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A study of unsteady wake-boundary layer interaction in turbomachines Gete, Zenebe 1996

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STUDY OF UNSTEADY W A K E - B O U N D A R Y LAYER INTERACTION IN T U R B O M A C H I N E S by Zenebe Gete B.Sc. 1985, Addis Ababa University, Ethiopia M.A.Sc. 1991 , The University of British Columbia A THESIS SUBMITTED IN PARTIAL FULFILLMENT O F T H E REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in T H E FACULTY O F GRADUATE STUDIES MECHANICAL ENGINEERING W e accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA October 1996 © Zenebe Gete, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada Date D f i : , l \ . WQt DE-6 (2/88) II A b s t r a c t The relative motions of rotor and stator blade rows in a turbomachine generate periodically unsteady flow subject to traveling wake perturbations. To better understand the attendant wake-boundary layer interactions, the study used an experimental two-dimensional physical model and a new calculation procedure. In the range of Reynolds number 0.144 x 10 s to 1.44 x 10 5 , measurements of instantaneous boundary layer velocity and wall shear stress were made on a tripped flat plate with a hot wire probe and a glue-on probe. Data was reduced through the process of ensemble averaging. For the reduced frequencies tested (ID = 0.33 to 9.33 ), skin friction coefficient was found to be frequency dependent and to have values up to twice that of the unperturbed boundary layers. Due to the nonlinear interactions with the boundary layer, the traveling discrete frequency wakes increased velocity profile fullness and increased random turbulent fluctuations. Unlike the response to stationary-wave-like external disturbances, for the traveling wake fluctuations the boundary layers exhibited increasing negative velocity phase shifts from 0° in the free stream to values of order 360° at the wall. Amplitude overshoots occurred close to the wall and undershoots were observed in the outer region of the profile. Calculated results of steady and unsteady velocity profiles and skin friction compare reasonably with experiment. Although the agreement in phase shift was rather poor, in both experiment and calculation the negative phase shift aspect of the traveling-wave fluctuations has been captured. Tab le of C o n t e n t s ABSTRACT ii LIST OF TABLES.. v LIST OF FIGURES vi NOMENCLATURE xii ACKNOWLEDGMENTS xv 1 . INTRODUCTION AND OBJECTIVES 1 2. LITERATURE REVIEW 8 2.1 INTRODUCTION 8 2 . 2 UNSTEADY LAMINAR BOUNDARY LAYERS 1 0 2 . 2 . 1 Laminar Boundary Layers Subject to Standing Wave Fluctuations 1 1 2 . 2 . 2 Laminar Boundary Layers Subject to Traveling Wave Fluctuations 1 3 2 . 3 UNSTEADY TURBULENT BOUNDARY LAYERS 1 8 2 . 3 . 1 Standing Wave Type Free Stream Fluctuation 1 9 2 . 3 . 2 Traveling Wave Type Free Stream Fluctuation 2 4 2 . 4 NONSTEADY F L O W S IN TURBOMACHINES 2 7 2 . 5 MEASUREMENT O F S K I N FRICTION IN TURBULENT BOUNDARY LAYERS 3 1 2 . 5 . 1 Structure Of Turbulent Boundary Layers 3 3 2 . 5 . 2 Wall Shear Stress Measurement Via Pressure Gradient 3 5 2 . 5 . 3 Wall Shear Stress Determination by Clauser Plots 3 5 2 . 5 . 4 Measurement of Wall Shear Stress by a Heated Element 3 6 3. EXPERIMENTAL APPARATUS, INSTRUMENTATION, MEASUREMENT TECHNIQUES AND DATA PROCESSING 45 3.1 INTRODUCTION 4 5 3 . 2 T H E W I N D TUNNEL AND R I G 4 5 3 . 3 H O T W I R E SUPPORT AND TRAVERSE MECHANISM 4 9 3 . 4 H O T W I R E ANEMOMETRY A N D CALIBRATION 5 0 3 . 5 DETERMINATION O F THE W A L L PROXIMITY EFFECT ON THE H O T W I R E 5 3 3 . 6 TRIGGERING METHOD FOR DATA ACQUISITION 5 4 3 . 7 DATA ACQUISITION A N D DATA PROCESSING 5 5 3 . 8 T H E TECHNIQUE O F ENSEMBLE AVERAGING 5 7 4. EXPERIMENTAL RESULTS AND DISCUSSION 71 4.1 INTRODUCTION 7 1 4 . 2 STEADY TURBULENT BOUNDARY LAYERS 7 3 4 . 3 DEVELOPMENT O F UNSTEADY TURBULENT BOUNDARY LAYERS 7 5 iv 4 . 3 . 1 Unsteady velocity Results 7 5 4 - 3 - 2 Random Fluctuations Under Periodic Perturbation 8 1 4 . 3 . 4 Amplitude Ratio Across the Boundary Layer 9 0 4 . 4 W A L L S H E A R S T R E S S U N D E R TRAVELING PERIODIC DISTURBANCES 9 1 4 . 4 . 1 Steady Turbulent Boundary Layer Wall Shear Stress 9 2 4 . 4 . 2 Unsteady Wall Shear Stress Results by Glue-on Probe Measurement 9 3 4 . 4 . 3 Evaluation of Wall Shear Stress from Velocity Curve Fits 9 7 4 . 4 . 4 Estimation of Skin Friction Coefficient Using Clauser Plots 9 9 4 . 5 CONCLUDING REMARKS ; 1 0 1 5. COMPUTATIONAL RESULTS AND DISCUSSION 148 5.1 INTRODUCTION 1 4 8 5 . 2 COMPUTATIONAL SCHEME 1 5 0 5 . 2 . 1 Equations of Motion 1 5 0 5 . 2 . 2 Numerical Formulation 1 5 1 5 . 2 . 3 Boundary & Initial Conditions 1 5 3 5 . 3 TRAVELING W A V E FREE STREAM OSCILLATION RESULTS 1 5 4 5 . 3 . 1 Turbulent Boundary Layer Velocity Profiles 1 5 4 5 . 3 . 2 Computational Skin Friction Coefficient Results 1 5 7 5 . 3 . 3 Comparisons of Calculated and Experimental Results of Skin Friction Coefficients 1 6 0 5 . 3 . 4 Calculated Shear Stress Profiles Across the Boundary Layer 1 6 2 5 . 4 CONCLUDING REMARKS 1 6 3 6. CONCLUSIONS AND RECOMMENDATIONS 180 6.1 SUMMARY 1 8 0 6 . 2 CONCLUSIONS 1 8 0 6 . 3 RECOMMENDATIONS 1 8 4 REFERENCES 186 APPENDIX A 191 APPENDIX B 194 APPENDIX C 197 V L is t of Tab les Table 4-1 Experimental Conditions for Velocity Measurements: I L = 3.0 m/s 73 Table 4-2 Values of the Reduced Frequency for Each Experimental Case 76 Table 4-3 Experimental Conditions for Wall Shear Stress Measurement 93 Table 5-1 Experimental Conditions and Results Applied to the computation: I L = 3.0 m/s 155 vi L is t of F igu res Fig. 1-1 A Cross-section of a Multi-stage Turbomachine Stage 5 Fig. 1-2: Schematic of a Turbomachinery Stage and the Velocity Triangles at Entrance and Exit of Rotor Blades 6 Fig. 1-3: Simplified Physical Modeling Process of a Turbomachine Stage Flow by a Two-Dimensional Wind Tunnel Flow 7 Fig. 2 -1 : Some Effects of Standing and Traveling Wave Free Stream Fluctuations on a Laminar Boundary Layer [ 9 ] 40 Fig. 2-2: Decomposition of a Typical Instantaneous Velocity in to its Components .... 41 Fig. 2-3: A Schematic of Directional and Magnitude Fluctuation of the Absolute Velocity In front of the Flat Plate 42 Fig. 2-4: Typical Turbulent Boundary Layer Regions on the Surface of a Flat Plate.... 43 Fig. 2-5: A Typical Turbulent Boundary Layer Velocity Profile 44 Fig. 3 -1 : The Wind Tunnel, Experimental Rig Assembly - Schematic 61 Fig. 3-2: A Scale Diagram of the Flat Plate Leading Edge, Trip Wire, Airfoil and Hot-Wire Probe 62 Fig. 3-2a A DISA 55P15 Boundary Layer Probe 63 Fig. 3-3: Unsteady Flow Generating Rig and Drive Mechanism 64 Fig. 3-4: Calibration Curve for a DISA 55P15 Probe for Velocity Measurement 65 Fig. 3-5: Schematic of the Preston Tube Used to Calibrate the Glue-on Probe 66 Fig 3-6 Schematic of a Glue-on Probe, Near Wall Velocity Profile and Thermal Boundary Layer Profile 66 Fig. 3-6a DISA 55P47 Glue-on Probe (Wall Shear Stress Sensor) 67 Fig. 3-7: Calibration Curve for a DISA 55P47 Glue-on Probe for Wall Shear Stress Measurement 68 Fig. 3-8: A Schematic of an Airfoil and Actual Angular Orientation of With Respect to the Relative Velocity 69 Fig.3-9: A Typical Ensemble Averaging Process and Resulting Velocity Components 70 VII Fig. 4 - 1 : Steady Turbulent Boundary Layer Profiles at Some Longitudinal Stations .. 103 Fig. 4-1a:Comparisons of Turbulent Boundary Layer Profiles at Constant Pressure Zero Incidence on Flat Plates 104 Fig. 4-2: Steady Turbulent Boundary Layer Random Fluctuations at Some Longitudinal Stations 105 Fig. 4.2a:Steady Turbulent Boundary Layer Random Fluctuations Compared to Previous Study 106 Fig. 4-3: A Typical History of Free Stream Velocity at X = 0.1 m, f = 20 Hz, S = 0.1 m 107 Fig. 4-4a: Periodic Velocity Fluctuations of the Free Stream: X = 0.1 m, S = 0.1 m, f = 20 Hz 108 Fig. 4-4b: Periodic Velocity Fluctuations of the Free Stream: X = 0.3 m, S = 0.1 m, f = 20 Hz 108 Fig. 4-4c: Periodic Velocity Fluctuations of the Free Stream: X = 0.1 m, S = 0.2 m, f = 1 0 H z 109 Fig. 4-4d: Periodic Velocity Fluctuations of the Free Stream: X = 0.3 m, S = 0.2 m, f = 1 0 H z 109 Fig. 4-5a: Velocity Profiles at Various Phases In a Single Cycle: Unsteady Flow, Ue = 3.0 m/s, Ur = 2.0 m/s, S = 0.1 m, f = 20 Hz 110 Fig. 4-5b: Velocity Profiles at Various Phases In a Single Cycle: Unsteady Flow, Ue = 3.0 m/s, Ur = 3.0 m/s, S = 0.1 m, f = 30 Hz 111 Fig. 4-5c: Velocity Profiles at Various Phases In a Single Cycle: Unsteady Flow, Ue = 3.0 m/s, Ur = 4.0 m/s, S = 0.1 m, f = 4 0 H z 112 Fig. 4-5d: Velocity Profiles at Various Phases In a Single Cycle: Unsteady Flow, Ue = 3.0 m/s, Ur = 2.0 m/s, S = 0.2 m, f = 10 Hz 113 Fig. 4-5e: Velocity Profiles at Various Phases In a Single Cycle: Unsteady Flow, Ue = 3.0 m/s, Ur = 3.0 m/s, S = 0.2 m, f = 15 Hz 114 Fig. 4-5f: Velocity Profiles at Various Phases In a Single Cycle: Unsteady Flow, Ue = 3.0 m/s, Ur = 4.0 m/s, S - 0.2 m, f = 20 Hz 115 VIII Fig. 4-6a: Comparisons of Average Turbulent Boundary Layer Velocity Profiles for Steady and Unsteady Flows With Different Disturbance Frequencies .. 116 Fig. 4-6b: Comparisons of Average Turbulent Boundary Layer Velocity Profiles for Steady and Unsteady Flows With Different Disturbance Frequencies .. 117 Fig. 4-7a: Time Histories of the Random Fluctuations Subject to Periodic Wake Disturbances at Some Transverse Locations: f = 20 Hz, S = 0.1 m, X = 0.1 m 118 Fig. 4-7b: Time Histories of the Random Fluctuations Subject to Periodic Wake Disturbances at Some Transverse Locations: f = 30 Hz, S = 0.1 m, X = 0.1 m 119 Fig. 4-7c: Time Histories of the Random Fluctuations Subject to Periodic Wake Disturbances at Some Transverse Locations: f = 40 Hz, S = 0.1 m, X = 0.1 m 120 Fig. 4-7d: Time Histories of the Random Fluctuations Subject to Periodic Wake Disturbances at Some Transverse Locations: f = 10 Hz, S = 0.2 m, X = 0.1 m 121 Fig. 4-8: Random Turbulent Fluctuation Profiles at Various Phases in a Single Period: Unsteady Flow, Ue = 3.0 m/s, Ur = 2.0 m/s, S = 0.1 m f = 20 Hz 122 Fig. 4-9: Random Turbulent Fluctuation Profiles at Various Phases in a Single Period: Unsteady Flow, Ue = 3.0 m/s, Ur = 3.0 m/s, S = 0.1 m f = 30 Hz 123 Fig. 4-10: Random Turbulent Fluctuation Profiles at Various Phases in a Single Period: Unsteady Flow, Ue = 3.0 m/s, Ur = 4.0 m/s, S = 0.1 m f = 40 Hz 124 Fig. 4-11: Profiles of Average Random Fluctuations Across the Boundary Layers at Different Downstream Locations: Unsteady Flow, Ue = 3.0 m/s, S = 0.1 m 125 Fig. 4-12: Profiles of Average Random Fluctuations at Various Downstream Locations: Unsteady Flow, Ue = 3.0 m/s, S = 0.1 m 126 Fig. 4-13a: Phase Shifts of the Unsteady Turbulent Boundary Layer Velocity With Varying Wake Passing Frequency at Some Downstream Stations, S = 0.1 m 127 Fig. 4-13b: Phase Shifts of the Unsteady Turbulent Boundary Layer Velocity at Varying Downstream Stations for Constant Frequency, S = 0.1 m 128 Fig. 4-14a: Phase Shifts of the Unsteady Turbulent Boundary Layer Velocity With Varying Wake Passing Frequency at Some Downstream Stations, S = 0.2 m.... 129 ix Fig. 4-14b: Phase Shifts of the Unsteady Turbulent Boundary Layer Velocity at Varying Downstream Stations for Constant Frequency, S = 0.1 m 130 Fig. 4-15a: Amplitude Fluctuation of the Periodic Fluctuation Across the Turbulent Boundary Layer: Same Frequency, Varying X, S = 0.1 m 131 Fig. 4-15b: Amplitude Fluctuation of the Periodic Fluctuation Across the Turbulent Boundary Layer: Same Station , Varying Frequency , S = 0.1 m . 132 Fig. 4-16a: Amplitude Fluctuation of the Periodic Fluctuation Across the Turbulent Boundary Layer: Same Frequency, Varying X, S = 0.2 m 133 Fig. 4-16b: Amplitude Fluctuation of the Periodic Fluctuation Across the Turbulent Boundary Layer: Same Frequency, Varying X, S = 0.2 m 134 Fig. 4-17: Comparison of Measured Steady Turbulent Boundary Layer Skin Friction Coefficient With Empirical Equations 135 Fig. 4-18a: Time History of Wall Shear Stress Coefficient Along the Flat Plate at Constant Frequency, S = 0.1 m 136 Fig. 4-18b: Time History of the Random Fluctuation of Wall Shear Stress Coefficient Along the Flat Plate at Constant Frequency 137 Fig. 4-18c: Time History of the Random Fluctuation of Wall Shear Stress Coefficient Along the Flat Plate at Constant Frequency 138 Fig. 4-19: Comparison of Measured Skin Friction Coefficients With Varying Disturbance Frequency: S = 0.1 m; Wall Shear Stress Normalized With the Local Free Stream Velocity 139 Fig. 4-20: Comparison of Measured Skin Friction Coefficients With Varying Disturbance Frequency: S = 0.2 m, Wall Shear Stress Normalized With the Local Free Stream Velocity 140 Fig. 4-21 a: Polynomial Curve Fit Velocity Profiles at Various Phases in a Single Cycle: Unsteady Flow, Ur = 2..0 m/s, S = 0.1 m.. f = 20 Hz 141 Fig. 4-21: Comparison of Steady Skin Friction Coefficients Using Data By Direct Measurement and Velocity Curve Fits 142 Fig. 4-22: Comparison of unsteady Skin Friction Coefficients Using Data By Direct Measurement and Velocity Curve Fits 143 Fig. 4-23: Determination of Skin Friction Coefficients for Turbulent Boundary Layers Along the Flat Plate by Clauser Plots: Steady Flow 144 Fig. 4-24: Determination of Skin Friction Coefficients for Turbulent Boundary Layers Along the Flat Plate by Clauser Plots: Unsteady Flow, S = 0.1 m, f = 20 Hz ... 145 Fig. 4-25: Determination of Skin Friction Coefficients for Turbulent Boundary Layers Along the Flat Plate by Clauser Plots: Unsteady Flow, S = 0.1 m, f = 30 Hz 146 Fig. 4-26: Determination of Skin Friction Coefficients for Turbulent Boundary Layers Along the Flat Plate by Clauser Plots: Unsteady Flow, S = 0.1 m, f = 40 Hz 147 Fig. 5 -1 : Schematic of a Two-Dimensional Computational Mesh [ 58 ] 152 Fig. 5-2: Calculated Turbulent Boundary Layer Velocity Profiles Along the Flat Plate for Steady Flow 164 Fig. 5-2a: Comparison of Experimental and Calculated Steady Turbulent Boundary Layer Velocity Profile 165 Fig. 5-3: Calculated Phase Velocity Profiles Under Traveling Wave Free Stream Fluctuation at Some Downstream Stations Along the Flat Plate: Unsteady Flow, f = 20 Hz 166 Fig. 5-3a: Comparison of Experimental and Calculated Turbulent Boundary Layer Velocity Profile on the Flat Plate; f = 20 Hz 167 Fig. 5-3b: Comparison of Experimental and Calculated Turbulent Boundary Layer Velocity Profile on the Flat Plate; f = 30 Hz 168 Fig. 5-3c: Comparison of Experimental and Calculated Turbulent Boundary Layer Velocity Profile on the Flat Plate; f = 40 Hz 169 Fig. 5-3d: Comparison of Experimental and Calculated Phase Shifts Across the Boundary Layer; f = 20 Hz 170 Fig. 5-4: Comparison of Skin Friction Coefficients Using Data Obtained by Direct Measurement, Velocity Curve Fit and Computation: Steady Turbulent 171 xi Fig. 5-5: Computed Average Skin Friction Coefficients at Different Free Stream Disturbance Frequencies 172 Fig. 5-6: Time History of Skin Friction Coefficients Along the Flat Plate in a Single Cycle 173 Fig. 5-7: Comparison of Wall Friction Coefficients Obtained by Direct Measurement, Velocity Curve Fit and Computation: f = 20 Hz, S = 0.1 m 174 Fig. 5-8: Comparison of Wall Friction Coefficients Obtained by Direct Measurement, Velocity Curve Fit and Computation: f = 40 Hz, S = 0.1 m 175 Fig. 5-8: Comparison of Wall Friction Coefficients Obtained by Direct Measurement, Velocity Curve Fit and Computation: f = 40 Hz, S = 0.1 m .... 176 Fig. 5-10: Calculated Shear Stress Profiles at Various Phases in a Cycle: f = 20 Hz 177 Fig. 5-11: Calculated Shear Stress Profiles at Various Phases in a Cycle: f = 30 Hz 178 Fig. 5-12: Calculated Shear Stress Profiles at Various Phases in a Cycle: f = 40 Hz 179 Fig. B-1: Generation of Constant Skin Friction Lines for Clauser Plots 196 Fig. C-1: Schematics of Viscous and Thermal Boundary Layers 197 Fig. C-2: Correlation of Preston Tube Reading to Wall Shear Stress 201 xii N o m e n c l a t u r e A Constant, hot wire calibration equation, Area ( m 2 ) Ai Constant in Clauser's plot equation a Constant B Periodic fluctuation amplitude ratio, calibration constant i B Calibration constant b Constant C, c Absolute velocity ( m/s ), constant Cf Skin friction coefficient Cq Flow coefficient D, d Diameter of duct ( m ) E Anemometer voltage output (Vo l ts ) Ea Anemometer voltage output at zero velocity (Volts ) Ffc Burst frequency ( H z ) / Functional relationship, dimensionless stream function, wake generation frequency ( Hz) / Dimensionless velocity profile / Wall friction parameter H Shape factor = 5*/e N Calibration constant, total number of cycles during data acquisition P Pressure ( P a ) p Pressure fluctuation ( Pa) AP Change in pressure along a tube, flat plate Q Traveling velocity of fluctuation ( m/s) Or Rate of heat transfer ( W ) xiii QwL Rate of heat transfer at the wall ( W ) Re,Re x Reynolds number = uex/v S Airfoil spacing ( m ), surface area ( m 2 ) T Period in a cycle ( sec ), Temperature t Time ( s e c ) t + Time scale K,U( t ) , Ui(t) Instantaneous boundary layer velocity ( m/s ), 1 = 1 ....N U Instantaneous velocity at time ti U Time averaged mean velocity ( m/s) <u> Ensemble average velocity in the free stream( m/s) U(t) Ensemble-averaged velocity u Average velocity of N records at time ti (u) Ensemble average velocity in the boundary layer ( m/s) ub,ur Blade ( ro to r ) velocity ( m/s ) u V Unsteady boundary layer velocity ( m/s ) u,up Organized velocity fluctuation component ( m/s ) Ue Instantaneous free stream velocity ( m/s) AUe Amplitude of free stream fluctuation velocity ( m/s) Average free stream velocity ( m/s) u Random fluctuation of velocity ( m / s ) Friction velocity = ^/T w 1 p ( m/s) + u Dimensionless velocity = u/ur V Time averaged mean velocity ( m/s) Vs Slip velocity ( m/s) V Boundary layer velocity in the y-direction ( m/s) W,w Relative velocities ( m/s) X,x Cartesian coordinate along the flat plate ( m ) y Cartesian coordinate in the transverse direction ( m ) + y Dimensionless transverse coordinate = yuT /v Y + Dimensionless coordinate in Clauser plots = yUoo Greek symbols a Angle of attack on airfoils P Angular direction of the relative velocity 5 Boundary layer thickness ( m or mm ) 5 * Displacement thickness 5 T Thermal Boundary Layer profile thickness e Eddy viscosity ( m 2/s ), perturbation parameter + 8 Dimensionless eddy viscosity 4> Phase angle v? Stream function n Dimensionless transverse coordinate = y(u0/vx) K Karman constant H Dynamic viscosity V Kinematic viscosity CO Fluctuation frequency ( rad /s ) ID Reduced frequency = fX/Ue 6 Momentum thickness P Density ( k g / m3 ) T Boundary layer shear stress T w Wall shear stress XV A c k n o w l e d g m e n t s I am greatly indebted to professor Evans for the guidance, support, encouragement and understanding I enjoyed throughout the course of my study at UBC. His trust and faith in me have been constant sources of strength and morale. I salute the opportunity bestowed on me to achieve