparison to the applied magnetic field. Thus, the applied magnetic field remains undisturbed by the flow, and magnetic induction equation can be omitted from the system of equations. Instead, a new set of governing equations are introduced, which approximates the electromagnetic ef- fects by adding source terms in fluid dynamics equations, thus resulting in an approximate approach called low magnetic Reynolds number approximation.

The governing equations for the unsteady flow of viscous fluid under low magnetic

Reynolds number formulation can be expressed in vector form as

ρ ρU→ 0 0

∂ ⎡ ⎤ ⎡ = ⎤ ⎡ ⎤ ⎡ = ⎤ ρU→ + → ρU→ U→+pI = →J B→ + → τ (3.21) ∂t ⎢ ⎥ ∇ · ⎢ ⊗ ⎥ ⎢ × ⎥ ⎢ ∇ · ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ = ⎥ ⎢ ⎥ ⎢ → ⎥ ⎢ → → ⎥ ⎢ → → → → ⎥ ⎢ ρet ⎥ ⎢ (ρet + p) U ⎥ ⎢ E J ⎥ ⎢ (U τ)+ q ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ · ⎥ ⎢ ∇· · ∇· ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ where 1 p ρe = ρU 2 + (3.22) t 2 γ 1 − and B→ and E→ are imposed magnetic and electric field, respectively. The generalized Ohm’s law is utilized for evaluating the current density →J. The source terms added to the Navier-

Stokes equations consist of Lorentz force in the momentum equation in addition to viscous

55 and pressure forces, whereas source terms in the energy equation contain work done by

Lorentz force and Joulean dissipation, which occur as the result of electric current flow through the fluid. Numerical solution of equations obtained with low magnetic Reynolds number formulation is relatively simple and efficient, even at substantially small values of magnetic Reynolds number and strong magnetic fields.

3.2 Formulation of Full MFD Equations for Numerical Simula-

tion

Equations of the two modeling approaches for solving an MFD problem were presented in vectorial form in Section 3.1. For simulating any flow fieldaroundanobject,itis necessary to express these equations in an appropriate coordinate system and rewrite them in the form convenient for the numerical technique. Furthermore, these models need to be nondimensionalized for a computationally efficient algorithm. Lastly, the equations must be transformed from physical space to computational space due to the finite difference technique utilized in the present investigation.

3.2.1 Cartesian Coordinate System

A three-dimensional Cartesian coordinate system has been selected to present MFD equations in their generalized form. The presence of a cross product in the governing equa- tions indicates that the MFD equations should be solved in a three-dimensional coordinate system, because the interaction of flow velocity U→ and magnetic field B→ generates a com- ponent U→ B→ normal to the plane of flow and magnetic field. Indeed, MFD problems ×

56 are doubtlessly three dimensional; however, to minimize the complexity of the problem and to reduce the computational efforts and cost associated with a numerical solution, on occassion, the MFD equations have been solved in extended two-dimensional coordinates systems, that is, the dependent variables have three components but are a function of two space variables. This makes magnetofluiddynamics problems remarkably different from classical fluid dynamics. Because while solving a two-dimensional problem in CFD, all field vectors are restricted to up to two-dimensional planes. On the contrary, two-dimensionality of MFD requires three components of field vectors to be solved in a two-spatial coordinate system. Thus, eight scalar equations must be solved even for a two-dimensional MFD problem. A full MFD system of eight equations has been provided in flux-vector form in three-dimensional coordinates as

∂Q ∂E ∂F ∂G ∂E ∂F ∂G + + + = v + v + v (3.23) ∂t ∂x ∂y ∂z ∂x ∂y ∂z where Q is the unknown state vector

T

Q = ρρuρvρwBx By Bz ρet (3.24) ∙ ¸ and the flux vectors are

57 ρu ⎧ ⎫ 2 1 ⎪ ρu + p∗ BxBx ⎪ ⎪ µeo ⎪ ⎪ − ⎪ ⎪ 1 ⎪ ⎪ ρuv BxBy ⎪ ⎪ µeo ⎪ ⎪ − ⎪ ⎪ 1 ⎪ ⎪ ρuw BxBz ⎪ ⎪ − µeo ⎪ E = ⎪ ⎪ (3.25) ⎪ ⎪ ⎨⎪ 0 ⎬⎪

⎪ uBy vBx ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ uB wB ⎪ ⎪ z x ⎪ ⎪ − ⎪ ⎪ 1 ⎪ ⎪ (ρet + p∗) u Bx (uBx + vBy + wBz) ⎪ ⎪ − µeo ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

ρv ⎧ ⎫ 1 ⎪ ρuv ByBx ⎪ ⎪ µeo ⎪ ⎪ − ⎪ ⎪ 2 1 ⎪ ⎪ ρv + p∗ ByBy ⎪ ⎪ µeo ⎪ ⎪ − ⎪ ⎪ 1 ⎪ ⎪ ρvw ByBz ⎪ ⎪ µeo ⎪ F = ⎪ − ⎪ (3.26) ⎪ ⎪ ⎪ vBx uBy ⎪ ⎨ − ⎬ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ vB wB ⎪ ⎪ z y ⎪ ⎪ − ⎪ ⎪ 1 ⎪ ⎪ (ρet + p∗) v By (uBx + vBy + wBz) ⎪ ⎪ − µeo ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

58 ρw ⎧ ⎫ 1 ⎪ ρuw BxBz ⎪ ⎪ µeo ⎪ ⎪ − ⎪ ⎪ 1 ⎪ ⎪ ρvw ByBz ⎪ ⎪ − µeo ⎪ ⎪ ⎪ ⎪ 2 1 ⎪ ⎪ ρw + p∗ BzBz ⎪ ⎪ µeo ⎪ G = ⎪ − ⎪ (3.27) ⎪ ⎪ ⎪ wBx uBz ⎪ ⎨ − ⎬ ⎪ wBy vBz ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ (ρet + p∗) w Bz (uBx + vBy + wBz) ⎪ ⎪ − µeo ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

0 ⎧ ⎫ ⎪ τ xx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ ⎪ ⎪ xy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ xz ⎪ ⎪ ⎪ Ev= ⎪ ⎪ (3.28) ⎪ ⎪ ⎨⎪ 0 ⎬⎪ ⎪ β ⎪ ⎪ yx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ βzx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uτ xx + vτxy + wτ xz + qx + qJx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

59 0 ⎧ ⎫ ⎪ τ yx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ ⎪ ⎪ yy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ yz ⎪ ⎪ ⎪ Fv= ⎪ ⎪ (3.29) ⎪ ⎪ ⎨⎪ βxy ⎬⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ βzy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uτ yx + vτyy + wτ yz + qy + qJy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

0 ⎧ ⎫ ⎪ τ zx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ ⎪ ⎪ zy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ zz ⎪ ⎪ ⎪ Gv= ⎪ ⎪ (3.30) ⎪ ⎪ ⎨⎪ βxz ⎬⎪

⎪ βyz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uτ zx + vτzy + wτ zz + qz + qJz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

60 and 1 p B2 + B2 + B2 ρe = ρ(u2 + v2 + w2)+ + x y z (3.31) t 2 γ 1 2µ − eo with 2 2 2 Bx + By + Bz p∗ = p + (3.32) 2µeo and magnetic stress is given by

1 ∂B ∂B β = i j (3.33) ij µ σ ∂x − ∂x eo e µ j i ¶

where E, F, and G are inviscid flux vectors, and Ev, Fv, and Gv are viscous flux vectors.

The viscous stress tensor for a Newtonian fluidwithStokeshypothesiscanbewrittenas

∂ui ∂uj 2 ∂uk τ ij = µ + δij µ (3.34) ∂x ∂x − 3 ∂x µ j i ¶ k

The heat transfer rate can be written as

∂T qi = k (3.35) ∂xi

The Joule heating may be expressed as

Bj ∂Bj ∂Bi qJ = νe (3.36) i µ ∂x − ∂x eo µ i j ¶

Subscripts i, j, and k are utilized for convenience in defining terms using Einstein summa- tion convention.

61 3.2.2 Modification to Full MFD Equations with → B→ = 0 Condi- ∇ · tion

Eigenvalues and eigenvectors of the previously stated system of full MFD equations need to be calculated for implementing TVD schemes to accurately capture the shock waves. Eigenvalues and eigenvectors can be determined by considering the convective portion of MFD equations. Mathematically, the ideal MFD equations can be written as

∂Q ∂E ∂F ∂G + + + =0 (3.37) ∂t ∂x ∂y ∂z

As a first step toward implementation, all inviscid flux vectors must be presented in terms of the unknown field vector Q through a linearization procedure as

∂Q ∂E ∂Q ∂F ∂Q ∂G ∂Q + + + =0 (3.38) ∂t ∂Q ∂x ∂Q ∂y ∂Q ∂z

∂E ∂F ∂G where the terms ∂Q , ∂Q , and ∂Q are referred to as flux Jacobiam matrices and are denoted by A, B, and C respectively. Mathematically,

∂Q ∂Q ∂Q ∂Q + A + B + C =0 (3.39) ∂t ∂x ∂y ∂z

Since the resulting system of ideal MFD equations is hyperbolic, therefore, its Jacobian matrices will have real eigenvalues and a complete set of both left and right eigenvectors.

However, it has been found that the flux Jacobian matrices A, B, and C have zero eigen-

62 values. That is, the eigenvalue associated with the normal component of the magnetic field has a null value because the Lorentz force is perpendicular to both the magnetic field in- tensity and the electric current. This will cause a singularity in the flux Jacobian matrices.

Along with this, the remaining seven eigenvalues of the ideal MFD equations may also locally degenerate to coincide with each other. This implies that the system of an ideal

MFD equation is not strictly hyperbolic.

To overcome the issue of singularity and eigenvalue degenerations, several studies have been performed [66, 108—111]. In the present work, the technique of Powell et al. [111] of introducing Gauss’s law of magnetism for modifying the Jacobian matrices has been implemented. This amendment results in an additional term in the flux vector form of

MFD equations that ultimately results in non-singular Jacobian matrices. Mathematically, it follows as

ρ ρU→ 0 = ⎡ → ⎤ ⎡ → → 1 2 1 → → ⎤ ⎡ 1 → ⎤ ∂ ρU ρU U+ p + 2µ B I µ B B µ B ⎢ ⎥ + → ⎢ ⊗ eo − eo ⊗ ⎥ + ⎢ eo ⎥ → B→ ∂t ⎢ ⎥ ∇ · ⎢ n o ⎥ ⎢ ⎥∇ · ⎢ B→ ⎥ ⎢ U→ B→ B→ U→ ⎥ ⎢ U→ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⊗ − ⊗ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 2 → 1 → → → ⎥ ⎢ 1 → → ⎥ ⎢ ρet ⎥ ⎢ ρet + p + 2µ B U µ B(U B) ⎥ ⎢ µ U B ⎥ ⎢ ⎥ ⎢ eo − eo · ⎥ ⎢ eo · ⎥ ⎣ ⎦ ⎣ n o ⎦ ⎣additional term⎦

0 | {z }

⎡ = ⎤ → τ = ⎢ ∇ · ⎥ (3.40) ⎢ → → → ⎥ ⎢ (νe B) ⎥ ⎢ −∇ × ∇ × ⎥ ⎢ = ⎥ ⎢ → → → → 1 → → 2 ⎥ ⎢ (U τ ) q + νe µ ( B) ⎥ ⎢ ∇ · · − ∇ · eo ∇ × ⎥ ⎣ ⎦

63 Equation 3.40 in flux vector form for 3D Cartesian coordinates can be shown with the additional term denoted by H as

∂Q ∂E ∂F ∂G ∂E ∂F ∂G + + + + H = v + v + v (3.41) ∂t ∂x ∂y ∂z ∂x ∂y ∂z with 0 ⎧ ⎫ ⎪ Bx ⎪ ⎪ µeo ⎪ ⎪ ⎪ ⎪ By ⎪ ⎪ ⎪ ⎪ µeo ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Bz ⎪ ⎪ µeo ⎪ → → H = ⎪ ⎪ B (3.42) ⎪ ⎪ ∇ · ⎨⎪ u ⎬⎪ ⎪ v ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ w ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uBx+vBy+wBz ⎪ ⎪ ⎪ ⎪ µeo ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪ It is interesting to note that the additional term is proportional to → B→, which is the ∇ · divergence of the magnetic field and theoritically has a zero value. Thus, the modification does not alter the physics of the problem and is purely mathematical in nature. This procedure results in non-conservation law form of the governing MFD equations.

3.2.3 Nondimensionalization

Nondimensionalization is performed for achieving several objectives. The most critical is for the numerical computation due to round-off error. In numerical calculations, values vary so dramatically that round-off error may affect the final results. Nonetheless, nondi-

64 mensionalization with respect to proper variables assures the range of numbers within the limit of zero and one. Hence, the viscous MFD equations are nondimensionalized using the following variables:

x y z Uref t u v w x∗ = ,y∗ = ,z∗ = ,t∗ = ,u∗ = ,v∗ = ,w∗ = , Lref Lref Lref Lref Uref Uref Uref

Bx By Bz p Bx∗ = ,By∗ = ,Bz∗ = ,p∗ = 2 , Uref √µeoρref Uref √µeoρref Uref √µeoρref ρref Uref

= ρ T e = τ L µ σ t ∗ ref eo e ρ∗ = ,T∗ = ,et∗ = 2 , τ = ,µeo∗ = ,σe∗ = , ρref Tref Uref Uref µref µeo σeref

νe µ k νe∗ = ,µ∗ = ,k∗ = , νeref µref kref

The resulting nondimensional parameters are the following:

Reynolds number:

ρref Uref Lref Reref = µref

Prandtl number:

µ cp Pr = ref kref

Magnetic Reynolds number:

Rmref = µeoσeref Uref Lref

65 Freestream Mach number:

Uref Mref = γpref Áρref p Magnetic Pressure number: 2 Bref Rbref = 2 ρref µeoUref

In subsequent sections, the asterisk notation designating nondimensional quantities will be dropped. Thus, all equations will be in nondimensional form, unless otherwise specified.

The nondimensional MFD equations in flux vector form are writtern as

∂Q ∂E ∂F ∂G ∂E ∂F ∂G + + + + H = v + v + v (3.43) ∂t ∂x ∂y ∂z ∂x ∂y ∂z where Q is the unknown state vector

T

Q = ρρuρvρwBx By Bz ρet (3.44) ∙ ¸ the flux vectors and the additional source term are

66 ρu ⎧ ⎫ 2 ⎪ ρu + p∗ Rbref BxBx ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ ρuv R B B ⎪ ⎪ bref x y ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ ρuw Rb BxBz ⎪ ⎪ − ref ⎪ E = ⎪ ⎪ (3.45) ⎪ ⎪ ⎨⎪ 0 ⎬⎪

⎪ uBy vBx ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ uB wB ⎪ ⎪ z x ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ (ρet + p∗) u Rb Bx (uBx + vBy + wBz) ⎪ ⎪ − ref ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

ρv ⎧ ⎫ ⎪ ρuv Rbref ByBx ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ρv + p∗ Rbref ByBy ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ ρvw R B B ⎪ ⎪ bref y z ⎪ F = ⎪ − ⎪ (3.46) ⎪ ⎪ ⎪ vBx uBy ⎪ ⎨ − ⎬ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ vB wB ⎪ ⎪ z y ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ (ρet + p∗) v Rb By (uBx + vBy + wBz) ⎪ ⎪ − ref ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

67 ρw ⎧ ⎫ ⎪ ρuw Rbref BxBz ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ρvw Rbref ByBz ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ ρw2 + p R B B ⎪ ⎪ ∗ bref z z ⎪ G = ⎪ − ⎪ (3.47) ⎪ ⎪ ⎪ wBx uBz ⎪ ⎨ − ⎬ ⎪ wBy vBz ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (ρet + p∗) w Rb Bz (uBx + vBy + wBz) ⎪ ⎪ − ref ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

0 ⎧ ⎫ ⎪ Rbref Bx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R B ⎪ ⎪ bref y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Rb Bz ⎪ ⎪ ref ⎪ → → H = ⎪ ⎪ B (3.48) ⎪ ⎪ ∇ · ⎨⎪ u ⎬⎪

⎪ v ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ w ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Rb (uBx + vBy + wBz) ⎪ ⎪ ref ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

68 0 ⎧ ⎫ ⎪ τ xx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ ⎪ ⎪ xy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ xz ⎪ ⎪ ⎪ Ev= ⎪ ⎪ (3.49) ⎪ ⎪ ⎨⎪ 0 ⎬⎪ ⎪ β ⎪ ⎪ yx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ βzx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uτ xx + vτxy + wτ xz + qx + Rb qJx ⎪ ⎪ ref ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

0 ⎧ ⎫ ⎪ τ yx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ ⎪ ⎪ yy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ yz ⎪ ⎪ ⎪ Fv= ⎪ ⎪ (3.50) ⎪ ⎪ ⎨⎪ βxy ⎬⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ βzy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uτ yx + vτyy + wτ yz + qy + Rb qJy ⎪ ⎪ ref ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

69 0 ⎧ ⎫ ⎪ τ zx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ ⎪ ⎪ zy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ zz ⎪ ⎪ ⎪ Gv= ⎪ ⎪ (3.51) ⎪ ⎪ ⎨⎪ βxz ⎬⎪

⎪ βyz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uτ zx + vτzy + wτ zz + qz + Rb qJz ⎪ ⎪ ref ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪ and 1 p B2 + B2 + B2 ρe = ρ(u2 + v2 + w2)+ + R x y z (3.52) t 2 γ 1 bref 2 − with B2 + B2 + B2 p = p + R x y z (3.53) ∗ bref 2 and nondimensional magnetic stress is

1 ∂B ∂B β = i j (3.54) ij R σ ∂x − ∂x mref e µ j i ¶

The viscous stress tensor can be written as

1 ∂ui ∂uj 2 ∂uk τ ij = µ + δij µ (3.55) Re ∂x ∂x − 3 ∂x ref ½ µ j i ¶ k ¾

70 Heat transfer rate can be written as

µ ∂T qi = 2 (3.56) Reref Prref (γ 1) M ∂xi − ref

Joulean dissipation is expressed as

Bj ∂Bj ∂Bi qJi = (3.57) Rm σe ∂xi − ∂xj ref µ ¶

3.2.4 Curvilinear Coordinate System

In the present efforts, a finite difference numerical technique has been utilized to ap- proximate the governing partial differential equations into an algebraic set of equations.

The resulting finite difference algebraic equations need to be solved in a rectangular com- putational domain of uniform spacing. However, a vast class of practical applications exists with non-rectangular physical domains, e.g., airfoil trailing edges, circular channels, and blunt bodies. Imposing a rectangular domain on such geometries is not only a difficult task but also highly inefficient. Therefore, a computational domain, where physical domain of any shape and non-uniform spacing can be transformed into uniformly spaced rectan- gulargrids,isamustforsolvingfinite difference algebraic equations. With the help of appropriate mapping functions, the physical domain can be transformed into a generalized curvilinear domain of equal spacing. If (x, y, z) are the coordinates in physical space, and

(ξ,η,ζ) are the coordinates in computational space, then mapping between the two can be performed as

71 dξ ξx ξy ξz dx ⎧ ⎫ ⎡ ⎤ ⎧ ⎫ = (3.58) ⎪ dη ⎪ ηx ηy ηz ⎪ dy ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎨⎪ ⎬⎪ ⎢ ⎥ ⎨⎪ ⎬⎪ ⎢ ⎥ dζ ⎢ ζx ζy ζz ⎥ dz ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎣ ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ where ξi,ηi, and ζi are the⎩ metrices⎭ of transformation,⎩ and the⎭ Jacobian of transformation is defined as

ξx ξy ξz ¯ ¯ J = ¯ ¯ (3.59) ¯ ηx ηy ηz ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ζx ζy ζz ¯ ¯ ¯ ¯ ¯ ¯ ¯ Subsequently, the nondimensionalized full¯ MFD governing¯ equation (3.43) in physical do- main (x, y, z) can be transformed to a generalized curvilinear domain (ξ,η,ζ) as

∂Q ∂E ∂F ∂G ∂E ∂F ∂G + + + + H = v + v + v (3.60) ∂t ∂ξ ∂η ∂ζ ∂ξ ∂η ∂ζ where Q Q = (3.61) J

1 E = ξ E + ξ F+ξ G (3.62) J x y z ¡ ¢ 1 F = η E + η F+η G (3.63) J x y z ¡ ¢ 1 G = ζ E + ζ F+ζ G (3.64) J x y z ¡ ¢ ∂B ∂B ∂B H = H x + y + z (3.65) M ∂ξ ∂η ∂ζ µ ¶ 1 B = ξ B + ξ B +ξ B (3.66) x J x x y y z z ¡ ¢

72 1 B = η B + η B +η B (3.67) y J x x y y z z ¡ ¢ 1 B = ζ B + ζ B +ζ B (3.68) z J x x y y z z ¡ ¢ 1 E = ξ E + ξ F +ξ G (3.69) v J x v y v z v ¡ ¢ 1 F = η E + η F +η G (3.70) v J x v y v z v ¡ ¢ 1 G = ζ E + ζ F +ζ G (3.71) v J x v y v z v ¡ ¢

E, F, and G represent the convective flux vectors, and Ev, Fv, and Gv represent the dif- fusion flux vectors, respectively. Each flux vector has eight components. The evaluation of convective flux terms is obtained from expressions (3.62) to (3.64). However, diffusion flux terms involve spatial derivatives of velocity and magnetic field; therefore, these derivatives must be transformed into computational domain, the procedure of which has been outlined by Hoffmann and Chiang [112].

Finally, the MFD equation (3.60) , by definition of the flux Jacobian matrices, can be rewritten as ∂Q ∂Q ∂Q ∂Q ∂E ∂F ∂G + A + B + C = v + v + v (3.72) ∂t ∂ξ ∂η ∂ζ ∂ξ ∂η ∂ζ where

∂E ∂Bx A = + HM = ξ A + ξ B+ξ C (3.73) ∂Q ∂Q x y z

∂F ∂By B = + HM = η A + η B+η C (3.74) ∂Q ∂Q x y z

∂G ∂Bz C = + HM = ζ A + ζ B+ζ C (3.75) ∂Q ∂Q x y z

73 ∂E ∂B A = + H x (3.76) ∂Q M ∂Q

∂F ∂B B = + H y (3.77) ∂Q M ∂Q

∂G ∂B C = + H z (3.78) ∂Q M ∂Q

A, B, and C are the flux Jacobian matrices in the generalized curvilinear domain. These equationshavebeenprovidedinthree-dimensionalcoordinatesandcanbeusedforatwo- dimensional coordinate system by dropping the terms that have derivatives with respect to z or ζ.

3.3 Boundary Conditions

For solving a well posed boundary value problem, a set of exclusive information with respect to the dependant variables and/or its derivatives must be specified along the do- main boundaries which are commonly known as boundary conditions. Boundary conditions are extremely important to achieve a physically correct solution of the problem. In com- putational fluid dynamics applications, boundary conditions become extremely important to guarantee a stable and converged solution. It is a requirement that must be satisfied by dependent variables and their derivatives at the domain’s boundary. The dependant vari- able or its gradient in a specific direction needs to be specified at the domain boundaries.

The application of boundary conditions depends on the physics of problem and operating conditions. Figure 3.1 shows different types of boundaries for a typical external flow. In this section, boundary conditions for full MFD equations under supersonic/hypersonic flow regimes are discussed.

74 Farfield (outflow)

Inflow Outflow

Mα > 1.0 Computational domain Mα > 1.0

Solid surface

Figure 3.1: Different types of boundary conditions for a typical external flow.

Inflow and Outflow Boundaries

The treatment of inflow and outflow boundary conditions is a critical task for achieving an accurate solution, because as the time marching scheme proceeds, any unsteady wave that exists within the domain of computation should leave the domain through outflow or farfield boundaries. Spurious numerical errors will occurr if outflow or farfield bound- ary conditions have been treated improperly, which may result in failure of the numerical scheme or non-physical solution. Fortunately, for supersonic/hypersonic flows, all eigen- values of the eight-wave structure of full MFD equations become positive, which means characteristic waves will enter into the domain at inflow, and if the outflow is supersonic, then all characteristic waves will leave the domain at the outflow. Thus, all primitive vari- ables are specified at the inflow; conversely, variables at the outflow must be evaluated from

75 the interior points as the solution evolves, because information cannot travel upstream to influence the interior points. It is worthwhile to mention that disturbances may propagate upstream in the subsonic portion of the boundary layer, but for the class of supersonic flows under consideration, the thickness of the boundary layer is found to be very small, and that portion remains confined to the solid wall and overall flow remains in the supersonic range.

Inflow and outflow boundary conditions are described as follows:

Inflow :

specified free-stream conditions for pressure, temperature, density, and velocity; applied magnetic field with respect to the selected distribution, e.g., dipole or uniform magnetic field distributions.

Outflow :

zero-order extrapolation of all primitive variables; applied magnetic field with re- spect to the selected distribution, e.g., dipole or uniform magnetic field distributions.

Body Surface

Specification of flow variables other than magnetic field at the body surface is not a difficult task for most flow regimes. However, for magnetic field components, it requires some extra care. The available literature on MHD flows indicates that most investigators have assumed either a perfectly conducting wall or an electrically insulating wall for hyper- sonic/supersonic aerodynamic flows over blunt bodies. Mathematically, it can be expressed as

d→B wall=0, for a perfectly conducting wall dn | → → B wall= Bspecified, for an electrically insulating wall | → where Bspecified is the prescribed value of the applied magnetic field.

76 Previous studies have shown that these boundary conditions work effectively for high magnetic Reynolds numbers and low strengths of magnetic field; however, at low magnetic

Reynolds numbers and strong magnetic fields, severe numerical difficulties in term of sta- bility have been observed in solving full MFD equations for blunt body flows. Therefore, in order to increase the stability of the numerical scheme and to achieve a physically ac- ceptable solution of the problem, a correct assessment for magnetic field components at the body surface is required. Porter and Cambel [37] first indicated that magnetic induction at the body surface would modify the magnetic field distribution; therefore, it is impor- tant to specify a boundary condition that consider the effects of magnetic induction at the surface to obtain a correct solution. Furthermore, a boundary condition that accounts for magnetic perturbations at the body surface is necessary to avoid numerical difficulties and to represent actual physical phenomenon.

The developed boundary conditions for magnetic components utilizes the following three equations at the wall:

Generalized Ohm’s law •

→ → → → J =σe(E + U B) (3.79) × where →J is the current density, E→ is the electric field potential, U→ is the flow velocity, and

B→ is the magnetic field.

77 Ampere-Maxwell equation with MFD assumption of neutral plasma that will • result in → B→ →J = ∇× (3.80) µeo

The constraint on magnetic field •

→ B→ = 0 (3.81) ∇·

For viscous flow at the wall, from the no-slip condition, U→ =0,andforflow past the blunt body configuration, one can easily set E→ =0. Therefore, from equation (3.79), →J = 0 at the body surface. Equation (3.80) can then be used to obtain two equations for the gradients of magnetic field components with respect to normal-to-the-wall. A third such gradient of magnetic field component is obtained from Equation (3.81). For example, if the normal-to-the-wall direction is the y-axis, then the three-derivative boundary conditions for Bx,By, and Bz can be derived according to the following process.

The combination of equations (3.79) and (3.80) at the wall results in

→ B→ =0 (3.82) ∇× which will yield two components of magnetic field along the y-direction

∂B ∂B x = y (3.83) ∂y ∂x ∂B ∂B z = y (3.84) ∂y ∂z

78 Subsequently, the third component can be obtained form equation (3.81) as

∂B ∂B ∂B y = x + z (3.85) ∂y − ∂x ∂z µ ¶

Equations (3.83) to (3.85) can be discretized at the wall according to the following proce- dure:

∂B B = B + ∆y y (3.86) xwall xinterior point ∂x ∂B B = B + ∆y y (3.87) zwall zinterior point ∂z

∂Bx ∂Bz By = By + (3.88) wall interior point − ∂x ∂z µ ¶

The right hand side of equations (3.86) to (3.88) are evaluated from the interior grid adjacent to the wall. Transformation of these equations into the computational domain needs to be performed for implementation. The following wall boundary conditions for other flow variables have been identified:

No-slip condition. • Adiabatic wall, zero normal gradient of pressure and density. • Evaluation of magnetic field components at body surface for blunt body flows. •

79 3.4 Formulation of Low Magnetic Reynolds Number Approach

for Numerical Simulation

As previously discussed, most aerodynamic applications have a very low magnetic

Reynolds number, which implies that any induced magnetic field will rapidly diffuse away.

