Thermoacoustically Induced and Acoustically Driven Flows and Heat Transfer in Enclosures A Thesis submitted to the faculty of Drexel University by Murat K. Aktas in partial fulfillment of the requirements for the degree of Doctor of Philosophy May 2004 © Copyright 2004 Murat K. Aktas. All Rights Reserved. ii ACKNOWLEDGMENTS I would like to express my gratitude to my advisor, Dr. Bakhtier Farouk, for his help, valuable guidance and encouragement throughout my studies at Drexel University. I would like to thank my thesis advisory committee members: Dr. Carr Everbach, Dr. Peter Lewin, Dr. David Miller, Dr. Howard Pearlman and Dr. David Wootton for their participation, time and valuable comments. I would also like to thank Dr. Elaine Oran from U.S. Naval Research Laboratory for her contributions. Lastly, I acknowledge the graduate assistantship and George Hill Jr. fellowship from the Department of Mechanical Engineering and Mechanics, Drexel University for the financial support provided for this study. iii TABLE OF CONTENTS LIST OF TABLES ..…………………………………………………………………… vi LIST OF FIGURES …………………………………………………………………… vii ABSTRACT …………………………………………………………………………… xii 1. INTRODUCTION ...………………………………………………………………… 1 1.1. Motivation and Background .………………………………………………… 1 1.2. Literature Review ..…………………………………………………………… 6 1.2.1. Thermoacoustic Waves and Thermoacoustic Convection ………… 7 1.2.2. Acoustic Streaming Induced Transport ……………………………… 9 1.3. Research Needs .…………………………………………………………… 20 1.4. Objectives …………………………………………………………………… 21 1.5. Outline of the Thesis ..……………………………………………………… 23 2. MATHEMATICAL AND COMPUTATIONAL MODELS ……………………… 24 2.1. Overview ..…………………………………………………………………… 24 2.2. Governing Equations .……………………………………………………… 24 2.3. Computational Model .……………………………………………………… 27 2.3.1. The Flux-Corrected Transport Algorithm ...………………………… 28 2.3.2. LCPFCT Solution Procedure ...……………………………………… 31 2.3.3. Boundary Conditions in LCPFCT …………………………………… 35 2.3.4. Boundary Conditions in the Present Navier-Stokes Solver ……… 36 iv 2.4. Computational Resources .………………………………………………… 38 3. INVESTIGATION OF THERMOACOUSTIC WAVES IN AN ENCLOSURE ..…………………………………………………………………… 39 3.1. Introduction ..………………………………………………………………… 39 3.2. Problem Description ...……………………………………………………… 39 3.3. Results and Discussion .…………………………………………………… 42 3.4. Summary ..…………………………………………………………………… 50 4. INTERACTION OF THERMOACOUSTIC WAVES AND BUOYANCY- INDUCED FLOWS IN AN ENCLOSURE ...………………………………… 51 4.1. Introduction ..………………………………………………………………… 51 4.2. Problem Description ...……………………………………………………… 51 4.3. Results and Discussion .…………………………………………………… 53 4.4. Summary ..…………………………………………………………………… 64 5. NUMERICAL SIMULATION OF OUTER ACOUSTIC STREAMING GENERATED BY STANDING WAVES IN AN ENCLOSURE ..…………… 65 5.1. Introduction ..………………………………………………………………… 65 5.2. Problem Description ...……………………………………………………… 65 5.3. Inner and Outer Streaming ………………………………………………… 67 5.4. Results and Discussion .…………………………………………………… 68 5.5. Summary ..…………………………………………………………………… 84 6. NUMERICAL SIMULATION OF THE FORMATION AND INTERACTION OF INNER/OUTER ACOUSTIC STREAMING STRUCTURES .…………… 85 6.1. Introduction ..………………………………………………………………… 85 v 6.2. Problem Description ...……………………………………………………… 85 6.3. Results and Discussion .…………………………………………………… 86 6.4. Summary …………………………………………………………………… 108 7. EXPERIMENTAL INVESTIGATION OF STREAMING FLOWS IN A RESONATOR ...………………………………………………………………… 109 7.1. Introduction ………………………………………………………………… 109 7.2. Experimental Apparatus ..………………………………………………… 109 7.3. Formation of a Standing Wave Field .…………………………………… 112 7.4. Flow Visualization Studies ..……………………………………………… 119 7.5. Summary …………………………………………………………………… 122 8. CONCLUDING REMARKS .…………………………………………………… 123 8.1. Summary and Research Contributions .………………………………… 123 8.2. Recommendations for Future Work ..…………………………………… 125 LIST OF REFERENCES ..………………………………………………………… 129 APPENDIX A: NOMENCLATURE .……………..…………………………………135 APPENDIX B: LIST OF PUBLICATIONS ..……………………………………… 137 VITA .………………………………………………………………………………… 139 vi LIST OF TABLES 5.1. Cases considered for acoustic streaming analysis .………………………… 69 6.1. Summary of cases considered for acoustic streaming analysis ...………… 87 6.2. Characteristic results for the cases studied ...……………………………… 106 vii LIST OF FIGURES 1.1. Acoustic compressor (Macrosonix Corp., Richmond, VA) ...……………… 3 1.2. Thermoacoustic refrigerator (Purdue University prototype) ……………… 4 1.3. First order velocity profiles at four different times ωt = 0, π/4, 3π/4, π .… 13 1.4. Rayleigh streaming formed between two parallel plates [38] (Maximum streaming velocity: vx″ = 0.76 mm/s, vy″ = 12.8 mm/s) .