one of my dreams. I am also thankful to professors Hill, Green and Lim for volunteering to serve on my advisory committee. The advice, follow-up, encouraging ideas, the challenge and apt critique of the study are honestly appreciated. The technical and office staff of the department have always been supportive. They have been cordial and friendly in the day-to-day interactions and acted professionally on official matters. To all I say thank you. Fellow students have been helpful in various ways. I want to thank Dr. Stefurak in particular for his friendship and various thoughtful suggestions. Finally, I am so grateful to my parents Ato (Mr.) Gete Desta and W/o (Mrs.) Asmarech Gebre Medhin for their love and support no matter where I am. Though formally uneducated and with no resources for themselves, they set me from the beginning to grasp the values of belief in God, education and hard work for a higher goal. Words of wisdom from my father like "a wise man never stops learning" continually inspire me to succeed. I humbly dedicate this work to them. Chapter 1 INTRODUCTION A N D OBJECTIVES Turbulent boundary layers in turbomachines are three-dimensional and unsteady. The flow becomes periodically unsteady due to the relative motion of the rotor and stator blades in a stage. Fig. 1.1 illustrates a cross-section of a multi-stage turbomachine and Fig. 1.2 depicts a schematic of a stage cascade. The rotor, with a tangential velocity Ubiade, sees the oncoming fluid with absolute velocity C, having a relative velocity W with respect to a moving frame of reference attached to the rotor itself. Fluid entering the rotor blades with a certain incident angle interacts with the blades, forms boundary layers and exits from each blade deflected from its original direction. As a result of viscous action a velocity defect is created behind each blade. The intensity of the velocity defect w affects both magnitude and direction of the absolute velocity C. A large velocity defect generates a varying incidence angle , and a fluctuating velocity magnitude at the entrance of the stator blades. In addition, such a velocity defect, or wake, travels downstream with a finite speed and intermittently perturbs the boundary layer on a downstream blade with a discrete frequency. This interaction influences the development of the boundary layers. Therefore, to estimate friction related losses, flow developments inside blade passages need to be understood. Turbomachine stages operate in the intermediate ranges ( 1.0 to 10.0 ) of a characteristic frequency known as the reduced frequency. The reduced frequency is defined as the ratio of the time it takes for a fluid particle of velocity U to travel a distance x to the time period for the frequency in one cycle. It can be expressed as m = cox/U where co is the frequency of oscillation. The periodic perturbation and the Chapter 1 Introduction and Objectives 2 consequent unsteadiness influences the aerodynamics and performance of the machine. The rotor-stator interaction is not limited to a single stage, but the influence persists throughout the downstream stages in a multi-stage machine. Periodic fluctuations in turbomachines are of the traveling-wave type. In contrast to standing-wave type fluctuations that have infinite convection velocity, traveling-waves have finite wave speeds. While standing-waves are dependent only on time, traveling-waves depend on both space and time. Mathematically, the two types are expressed as follows: 1/(0 = Be iM Standing-wave, U (x,t) = B eJt0 ^ ~  x 1 T r a v e l i n g - w a v e , where U is any dependent variable, B is the wave amplitude and Q is the wave speed. This study addresses several fundamental questions in the process of unsteady wake-boundary layer interaction. How does the turbulent boundary layer on a flat plate or stator react to a periodic disturbance generated upstream by rotor blades? What are the effects of traveling-wave disturbance frequency on the behavior of the boundary layer? Is there a correlation between periodic fluctuation frequency and skin friction? What is the significance of this fundamental study to flow conditions and performance of a turbomachine? Ultimately, from such an investigation one hopes to obtain clues that may help in designing an improved machine. In practice this usually means reducing drag and other losses through a better understanding of the phenomena of turbulence and unsteadiness, and by controlling flow separation. Therefore, the underlying motivation in this study is to understand the details of the flow regime inside the individual stages of turbomachines as a consequence of Chapter 1 Introduction and Objectives 3 wake-boundary layer interactions. However, several factors, like flow incidence and exit angles, secondary flows, blade twist, blade curvature, Mach number, end wall effects are simultaneously involved in a real machine. That makes it difficult to isolate and study the effects of periodic traveling wakes on the blade boundary layers. To this end, a simplified, representative model, manifesting the fundamentals of a single stage, (rotor-stator combination), was developed as schematically shown in Fig. 1.3. To investigate the wake-boundary layer interactions on velocity, random fluctuation 2T distributions, and on the skin friction coefficient (c* = — ^ - ), the flow condition in a real J pu£ machine needs to be simplified and replicated with a physical model. A two-dimensional experiment with the rotor represented by a set of moving blades (airfoils) and the stator replicated by a flat plate in a wind tunnel was designed and built. Dimensional analysis was used to limit the number of independent variables in the experiment. This method leads to functional relationships among selected nondimensional variables, the exact forms of which are determined based on experimental data. Appendix A shows the procedure of dimensional analysis used in this study. A calculation scheme was used to solve the two-dimensional unsteady boundary layer equations. The boundary layer was externally disturbed with a traveling-wave type sinusoidal fluctuations. The objectives of this study are limited as summarized below. 1. To develop a two-dimensional physical model to simulate wake-boundary layer interaction in turbomachines. Chapter 1 Introduction and Objectives 4 2. To examine the response of skin friction on a stator (flat plate) to external, periodic traveling fluctuations and make inference to boundary layer friction losses in turbomachines. 3. To investigate the turbulent boundary layer velocity that results from traveling wake -boundary layer interactions for comparison with an unperturbed layer. Determine whether the assumption of equivalence between steady and unsteady velocity profiles is valid. 4. To explore the impact of the external organized fluctuations on the random boundary layer fluctuations. Determine the extent of the coupling of these fluctuations and make recommendations about the inclusion of unsteadiness effects in mathematically modeling the Reynolds stresses. 5. To learn about and quantify the time response or phase shift of the boundary layer velocity due to traveling-wave periodic perturbations. In what follows, Chapter 2 presents a review of the published literature regarding the effects of externally imposed fluctuations on the development of boundary layers. In Chapter 3, the experimental apparatus, measurement techniques and data reduction used in this work are described. An experimental investigation into the development of stator blade (flat plate) turbulent boundary layer interaction with the traveling rotor wakes, including measurement of unsteady boundary layer velocities and wall shear, is presented in Chapter 4. Chapter 5 describes a calculation procedure used to predict traveling-wave wake-boundary layer interactions. The last chapter, Chapter 6, presents the conclusions with some suggestions for further study. Chapter 1 Introduction and Objectives 5 Chapter 1 Introduction and Objectives 6 Chapter 1 Introduction and Objectives 7 Fig. 1.3 A Simplified Physical Modeling Process of a Turbomachine Stage Flow by a Two-Dimensional Wind Tunnel Flow. 8 Chapter 2 LITERATURE REVIEW 2.1 Introduct ion When a solid body is immersed in a moving fluid, it experiences the effects of both inertia and viscous forces. Due to viscosity, a retarded region of fluid flow develops on the surface of the body. The retardation of the flow increases towards the surface of the solid body as one travel from the external (free stream) flow to the surface where it finally assumes the speed of the body on the wall. This layer of fluid, depending on the relative importance of inertia to viscous forces, is customarily known as a laminar, a transitional, or turbulent boundary layer. Besides the Reynolds number, the boundary layer also depends on several other factors such as free stream turbulence intensity, surface roughness, Mach number, pressure gradient and surface curvature. When the smooth flow nature of a fluid motion is no longer stable, it transforms to a turbulent flow phenomenon. Indeed turbulent flow appears in any flow where inertia forces are substantial compared to the prevailing viscous forces. The definition of turbulent flow has been modified throughout the years as the understanding of the flow regime becomes clearer. It has been defined by way of explaining its characteristic behaviors by various authors. Among others, according to von Karman [ 1 ] ; "Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they flow past solid surfaces or even when neighboring streams of the same fluid flow past or over one another." Later on, Hinze [2 ] proposed the following definition as a more descriptive one: Chapter 2 Literature Review 9 "Turbulence fluid motion is an irregular condition of flow in which the various quantities show random variation with time and space coordinates, so that statistically distinct average values can be discerned." From the above definitions, it is quite clear that turbulence is characterized as a nonsteady, three-dimensional, rotational, diffusive fluid flow containing a broad spectrum of frequencies, amplitudes and scales of motion where the structure consists of eddies and vortices of various sizes. In many engineering applications, and in many natural phenomena involving fluids, most flows are turbulent in character. Fluid machines like compressors, turbines, reciprocating engines, propulsion engines, aircraft surfaces, ships, rockets, etc., are mostly subjected to turbulent flows. It is therefore of great importance to understand the mechanisms of turbulent boundary layers in any fluid handling equipment or any surface of interest that comes in contact with a fluid in motion. From such an understanding, it is possible to establish a methodology of predicting the behavior of the flow field and performance of the desired machine. Laminar boundary layer studies are abundant in the literature. It seems that the problems are understood well and solutions to the governing equations are readily available. In current boundary layer studies, laminar boundary layer data is often used as a test case. The most challenging problems in fluid mechanics are, however, generally related to turbulent flows. The complexity and challenge come from the nonlinear and turbulent stress terms in the governing equations. Thus, for this flow, the equations of motion are not yet amenable to theoretical solutions. Even the rapid growth of computers, though very important, may not be able to cope with the huge storage requirement and very long computing times [ 55 ] . It casts doubt on whether the problem Chapter 2 Literature Review 10 of turbulent flow, in its entirety, can be handled in the foreseeable future. One has, therefore, to combine all the available methods of turbulent boundary layer studies including computational, theoretical and experimental methods. In the following sections, a survey of the literature with regard to boundary layers is presented. It concentrates on fundamental boundary layer studies related to turbomachines by means of experiments or computations. Findings are presented, discussed, and reviewed in reference to unsteady boundary layer flow conditions. 2.2 Unsteady Laminar Boundary Layers Much work has been conducted and documented on the theory of incompressible, Two-Dimensional, unsteady laminar boundary layers [3,4,5,6,61,62]. Among others, the response of the viscous part of the flow to fluctuations imposed on it externally has been exhaustively studied. Externally imposed fluctuations that give rise to unsteady boundary layer development may be classified into standing wave and traveling wave fluctuations. The classification fundamentally depends on the stationary or traveling nature of the organized external disturbances. Time-resolved free stream perturbations in velocity and/or pressure, that do not depend on any space coordinate, give rise to stationary wave types of fluctuations. For instance, a sudden change in the direction of a fluid motion on an airfoil, or fluctuations in the magnitude of the velocity of an oncoming stream relative to a body produce fluctuations in the external flow. This oscillation is classified as a standing wave type of fluctuation. In such cases the wave speed is deemed to be very large or approaching infinity compared to the mean free stream velocity. On the other hand, if the external disturbance is dependent on both space and time, and if it is transported with a finite Chapter 2 Literature Review 11 wave speed, the disturbance is classified as a traveling wave type of oscillation. In such cases the response of the boundary layer may be influenced by both frequency and the traveling speed of the perturbation. 2.2.1 Laminar Boundary Layers Subject to Standing Wave Fluctuations The problem of finding the response of a Two-Dimensional laminar boundary layer to standing wave type external fluctuations is believed to be solved. One of the earliest studies was conducted by Lighthill [4]. He developed analytical solutions by asymptotic expansions in powers of small amplitudes for a boundary layer with free stream velocity expressed as Ue(x,t) = Uo(x)[l + Be iwt], 2-1 where U e is the external instantaneous velocity, U 0 is the mean free stream velocity, B is the amplitude ratio, co is the frequency of oscillation, x is length in the flow direction and t is time. Separate solutions for low and high frequency conditions were developed. For low frequencies, the asymptotic solution showed phase advance at the wall and amplitude overshoot inside the boundary layer. Phase advance and amplitude overshoot is evidently due to the low inertia fluid near the wall being subjected to the same pressure gradient as that in the free stream. The low frequency approximation was, however, limited to a low frequency parameter, i.e., reduced frequency less than 0.6. The small amplitude assumption also restricts external fluctuations to less than 10% of the mean flow. For large reduced frequencies, the responses of the boundary layer are oscillations essentially of the shear layer type. The shear layer originates at the wall and its thickness is of the order Vv/co . Here V is the kinematic viscosity. The shear layer is Chapter 2 Literature Review 12 also found to be unaffected by the mean flow. Physically it means that the viscous flow has no time to respond to the velocity fluctuations imposed by the oscillations of the external pressure gradient. An extensive experimental and theoretical study of an oscillating laminar boundary layer of the standing wave type with and without adverse pressure gradient was reported by Hill and Stenning [5]. In the experimental study, they used an oscillating slide valve located downstream of the wind tunnel test section. The oscillating valve generated a purely time-dependent free stream velocity fluctuation. Over the frequency and amplitude ranges covered (w =0.1-10.0, AUe/ue up to 0.1), the mean velocity profile showed no change from that of steady flow cases. The insensitivity of the mean boundary layer velocity profile to the free stream fluctuation was confirmed by other laminar boundary layer studies as well [6, 9]. However, one has to be cautious regarding this conclusion for the correction to the mean flow appears in higher order terms of order e2 [7]. Here e is a perturbation parameter. For low frequencies, measurements under zero pressure gradient flow condition provided good agreements with the analytical solutions of Lighthill [4]. On the other hand, for high frequencies with the reduced frequency greater than 10.0, the experimental results agreed well with the shear wave solution. In addition, a new analytical solution that took into account the convective effects was developed. It covered the intermediate frequency range of the reduced frequency to between 1.0 and 10.0. In this range also, the lagging phase angle that was previously predicted either by the low frequency solution or the shear wave solution was indicated. This new solution yielded a good description of phase and amplitude behaviors in the intermediate range. The solution method is not restricted to small perturbations. An extension of Lighthill's low frequency Chapter 2 Literature Review 13 solution, the shear wave solution and the intermediate solution to the Howarth type boundary layer flow (adverse pressure gradient ) gave good agreements between experiment and prediction. A mean diffusion parameter of 0.1 was used to obtain the Howarth flow. In this case the flow frequency applicability was lowered to W equal to 0.45. The problem of stationary wave type perturbations of the free stream appears to be well understood. All the basic trends like velocity overshoot, phase advances or lags are almost accurately predicted by the analytical solution techniques. In general, with the availability of large capacity computers, the solution methods may not any more be bound with the restrictions and assumptions that are used to find a linear predictive technique. Higher order terms may be included in a computational scheme. But one has to be careful when large fluctuations of the free stream velocity and pressure exist, as flow reversal (separation) in the boundary layer may take place even if there is no positive (adverse) pressure gradient in the free stream. For example, for the designer of a turbine blade, whose design procedure usually depends on a steady flow model, the existence of unsteadiness, and particularly unsteady separation, are serious problems. If these flow phenomena exist and are significant, it is not sufficient to simply neglect them in predicting the performance of the blade. 