Thus, there is no need to solve the magnetic induction equation. This idea seems quite in- teresting, especially after reviewing the lengthy and complicated set of full MFD equations, their poor numerical sensitivity with respect to low ranges of magnetic Reynolds numbers, and hurdles in algorithm development associated with the additional term H,whichis proportional to the divergence of magnetic field and has been introduced to remove the singularity of flux Jacobian matrices.

By considering these issues, low magnetic Reynolds number formulation appears as a convenient tool for solving MFD problems with great efficiency. The low magnetic Reynold number equations in Cartesian coordinates, their nondimensionalization, and their trans- formation into computational space are discussed in the following section.

3.4.1 Cartesian Coordinate System

Equations for the low magnetic Reynolds number approach have been nondimension- alized, according to the variables used for full MFD equations in Section 3.2.3.Theasterik notation has been removed form all variables, and governing equations in flux vector form for the three-dimensional Cartesian coordinate system are written as

80 ∂Q ∂E ∂F ∂G ∂E ∂F ∂G + + + = v + v + v + S (3.89) ∂t ∂x ∂y ∂z ∂x ∂y ∂z MFD where Q is the unknown flux vector

T

Q = ρρuρvρwρet (3.90) ∙ ¸ and 1 p ρe = ρ(u2 + v2 + w2)+ (3.91) t 2 γ 1 −

E, F, and G are the inviscid flux vectors, and Ev, Fv, and Gv are the viscous flux vectors.

The additional source term is represented by SMFD. The non-dimensional flux vectors and source term are

ρu ⎧ ⎫ ⎪ ρu2 + p ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ E = ⎪ ρuv ⎪ (3.92) ⎪ ⎪ ⎨⎪ ⎬⎪ ρuw ⎪ ⎪ ⎪ ⎪ ⎪ (ρet + p) u ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

81 ρv ⎧ ⎫ ⎪ ρuv ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ F = ⎪ ρv + p ⎪ (3.93) ⎪ ⎪ ⎨⎪ ⎬⎪ ρuw ⎪ ⎪ ⎪ ⎪ ⎪ (ρet + p) v ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

ρw ⎧ ⎫ ⎪ ρuw ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ G = ⎪ ρvw ⎪ (3.94) ⎪ ⎪ ⎨⎪ ⎬⎪ ρw2 + p ⎪ ⎪ ⎪ ⎪ ⎪ (ρet + p) w ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

0 ⎧ ⎫ ⎪ τ ⎪ ⎪ xx ⎪ ⎪ ⎪ E = ⎪ ⎪ (3.95) v ⎪ τ xy ⎪ ⎪ ⎪ ⎨⎪ ⎬⎪ τ xz ⎪ ⎪ ⎪ ⎪ ⎪ uτ xx + vτxy + wτ xz + qx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

82 0 ⎧ ⎫ ⎪ τ ⎪ ⎪ yx ⎪ ⎪ ⎪ F = ⎪ ⎪ (3.96) v ⎪ τ yy ⎪ ⎪ ⎪ ⎨⎪ ⎬⎪ τ yz ⎪ ⎪ ⎪ ⎪ ⎪ uτ yx + vτyy + wτ yz + qy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

0 ⎧ ⎫ ⎪ τ ⎪ ⎪ zx ⎪ ⎪ ⎪ G = ⎪ ⎪ (3.97) v ⎪ τ zy ⎪ ⎪ ⎪ ⎨⎪ ⎬⎪ τ zz ⎪ ⎪ ⎪ ⎪ ⎪ uτ zx + vτzy + wτ zz + qz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

0 ⎧ ⎫ ⎪ (→J B→) ⎪ ⎪ x ⎪ ⎪ × ⎪ ⎪ → → ⎪ SMFD = Rmref ⎪ (J B) ⎪ (3.98) ⎪ y ⎪ ⎪ × ⎪ ⎨ → → ⎬ (J B)z ⎪ × ⎪ ⎪ → ⎪ ⎪ (→J B) U→+ 1 →J →J ⎪ ⎪ × · σe · ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

83 where the subscripts x, y, and z indicate vector components in their respective directions.

The expressions for viscous stress tensor and heat transfer rate are found to be same as that in the previous section for full MFD equations; therefore, they are not presented here.

Ohm’s law is utilized for evaluating the current density. It is noted that the governing equations for low magnetic Reynolds number approximation can also be obtained from the

full MFD equations of Section 3.2.3, discussed previously by setting Rbref =0;dropping

fifth, sixth, and seventh equations; and adding a source term SMFD.

3.4.2 Curvilinear Coordinate System

The governing equations for low magnetic Reynolds number approach have been trans- formed from physical space to computational space by utilizing the same mapping as that for full MFD equations, outlined in Section 3.2.4. The nondimensional governing equations in generalized curvilinear coordinates are

∂Q ∂E ∂F ∂G ∂E ∂F ∂G + + + = v + v + v + S (3.99) ∂t ∂ξ ∂η ∂ζ ∂ξ ∂η ∂ζ MFD where Q Q = (3.100) J

1 E = ξ E + ξ F+ξ G (3.101) J x y z ¡ ¢ 1 F = η E + η F+η G (3.102) J x y z ¡ ¢ 1 G = ζ E + ζ F+ζ G (3.103) J x y z ¡ ¢

84 1 E = ξ E + ξ F +ξ G (3.104) v J x v y v z v ¡ ¢ 1 F = η E + η F +η G (3.105) v J x v y v z v ¡ ¢ 1 G = ζ E + ζ F +ζ G (3.106) v J x v y v z v ¡ ¢ S S = MFD (3.107) MFD J

E, F, and G, represent the convective flux vectors, and Ev, Fv, and Gv represent the diffusion flux vectors, respectively. Each flux vector has five components. The additional source term SMFD models the MFD effects. When MFD equation (3.99) is rewritten by definition of the Jacobian matrices, a similar form to that of equation (3.72) is obtained with an addtional source term SMFD,as

∂Q ∂Q ∂Q ∂Q ∂E ∂F ∂G + A + B + C = v + v + v + S (3.108) ∂t ∂ξ ∂η ∂ζ ∂ξ ∂η ∂ζ MFD where ∂E A = = ξ A + ξ B+ξ C (3.109) ∂Q x y z

∂F B = = η A + η B+η C (3.110) ∂Q x y z

∂G C = = ζ A + ζ B+ζ C (3.111) ∂Q x y z

∂E A = (3.112) ∂Q

∂F B = (3.113) ∂Q

85 ∂G C = (3.114) ∂Q

It is worthwhile to note that the Jacobian matrices A, B, C, A, B, and C have been obtained by setting HM =0for the Jacobian matrices of full MFD equations.

3.5 Boundary Conditions

Similar boundary conditions as that of full MFD formulation have been implemented for the low magnetic Reynolds number approach for all flow variables except for magnetic

field. Since no induced magnetic field exists for this approximation, the magnetic field at the boundary does not need to be updated. Only the imposed magnetic field will exist at the boundaries and in the domain.

86 Chapter 4

DECOMPOSED FULL MFD

EQUATIONS

4.1 Introduction

The solution of classical full MFD equations with strong magnetic field strengths is neithertrivialnorasimpletaskforweaklyionizedflows, which occur in most aerodynamic applications. A significantly low level of electrical conductivity is observed in aerodynam- ics that results in the requirement of strong magnetic fields for generating a Lorentz force of considerable strength. Severe numerical difficulties encountered in solving full MFD equations under these circumstances are due primarily to the large magnitude of magnetic pressure and exceedingly high magnetic diffusivity.Thelargemagnitudeofmagneticpres- sure occurs as the result of proportionality of magnetic pressure to the square of the applied magnetic field, or mathematically as

1 2 Pmag = B 2µeo

87 With the application of a strong imposed magnetic field, the magnetic pressure can be several orders of magnitude larger than the aerodynamic pressure or the induced magnetic

field pressure. Thus, numerical errors associated with very large magnetic pressure term could have significant magnitude when compared with the relatively smaller fluid stress terms, potentially resulting in instability of the numerical scheme.

On the other hand, at low values of electrical conductivity, other difficulties in solving the magnetic induction equation occur because of an exceedingly high coefficient of mag- netic diffusion. In fact, it is the diffusion term in the magnetic induction equation that causes strong stiffness at low values of electrical conductivity. It is noteworthy to mention that low levels of electrical conductivity occur in most hypersonic flows; moreover, the values of electrical conductivity may decrease further in the free-stream region ahead of the shockwave and fairly downstream of the stagnation portion of the body. Eventually, a smaller value of electrical conductivity causes a significant increase in magnetic diffusivity, sincemagneticdiffusivity is inversely proportional to electrical conductivity that appears on the right side of the magnetic induction equation. Therefore, the order of magnitude of the diffusion term becomes several orders larger than the production term in the induction equation even in nondimensional form. Consequently, strong stiffness is generated, and an extremely small Courant-Friedrichs-Lewy (CFL) number is required to overcome this difficulty.

Thus, it is necessary to explore an alternate full MFD system of equations that not only account for the presence of an induced magnetic field but also provide an efficient way to avoid numerical degenerations associated with the difficulties identified above. The

88 alternate formulation of full MFD equations was devised by MacCormack and presented at the 34th Plasmadynamics and Laser Conference in 2003 [113]. It was introduced as

“Reduced MFD equations ” because the magnitude of the magnetic terms are reduced by removing the applied magnetic field component wherever possible. However, in the current efforts, the alternate formulation has been termed as “Decomposed Full MFD equations” since these equations have been derived based on the decomposition of the total magnetic

field into imposed and induced components. This terminology appears more generic and plausible, because the new formulation for MFD treats induced and imposed magnetic

fields separately. Details of this formulation are provided in the next section.

4.2 The Basic Concept

The concept of decomposition of total magnetic field is similar to the simplification in electromagnetic scattering where static and time-varying fields have been separated, and only the transient field is calculated while no loss has been observed with the separation of the two components. Since the total magnetic field consists of the sum of imposed and induced magnetic fields, decomposition of the total magnetic field is introduced as

→ → → Bt = Bi + Bo

where Bi is the induced magnetic field, Bo is the imposed magnetic field, and Bt is the total magnetic field. Following this strategy, the total magnetic field has been split into imposed and induced magnetic field components in all respective directions. The alternate form, the

89 decomposed full MFD (DFMFD) equations, utilizes the divergence and curl-free nature of the imposed magnetic field to eliminate the square of imposed magnetic field terms. For applications where the induced magnetic field is not negligible but considerably less than the applied magnetic field, the Lorentz force can be rewritten as

1 → → → Florentz = ( Bt) Bt µeo ∇ × ×

Subsequent to introducing the magnetic field splitting, it can be written as

1 → → → → → = ( Bi + Bo) Bt µeo ∇ × ∇ × ×

→ → Since the imposed magnetic field generated by currents outside the flow field, Bo =0 ∇ ×

1 → → → Florentz = ( Bi) Bt µeo ∇ × ×

Subsequently, magnetic stress terms in the induction equation and Joule heating ex- pression will become a function of the induced magnetic field, due only to the curl-free nature of the imposed magnetic field; therefore, mathematically,

1 → → 1 → → βij = Bt = Bi µeoσe ∇ × µeoσe ∇ ×

Upon introducing these relations into MFD equations, the order of magnetic pressure becomes similar to that of static pressure. Moreover, favorable reductions are achieved

90 in the magnetic induction equation where magnetic diffusion and advection terms become more equally balanced. Further details are discussed in the next section.

4.3 Mathematical Formulation

In this section, the procedure for deriving the governing equations of decomposed full

MFD formulation has been outlined. The generalized fluid dynamics equations with the

MFD assumption of neutral plasma along with magnetic induction equation can be rewrit- ten as

Continuity equation: ∂ρ + → (ρU→)=0 (4.1) ∂t ∇ ·

Momentum equation:

→ = ∂(ρU) → → → → = → + ρU U+pI = τ + Florentz (4.2) ∂t ∇ · ⊗ ∇ · h i

Induction equation:

→ ∂(Bt) → → → → → → → → + (U Bt Bt U)= νe Bt (4.3) ∂t ∇ · ⊗ − ⊗ −∇ × ∇ × ³ ´ Energy equation:

∂(ρet) → → → → = → → → νe → → 2 + (ρet + p) U = (U τ ) →q+Florentz U+ ( Bt) (4.4) ∂t ∇ · ∇ · · − ∇ · · µeo ∇ × h i

91 whereLorentzforceisgivenby

1 → → → Florentz = ( Bt) Bt µeo ∇ × ×

With the introduction of magnetic field splitting, the Lorentz force can also be written as

1 → → → 1 → → → Florentz = ( Bt) Bt ( Bo) Bo µeo ∇ × × − µeo ∇ × × or using Einstein notation, it can be written as

1 ∂Btk Btk 1 ∂Bti Btj 1 ∂Bok Bok 1 ∂Boi Boj Florentz = + + −2µ ∂xi µ ∂xj − −2µ ∂xi µ ∂xj ½ eo eo ¾ ½ eo eo ¾

By substituting the above expression for the Lorentz force into the momentum and energy equations, utilizing the divergence and curl-free nature of the imposed magnetic field, and adjusting the magnetic terms equivalent to fluid dynamics terms, the decomposed full MFD equations can be rearranged as

Continuity equation: ∂ρ + → (ρU→)=0 (4.5) ∂t ∇ ·

Momentum equation:

→ = → → → → ∂(ρU) → → → Bi Bt Bo Bi → = + ρU U+ p + pmag I ⊗ ⊗ = τ (4.6) ∂t ∇ · " ⊗ − µeo − µeo # ∇ · © ª

92 Induction equation:

→ ∂(Bi) → → → → → → → → + (U Bt Bt U)= νe Bi (4.7) ∂t ∇ · ⊗ − ⊗ −∇ × ∇ × ³ ´

Energy equation:

→ ∂(ρet) Bt = νe → → → → → → → → → → 2 + ρet + p + pmag U+ U Bi = (U τ ) q+ ( Bi) (4.8) ∂t ∇·" µeo · # ∇· · −∇· µeo ∇× ¡ ¢ ³ ´ where

1 2 1 → → pmag = Bi + Bi Bt (4.9) −2µeo µeo · and 1 p B2 ρe = ρU 2 + + i (4.10) t 2 γ 1 2µ − eo

It is interesting to note that subsequent to replacing the total magnetic pressure, 1 B2, 2µeo t

1 2 1 → → by the smaller quantity Bi + Bi Bt, the magnetic and aerodynamic pressures − 2µeo µeo · become similar in magnitude for strong imposed magnetic fields. Furthermore, magnetic

→ → stress terms become a function of induced magnetic fields due to the fact that νe( Bt)= ∇× → → νe( Bi), which eventually produces beneficial effects on the stability because not only ∇ × the production and diffusion terms of magnetic induction equation are similar, but also the

Joulean dissipation becomes a function of the induced magnetic field only. Subsequently, the equations can be represented in compact vector form as

93 ρ ρU→ = ⎡ → ⎤ ⎡ → → 1 → → 1 → → ⎤ ∂ ρU ρU U+ p + pmag I µ Bi Bt µ Bo Bi ⎢ ⎥ + → ⎢ ⊗ − eo ⊗ − eo ⊗ ⎥ ∂t ⎢ ⎥ ∇ · ⎢ ⎥ ⎢ B→ ⎥ ⎢ © U→ ªB→ B→ U→ ⎥ ⎢ i ⎥ ⎢ t t ⎥ ⎢ ⎥ ⎢ ⊗ − ⊗ ⎥ ⎢ ⎥ ⎢ → 1 → → → ⎥ ⎢ ρet ⎥ ⎢ ρet + p + pmag U µ Bt(U Bi) ⎥ ⎢ ⎥ ⎢ − eo · ⎥ ⎣ ⎦ ⎣ © ª ⎦ 0

⎡ = ⎤ → τ = ⎢ ∇ · ⎥ (4.11) ⎢ → → → ⎥ ⎢ (νe Bi) ⎥ ⎢ −∇ × ∇ × ⎥ ⎢ = ⎥ ⎢ → → → → νe → → 2 ⎥ ⎢ (U τ) q+ µ ( Bi) ⎥ ⎢ ∇ · · − ∇ · eo ∇ × ⎥ ⎣ ⎦ where

1 2 1 → → pmag = Bi + Bi Bt (4.12) −2µeo µeo ·

The imposed magnetic field cannot be eliminated entirely from the governing equations due to the nonlinear behavior of the magnetic field; however, no products of the imposed magnetic field will appear in the governing equations.

4.4 Cartesian Coordinate System

The system of decomposed full MFD equations can be represented in the flux-vector form of three-dimensional Cartesian coordinates as

∂Q ∂E ∂F ∂G ∂E ∂F ∂G + + + = v + v + v (4.13) ∂t ∂x ∂y ∂z ∂x ∂y ∂z

94 where Q is the unknown state vector

T Q = (4.14) ρρuρvρwBix Biy Biz ρet ∙ ¸ and flux vectors are

ρu ⎧ ⎫ 2 1 1 ⎪ ρu + p∗ µ Bix Btx µ Bix Box ⎪ ⎪ − eo − eo ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ρuv Biy Btx Bix Boy ⎪ ⎪ − µeo − µeo ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ρuw Biz Btx Bix Boz ⎪ ⎪ − µeo − µeo ⎪ E = ⎪ ⎪ (4.15) ⎪ ⎪ ⎨⎪ 0 ⎬⎪

⎪ uBty vBtx ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uBtz wBtx ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ (ρet + p∗) u Btx uBix + vBiy + wBiz ⎪ ⎪ − µeo ⎪ ⎪ ⎪ ⎪ ¡ ¢ ⎪ ⎩⎪ ⎭⎪

ρv ⎧ ⎫ 1 1 ⎪ ρuv µ Bix Bty µ Biy Box ⎪ ⎪ − eo − eo ⎪ ⎪ ⎪ ⎪ 2 1 1 ⎪ ⎪ ρv + p∗ Biy Bty Biy Boy ⎪ ⎪ − µeo − µeo ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ρvw Biz Bty Biy Boz ⎪ ⎪ µeo µeo ⎪ F = ⎪ − − ⎪ (4.16) ⎪ ⎪ ⎪ vBt uBt ⎪ ⎨ x − y ⎬ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ vBtz wBty ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ (ρet + p∗) v Bty uBix + vBiy + wBiz ⎪ ⎪ − µeo ⎪ ⎪ ⎪ ⎪ ¡ ¢ ⎪ ⎩⎪ ⎭⎪

95 ρw ⎧ ⎫ 1 1 ⎪ ρuw µ Bix Btz µ Biz Box ⎪ ⎪ − eo − eo ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ρvw Biy Btz Biz Boy ⎪ ⎪ − µeo − µeo ⎪ ⎪ ⎪ ⎪ 2 1 1 ⎪ ⎪ ρw + p∗ Biz Btz Biz Boz ⎪ ⎪ µeo µeo ⎪ G = ⎪ − − ⎪ (4.17) ⎪ ⎪ ⎪ wBt uBt ⎪ ⎨ x − z ⎬ ⎪ wB vB ⎪ ⎪ ty tz ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ (ρet + p∗) w Btz uBix + vBiy + wBiz ⎪ ⎪ − µeo ⎪ ⎪ ⎪ ⎪ ¡ ¢ ⎪ ⎩⎪ ⎭⎪

0 ⎧ ⎫ ⎪ τ xx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ ⎪ ⎪ xy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ xz ⎪ ⎪ ⎪ Ev= ⎪ ⎪ (4.18) ⎪ ⎪ ⎨⎪ 0 ⎬⎪ ⎪ β ⎪ ⎪ iyx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ βi ⎪ ⎪ zx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uτ xx + vτxy + wτ xz + qx + qJx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

96 0 ⎧ ⎫ ⎪ τ ⎪ ⎪ yx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ yy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ yz ⎪ Fv= ⎪ ⎪ (4.19) ⎪ ⎪ ⎪ β ⎪ ⎨⎪ ixy ⎬⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ βi ⎪ ⎪ zy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uτ yx + vτyy + wτ yz + qy + qJy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

0 ⎧ ⎫ ⎪ τ ⎪ ⎪ zx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ zy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ zz ⎪ Gv= ⎪ ⎪ (4.20) ⎪ ⎪ ⎪ β ⎪ ⎨⎪ ixz ⎬⎪ ⎪ β ⎪ ⎪ iyz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uτ zx + vτzy + wτ zz + qz + qJz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪ where 1 p B2 + B2 + B2 ρe = ρ(u2 + v2 + w2)+ + ix iy iz (4.21) t 2 γ 1 2µ − eo

97 and

1 2 1 → → p∗ = p Bi + Bi Bt (4.22) − 2µeo µeo · and magnetic stress is given by

1 ∂Bii ∂Bij βiij = (4.23) µ σe ∂xj − ∂xi eo µ ¶

E, F, and G are inviscid flux vectors, and Ev, Fv, and Gv are viscous flux vectors. Interest- ingly, only induced magnetic fieldcomponentshaveappearedintheunknownstatevector, in contrast to the classical full MFD formulation where total magnetic field components need to be calculated. The viscous stress tensor and expression for heat transfer rate will remain the same as that of the previous formulation in Section 3.2.1;thatis,

∂ui ∂uj 2 ∂uk τ ij = µ + δij µ (4.24) ∂xj ∂xi − 3 ∂xk µ ¶ and ∂T qi = k (4.25) ∂xi

However, Joule heating becomes a function of the induced magnetic field and may be expressed as

Bij ∂Bij ∂Bii qJ = νe (4.26) i µ ∂x − ∂x eo µ i j ¶

Although, an imposed magnetic field appears in the governing equations, it will remain in the background, since only an induced magnetic field will be computed in this approach.

98 4.5 Modification with → B→ = 0 condition ∇ · The implementation of total-variation-diminishing (TVD) schemes requires the evalu- ation of eigenvalues and eigenvectors of decomposed full MFD equations. Eigenvalues and eigenvectors can be determined by considering the convective part of DFMFD equations. It has been found that the system of DFMFD equations is not strictly hyperbolic. Similar to the full MFD system of equations, the DFMFD system of equations has eight scalar equa- tions with seven wave structures because the Jacobian matrix has a singularity associated with the normal component of the magnetic field. Therefore, a modification in DFMFD equations is required to remove the singularity. In the present work, a similar methodology as that of Powell et al. [111] has been utilized and implemented for the DFMFD equations.

This modification results in an additional term in the compact vector form of the DFMFD equation (4.11). Mathematically, the additional term can be presented as

ρ ρU→ 0 = ⎡ → ⎤ ⎡ → → 1 → → 1 → → ⎤ ⎡ 1 → ⎤ ρU ρU U+ p + pmag I Bi Bt Bo Bi Bt ∂ → µeo µeo µeo → → ⎢ ⎥ + ⎢ ⊗ − ⊗ − ⊗ ⎥ + ⎢ ⎥ Bi ∂t ⎢ ⎥ ∇ · ⎢ ⎥ ⎢ ⎥ ∇ · ⎢ B→ ⎥ ⎢ © U→ ªB→ B→ U→ ⎥ ⎢ U→ ⎥ ⎢ i ⎥ ⎢ t t ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⊗ − ⊗ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ → 1 → → → ⎥ ⎢ 1 → → ⎥ ⎢ ρet ⎥ ⎢ ρet + p + pmag U µ Bt(U Bi) ⎥ ⎢ µ U Bt ⎥ ⎢ ⎥ ⎢ − eo · ⎥ ⎢ eo · ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ © ª additional term 0 | {z } ⎡ = ⎤ → τ = ⎢ ∇ · ⎥ ⎢ → → → ⎥ ⎢ (νe Bi) ⎥ ⎢ −∇ × ∇ × ⎥ ⎢ = ⎥ ⎢ → → → → νe → → 2 ⎥ ⎢ (U τ ) q+ µ ( Bi) ⎥ ⎢ ∇ · · − ∇ · eo ∇ × ⎥ ⎣ ⎦ (4.27)

99 where

1 2 1 → → pmag = Bi + Bi Bt (4.28) −2µeo µeo ·

Similarly, equation (4.13) in the flux vector form for 3D Cartesian coordinates has been ammended with the additional term indicated by H

∂Q ∂E ∂F ∂G ∂E ∂F ∂G + + + + H = v + v + v (4.29) ∂t ∂x ∂y ∂z ∂x ∂y ∂z with 0 ⎧ ⎫ ⎪ Btx ⎪ ⎪ µeo ⎪ ⎪ ⎪ ⎪ Bty ⎪ ⎪ ⎪ ⎪ µeo ⎪ ⎪ ⎪ ⎪ Bt ⎪ ⎪ z ⎪ ⎪ µeo ⎪ → → H = ⎪ ⎪ Bi (4.30) ⎪ ⎪ ∇ · ⎨⎪ u ⎬⎪ ⎪ v ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ w ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uBtx +vBty +wBtz ⎪ ⎪ ⎪ ⎪ µeo ⎪ ⎪ ⎪ ⎪ ⎪ Worth noticing is the fact that⎩⎪ the additional term⎭⎪ H also becomes a function of the induced magnetic field due to the divergence-free nature of the imposed magnetic field;

→ → that is, Bo =0. Next, nondimensionalization of decomposed full MFD equations is ∇ · presented.

100 4.6 Nondimensionalization

The DFMFD equations are nondimensionalized according to the variables defined in

Section 3.2.3; the asterisk notation denoting nondimensional quantity has been dropped.

