Blade row and blockage modelling in an axial compressor throughflow code

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(1)Blade Row and Blockage Modelling in an Axial Compressor Throughflow Code by. Keegan D. Thomas. Thesis presented at the University of Stellenbosch in partial fulfilment of the requirements for the degree of. Master of Science in Mechanical Engineering. Department of Mechanical Engineering University of Stellenbosch Private Bag X1, Matieland, 7602, South Africa. Study leaders: Prof T.W. von Backström Prof G. D. Thiart. April 2005.

(2) Copyright © 2005 University of Stellenbosch All rights reserved..

(3) Declaration I, the undersigned, hereby declare that the work contained in this thesis is my own original work and that I have not previously in its entirety or in part submitted it at any university for a degree.. Signature: . . . . . . . . . . . . . . . . . . . . . . . . . . . K.D. Thomas. Date: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ii.

(4) Abstract The objective of the thesis is to improve the performance prediction of axial compressors, using a streamline throughflow method (STFM) code by modelling the hub and casing wall boundary layers, and additional flow mechanisms that occur within a blade row passage. Blade row total pressure loss and deviation correlations are reviewed. The effect of Mach number and the blade tip clearance gap are also reviewed as additional loss sources. An entrainment integral method is introduced to model the hub and casing wall boundary layers. Various 1-dimensional test cases are performed before implementing the integral boundary layer method into the STFM. The boundary layers represent an area blockage throughout the compressor, similar to a displacement thickness, but affects two velocity components. This effectively reduces the compressor flow area by altering the hub and casing radial positions at all stations. The results from the final STFM code with the integral boundary layer model, Mach number model and tip clearance model is compared against high pressure ratio compressor test cases. The blockage results, individual blade row and overall performance results are compared with published data. The deviation angle curve fits developed by Roos and Aungier are compared. There is good agreement for all parameters, except for the slope of deviation angle with incidence angle for low solidity. For the three compressors modelled, there is good agreement between the blockage prediction obtained and the blockage prediction of Aungier. The NACA 5-stage transonic compressor overall performance shows good agreement at all speeds, except for 90% of design speed. The NACA 10-stage subsonic compressor shows good agreement for low and medium speeds, but needs improvement at 90% and 100% of design speeds. The NACA 8-stage. iii.

(5) ABSTRACT. transonic compressor results compared well only at low speeds.. iv.

(6) Uitreksel Die doel van die tesis is om die voorspelling van die werkverrigting van aksiale kompressors wat ’n stroomlyndeurvloeimetode (SDVM) gebruik te verbeter. Dit word deur naaf- en huls-wandgrenslae en addisionele vloei meganismes wat in die lemry deurgange voorkom, te modelleer. Die lemry totale druk verlies- en afwykhoek-korrelasies is hersien. Die effek van Mach-getal en lempuntspeling is ook hersien sowel as addisionele verliesmeganismes. ’n Meesleur-integraal-metode is toegepas om die naaf- en huls- wandgrenslae te modelleer. Verskillende een-dimensionele toetsgevalle is gedoen voor die integraal-grenslaagmetode in die SDVM geimplementeer is. Die grenslae verteenwoordig ’n oppervlakte blokkasie deur die kompressor, soortgelyk aan ’n verplasingsdikte, maar wat twee snelheidskomponente beïnvloed. Die effek daarvan is om die kompressor deurvloeiarea te verminder, deur die naaf en huls se radiale posisie by alle stasies te verander. Die resultate van die finale SDVM-kode, met die integraal-grenslaag model, Mach-getalmodel en lemrymodel is met hoë drukverhouding kompressor-toetsgevalle vergelyk. Die blokkasieresultate, individuele lemry en totale werkverrigtingresultate is vergelyk met gepubliseerde data. Die afwykhoek kromme-passings ontwikkel deur Roos en Aungier is vergelyk. Daar was goeie ooreenstemming vir al die parameters, behalwe vir die helling van die afwykingshoek teenoor invalshoek grafiek vir lae soliditeit. Vir die drie kompressors gemodelleer, is daar goeie ooreenstemming tussen die blokkasie voorspelling verkry en die blokkasie voorspelling van Aungier. Die NACA 5-stadium transoniese kompressor se totale werkverrigting toon goeie ooreenstemming by alle spoede, met die eksperimentele data behalwe vir 90% van ontwerp spoed. Die NACA 10-stadium subsoniese kompressor toon goeie ooreenstemming vir lae en medium spoede, maar benodig verbetering by. v.

(7) UITREKSEL. vi. 90% en 100% van ontwerp spoed. Die voorspelling vir die NACA 8-stadium transoniese kompressor resultate het net by lae spoede goed vergelyk..

(8) Acknowledgements I would like to thank my supervisors, Professors T. W. von Backström and G. D. Thiart for their support. I thank Mr. T. H. Roos for his insight and passion in the field and towards my thesis. I thank Mr. R. H. Aungier for his willingness to supply much needed compressor data. Thank you to Armscor for their financial support for this research. I would also like to thank my family and friends for their support and understanding.. vii.

(9) Dedication To those who have assisted me in finding the truth... ROMANS 10:9 That if you confess with your mouth the Lord Jesus and believe in your heart that God raised Him from death, you will be saved.. viii.

(10) Contents Declaration. ii. Abstract. iii. Uitreksel. v. Acknowledgements. vii. Dedication. viii. Contents. ix. List of Figures. xii. List of Tables. xv. Nomenclature. xvii. Abbreviations. xx. 1 Introduction. 1. 1.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.2. Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 1.3. Layout of this thesis . . . . . . . . . . . . . . . . . . . . . . . .. 3. 2 Literature Review. 5. 3 Review of Blade Row Models in the Streamline Throughflow Code. 15. 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15. 3.2. STFM equation . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16. ix.

(11) x. CONTENTS. 3.3. Blade row modelling . . . . . . . . . . . . . . . . . . . . . . . .. 17. 3.4. Conclusion. 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4 Boundary Layer Modelling. 22. 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. 4.2. Flow 2200 of Coles and Hirst . . . . . . . . . . . . . . . . . . .. 23. 4.3. Flow 2300 of Coles and Hirst . . . . . . . . . . . . . . . . . . .. 26. 4.4. Flow 4000 of Coles and Hirst . . . . . . . . . . . . . . . . . . .. 27. 4.5. Conclusion. 28. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5 Additional Models. 32. 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32. 5.2. Compressor blockage modelling . . . . . . . . . . . . . . . . . .. 32. 5.3. Mach number model . . . . . . . . . . . . . . . . . . . . . . . .. 34. 5.4. Tip clearance model . . . . . . . . . . . . . . . . . . . . . . . .. 36. 5.5. Loss smoothing model . . . . . . . . . . . . . . . . . . . . . . .. 36. 6 Compressor Test Cases. 37. 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37. 6.2. NACA 5-stage transonic compressor . . . . . . . . . . . . . . .. 38. 6.2.1. Description . . . . . . . . . . . . . . . . . . . . . . . . .. 38. 6.2.2. Experiments . . . . . . . . . . . . . . . . . . . . . . . .. 39. 6.2.3. Modelling . . . . . . . . . . . . . . . . . . . . . . . . . .. 40. 6.2.4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41. NACA 10-stage transonic compressor . . . . . . . . . . . . . . .. 44. 6.3.1. Description . . . . . . . . . . . . . . . . . . . . . . . . .. 44. 6.3.2. Experiments . . . . . . . . . . . . . . . . . . . . . . . .. 44. 6.3.3. Modelling . . . . . . . . . . . . . . . . . . . . . . . . . .. 45. 6.3.4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46. NACA 8-stage transonic compressor . . . . . . . . . . . . . . .. 47. 6.4.1. Description . . . . . . . . . . . . . . . . . . . . . . . . .. 47. 6.4.2. Experiments . . . . . . . . . . . . . . . . . . . . . . . .. 48. 6.4.3. Modelling . . . . . . . . . . . . . . . . . . . . . . . . . .. 49. 6.4.4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51. 6.3. 6.4. 6.5. 7 Conclusions. 53.

(12) CONTENTS. xi. 8 Recommendations For Further Study. 55. A Input File. 57. B Deviation And Loss Correlations. 61. B.1 Howell deviation correlation . . . . . . . . . . . . . . . . . . . .. 61. B.2 Lieblein deviation correlation . . . . . . . . . . . . . . . . . . .. 64. B.3 Howell loss correlation . . . . . . . . . . . . . . . . . . . . . . .. 69. B.4 Lieblein loss correlation . . . . . . . . . . . . . . . . . . . . . .. 70. C Aungier Loss Models. 72. C.1 Blockage model . . . . . . . . . . . . . . . . . . . . . . . . . . .. 72. C.2 Mach number model . . . . . . . . . . . . . . . . . . . . . . . .. 76. C.3 Tip clearance model . . . . . . . . . . . . . . . . . . . . . . . .. 79. D Polynomial Fits of Lieblein Correlation. 81. E NACA 5-stage Experimental Tables. 86. F Blockage Results. 109. G NACA 5-Stage Results. 119. H NACA 10-Stage Results. 149. I. 157. NACA 8-Stage Results. List of References. 164.