2.2.2 Laminar Boundary Layers Subject to Traveling Wave Fluctuations A study regarding the case of free stream laminar boundary layer fluctuations generated by a system of vortices that travel downstream with a speed Q equal to 0.77U o was conducted by Patel [6]. He considered a laminar flat plate boundary layer with the free stream velocity expressed by Chapter 2 Literature Review 14 Ue(x,t) = Uo + Af/esincoO - x IQ) = Uo[\ + AUe I Uosinco(t - x IQ) 2 " 2 Ue = Uo + Ulei{» {t- xl® where U 0 is the time-mean free stream velocity, Id or AU e is the amplitude of the free stream fluctuation, co is the frequency and Q is the velocity of the traveling wave. This experiment covered reduced frequencies ranging from 0.314 up to 1.571 and free stream fluctuation amplitudes AUe/U0 ranging up to 0.14. The experimental results obtained demonstrated that the mean flow velocity profile was the same as the velocity profile found in steady flows. This conclusion is similar to the conclusion drawn in the case of standing wave free stream fluctuation results discussed earlier [4,5]. Patel [6] extended Lighthill's analysis to the case of traveling wave free stream fluctuation by linearizing the boundary layer equation and sought solutions in terms of the reduced frequency ro=wX/[/o. The size of the reduced frequency parameter is a measure of the unsteadiness in the flow. The reduced frequency m is the ratio of the time for a fluid particle to travel through a length x to the period of the free stream fluctuation wave. Qualitatively, it represents the ratio of the thickness of steady boundary layer to a diffusion distance 5 - Vv/co to which the viscous effect due to unsteadiness is felt in the flow [8]. For w « 1 . 0 , the problem reduces to the quasi-steady case where there is a phase lead whose value increased monotonically towards the surface. For w >>1.0, the unsteady response of the boundary layer was described by an extended shear wave type equation. Chapter 2 Literature Review 15 The response of the laminar boundary layer was dominantly influenced by the velocity of the traveling wave Q. Similar response to that of the purely time dependent oscillating flow was obtained for large Q/U 0 . The asymptotic solution (large traveling velocity) indicated a phase advance at the wall and velocity overshoot in the boundary layer. The overshoot occurred closer to the wall. However, reduction in Q/U D caused the amplitude of the velocity fluctuation and the phase advance at the surface to decrease. At Q/U 0 equal to 1.0, the phase advance at the wall dropped to 38 degrees while a 10 degrees phase lag appeared in the outer part of the boundary layer. According to the analytical results, as Q/U 0 decreased below 1.0, a wall shear phase lag ranging from 37 degrees at the lowest frequency to 122 degrees at the highest frequency resulted. There is a sharp contrast in the phase lead when compared with the phase shift observed for a boundary layer subjected to a stationary wave type free stream fluctuation. For standing waves of high frequency fluctuations, the result indicates a phase lead reaching a maximum value of 45 degrees at the wall. The experimental results for Q/U 0 = 0.77 showed reasonable agreement for low frequencies, while at higher reduced frequencies of up to 1.571, the agreement was not satisfactory. The explanation for the behavior of the boundary layer when Q < U0 lies in the response of the boundary layer to the driving pressure gradient. Under this circumstance, the pressure gradient acts in phase with the inertia of the fluid in the boundary layer that always resists fluctuations. This state of flow leads to the dramatic phase shift, from phase lead to phase lag, as the free stream perturbation changes from a stationary wave to a traveling wave. The role of inertia in the boundary layer and the pressure gradient across it is best understood by considering the free stream pressure gradient expressed by the unsteady Bernoulli equation. Pressure gradient in the free stream is given by Chapter 2 Literature Review 16 1 dp dUe TJ dUe — — = + Ue p dx dt dx 2-3 If the free stream velocity is expressed by Ue(x,t) = Uo + BsincoO - x IQ), time and space derivatives of the free stream velocity are expressed as: dUe 2-2 = coBcosto(f- x/Q) inertia term, cos co (t - x IQ) convective term. dt dUe tofitVo 2-4 2-5 dx Q Substituting the above time and space derivatives into the pressure gradient equation, after some rearrangement, we obtain the following expression: 1 dp = toBcosco p dx ( *) 1 - ( U°) ( u0) B — f x) t - — — sin to t -to to Q) 2-6 Thus by considering the relative magnitudes of certain quantities like B, U 0 and Q, it becomes clear in how various contributions to the phase shift are realized. For instance, if B < ~ 0 .1 , the pressure gradient felt across the whole boundary layer becomes: - - — = coBcostoO - x 10[1 - Uo IQ]. p dx 2-7 In this condition it is apparent that the free stream perturbation is a stationary wave type leading to a phase lead in the boundary layer. Evans [9] conducted a computational study of laminar boundary layer free stream fluctuations of both standing wave and traveling wave types. He used a code originally developed by Cebeci and Carr [10] for a stationary free stream perturbation and modified it to accommodate a traveling wave free stream fluctuating velocity. Results of Chapter 2 Literature Review 17 computations for the standing wave free stream disturbance predicted well the data obtained from other works [4, 5, 6]. Fig. 2.1 shows results and comparisons made by Evans [9]. As to the traveling wave free stream fluctuation condition, although there is very little experimental or theoretical information, the calculation provided a useful perspective into the response of the boundary layer when subjected to these types of fluctuations. Both the reduced frequency, w, and the ratio of the traveling wave velocity to the mean free stream velocity, Q/Uo, were the important parameters used to describe the unsteady laminar boundary layer development subjected to traveling wave free stream fluctuations. In the study mentioned above, with w =0.314 and Q/U 0 = 0.77, calculations show excellent agreement with the phase shift measurements of Patel [6] while the amplitude ratio was found to be under-predicted. Results for Q/U 0 -» °° confirm the phase lead of the inner part of the boundary layer in agreement with the analytical solution. When Q/Uo = 1.0, as the inertia and convective parts of the pressure gradient cancel out, there is very little phase shift expected and the amplitude ratio satisfies the imposed free stream boundary conditions. In the case where Q/U 0 is less than 1.0, an increase in phase lag was obtained as the values of Q/U 0 decrease. The amplitude ratio profiles remain similar to the standing wave cases under low frequency conditions. On the other hand, as the parameter Q/U 0 gets greater than 1.0, the phase lag decreases turning into a phase lead for larger values. When dealing with high frequencies where Q/U 0 = 0.77, the numerical procedure did not accurately predict the overshoot in the inner part of the boundary layer. Chapter 2 Literature Review 18 2.3 Unsteady Turbulent Boundary Layers It is well known that turbulent flow in general is remarkably different from laminar flow. Turbulent flow has much greater diffusivity than laminar flow. This behavior of a turbulent flow enables a boundary layer to flow without separation into regions of steeper adverse pressure gradients. Such a phenomenon is due to the diffusion of high-momentum flow from the outer edge inward, overcoming the retarding effect of the low inertia boundary layer fluid near the wall. Besides controlling flow separation of boundary layers to some degree, in some practical applications, the higher rate of diffusion of mass (mixing) in turbulent flow makes it more useful than a laminar flow. In spite of these benefits, a turbulent boundary layer induces a much higher skin friction coefficient or heat transfer coefficient than a laminar boundary layer. These coefficients are indicative of the losses associated with such flow regimes. Both coefficients are of considerable interest for one involved in a compressor or turbine study. The response of a fully turbulent boundary layer flow to an organized fluctuating upstream or free stream disturbance has been rarely studied on a fundamental level. Of those studies conducted, the work of Karlsson [11] pioneered the first important experimental investigation of a turbulent boundary layer when the free stream disturbance was purely time dependent. This flow disturbance is equivalent to a stationary or standing wave perturbation to the free stream flow. On the other hand, with respect to time- and space-dependent free stream disturbances, Patel [12], Yip and Evans [13] and Holland and Evans [72] conducted studies for limited cases of frequency and amplitude variation. In what follows, investigations reported in the literature regarding the responses of a turbulent boundary layer to a stationary wave and traveling wave free stream fluctuations are reviewed. Chapter 2 Literature Review 19 2.3.1 Standing Wave Type Free Stream Fluctuation A comprehensive review of unsteady turbulent boundary layer was presented by Carr [14]. The main thrust of the presentation was to provide documentation of the pertinent studies, the nature of different experimental facilities and results of the experiments. The difficulty of modeling unsteady turbulent flow problems in a laboratory setting has been apparent as one looks at the various unsteady turbulent boundary-layer flows documented. Experimentalists implemented different facilities and experimental conditions in their studies. In all cases, experiments were constrained by the respective experimental facilities implemented, and individual studies were limited in scope. Because of the novelties of the experimental designs, one should be cautious when comparing results obtained in such facilities. As mentioned earlier, Karlsson's study of unsteady turbulent boundary layers whose free stream was perturbed by a stationary wave, was among the basic studies conducted. His facility was an open return wind tunnel where the flow oscillation was produced by a set of rotating vanes installed near the exit of the tunnel. As the vanes rotate, they produce a pulsating flow due to the variable blockage of the tunnel, subjecting the boundary layer developing on the flat plate to a purely time dependent (stationary wave) free stream oscillations. In another technique, Brembati [15] installed a flexible section as part of the roof of an open return wind tunnel where it was possible to generate a combination of a fluctuating free stream velocity and an adverse pressure gradient. For detailed information about the various facilities, reference should be made to Carr's paper [14]. Making use of different techniques and facilities has its advantages and limitations, and obviously no one design is clearly better than another in such diverse investigations Chapter 2 Literature Review 20 of unsteady turbulent boundary layers. A broad spectrum of unsteady turbulent flows may be investigated by different methods and facilities where a single technique alone is restricted in its use and scope rendering limited results. One of the main conclusions from the unsteady turbulent boundary layer studies concerns the time-averaged mean velocity. Assessing all the experimental results available, the studies conducted demonstrate that the time-averaged mean velocity is nearly invariant even for high amplitude and different frequencies of organized fluctuations of the free stream flow. In other words, it means that the time mean velocity profile of the boundary layer remains unchanged, and appears as one would expect for a steady flow with a corresponding mean velocity of the oscillating outer flow. This result has been obtained and observed through various studies [11, 12, 16] over a wide range of frequencies and amplitudes. The same behavior of the mean velocity profile in unsteady laminar boundary layers has also been observed. Not only is the time mean velocity profile nearly invariant in simplified experimental studies, it was also demonstrated to hold true even in a complex experiment performed on a jet engine compressor as shown by Evans [17]. Although the time mean velocity may suffice in a general and preliminary design procedure, it is important to include the effects of periodicity that are otherwise camouflaged by the time averaging procedure during the detailed design processes. The unsteadiness effects are felt through second order terms of the free stream perturbation which in most linearized solution methods are not accounted for [ 7 ] . There are conditions where analysis of an unsteady flow problem is performed based on steady flow results. This may be sufficient if the need is just to obtain the mean performance characteristics of a system. The assumption that the mean unsteady Chapter 2 Literature Review 21 value is the same as the steady value can be misleading if the transient quantities and their interactions with the mean flow are not properly evaluated, and if the purpose of the study is to identify the fundamental nature of the unsteady flow field. Basic studies need to explore the flow beyond the time-mean quantities and evaluate the effects of the organized free stream fluctuation amplitude, frequency, and traveling wave velocity on the boundary layer. Besides global considerations, local unsteady effects must be examined as well. For instance, in some cases even though the time mean velocity profile in the unsteady turbulent boundary layers was the same as in steady flow case, flow reversal was observed on a flat plate as reported by Karlsson [11], and flow transformation from laminar to turbulence took place through one cycle on an axial-flow compressor stator blade as shown by Evans [17]. Indeed, the time averaging process can mask the unsteadiness effects and thus cannot be fully relied upon to predict the characteristics of the unsteady turbulent boundary layer. Regarding the unsteady turbulent intensity, studies [16] show the turbulent structure unaffected by the oscillation of the free stream flow field. This leads, under certain conditions, to the use of steady flow turbulence models to predict unsteady turbulent boundary layer behaviors. But there are cases where significant changes in turbulent intensity can be encountered. There may be a situation where the interaction between the oscillatory motion and the turbulence structure is significant as shown by Mizushina et al.[18]. The turbulent intensity in the study no longer had a similar pattern to the steady case when the oscillation was increased beyond a certain critical frequency. This frequency was associated with turbulent burst frequencies in the boundary layer. The velocity distribution obtained from ensemble averaging varied in such cases. Houdeville et al. [19] conducted experiments with amplitudes of oscillation of the order Chapter 2 Literature Review 22 of 30 to 40 percent of the mean outer flow and showed that nonlinear effects in the flow can be significant. The turbulence levels at a fixed point in space fluctuate with the same frequency as the outer flow. This implies that the random disturbances tend to follow the periodic variation of the outer flow fluctuations. The near wall flow responds quite readily to unsteadiness when oscillatory external velocities are imposed on the boundary layer flow. In some cases, the unsteady viscous reaction to the imposed fluctuations is totally confined to the Stokes' layer near the wall. This layer may be neglected if the goal of the study is to predict the global flow behavior of the unsteady flow as shown by Lyrio et al. [20], but when dealing with a problem that depends strongly on the Stokes' layer, prediction of the boundary layer behavior directly depends on the ability to associate the wall stress to the flow in the central region of the boundary layer. Simpson et al. [21] reported major phase changes near the wall demonstrating clearly that the flow near the wall can vary dramatically during free stream oscillations. Unsteady turbulent boundary layers can be generated by a wide range of variations of amplitude and frequency. Even low amplitude or low frequency fluctuations may not lead to quasi-steady flow behavior. In addition, all the effects of unsteadiness may not be represented by a single dimensionless number. This is because of the existence of different time scales for the wall region and the outer region. Moreover, there is a difference in response to temporal and spatial variations of velocity in the boundary layer. Usually, a reduced frequency parameter based on a streamwise dimension x, boundary layer thickness 5, displacement thickness 5*, etc., is used to characterize the unsteady nature of the flow. Another dimensionless parameter that has been used to describe the unsteady effects is the burst frequency. The burst frequency Chapter 2 Literature Review 23 (F b), defined as F b = Uo/55 for a flat plate, was developed from steady flow data, and acts as an indicator of the frequency at which the turbulent eddy structure will respond to the external forcing function. When the free stream oscillates at the burst frequency, some effects on the sublayer were observed by Acarya and Reynolds [22]. They were not observed, however, when the frequency was 60% of Fb. On the other hand, the largest phase shift occurred in the sublayer for frequencies less than 40% of F b as shown by Karlsson [11], 27% of F b as reported by Ramaprian et al. [23]. In many of the unsteady turbulent boundary layer experiments reported in the literature, measurements were made at only one downstream station. In other cases, the boundary layer was not tripped at the leading edge of the test surface. Thus, the task of identifying the local unsteady viscous effect may not be possible without data obtained at several downstream stations. When a certain frequency of oscillation is imposed on a turbulent boundary flow, one has to devise a system of identifying the response to the organized external oscillation and the random fluctuation of the turbulent field separately. Such a method to identify quantities obtained experimentally is a method known as the triple decomposition technique [19, 24]. A field quantity u is decomposed to its components as U = U + U + W 2-8 where U is the time average quantity, U is the organized fluctuating quantity that is derived by an ensemble-averaging process based on a conditional sampling technique, and u' is the random fluctuating quantity. The sum of U and U gives the ensemble average value <u>. This is the ensemble average of instantaneous values selected at specific phase value and summed over a statistically reasonable number of cycles. The quantity that represents the organized fluctuation is then defined as Chapter 2 Literature Review 24 u = (u) - u, 2-9 and the random fluctuation is expressed by subtracting the ensemble averaged velocity from the instantaneous velocity as follows: This technique of decomposition is depicted schematically as illustrated in Fig. 2.2. Details of the ensemble averaging process are given in the data processing section of the next chapter. 