The nondimensional decomposed full MFD equations in flux vector form are provided as

∂Q ∂E ∂F ∂G ∂E ∂F ∂G + + + + H = v + v + v (4.31) ∂t ∂x ∂y ∂z ∂x ∂y ∂z where T Q = (4.32) ρρuρvρwBix Biy Biz ρet ∙ ¸

ρu ⎧ ⎫ 2 ⎪ ρu + p∗ Rbref Bix Btx Rbref Bix Box ⎪ ⎪ − − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ρuv Rbref Biy Btx Rbref Bix Boy ⎪ ⎪ − − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ρuw Rb Biz Btx Rb Bix Boz ⎪ ⎪ − ref − ref ⎪ E = ⎪ ⎪ (4.33) ⎪ ⎪ ⎨⎪ 0 ⎬⎪

⎪ uBty vBtx ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uBtz wBtx ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (ρet + p∗) u Rb Btx uBix + vBiy + wBiz ⎪ ⎪ − ref ⎪ ⎪ ⎪ ⎪ ¡ ¢ ⎪ ⎩⎪ ⎭⎪

101 ρv ⎧ ⎫ ⎪ ρuv Rbref Bix Bty Rbref Biy Box ⎪ ⎪ − − ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ρv + p∗ Rbref Biy Bty Rbref Biy Boy ⎪ ⎪ − − ⎪ ⎪ ⎪ ⎪ ρvw R B B R B B ⎪ ⎪ bref iz ty bref iy oz ⎪ F = ⎪ − − ⎪ (4.34) ⎪ ⎪ ⎪ vBt uBt ⎪ ⎨ x − y ⎬ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ vBtz wBty ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (ρet + p∗) v Rb Bty uBix + vBiy + wBiz ⎪ ⎪ − ref ⎪ ⎪ ⎪ ⎪ ¡ ¢ ⎪ ⎩⎪ ⎭⎪

ρw ⎧ ⎫ ⎪ ρuw Rbref Bix Btz Rbref Biz Box ⎪ ⎪ − − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ρvw Rbref Biy Btz Rbref Biz Boy ⎪ ⎪ − − ⎪ ⎪ ⎪ ⎪ ρw2 + p R B B R B B ⎪ ⎪ ∗ bref iz tz bref iz oz ⎪ G = ⎪ − − ⎪ (4.35) ⎪ ⎪ ⎪ wBt uBt ⎪ ⎨ x − z ⎬ ⎪ wB vB ⎪ ⎪ ty tz ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (ρet + p∗) w Rb Btz uBix + vBiy + wBiz ⎪ ⎪ − ref ⎪ ⎪ ⎪ ⎪ ¡ ¢ ⎪ ⎩⎪ ⎭⎪

102 0 ⎧ ⎫ ⎪ Rbref Btx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Rbref Bty ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Rb Btz ⎪ ⎪ ref ⎪ → → H = ⎪ ⎪ Bi (4.36) ⎪ ⎪ ∇ · ⎨⎪ u ⎬⎪ ⎪ v ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ w ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Rb uBtx + vBty + wBtz ⎪ ⎪ ref ⎪ ⎪ ⎪ ⎪ ¡ ¢ ⎪ ⎩⎪ ⎭⎪

0 ⎧ ⎫ ⎪ τ xx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ ⎪ ⎪ xy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ xz ⎪ ⎪ ⎪ Ev= ⎪ ⎪ (4.37) ⎪ ⎪ ⎨⎪ 0 ⎬⎪ ⎪ β ⎪ ⎪ iyx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ βi ⎪ ⎪ zx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uτ xx + vτxy + wτ xz + qx + Rb qJx ⎪ ⎪ ref ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

103 0 ⎧ ⎫ ⎪ τ ⎪ ⎪ yx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ yy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ yz ⎪ Fv= ⎪ ⎪ (4.38) ⎪ ⎪ ⎪ β ⎪ ⎨⎪ ixy ⎬⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ βi ⎪ ⎪ zy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uτ yx + vτyy + wτ yz + qy + Rb qJy ⎪ ⎪ ref ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

0 ⎧ ⎫ ⎪ τ ⎪ ⎪ zx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ zy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ zz ⎪ Gv= ⎪ ⎪ (4.39) ⎪ ⎪ ⎪ β ⎪ ⎨⎪ ixz ⎬⎪ ⎪ β ⎪ ⎪ iyz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uτ zx + vτzy + wτ zz + qz + Rb qJz ⎪ ⎪ ref ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪ where 1 p B2 + B2 + B2 ρe = ρ(u2 + v2 + w2)+ + R ix iy iz (4.40) t 2 γ 1 bref 2 −

104 and

1 2 → → p∗ = p Rb B + Rb Bi Bt (4.41) − 2 ref i ref · and the nondimensional magnetic stress is

1 ∂Bii ∂Bij βiij = (4.42) Rm σe ∂xj − ∂xi ref µ ¶

The viscous stress tensor can be written as

1 ∂ui ∂uj 2 ∂uk τ ij = µ + δij µ (4.43) Re ∂xj ∂xi − 3 ∂xk ref ½ µ ¶ ¾

Heat transfer rate can be written as

µ ∂T qi = 2 (4.44) Reref Prref (γ 1) M ∂xi − ref and Joulean dissipation is

Bij ∂Bij ∂Bii qJi = (4.45) Rm σe ∂xi − ∂xj ref µ ¶

4.7 Curvilinear Coordinates System

Subsequently, nondimensional DFMFD equations have been transformed into the com- putational domain to obtain discretized finite difference equations for numerical solution.

The transformation of DFMFD equations from physical space (x, y, z) to generalized com-

105 putational space (ξ,η,ζ) was obtained using the procedure outlined in Section 3.2.4.The transformed equations are

∂Q ∂E ∂F ∂G ∂E ∂F ∂G + + + + H = v + v + v (4.46) ∂t ∂ξ ∂η ∂ζ ∂ξ ∂η ∂ζ where Q Q = (4.47) J

1 E = ξ E + ξ F+ξ G (4.48) J x y z ¡ ¢ 1 F = η E + η F+η G (4.49) J x y z ¡ ¢ 1 G = ζ E + ζ F+ζ G (4.50) J x y z ¡ ¢

∂Bix ∂Biy ∂Biz H = HM + + (4.51) Ã ∂ξ ∂η ∂ζ ! 1 B = ξ B + ξ B +ξ B (4.52) ix J x ix y iy z iz ¡ ¢ 1 B = η B + η B +η B (4.53) iy J x ix y iy z iz ¡ ¢ 1 B = ζ B + ζ B +ζ B (4.54) iz J x ix y iy z iz ¡ ¢ 1 E = ξ E + ξ F +ξ G (4.55) v J x v y v z v ¡ ¢ 1 F = η E + η F +η G (4.56) v J x v y v z v ¡ ¢ 1 G = ζ E + ζ F +ζ G (4.57) v J x v y v z v ¡ ¢

106 E, F, and G represent the convective flux vectors and Ev, Fv, and Gv represent the diffusion

flux vectors in computational space, respectively. Each flux vector has eight components.

Convective flux terms can be evaluated from expressions (4.48) to (4.50).However,diffusive

flux terms, equations (4.55) to (4.57) , must be transformed into computational space; since these terms involve spatial derivatives of velocity and magnetic field. Hoffmann and

Chiang [112] have described the procedure for transformation of diffusion flux terms from physical space to computational space for the Navier-Stokes equations.

Following the similar procedure outline in [112], the expressions of diffusion flux vec- tors have been obtained for the DFMFD formulation. Subsequently, the expressions of nondimensional diffusion flux vectors in 2D and 3D generalized coordinates are provided in Appendices E and F.

Finally, the DFMFD equation (4.46) can be rewritten using the definition of the flux

Jacobian matrices as

∂Q ∂Q ∂Q ∂Q ∂E ∂F ∂G + A + B + C = v + v + v (4.58) ∂t ∂ξ ∂η ∂ζ ∂ξ ∂η ∂ζ where

∂E ∂Bix A = + HM = ξ A + ξ B+ξ C (4.59) ∂Q ∂Q x y z

∂F ∂Biy B = + HM = η A + η B+η C (4.60) ∂Q ∂Q x y z

∂G ∂Biz C = + HM = ζ A + ζ B+ζ C (4.61) ∂Q ∂Q x y z

107 ∂E ∂B A = + H ix (4.62) ∂Q M ∂Q

∂F ∂B B = + H iy (4.63) ∂Q M ∂Q

∂G ∂B C = + H iz (4.64) ∂Q M ∂Q

A, B, and C are the flux Jacobian matrices in the generalized curvilinear domain, whereas

A, B, and C are the flux Jacobian matrices in the physical domain. Note that only induced magnetic field components need to be transformed because partial derivatives of imposed magnetic field components have vanished in the physical domain.

4.8 Eigenstructure of the System

The introduction of magnetic field decomposition has produced significant changes in the structure of MFD equations. Note that although the magnitude of magnetic terms are reduced by removing the imposed magnetic field products as often as possible, the number of terms is increased. Additional terms have appeard particularly in the convective portion of the alternate form of full MFD equations; therefore, it becomes necessary to determine the eigenstructure of the system for developing the numerical scheme. The eigenstructure is associated with the convective portion of DFMFD equations written as

∂Q ∂E ∂F ∂G + + + =0 (4.65) ∂t ∂x ∂y ∂z

The system of equations for three dimensions contains eight equations and will result in an 8 8 Jacobian matrix for each direction. This system is also known as an eight-wave ×

108 Riemann problem. Determination of an eigensystem is also important for a Riemann solver that is based on the eigenstructure of the Jacobian matrix. It should be noted that for a multidimensional problem, a sigularity exists in the 8 8 Jacobian matrices of complete × MFD equations, which will produce a zero eigenvalue and indicates a nonphysical system.

Therefore, the additional zero term H has been included as it was introduced by Powell et al. [111]. The modified eight-wave system for DFMFD formulation can be expressed as

∂Q ∂E ∂F ∂G + + + + H =0 (4.66) ∂t ∂x ∂y ∂z

In terms of flux Jacobian matrices, this can be written as

∂Q ∂Q ∂Q ∂Q + A + B + C =0 (4.67) ∂t ∂x ∂y ∂z

The eigenvalues of the flux Jacobian matrices A, B, and C indicate the wave propaga- tion speed in x, y, and z directions, respectively. Each Jacobian matrix has real eigenvalues and a set of right and left eigenvectors. If X, Y, and Z are the right eigenvector matrices

1 1 1 and X− , Y− , and Z− are the left eigenvector matrices of Jacobian matrices A, B, and

C, respectively, then the following relations will hold:

1 X− AX = DA (4.68)

1 Y− BY = DB (4.69)

1 Z− CZ = DC (4.70)

109 where DA, DB, and DC are the diagonal matrices composed of the eigenvalues of the respective Jacobian matrix system as the elements of the matrix.

The determination of eigenvalues and eigenvectors is relatively difficult when Jacobian matrices are expressed in term of conservative variables. Therefore, a simplification has been introduced by defining a primitive variable vector Q´as

T Q´= ρuvwBix Biy Biz p ∙ ¸

Subsequently, equation (4.66) is rewritten in terms of the primitive variable vector Q´ as ∂Q´ ∂Q´ ∂Q´ ∂Q´ + A´ + B´ + C´ =0 (4.71) ∂t ∂x ∂y ∂z with

1 ∂Q − ∂E ∂B A´ = + H ix (4.72) ∂Q´ ∂Q´ M ∂Q´ ∙ ¸ 1 ∙ ¸ ∂Q − ∂F ∂B B´ = + H iy (4.73) ∂Q´ ∂Q´ M ∂Q´ ∙ ¸ 1 ∙ ¸ ∂Q − ∂G ∂B C´ = + H iz (4.74) ∂Q´ ∂Q´ M ∂Q´ ∙ ¸ ∙ ¸ where A´, B´, and C´are the auxiliary flux Jacobian matrices. The auxiliary flux Jacobian matrices are simpler than the original flux Jacobian matrices; therefore, they are preferred for determining the eigenstructure. The mathematical details of equation (4.67) and matrix

A´are provided in Appendix A.

110 Now, consider matrix A´. Let Lx betheleftandRx be the right eigenvectors of Jacobian matrix A´, and suppose that DA´is the diagonal eigenvalues matrix of A´. Then by definition,

LxA´Rx=DA´ (4.75)

1 ∂Q − ∂E ∂B L + H ix R =D x ∂W ∂W M ∂W x A´ ∙ ¸ ∙ ¸ 1 ∂Q − ∂E ∂B ∂Q L + H ix R =D x ∂W ∂Q M ∂Q ∂W x A´ ∙ ¸ ∙ ¸ 1 ∂Q − ∂Q L A R =D (4.76) x ∂W ∂W x A´ ∙ ¸

The uniqueness of eigenvalues results in the same diagonal elements of matrices DA and DA´, or mathematically as

DA=DA´ (4.77)

Thus, comparing equations (4.68) and (4.75) , this can be written as

1 ∂Q ∂Q − X = R and X 1 = L (4.78) ∂W x − x ∂W ∙ ¸

Since the auxiliary flux Jacobian matrices are easy to determine, compared to the original flux Jacobian matrices, it is recommended to compute the eigenvalues based on the auxiliary form of the Jacobian matrices. The eigenvalues of matrix A´are

111 Entropy wave λex = u

Alfvén waves λax = u vax ± ±

Fast acoustic waves λfx = u vfx ± ±

Slow acoustic waves λsx = u vsx ± ±

Divergence wave λdx = u with the following relations for Alfvén wave, fast wave, and slow wave

Btx vax = √µeoρ

1 v = [(v2 + c2)+z ] fx 2 a s x r 1 v = [(v2 + c2) z ] sx 2 a s x r − where

2 2 2 2 2 zx = (v + c ) 4c v a s − s ax q B2 + B2 + B2 2 γp tx ty tz cs = and va = ρ s µeoρ

It is interesting to note that the same eigenvalues as that of the classical full MFD approach have been found for the proposed alternate formulation (decomposed full MFD

approach), since Btx ,Bty , and Btz indicate total magnetic field components in the x, y, and z directions respectively. Subsequently, the diagonal eigenvalue matrix DA can be expressed as

112 λex 0000000 ⎡ ⎤ 0 λax+ 000000 ⎢ ⎥ ⎢ ⎥ ⎢ 00λ 00000⎥ ⎢ ax ⎥ ⎢ − ⎥ ⎢ ⎥ ⎢ 00 0λfx+ 0000⎥ ⎢ ⎥ DA = ⎢ ⎥ ⎢ ⎥ ⎢ 00 0 0λfx 000⎥ ⎢ − ⎥ ⎢ ⎥ ⎢ 00 0 0 0λ 00⎥ ⎢ sx+ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 00 0 0 0 0λsx 0 ⎥ ⎢ − ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 00 0 0 0 0 0λdx ⎥ ⎢ ⎥ ⎣ ⎦ The right and left eigenvector matrices of A´are

X = rex rax+ rax rfx+ rfx rsx+ rsx rdx ∙ − − − ¸

T 1 X− = lex lax+ lax lfx+ lfx lsx+ lsx ldx ∙ − − − ¸

In the present investigation, the eigenvectors of DFMFD equations and classical full

MFD equations based on primitive variable vectors have been evaluated. It has been found that the eigenvectors of DFMFD equations are different than classical full MFD equations.

However, the original set of eigenvectors can still be implemented in the numerical scheme for solving DFMFD equations because the conservation form of flux-vector splitting is utilized. The eigenvalues and eigenvectors for the flux Jacobian matrices B´and C´can be obtained by replacing x with y and z, respectively.

113 The eigenstructures for the flux Jacobian matrix A in generalized computational space for two- and three-dimensional coordinates are provided in Appendices B and C, respec- tively.

4.9 Boundary Conditions

In this section, boundary conditions for supersonic/hypersonic flow regimes over blunt bodies are discussed. Since decomposed full MFD equations treats imposed and induced magnetic fields separately, at the surface boundary these two components have been spec- ified individually. A description of boundary conditions is provided as follows:

Inflow

Specified free-stream conditions for pressure, temperature, density, and velocity. Im- posed magnetic field according to the specified distribution, e.g., dipole or uniform magnetic

field distributions.

Outflow

Zero-order extrapolation of all primitive variables. Imposed magnetic field with respect toaspecified distribution, e.g., dipole or uniform magnetic field distributions.

Body Surface

No-slip condition. • Adiabatic wall, zero normal gradient of pressure and density. • Evaluation of induced magnetic field components at body surface. • Specified imposed magnetic field with respect to a particular distribution, e.g. dipole • or uniform magnetic field distributions.

114 Evaluation of induced magnetic field at the body surface for blunt body supersonic

/hypersonic flows was performed by a similar procedure as that in Section 3.3. The dis- cretized equations for magnetic field components at the wall are provided as

∂Biy Bi = Bi + ∆y (4.79) xwall xinterior point ∂x

∂Biy Bi = Bi + ∆y (4.80) zwall zinterior point ∂z

∂Bix ∂Biz Bi = Bi + (4.81) ywall yinterior point − ∂x ∂z µ ¶

Noteworthy is the fact that only induced magnetic field components need to be calculated at the surface for the present decomposed full MFD formulation, which will certainly help save a significant amount of computational time. Subsequently, these equations have been transformed into a curvilinear coordinates system.

4.10 Advantages

Decomposed full MFD equations are mathematically and physically equivalent to clas- sical full MFD equations and retain the conservation law form of the equations and their eigenvalues but are better posed for numerical solutions. It is important to mention that, although the number of terms in DFMFD equations has been increased, the sources of strong stiffness associated with the order of magnitude of magnetic pressure and diffu- sion terms have been reduced significantly when magnetic field splitting is introduced.

This has resulted in primarily two advantages when computations have been performed for flows within strong imposed magnetic fields and for low ranges of magnetic Reynolds

115 numbers. First, a substantial increase in the stability of the numerical scheme (especially for strong imposed magnetic fields) is achieved, and second, a relatively larger time step can be utilized to achieve the solution, which will certainly produce a substantial reduction in computational time compared with classical full MFD equations.

116 Chapter 5

NUMERICAL METHOD

The numerical procedure used to solve magnetofluiddynamics equations is discussed in this chapter. Once the partial derivatives are approximated by finite difference equations, the resulting algebraic equations must be solved at each discrete point in the computational domain. The proposed numerical method is explicit based on a fourth-order Runge-Kutta

(RK4) scheme because of its high-order accuracy, stability, low storage requirements, and simplicity of programming. Since approximation of convective fluxes is achieved through a central difference with second-order accuracy, the dispersion error will occur in the domain of computation. These errors may cause detrimental oscillations in the neighborhood of discontinuities; subsequently, the oscillations can grow in amplitude rapidly and may cause inaccuracies in the solution or failure of the scheme. Hence, a damping term is needed to increase stability by eliminating oscillations. Recently, modern high-resolution conservative numerical schemes [114—117] have gained in popularity in the scientificcommunityfor simulating flows of liquids, gases, or plasmas. These schemes are widely utilized because they provide a robust and accurate method for capturing discontinuities that arise as the result of nonlinearity of equations, i.e., at shocks, current sheets, and contact discontinuities.

117 A total-variation-diminishing (TVD) principle is an important element of such schemes.

Despite their expensive cost, TVD schemes have shown some reliability for damping out the fluctuations for domains with strong discontinuities. TVD schemes reduce the order of accuracy of spatial interpolation near discontinuities and extrema. While eliminating oscillation, a reduction in shock smearing is also achieved with this technique.

In this chapter, the RK4 scheme and the TVD model utilized for stabilizing the scheme is introduced. Local time-stepping for accelerating the convergence rate is addressed, and

finally a domain decomposed solution algorithm based on the RK4 scheme is proposed for solving supersonic backward-facing step-flow problem.

5.1 Four-Stage Modified Runge-Kutta Scheme

A fourth-order Runge-Kutta scheme augmented with a second-order TVD model in the post-processing stage at each time level has shown a good improvement in the stability and shock-capturing capability [118, 120—123]. The four-stage modified Runge-Kutta scheme can be expressed as

(0) (n) Qi,j,k = Qi,j,k (5.1)

(1) (n) t Q = Q 4 f¯(0) (5.2) i,j,k i,j,k − 4 i,j,k

(2) (n) t Q = Q 4 f¯(1) (5.3) i,j,k i,j,k − 3 i,j,k

(3) (n) t Q = Q 4 f¯(2) (5.4) i,j,k i,j,k − 2 i,j,k

(4) (n) Q = Q tf¯(3) (5.5) i,j,k i,j,k − 4 i,j,k

118 where

∂E ∂F ∂G ∂E ∂F ∂G f = + + + H v v v for full MFD equations ∂ξ ∂η ∂ζ − ∂ξ − ∂η − ∂ζ (5.6)

∂E ∂F ∂G ∂Ev ∂Fv ∂Gv f = + + + SMFD for low Rm equations ∂ξ ∂η ∂ζ − ∂ξ − ∂η − ∂ζ (5.7)

Figure 5.1 illustrates the modified RK4 scheme solution algorithm with the post-processing stage, which consists of a TVD model, discussed below.

5.1.1 Post-Processing Stage

The modified Runge-Kutta scheme stabilized with a TVD scheme for damping out the numerical fluctuations associated with convective fluxes is used to obtain the numerical solutions. The TVD formulation is designed in such a way that it adjusts the amount of damping by switching from second to first order in accuracy where needed. The TVD model is based on the characteristic values and vectors of the flux Jacobian matrices associated with the convective part of the equations. The procedure for determining the eigenstructure of flux Jacobian matrices associated with classical full MFD equations is described by

Damevin [119], where the details of characteristic values and characteristic vectors have been provided in the relevant appendices as well. For the low magnetic Reynolds number approximation, the eigenstructure is the same as that used for Euler equations, which is described by Hoffmann and Chiang [112].

119 START

Inputs • Free stream conditions • Geometrical parameters

Grid Generation Initialization and Boundary Conditions • Generation of computational grid • Fluid and electromagnetic properties • Evaluation of transformation metrics and Jacobian • Initialization of solution • Application of boundary conditions

Time Loop

Four Stage Modified Runge-Kutta Scheme with TVD model • Four stages of RK4 scheme n+1 n Qi, j ,k = Qi, j,k + ∆Qi, j,k • Post-processing stage n+1 n+1 n Qi, j ,k = Qi, j,k + TVDi, j ,k

Update the Boundary Conditions • Update the results of computation

False Convergence criterion

True Outputs Primitive variables (Q): density, pressure, temperature, velocity and magnetic field

STOP

Figure 5.1: Solution algorithm based on modified Runge-Kutta scheme.

120 The procedure for determining the eigenstructure of flux Jacobian matrices associated with decomposed full MFD equations is described in Chapter 4, the details of eigenvalues and eigenvectors in the generalized two-dimensional and three-dimensional domains have been provided in Appendices B and C, respectively.

The post-processing stage is the last stage of computation and consists of a TVD model. It is made by correcting the last calculated unknown vector of the numerical scheme. Mathematically, it can be expressed as

n+1 n+1 1 t n n n n Qi,j,k = Qi,j,k 4 Xi+1/2,j,kΦi+1/2,j,k Xi 1/2,j,kΦi 1/2,j,k − 2 ξ − − − 4 1 t ¡ n n n n ¢ 4 Yi,j+1/2,kΘi,j+1/2,k Yi,j 1/2,kΘi,j 1/2,k (5.8) − 2 η − − − 4 1 t ¡ n n n n ¢ 4 Zi,j,k+1/2Λi,j,k+1/2 Zi,j,k 1/2Λi,j,k 1/2 − 2 ζ − − − 4 ¡ ¢

It is important to mention that the characteristic values and characteristic vectors will differ for full MFD and low magnetic Reynolds number formulations; nonetheless, a similar post-processing stage will be utilized for each MFD formulation. The eigenvector matrix

X corresponds to the flux Jacobian matrix A in computational space, and is provided in

Appendices B and C, respectively. The eigenvector matrices Y and Z that correspond to the flux Jacobian matrices B and C can be obtained by replacing ξ with η and ζ, respectively. The discretization of convective terms can be expressed as

∂E Ei+1,j,k Ei 1,j,k = − − (5.9) ∂ξ 2 ξ µ ¶i,j,k 4

121 ∂F Fi,j+1,k Fi,j 1,k = − − (5.10) ∂η 2 η µ ¶i,j,k 4

∂G Gi,j,k+1 Gi,j,k 1 = − − (5.11) ∂ζ 2 ζ µ ¶i,j,k 4

∂B ∂B ∂B H = H x + y + z i,j,k Mi,j,k ∂ξ ∂η ∂ζ ∙µ ¶ µ ¶ µ ¶¸i,j,k

Bxi+1,j,k Bxi 1,j,k Byi,j+1,k Byi,j 1,k Bzi,j,k+1 Bzi,j,k 1 = H − − + − − + − − Mi,j,k 2 ξ 2 η 2 ζ " 4 4 4 # (5.12)

Second-order central difference approximations were utilized for the diffusion terms

∂Ev ∂Fv ∂Gv ∂ξ , ∂η , and ∂ζ , according to the procedure outlined by Hoffmann i,j,k i,j,k i,j,k ³ ´ ³ ´ ³ ´ and Chiang [112]. The expressions of nondimensional diffusion flux vectors in 2D and 3D generalized coordinates are provided in Appendices E and F, respectively. It is important to note that the components of the diffusion flux vectors are expressed as the sum of terms of the form: LMξ,LMη, and LMζ, where subscripts ξ, η, and ζ indicate partial derivative in the respective direction. Following generic expressions were obtained according to the procedure described by Hoffmann and Chiang [112].

∂ 1 (LMξ)= [(Li+1,j,k + Li,j,k)(Mi+1,j,k Mi,j,k) ∂ξ 2(∆ξ)2 −

(Li,j,k + Li 1,j,k)(Mi,j,k Mi 1,j,k)] (5.13) − − − −

122 ∂ 1 (LMη)= [(Li+1,j,k + Li,j,k)(Mi+1,j+1,k + Mi,j+1,k Mi+1,j 1,k Mi,j 1,k) ∂ξ 8∆ξ∆η − − − −

(Li,j,k + Li 1,j,k)(Mi,j+1,k + Mi 1,j+1,k Mi,j 1,k Mi 1,j 1,k)] − − − − − − − − (5.14)

∂ 1 (LMζ )= [(Li+1,j,k + Li,j,k)(Mi+1,j,k+1 + Mi,j,k+1 Mi+1,j,k 1 Mi,j,k 1) ∂ξ 8∆ξ∆ζ − − − −

(Li,j,k + Li 1,j,k)(Mi,j,k+1 + Mi 1,j,k+1 Mi,j,k 1 Mi 1,j,k 1)] − − − − − − − − (5.15)

∂ 1 (LMξ)= [(Li,j+1,k + Li,j,k)(Mi+1,j+1,k + Mi+1,j,k Mi 1,j+1,k Mi 1,j,k) ∂η 8∆η∆ξ − − − −

(Li,j,k + Li,j 1,k)(Mi+1,j,k + Mi+1,j 1,k Mi 1,j,k Mi 1,j 1,k)] − − − − − − − − (5.16)

∂ 1 (LMη)= [(Li,j+1,k + Li,j,k)(Mi,j+1,k Mi,j,k) ∂η 2(∆η)2 −

(Li,j,k + Li,j 1,k)(Mi,j,k Mi,j 1,k)] (5.17) − − − −

∂ 1 (LMζ)= [(Li,j+1,k + Li,j,k)(Mi,j+1,k+1 + Mi,j,k+1 Mi,j+1,k 1 Mi,j,k 1) ∂η 8∆η∆ζ − − − −

(Li,j,k + Li,j 1,k)(Mi,j,k+1 + Mi,j 1,k+1 Mi,j,k 1 Mi,j 1,k 1)] − − − − − − − − (5.18)

∂ 1 (LMξ)= [(Li,j,k+1 + Li,j,k)(Mi+1,j,k+1 + Mi+1,j,k Mi 1,j,k+1 Mi 1,j,k) ∂ζ 8∆ζ∆ξ − − − −

(Li,j,k + Li,j,k 1)(Mi+1,j,k + Mi+1,j,k 1 Mi 1,j,k Mi 1,j,k 1)] − − − − − − − − (5.19)

123 ∂ 1 (LMη)= [(Li,j,k+1 + Li,j,k)(Mi,j+1,k+1 + Mi,j+1,k Mi,j 1,k+1 Mi,j 1,k) ∂ζ 8∆ζ∆η − − − −

(Li,j,k + Li,j,k 1)(Mi,j+1,k + Mi,j+1,k 1 Mi,j 1,k Mi,j 1,k 1)] − − − − − − − − (5.20)

∂ 1 (LMζ )= [(Li,j,k+1 + Li,j,k)(Mi,j,k+1 Mi,j,k) ∂ζ 2(∆ζ)2 −

(Li,j,k + Li,j,k 1)(Mi,j,k Mi,j,k 1)] (5.21) − − − −

Harada et al. [118, 120] and Augustinus et al. [122] investigated a broad collection of

TVD limiters associated with the TVD schemes. In the present research work, the Davis-

Yee symmetric TVD scheme based on the one dimensional problem reported by Harada et al. [118] was selected. Details follow here.

