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