2.3.2 Traveling Wave Type Free Stream Fluctuation In the literature , most of the research related to unsteady turbulent boundary layers deals with stationary wave type free stream oscillatory disturbances. But , more often than not, in fluid machinery the free stream fluctuations are of the traveling wave type. This is in fact the prevalent flow condition, especially in turbomachinery stages where the relative motion of rotor and stator causes the unsteadiness. The rotation of blades gives rise to repeated wakes, and the mean stream flow transports the wakes downstream. These transported wakes in turn interact with the boundary layers on downstream blades. Despite the wide-spread existence of traveling-wave free stream disturbances in fluid machinery, the nature of the wake-boundary layer interaction and its impact on the wall shear stress, phase shifts, etc., are not clearly known. This gives rise to several questions: How does the frequency of the traveling wave influence the development of the boundary layers on the blades? How does the amplitude of the free stream fluctuation affect the boundary layer behavior? What is the influence of the traveling wave velocity on the characteristics of the turbulent boundary layer? Is it possible to W = u- (u) 2-10 Chapter 2 Literature Review 25 predict or include the unsteadiness effect during the design process? How are the various quantities correlated with the wall resistance? These and many other questions could be raised and need to be addressed so that one achieves a greater insight into the underlying fluid mechanics of transient and periodic flows and be in a position to improve the performance of turbines and compressors through representative models. In fact the whole range of flow problems in turbomachinery is very complex. It involves several parameters that are space and time dependent. Thus, the flow encountered is not amenable to simple theoretical prediction methods. Nevertheless, simplification of the relative importance of the parameters involved, and fundamental experimental studies, may provide a better understanding of the specific flow phenomenon under investigation. One such study on unsteady turbulent boundary layers disturbed by a traveling wave was undertaken by Patel [12]. He managed to generate traveling disturbances as a series of vortices starting from the entrance of the test section using flexible flaps attached to the mouth of the contraction section of the wind tunnel. The flaps oscillated sinusoidally inducing a series of traveling vortices creating the oscillatory velocity perturbation in the oncoming stream. The periodically shed vortices grow in amplitude as they are convected by the mean flow downstream along the flat plate. In comparison to Patel's experiment, free stream oscillations in fluid machinery are usually decaying rather than growing in amplitude. Frequencies in his study ranged from 4 to 12 Hz. The free stream velocity was 19.8 m/s while the free stream amplitude of fluctuations were up to 0.11 of the mean flow velocity. In this particular study, the flat plate was tripped at the leading edge. Chapter 2 Literature Review 26 Similar to the findings in unsteady laminar boundary layers and unsteady turbulent boundary layer with stationary free stream perturbation, Patel [12] found that the mean velocity profile was insensitive to the traveling wave disturbance. He also developed an approximate analysis based on his results. The conclusion drawn was that the traveling wave velocity exhibits a dominant influence on the turbulent boundary layer response to the free stream oscillations. The results obtained provide important information within the bounds of the limited studies performed. However, it should be pointed out that the linearization process employed in the analysis neglected higher order fluctuation amplitudes which in effect are responsible for the nonlinear interactions in the boundary layer. The linearization process assumes the decoupling of the oscillations and the turbulence (the random fluctuation) restricting the possible interactions with the mean flow. There are indications that the mean flow may be sensitive to the free stream oscillations contrary to prior conclusions [25]. Evans and Yip [13] studied a turbulent boundary layer perturbed by convected wakes in the free stream. Rods mounted on a rotating squirrel cage were implemented to generate periodic wakes. The squirrel cage contains the flat plate inside the system that carried the wake generators. A turbulent boundary layer was maintained by tripping the flat plate near the leading edge. When the result of the profile is compared with the undisturbed turbulent boundary layer profile, an increase in the fullness of the boundary layer profile near the wall was obtained. This result was due to the imposed fluctuations of the wakes interacting with the turbulent boundary layer. Maximum fluctuation in amplitude occurred in the inner half of the boundary layer. As anticipated, the amplitude decreased further downstream as a result of wake decay in the free stream. In addition, the inner part of the boundary layer Chapter 2 Literature Review 27 showed a phase lag. The phase angle increased in the downstream direction, due to the lower local convection velocity that carries the velocity defect of the wake in this region of the boundary layer. These results agreed with the findings of Patel [12] for traveling wave type disturbances. The study did not, however, find a correlation between phase lag and the reduced frequency. The study of unsteady boundary layers was not limited to purely laminar and fully turbulent flows. Some researchers have directed their attention to the exploration of the process of transition from laminar to turbulent flow under simulated rotor-stator interactions [26,27,28]. Simplified experimental models, representative of a turbomachinery stage, were designed and employed in the investigations. The influence of wake passing frequency on the boundary layer development, particularly on the transition process was examined [26, 28]. When the Reynolds' number was relatively low, the boundary layer was found to be turbulent early, underneath the free stream disturbance. The boundary layer became turbulent and laminar intermittently as the wake generators passed in front of the plate. But as the turbulent stripes underneath the disturbed free stream travel downstream and grow together, the embedded laminar flow region disappears and the boundary layer transforms to a fully turbulent flow. Under such conditions, the origin of transition moved up and downstream along the surface. Moreover, the origin moved further upstream as the frequency increased, thereby promoting early transition of the boundary layer. 2.4 Nonsteady Flows In Turbomachines Among the variety of unsteady flows observed in turbomachines, unsteadiness arising from rotor-stator interactions has been closely investigated. It is known that the unsteady Chapter 2 Literature Review 28 interactions of the rotating and stationary blade rows in an axial flow turbomachine affect many aspects of performance such as blade loading, stage efficiency, heat transfer, stall margin and noise generation [29,45]. Designers of turbomachinery blading make use of the results of steady flow analysis obtained from cascade tests. This procedure is equivalent to assuming that the blade rows of an actual axial flow machine are sufficiently far apart so that the flow is steady in both the stationary and rotating frames of reference. Even if this assumption is accepted and considered relevant in an experimental procedure, the flow in a real machine is nevertheless unsteady. Wakes shed from upstream row of blades are transported downstream interacting with subsequent rows of blades, and enhance flow unsteadiness. Effects of wakes are felt even in the far field regions of a flow [29]. Experiments performed by Laksminaryana et al. [30] on the decay of both rotor and stator wakes in an axial flow compressor indicate that the decay rate of wakes was more rapid in the near wake regions than in the far wake regions. It was also observed that the decay rates in a real machine are different from decay rates obtained from isolated blades in a rectilinear cascade test. As the spacing between blade rows is typically within the range of 1/2 a blade chord, the effect of wakes in a real machine is felt strongly in the row of blades immediately downstream. However, there is little information available for the designer that enables him to take into account during the design process the unsteadiness generated by upstream wakes. Convection and interaction of the wakes generated by turbomachine blades with downstream blade rows significantly influences the profile loss compared to that in cascade operations. For instance, the rotor profile loss in a single stage machine was Chapter 2 Literature Review 29 between two and four times greater than that for the same cascade model operating with steady inlet conditions [30]. It has been indicated that the decrease in efficiency was attributed to an increase in the profile loss manifested due to the effects of unsteadiness. The aerodynamic loading of a blade in a real machine varies with a change in incidence angle of the oncoming flow and the relative velocity associated with it [31]. The relative velocity at the exit of an upstream rotor exhibits a velocity deficit in the wake region. A change in the angle of attack of the flow from an upstream rotor impinging on a downstream stator is schematically illustrated in Fig. 2.3. Such fluctuations in the angle of attack and magnitude of the velocity as viewed from a stationary frame (stator) originate from the fluctuations of the relative velocity as seen from a rotating frame (rotor). Therefore, a downstream blade exposed to such fluctuations is subject to variations in aerodynamic loading. Detailed studies concerning the wake-passing effects on flows in compressor blades were performed by various researchers [32, 33, 34]. Studies dealing with turbine blades were conducted by Hodson [29]. Unlike compressor blades, where one is primarily concerned with the impact of unsteadiness on the aerodynamic performance, in the case of turbine blades, the influence of unsteadiness on the rate of heat transfer to the blade surface is of paramount importance. The response of a row of blades to the free stream unsteadiness is of fundamental importance in determining the heat transfer and/or aerodynamics of the stage considered. The works of Evans [32] and Walker [33] indicate how the wakes were responsible for the unsteady transition process on the stator located downstream of the rotor. Early transition to turbulent flow is attributed to the wakes impinging on the boundary layer developing on the stator. Chapter 2 Literature Review 30 Although these experiments in real machines were successful in contributing to a better understanding of the nature of these unsteady flows in practice, the detailed. process is not clearly known. Because contributions from other sources coexist along with that of the wakes, the unsteadiness information pertaining only to the wakes cannot be clearly identified. Various factors contribute to the generation of unsteadiness in turbomachinery flow. Among those factors are the free stream turbulence, periodic wakes shed and transported from upstream blades, upstream potential interaction due to the relative motion of blades, inlet flow distortions, rotating stall, end wall effects, leakage, Reynolds number, wall roughness, etc. It is difficult if not impossible to determine the relative importance and contributions of each factor when all prevail simultaneously in an actual machine. One has to go through a series of assumptions and simplifications to idealize and model the flow. From such a process it is possible to estimate the relative importance of a particular parameter. The desired parameter may be obtained via dimensional analysis. Then it may be feasible to conduct a basic experiment to get an insight into the fundamentals of the flow. In this study the concern is unsteadiness due to wake generation, transport and its interaction with a turbulent boundary layer. These effects were isolated by modeling a rotor-stator interaction using linearly traversing airfoils in front of a flat plate in a wind tunnel. The effects of the transported wakes on the development and characteristics of the turbulent boundary layer and the unsteady wall shear stress were then investigated. Chapter 2 Literature Review 31 2.5 Measurement Of Skin Friction In Turbulent Boundary Layers It is known that a body moving in a fluid experiences various types of forces and moments as a result of its interaction with the surrounding medium. Of practical interest in this particular case is the drag force. It resists the motion of the body and acts parallel to the axis of the immersed object. Depending on the shape of the body, the drag force may entirely be form drag (pressure drag), skin friction drag or a mixture of both. The tangential component of the surface force on the body emanates from the wall shear stress. Thus, the integrated skin friction on the surface of the body in direct contact with the moving fluid is of practical importance. For instance, streamlined (slender shaped) bodies like airplane wings and turbomachinery blades encounter significant amount of skin friction forces. Estimation of such forces may be obtained experimentally through velocity and wall shear stress measurements in the boundary layer of the flow, or via computations supported by semi-empirical closure equations for the basic boundary layer equations. The ultimate goal of many experimental or numerical studies of fluid flow is to determine surface distributions like skin friction, heat transfer and pressure. In particular, knowledge of the drag created by fluid flowing over a solid surface is essential in the understanding and design of many practical applications, whether it is a turbomachine, an aircraft or flow through a pipe. There has been, therefore, a considerable effort directed to the measurement of skin friction on solid bodies immersed in a fluid flow. Until recently however, experimental, theoretical and numerical methods were limited to steady or time-averaged flow fields. Some flows that are usually treated as steady actually have large components of unsteadiness that could appear in the form of random turbulent fluctuations, large scale Chapter 2 Literature Review 32 coherent flow structures, forced pulsation due to imposed forces and excitations, and upstream wake generation. It has thus become of great relevance to obtain time-resolved components to accurately predict flow fields and performance of fluid machinery. Skin friction and heat transfer gages have been traditionally used to disclose features of a flow in a system. Hot film probes, for instance, have been employed for the measurement of skin friction for more than half a century. There are also other methods that have been in use to determine wall shear stress. In general, methods of wall shear stress measurement may be classified as direct, indirect and heat transfer techniques. The indirect methods can be further divided into momentum balance technique and correlation methods. The correlation methods extract information about the wall shear stress from well-established relations. For example, the use of a Preston tube depends on the law of the wall that has been determined to hold true for a number of flow situations. Because of its dependence on a large part of the boundary layer, the results are global in nature. On the other hand, some devices used to measure the wall shear stress depend on the flow in the immediate vicinity of the sensor. These methods also relay on established correlations between the flow and the shear stress. The flush-mounted sensor is such an instrument. The momentum balance technique depends on the measurement of integral quantities to determine the wall shear stress. In this instance, knowledge of the velocity profile on the solid surface at a designated station is required. The ideal technique to measure wall shear stress is to employ the direct method. In this method, the force applied to the wall by the fluid is directly measured. The technique does not depend directly on the flow field or the fluid properties. The floating element Chapter 2 Literature Review 33 sensor is an example of such an instrument. Nevertheless, the advantage of using this instrument is outweighed by a number of mechanical problems . Details of the various techniques of wall shear stress measurements are given in the literature [35, 36, 53]. 