5.1.2 Davis-Yee Symmetric TVD Limiters

The TVD scheme is based on the eigenstructure of the convective flux Jacobian matri- ces. It is a second-order TVD formulation that adjusts the amount of damping by switching from second to first order in accuracy in the domain of computation. It remains second order in the smooth regions of the domain, but switches to first order where large gradi- ents occur to prevent any oscillations. Second-order TVD schemes can be developed by employing flux-limiter functions. Flux-limiter functions are given as

t 2 1 1 1 1 Φ = 4 λξ 1 g + ψ λξ 1 α g (5.22) i 2 ,j,k i ,j,k i 2 ,j,k i ,j,k i 2 ,j,k i 2 ,j,k ± − ξ ± 2 ± ± 2 ± − ± ∙4 ³ ´ ³ ´³ ´¸

t 2 1 1 1 1 Θ = 4 λη 1 h + ψ λη 1 β h (5.23) i,j 2 ,k i,j ,k i,j 2 ,k i,j ,k i,j 2 ,k i,j 2 ,k ± − η ± 2 ± ± 2 ± − ± ∙4 ³ ´ ³ ´³ ´¸

124 t 2 1 1 1 1 Λ = 4 λζ 1 i + ψ λζ 1 γ i (5.24) i,j,k 2 i,j,k i,j,k 2 i,j,k i,j,k 2 i,j,k 2 ± − ζ ± 2 ± ± 2 ± − ± ∙4 ³ ´ ³ ´³ ´¸

Eigenvalues λξ,λη, and λζ are provided in the following section. The following limiters are selected:

1 gi 1 ,j,k =minmod 2αi 1 ,j,k, 2αi 1 ,j,k, 2αi 3 ,j,k, αi 1 ,j,k + αi 3 ,j,k (5.25) ± 2 ∓ 2 ± 2 ± 2 2 ∓ 2 ± 2 ½ ³ ´¾

1 hi,j 1 ,k =minmod 2βi,j 1 ,k, 2βi,j 1 ,k, 2βi,j 3 ,k, βi,j 1 ,k + βi,j 3 ,k (5.26) ± 2 ∓ 2 ± 2 ± 2 2 ∓ 2 ± 2 ½ ³ ´¾ 1 ii,j,k 1 =minmod 2γi,j,k 1 , 2γi,j,k 1 , 2γi,j,k 3 , γi,j,k 1 + γi,j,k 3 (5.27) ± 2 ∓ 2 ± 2 ± 2 2 ∓ 2 ± 2 ½ ³ ´¾ Components α, β, and γ are defined in the generalized coordinate system as

1 − αi 1 ,j,k =2(X )i 1 ,j,k(Qi+1,j,k Qi,j,k)Á (Ji+1,j,k + Ji,j,k) (5.28) ± 2 ± 2 −

1 − βi,j 1 ,k =2(Y )i,j 1 ,k(Qi,j+1,k Qi,j,k)Á (Ji,j+1,k + Ji,j,k) (5.29) ± 2 ± 2 −

1 − γi,j,k 1 =2(Z )i,j,k 1 (Qi,j,k+1 Qi,j,k)Á (Ji,j,k+1 + Ji,j,k) (5.30) ± 2 ± 2 −

The entropy correction function is defined as

z for z δ | | | | ≥ ψ(z)=⎧ (5.31) 2 ⎪ z2 + δ Á2δ for z <δ ⎨ | | ⎩⎪ ¡ ¢ The δ term is known as the entropy correction parameter and usually has a range of 0 <δ ≤ 0.125. Stability and convergence rates are strongly dependent on the entropy correction

125 parameter δ in the TVD scheme. A significantly smaller value of δ may cause a slower convergence rate, whereas a larger value introduces numerical dissipation in the solution.

A constant value of δ can provide an economical and correct solution for simple aerodynamic geometries in supersonic flows. However, blunt body configuration at Mref > 2.5 shows a substantial deviation to the actual solution when a constant value of the entropy correction parameter is employed. The entropy correction parameter cannot be specified as a constant and must be locally computed to avoid nonphysical solutions; therefore, a variable value of δ is required to achieve a physical solution. For a steady-state hypersonic blunt-body

MFD flow regime, the solution will be stabilized and converged to a physical solution if δ is based on the flow velocity and magnitude of a fast magnetoacoustic wave. The expressions for calculating a local value of δ in the respective directions are given by

δi+ 1 ,j,k = δ Ui+ 1 ,j,k + Vi+ 1 ,j,k + Wi+ 1 ,j,k +(vfξ) 1 +(vfη) 1 +(vfζ) 1 2 2 2 2 i+ 2 ,j,k i+ 2 ,j,k i+ 2 ,j,k h¯ ¯ ¯ ¯ ¯ ¯ i ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ (5.32)

δi,j+ 1 ,k = δ Ui,j+ 1 ,k + Vi,j+ 1 ,k + Wi,j+ 1 ,k +(vfξ) 1 +(vfη) 1 +(vfζ) 1 2 2 2 2 i,j+ 2 ,k i,j+ 2 ,k i,j+ 2 ,k h¯ ¯ ¯ ¯ ¯ ¯ i ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ (5.33)

δi,j,k+ 1 = δ Ui,j,k+ 1 + Vi,j,k+ 1 + Wi,j,k+ 1 +(vfξ) 1 +(vfη) 1 +(vfζ) 1 2 2 2 2 i,j,k+ 2 i,j,k+ 2 i,j,k+ 2 h¯ ¯ ¯ ¯ ¯ ¯ i ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ (5.34)

U, V, and W denote the contravariant velocities, and vfξ ,vfη , and vfζ are the fast wave velocities in ξ, η, and ζ directions, respectively. Details of expressions (5.32) , (5.33) , and

(5.34) areprovidedbyAugustinusetal.[123].Theweighttermδ usually varies between

0.01 < δ<1.0. It is important to note that an optimum value of δ will result in fewer

126 iterations for achieving a converged solution with fair resolution of the outputs. Therefore, it is strongly recommended to use an optimum value of δ for achieving an economical and physically correct solution. Nonetheless, determination of an optimum value of δ is a cumbersome task that requires numerical experimentation with different values of δ,along with experience. Furthermore, the type of problem plays a significant role in determining an economically correct value of δ.

5.2 Local Time Stepping

In order to enhance the efficiency of a numerical scheme for steady-state solutions, evaluation of time at each grid point is recommended. This concept is known as local time stepping. For a steady-state problem, the solution history follows a path that has no physical meaning; therefore, it becomes possible to accelerate the convergence of the solution by introducing a local time step. The local time step strongly depends on the fastest characteristic wave of propagation and spatial parameter. A user-specified non-dimensional parameter known as the Courant-Friedrichs and Lewy (CFL) number is generally utilized for evaluating the local time at each grid point. The range of the CFL number is set between 0 and 0.2 for a typical explicit scheme. The time step for a three-dimensional

MFD problem can be computed as

∆t =min(∆tξ, ∆tη, ∆tζ )

∆ξ ∆η ∆ζ ∆tξ = CFL , ∆tη = CFL and ∆tζ = CFL (5.35) λξmax ληmax λζmax where λξ,λη, and λζ are the eigenvalues in the ξ, η, and ζ directions, respectively.

127 The eigenvalues for the classical full MFD equations are

λξ =max λξ ,λη =max λη and λζ =max λζ (5.36) max i=1..8 i max i=1..8 i max i=1..8 i ¡ ¢ ¡ ¢ ¡ ¢

λξ1 = Uλξ2 = U

λξ = U + vaξ λξ = U vaξ 3 4 − (5.37)

λξ = U + vfξ λξ = U vfξ 5 6 −

λξ = U + vsξ λξ = U vsξ 7 8 − where U is the contravriant velocity, vaξ is the Alfvén velocity, vfξ is the fast wave velocity, and vsξ is the slow wave velocity. These velocities in generalized computational space can be described as

U = ξxu + ξyv + ξzw

vaξ = ξxvax + ξyvay + ξzvaz (5.38) v2 = 1 [a (v2 + c2)] + z fξ 2 4 a s ξ

2 1 2 2 v = [a4 (v + c )] zξ sξ 2 a s − where v = Bx c2 = γP ax √ρ s ρ

v = By a = ξ2 + ξ2 + ξ2 ay √ρ 4 x y z (5.39) Bz 2 2 2 2 vaz = zξ = a4 a4 (v + c ) 4c v √ρ a s − s aξ q 2 2 2 2 £ ¤ va = vax + vay + vaz

The expressions for velocities in equation (5.39) will remain the same for decomposed full

MFD formulation except magnetic field components (Bx,By,Bz) , which will be replaced

by total magnetic field components Btx ,Bty ,Btz . ¡ ¢

128 The eigenvalues for a low magnetic Reynolds number formulation will become

λξ =max λξ ,λη =max λη and λζ =max λζ (5.40) max i=1..5 i max i=1..5 i max i=1..5 i ¡ ¢ ¡ ¢ ¡ ¢

λξ1 = Uλξ2 = U (5.41) λξ3 = Uλξ4 = U + vcsξ

λξ = U vcsξ 5 − where vcsξ = cs√a4. The eigenvalues in the η and ζ directions can be expressed in a similar waybyreplacingξ with η and ζ, respectively.

5.3 MultiblockStrategyforBackward-FacingStep

Having a physical domain that excludes the upstream region of the step for modeling supersonic backward-facing step flows may not provide accurate results, because even in a supersonic flow, disturbances within the subsonic portion of the boundary layer can travel upstream at the expansion corner and can affect the flow field. To avoid this negative aspect, it is recommended to include the upstream region of the step. A multiblock flow solver is one of the most powerful tools for handling this problem in structured grids. Some advantages of multiblocking include independent grid generation for each block, which alleviates several topological problems encountered in constructing the mesh, a greater control over grid refinement in the regions of interest, savings in computational cost, and economical memory allocation.

129 Thus, by adopting a multiblock approach in the present analysis, the global physical domain has been decomposed into two subdomains representing upstream and downstream regions with respect to step location. The grid topology for each subdomain remains con- sistent with the geometry. Subdomains have been connected through common boundaries with a single, overlapped interface. Information will be transferred through a common interface of blocks.

5.3.1 Multiblock Solution Algorithm

The developed solution algorithm is illustrated in Figure 5.2. First grids are generated and transformation metrics are evaluated for each domain; subsequently, initialization and application of boundary conditions are performed. The numerical solver is based on a modified Runge-Kutta scheme due to its higher-order accuracy and efficiency. As can be seen from the algorithm, the solutions for each domain are mutually dependent on each other; however, governing equations in each domain are solved locally. In the time integration loop for each iteration, once solution for first domain is achieved, the connecting face of the second domain is updated; subsequently, calculations for the second domain are performed, and finally the interface for the first domain is updated. The procedure toward a steady-state solution continues until a convergence criterion is achieved.

130 Domain # 1 Domain # 2

START

Inputs • Free stream conditions • Geometrical parameters

Grid Generation Initialization and Boundary Conditions • Generation of computational grid for Domain # 1 • Fluid and electromagnetic properties • Generation of computational grid for Domain # 2 • Initialization of solution • Evaluation of transformation metrics and Jacobian • Application of boundary conditions

Time Loop

Call RK4 Solver for Domain # 1

Update Interface for Domain # 2 Multiblock solver

Call RK4 Solver for Domain # 2

Update Interface for Domain # 1

False Convergence criterion

True Outputs Primitive variables (Q): density, pressure, temperature, velocity and magnetic field

STOP

Figure 5.2: Solution algorithm based on multiblock approach.

131 As discussed previously, the multiblocking technique utilizes information transfer at the common interface of blocks. For example, if IM1 is the maximum number of grid points in the I-direction for domain number one, then an updating interface for domain number two will take the form of

(Q1,j,k) =(QIM1 1,j,k) domain 2 − domain 1 where Q represents vector of unknown primitive variables. Similarly, the interface boundary condition for domain number one for the next time step will be updated according to

(QIM1,j,k)domain 1 =(Q2,j,k)domain 2

where j =1to JMAX and k =1to KMAX.

132 Chapter 6

RESULTS AND DISCUSSION

In this chapter, the three MFD formulations are utilized to develop two- and three- dimensional computer algorithms for each formulation. Flows over different geometries are simulated, and results are compared to the analytical and experimental data available in the literature.

The first step of implementation consists of development and validation of computer codes for the full MFD and low magnetic Reynolds number formulations. The MFD

Rayleigh problem was adopted for validation purpose because of its simplicity and avail- ability of the exact solution. Subsequently, validity of the low magnetic Reynolds number approximation of magnetofluiddynamic and its performance are discussed.

Next, the proposed decomposed full MFD equations are solved numerically by a mod- ified four-stage Runge-Kutta scheme with a TVD limiter. First, the DFMFD formulation is validated with the available closed form solution of magnetic Rayleigh problem; subse- quently, the solution is compared with the classical FMFD solution. In addition, a time- efficiency study is performed by considering a magnetic Rayleigh problem as a benchmark case for determining the computational performance of DFMFD, FMFD, and low magnetic

Reynolds number formulations for weak and strong imposed magnetic fields.

133 Subsequently, hypersonic flow over a blunt-body configuration is considered due to the complex nature of the problem. MFD analysis of hypersonic blunt-body flow with

FMFD equations is a challenging task because accumulation of an induced magnetic field may result in a significantly large amount of computational time, even for hypersonic flow regimes under a frozen state. Therefore, a blunt-body configuration subject to hypersonic

flow was selected as a second case to further validite DFMFD equations in comparison to full MFD (FMFD) equations.

Finally, supersonic flow over a backward-facing step is investigated under different types of applied magnetic field distributions for a low value of magnetic Reynolds number.

Low magnetic Reynolds number approximation was utilized, and results are presented for uniform and temperature-dependent electrical conductivity distributions.

6.1 Full MFD and Low Magnetic Reynolds Number Approaches

In this section the developed computer codes for full MFD and low magnetic Reynolds number formulations are investigated. The test case for MFD simulations is a magnetic

Rayleigh problem, which provides the key to unlocking various questions relative to the applicability of MFD modeling approaches and algorithm validation. The results from the solution of the low magnetic Reynolds number formulation is compared to the results obtained from the FMFD equations for different values of magnetic Reynolds numbers.

Errors in the results of both formulations are estimated over a range of magnetic Reynolds number and performance of each formulation has been assessed.

134 6.1.1 Magnetic Rayleigh Problem—Validation of Algorithms

The classical magnetic Rayleigh problem considers the unsteady motion of an incom- pressible viscous fluid in response to a flat plate suddenly set in motion along its own plane with the magnetic field imposed normal to the plate surface. In addition to the applied magnetic field, which remains constant, the velocity field induces a magnetic field that trav-

Bo els normal to the plate surface at a constant speed of Ao = (known as Alfvén wave). √µeo ρ Figure 6.1 illustrates the basic features of developed velocity profiles and applied magnetic

fields normal to the surface for a typical MFD Rayleigh problem at a particular time step.

This is a time-dependent problem that describes the propagation of disturbances due to the motion of the plate in the fluid. This problem provides an important benchmark for

MFD code validation due to the availability of an exact solution. The analytical solution available in the literature assumes same thicknesses of magnetic and velocity boundary layers, which results in a magnetic Prandtl number equal to one. The details of velocity and induced magnetic field expressions are given by [124].

Aoy Aoy u 1 − = 2.0 (erf (λ+)+erf(λ )) + e νe erfc (λ )+e νe erfc (λ+) U 4 − − − ∞ h i Aoy Aoy Bx 1 − = (erf (λ+) erf (λ )) + e νe erfc (λ ) e νe erfc (λ+) Bref 4 − − − − h i where

y Aot λ = ± ± 2√νet

Bref = U √µeoρ ∞

135 Y

Z X

Figure 6.1: Development of velocity profiles for MFD Rayleigh problem.

136 It should be noted that Bx is the induced magnetic fieldinthex-direction, and νe is the magnetic diffusivity. Symbols erf and erfc denote error function and complementary error function, respectively. The data for validation purposes is provided in Table 6.1.

Temperature has been calculated with the assumption of a magnetic Prandtl number equal one, whereas the magnetic Reynolds number is set equal to 2.5 based on the domain height.

Table 6.1: Operating conditions for MFD Rayleigh problem used for code validation. Operation conditions 3 Mach number M =1.63 10− ∞ × 5 3 Fluid density ρ =4.0 10− kg m− ∞ × Temperature T =940.0 K ∞ 7 1 Electrical conductivity σe =10/4πSiemensm− 4 Magnetic field strength By =1.449 10− T × Domain height h =2.5 m

In this investigation, two computer programs, 2DMFD and 2DLRMFD, were developed and validated for determining the range of applicability and efficiency of low magnetic

Reynolds approximation versus full MFD formulation. The computer code 2DMFD is based on classical FMFD equations whereas code 2DLRMFD is written by utilizing a low magnetic

Reynolds approach for modeling two dimensional (2-D) MFD problems. Subsequently, both codes were extended to a third dimension for simulating three-dimensional (3-D) MFD effects.

For the numerical solution of a magnetic Rayleigh problem, a rectangular domain of

10 150 grid spacing with clustering near the solid wall and a rectangular block of 10 150 5 × × × grids with clustering near the solid surface was generated for 2-D and 3-D versions of the

137 codes, respectively. It is important to mention that the numerical results obtained from

2-D and 3-D versions of the code are identical to each other; however, 3-D computer codes take substantially longer, compared to 2-D computer codes. For example, with the grid spacing specified above for the rectangular block, the computational time for a 3-D version of the MFD code is approximately five times larger than the 2-D version for the MFD

Rayleigh problem.

The initial and boundary conditions for the governing MFD equations for each approach must be specified for the flow and plasma variables. At the solid wall, adiabatic boundary condition for temperature, zero normal gradient of static and magnetic pressures, and uniform magnetic field normal to the plane are imposed. The plate, which is initially at rest, is suddenly set into motion with a constant speed of Uo in the x-direction. Consequently, computations were initiated at t =0.0 second and continued to the final time period at t =0.1 second for computation of the velocity and induced magnetic field profiles.

Figure 6.2 shows the non-dimensional velocity profiles for different time levels for the case of an electrically insulating wall (applied magnetic field at the wall). Excellent agree- ment between the exact and full MFD numerical solutions is apparent for all time levels.

For example, at a point where domain height (y)isequalto0.5m, the percentage er- ror between numerical solutions obtained from full MFD equations and the exact solution is negligible (˜0.16%) for all time levels. However, the error between the low magnetic

Reynolds number approach and the analytical solution is substantial. In fact, all velocity profiles obtained with the low magnetic Reynolds number approach collapse onto the same curve, and therefore, the wave propagation is not captured. The percentage error between

138 low magnetic Reynolds number formulation and the exact solution for the first-time level of t =0.02 second is found to be ˜84.2%, at a point where domain height (y)isequalto0.5m.

This significant value of error indicates that low magneticReynoldsnumberformulationis not valid in the high magnetic Reynolds number range.

2.5 X X

X X Exact solution 2 X Full MFD solution X Low Rm solution X -2 X X t=2×10 sec X X X X -2 XXX XXXXt=4×10 sec XXXX 1.5 X XX -2 t=6XX ×10X sec X X t=8×10-2 sec X -2

y(m) t=10×10 sec 1 X X

X X

X 0.5 X X X

X X

X 0 X 0 0.2 0.4 0.6 0.8 1

u/Uref

Figure 6.2: Comparison of numerical and analytical velocity distributions for different time intervals at Rm =2.5.

Figure 6.3 shows a build-up of induced magnetic field with respect to time; once again an agreement with analytical data has been demonstrated for all time levels. It is inter- esting to observe that equal thicknesses of velocity and magnetic boundary layers were computed from the full MFD approach (see Figures 6.2 and 6.3) which confirms the un- derlying assumption in the exact solution.

139 2.5 Exact solution X Full MFD solution -2 X t=2×10 sec X X X t=4×10-2 sec 2 -2 t=6×10 sec X t=8×10-2 sec X t=10×10-2 sec

X 1.5

X y(m) 1 X

X

0.5 X

X

0 X -0.5 -0.4 -0.3 -0.2 -0.1 0

Bix /Bref

Figure 6.3: Comparison of numerical and analytical induced magnetic field distributions for different time intervals at Rm =2.5.

3 Next, a set of calculations was performed for magnetic Reynolds number 2.5 10− . ×

This value of Rm was achieved by adjusting the electrical conductivity of the medium.

Velocity profiles for the two modeling approaches of magnetofluiddynamics are shown in

Figure 6.4 for different time levels. It is evident that low magnetic Reynolds number formulation provides similar results as that of full MFD equations. This is true, because

3 the induced magnetic field for Rm =2.5 10− is substantially smaller compared to the × high values of the induced magnetic field for magnetic Reynolds number of Rm =2.5.

140 2.5 X

X Low R solution 2 m Full MFD solution X -2 X t=2×10 sec X -2 X X t=4×10 sec X X X X -2 X XXX t=6×10 sec 1.5 X X X X X -2 X t=8×10 sec t=10×10-2 sec y(m) 1 X

X

0.5 X

X

0 0 0.2 0.4 0.6 0.8 1

u/Uref

Figure 6.4: Comparison of velocities obtained by Full MFD and low magnetic Reynolds 3 number formulations at Rm =2.5 10 . × −

This fact can be revealed by examining Figure 6.5 where a comparison of dimen- sionless induced magnetic fields for two magnetic Reynolds numbers was made. It is worthwhile to note that the induced magnetic field for a low magnetic Reynolds num-

3 ber (Rm =2.5 10− ) has very low strength and is practically negligible compared to the × induced magnetic fieldobtainedwithahighmagneticReynoldsnumber(Rm =2.5) for all time levels. For a low magnetic Reynolds number, any self-induced magnetic field will abruptly diffuse away. Thus, the strength of the induced magnetic field becomes substan- tially smaller than the applied magnetic field, there is no need for solving the full MFD equations, and low magnetic Reynolds number approximation can be utilized to achieve the solution.

141 2.5 -3 XX Rm =2.5×10 X Rm =2.5 -2 X X t=2×10 sec XX 2 t=6×10-2 sec t=10×10-2 sec XX

XX 1.5

XX y(m) 1 XX

X X

0.5 X X

X X

0 X -0.5 -0.4 -0.3 -0.2 -0.1 0

Bix/Bref

Figure 6.5: Comparison of induced magnetic field distributions obtained from Full MFD formulation for different values of magnetic Reynolds number.

Figure 6.6 illustrates the percentage of average error in the velocities obtained by the full

MFD and low magnetic Reynolds number formulations. Velocities for the two approaches at different time levels were probed along the y-direction and subsequently averaged based on the number of data points extracted. An increase in calculated error is observed with respect to the value of magnetic Reynolds number. Worth noticing is the fact that accumulation of percentage average error takes place as the time level increases. Also, the percentage average error has lower values for the range of magnetic Reynolds numbers near zero; for example, at Rm =0.125, the maximum error is less than 10% for all time steps. However, the percentage of average error begins to increase for relatively large magnetic Reynolds

142 numbers, as shown in Figure 6.6. This error estimation was performed to determine a cutoff value between full MFD and low magnetic Reynolds number approaches specifically for the MFD Rayleigh problem.

40 t=2×10-2 sec t=4×10-2 sec t=6×10-2 sec t=8×10-2 sec 30 t=10×10-2 sec

20

10 aeaeerrin%average error velocities

0 00.511.522.5

Magnetic Reynolds number (Rm)

Figure 6.6: Percentage average error in velocities obtained by Full MFD and low magnetic Reynolds number formulations for different values of magnetic Reynolds number.

Figure 6.7 shows computational time in terms of wall clock time for the two MFD modeling approaches to reach a time step of t =0.02 second for the MFD Rayleigh problem.

Time was calculated on an Intel Pentium D, dual-core microprocessor of 3.2 GHz speed with 3.0 gigabytes of RAM. Furthermore, the optimization option of the compiler was utilized to increase the execution speed. It is evident that low magnetic Reynolds number

143 approximation requires substantially less time compared to a full MFD set of equations for values of magnetic Reynolds number near zero. However, with an increase in the magnetic

Reynolds number, the full MFD approach showed stability, resulting in a relatively larger step size and less time to reach the desired solution. It is interesting to observe that as the magnetic Reynolds number approaches one, both schemes require a similar amount of time.

Once again, this time study was conducted specifically for the MFD Rayleigh problem to evaluate the performance of full MFD and low magnetic Reynolds number formulations over a range of magnetic Reynolds numbers.

8

6 Full MFD solution

Low Rm solution

4

Wall clock Time (hours) 2

0 00.250.50.751

Magnetic Reynolds number (Rm)

Figure 6.7: Wall clock time for full MFD and low magnetic Reynolds number formula- tions.

144 6.2 Decomposed Full MFD Approach

In this segment of the investigation, the proposed third formulation of magnetofluid- dynamics that utilizes the decomposition of total magnetic field into imposed and induced components was implemented numerically. A computer code called DECOMFD was writ- ten for implementation of decomposed full MFD equations. Both 2-D and 3-D versions of the code were developed for investigating the validity of DFMFD equations. After vali- dating the DFMFD approach, the computational efficiency of all three MFD formulations was compared for weak and strong strengths of the imposed magnetic field by considering the MFD Rayleigh problem as a reference case. Following details of code validation and comparison of results have been provided.

6.2.1 Magnetic Rayleigh Problem—Validation of Algorithm

Because of its simplicity and availability of the closed form solution, the magnetic

Rayleigh problem was selected once again for validating the developed computer algorithm based on the DFMFD approach to magnetofluiddynamics. The operating conditions were selected in accordance with the data presented in Table 6.1 for numerical simulations. Do- main dimensions and computational grid were adjusted similarly to the previous validation cases for FMFD and low magnetic Reynolds number formulations. Results of the numerical computations obatined from 2-D and 3-D versions of the computer code were found to be in good agreement with the exact solution; however, computational time associated with the 3-D code was large (five times larger than the 2-D version). Therefore, in all subsequent calculations, the 2-D version of the code was utilized.

145 Figure 6.8 illustrates the comparison of the non-dimensional velocity profiles obtained from the exact solution and numerical DFMFD approach between time intervals of 0.02 to 0.1 second. Development of the velocity profiles with respect to time for the DFMFD solution was found to be in accordance to the exact solution. In fact, a superb agreement between the solutions obtained from DFMFD equations and the exact relation was demon- strated for all time levels. The value of interaction parameter was equal to Qref =1043.84 for present analysis.

2.5 X

X Exact solution 2 DFMFD solution -2 X X t=2×10 sec t=4×10-2 sec X X X -2 X XXX XXX t=6×10 sec XXXX XXX XX -2 1.5 X t=8XX ×X10 sec -2 X t=10×10 sec y(m) 1 X

X

0.5 X

X

0 X 0 0.2 0.4 0.6 0.8 1

u/Uref

Figure 6.8: Comparison of velocities obtained from exact solution and DFMFD formu- lation at different time intervals.

Similarly, an excellent agreement between numerical and analytical solutions was ob- served for the induced magnetic fieldinthex-direction for all time levels, as shown in

146 Figure 6.9. Furthermore, equal thicknesses of velocity and magnetic boundary layers were computed from the DFMFD approach which justifies the basic assumption in the exact solution.

2.5 Exact solution X DFMFD solution -2 X t=2×10 sec X X X t=4×10-2 sec 2 -2 t=6×10 sec X t=8×10-2 sec X t=10×10-2 sec

X 1.5

X y(m) 1 X

X

0.5 X

X

0 X -0.5 -0.4 -0.3 -0.2 -0.1 0

Bix /Bref

Figure 6.9: Comparison of induced magnetic fields obtained from exact solution and DFMFD formulation at different time intervals.

Subsequent to validating the developed computer code for DFMFD equations with the

3 analytical model, a case at significantly lower magnetic Reynolds numbers (Rm =2.5 10− ) × was chosen for attaining further confidence over DFMFD formulation. Electrical conduc- tivity of the medium was adjusted to achieve a low magnetic Reynolds number and the corresponding value of interaction parameter was Qref =1.044. Figure 6.10 represents non-dimensional velocity profiles obtained from the three available modeling approaches

147 3 of magnetofluiddynamic for Rm =2.5 10− . ItisobviousthattheDFMFDformulation × also provides similar results as compared to classical full MFD equations and low mag- netic Reynolds number approximation for a substantially low range of magnetic Reynolds numbers. Note that all three formulations predicted similar results because the induced

3 magnetic field for Rm =2.5 10− is substantially smaller than the high values of an × induced magnetic field for magnetic Reynolds numbers of Rm =2.5.

2.5 X

X Low R solution 2 m Full MFD solution X DFMFD solution -2 X X X t=2×10 sec X X X X -2 X XX t=4×10 sec 1.5 XX X X X -2 X t=6×10 sec t=8×10-2 sec -2 y(m) t=10×10 sec 1 X

X

0.5 X

X

0 0 0.2 0.4 0.6 0.8 1

u/Uref

Figure 6.10: Comparison of velocities obtained by DFMFD, FMFD, and low magnetic 3 Reynolds number formulations at Rm =2.5 10 . × −

Furthermore, the non-dimensional induced magnetic field along the x-direction obtained from the two MFD formulations for different time levels is compared in Figure 6.11. The induced magnetic field distributions obtained from the DFMFD equations compare well

148 with the solution of FMFD equations. Observe that the strength of induced magnetic

field is substantially low, which indicates that dissipation supersedes the advection of the magnetic field for low ranges of magnetic Reynolds numbers.

2.5 Full MFD solution DFMFD solution -2 X t=2×10 sec X t=4×10-2 sec 2 t=6×10-2 sec t=8×10-2 sec t=10×10-2 sec

X 1.5 y(m) 1 X

0.5 X

0 X -0.0075 -0.005 -0.0025 0

Bix/Bref

Figure 6.11: Comparison of induced magnetic fields obtained from FMFD and DFMFD formulations for different time intervals at Rm =2.5.