(13) List of Figures 2.1. Blade passage (Cumpsty, 1989) . . . . . . . . . . . . . . . . . . . .. 7. 2.2. Diagram of leakage jet (Khalsa, 1996) . . . . . . . . . . . . . . . .. 8. 2.3. Jet-freestream interaction control volume . . . . . . . . . . . . . .. 9. 3.1. Flow diagram of Roos code . . . . . . . . . . . . . . . . . . . . . .. 16. 3.2. Zero camber design incidence angle . . . . . . . . . . . . . . . . . .. 18. 3.3. Design incidence angle slope factor . . . . . . . . . . . . . . . . . .. 19. 3.4. Off-design deviation slope . . . . . . . . . . . . . . . . . . . . . . .. 19. 3.5. Comparison of Howell and Lieblein loss prediction for NACA 65 cascade data (Roos, 1999) . . . . . . . . . . . . . . . . . . . . . . .. 21. 4.1. Velocity profile vs x for flow 2200 of Coles and Hirst . . . . . . . .. 24. 4.2. Momentum thickness vs x for flow 2200 of Coles and Hirst . . . . .. 24. 4.3. Shape factor vs x for flow 2200 of Coles and Hirst. . . . . . . . . .. 25. 4.4. Displacement thickness vs x for flow 2200 of Coles and Hirst . . . .. 25. 4.5. Velocity profile vs x flow 2300 of Coles and Hirst . . . . . . . . . .. 26. 4.6. Momentum thickness vs x for flow 2300 of Coles and Hirst . . . . .. 27. 4.7. Shape factor vs x for flow 2300 of Coles and Hirst. . . . . . . . . .. 27. 4.8. Displacement thickness vs x for flow 2300 of Coles and Hirst . . . .. 28. 4.9. Velocity profile vs x flow 4000 of Coles and Hirst . . . . . . . . . .. 29. 4.10 Momentum thickness vs x for flow 4000 of Coles and Hirst . . . . .. 29. 4.11 Shape factor vs x for flow 4000 of Coles and Hirst. . . . . . . . . .. 30. 4.12 Displacement thickness vs x for flow 4000 of Coles and Hirst . . . .. 30. 5.1. Blockage model addition to Roos code . . . . . . . . . . . . . . . .. 33. 5.2. Additional loss models implemented in Roos code . . . . . . . . . .. 35. 6.1. NACA 5-stage compressor computational grid . . . . . . . . . . . .. 40. xii.

(14) LIST OF FIGURES. xiii. 6.2. NACA 5-stage compressor Aungier vs current blockage comparison. 42. 6.3. Rotor Total Pressure Ratios vs Radius (100% Speed) . . . . . . . .. 43. 6.4. NACA 5-stage compressor total pressure ratio vs flow rate . . . . .. 43. 6.5. NACA 10-stage compressor computational grid . . . . . . . . . . .. 45. 6.6. NACA 10-stage compressor Aungier vs current blockage comparison 46. 6.7. NACA 10-stage compressor total pressure ratio vs flow rate . . . .. 47. 6.8. NACA 8-stage compressor computational grid . . . . . . . . . . . .. 49. 6.9. NACA 8-stage compressor Aungier vs current blockage comparison. 50. 6.10 NACA 8-stage compressor total pressure ratio vs flow rate . . . . .. 51. B.1 Howell’s cascade nominal deflection correlation . . . . . . . . . . .. 62. B.2 Howell’s off-design cascade correlation . . . . . . . . . . . . . . . .. 63. D.1 Zero camber design incidence angle . . . . . . . . . . . . . . . . . .. 82. D.2 Design incidence angle slope factor . . . . . . . . . . . . . . . . . .. 82. D.3 Thickness correction factor . . . . . . . . . . . . . . . . . . . . . .. 83. D.4 Zero camber design deviation angle . . . . . . . . . . . . . . . . . .. 83. D.5 Design deviation angle slope factor . . . . . . . . . . . . . . . . . .. 84. D.6 Design deviation angle exponent . . . . . . . . . . . . . . . . . . .. 84. D.7 Thickness correction factor . . . . . . . . . . . . . . . . . . . . . .. 85. D.8 Off-design deviation slope . . . . . . . . . . . . . . . . . . . . . . .. 85. F.1 NACA 5-stage compressor blockage comparison for different speeds 110 F.2 NACA 5-stage compressor hub and casing blockage comparison . . 111 F.3 NACA 5-stage compressor Aungier vs current blockage comparison 112 F.4 NACA 10-stage compressor blockage comparison for different speeds 113 F.5 NACA 10-stage compressor hub and casing blockage comparison . 114 F.6 NACA 10-stage compressor Aungier vs current blockage comparison 115 F.7 NACA 8-stage compressor blockage comparison for different speeds 116 F.8 NACA 8-stage compressor hub and casing blockage comparison . . 117 F.9 NACA 8-stage compressor Aungier vs current blockage comparison 118 G.1 NACA 5-stage compressor diagram . . . . . . . . . . . . . . . . . . 120 G.2 Rotor total temperature ratios vs radius (70% speed) . . . . . . . . 122 G.3 Stator total temperature ratios vs radius (70% speed) . . . . . . . 123 G.4 Rotor total pressure ratios vs radius (70% speed) . . . . . . . . . . 124 G.5 Stator total pressure ratios vs radius (70% speed) . . . . . . . . . . 125 G.6 Rotor static pressure ratios vs radius (70% speed) . . . . . . . . . . 126.

(15) LIST OF FIGURES. xiv. G.7 Stator static pressure ratios vs radius (70% speed) . . . . . . . . . 127 G.8 Rotor total temperature ratios vs radius (80% speed) . . . . . . . . 128 G.9 Stator total temperature ratios vs radius (80% speed) . . . . . . . 129 G.10 Rotor total pressure ratios vs radius (80% speed) . . . . . . . . . . 130 G.11 Stator total pressure ratios vs radius (80% speed) . . . . . . . . . . 131 G.12 Rotor static pressure ratios vs radius (80% speed) . . . . . . . . . . 132 G.13 Stator static pressure ratios vs radius (80% speed) . . . . . . . . . 133 G.14 Rotor total temperature ratios vs radius (90% speed) . . . . . . . . 134 G.15 Stator total temperature ratios vs radius (90% speed) . . . . . . . 135 G.16 Rotor total pressure ratios vs radius (90% speed) . . . . . . . . . . 136 G.17 Stator total pressure ratios vs radius (90% speed) . . . . . . . . . . 137 G.18 Rotor static pressure ratios vs radius (90% speed) . . . . . . . . . . 138 G.19 Stator static pressure ratios vs radius (90% speed) . . . . . . . . . 139 G.20 Rotor total temperature ratios vs radius (100% speed) . . . . . . . 140 G.21 Stator total temperature ratios vs radius (100% speed) . . . . . . . 141 G.22 Rotor total pressure ratios vs radius (100% speed) . . . . . . . . . 142 G.23 Stator total pressure ratios vs radius (100% speed) . . . . . . . . . 143 G.24 Rotor static pressure ratios vs radius (100% speed) . . . . . . . . . 144 G.25 Stator static pressure ratios vs radius (100% speed) . . . . . . . . . 145 G.26 NACA 5-stage compressor total pressure ratio vs flow rate . . . . . 146 G.27 NACA 5-stage compressor total temperature ratio vs flow rate . . 147 G.28 NACA 5-stage compressor efficiency vs flow rate . . . . . . . . . . 148 H.1 NACA 10-stage compressor diagram . . . . . . . . . . . . . . . . . 150 H.2 NACA 10-stage compressor total pressure ratio vs flow rate . . . . 154 H.3 NACA 10-stage compressor total temperature ratio vs flow rate . . 155 H.4 NACA 10-stage compressor efficiency vs flow rate . . . . . . . . . . 156 I.1. NACA 8-stage compressor diagram . . . . . . . . . . . . . . . . . . 158. I.2. NACA 8-stage compressor total pressure ratio vs flow rate . . . . . 161. I.3. NACA 8-stage compressor total temperature ratio vs flow rate . . 162. I.4. NACA 8-stage compressor efficiency vs flow rate . . . . . . . . . . 163.