2.5.1 Structure Of Turbulent Boundary Layers The structure of the turbulent boundary layer plays an important role in the measurement of the wall shear stress. It has been accepted that the velocity profiles of turbulent boundary layers, at least in moderate pressure gradients, have inner layers for which the velocity scale is the friction velocity u T [37]. The law of the wall and the law of the wake [38] are valid for various flow conditions and configurations. The inner part of the velocity profile consists of the viscous sublayer adjacent to the wall, a buffer zone and a logarithmic region. A schematic of turbulent boundary layer regions is shown in Fig. 2.4. Assuming a constant shear stress in the vicinity of the wall, the part of the velocity profile adjacent to the wall can be represented as where u is the velocity in the boundary layer, y is the distance from the wall, v is the 2-11 kinematic viscosity, u T is the friction velocity defined by Uj = JTW/p , T w is the wall shear stress and p is the density of the flow medium. For the viscous sublayer the equation reduces to a linear equation as u _yuj Itj. v 2-12 In the logarithmic region, the equation for the velocity profile becomes: 2-13 Chapter 2 Literature Review 34 A and B are universal constants. A typical turbulent boundary layer is schematically shown in Fig. 2.5. The viscous or linear region extends from the wall to y + « 5, the buffer zone extends from y + « 5 to y + « 45, the logarithmic region spans from y + « 45 to y/5 « 0.2 while the wake region extends from y/5 « 0.2 to the free stream where 5 is the boundary layer thickness. The boundary layer is characterized by large scale eddies in the outer region. Characteristic length and time scales are proportional to 6 and 5 /U 0 respectively. The eddies have long life times because of the low shear stress environment in which they reside. Thus, disturbances introduced in the outer region of the boundary layer take a very long time to die out. The logarithmic region is marked with eddies having length scale proportional to y + and time scale proportional to t + = v/u T2 . Due to the interaction of the outer and near wall flow, the energies possessed by the eddies are higher than those found in the outer region. Because of the prevailing high stress, the logarithmic region tends to adjust to the flow conditions more rapidly. Yet in the buffer zone, turbulent eddies become more energetic. It is in this region that most of the random turbulent producing mechanisms emanate. This region is bound to play a very significant role in the overall structure of the turbulent boundary layer. The viscous region is also characterized by length and time scales that depend on the wall units (y + and t*) as shown by Haritonidis [36]. The distinguishing feature of this region is its ability to adjust rapidly to changes of the wall conditions. This unique behavior makes it a suitable portion of the boundary layer to infer some flow characteristics to the wall conditions. In particular, it may be used to determine the instantaneous wall-shear stress. Chapter 2 Literature Review 35 2.5.2 Wall Shear Stress Measurement Via Pressure Gradient Average wall-shear stress data can be obtained from pressure gradient measurements for flows in constant area ducts [50,51]. By simply integrating the shear stress over the wetted surface of a given length of the duct, the pressure drop over the same length of the duct is directly balanced. Assuming radial symmetry and neglecting pressure drops across the fully developed flow, at any duct position, the pressure will be the same at any azimuthal location. For a duct having arbitrary cross sectional area A, the pressure drop Ap over a length L is expressed by where S is the surface area on which the stress acts. For a circular duct of internal diameter D, the above equation is simplified to This method of determining the mean wall shear stress is a reliable technique, assuming that L is long enough to furnish an accurate measurement of the pressure drop and that the flow is fully developed. Thus, with the exception of the pressure gradient, all streamwise gradients of the flow quantities are zero. The method is ideal for calibration purposes rather than for wide applications during boundary layer measurements. 2.5.3 Wall Shear Stress Determination by Clauser Plots To apply this technique (wall-shear stress determination), it is necessary to measure accurately the velocity profile across a flow section. Either a Hot-Wire or Laser-Doppler anemometer is a suitable instrument to measure the velocity in the boundary layer. The 2-14 s 2-15 Chapter 2 Literature Review 36 Clauser plot is based on the logarithmic region of the law of the wall. Once the friction velocity is obtained, it becomes possible to determine the wall shear stress. The velocity profile is expressed by a series of lines giving u/LL as a function of y lL /v using ^Cy 12 as a parameter [46]. C f is the skin friction coefficient. The chart obtained this way is superposed on the measured velocity profile where the value of the skin friction coefficient is then evaluated by a matching process. Details of the Clauser plot calculation technique are given in Appendix B. 2.5.4 Measurement of Wall Shear Stress by a Heated Element When a solid body is immersed in a flow, the effect of viscousity diffuses into the flow and forms the boundary layer. If the surface temperature of a body is also raised, diffusion of heat into the flow takes place generating a thermal boundary layer. This phenomenon occurs with continuous conduction of heat and vorticity from the wall into the flow where they are simultaneously removed downstream by the flowing medium. The rate of local heat transfer from the solid surface to the moving fluid is related to the local skin friction. Accepting the general velocity law of the wall for the boundary layer, a wall shear stress measurement can be inferred from the heat transfer from a small element embedded in the surface of a body. Since thin films have very short response time, they are suited for dynamic measurements. In proximity to the wall, a viscous sublayer forms where the shear stress essentially remains constant, equal to the wall shear value. A thermal boundary layer develops within a laminar or turbulent boundary layer over the heated film embedded or glued on the wall. The thermal boundary layer is generated by a sudden jump in surface temperature due to the presence of the heated element. In order to neglect the effect of Chapter 2 Literature Review 37 turbulent diffusion in the case of a turbulent boundary layer, it is assumed that the thermal boundary layer is completely submerged within the viscous sublayer. In steady laminar flow with negligible longitudinal pressure gradient, the heat transfer rate Q w to the fluid is related to the wall shear stress T w by T W O C G M ; 3 2 _ 1 6 Heat is supplied to the probe by a high gain feedback amplifier maintaining a constant temperature above the ambient fluid temperature. In the mean time, heat is continuously conducted from the film to the fluid. This method has been applied to determine the time average wall shear stress in turbulent boundary layer flows under the assumption that the thermal boundary layer generated by the probe is smaller than the viscous sublayer thickness of the hydrodynamic turbulent boundary layer. In its early stages, the technique was developed and utilized by Ludwieg [39,52], Liepmann & Skinner [40], and Bellhouse & Schulz [41]. Later on the technique was further developed [42, 67], and now, besides custom made sensors, there are commercially available sensors in the market as well. Application of hot films has been mostly limited to steady flow shear stress measurements. It has been noticed that the simple wall shear-heat transfer rate relationship may break down at higher frequencies of oscillation in unsteady flows [43]. Bellhouse et al. [41] tried to make an estimation of the frequency spectrum of wall shear stress fluctuations from measurements of instantaneous rate of wall heat transfer. The actual relationship between the amplitude of the wall shear stress and the heat transfer rate (as a function of frequency) was determined from a calibration in a periodic laminar boundary layer on a flat plate oscillating longitudinally. They attempted to extrapolate and estimate the effects of the higher frequency end of the spectrum from the lower Chapter 2 Literature Review 38 frequency calibrations. Despite their effort, an appropriate theory for unsteady flows was not developed. Menendez et al. [42] presented a theory for the measurement of skin friction in unsteady flows using hot film probes. They tested the accuracy of their theory for cases of turbulent and laminar boundary layer flows subjected to a free stream velocity of the standing wave type ( eqn. 2-2). The wall shear stress is then given as: A l Q w L i + ^£kA3?QzL Qwl dt dt 2-17 where A 1 p B and A 3 are constants. This equation reduces to the steady flow relationship when U e and Q W L are independent of time. Their finding suggests that conventional calibration procedure for flush mounted hot film skin friction measurement fails in high frequency periodic flows. Nonetheless, it may be useful in the lower ranges of fluctuating flow frequencies. The measurement of fluctuating skin friction in air with heated thin film gages were performed by Bellhouse et al. [43]. Their probe was mounted flush with the surface of a flat plate that oscillated in its plane in the direction of the air stream. They employed Lighthill's [4] technique to obtain a relationship between fluctuating heat transfer from the film and fluctuating skin friction. The frequencies tested ranged from 212 Hz to 1181 Hz. Assuming that the laminar sublayer of a turbulent boundary layer flow is identical with an oscillating laminar boundary layer, it was possible to make use of a heated element in the study of turbulent boundary layer skin friction. But if the thermal boundary layer extends beyond the linear viscous sublayer, then there is no unique correspondence between the wall shear stress and the heat transfer from the probe. Chapter 2 Literature Review 39 In practice, wall shear stress probes should be calibrated. The important aspects of hot elements in heat transfer operation are their sensitivity and frequency response. It is believed that in a fluctuating shear stress environment, the probe responds very quickly to the shear stress fluctuations making it useful to measure instantaneous values from which ensemble averaged quantities could be extracted. Bellhouse et al. [43] indicate that at frequencies below 200 Hz, the thin film outputs would be too high due to overcompensation of thermal feed back, and at higher frequencies the linearity of the velocity profile is questionable. Literature regarding perturbation of turbulent boundary layers by a free stream periodic fluctuations of the traveling wave type, and the response of the boundary layer to such disturbances is rarely available. Specifically, to the best of the author's knowledge, there are not works performed on the interactions and correlation of wall shear stress to traveling wave free stream fluctuations. It is, therefore, one of the tasks undertaken in this study to explore the response of the wall shear stress to perturbations of the traveling wave type. Chapter 2 Literature Review 40 Fig. 2.1 Some Effects of Standing- and Traveling-Wave Free Stream Fluctuations on a Laminar Boundary Layer [9 ] Chapter 2 Literature Review 41 u /V/v i u t) i i <t h — — I t u(t) - INSTANTANEOUS MEASURED VELOCITY u - TIME AVERAGED MEAN VELOCITY < u > - ENSEMBLE AVERAGED VELOCITY u p - PERIODIC COMPONENT OF VELOCITY u(t) = u + u p + u' u =~ f T u(t)dt T o 1 N <u(t)> = 77 I u(t + nr) W n=0 u p (t) = <u(t ]> - Q u' - RANDOM FLUCTUATIONS OF VELOCITY u'(t) = u(t) - <u{ t )> Fig. 2.2 Decomposition of a Typical Instantaneous Velocity in to its Components [14], taken from Evans. Chapter 2 Literature Review 42 Chapter 2 Literature Review 43 Outer Turbulent Layer Overlap Layer Viscous Wall Layer Fig. 2.4 A Typical Turbulent Boundary Layer Regions on the Surface of a Flat Plate 45 Chapter 3 E X P E R I M E N T A L A P P A R A T U S , INSTRUMENTATION, M E A S U R E M E N T TECHNIQUES A N D DATA PROCESSING 3.1 Introduct ion As mentioned previously, the task of this research was to conduct a fundamental study of the turbulent boundary layer development on a flat plate while the upstream flow is perturbed by a set of moving wakes. Since flows in compressors and turbines are of this nature, the idea here was to simplify the flow and represent it by a rotating mechanism in a wind tunnel where the periodicity variable is isolated and investigated To study the wake-turbulent boundary layer interaction near and far downstream from the leading edge of a flat plate, an existing low-speed wind tunnel was chosen for this work. A mechanism of moving airfoil blades was designed and installed as part of the wind tunnel to simulate the action of a rotor in a real machine. This chapter presents the apparatus, the instrumentation, the measurement techniques and data processing methods used in the experiments. 3.2 The Wind Tunnel and Rig A low-speed wind tunnel located in the aerodynamics laboratory, in the Department of Mechanical Engineering, at the University of British Columbia was used for this study. Holland [44] used this tunnel earlier for an investigation of unsteadiness in the boundary layer with the same fundamental features of the rig. The test section of the wind tunnel has a 400 mm X 250 mm cross-section, and the maximum free stream velocity available is about 20 m/s. Prior to the installation of the rotating mechanism, the Chapter 3 Experimental Apparatus, Instrumentation, Measurement 46 Techniques and Data Acquisition free stream turbulence intensity was 0.5%. With the unsteady rig in place, the turbulence intensity increased modestly to about 0.7%. Therefore, the existence of the test rig had a negligible effect on the levels of turbulence in the free stream air flow. The flow in the test section has been checked for two-dimensionality before any test was conducted. The effect of the tunnel bounding walls was limited to a small portion of the test cross-section. The two-dimensional aspect of the flow was ensured in about 80% of the central portion of the test cross-section. Indeed, since the experiments were conducted at about the center of the wind tunnel where the flow is not affected by the walls of the wind tunnel (because of the boundary layers formed on the boundary walls), the flow was certainly two-dimensional. The major modification work done in this experimental apparatus was to incorporate into the existing tunnel configuration a mechanism that simulates the presence of periodic wakes like those encountered in real fluid machines. A test rig that includes moving belts carrying cylindrical rods was originally designed by Holland [44]. The rig was modified and installed in the tunnel immediately upstream of the test section. The moving mechanism has airfoils attached to rotating synchronized gear belts thereby generating periodic wake disturbances in the oncoming air flow. The wind tunnel along with the unsteady rig is illustrated in Fig. 3 .1. A scaled drawing of the flat plate leading edge, hot wire probe, the trip wire and the airfoil are given in Fig. 3.2. Fig. 3.2a shows further details of the probe. Instead of rotating bars that were used in the earlier work [44], where the flow structure in the cylinder wake was dominated by discrete vortices, rotating airfoils were used in this case for better representation of blades and the ability to adjust the angle of attack of the incoming air flow. However, in this case, the flow structure in the airfoil wake has not been investigated. Chapter 3 Experimental Apparatus, Instrumentation, Measurement 47 Techniques and Data Acquisition An NACA 024 blade profile was selected with a chord length of 50 mm and a span of 390 mm. The airfoils were made from Styrofoam, cut to the design shape by means of a hot wire. Conformity of the airfoils to the selected profile was controlled during the manufacturing process. The airfoils were then prepared (trimmed and smoothed by fine grade sand paper) , coated with fiber glass and put in a vacuum plastic bag. The plastic bag was sealed till it became air-tight and subsequently subjected to atmospheric pressure for about 24 hours until the airfoils were cured. The airfoils were further prepared for uniformity and symmetry. These airfoils are very light, minimizing the centrifugal force acting on them while traveling along the rim of the gear pulleys, and they are strong enough to withstand the fatigue that would result over time due to rotation. This greatly reduced the risk of airfoil breakage. The angle of attack of an airfoil to the incident flow is adjustable on the desired relative velocity. A simplified schematic of the unsteady rig is depicted in Fig. 3.3. It consists of two 4.32 m long by 25.4 mm wide double-sided gear belts rotating around a simple gear pulley system. Two stainless steel shafts of 25.4 mm diameter, located 2.1 m apart, carry two 190 mm diameter gear pulleys each. The shafts were mounted on ball bearings that were bolted to vertical aluminum posts on each side of the tunnel. The upper parts of the bearings were assembled with take-up units that help tightening or loosening of the gear belts. The gear pulleys on each shaft were separated by a centerline distance of 0.46 m. Each gear belt was supported by two gear pulleys on each shaft on opposing ends as indicated in the diagram. Two idler pulleys are also installed (one on each side of the gear belts), between the upper and lower main shafts, to reduce the slackness and vibration of the belt-airfoil system. A cascade of seven airfoils were attached across the gear belts so that their vertical plane of motion is perpendicular to the free stream air Chapter 3 Experimental Apparatus, Instrumentation, Measurement 48 Techniques and Data Acquisition flow entering the test section. Another gear pulley is mounted on one end of the lower shaft and is driven by a variable-speed, 1.5 HP DC motor via a small gear belt. As the system is set in motion, the belt and the airfoils travel with a specified rotor velocity along a vertical plane upstream of the flat plate. Consequently, the airfoils shed traveling wakes downstream of the trailing edge of each airfoil in the same way that rotor blades would shed traveling wakes on the stator blades located downstream. The rotor (belt-airfoil system) speed was limited to a maximum velocity of 4.5 m/s, while in the experiments, the maximum velocity used was 4.0 m/s. Due to the direction of the shaft rotation, the airfoils travel upwards through the wind tunnel on the downstream side of the rig and travel downwards through the wind tunnel on the upstream side of the apparatus. Due to spacing on the belt, only single row of blades pass in front of the flat plate at any one time. By arranging the airfoils at specified distances and selecting a certain linear velocity for the belt, it was possible to adjust the wake passing frequencies ( f = U r / S) in front of the flat plate. When the frequencies are all normalized with the free stream velocity and the downstream length x, a range of intermediate reduced frequencies were produced. The wakes at a particular frequency are convected downstream by the local free stream velocity. They interact with the turbulent boundary layer on the flat plate located downstream. Experimental measurements were then taken at various downstream locations, across the boundary layer, as the airfoils passed upwards through the wind tunnel. For the simulation of the system to be consistent with a rotor-stator condition, the speed of the airfoils ought to be selected to provide a reasonable range of flow coefficient. To meet this condition, along with the selected linear rotor velocities, the undisturbed free stream velocity was set to 3.0 m/s for all experiments. Chapter 3 Experimental Apparatus, Instrumentation, Measurement 49 Techniques and Data Acquisition The stator blade in a real machine was represented by the flat plate in the wind tunnel. The flat plate measured 400 mm wide, 2.5 m long with a thickness of 12 mm and was made from Plexiglas. To prevent separation of the incident air flow, an elliptical Plexiglas unit was attached to the leading edge. The flat plate, supported by thin aluminum brackets (spaced 150 mm on both sides of the flat plate and rest on the floor of the wind tunnels by the side walls), was positioned at a height of 110 mm from the bottom of the tunnel wall. Throughout all the experiments, the leading edge of the flat plate was located at 60 mm