Finally, a comparison of computational time to reach the time interval of t =0.02 second from the three modeling approaches is illustrated in Figure 6.12. Time was estimated on an Intel Pentium D, dual-core processor at 3.2 GHz speed and 3.0 gigabytes of RAM with optimization instructions provided to the compiler for enhancing the execution speed. In fact, Figure 6.12 represents a generalized version of Figure 6.7 with the inclusion of wall clock time taken by DFMFD formulation under the operating conditions of Table 6.1 where

149 a substantially low strength of applied magnetic field is utilized. It is obvious that for a range of magnetic Reynolds numbers near zero, low magnetic Reynolds number formulation requires significantly less time than that of a complete set of MFD equations.

8

6 Full MFD solution

Low Rm solution DFMFD solution

4

Wall clock Time (hours) 2

0 0 0.25 0.5 0.75 1

Magnetic Reynolds number (Rm)

Figure 6.12: Wall clock time for FMFD, DFMFD, and low magnetic Reynolds number formulations.

3 For example, at Rm =2.5 10− , the difference in wall clock time is around 50.0 hours, × which is very large, and may increase further for complex MFD flows. Nonetheless, with the increase of magnetic Reynolds numbers, stability of both MFD formulations is increased, thereby resulting in the requirement of less computational time. Note that for magnetic

Reynolds numbers close to one, the difference in wall clock time using the low magnetic

Reynolds number approach and both full MFD formulations is significantly decreased. In

150 fact, the difference in wall clock time between the low magnetic Reynolds number approach and both full MFD formulations is found in the neighborhood of ˜0.08 hour for the value of magnetic Reynolds number equal to one.

It should be noted that the wall clock time presented in Figure 6.12 was computed for

4 a substantially low strength of the applied magnetic field (By =1.449 10− T ),anditwas × found that DFMFD equations require a similar amount of time as that of FMFD equations for different ranges of magnetic Reynolds numbers for the MFD Rayleigh problem. How- ever, this argument remains valid only for the weak strength of an applied magnetic field; further investigations with strong applied magnetic fieldstrengthhaveshownasubstantial reduction in computational time when the DFMFD approach is utilized as compared to the

FMFD approach. Because magnetic pressure is directly proportional to the square of mag- netic field intensity, when a strong magnetic field is imposed, the magnitude of magnetic pressure may become several orders higher than the static pressure, which may cause in- stability in the numerical scheme of FMFD equations. Consequently, a smaller time step is required to overcome this issue, which will ultimately increase the amount of time to achieve the solution from FMFD formulation. On the other hand, DFMFD equations can provide an alternative approach to solve full MFD equations within a permissible time frame with strong imposed magnetic fields. The results obtained with strong imposed magnetic fields is discussed in the following section.

151 6.2.1.1 Analysis of Strong Magnetic Field Strengths

Decomposed full MFD equations have shown substantial stability when computations have been performed with strong imposed magnetic fields. As mentioned, during the formu- lation of DFMFD equations, one of the important purposes of introducing decomposition of the magnetic field is to reduce the order of magnitude of the magnetic pressure for ap- plications where strong magnetic field is imposed. This reduction in the order of magnetic pressure will produce stability in the numerical scheme for a relatively large time step and ultimately lessen the requirement of a large amount of computational time. With this pro- posed aim, computations were performed with relatively strong strengths of the imposed magnetic field for an MFD Rayleigh problem to demonstrate efficieny of the DFMFD for- mulation compared to the FMFD formulation. Once again, the fluid dynamics conditions provided in Table 6.1 were utilized with higher magnetic field strengths for different mag- netic Reynolds numbers. A dual-core processor with previously described specifications was selected for all time-efficiency investigations discussed in this section.

The first case of calculations was conducted with a magnetic Reynolds number equal to

0.125, which was obtained by adjusting the electrical conductivity of the medium. A small time-period scale in the order of milliseconds was selected because application of a strong magnetic field can cause wave propagation at a faster rate. Figures 6.13 and 6.14 illustrate the wall clock time required by the two full MFD approaches for magnetic field strengths of

3 2 1.422 10− tesla and 1.3944 10− tesla respectively, for different time levels. It is worth- × × 3 while to notice that for relatively smaller magnetic field strength (By =1.422 10− T) , × both FMFD and DFMFD equations take a similar amount of time, as shown in Figure

152 2 6.13. However, as the order of magnetic field intensity increased (By =1.3944 10− T) , a × significant rise in the wall clock time taken by FMFD equations was observed.

0.02

FMFD solution 0.015 DFMFD solution

0.01

Wall clock Time (hours) 0.005

0 0 0.05 0.1 0.15 0.2 0.25 Time period (millisecond)

Figure 6.13: Comparison of wall clock times taken by FMFD and DFMFD formulations 3 for magnetic field strength of 1.422 10 TatRm =0.125. × −

In contrast, the wall clock time for DFMFD equations did not change and remained similartothelowermagneticfield strength case, as depicted in Figure 6.14. Noteworthy is thefactthatabout90 percent reduction in the wall clock time taken by FMFD equations for each time interval was observed when DFMFD equations were utilized with a magnetic

2 field strength of By =1.3944 10− T, whereas the difference in the wall clock time between × 3 the two approcahes is not significant for a magnetic field strength of By =1.422 10− T. ×

153 0.2

FMFD solution 0.15 DFMFD solution

0.1

Wall clock Time (hours) 0.05

0 0 0.05 0.1 0.15 0.2 0.25 Time period (millisecond)

Figure 6.14: Comparison of wall clock times taken by FMFD and DFMFD formulations 2 for magnetic field strength of 1.3944 10 TatRm =0.125. × −

154 Subsequently, a lower value of magnetic Reynolds number equal to 0.025 was selected for investigating the characteristic behavior of wall clock time for the two full MFD modeling approaches. For this smaller value of magnetic Reynolds number, the wall clock time for

3 FMFD and DFMFD equations with magnetic field strengths of 1.422 10− tesla and × 2 1.3944 10− tesla are shown in Figures 6.15 and 6.16, respectively, for different time × levels.

0.1

FMFD solution DFMFD solution

0.05 Wall clock Time (hours)

0 0 0.05 0.1 0.15 0.2 0.25 Time period (millisecond)

Figure 6.15: Comparison of wall clock times taken by FMFD and DFMFD approaches 3 2 for magnetic field strength of 1.422 10 TatRm =2.5 10 . × − × −

It is important to note that similar characteristic behaviors of the wall clock time, as that of the previous case with Rm =0.125, were observed for both values of magnetic field

3 intensities. That is, for By =1.422 10− T, FMFD and DFMFD approaches take an ×

155 almost similar amount of time, whereas a substantial reduction in the neighborhood of 99 percent has occurred when DFMFD equations are utilized for magnetic fieldstrengthof

2 By =1.3944 10− T. ×

9

FMFD solution DFMFD solution

6

3 Wall clock Time (hours)

0 0 0.05 0.1 0.15 0.2 0.25 Time period (millisecond)

Figure 6.16: Comparison of wall clock times taken by FMFD and DFMFD approaches 2 2 for magnetic field strength of 1.3944 10 TatRm =2.5 10 . × − × −

The percentage of reduction in the wall clock time for each time interval achieved by utilizing the DFMFD formulation is presented in Table 6.2 for different magnetic Reynolds numbers with respect to the magnitude of applied magnetic fields. It is interesting to note

2 that for a strong imposed magnetic field of By =1.3944 10− T, a large amount of time × was saved, as depicted in the second column of the table. That is, when DFMFD equations are utilized with higher magnetic field intensity, significant reductions in wall clock time

156 in the range of 90 percent to 99 percent were achieved. However, for a lower magnetic

3 strength of By =1.422 10− T, DFMFD equations lose their efficieny and take slightly × longer than FMFD equations, as indicated by the negative signs in the first column of the table. However, this loss is negligible and can be ignored with respect to the large savings of computational time when simulations were conducted for a strong imposed magnetic

2 field (By =1.3944 10− T) with DFMFD equations. ×

Table 6.2: Percentage reduction in wall clock time for two strengths of magnetic fields versus Magnetic Reynolds number. 3 2 Magnetic Reynolds number By =1.422 10− T By =1.3944 10− T × × 0.125 0.5% 90% − 0.025 3% 99% −

Figure 6.17 shows a comparison of wall clock time of three MFD solution approaches for different values of magnetic Reynolds numbers with a relatively stronger imposed mag-

2 netic field intensity (By =1.3944 10− T). Wall clock time was computed for attaining a × time interval of t =0.02 second. For low ranges of magnetic Reynolds numbers, DFMFD formulation requires less computation time than FMFD formulation. For example, at

Rm =0.125, the computation time of FMFD equations is ˜10 times greater than the com- putation time for DFMFD equations. Note that the difference in computation time of the two full MFD equations significantly increases when values of magnetic Reynolds number are further decreased. It has been observed that the numerical scheme for the FMFD ap- proach shows great instabilities when a strong magnetic field at low magnetic Reynolds

157 numbers is applied; therefore, a smaller time step becomes an essential requirement. How- ever, the numerical scheme based on DFMFD equations has shown improvement in stability with a strong imposed magnetic field at low magnetic Reynolds numbers. Thus, a rela- tively larger time step can be utilized to obtain the required solution. Subsequently, the differences in wall clock time for both full MFD approaches begin to diminish as magnetic

Reynolds numbers begin to increase. Consistent with the previous conclusions, low mag- netic Reynolds number approximation has been found to take the least amount of procesing time as compared to other MFD modeling approaches.

25

20 Full MFD solution

Low Rm solution DFMFD solution 15

10 Wall clock Time (hours) 5

0 00.250.50.751

Magnetic Reynolds number (Rm)

Figure 6.17: Wall clock time for FMFD, DFMFD, and low magnetic Reynolds number formulations.

158 6.3 Blunt-Body Hypersonic Flows

One of the most critical geometries of high-speed aerospace vehicles that are exposed to hypersonic flows is the blunt-body configuration. Blunt-body shapes are frequently utilized to reduce the heat transfer rate for hypersonic regimes. High temperature gradients, strong detached bow shock wave in front of the nose, and mixed subsonic-supersonic flow inthepost-shockregionmakethisproblemquitedifficult to solve numerically. Therefore, ablunt-bodyconfiguration has been selected as the second case for current numerical investigation. Hypersonic flow over a two-dimensional blunt body with different types of applied magnetic field distributions was considered. Geometry and free-stream conditions are provided in Table 6.3 for numerical simulation. Free-stream conditions are specified at the inflow boundary and for initializing the numerical solution. At the body surface, a no-slip velocity, zero-normal gradient pressure, and adiabatic boundary conditions have been imposed. Zero-order extrapolation of all primitive variables is utilized at the outer boundaries.

Table 6.3: Operating conditions for blunt-body hypersonic flow problem. Operation Conditions Mach number M =10.6 ∞ Pressure P =36.6 Pa ∞ Temperature T =294.0 K ∞ Body radius rb =0.1395 m Deflection angle θ =15o

Reference length Lref =0.1395 m

159 It is worthwhile to mention that computational time may increase significantly when complete MFD equations are utilized to solve this complex class of flows; therefore, the requirement of an economical grid that results in a correct converged solution within a permissible limit of time becomes crucial. A computational grid of 90 70 spacing for × blunt-body flow simulations was selected for computations and is depicted in Figure 6.18.

Clustering near the body surface was enforced to capture viscous effects. This grid was selected after several grid independence tests, details of which are discussed in the following section.

Figure 6.18: Computational grid system for blunt body flow.

160 6.3.1 Navier-Stokes Analysis

In order to obtain an economical grid for the computations, a grid independence test was performed before conducting MFD analysis. Figure 6.19 represents nondimensional surface pressure along the body surface for Navier-Stokes analysis; dynamic pressure was utilized to nondimensionalize the values. It is evident from the figure that an increase in thenumberofgridpointsupto150 110 did not affect the pressure distribution, and a × convergence in the pressure values is obvious. Since results obtained with mesh sizes of

90 70 are practically identical with the mesh of 150 110,agridof90 70 was selected × × × for all subsequent analyses.

2

Normal Shock Data Newtonian Pressure 1.5 Navier-Stokes (90×70) Navier-Stokes (150×110) ∞ 1 P/q

0.5

0 0 0.1 0.2 0.3 0.4 0.5 Surface length (x)

Figure 6.19: Surface pressure for Navier-Stokes analysis.

161 Furthermore, numerical results were validated with the exact solution. Figure 6.19 shows a comparison of numerical surface pressure with normal shock data and Newtonian pressure available in the literature. It can be seen that the numerical pressure calculated for the Navier-Stokes analysis agrees very well with the Newtonian theory and normal shock data near the stagnation region (only 0.7 percent error in stagnation pressure with respect to normal shock data is observed).

Moreover, pressure contours for the Navier-Stokes analysis are depicted in Figure 6.20 for a converged solution. Presence of a strong shock wave is obvious at the blunt section of the body. Pressure near the stagnation region has a maximum value, which is consistent with the theory, and has been found in good agreement with the available exact relations.

162

Level P 1 10 5000 3 2 94500

4 6 84000 8 73500 10 63000

8 52500 6 42000

4

2 31500 3 21000 1 1500

Figure 6.20: Pressure contours for Navier-Stokes computation.

163 6.3.2 Magnetofluiddynamic Analysis

Subsequently, magnetofluiddynamic analysis was performed with a uniform magnetic

field imposed along the y-direction, as shown in Figure 6.21. This type of magnetic field dis- tribution can be generated by placing magnets at the body surface. Laminar flow of air with calorically perfect assumption was considered for the present analysis. Strong ionization of the air was assumed to exist in the region between the body and shock wave due to high temperature values in the post-shock region; however, weak ionization (two orders less than the post-shock region) was considered in the free-stream region ahead of the shock wave.

A uniform distribution of electrical conductivity was assumed for the present study. The value of electrical conductivity was adjusted to achieve a magnetic Reynolds number equal to 1.788. The magnetic Reynolds number was computed on the basis of a reference length and the free-stream operating conditions. The decomposed full MFD equations, full MFD equations, and low magnetic Reynolds number approximation were utilized for modeling hypersonic flow over a blunt body with applied magnetic fields. Results of computations obtained from these MFD formulations are compared in the following section.

6.3.2.1 Comparison of FMFD and DFMFD Formulations

The primary objective of this section is to validate the proposed decomposed full MFD form of equations with the classical full MFD equations for high-speed blunt-body flow.

The developed boundary conditions for magnetic field components at the body surface was utilized to model the magnetic effects resulting from the magnetic field perturbations.

Other boundary conditions at the surface included a no-slip condition for velocities, zero-

164 normal gradients of static and magnetic pressures, and a zero-normal temperature gradient.

At the inflow boundary, free-stream conditions were prescribed. A uniform magnetic field of By =0.02T along the y-direction was specified in the domain of computation and at the boundaries for MFD analysis. Operating conditions of Table 6.3 were utilized for other

flow variables.

Figure 6.21: Imposed magnetic field along y-direction.

165 Figure 6.22 indicates the pressure contours obtained from MFD analysis by using FMFD and DFMFD formulations. It is interesting to observe that both MFD formulations provide a similar shock structure for present MFD computations. The effects of Lorentz force are evident on ionized gas, which has resulted in enlargement of the shock envelope, as com- pared to the Navier-Stokes analysis. However, enlargement in the shock structure is high at the body shoulder as compared to the stagnation region, which is mainly due to relatively larger strength of induced magnetic field away from the stagnation portion. It is important to mention that magnitude of the induced magnetic field increases to substantially larger values than the initially applied field at the region away from the stagnation point and may dominate the fluid stresses. Large values of magnetic field strength result in strong Lorentz force denoted by F→ = →J B→,where→J is the current density and B→ is the magnetic field × vector. Lorentz force, also known as electromagnetic force, was generated by the interaction of ionized fluid particles with the magnetic field. Consequently, strong interaction occurred at the body shoulder, thus causing larger displacement of the shock-wave structure at the shoulder for the applied uniform magnetic field distribution.

Subsequently, nondimensional surface pressures along the body surface obtained from full MFD and decomposed full MFD equations are presented in Figure 6.23. It has been verified that DFMFD formulation predicts a similar change in pressure distribution as that of FMFD formulation for magnetofluiddynamic analysis. A comparison with the

Navier-Stokes analysis shows that surface pressure decreases slightly near the stagnation point region after the application of magnetic field; however, away from the stagnation region, an increase in surface pressure is observed. The increase in surface pressure at the

166

0.8 0.8 0.8

0.6 0.6 0.6

1 1 0.4 0.4 0.4 1 5 5 2

4 4 6 4

6 0.2 6 0.2 0.2

8 8 8

1 10 1 10 1 10 Y 0 Y 0 Y 0

8 8 8

6

-0.2 -0.2 6 -0.2 4 4 6 4

2 5 5 1 -0.4 -0.4 -0.4 1 1

-0.6 -0.6 -0.6

-0.8 -0.8 -0.8 -0.2 0 0.2 -0.2 0 0.2 -0.2 0 0.2 X X X Navier-Stokes solution DFMFD solution FMFD solution

Level 1 2 3 4 5 6 7 8 9 10 P: 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Figure 6.22: Comparison of pressure contours obtained from the Navier-Stokes, DFMFD and FMFD analyses.

167 2

Navier-Stokes 1.5 FMFD DFMFD ∞ 1 P/q

0.5

0 0 0.1 0.2 0.3 0.4 0.5 Surface length (x)

Figure 6.23: Surface pressure for DFMFD and FMFD formulations. body shoulder indicates a flow-compression phenomenon, which is taking place primarily due to strong magnetic interaction. It should be noted that the magnetic interaction parameter is proportional to the square of magnetic field intensity and reciprocal of flow velocity. Augmentation of induced magnetic fieldstrengthoveraperiodoftimeanda significant increase in boundary layer thickness away from the stagnation portion result in high values of the magnetic interaction parameter; therefore, considerable flow compression has occurred at the shoulder after the application of a magnetic field.

It is important to mention that the computational time required by the proposed

DFMFD equations was smaller, as compared to conventional FMFD equations for the

168 present blunt-body flow computation with a relatively small value of an imposed magnetic

field. Further computations with a strong imposed magnetic field have shown that a sig- nificant amount of computational time can be saved when DFMFD formulation is utilized for hypersonic blunt-body flows.

Figure 6.24 represents the induced magnetic field lines for imposed uniform magnetic intensity of By =0.02T. Note that the induced magnetic field lines obtained from DFMFD and FMFD formulations are alike. Also, note that the induced magnetic field drapes around the blunt body subsequent to achieving a steady-state condition. Also, it is important to state that density of the lines is proportional to the magnetic flux intensity: dense lines point to relatively stronger values of the induced magnetic field, and vice versa. Thus, dense lines in the post-shock region indicate that induced magnetic fluxhasasstrongan intensity within the shock layer as that in the free-stream region; furthermore, the relatively crowded zone within the shock layer at the body shoulder away from the stagnation region shows relatively high strengths of induced magnetic field around the shoulder.

Figure 6.25 compares velocity fields for the Navier-Stokes and MFD analyses. Half of thegeometryisshowninthefigure due to symmetry of the flow field with respect to the x-axis. The velocity fields obtained from FMFD and DFMFD formulations are found to be similar to each other; thus, only the velocity field of FMFD computation is presented along with the Navier-Stokes result. It is worthwhile to note that a significant increase in the boundary layer thickness away from the stagnation region at the shoulder is evident for MFD calculation, thereby providing evidence of a strong magnetic interaction at this portion of the body.

169

1.5 1.5

1 1

0.5 0.5

Y 0 Y 0

-0.5 -0.5

-1 -1

-1.5 -1.5 -0.8 -0.4 0 0.4 -0.8 -0.4 0 0.4 X X

Induced magnetic field (Bi)-FMFD Induced magnetic field (Bi)-DFMFD

Figure 6.24: Induced magnetic field streamlines’ distributions obatined from DFMFD and FMFD formulations.

170

0.6 0.6

0.4 0.4 Y Y

0.2 0.2

0 0 -0.2 0 0.2 -0.2 0 0.2 X X Navier-Stokes formulation Full MFD formulation

Figure 6.25: Velocity field for the Navier-Stokes and MFD analyses.

171 Likewise, the proof of strong magnetic interaction becomes obvious by examining the total magnetic field vectors for a converged MFD solution, as depicted in Figure 6.26 along with the corresponding velocity field. The total magnetic field consists of induced plus initially imposed magnetic fields; the increase in magnetic field strength is apparent in the post-shock region especially at the shoulder. In fact, an increase in induced magnetic field has caused an increase in the magnitude of total magnetic flux. Away from the stagnation point region, the induced magnetic field has shown a maximum increase in magnitude.

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

Y 0 Y 0

-0.2 -0.2

-0.4 -0.4

-0.6 -0.6

-0.8 -0.8 -0.4 -0.2 0 0.2 -0.4 -0.2 0 0.2 X X

Velocity field Total magnetic field (B t)

Figure 6.26: Velocity field and total magnetic field vectors.

172 6.3.2.2 Validity of Low Magnetic Reynolds Number Formulation

Subsequent to verifying the DFMFD form of equations, an attempt was made to check the validity of the low magnetic Reynolds number approach for the different ranges of magnetic Reynolds numbers under consideration.

Figure 6.27 shows a comparison of the pressure contours for a low magnetic Reynolds number approach with FMFD and DFMFD formulations. Similar to the previous section, the value of the magnetic Reynolds number is set to 1.788. However, a relatively lower strength of uniform magnetic field along the y-direction was utilized because, at By =

0.02T, a low magnetic Reynolds number approach was predicting a significantly large shock wave standoff distance, which was practically difficult to capture with the domain size under consideration, and the requirement of a relatively larger computational domain was essential. For that reason, the magnetic field strength was reduced to By =0.015Tfor present comparison. Worth noticing is the fact that both MFD formulations predicted similar shock structure for this value of applied magnetic field; however, a low magnetic

Reynolds number approximation did not provide the correct result for predicting a relatively large shock envelope. Hence, it is shown that low magnetic Reynolds number approximation does not provide correct results for high magnetic Reynolds numbers.

Next, nondimensional surface pressures, obtained along the body surface, from the three MFD formulations are compared in Figure 6.28. As expected, low magnetic Reynolds number approximation did not predict a similar result as that of full MFD formulations.

The drop in stagnation pressure for a low magnetic Reynolds number approach is 20 percent with respect to FMFD approach. Furthermore, the increase in surface pressure away from

173

0.8 0.8 0.8

0.6 0.6 0.6

1 1 0.4 0.4 0.4 1 4

4

4 3 3 6

6 0.2 6 0.2 0.2 3

8 8 8

10 9 Y 0 Y 0 10 Y 0

8 8 8

-0.2 6 -0.2 6 -0.2 3 3 6 4 3

4

4

-0.4 -0.4 -0.4 1 1 1

-0.6 -0.6 -0.6

-0.8 -0.8 -0.8 -0.2 0 0.2 -0.2 0 0.2 -0.2 0 0.2 X X X

DFMFD solution FMFD solution Low Rm solution

Level 1 2 3 4 5 6 7 8 9 10 P: 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Figure 6.27: Comparion of DFMFD, FMFD and low magnetic Reynolds number for- mulations.

174 2

Navier-Stokes 1.5 FMFD DFMFD

LowRm ∞ 1 P/q

0.5

0 0 0.1 0.2 0.3 0.4 0.5 Surface length (x)

Figure 6.28: Comparison of FMFD, DFMFD and Low Rm formulations for Rm =1.79. the stagnation region is not consistent with the FMFD or DFMFD approaches. In fact, both

MFD formulations predict a uniform flow compression with a maximum rise of less than

60 percent in the surface pressure at the shoulder with respect to the Navier-Stokes result.

However, low magnetic Reynolds number approximation predicted a different characteristic behavior and resulted in a maximum increase of 105 percent in surface pressure at the shoulder with respect to the Navier-Stokes analysis.

175 A dipolar magnetic field distribution was utilized to investigate the validity of low magnetic Reynolds number approximation. Figure 6.29 illustrates the pressure contours obtained from DFMFD and low magnetic Reynolds number approaches for a magnetic

Reynolds number equal to 2.5(Rm =2.5). A dipole at the body center was placed for generating the magnetic field. Here, only decomposed full MFD formulation was utilized, since this alternative form of full MFD equations is more stable and predicts the same results within permissible limit of time as compared to the classical full MFD equations.

When the FMFD formulation was used for a strong imposed magnetic field or for

flows with high levels of electrical conductivity, the buildup of magnetic field increased and required longer for the accumulated magnetic field effects to dissipate, that is, the rate of dissipation of accumulated magnetic field decreased. Therefore, DFMFD equations are recommended. In all subsequent computations comparing the low magnteic Reynolds number approach with full MFD equations, the DFMFD formulation was utilized.

Consequently, the results obatined from DFMFD formulation are compared with the low magnetic Reynolds number approximation in Figure 6.29 for a magnetic field strength of Bo =0.2T, generated at the stagnation point with a dipolar distribution. It is in- teresting to note that although DFMFD and low magnetic Reynolds number approaches provide almost similar shock standoff distance, a close examination of Figure 6.29 indicates substantial differences in the contour levels within the post-shock region, which indicates disparity between the results obtained from the DFMFD and low magnetic Reynolds num- ber formulations.

176

1 1

0.8 0.8

1 1

0.6 0.6 4

5 3 5 0.4 0.4

7 2 7 0.2 0.2 9 9

9 10 Y 0 Y 0

9 9 -0.2 -0.2 2 7

7 -0.4 -0.4 5 3

5

4

-0.6 1 -0.6 1

-0.8 -0.8

-1 -1

-0.6 -0.4 -0.2 0 0.2 -0.6 -0.4 -0.2 0 0.2 X X

DFMFD solution Low Rm solution

Level 1 2 3 4 5 6 7 8 9 10 P: 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Figure 6.29: Pressure contours obatined from DFMFD and Low Rm formulations for Rm =2.5.

177 The difference in pressures obtained from DFMFD and low magnetic Reynolds number approaches becomes obvious when surface pressures along the surface length are compared, as shown in Figure 6.30. It is evident that low magnetic Reynolds number approximation does not predict the correct result. The maximum value of percentage error between the low magnetic Reynolds number approach and DFMFD formulation is found to be 104 percent at the body shoulder, which indicates that the induced magnetic field affects the result and can not be neglected for relatively high magnetic Reynolds numbers.

2

Navier-Stokes 1.5 DFMFD

LowRm ∞ 1 P/q

0.5

0 0 0.1 0.2 0.3 0.4 0.5 Surface length (x)

Figure 6.30: Comparison of surface pressures obtained from DFMFD and Low Rm for- mulations for Rm =2.5.

178 2 Finally, a significantly low value of magnetic Reynolds number equal to 2.5 10− was × selected to examine the validity of low magnetic Reynolds number approximation. A dipole- type magnetic field distribution having a magnetic field strength of Bo =1.0Twasimposed

2 for the lower bound value of magnetic Reynolds numbers (Rm =2.5 10− ). The pressure × contours for the two MFD formulations with the applied magnetic field distribution are shown in Figure 6.31. Note that low magnetic Reynolds number approximation predicts a similar shock structure as that of DFMFD formulation. However, a small amount of shock smearing in front of the subsonic region of DFMFD solution has appeared, which is primarily due to the weight term of the entropy correction parameter. Nonetheless, solutions of both approaches have been found in good agreement with each other.

Subsequently, Figure 6.32 represents the nondimensional surface pressure along the sur- face for low magnetic Reynolds number approximation and DFMFD solution. It is worth- while to notice that surface pressure distribution obtained from low magnetic Reynolds formulation compares well with the solution of DFMFD approach. This has been con-

firmed by calculating the difference in pressures from the two approaches; that is, the maximum difference between the pressures of two modeling approaches is about 9 percent.

The increase in pressure at the surface away from the stagnation region occurrs mainly due to Lorentz force, which results in magnetic interaction within the boundary layer.

Furthermore, the presence of a thick boundary layer enhances the magnetic interaction.

In fact, magnetic interaction behaves inversely proportional to the velocity, since velocity within the boundary layer decreases and magnetic interaction increases, thereby causing

flow compression at the shoulder.

