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The Conversion of Volume Integrals to Surface Form University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 5-2005 Industry Motivated Advancements of Current Combustion Instability Model: The Conversion of Volume Integrals to Surface Form Sean Robert Fischbach University of Tennessee, Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Aerospace Engineering Commons Recommended Citation Fischbach, Sean Robert, "Industry Motivated Advancements of Current Combustion Instability Model: The Conversion of Volume Integrals to Surface Form. " Master's Thesis, University of Tennessee, 2005. https://trace.tennessee.edu/utk_gradthes/4556 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Sean Robert Fischbach entitled "Industry Motivated Advancements of Current Combustion Instability Model: The Conversion of Volume Integrals to Surface Form." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Master of Science, with a major in Aerospace Engineering. Joseph Majdalani, Major Professor We have read this thesis and recommend its acceptance: Gary Flandro, Kenneth Kimble Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) To the Graduate Council: I am submitting herewitha thesiswritten by SeanRobert Fischbach entitled "Industry Motivated Advancementsof CurrentCombustion Instability Model: The Conversion of Volume Integrals to Surface Form." I have examined the finalpaper copy of thisthesis for formand content andrecommend thatit be· accepted in partialfulfillment of the requirements for the degree of Master of Science, witha major in Aerospace Engineering. Dr. Joseph Majdalani,Major Professor We have read thisthesis ::;z;;;�Dr. GaryFlandro Dr. KennethKimble Deanof Graduate Studies Industry Motivated Advancements of Current Combustion Instability Model: The Conversion of Volume Integrals to Surface Form A Thesis Presented for the Masterof Science Degree University of Tennessee, Knoxville Sean RobertFischbach May 2005 Acknowledgements I would like to express a mighty 'thank you' to my parents, Robert and Denice Fischbach. They have granted me every possible opportunity to excel, unconditionally. Devoid of their love and support, this journey could not have come to fruition. I would also like to thank Dr. Flandro. His eternal knowledge of Combustion Instability has resonated in this work and his selfless dedication to teaching and research has inspired me. Dr. Majdalani equally deservesmy immense gratitude. As my advisor he has helped me every step of the way, never allowing me to settle for 'O.K.' or 'close enough.' He has repeatedly offered me opportunities to participate in exciting side projects that have only deepened my knowledge and abilities. His talents as an editor surpass those of any college English professor that I have known. I would like to thank Dr. Kimble for participating on my committee, and for his valuable editorial comments. Next, I would like to thank Dr. French, of Software and Engineering Associates. His technical advice has helped me enormously. He has worked tirelessly at expediting the SSP confirmation for this study. Working with him has taught me a valuable lesson regarding the differences between research and industry. Last of all I would like to thank God, without whose blessing and subtle intercessions, this project, as formidable as once it seemed, would have lacked a happy ending. 11 Abstract This study reports a current development in the widely used combustion stability algorithm employed by propulsion industries as a predictive tool for the design of large combustors. It has been recently demonstrated that, by incorporating unsteady rotational sources and sinks in the acoustic energy assessment, a more precise formulation of the acoustic instability in rocket motors can be achieved. The new algorithm, when applied to the linear stability formulation, leads to ten growth rate terms. In this thesis, these ten stability corrections are converted from volumetric to surface integral form. They are further converted to an acoustic form that is directly amenable to implementation in the Standard Stability Prediction code. The reduction to surface form greatly facilitates the evaluation of individual stability growth rates as they become function of quantities distributed along the chamber's control surface. This will preclude the need to carry out a rotational flow analysis inside the motor. Only surface quantities will be needed and these will be converted to acoustic form whenever possible using the no slip condition or other applicable response functions. Effectively, all needed information will be obtainable directly from the acoustic field. By precluding the need to evaluate the rotational field (which can be highly uncertain in arbitrary geometry), the evaluation of acoustic stability integrals is made possible in practical motors with variable grain perforation. The analysis entails acquiring and applying several vortico-acoustic and vector identities, the most notable of which being the Gaussian divergence theorem. iii Table of Contents 1. INTRODUCTION................................................................................................. - 1 - 1.1. ORIGlN OF COMBUSTION INSTABILITY ....•......................................•.................- 1- 1.2. COMBUSTION INST ABILITY CHARACTERISTICS IN SRMs................................. - 3 - 1.3. COMBUSTION INSTABILITY IN LIQUID ROCKETS .............................................•- 4- 1.4. COMBUSTION INSTABILITY CHARACTERISTICS IN GAS TURBINES ...................- 5 - 1.5. COMBUSTION INSTABILITY CLASSIFICATION ...................................................- 5 - 1.6. COMBUSTION INST ABILITY CONSEQUENCES ....................................................- 6 - 1. 7. HISTORICAL EXAMPLES ...................................................................................- 7 - 2. COMBUSTION INST ABILITY MODELS ..................................................... - 12 - 2.1. CULICK'S CONTRIBUTIONS ............................................................................- 12- 2.2. FLANDRO'S CONTRIBUTIONS .........................................................................- 15- 3. MULTIDIMENSIONAL ENERGY BALANCE ............................................. - 18 - 3.1. BASIC FORMULATION ....................................................................................- 18 - 3 .2. ENERGY ASSESSMENT ....................................................................................- 22 - 3 .3. DEFINITION OF FLOW VARIABLE ...................................................................- 23 - 4. VOLUME TO SURFACE CONVERSION OF ROCKET STABILITY INTEGRALS .•........•.........................................•......•........................•...•.•..•...............•• - 26 - 4.1. FIRST FACTOR: EXTENDED PRESSURE COUPLING ..........................................- 26- 4.2. SECOND FACTOR: DILATA TIONAL ENERGY CORRECTION ..............................- 31 - 4.3. THIRD FACTOR: ACOUSTIC MEAN FLOW CORRECTION ..................................-32- lV 4.4. FOURTH FACTOR: FLOW TURNING CORRECTION ........................................... -33 - 4.5. FIFTH FACTOR: ROTATIONAL FLOW CORRECTION ......................................... -36- 4.6. SIXTH FACTOR: MEAN VORTICAL CORRECTION ............................................ -37 - 4.7. SEVENTH FACTOR: VISCOUS CORRECTION .................................................... -41 - 4.8. EIGHTH FACTOR: PSEUDO ACOUSTICAL CORRECTION ................................... -46 - 4.9. NINTH FACTOR: PSEUDO-ROTATIONAL CORRECTION.................................... -49 - 4.10. TENTH FACTOR: UNSTEADY NOZZLE CORRECTION ................................... -50 - 4.11. CONVERSION SUMMARY ............................................................................ -51 - 5. SURFACE INTEGRAL EVALUATION AND VERIFICATION ................. - 54 - 5.1. SURFACE INTEGRAL EVALUATION ................................................................. -54 - 5.2. NUMERICAL COMPARISON ............................................................................. - 60 - 5.3. SSP COMPARISON .......................................................................................... -62- 6. CONCLUSION ................................................................................................... - 64 - REFERENCES ............................................................................................................ - 67 - APPENDIX .................................................................................................................. - 74 - A. l VECTOR IDENTITIES ....................................................................................... -75 - A.2 THEOREMS ..................................................................................................... -76 - A.3 TIME AVERAGING .......................................................................................... -76 - VITA...............................................................
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