(16) List of Tables A.1 5-stage input file example . . . . . . . . . . . . . . . . . . . . . . .. 59. B.1 Slope factor n constants . . . . . . . . . . . . . . . . . . . . . . . .. 66. B.2 Blade section shape correction factor for deviation . . . . . . . . .. 67. B.3 Blade section shape correction factor for incidence . . . . . . . . .. 68. E.1 70% speed table . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87. E.2 70% speed table (cont.) . . . . . . . . . . . . . . . . . . . . . . . .. 88. E.3 70% speed table . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 89. E.4 70% speed table . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 90. E.5 80% speed table . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 91. E.6 80% speed table (cont.) . . . . . . . . . . . . . . . . . . . . . . . .. 92. E.7 80% speed table . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 93. E.8 80% speed table (cont.) . . . . . . . . . . . . . . . . . . . . . . . .. 94. E.9 80% speed table . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95. E.10 80% speed table (cont.) . . . . . . . . . . . . . . . . . . . . . . . .. 96. E.11 90% speed table . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97. E.12 90% speed table (cont.) . . . . . . . . . . . . . . . . . . . . . . . .. 98. E.13 90% speed table . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99. E.14 90% speed table (cont.) . . . . . . . . . . . . . . . . . . . . . . . . 100 E.15 90% speed table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 E.16 90% speed table (cont.) . . . . . . . . . . . . . . . . . . . . . . . . 102 E.17 100% speed table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 E.18 100% speed table (cont.) . . . . . . . . . . . . . . . . . . . . . . . . 104 E.19 100% speed table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 E.20 100% speed table (cont.) . . . . . . . . . . . . . . . . . . . . . . . . 106 E.21 100% speed table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107. xv.

(17) LIST OF TABLES. xvi. E.22 100% speed table (cont.) . . . . . . . . . . . . . . . . . . . . . . . . 108 G.1 5-stage rotor blade table . . . . . . . . . . . . . . . . . . . . . . . . 121 G.2 5-stage stator blade table . . . . . . . . . . . . . . . . . . . . . . . 121 H.1 10-stage rotor blade table . . . . . . . . . . . . . . . . . . . . . . . 151 H.2 10-stage stator blade table . . . . . . . . . . . . . . . . . . . . . . . 152 H.3 10-stage exit guide vane table . . . . . . . . . . . . . . . . . . . . . 153 I.1. 8-stage rotor blade table . . . . . . . . . . . . . . . . . . . . . . . . 159. I.2. 8-stage stator blade table . . . . . . . . . . . . . . . . . . . . . . . 160. I.3. 8-stage exit guide vane table . . . . . . . . . . . . . . . . . . . . . . 160.

(18) Nomenclature a. position of maximum thickness. B. fractional area blockage factor. c. chord length. CD. drag coefficient. CDa. annulus drag. CDp. profile drag. CDs. secondary loss drag. Cf. skin friction coefficient. CL. lift coefficient. Cp. specific heat. Deq. equivalent diffusion ratio. E. entrainment function. f. blade force. g. gravitational constant. h. blade height/enthalpy. H. boundary layer streamwise shape factor. Hi. entrainment shape factor. H1. boundary layer meridional shape factor. H2. boundary layer tangential shape factor. i. incidence angle. K. velocity profile parameter. Ksh. blade shape parameter. m. tangential velocity profile exponent. m ˙. mass flow rate. Ma. Mach number. n. meridional velocity profile exponent. n+. normalized distance from wall. xvii.

(19) NOMENCLATURE. N. rpm/no. of blades. Nrow blade row number P. pressure. r. radial coordinate. R. gas constant. R0. blade defect force. Re. Reynolds number. s. pitch/entropy. t. maximum thickness. T. temperature. V. velocity. x. axial coordinate. y. distance normal to wall. Greek symbols α. absolute flow angle. β. relative flow angle/wake depth parameter 0. absolute blade angle. 0. relative blade angle. α β δ. deviation angle/boundary layer displacement thickness. δc. tip clearance. δ∗. boundary layer streamwise displacement thickness. δ1∗. boundary layer meridional displacement thickness. δ2∗. boundary layer tangential displacement thickness. ∆. differential. ². deflection angle. γ. blade stagger angle. η. efficiency. θ. camber angle/momentum thickness. µ. fluid viscosity. ν. kinematic viscosity/blade force defect thickness. ξ. vorticity. ρ. density. σ. solidity. τ. tip clearance. xviii.

(20) NOMENCLATURE. ω. loss coefficient. φ. stagger angle parameter. ψ. stream function. Subscripts 0. zero camber. 1. inlet/meridional. 2. outlet/tangential. 10. 10% thickness. ∗. sonic flow condition. ∞. free stream. δ. deviation. θ. tangential direction. abs. absolute. c. negative stall angle/corrected. e. edge of boundary layer. FS. freestream. i. incidence. jet. leakage jet. m. meridional/minimum loss/mean. max. maximum. o. stagnation property. ps. pressure surface. r. radial direction. ref. reference. rel. relative. s. positive stall angle. sh. shape. ss. suction surface. sw. streamwise. t. thickness. x. axial direction. w. wall. xix.

(21) Abbreviations AWBL. annulus wall boundary layer. CFD. computational fluid dynamics. DCA. double circular arc. EGV. exit guide vanes. EWBL. endwall boundary layer. IGV. inlet guide vanes. MTFM. matrix throughflow method. NACA. National Advisory Committee for Aeronautics. NASA. National Aeronautics and Space Administration. RM. research memorandum. SCM. streamline curvature method. STFM. streamline throughflow method. SUCC. Stellenbosch university compressor code. xx.

(22) Chapter 1. Introduction 1.1. Background. Although there have been significant developments in computational fluid dynamic (CFD) methods in the recent past, throughflow methods are still required for the design and modelling of axial compressors. CFD methods are fully three-dimensional, needing extensive computer memory and processing power. Throughflow methods are two-dimensional inviscid methods that solve axisymmetric flow in the axial-radial meridional plane. Their memory requirements and complexity are significantly lower than those of CFD methods. Throughflow codes model blade rows using empirical cascade correlations for total pressure loss within a passage, and the deflection of the flow at the trailing edge of a blade row. When applied to axial compressors, these methods generate compressor maps that display the performance of the compressor for different operating conditions and speeds. They predict the velocity, temperature and pressure profiles in a compressor for specified initial and operating conditions. Throughflow methods, when used as an application tool, can generate the performance maps of a compressor. Three methods are used to predict the performance of turbomachines: the streamline curvature method (SCM) (Novak, 1967), the matrix throughflow method (MTFM) (Marsh, 1968) and the streamline throughflow method (STFM) (Von Backström and Roos, 1993; Roos, 1995). Davis & Millar (1975) compare the SCM and MTFM, highlighting the advantages and disadvantages of each. Gannon & Von Backström (2000) also. 1.

(23) CHAPTER 1. INTRODUCTION. 2. give a comparison of the two methods. The SCM has the advantage of simulating individual streamlines making it easier to program because properties are conserved along streamlines. However, the SCM has a slower convergence time compared with the MTFM. The MTFM uses a fixed geometrical grid. The condition of properties being conserved along a streamline cannot be applied. The stream function value has to be interpolated between the geometric grid points. The MTFM is also more stable than the SCM (Davis and Millar, 1975). The STFM is a combination of the SCM and the MTFM. It combines the advantages of conservation of properties along streamlines in the SCM and the stability of the MTFM, and was developed using a transformation proposed by Boadway (1976). Roos (1995) applied the transformation of Boadway to the MTFM equation to develop the STFM, resembling the SCM with the stream function being the dependent variable, but having a formulation similar to the MTFM. Two forms of the STFM equation have been developed locally. Roos (1995) developed an incompressible code using an extended version of the STFM equation. This code was limited to prediction of low-pressure ratio compressors. Gannon (1996) then developed a STFM code using the compact version of the throughflow equation. It was a compressible flow formulation that could simulate low-pressure ratio compressors accurately, but it was not applied to high-pressure ratio compressors. Roos (1999) reported that the compressible flow STFM over-predicted the overall pressure ratio of a high-pressure ratio compressor illustrating a deficiency in the prediction method. The method did not include accurate loss and deviation correlations, and it did not model some significant losses, such as shock and boundary layer losses, relevant in high-pressure ratio compressors. Roos (1999) then developed a compressible flow STFM using the loss and deviation correlation of Lieblein (1960) which predicts the deviation more accurately, but still with a crude prescribed boundary layer blockage model. The boundary layer may be simulated as a blockage factor in throughflow codes and is given as an input to the current Roos code. The estimate of Cumpsty (1989) of the boundary layer growth is used in the local codes. Cumpsty suggested a blockage factor to model the growth of a boundary layer through a compressor. The blockage factor alters the geometry of the compressor by changing the hub and casing radii. The altered compressor geometry simulates the effective loss by the endwall boundary layer..