179

1 1

0.8 0.8

0.6 0.6

1 1

0.4 3 0.4 3

4

4

6 0.2 6 0.2 3 8 3 8

1 0 10 Y 0 Y 0

8 3 8 3 -0.2 -0.2 6 6

4 4

-0.4 3 -0.4 3 1 1 -0.6 -0.6

-0.8 -0.8

-1 -1

-0.6 -0.4 -0.2 0 0.2 -0.6 -0.4 -0.2 0 0.2 X X

DFMFD solution Low Rm solution

Level 1 2 3 4 5 6 7 8 9 10 P: 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Figure 6.31: Comparison of pressure contours obatined from DFMFD and Low Rm 2 formulations for Rm =2.5 10 . × −

180 2

Navier-Stokes 1.5 DFMFD

LowRm ∞ 1 P/q

0.5

0 0 0.1 0.2 0.3 0.4 0.5 Surface length (x)

Figure 6.32: Comparison of surface pressures obtained from DFMFD and Low Rm for- 2 mulations for Rm =2.5 10 . × −

181 Figure 6.33 illustrates dimensionless shock standoff distances for MFD analysis obtained from the low magnetic Reynolds number approximation and decomposed full MFD formu- lation. Body radius is utilized to nondimensionalize the result. A comparison with exact value for the Navier-Stokes analysis was made, and only a percentage error of ˜7 percent was calculated between the exact and the Navier-Stokes computations. It was found that the flow structure significantly depends on the strengths of applied magnetic field. That is, stronger Lorentz force causes a larger increase in the shock standoff distance. Moreover, increments in shock standoff distance with the increase of magnetic field strength are evi- dent for both approaches and are in good agreement with each other. As expected, both approaches showed similar characteristic behavior for shock standoff distances. A reason- ably good agreement was observed between the solutions of the two approaches. Maximum percentage error of ˜6 percent was found between the results of DFMFD and low magnetic

Reynolds number formulations for a magnetic field strength of 2.0T.

182 2

Navier-Stokes DFMFD

1.5 LowRm

/r 1 ∆

0.5

0 00.511.52 B(T)

Figure 6.33: Comparison of shock stand off distances obtained from DFMFD and 2 LowRm formulations for Rm =2.5 10 . × −

183 6.4 Supersonic Flow over a Backward-Facing Step

Validation of the low magnetic Reynolds number approximation for a low range of magnetic Reynolds numbers has been utilized to explore the effectsofanappliedmagnetic

field for supersonic flows over a backward-facing step. A backward-facing step is one of the critical classes of geometries used to model internal as well as external separated flows around various geometries. Some applications of particular interest include engine combus- tors, supersonic inlets, and flow-behind projectiles. Supersonic backward-facing step flows involve flow seperation, formation of an expansion fan, flow reattachment, and formation of an blique shock wave in the downstream region. Along with this, a recirculation zone at the corner and formation of a free shear layer that seperates the recirculation zone from the expanded free-stream flow exist. The multifarious nature of flow field causes instability and may result in the failure of numerical scheme for full MFD equations. Therefore, low magnetic Reynolds number approximation was utilized for determining the effect of Lorentz force on this complex class of flows. A multiblock strategy was adopted for generating the grid for numerical analysis. A computer code named 2DBFS based on the multiblock solu- tion technique was developed and successfully implemented. Details of the computational grid and validation of the code are discussed in the following sections.

6.4.1 Multiblock Numerical Gird

Complexity of the flow field causes spurious numerical difficulties and affects solution convergence that may not provide correct results. Therefore, an accurate and efficient grid that results in a correct converged solution within a permissible limit of time is necessary.

184 Figure 6.34 shows a two-dimensional multiblock computational grid obtained after several grid independent tests. Grid-clustering is enforced at the base and near the wall regions to accurately capture the large-flow gradients. The domains upstream and downstream of the step have been specified according to the dimensions provided by Smith [86]. It is important to mention that the difference in pressure values was large at the beginning of the recompression region (closer to the base and near the recirculation corner) for different grid spacing. Thus, more grid points were introduced at this region to attain better accuracy.

Finally, grid spacing of 112 120 for the downstream domain and 56 61 for the upstream × × domain were chosen for all subsequent analyses. The data for investigating the validity of the developed model was obtained from the experimental work of Smith [86]. Details of the geometry and operating conditions are provided in Table 6.4.

It should be noted that experimental data was provided in a foot-pound-second (fps) system [86], which was converted into an SI system of units for the present computations.

The current numerical analysis was performed for flow exposed to an unconfined envi- ronment. The term ‘unconfined environment’ refers to the domain of flow that does not contain an upper wall; therefore, an extrapolating boundary condition is required at the upper wall for numerical computation. However, the term ‘confined environment’ considers the upper wall of the domain; thus, a no-slip boundary condition is usually specified for the numerical simulation. The developed model was validated with the experimental data for the Navier-Stokes calculation; subsequently, numerical MFD analysis was conducted for different magnetic field distributions.

185

Figure 6.34: Multiblock computational grid for backward-facing step flow.

Table 6.4: Geometry and operating conditions for supersonic backward-facing step flow.

Operating Conditions step height h =11.252 mm upstream length 101.6 mm downstream length 304.8 mm upstream height 147.78 mm Mach number M =2.5 ∞ stagnation pressure Pstag =127.553 kPa stagnation temperature Tstag =362.44 K Reynolds number Re =1.005 104/mm ∞ ×

186 6.4.2 Navier-Stokes Analysis

Figure 6.35 illustrates pressure contours along with streamlines obtained for the Navier-

Stokes computation. Formation of the expansion fan is evident, with the leading expansion wave at approximately 23.8 degrees, compared to a value of 23.58 degrees obtained from the analytical solution for an inviscid, supersonic flow. A circulation region at the corner that contains a complex structure of vortices is also obvious in the figure; approximately uniform pressure has been observed in this region. Flow separation and reattachment can be observed by considering the streamlines; in fact, a shear layer separates the circulation region from the flow downstream of the expansion fan. Finally, appearance of an oblique shock wave through coalescence of compression waves further downstream has occurred, which results in turning of the flow parallel to the free-stream direction. Moreover, a comparison of turning angle measured between the shear layer and the wall of the test section was made with the experimental value. The flow-turning angle from the present numerical computation was found to be approximately 17 degrees, which is in agreement with the measured value of 17 degrees obtained from the shadowgraph of the experiment reported by Smith [86].

The computed pressure distribution along the horizontal surface downstream of the step is compared to the experimental values of Smith [86] in Figure 6.36. A reasonably good agreement between the experimental and numerical values of the pressure was achieved; free-stream pressure was utilized to nondimensionalize the values.

187 0.1 0.2 0.3 Axial distance (m)

Figure 6.35: Pressure contours obatined from the Navier-Stokes analysis.

1.2

0.8

∞ Experiment [86] Computation P/P 0.4

0 0.10.150.20.25 Axial distance (m)

Figure 6.36: Comparison of pressure distribution with experimental data of [86].

188 6.4.3 Magnetofluiddynamic Analysis

Subsequent to validating the developed model, computations were initiated to investi- gate the effects of an applied magnetic field for this complex class of flows. Air was assumed to be calorically perfect under laminar flow conditions. Electrical conductivity of gas in the upstream domain was set to zero, while a value of electrical conductivity equal to 1142

Siemens/m was selected in the downstream domain. This value of electrical conductivity was chosen in order to achieve a low magnetic Reynolds number, so that low magnetic

Reynolds number approximation could be utilized. With this value of electrical conductiv- ity, the magnetic Reynolds number based on the step height was equal to 0.01.Theload factor was zero; that is, there was no applied electric field. Three types of magnetic field distributions with varying strengths were implemented for MFD flow control analyses.

6.4.3.1 Uniform Magnetic Field in Y-Direction

The first set of numerical experiments for MFD consisted of the application of a uniform magnetic field in the y-direction perpendicular to the free-stream flow. The magnetic

field was attenuated far downstream before exit to prevent flow separation at the exit, which would certainly violate the extrapolating boundary condition and may cause solution divergence. The applied magnetic field had a uniform value from x =0.1016 m to x =

0.1778 m and decreased linearly to zero from x =0.1778 m to x =0.3302 m. Figure 6.37 illustrates the applied magnetic field distribution along with pressure contours obtained with magnetic field strengths of 0.15Tand0.25T. Comparison with the Navier-Stokes analysis indicates a significant increase in the size of the circulation zone that causes a

189

0.1 0.2 0.3 (a) Navier-Stokes

0.1 0.2 0.3 (b) Magnetic field distribution

0.1 0.2 0.3

(c) By =0.15T

0.1 0.2 0.3

(d) By = 0.25T

Figure 6.37: Uniform magnetic field distribution along with pressure contours obtained with different strengths of magnetic field.

190 displacement in the oblique shock-wave location. Furthermore, following the application of a magnetic field, the oblique shock wave becomes steeper and thicker compared to the

Navier-Stokes prediction.

Figure 6.38 represents pressure distributions obtained along the horizontal surface downstream of the step for different magnetic field strengths along with the Navier-Stokes calculation. Comparison with the Navier-Stokes analysis reveals an overall increase in pres- sure levels and displacement in the shock location toward the exit section with the increase of applied magnetic field intensity. The magnetic field caused high pressure downstream of the step over the entire domain; in fact, it was the Lorentz force which resulted in flow com- pression and shock standoff in this region. Nonetheless, the pressure difference upstream and downstream of the shock slightly decreased, as observed in Figure 6.38. It is interesting to note that although pressure increased after application of the magnetic field, it remained uniform in the recirculation region. Furthermore, a decrease in flow velocity was also ob- served with the application of a magnetic field. This flow retardation strongly depends on the magnitude of the magnetic field and occurs because of Lorentz force generated as the result of the interaction of plasma and the magnetic field.

191 2

1.5

X X X X X X X X X X ∞ 1 X

P/P X

B =0.0T X y X B =0.15T 0.5 y X By =0.20T XXXXXXXXXXXXX X X X X X By =0.25T

By =0.30T

0 0.1 0.15 0.2 0.25 0.3 0.35 Axial distance (m)

Figure 6.38: Pressure distributions along the horizontal surface for different values of uniform magnetic field strengths.

192 6.4.3.2 Magnetic Field Generated by A Dipole Aligned with X-axis

The second set of MFD analyses employed two cases of magnetic field distribution generated by placing dipoles aligned with the positive x-axis at two different locations of

0.1397 m and 0.2032 m in the downstream domain. Figure 6.39 shows pressure contours along with magnetic field distributions for both dipole locations. For the dipole located at x =0.1397 m, the application of Bo =1.0Tmagneticfield causes a movement of shock toward the downstream section and a small smearing of the shock. Increase of the magnetic

field strength to a value of Bo =2.0T further pushes the shock wave downstream toward the exit section, causing substantial shock smearing and reducing the expansion fan angle.

For the dipole positioned at x =0.2032 m, flow structure remained mostly similar to the Navier-Stokes flow field after the application of magnetic field Bo =1.0T, except some compression waves were observed at the dipole location. Nonetheless, further increase of magnetic field strength (Bo =2.0T) not only generated the strong compression waves at the dipole location but also resulted in the increase of a recirculation zone, thus causing a shock displacement toward the exit section. Moreover, shock smearing, reduction in the expansion fan angle, and a slight increase in shock wave angle were observed after the application of magnetic field Bo =2.0T.

Figure 6.40 depicts pressure distributions along the horizontal surface downstream of the step for the two locations of dipoles, shown previously in Figure 6.39. A gradual increase in pressure distributions at the circulation region and the movement of shock toward the exit section are evident for MFD analysis with the dipole located at x =0.1397 m.

193

0.1 0.2 0.3 Navier-Stokes

0.1 0.2 0.3 0.1 0.2 0.3 Dipole at x = 0.1397 m Dipole at x = 0.2032 m

0.1 0.2 0.3 0.1 0.2 0.3

Bo =1.0T Bo =1.0T

0.1 0.2 0.3 0.1 0.2 0.3

Bo =2.0T Bo =2.0T

Figure 6.39: Magnetic field distributions generated by dipoles aligned with the x-axis and corresponding pressure contours obtained with different magnetic field strengths.

194 1.5

1 ∞

P/P Bo =0.0T

Bo = 1.0T, x = 0.1397 m

0.5 Bo = 2.0T, x = 0.1397 m

Bo = 1.0T, x = 0.2032 m

Bo = 2.0T, x = 0.2032 m

0 0.10.150.20.250.30.35 Axial distance (m)

Figure 6.40: Pressure distributions along the horizontal surface for different magnetic field strengths of dipoles aligned with the x-axis.

195 In contrast, for the dipole positioned at x =0.2032 m, application of Bo =1.0T magnetic field, neither causes an increase in pressure at the recirculation region nor results in movement of the shock. However, for magnetic field strength Bo =2.0T, a significant rise in pressure at the recirculation zone and displacement of the shock are observed. Moreover,

flow compression phenomenon at the dipole location of x =0.2032 m is also evident for

MFD computations. That is, for Bo =1.0T, the pressure distribution remains similar to the Navier-Stokes analysis from the base to x ∼= 0.15 m; however, a rise in pressure in the neighborhood of the dipole position has occurred. Further enlargement in pressure hills were observed for magnetic intensity of Bo =2.0T. The appearance of pressure hills indicates that magnetic interaction becomes important within the boundary layer at the position where a strong magnetic field is present. In fact, MFD interaction is inversely proportional to the velocity; since velocity within the boundary layer decreases, magnetic interaction increases. Finally, pressure recovery is achieved for all values of magnetic field strengths at the exit section for both dipole positions. Pressure distributions along the horizontal surface for different magnetic field strengths of dipoles aligned with the x-axis.

6.4.3.3 Magnetic Field Generated by Dipole Pointed in Direction of Y-axis

Finally, dipoles with their axes perpendicular to the surface were placed at two differ- ent locations for magnetofluiddynamic computations. Figure 6.41 indicates the magnetic

field generated by dipoles and the corresponding pressure distributions for different field strengths. For the dipole positioned at 0.1397 m in the downstream domain, a displacement of shock toward the exit section and some smearing of the oblique shock wave are observed

196 with a magnetic field of Bo =1.0T. Furthermore, an increase of magnetic field strength

(Bo =2.0T) causes considerable shock smearing, shock movement toward the exit section, and reduction in the expansion fan angle.

Similar to what was discussed in the previous case in section 6.4.3.2,forthedipole located at x =0.2032 m, the application of a magnetic field of Bo =1.0Tresultedin formation of compression wave at the dipole position without causing any changes in oblique shock standoff.However,forBo =2.0T, considerable shock displacement toward the exit section and strong compression waves were generated at the dipole location. In addition, shock smearing, reduction in the expansion fan angle, and a small increment in oblique shock wave angle occurred for the higher strength of magnetic field.

Figure 6.42 shows pressure distributions along the surface for different dipole locations with different strengths of magnetic field. Shock displacements for both dipole locations discussed earlier are clearly visible in this figure. For the dipole located at x =0.1397 m, the increase in pressure levels at the circulation region took place for all values of magnetic fields. Interestingly, the flow expansion that occurred before the shock for the

Navier-Stokes computation did not occur for MFD computation with a magnetic field strength of Bo =1.0T. Moreover, this small expansion for the Navier-Stokes calculation mutates to flow compression with further increase of magnetic field strength (Bo =2.0T).

Nevertheless, recovery in all pressure levels is achieved downstream of the oblique shock region. For the dipole positioned at x =0.2032 m, pressure levels in the recirculation zone and the shock location remains similar to the Navier-Stokes analysis, except a hill has appeared at the dipole location after crossing the shock wave for the magnetic field of

197

0.1 0.2 0.3 Navier-Stokes

0.1 0.2 0.3 0.1 0.2 0.3 Dipole at x = 0.1397 m Dipole at x = 0.2032 m

0.1 0.2 0.3 0.1 0.2 0.3

Bo =1.0T Bo =1.0T

0.1 0.2 0.3 0.1 0.2 0.3

Bo =2.0T Bo =2.0T

Figure 6.41: Magnetic field distributions generated by dipoles pointed in the direction of y-axis and corresponding pressure contours obtained with different mag- netic field strengths.

198 1.5

1 ∞ P/P

B =0.0T 0.5 o Bo =1.0T,x =0.1397m

Bo =2.0T,x =0.1397m

Bo =1.0T,x =0.2032m

Bo =2.0T,x =0.2032m

0 0.1 0.15 0.2 0.25 0.3 0.35 Axial distance (m)

Figure 6.42: Pressure distributions along the horizontal surface for different magnetic field strengths of dipoles pointed in the direction of y-axis.

Bo =1.0T. However, for Bo =2.0T, an increase in pressure at the recirculation region and significant shock displacement occurred. Furthermore, a substantial rise in the pressure hill took place at the dipole location for higher magnetic field intensity, which confirms that magnetic interaction has increased in the boundary layer. Again, pressure is becoming uniform downstream of the oblique shock wave for all magnetic field strengths.

6.4.3.4 Effect of Temperature Dependency of Electrical Conductivity

In the last part of MFD analysis, the effect of variable electrical conductivity on MFD control of flow over a backward-facing step has been explored. The expression for electrical conductivity distribution is [54]

199 T 4.0 σ = σ e o T µ stag ¶ where σo is the constant value of electrical conductivity, and Tstag is the stagnation tempera- ture. Figure 6.43 shows pressure contours obtained with uniform magnetic field distribution by considering variable electrical conductivity of the medium. Note that a magnetic field strength of 0.25T does not produce considerable change in the flow structure, as observed previously for constant electrical conductivity case (see Figure 6.37d). However, with the increase of magnetic field intensity (0.5T) , enlargement of the recirculation zone, shock smearing, and movement of the shock toward the exit section have occurred, as depicted in Figure 6.43d.

Figure 6.44 represents pressure distributions obtained along the horizontal surface downstream of the step for constant and variable electrical conductivities along with the

Navier-Stokes analysis. It is worthwhile to observe that for constant electrical conductivity computation, the magnetic field of 0.25T strength has caused significant rise in the pres- sure distribution and considerable displacement in the shock location, whereas for variable electrical conductivity case, neither the rise in pressure nor the shock displacement are sig- nificant, as compared to the Navier-Stokes calculations. The comparison of results obtained with constant and variable electrical conductivities indicates that a decrease in magnetic interaction has been observed for variable electrical conductivity distribution, which will reduce the effectiveness of MFD control.

200 0.1 0.2 0.3 (a) Navier-Stokes

0.1 0.2 0.3 (b) Magnetic field distribution

0.1 0.2 0.3

(c) By =0.25T

0.1 0.2 0.3

(d) By =0.5T

Figure 6.43: Pressure contours obtained with variable electrical conductivity for differ- ent strengths of uniform magnetic field distribution.

201 2

X X X X X 1.5 X X

X

X

∞ 1 P/P X XXXXXXXXXXXXX XX X X X X X B =0.0T X y X B =0.25T(constantσ ) 0.5 y e By =0.25T(variableσe )

By =0.50T(variableσe )

0 0.1 0.15 0.2 0.25 0.3 0.35 Axial distance (m)

Figure 6.44: Pressure distributions along the horizontal surface for constant and vari- able electrical conductivity distributions.

202 6.4.4 Velocity Distributions

Figure 6.45 illustrates streamline patterns for Navier-Stokes and MFD flow compu- tations for different magnetic field distributions. The results of highest magnetic field strength for each distribution have been provided in order to explicitly discuss the influ- ence of magnetic field. Since these patterns have qualitative representations of the flow

field, an enlarged region near the step corner has been presented to provide a better under- standing of the circulation region. Streamlines through a Prandtl-Mayer expansion turn toward the lower surface, and attainment of a free-stream direction through the oblique shock wave is visible in all cases. As can be seen from the figure, a complex structure of circulation in the separation bubble is evident in all cases–each one having a distinct shape from the other. A combination of upper and lower vortices has been observed for the present Navier-Stokes and MFD calculations. For convenience, the lower vortex is referred to as the primary vortex, and the upper vortex is referred to as the secondary vortex.

It is interesting to observe that the separation bubble is smaller in size in the Navier-

Stokes analysis; application of the magnetic field has caused enlargement in the size of the separation bubble for all distributions. Furthermore, the magnetic effects have changed the structure of the vortices within the bubble, as compared to the Navier-Stokes prediction.

Relatively smaller vortices are observed for the Navier-Stokes calculations, whereas large vortices have occurred for MFD flow computations, although size and shape of these vortices strongly depend on the orientation and strength of the magnetic field vectors.

For example, in the case of a uniform magnetic field distribution along the y-direction, enlargement of the separation bubble and a combination of vortices are evident in Figure

203 0.10 0.15 0.20 0.10 0.15 0.20

(a) Navier-Stokes (b) Uniform magnetic field along y-axis (By = 0.25T)

0.10 0.15 0.20 0.10 0.15 0.20

(c) Dipole pointed along the x-axis at x = 0.1397 m (Bo =2.0T) (d) Dipole pointed along the x-axis at x = 0.2032 m (Bo =2.0T)

0.10 0.15 0.20 0.10 0.15 0.20

(e) Dipole pointed along the y-axis at x = 0.1397 m (Bo =2.0T) (f) Dipole pointed along the y-axis at x = 0.2032 m (Bo =2.0T)

Figure 6.45: Streamline patterns for the Navier-Stokes and MFD computations with different magnetic field distributions.

204 6.45b. Moreover, the primary vortex located under the secondary vortex, has been stretched toward the reattachment point downstream of the step.

Similar characteristic behavior of this vortex topology remains persistent for both ori- entations of dipoles located at x =0.1397 m and x =0.2032 m in the downstream domain.

For the dipole positioned at x =0.1397 m, the magnetic effects have resulted in contraction of the primary vortex confined to the corner and further enlargement of the secondary vor- tex extended up to the reattachment point, as shown in Figures 6.45c and 6.45e, for both dipole orientations along the x and y axes. In contrast, for the dipole located at x =0.2032 m,themagneticfield has resulted in an expansion and stretching of the primary vortex and a small contraction in the secondary vortex, as depicted in Figures 6.45d and 6.45f for both dipole orientations along the x and y axes.

205 Chapter 7

CONCLUSIONS

This dissertation focused on the development of numerical algorithms to implement different magnetofluiddynamic formulations for aerodynamic applications. First, the basic formulations of MFD, known as full MFD equations, and low magnetic Reynolds approxi- mation were investigated numerically. Validity of low magnetic Reynolds number approx- imation were examined for a wide range of magnetic Reynolds numbers. Subsequently, a third formulation based on decomposition of total magnetic field into imposed and induced components was developed and successfully implemented for an explicit solver. A good comparison of results obtained from all MFD formulations was achieved for specificoper- ating conditions. Finally, the performance of each MFD modeling approach was evaluated for different ranges of magnetic Reynolds numbers with weak and strong imposed magnetic

fields.

The governing partial differentiation equations were transformed into a computational domain where finite difference approximation is utilized to obtain the algebraic equations.

Flux vector splitting for the convective terms and central differencing for the diffusion terms were utilized. Afterward, the resulting finite difference algebraic equations were solved numerically. A time-explicit numerical scheme based on the fourth-order multistage Runge-

206 Kutta method for time integration was utilized to achieve the solution. A second-order total variation diminishing model based on Davis-Yee symmetric limiters was implemented in the post-processing stage for enhancing the stability and shock-capturing capability of the numerical scheme.

Two-dimensional and three-dimensional versions of computer codes were developed for each MFD modeling approach. The developed MFD solvers were successfully validated with the exact solutions available for velocity and induced magnetic fields in the literature for the magnetic Rayleigh problem. Subsequently, a complex class of flows in hypersonic regime was considered for further algorithm validation. A blunt-body configuration subjected to hypersonic flow was selected for investigating the developed codes for the Navier-Stokes and MFD analyses.

Subsequent to verifying the developed codes, the validity and range of applicability of available approximate governing equations of MFD, known as the low magnetic Reynolds number formulation, was explored. Recently, some concerns have been raised with regard to the corresponding governing equations; therefore, accuracy and validity of low magnetic

Reynolds number approximation with respect to full MFD equations were examined for different ranges of magnetic Reynolds numbers. Based on the magnetic Rayleigh problem, the results obtained from the present investigation indicate that the low magnetic Reynolds number approach can be utilized to model an MFD problem when the magnetic Reynolds number is in the range of ˜0.0 0.5. For such low ranges of magnetic Reynolds numbers, − substantially smaller values of induced magnetic field have been observed; thus, the induced magnetic field will not affect the flow field and a set of governing equations (known as low

207 magnetic Reynolds number formulation), which approximate the electro-magnetic effects by incorporating a magnetic source term in fluid dynamics equations can be utilized for modeling an MFD problem. Furthermore, the cutoff value of a magnetic Reynolds number is highly problem- and operating-condition dependent, although for the present Rayleigh case, it was set to approximately 0.125, due to the existence of less than 10 percent average error in the velocities obtained from full MFD and low magnetic Reynolds number formulations.

Subsequently, performance of the low magnetic Reynolds number approach and full

MFD equations were evaluated by comparing the computational time required by each formulation. Once again, the magnetic Rayleigh problem was selected for this investigation.

It was found that for low values of magnetic Reynolds numbers, the full MFD equations require a substantially large amount of time, as compared to the low magnetic Reynolds number approach. For extremely small values of magnetic Reynolds numbers, such as

0.01 or less, it is highly advisable to use the low magnetic Reynolds number formulation compared to the full MFD approach due to the excessive computational requirement of the full MFD approach. The numerical solution of equations obtained with low magnetic

Reynolds number approximation is relatively simple and efficient, even for significantly smaller magnetic Reynolds numbers and with strong imposed magnetic fields.

Next, the proposed alternate form of full MFD equations, decomposed full MFD equa- tions, was developed for the explicit solver and numerically investigated. After validating the alternate form with the exact solution of the magnetic Rayleigh problem, further jus- tification with the classical full MFD and low magnetic Reynolds number approaches were demonstrated. It was shown that the alternate form, DFMFD equations, predicts the

208 same results as that of classical full MFD equations and low magnetic Reynolds number approximation within the appropriate regime.

The basic concept for alternate formulation is to develop a tool for efficient computation of magnetoaerodynamic flows at low levels of electrical conductivity and strong imposed magnetic fields using the full MFD approach. Therefore, performance of the DFMFD approach was evaluated for weak and strong imposed magnetic fields. It was found that for a weak strength of applied magnetic field, the two full MFD formulations require a similar amount of computational time for a wide range of magnetic Reynolds numbers. However, with a relatively strong applied magnetic field, significant reduction in the computation time was achieved when DFMFD equations were utilized for plasma flow at low values of magnetic Reynolds number. For example, at a magnetic Reynolds number of 0.125, the computational time required by full MFD equations is approximately 10 times greater than the computational time of alternate formulation (DFMFD equations) based on the magnetic Rayleigh problem to achieve a time level of 0.02 second. Consequently, it can be concluded that for a relatively strong imposed magnetic field, DFMFD equations require substantially less computational time compared to the full MFD equations for low values of magnetic Reynolds numbers.

Following the magnetic Rayleigh problem, electromagnetic control of hypersonic flow over a blunt body was examined for uniform and dipole types of magnetic field distrib- utions. Enlargement of the shock structure and flow compression at the body shoulder were observed for MFD analysis. As expected, DFMFD formulation predicted the same results as that of FMFD formulation. Further numerical experiments with strong imposed

209 magnetic field showed that either full MFD equations revealed great instabilities or re- quired a substantially large amount of time for achieving the converged solution; however,

DFMFD equations provided a converged solution within permissible limit of time, even for flow regimes with significantly smaller magnetic Reynolds numbers, for example, in the

2 neighborhood of the order of 1 10− . × Furthermore, validity of low magnetic Reynolds number approximation was examined for hypersonic blunt-body flow regime. The numerical scheme for FMFD equations became highly unstable for low values of magnetic Reynolds number; therefore, an alternate form,

DFMFD equations, was utilized for attaining the solution at low values of Rm and with astrongimposedmagneticfield. It was found that flow computations may significantly be affected by the existence of ionized air in the domain. With uniform distribution of electrical conductivity, in the region of flow where ionization is confined between the body and the shock wave, the low magnetic Reynolds approach provides similar results as that of DFMFD formulation. However, for most experimental facilities where ionized air is dispersed in the entire domain, the results obtained from low magnetic Reynolds number approximation may not compare well with the results obtained from DFMFD equations.