………………… 14 1.5. Schlichting streaming formed on a flat plate [38] (Maximum streaming velocity: vx″ = 1.1 mm/s, vy″ = 0.01 mm/s) ………………………………… 15 1.6. Secondary flow in the neighborhood of a cylinder (Andrade [55]) ……… 19 2.1. Flow chart of the LCPFCT procedure ...…………………………………… 34 2.2. Ghost cell approach for a boundary at the cell interface ………………… 37 3.1. Geometry and the boundary conditions of the problem .………………… 40 3.2. Variation of pressure with time (experimental and numerical results) .… 41 3.3. Variation of (a) pressure (b) temperature along the horizontal mid-plane of the enclosure at four different times, (A = 1/3) .…………… 43 3.4. Velocity vectors in the enclosure at t = 0.25τc (A = 1/3) .………………… 45 3.5. Variation of density along the horizontal mid-plane of the enclosure at four different times, (A = 1/3) .…………………………………………… 45 3.6. Pressure variation at the midpoint of the enclosure with time, (A=1/3) ... 46 3.7. Frequency spectra of the pressure variations at the midpoint of the enclosure with time, (A=1/3) ………………………………………… 46 3.8. Variation of local heat flux with time at the mid-point of the left and right wall, (A = 1/3) ...………………………………………………………… 48 3.9. Variation of pressure along the horizontal mid-plane of the enclosure at four different times in case of symmetric heating (A = 1/3) ...………… 49 viii 3.10. Variation of pressure along the horizontal mid-plane of the enclosure at four different times in case of asymmetric heating/cooling (A = 1/3) ... 49 4.1. Geometry and the boundary conditions of the problem .………………… 52 4.2. Variation of pressure with time at the midpoint of the enclosure for two different overheat ratio .……………………………………………………… 53 4.3. Effect of the rapidity of the left wall heating process on the -6 midpoint pressure of the enclosure (τc = 36.82×10 s, A = 1/3) ...……… 55 4.4. Variation of the x- component of the velocity vector along the horizontal mid-plane of the enclosure at t = 0.025 ..……………………… 56 4.5. Variation of the y- component of the velocity vector along the horizontal mid-plane of the enclosure at t = 0.025s ……………………… 57 4.6. Variation of the velocity vectors and flow field depending on the rapidity of the wall heating process at t = 0.025 s. a) Impulsive heating, b) Gradual heating (τc), c) Gradual heating (5τc), d) Gradual heating (20τc) …………………………………………………… 60 4.7. Effect of thermoacoustic wave motion on a) midpoint pressure and b) Nusselt number at the right wall of the enclosure for two different gases ...………………………………………………………… 62 4.8. Distribution of the velocity vectors in the flow field for two different gases at t= 0.01 s. a) Nitrogen, b) Helium ………………………………… 63 5.1. Schematic of the problem ...………………………………………………… 66 5.2. The computational grid ……………………………………………………… 67 5.3. Variation of (a) pressure (b) u velocity along the symmetry axis of the enclosure at four different instant (ωt = 0, π/2, π, 3π/2) during the acoustic cycle starting at t = 0.350 s (Case-1) .…………………………… 70 5.4. Time evolution of right wall pressure (Case-1) …………………………… 72 5.5. Variation of oscillatory flow field in the enclosure at four different instant (ωt = 0, π/2, π, 3π/2) during the acoustic cycle starting at t = 0.350 s (Case-1) .………………………………………………………… 73 5.6. Streamlines in the enclosure based on the time averaged flow field for (Case-1) ...………………………………………………………………… 75 ix 5.7. Variation of (a) pressure (b) u velocity along the symmetry axis of the enclosure at four different instant (ωt = 0, π/2, π, 3π/2) during the acoustic cycle starting at t = 0.250 s (Case-2) ..……………………… 76 5.8. Variation of oscillatory flow field in the enclosure at four different instant (ωt = 0, π/2, π, 3π/2) during the acoustic cycle starting at t = 0.350 s (Case-2) .………………………………………………………… 78 5.9. Streamlines in the enclosure based on the time averaged flow field (Case-2) .……………………………………………………………………… 79 5.10. Streamlines in the enclosure based on the time averaged flow field (Case-3) .……………………………………………………………………… 80 5.11. Time evolution of the right wall pressure (Case-1, 2, and 3) .…………… 81 5.12. Frequency spectra of the pressure fluctuations at the right wall (Case-1, 2, and 3) …………………………………………………………… 83 6.1. Schematic of the problem ...………………………………………………… 85 6.2. Variation of (a) pressure (b) u- velocity along the symmetry axis of the enclosure at four different instant (ωt = 0, π/2, π, 3π/2) during the acoustic cycle # 100 (Case A) ……………………………………………… 88 6.3. Frequency spectra of the pressure fluctuations at the left wall (Case A) ..…………………………………………………………… 90 6.4. Flow field in the enclosure at t = 0.005 s (Case A) .……………………… 90 6.5. Variation of the streaming velocity along the semi-height of the enclosure (Case A), (a) u- component of the streaming velocity at x = 3L/4 and (b) v- component of the streaming velocity at x = L/2 .…… 92 6.6. Variation of the pressure along the symmetry axis of the enclosure at four