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Incompressible Skewed Turbulent Boundary Layer on an End Wall of a Curved Two-Dimensional Diffuser Marvin Dale Smith Iowa State University

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Incompressible Skewed Turbulent Boundary Layer on an End Wall of a Curved Two-Dimensional Diffuser Marvin Dale Smith Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1970 Incompressible skewed turbulent boundary layer on an end wall of a curved two-dimensional diffuser Marvin Dale Smith Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mechanical Engineering Commons Recommended Citation Smith, Marvin Dale, "Incompressible skewed turbulent boundary layer on an end wall of a curved two-dimensional diffuser " (1970). Retrospective Theses and Dissertations. 4199. https://lib.dr.iastate.edu/rtd/4199 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. 70-18,908 SMITH, Marvin Dale, 1935- INCOMPRESSIBLE SKEWED TURBULENT BOUNDARY LAYER ON AN END WALL OF A C'JRVSD TWO-DIMENSIONAL DIFFUSER. Iowa State University, Ph.D., 1970 Engineering, mechanical I University Microfilms, A XEROXCompany, Ann Arbor, Michigan j. : Art T»T^oT^xtrr»T\ INCOMPRESSIBLE SKEWED TURBULENT BOUNDARY LAYER ON AN END WALL OF A CURVED TWO-DIMENSIONAL DIFFUSER by Marvin Dale Smith A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of Tlie Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subjects; Mechanical Engineering Aerospace Engineering Approved; Signature was redacted for privacy. In Charge of Major Work Signature was redacted for privacy. Heads of Major Departments Signature was redacted for privacy. DeaU of Gradual^ C^lege Iowa State University Of Science and Technology Ames, Iowa 1970 11 TABLE OF CONTENTS Page SYMBOLS AND NOTATION Iv INTRODUCTION 1 REVIEW OF LITERATURE 9 The Two-Dimenslonal Turbulent Boundary Layer 9 The Skewed Turbulent Boundary Layer 28 THEORETICAL DEVELOPMENT 50 Similarity Parameters 50 Order of Magnitude Analysis 68 EXPERIMENTAL APPARATUS 74 EXPERIMENTAL PROCEDURES 85 Instrumentation Calibration 85 Preliminary Procedures 96 Data Acquisition 100 EXPERIMENTAL DATA REDUCTION 109 Calculation of Basic Flow Variables 109 Parameter Calculations 125 Data Uncertainty 128 ANALYSIS AND DISCUSSION OF DATA 140 Diffuser Data 140 Compiled Three-Dimensional Boundary Layer Data 160 Remarks on a Double-Skewed Turbulent Boundary Layer 184 Remarks on Velocity Profile Models 188 CONCLUSIONS 206 LITERATURE CITED 211 TABLE OF CONTENTS (Continued) Page ACKNOWLEDGMENTS 218 APPENDIX A. SELECTED VELOCITY MODELS FOR THREE-DIMENSIONAL 219 TURBULENT BOUNDARY LAYERS APPENDIX B. TURBULENT FLOW EQUATIONS 225 APPENDIX C, HOT-WIRE INSTRUMENTATION 236 APPENDIX D. DIFFUSER EXPERIMENTAL DATA 264 APPENDIX E. DIFFUSER SIDE WALL COORDINATE SYSTEM 275 APPENDIX F. SELECTED EXPERIMENTAL DATA 277 iV 1 SYMBOLS AND NOTATION a least squares coefficient (page 118) A , A , A parameters in hot-wire relations (page 250) u v' w b least squares coefficient (page 118) Bu» B^, parameters in hot-wire relations (page 251) C-, C„, C„ coefficients in second order least squares equation (page 92) parameters in hot-wire relations (page 251) w fluctuating voltage associated with wires 1, 2 and 3, u' V w volts (page 237) E instantaneous voltage, volts (246) ¥ mean voltage, volts (page 246) bridge voltage, volts (page 92) £,£.,£ instantaneous voltage associated with wires 1, 2 and 3 ° Y at angles a, S and y respectively, volts (page 237) Ah Pitot-static differential pressure, in, of H^O h wall static pressure, in, of H^O h^, h^, h^ coordinate scale factors (page 227) or H-1 streamwise shape factor (page 127) or H-2 crossflow shape factor (page 127) k velocity potential proportionality constant, ft/sec (page 55) k * constant in hot-wire relations (pages 240, 241) k , k , k constant k which corresponds to wires 1,2 and 3 " ^ " respectively (page 237) ^Symbols and notation not in this list are defined and used locally within the text. Kj^, K^, coefficients in Kings equation (pages 91, 246) L characteristic body length in Ç-direction, ft (page 57) t characteristic body length in crossflow direction, ft (page 57) MP modified electrical power parameter, watts (page 112) MPR modified power ratio (page 113) MP , MPqfx modified power values corresponding to an arbitrary angle 0 between the wire and the mean velocity direction, also shown in 8 = 90°, watts MP , MP modified power values corresponding to the particular angular definitions, watts (page 239) N exponent in Kings equation (page 91) P, p", P' instantaneous, mean and fluctuating fluid static pres­ sure (page 51) * P normalized static pressure (page 58) q # q_# q fluctuating velocity values used in hot-wire equations, ® ^ ^ ft/sec (page 238) q q q *9 q» Reynolds stress velocity terms from hot-wire calculations, s t s n n t (ft/sec)2, (pages 254, 255) total fluctuating velocity, (ft/sec)2 (page 183) Q velocity