Finally, the low magnetic Reynolds number approach was utilized to numerically sim- ulate the effects of applied magnetic field on supersonic separated flows over a backward- facing step, one of the critical classes of geometries used to model internal as well as ex- ternal separated flows around various geometries. Using a mutliblock solution strategy, an explicit solver based on a four-stage modified Runge-Kutta scheme augmented with TVD limiters was developed for solving a viscous, compressible MFD set of equations. Different

210 types of magnetic field distributions were implemented for the MFD computations. Several magnetic field configurations under consideration caused an increase in the size of the sep- aration bubble, movement of shock toward the exit section, increase in the oblique shock wave angle, and decrease in flow velocity in the plasma domain. Moreover, the applied magnetic field altered the vortex patterns for all types of magnetic field distributions. A uniform magnetic field caused an increase in pressure levels downstream of the step over the entire domain. A further increase in the strength of the magnetic field thickened the oblique shock wave. For the dipole aligned with the positive x-axis, flow compression at the recirculation region and shock—smearing were observed for x =0.1397 m location. For the dipole positioned at x =0.2032 m, formation of pressure hills in the neighborhood of the dipole position took place for MFD computations; however, flow compression at the recirculation region occurred only for the higher strength of magnetic field. For the dipole pointed in the positive y-axis direction, an increase in pressure at the circulation zone and shock—smearing occurred for the x =0.1397 m position,whereas,forthelocation of x =0.2032 m, an increase in pressure levels at the circulating region and formation of a pressure hill after crossing the shock at the dipole location were observed with an increase in magnetic field intensity. Computations performed with temperature-dependent electrical conductivity distribution showed a reduction in MFD effectsascomparedtothe constant-conductivity case. Interaction of the applied magnetic field with the flow was strongly dependent on distribution of electrical conductivity in the domain, the strength and type of magnetic field distributions, locations, and orientations.

211 A critical accomplishment of the present research activity is the simulation of flows with strong imposed magnetic field for aerodynamic flow regime of low level of electrical conductivity by utilizing the alternate form of full MFD equations (DFMFD equations), which is significantly important for investigating the validity of low magnetic Reynolds number approximation, because the numerical scheme of FMFD formulation has shown great instabilities at low levels of electrical conductivity with strong imposed magnetic

field. For this situation, achieving a converged solution by FMFD equations becomes almost impossible, and consequently, no comparison between the approximate formulation and full MFD equations could be made.

This dissertation attempted to advance the basic understanding of MFD modeling formulations and presents an alternate formulation of full MFD equations–the decom- posed full MFD equations–to remove the numerical difficulties. The potential of DFMFD formulation to model complex MFD flows with strong imposed magnetic field is highly appreciable. Further investigation in this area is highly encouraged.

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225 APPENDICES

226 Appendix A

IdealEight-WaveMFDEquationBasedonPrimitiveVariableVector

The ideal DFMFD equations in term of flux vectors are rewritten here as

∂Q ∂E ∂F ∂G + + + + H =0 (A.1) ∂t ∂x ∂y ∂z where E, F, and G are inviscid flux vectors, and H is the additional term introduced for removing the singularity assocoated with an eight-wave system of DFMFD equations.

Moreover, a simplification can be introduced by defining a primitive variable vector Q´as

T Q´= (A.2) ρuvwBix Biy Biz p ∙ ¸

Now, equation (A.1) is reconstructed in terms of Q´as

∂Q ∂Q´ ∂E ∂Q´ ∂F ∂Q´ ∂G ∂Q´ + + + ∂Q´ ∂t ∂Q´∂x ∂Q´∂y ∂Q´∂z

∂B ∂Q´ ∂B ∂Q´ ∂B ∂Q´ + H x + y + z =0 (A.3) M ∂Q´ ∂x ∂Q´ ∂y ∂Q´ ∂z µ ¶ 1 1 ∂Q´ ∂Q − ∂E ∂Q´ ∂B ∂Q´ ∂Q − ∂F ∂Q´ ∂B ∂Q´ + + H x + + H y ∂t ∂Q´ ∂Q´∂x M ∂Q´ ∂x ∂Q´ ∂Q´∂y M ∂Q´ ∂y ∙ ¸ µ ¶ ∙ ¸ µ ¶ 1 ∂Q − ∂G ∂Q´ ∂B ∂Q´ + + H z =0 (A.4) ∂Q´ ∂Q´∂z M ∂Q´ ∂z ∙ ¸ µ ¶

227 1 1 ∂Q´ ∂Q − ∂E ∂B ∂Q´ ∂Q − ∂F ∂B ∂Q´ + + H x + + H y ∂t ∂Q´ ∂Q´ M ∂Q´ ∂x ∂Q´ ∂Q´ M ∂Q´ ∂y ∙ ¸ ∙ ¸ ∙ ¸ ∙ ¸ 1 ∂Q − ∂G ∂B ∂Q´ + + H z =0 (A.5) ∂Q´ ∂Q´ M ∂Q´ ∂z ∙ ¸ ∙ ¸ which can be written as

∂Q´ ∂Q´ ∂Q´ ∂Q´ + A´ + B´ + C´ =0 (A.6) ∂t ∂x ∂y ∂z with

1 ∂Q − ∂E ∂B A´ = + H ix (A.7) ∂Q´ ∂Q´ M ∂Q´ ∙ ¸ 1 ∙ ¸ ∂Q − ∂F ∂B B´= + H iy (A.8) ∂Q´ ∂Q´ M ∂Q´ ∙ ¸ 1 ∙ ¸ ∂Q − ∂G ∂B C´= + H iz (A.9) ∂Q´ ∂Q´ M ∂Q´ ∙ ¸ ∙ ¸ where A´, B´,andC´are the auxiliary flux Jacobian matrices. It is convenient to utilize the auxiliary form for determining the eigenstructure of the system because of its simplicity, as compared to the original form of flux Jacobian matrices.

Subsequently, the matrix A´has been evaluated as

228 uρ 00 0 000 ⎡ ⎤ Bt B ⎢ 0 u 00 0 y tz 1 ⎥ ⎢ ρ ρ ρ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 00 u 00Btx 00⎥ ⎢ ρ ⎥ ⎢ − ⎥ ⎢ ⎥ ⎢ Btx ⎥ ⎢ 00 0 u 00ρ 0 ⎥ ⎢ − ⎥ A´= ⎢ ⎥ ⎢ ⎥ ⎢ 00 0 0 u 000⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 Bty Btx 00u 00⎥ ⎢ − ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 Bt 0 Bt 00u 0 ⎥ ⎢ z − x ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 γp 00(γ 1) −→U −→B o 00u ⎥ ⎢ − · ⎥ ⎣ ⎦ The eight eigenvalues of matrix A´are found as

Entropy wave λex = u

Alfvén waves λax = u vax ± ±

Fast acoustic waves λfx = u vfx ± ±

Slow acoustic waves λsx = u vsx ± ±

Divergence wave λdx = u with the following relations for Alfvén wave, fast acoustic wave, and slow acoustic wave:

229 Btx νax = √µeoρ 1 ν = [(ν2 + c2)+z ] fx 2 a s x r 1 ν = [(ν2 + c2) z ] sx 2 a s x r − where

2 2 2 2 2 zx = (ν + c ) 4c ν a s − s ax q 2 2 2 2 γp Btx + Bty + Btz cs = and va = ρ s µeoρ

Btx, Bty,andBtz indicate total magnetic fieldcomponentsinthex, y,andz directions, respectively. It is important to mention that the eigenvalues remain the same as that for

FMFD formulation.

Similarly, the matrices B´and C´and their eigenvalues can be evaluated by utilizing the equations (A.8) and (A.9), respectively.

The eigenvectors for an eight-wave structure of DFMFD equations based on primitive variable vector Q´were evaluated. It was found that the eigenvectors of DFMFD equations are different than the FMFD equations. However, the original set of eigenvectors based on

FMFD approach can be utilized in the solution scheme because the conservation form of the flux-vector splitting is used.

230 Appendix B

Eigenstructure in Generalized 2-D Computational Domain

The eigenvalues of flux Jacobian matrix A for the eight-wave structure of decomposed full MFD equations in generalized 2-D coordinates are provided as

λeξ = U (B.1)

λaξ = U νaξ (B.2) ± ±

λfξ = U νfξ ± ±

λsξ = U νsξ ± ±

λdξ = U where

U = ξxu + ξyv

νaξ = ξxνax + ξyvay

1 ν2 = a ν2 + c2 + z fξ 2 4 a s ξ £ ¡ ¢ ¤ 2 1 2 2 ν = a4 ν + c zξ sξ 2 a s − £ ¡ ¢ ¤

231 with

Btx νax = √µe0ρ

Bty νay = √µe0ρ

Btz νaz = √µe0ρ

2 2 2 2 νa = νax + vay + vaz

p c2 = γ s ρ

2 2 2 2 2 zξ = a4 a4 (ν + c ) 4c ν a s − s aξ q £ ¤ 1 The diagonal eigenvalue matrix, DA = X− AX is

λeξ 0000000 ⎡ ⎤

⎢ 0 λaξ+ 000000⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 00λ 00000⎥ ⎢ aξ ⎥ ⎢ − ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 00 0λfξ+ 0000⎥ ⎢ ⎥ DA = ⎢ ⎥ ⎢ ⎥ ⎢ 00 0 0λfξ 000⎥ ⎢ − ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 00 0 0 0λsξ+ 00⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 00 0 0 0 0λsξ 0 ⎥ ⎢ − ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 00 0 0 0 0 0λdξ ⎥ ⎢ ⎥ ⎣ ⎦ The right and left eigenvectors are expressed as

232 X = reξ raξ+ raξ rfξ+ rfξ rsξ+ rsξ rdξ ∙ − − − ¸

1 X− = leξ laξ+ laξ lfξ+ lfξ lsξ+ lsξ ldξ ∙ − − − ¸

It should be noted that the eigenvectors of DFMFD equations are changed. However, the eigenvectors of FMFD equations can be utilized because the conservation form of flux- vector splitting is used. The eigenvectors of Jacobian matrix A basedontheflux vector Q for FMFD equations in 2-D computational domain are

1 ⎧ ⎫ ⎪ ⎪ ⎪ u ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ v ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ w ⎪ ⎪ ⎪ reξ = ⎪ ⎪ ⎪ ⎪ ⎨⎪ 0 ⎬⎪

⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ U 2 ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

233 0 ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ρß1ξ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ρß2ξ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ρß ⎪ ⎪ 3ξ ⎪ raξ = ⎪ ⎪ ± ⎪ ⎪ ⎪ ⎪ ⎨ √µ ρß1ξ ⎬ ∓ eo ⎪ ⎪ ⎪ µ ρß ⎪ ⎪ √ eo 2ξ ⎪ ⎪ ∓ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ √µeoρß3ξ ⎪ ⎪ ∓ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ρuß1ξ + ρvß2ξ + ρwß3ξ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

ρCf ⎧ ⎫ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρuCf Ef1 ⎪ ⎪ νfξ ⎪ ⎪ ± ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρvCf Ef2 ⎪ ⎪ νfξ ⎪ ⎪ ± ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρwCf Ef3 ⎪ ⎪ νfξ ⎪ ⎪ ± ⎪ ⎪ ⎪ r = ⎪ ⎪ fξ ⎪ E4 ⎪ ± ⎪ ⎪ ⎨⎪ ⎬⎪

E5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ E ⎪ ⎪ 6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 2 γp 1 ⎪ ⎪ ρU Cf + Cf (uEf1 + vEf2 + wEf3) ⎪ ⎪ 2 γ 1 νfξ ⎪ ⎪ ⎛ − ± ⎞ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎜ + µ (BtxE4 + BtyE5 + BtzE6) ⎟ ⎪ ⎪ ⎜ eo ⎟ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ ⎝ ⎠ ⎪ ⎩⎪ ⎭⎪

234 ρCs ⎧ ⎫ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρuCs Es1 ⎪ ⎪ νsξ ⎪ ⎪ ± ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρvCs Es2 ⎪ ⎪ νsξ ⎪ ⎪ ± ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρwCs Es3 ⎪ ⎪ νsξ ⎪ ⎪ ± ⎪ ⎪ ⎪ r = ⎪ ⎪ sξ ⎪ E4 ⎪ ± ⎪ ⎪ ⎨⎪ ⎬⎪

E5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ E ⎪ ⎪ 6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 2 γp 1 ⎪ ⎪ ρU Cs + Cs (uEs1 + vEs2 + wEs3) ⎪ ⎪ 2 γ 1 νsξ ⎪ ⎪ ⎛ − ± ⎞ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎜ + µ (BtxE4 + BtyE5 + BtzE6) ⎟ ⎪ ⎪ ⎜ eo ⎟ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ ⎝ ⎠ ⎪ ⎩⎪ ⎭⎪ 0 ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ rdξ = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ξx ⎬ ⎪ ⎪ ⎪ ξ ⎪ ⎪ y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ Btξ ⎪ ⎪ µeo ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

235 T 1 2 1 2γpρU (γ 1) ⎧ − − ⎫ ⎪ ⎪ ⎪ 1 ρu (γ 1) ⎪ ⎪ γp ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρv (γ 1) ⎪ ⎪ γp − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ρw (γ 1) ⎪ ⎪ γp − ⎪ leξ = ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎨ ρBtx (γ 1) ⎬ µeoγp − ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρBty (γ 1) ⎪ ⎪ µeoγp ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ 1 ρB (γ 1) ⎪ ⎪ µ γp tz ⎪ ⎪ eo − ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρ (γ 1) ⎪ ⎪ − γp − ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪ T

[(Ef2Es3 Ef3Es2) u +(Ef3Es1 Ef1Es3) v ⎧ ⎛ − − ⎞ ⎫ ⎪ ⎪ ⎪ ⎜ +(Ef1Es2 Ef2Es1) w]/2ρθ1 ⎟ ⎪ ⎪ ⎜ − ⎟ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ ⎝ ⎠ ⎪ ⎪ 1 ⎪ ⎪ (Ef2Es3 Ef3Es2) ⎪ ⎪ 2ρθ1 ⎪ ⎪ − − ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ (Ef3Es1 Ef1Es3) ⎪ ⎪ − 2ρθ1 − ⎪ ⎪ ⎪ ⎪ ⎪ l = ⎪ 1 ⎪ aξ ⎪ (Ef1Es2 Ef2Es1) ⎪ ± ⎪ − 2ρθ1 − ⎪ ⎨⎪ ⎬⎪ 1 − ξyE6 ⎪ 2 µ ρθ2 ⎪ ⎪ ∓ √ eo ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ξ E6 ⎪ ⎪ 2 µ ρθ x ⎪ ⎪ ± √ eo 2 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ξ E4 ξ E5 ⎪ ⎪ 2 µ ρθ y x ⎪ ⎪ ± √ eo 2 − ⎪ ⎪ ⎪ ⎪ ¡ ¢ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

236 T

νfξ [(Es2ß3ξ Es3ß2ξ) u +(Es3ß1ξ Es1ß3ξ) v ⎧ ⎛ ± 2θ1 − − ⎞ ⎫ ⎪ ⎪ ⎪ 1 1 1 2 ⎪ ⎪ ⎜ +(Es1ß2ξ Es2ß1ξ) w]+ U (γ 1) ⎟ ⎪ ⎪ ⎜ 2 Cf Cs 2γp ⎟ ⎪ ⎪ ⎜ − − − ⎟ ⎪ ⎪ ⎝ ⎠ ⎪ ⎪ νfξ 1 1 1 ⎪ ⎪ (Es2ß3ξ Es3ß2ξ) u (γ 1) ⎪ ⎪ 2θ1 2 Cf Cs γp ⎪ ⎪ ∓ − − − − ⎪ ⎪ ⎪ ⎪ νfξ 1 1 1 ⎪ ⎪ (Es3ß1ξ Es1ß3ξ) v (γ 1) ⎪ ⎪ 2θ1 2 Cf Cs γp ⎪ ⎪ ∓ − − − − ⎪ ⎪ ⎪ l = ⎪ νfξ 1 1 1 ⎪ fξ ⎪ (Es1ß2ξ Es2ß1ξ) w (γ 1) ⎪ ± ⎪ ∓ 2θ1 − − 2 Cf Cs γp − ⎪ ⎨⎪ − ⎬⎪ 1 1 1 1 θ4Cs Btx (γ 1) ⎪ 2 2 Cf Cs µeoγp ⎪ ⎪ − − − ⎪ ⎪ ⎪ ⎪ 1 1 1 1 ⎪ ⎪ θ5Cs Bty (γ 1) ⎪ ⎪ 2 2 Cf Cs µ γp ⎪ ⎪ − − eo − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 θ C 1 1 1 B (γ 1) ⎪ ⎪ 2 6 s 2 C Cs µ γp tz ⎪ ⎪ − f − eo − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 1 (γ 1) ⎪ ⎪ 2 Cf Cs γp ⎪ ⎪ − − ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪ T νsξ [(Ef2ß3ξ Ef3ß2ξ) u +(Ef3ß1ξ Ef1ß3ξ) v ⎧ ⎛ ± 2θ1 − − ⎞ ⎫ ⎪ ⎪ ⎪ 1 1 1 2 ⎪ ⎪ ⎜ +(Ef1ß2ξ Ef2ß1ξ) w]+ U (γ 1) ⎟ ⎪ ⎪ ⎜ 2 Cs Cf 2γp ⎟ ⎪ ⎪ ⎜ − − − ⎟ ⎪ ⎪ ⎝ ⎠ ⎪ ⎪ νsξ 1 1 1 ⎪ ⎪ (Ef2ß3ξ Ef3ß2ξ) u (γ 1) ⎪ ⎪ 2θ1 2 Cs Cf γp ⎪ ⎪ ± − − − − ⎪ ⎪ ⎪ ⎪ νsξ 1 1 1 ⎪ ⎪ (Ef3ß1ξ Ef1ß3ξ) v (γ 1) ⎪ ⎪ 2θ1 2 Cs Cf γp ⎪ ⎪ ± − − − − ⎪ ⎪ ⎪ l = ⎪ νsξ 1 1 1 ⎪ sξ ⎪ (Ef1ß2ξ Ef2ß1ξ) w (γ 1) ⎪ ± ⎪ ± 2θ1 − − 2 Cs Cf γp − ⎪ ⎨⎪ − ⎬⎪ 1 1 1 1 θ4Cf Btx (γ 1) ⎪ 2 2 Cs Cf µeoγp ⎪ ⎪ − − − ⎪ ⎪ ⎪ ⎪ 1 1 1 1 ⎪ ⎪ θ5Cf Bty (γ 1) ⎪ ⎪ 2 2 Cs Cf µ γp ⎪ ⎪ − − eo − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 θ C 1 1 1 B (γ 1) ⎪ ⎪ 2 6 f 2 Cs C µ γp tz ⎪ ⎪ − − f eo − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 1 (γ 1) ⎪ ⎪ 2 Cs Cf γp ⎪ ⎪ − − ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

237 T 0 ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ldξ = ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎨ (E5ß3ξ E6ß2ξ) ⎬ θ2 − ⎪ ⎪ ⎪ 1 ⎪ ⎪ (E6ß1ξ E4ß3ξ) ⎪ ⎪ θ2 ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ (E4ß2ξ E5ß1ξ) ⎪ ⎪ θ2 ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ where ⎩⎪ ⎭⎪

Btξ = ξxBtx + ξyBty

ß1ξ = ξyBtz

ß2ξ = ξ Btz − x

ß3ξ = ξxBty + ξyBtx

2 2 Cf =4πρ ν a4v fξ − a ¡ 2 2 ¢ Cs =4πρ ν a4v sξ − a ¡ 2 ¢ Ef1 = γp 4πρν ξ a4BtξBtx fξ x − ¡ 2 ¢ Es1 = γp 4πρν ξ a4BtξBtx sξ x − ¡ 2 ¢ Ef2 = γp 4πρν ξ a4BtξBty fξ y − ¡ 2 ¢ Es2 = γp 4πρν ξ a4BtξBty sξ y − ¡ ¢ Ef3 = γpa4BtξBtz −

238 Es3 = γpa4BtξBtz −

E4 = 4πγpξ ß3ξ − y

E5 =4πγpξxß3ξ

E6 =4πγpa4Bz

θ1 =ß1ξ (Ef3Es2 Ef2Es3)+ß2ξ (Ef1Es3 Ef3Es1)+ß3ξ (Ef2Es1 Ef1Es2) − − − 2 2 θ2 = ξ (BtyE5 + BtzE6)+ξ (BtzE6 + BtxE4) ξ ξ (BtxE5 + BtyE4) x y − x y

θ3 =(Cs Cf ) 2a4 (BtxE4 + BtyE5 + BtzE6) ξ ξ (BtxE5 + BtyE4) − − x y 1 £ ¤ θ4 = E4 µeoγpθ3

1 θ5 = E5 µeoγpθ3

1 θ6 = E6 µeoγpθ3 The eigenvalues and eigenvectors for the flux Jacobian matrix B are obtained by sub- stituting ξ with η.Themetricscoefficients for 2-D formulation are provided in Appendix

F.

239 Appendix C

Eigenstructure in Generalized 3-D Computational Domain

The eigenvalues of flux Jacobian matrix A for the eight-wave structure of decomposed full MFD equations in generalized 3-D coordinates are

λeξ = U (C.1)

λaξ = U νaξ (C.2) ± ±

λfξ = U νfξ ± ±

λsξ = U νsξ ± ±

λdξ = U where

U = ξxu + ξyv + ξzw

νaξ = ξxνax + ξyvay + ξzvaz

1 ν2 = a ν2 + c2 + z fξ 2 4 a s ξ £ ¡ ¢ ¤ 2 1 2 2 ν = a4 ν + c zξ sξ 2 a s − £ ¡ ¢ ¤

240 with

Btx νax = √µe0ρ

Bty νay = √µe0ρ

Btz νaz = √µe0ρ

2 2 2 2 νa = νax + vay + vaz

p c2 = γ s ρ

2 2 2 2 2 zξ = a4 a4 (ν + c ) 4c ν a s − s aξ q £ ¤ 1 The diagonal eigenvalue matrix, DA = X− AX is

λeξ 0000000 ⎡ ⎤

⎢ 0 λaξ+ 000000⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 00λ 00000⎥ ⎢ aξ ⎥ ⎢ − ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 00 0λfξ+ 0000⎥ ⎢ ⎥ DA = ⎢ ⎥ ⎢ ⎥ ⎢ 00 0 0λfξ 000⎥ ⎢ − ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 00 0 0 0λsξ+ 00⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 00 0 0 0 0λsξ 0 ⎥ ⎢ − ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 00 0 0 0 0 0λdξ ⎥ ⎢ ⎥ ⎣ ⎦ The right and left eigenvectors are expressed as

241 X = reξ raξ+ raξ rfξ+ rfξ rsξ+ rsξ rdξ ∙ − − − ¸

T 1 X− = leξ laξ+ laξ lfξ+ lfξ lsξ+ lsξ ldξ ∙ − − − ¸

It should be noted that the eigenvectors of DFMFD equations are changed. However, the eigenvectors of FMFD equations can be utilized because the conservation form of flux- vector splitting is used. The eigenvectors of Jacobian matrix A basedontheflux vector Q for FMFD equations in 3-D computational domain are

1 ⎧ ⎫ ⎪ ⎪ ⎪ u ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ v ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ w ⎪ ⎪ ⎪ reξ = ⎪ ⎪ ⎪ ⎪ ⎨⎪ 0 ⎬⎪

⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ U 2 ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

242 0 ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ρß1ξ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ρß2ξ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ρß ⎪ ⎪ 3ξ ⎪ raξ = ⎪ ⎪ ± ⎪ ⎪ ⎪ ⎪ ⎨ √µ ρß1ξ ⎬ ∓ eo ⎪ ⎪ ⎪ µ ρß ⎪ ⎪ √ eo 2ξ ⎪ ⎪ ∓ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ √µeoρß3ξ ⎪ ⎪ ∓ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ρuß1ξ + ρvß2ξ + ρwß3ξ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

ρCf ⎧ ⎫ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρuCf Ef1 ⎪ ⎪ νfξ ⎪ ⎪ ± ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρvCf Ef2 ⎪ ⎪ νfξ ⎪ ⎪ ± ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρwCf Ef3 ⎪ ⎪ νfξ ⎪ ⎪ ± ⎪ ⎪ ⎪ r = ⎪ ⎪ fξ ⎪ E4 ⎪ ± ⎪ ⎪ ⎨⎪ ⎬⎪

E5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ E ⎪ ⎪ 6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 2 γp 1 ⎪ ⎪ ρU Cf + Cf (uEf1 + vEf2 + wEf3) ⎪ ⎪ 2 γ 1 νfξ ⎪ ⎪ ⎛ − ± ⎞ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎜ + µ (BtxE4 + BtyE5 + BtzE6) ⎟ ⎪ ⎪ ⎜ eo ⎟ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ ⎝ ⎠ ⎪ ⎩⎪ ⎭⎪

243 ρCs ⎧ ⎫ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρuCs Es1 ⎪ ⎪ νsξ ⎪ ⎪ ± ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρvCs Es2 ⎪ ⎪ νsξ ⎪ ⎪ ± ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρwCs Es3 ⎪ ⎪ νsξ ⎪ ⎪ ± ⎪ ⎪ ⎪ r = ⎪ ⎪ sξ ⎪ E4 ⎪ ± ⎪ ⎪ ⎨⎪ ⎬⎪

E5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ E ⎪ ⎪ 6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 2 γp 1 ⎪ ⎪ ρU Cs + Cs (uEs1 + vEs2 + wEs3) ⎪ ⎪ 2 γ 1 νsξ ⎪ ⎪ ⎛ − ± ⎞ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎜ + µ (BtxE4 + BtyE5 + BtzE6) ⎟ ⎪ ⎪ ⎜ eo ⎟ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ ⎝ ⎠ ⎪ ⎩⎪ ⎭⎪ 0 ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ rdξ = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ξx ⎬ ⎪ ⎪ ⎪ ξ ⎪ ⎪ y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ξz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ Btξ ⎪ ⎪ µeo ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

244 T 1 2 1 2γpρU (γ 1) ⎧ − − ⎫ ⎪ ⎪ ⎪ 1 ρu (γ 1) ⎪ ⎪ γp ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρv (γ 1) ⎪ ⎪ γp − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ρw (γ 1) ⎪ ⎪ γp − ⎪ leξ = ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎨ ρBtx (γ 1) ⎬ µeoγp − ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρBty (γ 1) ⎪ ⎪ µeoγp ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ 1 ρB (γ 1) ⎪ ⎪ µ γp tz ⎪ ⎪ eo − ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ρ (γ 1) ⎪ ⎪ − γp − ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪ T

[(Ef2Es3 Ef3Es2) u +(Ef3Es1 Ef1Es3) v ⎧ ⎛ − − ⎞ ⎫ ⎪ ⎪ ⎪ ⎜ +(Ef1Es2 Ef2Es1) w]/2ρθ1 ⎟ ⎪ ⎪ ⎜ − ⎟ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ ⎝ ⎠ ⎪ ⎪ 1 ⎪ ⎪ (Ef2Es3 Ef3Es2) ⎪ ⎪ 2ρθ1 ⎪ ⎪ − − ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ (Ef3Es1 Ef1Es3) ⎪ ⎪ − 2ρθ1 − ⎪ ⎪ ⎪ ⎪ ⎪ l = ⎪ 1 ⎪ aξ ⎪ (Ef1Es2 Ef2Es1) ⎪ ± ⎪ − 2ρθ1 − ⎪ ⎨⎪ ⎬⎪ 1 − ξzE5 ξyE6 ⎪ 2 µ ρθ2 ⎪ ⎪ ± √ eo − ⎪ ⎪ ⎪ ⎪ 1 ¡ ¢ ⎪ ⎪ (ξ E6 ξ E4) ⎪ ⎪ 2 µ ρθ x z ⎪ ⎪ ± √ eo 2 − ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ξ E4 ξ E5 ⎪ ⎪ 2 µ ρθ y x ⎪ ⎪ ± √ eo 2 − ⎪ ⎪ ⎪ ⎪ ¡ ¢ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