(24) CHAPTER 1. INTRODUCTION. 1.2. 3. Objectives. This thesis develops the Roos (1999) code using the expanded STFM equation. The throughflow code attempts to predict the performance of high-pressure ratio compressors accurately. The main objective of this thesis is to predict the blockage of axial compressors using a boundary layer method. A secondary objective is to improve the performance prediction of high-speed axial compressors after reviewing the loss mechanisms within a blade passage. A list of tasks to be performed follows. • Reviewing and comparing different deviation and pressure loss correlations. • Introducing endwall boundary layer models for the compressor hub and casing walls. • Verifying and using an accurate blade row correlation and pressure loss correlation for performance prediction. • Using a throughflow method, with additional models, to improve the prediction of blade row and compressor performance. • Tabulating individual blade row and compressor performance data. • Comparing the existing prediction method with the improved method. • Comparing the effect of different loss models on performance prediction. • Comparing numerical blade row performance and overall performance results with experimental results.. 1.3. Layout of this thesis. This thesis consists of the following sections. The blockage models researched for the baseline Roos code are summarised in Chapter 2. A description is given of the complex flow within a blade passage, with each process being explained briefly. Blockage methods and additional loss mechanisms are discussed and an explanation is given for the methods used to help improve the performance prediction modelling..

(25) CHAPTER 1. INTRODUCTION. 4. The compressible flow STFM developed by Roos is the baseline code used for improving the modelling of axial compressors. The basic code, its structure and the models already available in this code are explained in Chapter 3. A boundary layer method is reviewed with 1-dimensional test cases to see how well the method can predict the flow properties such as displacement thickness and shape factor. This is described in Chapter 4. The improvements to the code and the reasons for their use are explained in Chapter 5. The performance of three axial compressors is simulated for the verification of the code. A brief description of each of the compressors is given in Chapter 6. A process for modelling each compressor numerically is also described. The results of the predictions are compared with experimental results. A summary of the results obtained from the research is given in Chapter 7. The extent to which the objectives of the thesis have been met and how the modelling of axial compressors has been improved are also discussed. Recommendations as to how the numerical modelling can further be improved, are made in Chapter 8..

(26) Chapter 2. Literature Review Locally two forms of the throughflow equation have been programmed. There is a compact and an expanded form of the throughflow equation. Roos (1995) uses the expanded form of the equation, equation 2.0.2, for his code, verifying and comparing numerical and experimental results of the low-pressure ratio 3-stage Rofanco compressor. Gannon (1996) used the compact form, equation 2.0.1. His research on the low-pressure ratio Cheetah compressor, which has ten stages, showed that compressors with so many stages could be modelled accurately. The expanded form of the throughflow equation only becomes inaccurate for compressors with hub-tip ratios less than 0.4. The compressors tested for this thesis all have hub-tip ratios greater than 0.4. Essentially the expanded form of the equation, equation 2.0.2, will give as accurate results as the compact form, equation 2.0.1, for these compressors. After the code of Gannon was found by Roos (1999) to be inaccurate for high pressure ratio compressors by , Roos added compressibility to his code and used more accurate blade row modelling. This method and these models form the basic code for this thesis.. ∂r ∂r ∂ 2 r − 2 ∂ψ ∂z ∂ψ∂z. Ã 1+. ½. ∂r ∂ψ. ¾2 !. 1 ∂ ρr ∂ψ. µ 2. = (ρr). ∂r ∂ψ. µ ¶ µ ¶2 µ ¶ ∂r ∂r ∂ 1 ∂r ρr − ρr ∂ψ ∂ψ ∂z ρr ∂z. ¶3 ·. 5. ∂ho ∂s Vθ ∂(rVθ ) −T − ∂ψ ∂ψ r ∂ψ. ¸ (2.0.1).

(27) 6. CHAPTER 2. LITERATURE REVIEW. ∂r ∂r ∂ 2 r 2 − ∂ψ ∂z ∂ψ∂z µ 2. (ρr). ∂r ∂ψ. ¶3 ·. Ã 1+. ½. ∂r ∂z. ¾2 !. ∂2r − ∂ψ 2. µ. ∂r ∂ψ. ¶2. ∂2r 1 − ∂z 2 r. µ. ∂r ∂ψ. ¶2 =. " ½ ¾ ( ¸ µ ¶2 )# ∂ho ∂s Vθ ∂(rVθ ) 1 ∂r ∂ρ 2 ∂ρ ∂ρ ∂r ∂r −T − − − 1+ ∂ψ ∂ψ r ∂ψ ρ ∂z ∂ψ ∂z ∂ψ ∂ψ ∂z (2.0.2). To complete the code, more of the flow mechanisms that occur in a highspeed compressor needed to be modelled. Various models used in the code also needed to be improved. The boundary layer blockage model of Cumpsty (1989) is conservative, and the blade row correlation of Howell (1942), used to calculate the deviation and pressure loss, could be replaced with a more accurate model. Development of a compressible flow code with improved flow modelling ensued. The flow through blade rows is difficult to predict in compressors. A theoretical method based on two-dimensional cascades, can be used to predict the flow, but is unable to predict all losses that occur in the passage satisfactorily, especially at far off-design conditions. Empirical correlations have been developed based on the experimental data of blade effects and from cascade tests of standard blades. These correlations use common parameters to calculate losses associated with a certain flow process that occurs in the blade passage. Correlations have been developed for deviation angle, total pressure loss, tip clearance loss and shock loss, to name a few. Howell (1942) and Lieblein (1960) both developed their own blade row total pressure loss and deviation models that are used in the Roos code. The deviation angle is the difference in angle between the flow angle and the blade angle at the trailing edge of the blade. The deviation helps show the amount of turning imparted by a blade row and therefore the amount of energy being added to the flow in a rotor. Figure 2.1 (Cumpsty, 1989) shows that various processes within a blade passage influence the flow. There is a tip clearance vortex that develops because of the gap between the blade tip and the annulus wall. This vortex propagates through the blade passage and also affects the primary passage flow. Another phenomenon that is formed as it propagates downstream is a.

(28) CHAPTER 2. LITERATURE REVIEW. 7. wall boundary layer, not shown in the diagram. The boundary layer creates a displacement thickness on the hub and casing walls that alters the throughflow velocity distribution of the flow. Most compressors have a tip clearance at the outer annulus wall in their rotor blade rows. A ’scraping’ vortex, which is a pile-up of boundary layer fluid at the tip, is created by the incoming flow collecting near the leading edge of the pressure surface at the tip (Cumpsty, 1989). On the suction side, a clearance vortex is created by the flow escaping through the tip clearance. These vortices contribute to the loss in performance experienced in the compressor. Only the tip clearance vortex is modelled. There is a similar phenomenon at the stator hub.. Figure 2.1: Blade passage (Cumpsty, 1989). These processes need to be simulated for an accurate prediction method to be developed. The prediction of the endwall boundary layer is the main topic of this thesis. Three methods have been researched; Khalid et al. (1999), Dunham (1995) and Aungier (2003b). Blockage refers to a reduction in flow area due to local velocity defects. From their research on a low-speed compressor, Navier-Stokes computations and wind tunnel tests, Khalid et al. (1999) were able to identify consistent trends of blockage variation with a defined loading parameter. They then developed a correlation that relates the blockage to the flow through the tip clearance area. This method illustrates the role of tip clearance in blockage growth and the trends observed from experiments. For this analysis, it is.

(29) 8. CHAPTER 2. LITERATURE REVIEW. assuming that the blockage at the exit of a blade row is taken as the summation of all velocity defect regions along the blade chord. Also it is assumed that the four processes occur sequentially for tip clearance blockage. A leakage jet, figure 2.2 (Khalsa, 1996), is a jet of fluid that passes from the pressure surface of a blade to the suction surface through the tip region, and then flows into the adjacent blade passage. It is said to be pressure driven and not viscosity driven.. Figure 2.2: Diagram of leakage jet (Khalsa, 1996). A leakage jet velocity, Vjet , across the blade is calculated (Khalsa, 1996) using Bernoulli’s equation. Storer & Cumpsty (1991) assume that the streamwise component of the fluid entering the clearance gap, Vsw , can be approximated by also applying Bernoulli’s equation and assuming that the total pressure is the same as the freestream total pressure, and the appropriate static pressure is the blade pressure surface pressure just inboard of the tip. From the two equations for leakage and freestream velocity, an equation for the leakage angle can then be calculated. tan α =. Pps − Pss Po − Pps. (2.0.3). This leakage phenomenon exists all along the blade. A mass-averaged leakage angle is then calculated across the blade chord. The leakage fluid that exits the blade tip and propagates into the adjacent blade passage at the leakage angle, is the vortex jet. As the vortex jet prop-.

