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Asymptotic Solutions for the Fundamental Isentropic Relations in Variable Area Duct Flow

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Asymptotic Solutions for the Fundamental Isentropic Relations in Variable Area Duct Flow University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 5-2006 Asymptotic Solutions for the Fundamental Isentropic Relations in Variable Area Duct Flow Esam M. Abu-Irshaid University of Tennessee, Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Aerospace Engineering Commons Recommended Citation Abu-Irshaid, Esam M., "Asymptotic Solutions for the Fundamental Isentropic Relations in Variable Area Duct Flow. " Master's Thesis, University of Tennessee, 2006. https://trace.tennessee.edu/utk_gradthes/4479 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 Esam M. Abu-Irshaid entitled "Asymptotic Solutions for the Fundamental Isentropic Relations in Variable Area Duct Flow." 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 herewith a thesis written by Esam M. Abu-Irshaid entitled "Asymptotic Solutions forthe Fundamental IsentropicRelations in Variable Area Duct Flow." I have examined the finalpaper copy of this thesis forform and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Aerospace Engineering. � ��alani, Major Profe We have read this thesis an '.�� Dr. Gary Flandro Dr. Kenneth Kimble Accepted forthe Council: ViceG=--� Chancellor and \s- Dean of Graduate Studies Asymptotic Solutions for the Fundamental Isentropic Relations in Variable Area Duct Flow A Thesis Presented forthe Master of Science Degree University of Tennessee, Knoxville Esam M. T. Abu-Irshaid May2006 Dedication All praise and gratitude is due to God who has guided me through this accomplishment as I would not have ultimately gotten this far had it not been for his guidance. During the writing of my thesis, I have been assisted morally and academically. I thus would like to dedicate my thesis to my family: My parents who have sacrificed every possible event and opportunity in order for me to reach this point in my life and who have unconditionally supported me throughout this journey; my brother, Osama, whose wisdom I have not seen the likes of which and whose opinions have inspired my thoughts; the rest of my family who have never turned me down and have always been there for me; my fiancee, Lila, whose compassion has brought meaning to my life and who has always been an infinite source of motivation. To you all, I dedicate my thesis. 11 Acknowledgements I would like to express my most sincere gratitude to my advisor, Dr. Joseph Majdalani, whose help in this project has been indispensable. The idea of me pursuing another M.S. was actually his. In fact, Dr. Majdalani's insights and talents in perturbation theory have been instrumental to my success. He has always been with me at every step of the way until my thesis was fully completed; Dr. Gary Flandro who has always been a guiding hand and whose assistance, academically, has contributed to the improvement of my work; and Dr. Kimble who has kindly accepted to participate on my committee and forhis contributing suggestions; to all of my friends and beloved ones for their continuous thoughtful and warm wishes. I would also like to especially thank Sean Fischbach forhis efforts with the final touches in writing this thesis. So to all of those who have been there forme, thank you. 111 Abstract We consider the isentropic flow equations that relate pressures, temperatures, densities, and critical back pressure to their total quantitiesfor a given flowMach number and nozzle area ratio. Using proper substitutions, we bypass the Mach numbe� and link these fundamental commodities directly to the local area expansion ratio. The transcendental equations we obtain are then solved asymptotically and presented in closed formto arbitrary order. Both subsonic andsupersonic branchesof the solution are identified and suitably treated. Two representations of the isentropic variables are introduced for both subsonic and supersonic cases. The first representation defines the isentropic variable as the total quantity divided by its local, while the second representation is described as the reciprocal of the first one. Expressions forthe truncated errors forthe supersonic case are firstdetermined. Then, Bosley's technique is applied to confirm the truncation order for each approximation. The finalexpressions provide direct solutions for the main thermodynamic properties as function of the area expansion and gas compression ratios. This enables us to calculate these widely used quantities without resorting to numerical iteration, intermediate Mach number calculations, guesswork, or trial. The work increases our repertoire of isentropic flow approximations that are ubiquitously used in the propulsion andpower generation industries. IV Table of Contents 1. Introduction ........•.•...•....•................•.••.............•••.•.....................••...•......•............... - 1 - 2. Fundamental Equations ...•................•....•.••...•......................•.......•....................... - 4 - 3. Inversion of Subsonic Relations .......................................................................... - 8 - 4. Inversion of Supersonic Relations and Error Analysis ................................... - 12 - 4.1. Derivation of Corrections................................................... .........- 12 - 4.2. Error Analysis of Supersonic Relations..................................... ......- 14 - 5. Results and Discussion ...................................................................- 19 - 5.1. Subsonic Case ........................................................................- 19 - 5.2. Supersonic Case .................................................... ..................- 23 - 6. Conclusion .•....•..........................•..•..•.•..•••...•......•........................•..•••................... - 39 - References ••••••.••.•••...•••.•........••••••••...............•....•.••.••••••..•.....•.•.••••.........•••....•............... - 42 - References ••••........•.................••••••••..................•.•..•..••••....•.•......••........•.•••...••...•........... - 43 - Vita .•...•••..••.•••...•...•.....•.............•••.••...•...•.•.•.••••.•......••..................•........•.••.•..........•.•..... - 44 - V List of Tables Table 1. Truncation error index k(n) .......................................................................-18 - Table 2. Order of asymptotic solutions (r) compared to least-squares estimates (y =1.2, s S 35 - = o I s(x) >·······················································································--- Table 3. Order of asymptotic solutions (r) compared to least-squares estimates (y=l.4, s=sols(x)) ............................................................................................-35- Table 4. Order of asymptotic solutions (r) compared to least-squares estimates (y=l.2, s =s(x)lso) ............................................................................................- 35 - Table 5. Order of asymptotic solutions (r) compared to least-squares estimates (y = 1.4, s= s( x) S ) ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••- 35 - I o Table 6. General subsonic solutions for the isentropic relations, q = ij(x)/ ij0 ••••• - 41 - Table 7. General supersonic solutions forthe isentropic relations, s = s(x) I s0 •• - 41- VI List of Figures Figure 1. Comparison between numerical solutions and subsonic a) pressure, b) temperature, c) density, and ratios at different orders. Everywhere, we use Q="iiol'if(x) and y = 1.2.. ......................................................................................... - 20 - Figure 2. Comparison between numerical solutions and subsonic a) pressure, b) temperature, c) density, and d) back pressure ratios at different orders. Everywhere,we use q = q(x)lq0 and y = 1.2 ••••••••• -.............................................. - 21 - Figure 3. Comparison between numerical solutions and subsonic a) pressure, b) temperature, c) density ratios at different orders. Everywhere,we use Q=q0 /q(x) and y = 1.4.. .......................................................................................................... - 22 - Figure 4. Comparison between numerical solutions and subsonic a) pressure, b) temperature, c) density, and d) back pressure ratios at different orders. q Everywhere,we use = q(x)lq0 and y = 1.4................... --................................ - 23 - Figure 5. Comparison between numerical solutions and supersonic a) pressure, b) temperature, c) density, at differentorders. Everywhere, we use s = s0 /s(x) and y = 1.2 . ....................................................................................................................
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