245 T

νfξ [(Es2ß3ξ Es3ß2ξ) u +(Es3ß1ξ Es1ß3ξ) v ⎧ ⎛ ± 2θ1 − − ⎞ ⎫ ⎪ ⎪ ⎪ 1 1 1 2 ⎪ ⎪ ⎜ +(Es1ß2ξ Es2ß1ξ) w]+ U (γ 1) ⎟ ⎪ ⎪ ⎜ 2 Cf Cs 2γp ⎟ ⎪ ⎪ ⎜ − − − ⎟ ⎪ ⎪ ⎝ ⎠ ⎪ ⎪ νfξ 1 1 1 ⎪ ⎪ (Es2ß3ξ Es3ß2ξ) u (γ 1) ⎪ ⎪ 2θ1 2 Cf Cs γp ⎪ ⎪ ∓ − − − − ⎪ ⎪ ⎪ ⎪ νfξ 1 1 1 ⎪ ⎪ (Es3ß1ξ Es1ß3ξ) v (γ 1) ⎪ ⎪ 2θ1 2 Cf Cs γp ⎪ ⎪ ∓ − − − − ⎪ ⎪ ⎪ l = ⎪ νfξ 1 1 1 ⎪ fξ ⎪ (Es1ß2ξ Es2ß1ξ) w (γ 1) ⎪ ± ⎪ ∓ 2θ1 − − 2 Cf Cs γp − ⎪ ⎨⎪ − ⎬⎪ 1 1 1 1 θ4Cs Btx (γ 1) ⎪ 2 2 Cf Cs µeoγp ⎪ ⎪ − − − ⎪ ⎪ ⎪ ⎪ 1 1 1 1 ⎪ ⎪ θ5Cs Bty (γ 1) ⎪ ⎪ 2 2 Cf Cs µ γp ⎪ ⎪ − − eo − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 θ C 1 1 1 B (γ 1) ⎪ ⎪ 2 6 s 2 C Cs µ γp tz ⎪ ⎪ − f − eo − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 1 (γ 1) ⎪ ⎪ 2 Cf Cs γp ⎪ ⎪ − − ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪ T νsξ [(Ef2ß3ξ Ef3ß2ξ) u +(Ef3ß1ξ Ef1ß3ξ) v ⎧ ⎛ ± 2θ1 − − ⎞ ⎫ ⎪ ⎪ ⎪ 1 1 1 2 ⎪ ⎪ ⎜ +(Ef1ß2ξ Ef2ß1ξ) w]+ U (γ 1) ⎟ ⎪ ⎪ ⎜ 2 Cs Cf 2γp ⎟ ⎪ ⎪ ⎜ − − − ⎟ ⎪ ⎪ ⎝ ⎠ ⎪ ⎪ νsξ 1 1 1 ⎪ ⎪ (Ef2ß3ξ Ef3ß2ξ) u (γ 1) ⎪ ⎪ 2θ1 2 Cs Cf γp ⎪ ⎪ ± − − − − ⎪ ⎪ ⎪ ⎪ νsξ 1 1 1 ⎪ ⎪ (Ef3ß1ξ Ef1ß3ξ) v (γ 1) ⎪ ⎪ 2θ1 2 Cs Cf γp ⎪ ⎪ ± − − − − ⎪ ⎪ ⎪ l = ⎪ νsξ 1 1 1 ⎪ sξ ⎪ (Ef1ß2ξ Ef2ß1ξ) w (γ 1) ⎪ ± ⎪ ± 2θ1 − − 2 Cs Cf γp − ⎪ ⎨⎪ − ⎬⎪ 1 1 1 1 θ4Cf Btx (γ 1) ⎪ 2 2 Cs Cf µeoγp ⎪ ⎪ − − − − ⎪ ⎪ ⎪ ⎪ 1 1 1 1 ⎪ ⎪ θ5Cf Bty (γ 1) ⎪ ⎪ 2 2 Cs Cf µ γp ⎪ ⎪ − − − eo − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 θ C 1 1 1 B (γ 1) ⎪ ⎪ 2 6 f 2 Cs C µ γp tz ⎪ ⎪ − − − f eo − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 1 (γ 1) ⎪ ⎪ 2 Cs Cf γp ⎪ ⎪ − − ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎭⎪

246 T 0 ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ldξ = ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎨ (E5ß3ξ E6ß2ξ) ⎬ θ2 − ⎪ ⎪ ⎪ 1 ⎪ ⎪ (E6ß1ξ E4ß3ξ) ⎪ ⎪ θ2 ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ (E4ß2ξ E5ß1ξ) ⎪ ⎪ θ2 ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ where ⎩⎪ ⎭⎪

Btξ = ξxBtx + ξyBty + ξzBtz

ß1ξ = ξ Btz ξ Bty y − z

ß2ξ = ξ Btx ξ Btz z − x

ß3ξ = ξxBty + ξyBtx

2 2 Cf =4πρ ν a4v fξ − a ¡ 2 2 ¢ Cs =4πρ ν a4v sξ − a ¡ 2 ¢ Ef1 = γp 4πρν ξ a4BtξBtx fξ x − ¡ 2 ¢ Es1 = γp 4πρν ξ a4BtξBtx sξ x − ¡ 2 ¢ Ef2 = γp 4πρν ξ a4BtξBty fξ y − ¡ 2 ¢ Es2 = γp 4πρν ξ a4BtξBty sξ y − ¡ 2 ¢ Ef3 = γp 4πρν ξ a4BtξBtz − fξ z − ¡ ¢

247 2 Es3 = γp 4πρν ξ a4BtξBtz − sξ z − ¡ 2 2 ¢ E4 = 4πγp ξ + ξ Bx ξ ξ By + ξ Bz − y z − x y z £¡2 2 ¢ ¡ ¢¤ E5 =4πγp ξ + ξ By ξ (ξ Bz + ξ Bx) z x − y z x £¡ 2 2¢ ¤ E6 =4πγp ξ + ξ Bz ξ ξ Bx + ξ By x y − z x y £¡ ¢ ¡ ¢¤ θ1 =ß1ξ (Ef3Es2 Ef2Es3)+ß2ξ (Ef1Es3 Ef3Es1)+ß3ξ (Ef2Es1 Ef1Es2) − − − 2 2 2 θ2 = ξx (BtyE5 + BtzE6)+ξy (BtzE6 + BtxE4)+ξz (BtxE4 + BtyE5)

ξ ξ (BtxE5 + BtyE4) ξ ξ (BtyE6 + BtzE5) ξ ξ (BtzE4 + BtxE6) − x y − y z − z x

θ3 =(Cs Cf )[2a4 (BtxE4 + BtyE5 + BtzE6) −

ξ ξ (BtxE5 + BtyE4) ξ ξ (BtyE6 + BtzE5) ξ ξ (BtzE4 + BtxE6)] − x y − y z − z x 1 θ4 = E4 µeoγpθ3

1 θ5 = E5 µeoγpθ3

1 θ6 = E6 µeoγpθ3 The eigenvalues and eigenvectors for the flux Jacobian matrix B and C are obtained by substituting ξ with η and ζ, respectively. The metrics coefficients for 3-D formulation are provided in Appendix G.

248 Appendix D

Diffusion Flux Vectors in Two Dimensions

The non-dimensional DFMFD equations were transformed form physical space to gen- eralized computational spcae. The transformed diffusion flux vectors for two-dimensional computational space are

1 E = ξ E + ξ F (D.1) v J x v y v 1 ¡ ¢ F = η E + η F (D.2) v J x v y v ¡ ¢ where Ev and Fv are the diffusion-flux vectors in physical space along the x and y directions, respectively.

The diffusion-flux terms involve spatial derivatives of velocity and magnetic field; there- fore, these derivatives must be transformed into the computational domain, as outlined by

Hoffmann and Chiang [112]. For uniform electrical conductivity distribution, the flux vec- tors are expressed as

249 0 ⎧ ⎫ ⎪ ⎪ ⎪ µ ⎪ ⎪ Re (a1uξ + a3vξ + c1uη + c3vη) ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ µ (a u + a v + c u + c v ) ⎪ ⎪ Re 3 ξ 2 ξ 4 η 2 η ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ µ ⎪ ⎪ Re (a4wξ + c5wη) ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ a4 (Bx) + c5 (Bx) ⎪ ⎪ Rm σe ξ η ⎪ ⎪ ∞ ⎪ ⎪ ⎪ 1 ⎪ h i ⎪ E = ⎪ 1 ⎪ (D.3) v ⎪ a4 (By) + c5 (By) ⎪ J ⎪ Rm σe ξ η ⎪ ⎪ ∞ ⎪ ⎨ h i ⎬ 1 a4 (Bz) + c5 (Bz) ⎪ Rm σe ξ η ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ h i ⎪ ⎪ µ [ 1 a (u2) + 1 a (v2) + 1 a (w2) + a (uv) + 1 a T ⎪ ⎪ Re 2 1 ξ 2 2 ξ 2 4 ξ 3 ξ Pr(γ 1)M 2 4 ξ ⎪ ⎪ ∞ − ∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 1 c (u2) + 1 c (v2) + 1 c (w2) + c uv + c vu + 1 c T ] ⎪ ⎪ 2 1 η 2 2 η 2 5 η 3 η 4 η Pr(γ 1)M2 5 η ⎪ ⎪ − ∞ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ + [ a4Bx ξ Bξ (Bx) + a4By ξ Bξ (By) + a4Bz (Bz) ⎪ ⎪ Rm σe x ξ y ξ ξ ⎪ ⎪ ∞ − − ⎪ ⎪ ⎪ ⎪ ¡ ¢ ¡ ¢ ⎪ ⎪ c5Bx ξxBη (Bx)η + c5By ξyBη (By)η + c5Bz (Bz)η] ⎪ ⎪ − − ⎪ ⎪ ⎪ ⎪ ¡ ¢ ¡ ¢ ⎪ ⎩⎪ ⎭⎪

250 0 ⎧ ⎫ ⎪ ⎪ ⎪ µ ⎪ ⎪ Re (c1uξ + c4vξ + b1uη + b3vη) ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ µ (c u + c v + b u + b v ) ⎪ ⎪ Re 3 ξ 2 ξ 3 η 2 η ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ µ ⎪ ⎪ Re (c5wξ + b4wη) ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ c5 (Bx) + b4 (Bx) ⎪ ⎪ Rm σe ξ η ⎪ ⎪ ∞ ⎪ ⎪ ⎪ 1 ⎪ h i ⎪ F = ⎪ 1 ⎪ v ⎪ c5 (By) + b4 (By) ⎪ J ⎪ Rm σe ξ η ⎪ ⎪ ∞ ⎪ ⎨ h i ⎬ 1 c5 (Bz) + b4 (Bz) ⎪ Rm σe ξ η ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ h i ⎪ ⎪ µ [ 1 c (u2) + 1 c (v2) + 1 c (w2) + c vu + c uv + 1 c T ⎪ ⎪ Re 2 1 ξ 2 2 ξ 2 5 ξ 3 ξ 4 ξ Pr(γ 1)M 2 5 ξ ⎪ ⎪ ∞ − ∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 1 b (u2) + 1 b (v2) + 1 b (w2) + b (uv) + 1 b T ] ⎪ ⎪ 2 1 η 2 2 η 2 4 η 3 η Pr(γ 1)M 2 4 η ⎪ ⎪ − ∞ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ + [ c5Bx η Bξ (Bx) + c5By η Bξ (By) + c5Bz (Bz) ⎪ ⎪ Rm σe x ξ y ξ ξ ⎪ ⎪ ∞ − − ⎪ ⎪ ⎪ ⎪ ¡ ¢ ¡ ¢ ⎪ ⎪ b4Bx ηxBη (Bx)η + b4By ηyBη (By)η + b4Bz (Bz)η] ⎪ ⎪ − − ⎪ ⎪ ⎪ ⎪ ¡ ¢ ¡ ¢ ⎪(D.4) ⎩⎪ ⎭⎪ where

Bξ = ξxBx + ξyBy

Bη = ηxBx + ηyBy

The metrics coefficients for 2-D formulation are provided in Appendix F.

251 Appendix E

Diffusion Flux Vectors in Three Dimensions

The nondimensional DFMFD equations were transformed from physical space to gen- eralized computational space. The transformed diffusion flux vectors for three-dimensional computational space are

1 E = ξ E + ξ F + ξ G (E.1) v J x v y v z v 1 ¡ ¢ F = η E + η F + η G (E.2) v J x v y v z v 1 ¡ ¢ G = ζ E + ζ F + ζ G (E.3) v J x v y v z v ¡ ¢ where Gv, Ev,andFv are the diffusion-flux vectors in physical space along the x, y, and z directions, respectively.

The diffusion-flux terms involve spatial derivatives of velocity and magnetic field; there- fore, these derivatives must be transformed into the computational domain, as outlined by

Hoffmann and Chiang [112]. For uniform electrical conductivity distribution, the flux vec- tors are expressed as

252 0 ⎧ ⎫ ⎪ ⎪ ⎪ µ ⎪ ⎪ Re (a1uξ + a5vξ + a7wξ + d1uη + d7vη + d9wη + e1uζ + e7vζ + e9wζ ) ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ µ ⎪ ⎪ Re (a5uξ + a2vξ + a6wξ + d5uη + d2vη + d10wη + e5uζ + e2vζ + e10wζ) ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ µ (a u + a v + a w + d u + d v + d w + e u + e v + e w ) ⎪ ⎪ Re 7 ξ 6 ξ 3 ξ 6 η 8 η 3 η 6 ζ 8 ζ 3 ζ ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ a4 (Bx) + d4 (Bx) + e4 (Bx) ⎪ ⎪ Rm σe ξ η ζ ⎪ ⎪ ∞ ⎪ ⎪ h i ⎪ ⎪ 1 ⎪ ⎪ a4 (By) + d4 (By) + e4 (By) ⎪ ⎪ Rm σe ξ η ζ ⎪ ⎪ ∞ ⎪ ⎪ h i ⎪ ⎪ 1 ⎪ ⎪ a4 (Bz) + d4 (Bz) + e4 (Bz) ⎪ ⎪ Rm σe ξ η ζ ⎪ ⎪ ∞ ⎪ 1 ⎪ h i ⎪ ⎪ µ 1 1 1 ⎪ Ev = ⎪ [ a (u2) + a (v2) + a (w2) + a (uv) + a (vw) + a (uw) ⎪ J ⎪ Re 2 1 ξ 2 2 ξ 2 3 ξ 5 ξ 6 ξ 7 ξ ⎪ ⎪ ∞ ⎪ ⎨⎪ ⎬⎪ + 1 a T + 1 d (u2) + 1 d (v2) + 1 d (w2) + d vu ⎪ Pr(γ 1)M 2 4 ξ 2 1 η 2 2 η 2 3 η 5 η ⎪ ⎪ − ∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +d wu + d uv + d wv + d uw + d vw + 1 d T ⎪ ⎪ 6 η 7 η 8 η 9 η 10 η Pr(γ 1)M 2 4 η ⎪ ⎪ − ∞ ⎪ ⎪ ⎪ ⎪ 1 2 1 2 1 2 ⎪ ⎪ + e1 (u )ζ + e2 (v )ζ + e3 (w )ζ + e5vuζ + e6wuζ ++e7uvζ ⎪ ⎪ 2 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +e wv + e uw + e vw + 1 e T ] ⎪ ⎪ 8 ζ 9 ζ 10 ζ Pr(γ 1)M 2 4 ζ ⎪ ⎪ − ∞ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ + [ a4Bx ξ Bξ (Bx) + a4By ξ Bξ (By) + a4Bz ξ Bξ (Bz) ⎪ ⎪ Rm σe x ξ y ξ z ξ ⎪ ⎪ ∞ − − − ⎪ ⎪ ⎪ ⎪ ¡ ¢ ¡ ¢ ¡ ¢ ⎪ ⎪ d4Bx ξxBη (Bx)η + d4By ξyBη (By)η + d4Bz ξzBη (Bz)η ⎪ ⎪ − − − ⎪ ⎪ ⎪ ⎪ ¡ ¢ ¡ ¢ ¡ ¢ ⎪ ⎪ e4Bx ξ Bζ (Bx) + e4By ξ Bζ (By) + e4Bz ξ Bζ (Bz) ] ⎪ ⎪ − x ζ − y ζ − z ζ ⎪ ⎪ ⎪ ⎪ ¡ ¢ ¡ ¢ ¡ ¢ (E.4)⎪ ⎩⎪ ⎭⎪

253 0 ⎧ ⎫ ⎪ ⎪ ⎪ µ ⎪ ⎪ Re (d1uξ + d5vξ + d6wξ + b1uη + b5vη + b7wη + f1uζ + f7vζ + f9wζ ) ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ µ (d u + d v + d w + b u + b v + b w + f u + f v + f w ) ⎪ ⎪ Re 7 ξ 2 ξ 8 ξ 5 η 2 η 6 η 5 ζ 2 ζ 10 ζ ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ µ (d u + d v + d w + b u + b v + b w + f u + f v + f w ) ⎪ ⎪ Re 9 ξ 10 ξ 3 ξ 7 η 6 η 3 η 6 ζ 8 ζ 3 ζ ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ d4 (Bx) + b4 (Bx) + f4 (Bx) ⎪ ⎪ Rm σe ξ η ζ ⎪ ⎪ ∞ ⎪ ⎪ h i ⎪ ⎪ 1 ⎪ ⎪ d4 (By) + b4 (By) + f4 (By) ⎪ ⎪ Rm σe ξ η ζ ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ h i ⎪ ⎪ 1 d (B ) + b (B ) + f (B ) ⎪ ⎪ Rm σe 4 z ξ 4 z η 4 z ζ ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ h i ⎪ ⎪ µ 1 2 1 2 1 2 ⎪ 1 ⎪ Re [ 2 d1 (u )ξ + 2 d2 (v )ξ + 2 d3 (w )ξ + d5uvξ + d6vwξ + d7vuξ ⎪ ⎪ ∞ ⎪ Fv = ⎪ ⎪ J ⎪ ⎪ ⎪ 1 ⎪ ⎨ +d8vwξ + d9wuξ + d10wvξ + Pr(γ 1)M 2 d4Tξ ⎬ − ∞ ⎪ ⎪ ⎪ + 1 b (u2) + 1 b (v2) + 1 b (w2) ⎪ ⎪ 2 1 η 2 2 η 2 3 η ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ b uv b vw b uw 1 b T ⎪ ⎪ + 5 ( )η + 6 ( )η + 7 ( )η + Pr(γ 1)M 2 4 η ⎪ ⎪ − ∞ ⎪ ⎪ ⎪ ⎪ 1 2 1 2 1 2 ⎪ ⎪ + f1 (u ) + f2 (v ) + f3 (w ) ⎪ ⎪ 2 ζ 2 ζ 2 ζ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +f vu + f wu + f uv + f wv + f uw + f vw + 1 f T ] ⎪ ⎪ 5 ζ 6 ζ 7 ζ 8 ζ 9 ζ 10 ζ Pr(γ 1)M 2 4 ζ ⎪ ⎪ − ∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 1 [ d B η B (B ) + d B η B (B ) + d B η B (B ) ⎪ ⎪ Rm σe 4 x x ξ x ξ 4 y y ξ y ξ 4 z z ξ z ξ ⎪ ⎪ ∞ − − − ⎪ ⎪ ⎪ ⎪ ¡ ¢ ¡ ¢ ¡ ¢ ⎪ ⎪ b B η B (B ) + b B η B (B ) + b B η B (B ) ⎪ ⎪ 4 x x η x η 4 y y η y η 4 z z η z η ⎪ ⎪ − − − ⎪ ⎪ ⎪ ⎪ ¡ ¢ ¡ ¢ ¡ ¢ ⎪ ⎪ f4Bx ηxBζ (Bx)ζ + f4By ηyBζ (By)ζ + f4Bz ηzBζ (Bz)ζ] ⎪ ⎪ − − − ⎪ ⎪ ⎪ ⎪ ¡ ¢ ¡ ¢ ¡ ¢ (E.5)⎪ ⎩⎪ ⎭⎪

254 0 ⎧ ⎫ ⎪ ⎪ ⎪ µ ⎪ ⎪ Re (e1uξ + e5vξ + e6wξ + f1uη + f5vη + f6wη + c1uζ + c5vζ + c7wζ ) ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ µ (e u + e v + e w + f u + f v + f w + c u + c v + c w ) ⎪ ⎪ Re 7 ξ 2 ξ 8 ξ 7 η 2 η 8 η 5 ζ 2 ζ 6 ζ ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ µ (e u + e v + e w + f u + f v + f w + c u + c v + c w ) ⎪ ⎪ Re 9 ξ 10 ξ 3 ξ 9 η 10 η 3 η 7 ζ 6 ζ 3 ζ ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ e4 (Bx) + f4 (Bx) + c4 (Bx) ⎪ ⎪ Rm σe ξ η ζ ⎪ ⎪ ∞ ⎪ ⎪ h i ⎪ ⎪ 1 ⎪ ⎪ e4 (By) + f4 (By) + c4 (By) ⎪ ⎪ Rm σe ξ η ζ ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ h i ⎪ ⎪ 1 e (B ) + f (B ) + c (B ) ⎪ ⎪ Rm σe 4 z ξ 4 z η 4 z ζ ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ h i ⎪ ⎪ µ 1 2 1 2 1 2 ⎪ 1 ⎪ Re [ 2 e1 (u )ξ + 2 e2 (v )ξ + 2 e3 (w )ξ + e5uvξ + e6vwξ + e7vuξ ⎪ ⎪ ∞ ⎪ Gv = ⎪ ⎪ J ⎪ ⎪ ⎪ 1 ⎪ ⎨ +e8vwξ + e9wuξ + e10wvξ + Pr(γ 1)M2 e4Tξ ⎬ − ∞ ⎪ ⎪ ⎪ + 1 f (u2) + 1 f (v2) + 1 f (w2) + f uv + f uw + f vu ⎪ ⎪ 2 1 η 2 2 η 2 3 η 5 η 6 η 7 η ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ f vw f wu f wv 1 f T ⎪ ⎪ + 8 η + 9 η + 10 η + Pr(γ 1)M2 4 η ⎪ ⎪ − ∞ ⎪ ⎪ ⎪ ⎪ 1 2 1 2 1 2 ⎪ ⎪ + c1 (u ) + c2 (v ) + c3 (w ) ⎪ ⎪ 2 ζ 2 ζ 2 ζ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +c (uv) + c (vw) + c (uw) + 1 f T ] ⎪ ⎪ 5 ζ 6 ζ 7 ζ Pr(γ 1)M 2 4 ζ ⎪ ⎪ − ∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 1 [ e B ζ B (B ) + e B ζ B (B ) + e B ζ B (B ) ⎪ ⎪ Rm σe 4 x x ξ x ξ 4 y y ξ y ξ 4 z z ξ z ξ ⎪ ⎪ ∞ − − − ⎪ ⎪ ⎪ ⎪ ¡ ¢ ¡ ¢ ¡ ¢ ⎪ ⎪ f B ζ B (B ) + f B ζ B (B ) + f B ζ B (B ) ⎪ ⎪ 4 x x η x η 4 y y η y η 4 z z η z η ⎪ ⎪ − − − ⎪ ⎪ ⎪ ⎪ ¡ ¢ ¡ ¢ ¡ ¢ ⎪ ⎪ c4Bx ζxBζ (Bx)ζ + c4By ζyBζ (By)ζ + c4Bz ζzBζ (Bz)ζ ] ⎪ ⎪ − − − ⎪ ⎪ ⎪ ⎪ ¡ ¢ ¡ ¢ ¡ ¢ (E.6)⎪ ⎩⎪ ⎭⎪ where

Bξ = ξxBx + ξyBy + ξzBz

255 Bη = ηxBx + ηyBy + ηzBz

Bζ = ζxBx + ζyBy + ζzBz

The metrics coefficients for 3-D formulation are provided in Appendix G.

256 Appendix F

Metrices Coefficients in Two Dimensions

4 2 2 2 4 2 a1 = 3 ξx + ξy (F.1) a2 = ξx + 3 ξy (F.2)

1 2 2 a3 = 3 ξxξy (F.3) a4 = ξx + ξy (F.4)

4 2 2 2 4 2 b1 = 3 ηx + ηy (F.5) b2 = ηx + 3 ηy (F.6)

1 2 2 b3 = 3 ηxηy (F.7) b4 = ηx + ηy (F.8)

4 4 c1 = 3 ηxξx + ξyηy (F.9) c2 = ηxξx + 3 ξyηy (F.10)

2 2 c3 = η ξ ξ η (F.11) c4 = ξ η ξ η (F.12) x y − 3 x y x y − 3 y x c5 = ξxηx + ξyηy (F.13)

257 Appendix G

Metrices Coefficients in Three Dimensions

4 2 2 2 2 4 2 2 a1 = 3 ξx + ξy + ξz (G.1) a2 = ξx + 3 ξy + ξz (G.2)

2 2 4 2 2 2 2 a3 = ξx + ξy + 3 ξz (G.3) a4 = ξx + ξy + ξz (G.4)

1 1 a5 = 3 ξxξy (G.5) a6 = 3 ξyξz (G.6)

1 a7 = 3 ξxξz (G.7)

4 2 2 2 2 4 2 2 b1 = 3 ηx + ηy + ηz (G.8) b2 = ηx + 3 ηy + ηz (G.9)

2 2 4 2 2 2 2 b3 = ηx + ηy + 3 ηz (G.10) b4 = ηx + ηy + ηz (G.11)

1 1 b5 = 3 ηxηy (G.12) b6 = 3 ηyηz (G.13)

1 b7 = 3 ηxηz (G.14)

4 2 2 2 2 4 2 2 c1 = 3 ζx + ζy + ζz (G.15) c2 = ζx + 3 ζy + ζz (G.16)

2 2 4 2 2 2 2 c3 = ζx + ζy + 3 ζz (G.17) c4 = ζx + ζy + ζz (G.18)

1 1 c5 = 3 ζxζy (G.19) c6 = 3 ζyζz (G.20)

1 c7 = 3 ζxζz (G.21)

258 4 4 d1 = 3 ξxηx + ξyηy + ξzηz (G.22) d2 = ξxηx + 3 ξyηy + ξzηz (G.23)

4 d3 = ξxηx + ξyηy + 3 ξzηz (G.24) d4 = ξxηx + ξyηy + ξzηz (G.25)

2 2 d5 = ξ η ξ η (G.26) d6 = ξ η ξ η (G.27) x y − 3 y x x z − 3 z x

2 2 d7 = ξ η ξ η (G.28) d8 = ξ η ξ η (G.29) y x − 3 x y y z − 3 z y

2 2 d9 = ξ η ξ η (G.30) d10 = ξ η ξ η (G.31) z x − 3 x z z y − 3 y z

4 4 e1 = 3 ξxζx + ξyζy + ξzζz (G.32) e2 = ξxζx + 3 ξyζy + ξzζz (G.33)

4 e3 = ξxζx + ξyζy + 3 ξzζz (G.34) e4 = ξxζx + ξyζy + ξzζz (G.35)

2 2 e5 = ξ ζ ξ ζ (G.36) e6 = ξ ζ ξ ζ (G.37) x y − 3 y x x z − 3 z x

2 2 e7 = ξ ζ ξ ζ (G.38) e8 = ξ ζ ξ ζ (G.39) y x − 3 x y y z − 3 z y

2 2 e9 = ξ ζ ξ ζ (G.40) e10 = ξ ζ ξ ζ (G.41) z x − 3 x z z y − 3 y z

4 4 f1 = 3 ηxζx + ηyζy + ηzζz (G.42) f2 = ηxζx + 3 ηyζy + ηzζz (G.43)

4 f3 = ηxζx + ηyζy + 3 ηzζz (G.44) f4 = ηxζx + ηyζy + ηzζz (G.45)

2 2 f5 = η ζ η ζ (G.46) f6 = η ζ η ζ (G.47) x y − 3 y x x z − 3 z x

2 2 f7 = η ζ η ζ (G.48) f8 = η ζ η ζ (G.49) y x − 3 x y y z − 3 z y

2 2 f9 = η ζ η ζ (G.50) f10 = η ζ η ζ (G.51) z x − 3 x z z y − 3 y z

259