downstream from the 1/4th chord length of the airfoils (the location at which they are attached to the gear belts). The distance between the airfoil trailing edge and flat plate was in the order of 40 mm. Before any flow disturbance due to the moving blades was imposed on the boundary layer downstream of the leading edge, the flow was intended to be turbulent starting from the leading edge. This was ensured by making use of a 0.8 mm diameter steel trip-wire that was glued to the top part of the plate at 20.0 mm from the leading edge. 3.3 Hot Wire Support and Traverse Mechanism Instantaneous velocity or wall shear stress measurements in turbulent flows require very high frequency response sensors. For velocity measurement a boundary layer type DISA 55P15 Hot-Wire probe (Fig 3.2a) was selected. The support for this probe is a 90° bend which was aligned parallel to the longitudinal direction. To measure the vertical traverse and position of the hot wire probe accurately, a micrometer mechanism was used. This traverse mechanism consists of a micrometer with a smallest division of 0.1 mm, a probe holding tube and a support system. The Chapter 3 Experimental Apparatus, Instrumentation, Measurement 50 Techniques and Data Acquisition support system in turn includes four track wheels attached to axles that carry the main body of the micrometer assembly and two aluminum tracks on either side of the wind tunnel test section. A longitudinal slot on the roof of the wind tunnel allowed an easy horizontal displacement of the traverse mechanism. This permits velocity measurements at various longitudinal positions on the flat plate. At each streamwise measurement position, the probe was first allowed to gently rest on the flat plate to determine zero offset. Then by simply subtracting the zero offset from the micrometer reading, the vertical coordinate of the sensor could be read easily. To avoid probe breakage, cautious probe positioning, adjustment of the micrometer, and visual reference were required. This ensured that the probe was touching the flat plate and it helped verify that both prongs of the hot wire probe were aligned horizontally in the direction of the free stream flow. 3.4 Hot Wire Anemometry and Calibration The unsteady nature of turbulent flows requires high frequency instantaneous velocity and stress measurements. A Hot Wire Anemometer (HWA) instrument guarantees the process of such data collection. During the entire experimentation, a DISA TYPE 55D10 constant temperature anemometer was employed. A single sensor hot wire probe served to measure all the velocity data. The same probe has been used throughout the experiment to measure velocity across the boundary layer at any downstream station on the flat plate. The sensor was a 5 pm diameter platinum plated tungsten wire with a 2 mm length. The wire is welded to two prongs whose stem part is coated with ceramic. The other end of the probe has two plugs that are used to connect it to the probe support. The lead wires from the probe support are then connected to a 5 m coaxial Chapter 3 Experimental Apparatus, Instrumentation, Measurement 51 Techniques and Data Acquisition cable. This cable in turn is plugged to the Hot Wire Anemometer instrument making it one of the four legs of the Wheatstone bridge resistors. The bridge used was a general purpose DISA C16 with a frequency response of 100 kHz. A sampling frequency of about 10 kHz was found sufficient for all experimental cases. It deemed to provide a high degree of resolution of the instantaneous velocity/stress output. The anemometer output (analog reading) is given in terms of voltages. Voltage output from the probe and the corresponding flow velocity in the test section are expressed by the calibration equation given by E 2 = A 2 + BU N 3-1 where E is the anemometer output voltage and U is the corresponding velocity. A , B and N are calibration constants. The calibration constants were determined against a Pitot tube inserted in the flow. With the probe set in place, several voltage readings were recorded and the reference velocity obtained from Pitot tube was entered correspondingly. This procedure was repeated for several velocities within the limit of the tunnel to obtain the calibration constants. Selecting the proper exponent for the velocity that results in the best fit ( N = 0.5 in this case), a least squares curve fitting procedure was performed to get A and B. Repeating the procedure indicates that the standard deviation is in the order of 0.06. Fig. 3.4. shows the linear curve resulted from calibration. The distortion due to measurement at the low velocity end was checked by superposing another curve fit line by omitting the smallest datum point. The effect was found to be about 3% variation on the actual velocity obtained via the two curves. Thus no significant error would be admitted when the original curve fit was used. Knowledge of the wall shear stress is of both fundamental and practical importance. While the mean stress indicates the overall state of the flow over a given Chapter 3 Experimental Apparatus, Instrumentation, Measurement 52 Techniques and Data Acquisition surface, the fluctuating stress represents the individual processes that transfer momentum to the wall. As one of the tasks in this research was to investigate the response of the wall shear stress on the flat plate to the upstream rotor wake stimulus, a measurement technique was required to measure fluctuating shear stress. Hence a DISA 55P47 glue-on shear stress probe was selected. Fig. 3.6 shows a schematic of such a glue-on probe and thermal boundary layer. A scale diagram of the sensor is given in Fig. 3.6a. The sensor is 0.1 x 0.9 mm Nickel film deposited on a 0.05 mm thick polyimid foil carrying a 0.05 |um quartz coating. Two 0.1 mm diameter copper leads connect the probe to the coaxial cable whose other end was connected to the Hot Wire Anemometer bridge unit. The concept of a hot element as a sensor to measure skin friction is based on the relation between local skin friction and the rate of heat transfer from the small heated element mounted on the surface ( of the stator in this case) . The probe is glued on the surface in such a way that the longer dimension is normal to the direction of the free stream flow. As in the case of the hot wire probe, the glue-on probe acts as one of the arms of the bridge. Once the glue- on probe is mounted on the surface of the flat plate, heat transfer is related to the gradient of the velocity. The electrical power, E, is related to the wall shear stress, T w , by the equation E 2 - EQ 2 + B T w1 / 3 , 3-2 where E0 is the electrical power at zero air flow and B is a calibration constant. The details of the derivation of equation 3.2 and the calibration procedures are given in Appendix C. Chapter 3 Experimental Apparatus, Instrumentation, Measurement 53 Techniques and Data Acquisition Since one measures the power supply to the probe, operation of the skin friction gage is the same as that of a hot wire. The anemometer senses any change in the resistance of the sensor and in response supplies the appropriate current to maintain the probe at constant temperature. Thus the procedure of calibration is similar to that of the hot wire discussed above. The wall shear stress probe was calibrated against results obtained by using a Preston tube [47,50]. For details reference can be made to Appendix C. A schematic of a Preston tube is given in Fig. 3.5. The difference between the pressure recorded by the Preston tube (a total Pitot tube) and the undisturbed static pressure on the wall is associated with the wall shear stress. Thus, to calibrate the glue on probe, static taps on the flat plate were drilled. A 2 mm diameter Preston tube was used as a standard to obtain wall shear stress at various air flow velocities. The electrical power supplied to the sensor was therefore indirectly related to the wall shear stress via the Preston tube readings of the pressure difference. A least square fitting procedure applied to the data 1/3 collected resulted in a linear relationship between E and Tw as depicted in Fig. 3.7. 3.5 Determination of the Wall Proximity Effect on the Hot Wire. Wall proximity corrections were required for the hot wire sensor. A hot wire transmits heat to a solid surface at different rates when it gets close to the wall. These changing rates of heat transfer are due to changes in convection and radiation when a measurement is taken in a flow. Therefore, the presence of the wall alters the heat transfer conditions in the probe when it is brought in proximity to the surface thereby affecting the calibration constants. Chapter 3 Experimental Apparatus, Instrumentation, Measurement 54 Techniques and Data Acquisition There are different methods by which the wall proximity effect can be taken into consideration [44,48,49]. Among them, the still air correction method is a relatively simple and practical technique. Fundamentally, the assumption for the still air correction method is that the increase of heat transfer in the flow medium due to the presence of the wall is the same as that obtained in the still air condition (i.e., no flow situation) at similar transverse locations from the wall. Setting the wind tunnel velocity to zero, hot wire measurements were taken at different vertical distances from the flat plate to determine the amount of correction required at the respective heights. The difference in heat transfer between those measured in flowing air and still air can be expressed as E 2(y)-E 2(™). ' 3-3 This expression was used to correct the voltage record obtained during all times that velocity measurement experiments were conducted. It was assumed to be valid at any wind tunnel free stream velocity and the corresponding height above the flat plate. Assuming that this expression has universal validity, the increment is subtracted from the current voltage output during the process of converting voltage to velocity by using the calibration equation. Mathematically, the calibration equation is expressed by E 2(y) -A 2- (E02(y) - £02(°°)) = BU N 3-4 The correction was found to be relevant in the neighborhood of the flat plate at heights less than 3 mm from the wall. 3.6 Triggering Method for Data Acquisi t ion The rotation of the belt-airfoil system required a way of signaling the data acquisition system when to begin registering data. The method chosen to provide the signal of the Chapter 3 Experimental Apparatus, Instrumentation, Measurement 55 Techniques and Data Acquisition start of data collection was a Hall effect switch. A Hall effect sensor provides a continuous voltage output that increases with a strong magnetic field and decreases with a weak magnetic field intensity. Power to the Hall effect switch was supplied by a 9 Volt battery. The switch output was connected to one of the channels in an Analog to Digital (A/D) converter board that was installed in a computer used to acquire data automatically. As the gearbelt-airfoil system rotates, a small magnet attached to the gearbelt induces the switch to open when it comes very close to the stationary switch as it passes by. A predetermined voltage must be set in the data acquisition program for the switch to function properly and send the signal to the board. 3.7 Data Acquis i t ion and Data Processing All of the data from the hot wire and glue-on probe measurements were acquired with a dedicated 486DX-50 MHz personal computer. It was equipped with a 12 bit, 32 channel, CIO-AD08 Analog to Digital (A/D) converter board. The A/D converter board was configured to accept anemometer output voltages that range from zero to ten volts in 4096 (2 1 2 ) increments. The time required to register data either for velocity or wall shear stress measurements covers the time needed for the seven airfoils attached to the belts to pass through the wind tunnel upwards, excluding any effect that might be generated from the downward moving airfoils as they pass through the upstream section. A data sampling frequency on the order of 10 kHz was found sufficient for all experiments conducted. The rate of data conversion was set to give 5000 to 8000 data points during a single cycle. Two sets of airfoil spacings were considered in this study. The airfoil spacing along with the rotor speeds selected provided a range of wake passing frequencies. To avoid very low and very high frequency noise, the system was set to Chapter 3 Experimental Apparatus, Instrumentation, Measurement 56 Techniques and Data Acquisition provide a two Hz high pass and a ten kHz low pass frequency during actual data collection. The whole operation of the experiment was controlled and directed via QuickBasic programs. These programs were developed and adapted to suit all the experiments conducted. Once the external conditions are set up, the computer is switched to data acquisition mode, and as the programs run, data collection proceeds automatically for the number of cycles (100 for velocity and 80 for wall shear measurements) input to the program. Collection of data begins as soon as the magnetic pick-up is switched on. It is then followed by acquisition of instantaneous velocity or wall shear stress until the required number of data points are collected at the specified rate. During velocity measurements, the vertical position of the hot wire had to be changed intermittently after data collection for the specified number of at each location. The entire operation was also closely monitored for any mechanical failure and for other safety precautions. Uniformity and consistency in the mechanical system of the experiment was ensured by constant check up of the belt-airfoil system for slackness, airfoil breakage and any possible deviation from the preset value of the angle of attack of the airfoils. A non-zero angle of attack was set because the airfoils were relatively thin. Thus, it was desired to produce wider wakes by adjusting the angle of attack to about 10°. First, however, the airfoils had to be oriented in the direction of the respective relative velocities. A schematic of the airfoils angular orientation is illustrated in Fig. 3.8. To monitor the hot wire and glue on probe signals visually, an oscilloscope was used along with the rest of the data acquisition system. The data from each experiment were temporarily stored on the hard disk of the computer. Once all the velocity measurements were taken across the boundary layer, Chapter 3 Experimental Apparatus, Instrumentation, Measurement 57 Techniques and Data Acquisition further data processing was conducted before any other downstream location was chosen for another measurement. In the case of wall shear stress measurements, there was no need to traverse the probe across the boundary layer, therefore, data processing could proceed as soon as the required number of samples were collected at each longitudinal station. In all cases the primary data reduction procedure employed was the technique of ensemble averaging. 3.8 The Technique of Ensemble Averaging Data registered during any set of experiments were processed by the technique of ensemble averaging. From the ensemble averages, different components of the boundary layer are extracted. Since some of the terms used in ensemble averaging will be used in the following chapters, the definition and significance of each term is given below. Velocity measurement data are used to elucidate the process. Instantaneous velocity measured in the boundary layer is composed of the time mean velocity, the organized unsteady velocity due to the periodic wake disturbance, and the random fluctuating velocity. This triple decomposition of the instantaneous velocity brings out some information about the roles each component may play in the boundary layer as well as the relative significance of each component in the velocity field. Since accurate representation of the unsteady velocity phenomena in the boundary layer cannot be obtained from a single velocity record, a statistically acceptable number of sampling cycles is required. It is over this full range of cycles that the ensemble averaging is performed. It is necessary that each cycle begins at the same time in a period of rotor revolution, and that the same experimental conditions are maintained. Chapter 3 Experimental Apparatus, Instrumentation, Measurement 58 Techniques and Data Acquisition Typical raw velocity signal traces in time and the various components of the resulting ensemble-averaged velocity are given in Fig. 3.9. The figures illustrate instantaneous velocity measurements during a given cycle, the time mean velocity, the periodic fluctuation due to wakes (in this case) and the random turbulent velocity. The unsteady velocity uy at any time in any cycle can be expressed as utj = <t7> + u'. 3-5 uij =U + u + u' 3-6 where C/ is average velocity, U is the periodic fluctuation, u' is the turbulent velocity and < U > is the ensemble average velocity. Given N number of cycles at any station in the boundary layer, the ensemble-averaged velocity is determined by sampling at constant phase according to the equation 1 N (U)j = Lim — Yuij. 3-7 Theoretically N should approach infinity, however, it is always finite in practice and the error is small when N is sufficiently large. A sample number N equal to 100 was chosen in this study for the ensemble averaging process because the ensemble average velocity and turbulence showed insignificant change for higher values through a number of test runs. Once the ensemble averaged velocity is available, the ensemble-averaged turbulent velocity is obtained from the equation: 1 N <u' 2 > = Lim — V (w - <c7» l 72 . 3-8 N^y°°Ni=l J The time mean velocity is also calculated by averaging over the number of data points in the ensemble averaged velocity according to the equation: Chapter 3 Experimental Apparatus, Instrumentation, Measurement 59 Techniques and Data Acquisition i M 1 M N ^ = ^ L < ^ = T 7 ^ I L " i / . 3-9 M ..  