(30) CHAPTER 2. LITERATURE REVIEW. 9. agates further into the blade passage, a shear layer between the jet and the free stream develops. The growth of the shear layer is calculated using the method of Schlichting (1960). The distance between the blade surface and the interface region is defined as the mixing length. A zero pressure gradient is assumed between the clearance exit and the jet-free stream interaction region. Each section of the jet interacts with the free stream resulting in flow in the vortex direction. A control volume analysis, shown in figure 2.3 (Khalsa, 1996), allows the calculation of the angle of the vortex direction, and the width and depth of the velocity defect. Loss is generated at the jet-free stream interface by the dissipation of the kinetic energy of the velocity components normal to the vortex direction.. Figure 2.3: Jet-freestream interaction control volume. The angle of the interaction zone, between the free stream and jet, and the blade stagger angle is predicted using the theory of Martinez-Sanchez & Gauthier (Khalsa, 1996). The theory predicts the location of the jet-free stream interaction zone for a situation where the total pressures of the jet and free stream are unequal. The defect formed at the jet-free stream interaction region is assumed to behave in a manner similar to the two-dimensional wakes described by Hill et al. (1963). Calculation of the mixing of a velocity defect in a pressure gradient is performed and a cosine shaped radial profile presumed. The wake depth.

(31) CHAPTER 2. LITERATURE REVIEW. 10. and height parameters are found by simultaneously solving the conservation of mass and conservation of vortex-trajectory direction momentum equations. The initial conditions for the wake are set by the total pressure loss incurred by the mixing of the leakage flow with the free stream. Once the wake parameters have been found, the blockage can be calculated. Dunham (1995) uses a different approach to model the flow in the endwall region. Dunham (1993) describes four processes that occur in a blade passage and develops a separate model for each process. The four models are the annulus wall boundary layer (AWBL) model, the secondary flow model, tip clearance model and the spanwise mixing model. The complex three-dimensional boundary layer that develops along the hub and casing walls is modelled as a circumferentially averaged boundary layer in throughflow codes (Dunham, 1995). This is similar to a displacement thickness, but calculated for two-dimensional flow. The AWBL model used by Dunham to model the endwall boundary layer is that of Hirsch & de Ruyck (1981), which is an integral boundary layer method. Integral methods are ordinary differential equations, averaged across the boundary layer and are much easier to program than differential methods, but still give accurate results (White, 1991). The axial and tangential momentum equations, and the entrainment equation need to be integrated from one axial calculation plane to the next, and are given in equations 2.0.4 to 2.0.6. The entrainment method of Head (Cebeci and Cousteix, 1999) is used by Dunham to calculate the boundary layer thickness. Initial estimates are required of the various defect thicknesses associated with each of the equations. Equations 2.0.4 to 2.0.6 are integrated stepwise by the Euler-Cauchy method..

(32) 11. CHAPTER 2. LITERATURE REVIEW. d dVxe 2 2 (ρrVxe θxx ) + ρrVxe δx∗ − ρVxe sin λ(δθ∗ tan α + θθθ ) = rτx + rRx0 dx dx (2.0.4) d dVθe 2 2 (ρrVxe θθx ) + ρrVxe δx∗ + ρVxe sin λ(δx∗ tan α + θθx ) = rτθ + rRθ0 dx dx (2.0.5) d [ρrVxe (δ − δx∗ )] = ρrVxe sec αE dx (2.0.6) The definitions of the various boundary layer thicknesses are given in equations 2.0.7 to 2.0.11. Z ρe Vxe δ1∗. =. 2 ρe Vxe θ11 =. (ρe Vxe − ρVx )dy. (2.0.7). ρVx (Vxe − Vx )dy. (2.0.8). ρVx (Vθe − Vθ )dy. (2.0.9). (ρe Vθe − ρVθ )dy. (2.0.10). ρVθ (Vθe − Vθ )dy. (2.0.11). Z Z. ρe Vxe Vθe θ12 = Z ρe Vθe δ2∗ = 2 ρe Vθe θ22 =. Z. Dunham assumes profiles for the axial and tangential velocity given by equations 2.0.12 and 2.0.13. b, B and N are parameters that describe the shape of the boundary layer profile, and need to be calculated. ½ ¾ ³ n Ń Vx = 1−b 1− [1 − e−Kn+ ] Vxe δ ½ ¾ ³ n Ń Vθ = tan α − B 1 − [1 − e−Kn+ ] Vxe δ. (2.0.12) (2.0.13). The density ratio is approximated using the expressions that follow..

(33) 12. CHAPTER 2. LITERATURE REVIEW. µ ¶ µµ ¶¶2 ρ Vx Vx = 1 − 2a 1 − +a 1 + 4a)(1 − ρe Vxe Vxe µ ¶ γ−1 a = 0.89 M a2 2. (2.0.14) (2.0.15). The Head entrainment method (Head, 1958) uses a known velocity distribution to obtain the boundary layer development on a two-dimensional body. Entrainment is the process of turbulent mixing between two flow regimes. The neighbouring flow to a turbulent flow region acquires the general motion of the turbulent flow. Entrainment is the controlling factor in the development of the turbulent boundary layer. Head uses several sets of experimental data to develop a correlation between the entrainment, E, and the shape factor, H. A new shape factor, Hi , is defined, called the entrainment shape factor, dependent on the H value. The skin friction equation of Ludwieg and Tillmann (White, 1991) is used to calculate the shear stress components. Algebraic expressions for the defect forces, Rx0 and Rθ0 are used with empirical constants fitting available experimental data. The secondary flow model calculates the secondary flow field induced by the streamwise vorticity. A circulation develops in the throughflow direction within a blade passage due to the difference in pressure on the pressure and suction surfaces of two adjacent blades. There is also a vorticity that develops at the trailing edge of a blade. Vortices induce velocity components and alter the flow angles. Therefore there is need to apply such a model to predict the overturning that occurs midstream and the underturning close to the wall but outside the boundary layer. The first step is to calculate the streamwise vorticity. Marsh (1968) derives an equation for the passage secondary vorticity and Came & Marsh (1974) derive equations for the trailing vorticity. From the sum of these equations, equation 2.0.16 is derived. The secondary flow field can now be calculated from the streamwise vorticity distribution. ξsw = −vx2 sec α2. dα2 dr. (2.0.16). A bound vortex at the tip of each blade is shed into the blade passage. This vortex is modelled at the tip clearance radius only and contributes to the streamwise vorticity induced by secondary flows..

(34) CHAPTER 2. LITERATURE REVIEW. 13. The spanwise mixing model solves the viscous form of the streamline curvature equations to enable spanwise mixing effects due to turbulent diffusion (Dunham, 1993). Aungier uses an approach related to that of Dunham, with separate models to simulate the endwall boundary layer and tip clearance. The boundary layer model is similar to Dunham’s. However, Aungier’s approach differs from Dunham’s approach when calculating the momentum and displacement thicknesses for the conservation equations. Aungier uses the following power-law velocity profile assumptions and also assumes a constant density. ³ y ń Vm = Vme δ ³ y ´m Vθ = Vθe δ. (2.0.17) (2.0.18). This gives the following relationships between the displacement thickness, shape factor and velocity power exponents; these equations are somewhat less complicated to program than Dunham’s.. n=. θ11 δ − δ1∗ − 2θ11. (2.0.19). m=. θ12 (n + 1)2 δ − θ12 (n + 1). (2.0.20). H1 =. δ1∗ = 2n + 1 θ11. (2.0.21). 2H1 θ11 H1 − 1. (2.0.22). δ − δ1∗ =. δ1∗ n = δ n+1 H2 =. δ2∗ = 2m + 1 θ22. δ2∗ m = δ m+1. (2.0.23) (2.0.24) (2.0.25). Both Dunham and Aungier’s method use the Ludwieg-Tillmann skin fric-.

(35) CHAPTER 2. LITERATURE REVIEW. 14. tion equation to calculate the shear stress and blade force. However, for the blade force component calculations Dunham uses constants based on specific experimental data. In the paper by Dunham (1993), only low-speed compressor test cases have been done. He suggests further refinement of these empirical constants for high-speed compressors. Aungier (2003b) states there are contradictory opinions on the calculation of the blade force defect. However, Aungier developed a method to calculate the blade force defect where it is continually updated as the governing equations are iterated. This seems to be more acceptable than Dunham’s force calculations. In this chapter the methods of Khalid et. al., Dunham and Aungier were discussed. The boundary layer model used for this thesis is the method developed by Aungier (2003b). Firstly, a momentum integral method was thought to be more accurate than a correlation. This left a choice between Dunham’s method or Aungier’s method. The Aungier method was chosen over the Dunham method because the calculation of the boundary layer thicknesses would be easier to program. The programming is simplified by not needing to integrate the displacement thicknesses equations. A momentum integral method is also simpler to program than a finite difference method and still gives accurate results for the boundary layer parameters. A 1-dimensional integral method was researched before implementing the Aungier method..