at the boundary layer edge, ft/sec (page 125) Q total instantaneous velocity at any point, ft/sec (page 239) Qg total mean velocity at any point, ft/sec R Reynolds number based on h^Ç (page 69) "hi? Rsh»Rgc»^g resistances, ohms (page 88) R ,R , R , ,R , resistances, ohms (page 112) c* p' p.h; cab ' vi resistance, ohms (page 92) R overheat ratio (page 91) OU R ,R ,R radii of the pressure wall, ceriterline and suction p.w. ctr s.w. wall, in. (page 107) Tsh» Tg sensor temperatures, °F (page 88) AT temperature difference, °F (page 88) u, u, u' streamwise instantaneous, mean and fluctuating velocity u'v' Reynolds stress velocity terms, (ft/sec) u'w' Reynolds stress velocity terms, (ft/sec) U normalizing velocity, ft/sec (page 57) U equivalent velocity, ft/sec (page 240) (U/Q)„ ^ or velocity ratio at the edge of the wall collateral ' * layer (page 191) UQCL u* normalized velocity (page 57) u^ boundary layer parameter (page 11) u^ shearing velocity, ft/sec (page 11) V, V, v' normal instantaneous, mean and fluctuating velocity components, ft/sec (page 51) v'w' Reynolds stress velocity terms, (ft/sec) V equivalent velocity, ft/sec (page 242) V normalizing velocity, ft/sec (page 57) V normalized velocity (page 57) Vp Head's entrainment velocity, ft/sec (page 27) w, w, w' crossflow instantaneous, mean and fluctuating velocity components, ft/sec (page 51) vil W equivalent velocity, ft/sec (page 242) W normalizing velocity, ft/sec (page 57) w normalized velocity (page 57) ~ or WC/WCL ratio of the triangular crossflow velocity to the cross- C,L, flow velocity at the wall collateral layer edge (page 190) or WQCL ratio of the crossflow velocity to the boundary layer mC.L* edge velocity at the wall collateral layer edge (page 191) w/Q a particular crossflow velocity ratio (page 198) wc' /Q X cartesian coordinate in the streamwise direction, ft y cartesian coordinate in the direction normal to a surface, ft y"*" boundary layer parameter (page 11) z cartesian coordinate which is orthogonal to x and y, ft a thermal resistivity, °F ^ (page 88) a, a instantaneous and mean angle between the total mean velocity vector and sensor number respectively, deg (page 239) a angle between the boundary layer edge velocity vector and any local boundary layer velocity vector, deg "c.L. the value of a at the wall collateral layer edge, deg ALPHA-CL (page 125) 3,3 Instantaneous and mean angle between the total mean velocity vector and sensor number 2 respectively, deg (page 239) 3 Y specific weight of manometer fluid, Lb^/ft Y, Y Instantaneous and mean angle between the total mean velocity vector and sensor number 3 respectively, deg (page 239) vili 6 or B.L. THK boundary layer thickness, in. (page 125) d* or DISP, THK 1 boundary layer displacement thickness, in, (page 126) 6* or DISP, THK 2 boundary layer displacement thickness, in, (page 126) e the value of Tan a C.L. parameters in the hot-wire relations (page 251) uu' vv* parameters in the hot-wire relations (page 253) uV uw* vw Ç curvilinear coordinate in the crossflow direction (page 225) n curvilinear coordinate perpendicular to the surface (page 225) 9 centerline turning angle, deg 0 maximum centerline turning angle or total turning angle of the duct, deg e^^or Mora, THK 11 streamwise momentum thickness, in, (page 126) Bg^or Mom, THK 21 interaction momentum thickness, in, (page 127 BggOr Mom, THK 22 crossflow momentum thickness, in, (page 127) Mom, THK 12 interaction momentum thickness, in, (page 127 K^, parameters in hot-wire relations, (page 251) 2 X bulk viscosity, Lb^-sec/ft 2 y viscosity, Lb^-sec/ft 2 V kinematic viscosity, ft /sec eddy viscosity, ft 2/sec Ç curvilinear coordinate in the streanwise direction (page 225) Ç an angle in the hot-wire relations, deg (page 239) ix 3 p fluid density» slugs/ft maximum shear stress in the boundary layer, Lb^/ft 2 wall shear stress, Lb^/ft 2 2 (|) velocity potential, ft /sec (page 55) 1 INTRODUCTION The three-dimensional turbulent boundary layer developing in an incompressible Newtonian fluid flow on the end wall of the diffuser shown in Figure 1 is the subject of concern in this dissertation* A number of experimental investigations have been concerned with the development of the boundary layer with a curved free stream streamline. The primary differences between this experimental investigation and others known to the author are that this work entails the development of a thin boundary layer in a curved diffusing passage and that the passage is considerably smaller than the others. Boundary layers developing in a curved passage have a varying pres­ sure gradient (both in magnitude and direction) and because of this an equilibrium boundary layer is not expected to exist, since two-dimensional equilibrium turbulent boundary layers require a specific relationship between the pressure gradient and the wall shear stress.
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