J MxN I ,  J where M is the number of data points in the ensemble-averaged quantity. Then the organized fluctuation velocity U is determined by simply subtracting the time mean velocity from the ensemble-averaged velocity by the following expression: u = (U)-U. 3-10 Other boundary layer properties such as the displacement thickness 5*, the momentum thickness 0, and the shape factor H, are then determined readily from the profiles of the ensemble averaged velocity by the following equations: Displacement thickness: <6*>= J 0 Momentum thickness: <5> <e>= / o (u) (ue) V <Ue)j dy, 3-12 Shape factor: 3 - 1 3 Polynomial curves were fit to the ensemble-averaged velocity data at the respective downstream stations and equations were formulated for each phase. From these equations of the velocity profiles the wall shear stress could be deduced by taking the derivative of the respective equations and evaluating them at the surface of the plate. Comparison could then be made with the results obtained by direct wall shear Chapter 3 Experimental Apparatus, Instrumentation, Measurement 60 Techniques and Data Acquisition stress measurement. Clauser plots may also be applied to the velocity data under the assumption that the "law of the wal l " holds for unsteady flow conditions. Determination of periodic fluctuation amplitude and phase lag or lead of the velocity across the boundary layer, relative to the free stream flow velocity, followed from the ensemble-averaged velocity data. For one complete cycle of the ensemble-averaged velocity record, the amplitude fluctuation results from the difference between the ensemble-averaged velocity and the time mean velocity for a particular station under consideration. One complete cycle represents the passage of a single airfoil. Normalizing the amplitude either with the local free stream mean velocity or with the local amplitude of the organized fluctuation at the free stream provides a profile of the amplitude ratio variation across the boundary layer. The phase of the velocity profile across the boundary layer relative to the reference velocity at the free stream could be extracted from the ensemble averaged velocities at a designated longitudinal location. The procedure depends on the following premises. Since the passage of an airfoil generates periodic velocity perturbation on the flat plate boundary layer, each complete passing of a single airfoil velocity profile (wake) is assigned phases ranging from zero to 360 degrees. Any instant in the cycle is therefore associated with a specific phase angle. Taking the free stream velocity profile to be at zero phase, the phase lead or lag of the remaining velocity records across the boundary layer (at varying vertical distances above the flat plate) follows. The difference in angle between the reference velocity record and that at any vertical position, at the same downstream location, indicates whether there exists a phase lead or lag. A positive angle is indicative of phase lead by the measured velocity record while a negative angle corresponds to a phase lag with respect to the reference velocity record. Chapter 3 Experimental Apparatus, Instrumentation, Measurement 61 Techniques and Data Acquisition >-15 C L £ co •£ e o « C L S. Q . O S 4) CO u s; « co <° « C L 2 J CO (0 C5 C5 I— ~ . . . O r - CM S 03 ffl r - ^ r -U0 o o E 13 c c 3 u -a E n o c E u > CO u o o CO Lt DC < I - W n V W U 3 u «3 E u • 00 CO DC E *c m CL X LU 13 C 13 c c •a c CNJ i i i u n i i r r kmaqp Chapter 3 Experimental Apparatus, Instrumentation, Measurement Techniques and Data Acquisition 2.1 m Fig. 3.3 Unsteady Flow Generating Rig and Drive Mechanism. Chapter 3 Experimental Apparatus, Instrumentation, Measurement 65 Techniques and Data Acquisition Chapter 3 Experimental Apparatus, Instrumentation, Measurement Techniques and Data Acquisition Fig. 3.6 Schematic of a Glue-on Probe, Near Wall Velocity Profile and Thermal Boundary Layer Profile Chapter 3 Experimental Apparatus, Instrumentation, Measurement 67 Techniques and Data Acquisition T3 LULU 8 Chapter 3 Experimental Apparatus, Instrumentation, Measurement 68 Techniques and Data Acquisition Chapter 3 Experimental Apparatus, Instrumentation, Measurement Techniques and Data Acquisition Fig. 3.8 A Schematic of an Airfoil and Actual Angular Orientation With Respect to the Relative Velocity. Chapter 3 Experimental Apparatus, Instrumentation, Measurement 70 Techniques and Data Acquisition £ 7 , ( / ) UN[t) T ime , t T i m e , / I n s t a n t a n e o u s v e l o c i t y , IS, . a \ t i m e , 11 Average veloc i ty U of N records at t ime , t, U ( / ) , i n s t a n t a n e o u s v e l o c i t y 0(f), e n s e m b l e -a v e r a g e d v e l o c i t y U m e a n v e l o c i t y Time, / ( b ) Fig. 3.9 A Typical Ensemble Averaging Process and Resulting Velocity Components [ 1 3 ] . 7 1 Chapter 4 EXPERIMENTAL RESULTS A N D DISCUSSION 4.1 Introduct ion This chapter presents experimental results of unsteady turbulent boundary layer measurements on a flat plate. The results include measurements of turbulent boundary layer velocities and wall shear stress at some locations on the surface of the flat plate. Various boundary layer characteristics are evaluated from the respective instantaneous measurements. Although the nature of any turbulent flow is inherently unsteady, the term unsteadiness in this presentation is reserved for a time-resolved flow characterized by periodic fluctuations imposed by an external mechanism. Such a flow is essentially different from unsteadiness due to turbulent flow. The frequency of disturbance is discrete and low in magnitude when compared to frequencies of the random turbulent fluctuations. A detailed description of the experimental facility, data acquisition and processing techniques are given in Chapter 3. The system is a simplified, two-dimensional representation of a real turbomachine stage. Various complexities found in a real machine such as three-dimensional flow, surface curvature, blade entrance angle, blade twist, end wall effects, secondary flows etc. were not considered in this study. Attention was rather paid to conducting a fundamental investigation on the effects of traveling organized fluctuations (wakes in this case) with discrete frequencies on the boundary layer development on the flat plate. Chapter 4 Experimental Results and Discussion 72 In practical engineering applications, periodic unsteadiness could be the source of vibrations, noises, structural failures, reduced performance due to boundary layer growth and separation, and time dependent heat transfer processes. For instance, in a turbomachine stage, the growth of the unsteady boundary layer causes reduction in stagnation pressure, and in general oscillatory flows may lead to unstable operating conditions. The primary objects of measurement were the instantaneous boundary layer velocity and the wall shear stress on the flat plate. Several quantities were predetermined and set to fixed values: the free stream velocity was 3.0 m/s throughout the whole experiments; the angle of attack on the rotating airfoils was about 10°, the gap between the trailing edges of the airfoils and the leading edge of the flat plate depended on the rotor speed, and the gaps ranged from 36 mm to 45 mm. Several cases were obtained by varying the rotor speed and the airfoil spacing, so that the ratio of upstream velocity to rotor speeds resulted in different flow coefficients. Velocity measurements took place at four downstream locations and at several transverse locations across the boundary layers, while wall shear stress measurements were taken at nine downstream stations on the surface of the flat plate. Table 4.1 shows the experimental cases considered for velocity measurements In this chapter, steady turbulent boundary layer results are first presented. This is followed by unsteady turbulent boundary layer results and wall shear stress measurements. Comparisons of boundary layer profiles and wall shear stress for different cases are then presented, and finally the effects of disturbance frequency, or the reduced frequency, are discussed. Chapter 4 Experimental Results and Discussion 73 4.2 Steady Turbulent Boundary Layers It is known that all turbulent boundary layer flows are unsteady in nature. However, if the turbulent flows evolve from "a natural transition" process, the flows can be treated as steady flows through a time-averaging scheme. In such kinds of flows there are no externally imposed fluctuations. One then assumes that the free stream velocity is free from random fluctuations. In the wind tunnel employed in this work, the free stream turbulence intensity was about 0.5%. The flat plate boundary layer was tripped at 20 mm Table 4 .1 . Experimental Condit ions for Velocity Measurements: U 0 - 3.0 m/s Case rotor vel. Spacing Frequency Station Flow Coeff. Ur (m/s) S(m) f (Hz) X(m) c e 1 0.0 — 0.0 0.1, 0.3, 0.5, 0.7 0.0 2 2.0 0.1 20 0.1, 0.3, 0.5, 0.7 1.5 3 3.0 0.1 30 0.1, 0.3, 0.5, 0.7 1.0 4 4.0 0.1 40 0.1, 0.3, 0.5, 0.7 0.75 5 2.0 0.2 10 0.1, 0.3 1.5 6 3.0 0.2 15 0.1, 0.3 1.0 7 4.0 0.2 20 0.1, 0.3 0.75 from the leading edge to generate a turbulent flow on the plate. With the flat plate in place, free stream velocity measurements indicated a zero mean pressure gradient along the longitudinal direction. The flow along the center of the test section was checked for two-dimensionality, and no interference from the bounding walls of the tunnel was detected in the measurement region. Chapter 4 Experimental Results and Discussion 74 Steady turbulent boundary layer velocities were measured at 0.1m, 0.3m, 0.5m and 0.7m from the leading edge. The time-mean velocity profiles at the respective stations are shown in Fig. 4 . 1 . Note that all the profiles were obtained in the absence of the unsteady flow generator. The velocities are normalized with the free stream velocity. The boundary layers are turbulent except at x = 0.1 m where the profile appears to be transitional. This may be attributed to possible flow separation at the trip wire and low operating Reynolds number. The fullness of the profiles increase in the downstream direction. Fig. 4.1a illustrates a comparison of experimental steady boundary layer velocity profiles with Blasius and the one-seventh profiles, which are known in the literature [37,69]. The measured velocity profile at x = 0.1 m is indeed transitional while the profiles at the subsequent stations are turbulent. The observed discrepancy is characteristic of turbulent boundary layer profiles at different Reynolds numbers. Clauser [46] observes that the belief that turbulent boundary layers can be represented by a single one-seventh power curve is not justified. Experimental data show [1,46] that the exponent can vary from one-third to one-tenth. When plotted with y/b as one of the coordinates, the profiles did not collapse to a single curve as in the case of laminar boundary layers. This non-similarity is, of course, because of the different scales in the inner and outer regions of the turbulent boundary layer [1]. Fig. 4.2 illustrates the random fluctuation profiles of the steady turbulent boundary layers at the same longitudinal stations shown in Table 4 .1 . Fig. 4.2a compares the fluctuations with a similar work in the literature [72]. The profiles show the decay of turbulence at the wall and in the outer edge of the boundary layers. The maximum turbulence intensity is located closer to the wall in all cases. It is in this region that high levels of turbulence are generated. The purpose of steady turbulent boundary layer Chapter 4 Experimental Results and Discussion 75 measurements was to ensure the validity of probe outputs and for comparison with unsteady boundary layer results. 4.3 Development of Unsteady Turbulent Boundary Layers 4.3.7 Unsteady velocity Results In contrast to transient flows, there is a class of unsteady turbulent boundary layers that are periodically unsteady at all times during the operation of a system. An example of such a flow is found in a turbomachine stage. The organized nature of the disturbance also contrasts with the fluctuations in steady turbulent boundary layers discussed above. In the present experiment incident wakes from the rotor blades impinge on the turbulent boundary layer on the flat plate. The interaction of such wakes with the resident boundary layer alters the structure of the profile at any station. Since the wakes are also traveling with a finite velocity in the flow direction, the response of the boundary layer is affected in a fashion differently from the response of any other types of unsteadiness. A sample time history of an ensemble-averaged velocity in the free stream is shown in Fig. 4.3. The figure clearly indicates the wakes created behind the airfoils so that the free stream flow is suddenly subjected to periodically oscillating disturbances. The frequency of disturbance is estimated from the frequency of wake generation by the rotor elements. To see the periodic nature of the wakes alone clearly, the organized fluctuation is separated from the time-mean part of the ensemble average velocity. This has been accomplished via the triple decomposition technique explained in Chapter 3. Typical periodic fluctuations in the free stream flow are depicted in Figs. 4.4a and 4.4b for a wake frequency of 20 Hz at X = 0.1 m and X = 0.3 m for blade spacing S = 0.1 m. Figs. 4.4c and 4.4d are similar periodic fluctuations at the same locations but different Chapter 4 Experimental Results and Discussion 76 blade spacing S = 0.2 m. In all cases, the spacing of the airfoils on the belts guaranteed data collection only from one row of airfoils at any time in the test section. The figures indicate fluctuations from the seven airfoils of the rotor while a single dip in the time history represents the velocity deficit during the passage of a single airfoil. Average values of the fluctuation amplitudes were subsequently determined from such figures for individual cases. The fluctuation amplitudes were in the order of 10% of the free stream velocity. The first set of unsteady turbulent boundary layer experiments dealt with cases where the airfoil spacing was 0.1 m. The selected rotor velocities ( 2.0, 3.0 & 4.0 m/s) along with the airfoil spacing generated fluctuation frequencies of 20, 30 & 40 Hz respectively. In the other set of experiments, the airfoil spacing was changed to 0.2 m. In these cases the rotor speeds gave oscillation frequencies of 10, 15, and 20 Hz. When each fluctuation frequency was made nondimensional with a combination of the longitudinal axis and the corresponding free stream velocity, a reduced frequency is obtained. Table 4-2 shows values of the reduced frequency for each case. Table 4.2 Values of the Reduced Frequency for Each Experimental Case. X [ m ] S = 0.1 m S = 0 .2m 20 Hz 30 Hz 40 Hz 10 Hz 15 Hz 20 Hz 0.1 0.67 1.0 1.33 0.33 0.5 0.67 0.3 2.0 3.0 3.99 0.99 1.5 2.0 0.5 3.33 5.0 6.66 0.7 4.67 7.0 9.33 Chapter 4 Experimental Results and Discussion 77 One way of observing how the organized fluctuation frequency influences the boundary layer development is to look at the nature of the velocity profile. Fig. 4-5a illustrates velocity profiles of the unsteady turbulent boundary at different phases in a single period for case 2. All the velocity profiles are for the central airfoil during the passage of one period. The plots also successively show the behavior of the unsteady velocity profiles at different stations along the flat plate. These superposed profiles indicate the history and state of fluid motion across the layer as it interacts with the organized disturbances. It is clear from the figures that the boundary layer is sensitive to the external perturbation. The response, however, varies across the boundary layer and in the flow direction. Consider the phase-averaged velocity profiles at the first station ( X = 0.1 m) in the above figure. The external and internal regions of the turbulent boundary layer react differently throughout the cycle. Fluctuations in the external region are visibly enhanced in response to the external disturbance. It is apparent that the profiles differ significantly at each phase angle. Maximum fluctuations of up to 8% are obtained. The mean velocity profile is included for comparison in each chart. Following the velocity profiles from zero phase angle to the end of one cycle, the profile fullness decreases as the flow decelerates until it reaches the point of minimum velocity of the wake. The profile fullness then increases as the flow accelerates. The unsteady effects are significant particularly in the outer region of the boundary layer. On the other hand, near wall profiles show no perceptible fluctuations. The fact that greater fluctuations were observed in the outer region was not surprising. It is known that shear stress diminishes towards the outer region of the profile. This reduces the influence of viscous forces in the external region thereby Chapter 4 Experimental Results and Discussion 78 rendering it more reactive to the effects of external disturbances. Therefore the impressed pressure gradient across the boundary layer appears to cause greater fluctuations in the outer region than it does in the inner region. In the case of boundary layers subject to standing waves, the pressure gradient across the layer opposes the inertia forces in the inner region. In this case, however, where there is a traveling wave disturbance, the pressure gradient reinforces the inertia forces. This phenomenon makes it difficult for the inner region to respond readily, in terms of fluctuations, compared to the external region. However, compared to steady flow velocity profiles the velocity in the inner region is higher for the periodically disturbed profiles. Observation of the phase-averaged profiles at several stations along the flat plate also shows how the wake-boundary layer interaction evolves in the downstream direction. Irrespective of disturbance frequency, the response of the external profile was strong at X = 0.1 m from the leading edge. The velocity fluctuations diminish in the downstream direction. This is because the disturbing wakes dissipate their energy as they are convected downstream and more mixing and interaction with the boundary layer takes place through the passage of time. Figs. 4.5b and 4.5c show additional velocity profiles at different times in a single period for cases 3 and 4 while Figs. 4.5d to 4.5f are for cases 5, 6 and 7. The velocity profiles become more full along the longitudinal direction as was observed by Yip and Evans [13], and this trend increases with increasing frequency. A valuable clue about the periodically disturbed velocity profiles as compared to steady boundary layer profiles may be obtained by looking into the time-averaged profiles for the respective cases. Hence, time-averaged velocity profiles obtained from the phase velocity profiles in a single cycle along with average steady velocity profiles Chapter 4 Experimental Results and Discussion 79 are presented subsequently. Figs. 4.6a and 4.6b show curves of average velocity profiles at different disturbance frequencies at successive longitudinal locations. One of the interesting results from this comparison is the increase in the fullness of the profiles with increasing frequency. This suggests greater momentum exchange closer to the wall with increasing frequency. The other observation is the difference in the average velocity profiles in unsteady flows as compared to steady boundary layer profiles. Previous studies for both laminar and turbulent boundary layers exposed to standing wave disturbances [4,5,11], and boundary layers perturbed by traveling waves [6,9], showed an insensitivity of the mean boundary layer profile to fluctuation