(36) Chapter 3. Review of Blade Row Models in the Streamline Throughflow Code 3.1. Introduction. The streamline throughflow method (STFM) code used was programmed by Roos in Fortran77. Before the code could be used for prediction purposes, it was converted to Fortran95. There is a significant difference in the syntax between the two versions of Fortran. After the code was converted to Fortran95, a flow diagram of the program was created, showing each of the modules used in the code. The main program in which all modules are called is named "Program Testing". This is shown in figure 3.1. Each block represents a module, which performs a specific function for the code when called. The first module, "read_name", identifies the name of the input file which needs to be read. The "read_input_file" module then loads the input file parameters to the allocated memory. The input file contains the initial and operating conditions, and the compressor and blade row geometries. The layout of the input file parameters is given in appendix A. The blade rows are then indexed to identify their position relative to the compressor axial stations in the "create_blade_index" module. Also given in the input file is an area blockage factor for each axial station. The "apply_blockage" module applies the blockage factor to the compressor hub and casing equally, essentially changing the hub and casing radial posi15.

(37) CHAPTER 3. REVIEW OF BLADE ROW MODELS IN THE STREAMLINE THROUGHFLOW CODE 16. tions. The streamline positions and the axial-radial grid are then created by the "re_init_streamlines" module. Stream function values increase from 0.0 at the hub to 1.0 at the casing, in constant increments. The initial conditions are then applied across the entire grid as an input for the first iteration in the "initialise_cz_cr_m_T" module. All the blade row correlations and performance evaluation are calculated in the "calculate_everything" module. The blade row correlations used are described in section 3.3. The "sweep_2r" module is essentially the solution of the STFM equation, which will be explained in section 3.2. Finally all relevant performance variables are exported to a file by the "output" module.. Figure 3.1: Flow diagram of Roos code. 3.2. STFM equation. The two throughflow methods commonly used for modelling axial compressors are the streamline curvature method (SCM) and matrix throughflow method (MTFM). Oates (1988) derived the MTFM equation, equation 3.2.1, used for turbomachines. This equation is used to develop the streamline throughflow.

(38) CHAPTER 3. REVIEW OF BLADE ROW MODELS IN THE STREAMLINE THROUGHFLOW CODE 17. method (STFM). To simplify the derivation of the STFM the following assumptions are made: • An inviscid fluid has negligible viscosity • All gradients in tangential direction are zero • Blade rows are modelled as actuator discs. · ¸ · ¸ · ¸ ∂ ρo ∂ψ ∂ ρo ∂ψ ρ ∂ho ∂s Vθ ∂(rVθ ) + = r −T − ∂z ρr ∂z ∂r ρr ∂r ρo ∂ψ ∂ψ r ∂ψ. (3.2.1). After Boadway’s transformation has been applied to cylindrical coordinates, the expanded form of the STFM equation is obtained, equation 2.0.2. For this method a linear stream function distribution is used and the radial positions of the streamlines are calculated through the compressor. Equation 2.0.2 is discretised using the method of Greyvenstein (1981) for an orthogonal grid and then solved numerically.. 3.3. Blade row modelling. The blade row correlations of Howell (1942) and Lieblein (1960) presented in appendix B and that are used to calculate deviation and total pressure loss, are two options available in the Roos code. The Howell and Lieblein deviation correlations are based on experimental cascade data. Equations were developed by both authors to calculate design reference conditions and then to account for off-design conditions to calculate the deviation angle using linear slope factors. These two models are explained in appendix B. The Lieblein correlation is based on the NACA65 cascade data. The data is presented graphically in Johnson (1965) and is a function of the blade solidity, maximum thickness to chord ratio and inlet flow angle. Both Roos (1999) and Aungier (2003b) developed equations to fit the graphs of Lieblein. This was done to check if the Roos curve fits are acceptable for use in the STFM code when compared with the Aungier curve fits. Figures 3.2 to 3.4 show the comparison between the Roos and Aungier curve fits and the experimental data..

(39) CHAPTER 3. REVIEW OF BLADE ROW MODELS IN THE STREAMLINE THROUGHFLOW CODE 18. Figure 3.2: Zero camber design incidence angle. The only major discrepancies between the two are for the design incidence slope factor for low solidities and inflow angles, figure 3.3, and the slope of deviation angle versus incidence angle at low solidities, figure 3.4. A comparison for all parameters is given in appendix D showing the experimental data and the curve fits of both authors. A third total pressure loss option was developed by Roos using the Howell and Lieblein correlations. A loss model able to predict both positive and negative stall is required (Roos, 2001). Figure 3.5 shows the loss calculated for a NACA-65 series cascade for different stagger angles ranging from 20 to 55 degrees. The blade series has a camber of 20 degrees and a solidity equal to 1.5. The different predictions of Howell and Lieblein can clearly be seen (Roos, 2001). Lieblein’s correlation does not predict negative stall at all, with Howell’s correlation not capturing positive stall for low stagger angles. In the low loss region Howell’s correlation is less accurate than Lieblein’s. The developed correlation uses a combination of weighting factors to select the amount of loss applied by each correlation. Each weighting factor ranges from 0 to 1. It is based on the equivalent diffusion factor and incidence angle. The first weighting factor is calculated using the equivalent diffusion factor. The diffusion factor is calculated by the Lieblein correlation, appendix B.4. Limits are put on the value of the diffusion factor from which the first weighting.

(40) CHAPTER 3. REVIEW OF BLADE ROW MODELS IN THE STREAMLINE THROUGHFLOW CODE 19. Figure 3.3: Design incidence angle slope factor. Figure 3.4: Off-design deviation slope. factor is calculated. If the diffusion factor is within these limits, equation 3.3.3 is used. When the diffusion factor is in the low loss region, Deq < 1.8, emphasis is put on the Lieblein correlation, and when it is near stall, Deq > 2.0, the emphasis is towards Howell’s correlation (Roos, 2001)..

(41) CHAPTER 3. REVIEW OF BLADE ROW MODELS IN THE STREAMLINE THROUGHFLOW CODE 20. Deq > 2.0 =⇒ WF1 = 1. (3.3.1). Deq < 1.8 =⇒ WF1 = 0. (3.3.2). 1.8 < Deq < 2.0 =⇒ WF1 =. Deq − 1.8 0.2. (3.3.3). A similar approach is taken with the second weighting factor. The Lieblein correlation is preferred in the low loss region with Howell’s correlation emphasized at positive and negative stall. It is calculated based on the incidence angle. The weighted factor depends on the incidence being either negative or positive. If the incidence is negative, equation 3.3.4 is used. If the incidence is positive, equation 3.3.5 is used. If WF2 is less than 0, it is limited to 0, and if it is greater than 1 it is limited to 1.. i > 0 =⇒ WF2 =. |i| − 0.1 0.5. (3.3.4). i < 0 =⇒ WF2 =. |i| − 0.25 0.55. (3.3.5). The final weighting factor uses the other two weighting factors to calculate the weighting for the amount of loss that will be calculated from the Lieblein correlation and the Howell correlation, equations 3.3.6 and 3.3.7. The "blend_corr" subroutine is a model that can capture both negative and positive stall.. WF3 = 1 − (1 − WF1)(1 − WF2) CD = (1 − WF3) CD,Lieblein + WF3 CD,Howell. 3.4. (3.3.6) (3.3.7). Conclusion. In this chapter the STFM equation used to calculate the radial positions of the streamlines was presented. The blade row models of Howell and Lieblein that calculate the deviation and total pressure loss wihin a blade passage were reviewed. The Lieblein curve fits were compared with the NACA 65 cascade.

(42) CHAPTER 3. REVIEW OF BLADE ROW MODELS IN THE STREAMLINE THROUGHFLOW CODE 21. Figure 3.5: Comparison of Howell and Lieblein loss prediction for NACA 65 cascade data (Roos, 1999). experimental data and the curve fits of Aungier. A blending loss model of Roos dependent on the Howell and Lieblein loss models was also reviewed..

(43) Chapter 4. Boundary Layer Modelling 4.1. Introduction. Before any blockage model could implemented in the STFM code, different boundary layer methods were investigated and test cases conducted for 1dimensional flow problems. From turbulent boundary layer theory, an integral method was chosen to model boundary layers. The boundary layer method of Aungier (2003b) is also an integral entrainment method, but uses both the axial and tangential velocity components. Integral methods are easier to program than finite difference methods, as they are ordinary differential equations averaged across the boundary layer (White, 1991). Research was first done on trying to model a boundary layer in an adverse pressure gradient using an integral entrainment method. The Kármán momentum integral relation was used, equation 4.1.1. Cf θ ∂Vxe ∂θ + (2 + H) = ∂x Vxe ∂x 2. (4.1.1). Given the velocity profile, the three unknowns were the skin friction, Cf , momentum thickness, θ, and shape factor, H. Therefore, two other equations were required. The Ludwieg-Tillman skin friction equation, equation 4.1.2, is used by Aungier in his compressor boundary layer loss model. This equation was used. µ Cf = 0.246exp(−1.561H). ρe Ve θ µ. ¶−0.268 (4.1.2). The Head entrainment equation, equation 4.1.3, was chosen. The Head 22.

(44) CHAPTER 4. BOUNDARY LAYER MODELLING. 23. entrainment equation is another differential equation correlating the shape factor, H, and the entrainment, E. Aungier developed his own equation to calculate the entrainment, equation 4.1.4.. d (Vxe θH) = Vxe E dx E = 0.025(H − 1). (4.1.3) (4.1.4). Three test cases were modelled. They are flow 2200 of Coles & Hirst, flow 2300 of Coles & Hirst and flow 4000 of Coles & Hirst. All three test cases have a boundary layer in an adverse pressure gradient. Flow 4000 of Coles & Hirst is an axisymmetric cylinder in an axially symmetric flow. This could represent the casing of an axial compressor. The test cases are discussed individually in sections 4.2 to 4.4. For each test case a velocity profile is given, with initial values used being two of Cf , H and θ. A fourth order Runga-Kutta integration method is used to calculate the properties at the downstream point.. 4.2. Flow 2200 of Coles and Hirst. Flow 2200 of Coles & Hirst represents experiments done by Clauser on a turbulent boundary layer in an adverse pressure gradient. Velocity, displacement and momentum thickness data are given by Coles & Hirst (1968). The shape factor is calculated from the displacement and momentum thickness. A curve fit of the velocity points is developed from the points at which the experiments are performed and used in the boundary layer code. The same water properties as given for flat plate flow are used as initial values. The turbulent power law equation developed by Prandtl is used to calculate the displacement thickness.. 0.37 δ ≈ 1/5 θ Rex. (4.2.1). Simulations are run to see how well the method predicts the flow parameters. The simulation is run for four different integration step sizes to see the influence of the integration step size. Comparisons are given graphically for the momentum thickness, displacement thickness and shape factor..

(45) CHAPTER 4. BOUNDARY LAYER MODELLING. 24. Figure 4.1: Velocity profile vs x for flow 2200 of Coles and Hirst. Figure 4.2: Momentum thickness vs x for flow 2200 of Coles and Hirst. The influence of the integration step size can be seen in figures 4.2 to 4.4. The experimental and numerical momentum thicknesses match well at the final data point for 1000 steps. The numerical shape factor is over-predicted at the final axial point, which results in an over-prediction in the displacement.

(46) CHAPTER 4. BOUNDARY LAYER MODELLING. 25. Figure 4.3: Shape factor vs x for flow 2200 of Coles and Hirst. Figure 4.4: Displacement thickness vs x for flow 2200 of Coles and Hirst. thickness, figure 4.4. Figure 4.4 shows an improvement in the prediction of the displacement thickness at the final data point as the number of integration.

(47) CHAPTER 4. BOUNDARY LAYER MODELLING. 26. steps is increased from 20 to 1000, but worsens at the initial data points.. 4.3. Flow 2300 of Coles and Hirst. Flow 2300 of Coles & Hirst is also performed by Clauser. Experiments on a turbulent boundary layer in an adverse pressure gradient are performed, with resulting data given of the velocity, momentum thickness and displacement thickness (Coles and Hirst, 1968). A curve fit of the velocity is developed from the experimental data points and used in the numerical prediction, shown in figure 4.5. Initial conditions used are θ = 0.0154 and H = 1.788.. Figure 4.5: Velocity profile vs x flow 2300 of Coles and Hirst. The simulation is run for four different integration step sizes, 20 steps, 40 steps, 100 steps and 1000 steps. The experimental versus numerical comparisons are given in figures 4.6 to 4.8. In figure 4.6, the momentum thickness prediction is best for 20 integration steps at the final experimental data point. The prediction worsens as the number of steps is increased from 100 steps to 1000 steps. Figure 4.7 shows that the prediction of the shape factor improves with the reduction of the integration step size. Figure 4.8 shows the 20 step numerical results over-predicting the displacement thickness, with the 100 step run predicting the final data point the best..

(48) CHAPTER 4. BOUNDARY LAYER MODELLING. 27. Figure 4.6: Momentum thickness vs x for flow 2300 of Coles and Hirst. Figure 4.7: Shape factor vs x for flow 2300 of Coles and Hirst. 4.4. Flow 4000 of Coles and Hirst. Flow 4000 of Coles & Hirst represents experiments done by Moses on a turbulent boundary layer of a cylinder with an axially symmetric flow and adverse pressure gradient. Integral thicknesses are computed from the experiments.

(49) CHAPTER 4. BOUNDARY LAYER MODELLING. 28. Figure 4.8: Displacement thickness vs x for flow 2300 of Coles and Hirst. and tabulated with the experimental velocity points in Coles & Hirst (1968). A 5th order polynomial of the velocity is developed from the experimental data points (White, 1991) and used in the integral prediction method. The polynomial fit is shown in figure 4.9. The initial conditions used were θ = 0.000348 and H = 1.62. The simulation is run for four different integration steps sizes, 20 steps, 40 steps, 100 steps and 1000 steps. The experimental and numerical are compared in figures 4.10 to 4.12. The momentum thickness prediction improves as the integration step size is reduced, but still under-predicts between x = 0.5 m and x = 0.7 m. Figure 4.11 shows that all three predictions follow the trend of the experimental data, with the 20 step run being closest to the experimental data. In figure 4.12, the 100 and 1000 step runs have good agreement at the initial data points, but cannot predict the downward trend after the maximum displacement thickness. However of the three runs, the 20 step run best predicts the final displacement thickness.. 4.5. Conclusion. The predictions of the displacement thickness, equivalent to a blockage, vary for different integration step sizes. This could be due to the range over which.

(50) CHAPTER 4. BOUNDARY LAYER MODELLING. 29. Figure 4.9: Velocity profile vs x flow 4000 of Coles and Hirst. Figure 4.10: Momentum thickness vs x for flow 4000 of Coles and Hirst. the experimental data is documented. Flow 2200 starts at 2 m and ends at 10 m. This is a range of 8 m. The final data point displacement thickness is 0.0623 m. When taking the displacement thickness as a percentage of the range, the displacement thickness is 0.77%.

(51) CHAPTER 4. BOUNDARY LAYER MODELLING. 30. Figure 4.11: Shape factor vs x for flow 4000 of Coles and Hirst. Figure 4.12: Displacement thickness vs x for flow 4000 of Coles and Hirst. of the range. When done for flows 2300 and 4000, the percentages are 2.5% and 0.72% respectively. This could indicate the number of integration steps required. This could be a reason for flow 4000 showing good agreement for displacement thickness at the 20 step run..

(52) CHAPTER 4. BOUNDARY LAYER MODELLING. 31. For flow 2200, all three runs over-predict the shape factor. However, the momentum thickness and displacement thickness predictions are improved to within the range of the experimental data when the step size is reduced further. For flow 2300, the 20 step run predicts the momentum thickness the best. The shape factor prediction improves as the number of steps is increased, but for the displacement thickness the 100 step run best predicts the final data point. Flow 4000 is a much more complex flow to model. The prediction of momentum thickness improves as the step size is reduced, with the 20 step run showing the best results for the shape factor and displacement thickness. This complex flow can be modelled even with a few steps, showing that it may be possible to model a compressor with the same number of stations as the free stream flow..

(53) Chapter 5. Additional Models 5.1. Introduction. Four models are programmed into the Roos (1995) STFM code. They are the boundary layer, Mach number, tip clearance and smoothing models. The first module investigated is the boundary layer model, then the tip clearance model, Mach number model and smoothing model.. 5.2. Compressor blockage modelling. Once the 1-dimensional boundary layer test cases are completed, the 2-dimensional boundary layer blockage model of Aungier (2003b) is implemented into the STFM code. After the code structure is studied, the position of the blockage model is identified within the code. The blockage model loops over from the initialization of the streamlines to the final calculation of the flow properties after the global tolerance is reached. This is shown in figure 5.1. The blockage is then iterated until the tolerance has been reached. The solution procedure is outlined here (Aungier, 2003b), with the model itself given in appendix C.1. The boundary layer model is limited to turbulent flow. All Reynolds numbers based on momentum thickness are controlled to be greater than 250. A corrected momentum thickness is then calculated based on the Reynolds number. Initial meridional and tangential shape factors, H1 and H2 , and momentum thicknesses, θ11 and θ12 , are required at station 1 of the modelled compressor. A flat plate flow value for H1 of 1.4 is used, with a one-seventh power-law tangential velocity profile, m = 1/7. An initial fractional area block-. 32.

(54) CHAPTER 5. ADDITIONAL MODELS. 33. Figure 5.1: Blockage model addition to Roos code. age factor, B, split equally between hub and casing, is used. The displacement thickness can then be calculated for an end-wall, using equation 5.2.1 1 ˙ 2πρe Vme δ1∗ = B m 2. (5.2.1). The analysis begins at station two of the modelled compressor and marches downstream, computed for both the hub and casing endwalls. The momentum thickness values are first estimated using equations 5.2.2 and 5.2.3.. ∂ 2 [rρe Vme θ11 ] = 0 ∂m. (5.2.2). ∂ 2 [r ρe Vme Vθe θ12 ] = 0 ∂m. (5.2.3). The upstream values of H 1 and H 2 are used. All the other parameters are estimated using equations C.1.15 to C.1.21. The entrainment and shear stress, and blade force if necessary, are then calculated. Equations C.1.1 to C.1.3 are then integrated to calculate the momentum.

(55) 34. CHAPTER 5. ADDITIONAL MODELS. thicknesses, θ11 and θ12 , and (δ − δ1∗ ). The values of n and δ are computed from equations C.1.15 and C.1.19, with equation 5.2.4 as a limit. Then m is computed using equation C.1.16. θ12 ≤ 0.99. δ n+1. (5.2.4). The other parameters are calculated using equations C.1.17 to C.1.21. If there is convergence on θ11 , H 1 and ν, the steps are repeated from computation of the entrainment and shear stress calculations. Once convergence is achieved, the steps are repeated from the initial estimate of momentum thicknesses for the next meridional station until all stations are complete. Once the boundary layer analysis is complete for the hub and casing, the fractional area blockage is calculated using equation 5.2.5. B=. Shroud X. 2πrρe Vme δ1∗ /m ˙. (5.2.5). Hub. 5.3. Mach number model. The Mach number and tip clearance loss models are implemented in the code once the boundary layer model is verified. These two models add to the total pressure loss of the blade rows. As is shown in figure 5.2, the loss models were added to the "calculate_everything" subroutine. The Mach number model is described in appendix C.2. The input to this subroutine is the number of streamlines, number of axial stations, mass flow and initialized fluid properties. The subroutine starts by calling the "findMukCpPr" subroutine, which calculates the specific heat properties for the given grid point temperature. The "calculate_everything" subroutine calculates properties for each streamline at all stations by advancing from the compressor inlet station to the final axial station. The code first needs to identify if there is a blade row at a specific station. A blade row is specified in the code at the trailing edge. The position of the trailing edge of a blade row is indexed to a specific axial station. If there is no blade row, conservation of angular momentum is applied. If there is a blade row, the properties of the blade row are calculated based on the specific radial position. For each blade row index, the blade type is also specified. The blade.

(56) CHAPTER 5. ADDITIONAL MODELS. 35. type can be either a rotor or stator. An inlet guide vane (IGV) or exit guide vane (EGV) is modelled as a stator, as they are stationary blade rows. Once the blade type is known, the loss and deviation models are applied. The Mach number and tip clearance models are applied to give the total pressure loss. The fluid properties are then calculated for the trailing edge, taking into account the loss effects.. Figure 5.2: Additional loss models implemented in Roos code. However, the Mach number model has a significant effect on the total pressure loss. The total pressure loss correlations of Howell and Lieblein are for low speed subsonic compressor flow. The effect of higher throughflow velocities needs to be taken into account. As the Mach number increases, the incidence range is reduced. The flow is limited by choking, which is also accounted for in the model. The Mach number model includes the off-design performance correlations. The combination of these two models is documented in appendix C.2..

(57) CHAPTER 5. ADDITIONAL MODELS. 5.4. 36. Tip clearance model. The tip clearance module models the flow that passes through the clearance gap at a rotor tip and a stator hub section. In the code it is positioned at the streamline next to the wall. The influence of the tip clearance on the pressure loss was shown to be insignificant. This model is documented in appendix C.3.. 5.5. Loss smoothing model. Aungier (2003b) suggests applying a model to correct the total pressure loss calculated at each streamline position for an axial station. The total pressure loss is calculated at each streamline position, ω ¯ i . A pressure loss differential is then calculated in the relative frame of reference using equation 5.5.1 given below. (∆Pt0 )i = ω ¯ i (Pti0 − Pi0 )in. (5.5.1). A corrected pressure loss differential is then applied at each streamline position using a trapezoidal rule approximation.. (∆Pt0 )i,c = [(∆Pt0 )i−1 + 2(∆Pt0 )i + (∆Pt0 )i+1 ]/4. (5.5.2). (∆Pt0 )1,c = 2(∆Pt0 )2,c − (∆Pt0 )3,c. (5.5.3). (∆Pt0 )N,c = 2(∆Pt0 )N −1,c − (∆Pt0 )N −3,c. (5.5.4). This smoothing is rarely required, except when the flow profiles are highly distorted, such as when choke is encountered in the blade passage (Aungier, 2003b)..

(58) Chapter 6. Compressor Test Cases 6.1. Introduction. Three compressors were chosen to validate the numerical predictions obtained from the improved code against experimental data. The effect of the different loss models added to the STFM code, on the performance prediction of compressors, was also investigated. The NACA 5-stage compressor, the NACA 8-stage compressor and the NACA 10-stage compressor were chosen. These three compressors were chosen because there is substantial performance data available, and because there are published blockage predictions of these compressors. Aungier (2003b) uses these three compressors as test cases for both overall performance as well as blockage comparisons. The local STFM has already been validated for low Mach number compressors (Roos, 1995; Gannon, 1996), but not for high Mach number compressors. The NACA 5-stage compressor and the NACA 8-stage compressor are both transonic compressors having higher loading on each blade row than the NACA 10-stage subsonic compressor. All three compressors were built in the 1950’s as research units. They have NACA65 and double circular arc (DCA) blade profiles. Further detail concerning the geometry, blades and modelling of each of these compressors is presented in the relevant sections below. To model a compressor numerically, initial and inlet conditions, geometry and blade data are required. The initial conditions set include the number of streamlines, the gas properties, mass flow and speed. The gas properties are set for the initial iteration of the code, after which they are calculated as a function of temperature. The inlet conditions required are the stagnation. 37.

(59) CHAPTER 6. COMPRESSOR TEST CASES. 38. pressure and temperature. Stations given by axial and radial positions of the hub and annulus profiles represent the geometry of a compressor. For each station an initial blockage percentage estimate is given. The blockage percentage is a representation of the boundary layer on the hub and casing walls as they develop in the compressor. Numerically, the blockage percentage shifts the radial position of the hub and casing walls. From Cumpsty (1989), an initial estimate of the growth of a boundary layer in a compressor of 0.5% per blade row was used. However, Cumpsty states a conservative maximum of 4%. At higher speeds, this maximum is ignored. The percentage is increased with the same increment until the last blade row. Thereafter, it is kept constant at the final value of the last blade row. Blade parameters are given at a number of radii for each blade row. Each table gives the number of blades, radius ratio, chord length, camber, stagger, solidity, point of maximum thickness and maximum thickness for each stage. The radius ratio values are all referenced to the casing radius for each compressor. In all three compressors the casing radius is 10" (0.254 m). The chord length is given in inches, the camber either as a lift coefficient or angle in degrees. The stagger is given as an angle in degrees and the maximum thickness is given as a percentage. From this, polynomials are developed to represent these parameters as a function of radius. A template of the input file is given in appendix A.. 6.2 6.2.1. NACA 5-stage transonic compressor Description. The NACA 5-stage compressor is an axial-flow compressor that operates with transonic relative inlet Mach numbers for all rotor blade rows. It was designed as a research compressor to study the problems that may arise for transonic stages (Sandercock et al., 1954). By increasing the inlet relative Mach number to the rotor, the stage pressure rise is increased. The number of stages required to produce the desired pressure ratio, is reduced. This results in a reduction in compressor size and weight. The compressor is designed to produce an overall total pressure ratio of 5.0 at a rotor tip speed of 1100 ft/s (335.3 m/s). The casing of the compressor is constant with a diameter of 20" (0.508 m), and at the inlet to the first rotor,.

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