frequencies. It is usually assumed that the average unsteady and steady velocity profiles are approximately the same. However, the experimental results, for the disturbance frequencies tested in this study, show a variation in the average velocity profiles in unsteady flows. The profiles in the unsteady flow cases have steep velocity gradients close to the wall, and this is true for all frequencies considered and at all downstream stations indicated. Sensitivity of the average unsteady turbulent boundary layers to imposed fluctuation frequencies has practical significance in turbomachine blade design. Usually, during the design process, the mean turbulent velocity obtained from steady flow is assumed sufficient. The variations of the averaged velocity profiles observed here lead to some questions. Why do we observe differences in the average profiles for the respective cases? Are flow mechanisms in steady and unsteady flows equivalent? What are the reasons for the discrepancies stated above? The dynamics of turbulent boundary layers in either case need to be evaluated to shed light on the possible physical mechanisms that may prevail in the respective flows. Chapter 4 Experimental Results and Discussion 80 In general, the behavior of turbulent boundary layer flows is significantly different from laminar boundary layer flows. In steady turbulent boundary layers, the random fluctuations play greater roles in changing the structure of the boundary layer. The process of mixing, momentum and energy transfers across a layer reduce the effective diffusion length of viscosity to a smaller transverse distance. High momentum fluid particles from the outer region penetrate deeper to the inner part of the boundary layer. Because of the exchange of momentum, slowly moving fluid particles close to the wall are energized to overcome inertia. It is such an interaction that results in the fullness of the boundary layer profiles compared to laminar boundary layer profiles. The force due to the correlation of the random fluctuations is orders of magnitude larger than the contribution of the viscous force. The inclusion of periodic fluctuations on top of random fluctuations in a turbulent boundary layer brings in extra forces to the system. These forces are, similar to the Reynolds shear stress terms, formed due to the correlation of the periodic fluctuations and the nonlinear interactions with the mean and turbulent shear stress. Such forces further affect the structure of the turbulent boundary layer on top of the Reynolds shear stress terms. Two effects are noticed from the interaction of the organized fluctuations with the turbulent boundary layer. One of the effects is manifested in the increased fullness of the average turbulent boundary layer velocity profile and the oscillation of the instantaneous boundary layer velocity as was shown in the phase velocity profiles discussed earlier. The other significant effect of the organized fluctuation interaction is seen in the behavior of the random fluctuations. Experimental results obtained in this work mark an increase in the random fluctuations whenever wakes from the upstream airfoils interacted with the boundary layer. A free stream flow, which otherwise would Chapter 4 Experimental Results and Discussion 81 have negligible random fluctuations (0.5%) jumps to higher levels of turbulence (in the order of 10%) whenever it senses the organized fluctuations. The influence of the organized fluctuations is not limited to the free stream alone, but diffuses towards the wall across the layer. High momentum fluid "chunks" are transported inward, and because of this transport phenomena structural changes are promoted in the boundary layer. Similar to the steady turbulent boundary layer, additional terms in the governing equations emerge when a boundary layer develops under the influence of traveling organized fluctuations of discrete frequencies. Nonlinear interactions between the time-mean and the organized fluctuations, and between the organized and the random fluctuations are among the significant factors that effect structural changes. The boundary layer developing under these conditions is bound to be different from a steady turbulent boundary layer formed in the absence of the organized fluctuations. Thus the extra fullness of the boundary layer observed is a consequence of such nonlinear interactions. In light of the above discussion, it seems clear that external disturbances of discrete frequencies have an important role to play in changing the behavior of the turbulent boundary layer. It is therefore important that the effects of organized fluctuations on the average turbulent boundary layer be taken into account, instead of the steady value, during the design process of a turbomachine blade. In what follows the effect of the organized fluctuation on the random fluctuations will be presented in detail. 4-3-2 Random Fluctuations Under Periodic Perturbation The effect of the traveling wake disturbance on the random fluctuation can be seen by looking at the respective time-histories of the random fluctuations at a particular station across the boundary layer. Figs. 4-7a to 4.7c show the time histories at selected Chapter 4 Experimental Results and Discussion 82 transverse locations across the boundary layer for cases 2, 3 and 4 respectively. It is apparent that the impingement of wakes changes the nature and magnitude of the random fluctuations dramatically in the free stream as well as across the profile. The free stream random fluctuations appear to anticipate the periodic fluctuations. The amplitude of the fluctuations due to periodicity fades towards the wall. Fig. 4.7d is a similar figure for blade spacing S = 0.2 m. In this case, the amplitude of the random fluctuations appear to be visible even close to the wall. Profiles of the turbulent random fluctuations across the boundary layer reveal transverse locations where significant momentum and energy transfers are realized, but the presence of the wall forces the flow to abide by the boundary conditions existing on the surface. Hence, the mean velocity and the random fluctuations maintain zero relative velocity on the surface. Phase-averaged profiles of random fluctuations at particular stations at different frequencies for cases 2, 3 and 4 are given in Figs. 4.8 to 4.10. The following observations can be made from the figures. 1. The profiles of the random fluctuations vary during a period of one cycle. 2. The trend of the profiles is to increase to a maximum near the wall and decrease towards the outer region. 3. Variations of the random fluctuations are higher in the outer region than in the inner region. 4. The point of maximum fluctuation shifts towards the wall as one progresses along the flat plate for a particular disturbance frequency. 5. The shift of the maximum fluctuation location increases with increasing perturbation frequency. Chapter 4 Experimental Results and Discussion 83 The region where maximum fluctuations are recorded experiences intense turbulent activity. As one travel towards the free stream, the activities of the random fluctuations diminish thereby reducing the profile to the free stream periodic fluctuation condition at the external edge of the boundary layer. The significance of variation in the random fluctuations diminish in the flow direction. Whether the steady turbulent random fluctuation profile is affected when an external periodic fluctuation is imposed was further investigated. A high level of free stream turbulence intensity is known to promote turbulence in the boundary layer [70]. In a similar way, the presence of a wake disturbance with discrete frequency and amplitude are found to introduce increased turbulence intensity in the boundary layer. The increment is stronger in the external region of the boundary layer as shown in the above figures. The response of the boundary layer to the discrete disturbance frequencies was obtained by isolating the random part of the fluctuation from the ensemble-averaged component of the instantaneous velocity during the data reduction process. The influence of the organized fluctuation on the random fluctuation is clearly seen when comparison is made between the steady turbulent boundary layer and the layers subject to the influence of periodic fluctuations. As was shown in Figs. 4.7a to 4.7d, it is apparent that the free stream random fluctuation is affected by the interaction of the discrete disturbance frequencies. The degree of such interactions within the boundary layers is illustrated in Figs. 4.11 and 4.12 for cases 2, 3 and 4. The steady profiles (f=0 Hz) exhibits the expected behaviors. The profile starts from zero at the wall, reaches a maximum within the boundary layer and recedes back to the free stream conditions at the outer edge. With the passage of the airfoils in front of the flat plate, the levels of turbulence increased for all frequencies considered. A trend shown by Fig. 4.11 is the Chapter 4 Experimental Results and Discussion 84 growth of the turbulence intensity with increasing disturbance frequency. For instance, at X = 0.1 m from the leading edge, the levels of turbulence at the free stream are 0.7% for the steady flow (0 Hz), about 3% for unsteady flow with a frequency of 20 Hz, about 6% for 30 Hz and about 8% for 40 Hz. Although the variation is visible in the outer region, the effect of the wakes is also felt in the near wall regions. The decay of the turbulence levels with downstream distance is not unexpected, for both the wakes and the underlying turbulent boundary layer dessipate some of their energy in the streamwise direction. Furthermore, the location of maximum intensity shifts towards the wall as the frequency is increased. The flow appears to gain momentum as the frequency is increased thereby enabling more fluid particles to penetrate deeper to the boundary layer. The shift of maximum turbulence intensity along the flat plate may also be clearly seen. Fig. 4.12 shows a comparison of the unsteady random fluctuations at particular frequencies along the various measurement locations on the plate. The flow had increasing relative velocity with increasing rotor speed in front of the flat plate. Therefore, the flow is presumed to be perturbed with more energetic fluid with wakes of shorter period and inter-blade wake profile mixing. The increase in turbulence intensity is certainly due to the effects of wake interaction with the turbulent boundary layer. The periodically unsteady wakes generate additional sources of turbulence. Although the actual mechanism of interaction of the organized fluctuations and the random fluctuations are not known, the results are indicative of their coupling and the ensuing nonlinear interactions between them. The results therefore, suggest that steady turbulent flow random fluctuation profiles do not provide a good Chapter 4 Experimental Results and Discussion 85 estimate of the averages of the random fluctuations when there is periodic wake -boundary layer interaction. Within the bounds of the experiment, the observed phenomena imply higher turbulence production in higher disturbance frequency regions. The variation in the rate of growth of the turbulent intensity at a station in the inner regions was not significantly sensitive to frequency variation. There are, however, visible differences in the slopes of the turbulence intensity curves in the inner region when results at different longitudinal locations are compared for the same frequency. The location of the maximum turbulent intensity shifts towards the wall as one travel downstream with the flow. A look at Fig. 4.12 verifies this state of maximum intensity shift. For instance, at X = 0.1 m, maximum values of the turbulent intensity lie between y/5 * 0.2 to 0.26, at X = 0.3 between y/6 - 0.12 to 0.16, at X = 0.5 between y/5 - 0.06 to 0.12, and further downstream at X = 0.7 m, between y/5 « 0.08 to 0.1. These results illustrate how close to the wall the effects of wake interaction have penetrated across the boundary layer. It is such penetration and mixing that resulted in fuller boundary layer profiles along the streamwise direction. The experiments have shown that perturbing a turbulent boundary layer with traveling wakes changes the structure of the boundary layer significantly. Not only does it affect the fullness of the average velocity profiles, but also the magnitudes of the random fluctuations too. It has been found that wake-boundary layer interaction^ promotes momentum and energy transfers in the boundary layer. The transfer mechanisms may be ascribed to the nonlinear interactions or correlation of the organized fluctuation with the mean and random fluctuations of the flow. As shown in the figures above, the random fluctuations extract additional energy from the periodic fluctuation components. Chapter 4 Experimental Results and Discussion 86 In addition, the extraction of momentum was faster and easier in the outer region because of the decreasing influence of viscosity (shear stress). On the other hand, as the wake effect diffuses towards the wall, its influence on the random fluctuation appears to be less. Traveling periodic fluctuations in the free stream behave as sources of turbulence and trigger higher random fluctuations. This phenomenon implies an increase of the Reynolds stress terms in the boundary layer. As shown by the experimental results, the significance of the nonlinear interactions calls for closer attention when dealing with unsteady turbulent boundary layers. Therefore, in modeling closure equations in a computational scheme, provision must be made to account for the effects of the organized fluctuation both on the mean velocity and random fluctuation equations. In this way, the problem of unsteadiness may be approximated and accounted for rationally in the analysis and design of turbomachinery blades. 4.3.3 Velocity Phase Shifts Across the Boundary Layer It is clear that turbulent boundary layers respond to imposed periodic fluctuations as shown in the previous sections. The response of the boundary layer is not, however, instantaneous across the profile. Based on the nature of the external disturbance, the boundary layer reacts either ahead of time or at a latter time relative to the free stream flow condition. For example, laminar boundary layers are known to exhibit a phase lead at the wall relative to the external flow when subjected to external fluctuating disturbances of the stationary wave type [4,5,9]. On the other hand, when the boundary layer is perturbed by external disturbances of the traveling wave type, the inner layer lags in response with reference to the free stream flow, as shown by Patel [6,12] and Evans [9]. Chapter 4 Experimental Results and Discussion 87 In this study the phase shift across the turbulent boundary layer was determined experimentally using the ensemble averaged velocity data. At any longitudinal station, all velocity records were referenced to the respective free stream velocities. The time history of the central airfoil wakes were used to determine whether the velocity at a particular location across the layer was leading or lagging the free stream velocity. Phase angles determined from the ensemble averaged velocities in this way are presented in Figs. 4.13a and 4.13b for cases 2, 3 and 4 while Figs. 4.14a and 4.14b illustrate cases 5, 6 and 7. Each figure illustrates phase angles at specific downstream locations with varying frequencies. Positive values of the phase angle represent a phase lead with respect to the free stream velocity, while negative values represent a phase lag. All cases share a similar response across the boundary layer profile, with a phase lag when the traveling wakes interact with the turbulent boundary layer. For each case the phase lag increases towards the wall with maximum phase lag very close to the wall. However, as a result of mixing, viscous actions, and other interactions of the disturbance and the boundary layer, the identity of a profile of an airfoil near the surface of the flat plate would not be as distinctive as in the external region. Therefore, there is a certain degree of uncertainty in the actual magnitudes of the phase lags very close to the wall. The uncertainty was estimated to be in the range of 10% to 15%. Nevertheless, for all frequencies tested, the results exhibit phase lags across the boundary layer. The profile responses take place later in time after the free stream has been perturbed. Depending on the frequency and location on the flat plate, phase lags on the order of one periods were recorded close to the wall. The trend of the phase lags with frequency variation is not clearly determined. In most cases, it appears that the phase lag increases with frequency. The highest Chapter 4 Experimental Results and Discussion 88 frequency tested (40 Hz) consistently exhibited lower phase lags than the other frequencies at all downstream locations, however. At the last longitudinal station the phase lag was found to decrease with increasing frequency. It is important therefore to investigate why the velocities in the boundary layer lag the free stream velocity under traveling wave disturbances. In this study it has been found that the free stream oscillations (wakes) travel in the flow direction with a velocity of approximately 0.8 times the upstream velocity. Understanding the process by which an external disturbance is transmitted across the layer may help explain the behavior of the phase angles. Insight to this process may be gained by considering the interactions of relevant forces across the profile. In steady turbulent flows the mean pressure gradient in the transverse direction would be constant, that is, the same pressure gradient acts from the wall to the free stream. But in unsteady flows, the profile is exposed to additional oscillating pressure gradients. The frequency of oscillation and the speed of the oscillation have important roles to play in the process. This process is best shown by following Patel's [12] analysis of the free stream pressure gradient impressed on the boundary layer. Let the free stream velocity be expressed as U(x,t) = U0+ Ul{x)ei^ t~ x,&, 4-1 and the boundary layer momentum equation for the ensemble averaged velocity be du du du 1 dp d u d , , , . — + u— + v — = - — - + v — T - — ( w ' v ' ) . 4-2 dt dx dy p dx Qy z dy The free stream pressure gradient is expressed in terms of the local unsteady acceleration and the convective acceleration as follows; Chapter 4 Experimental Results and Discussion 89 -L<tJJL + u™. 4-3 p dx dt dx Making use of eqn. 4-1 and performing the proper algebra, the imposed pressure gradient assumes the following form: dt dx TT  d U l •  U 0 T r eiw(t-x/Q) TT dUi . Ux2 Ui—— - ico 4-4 2iu>(t-x/Q) dx Q The first term in the right hand side is the local acceleration. The rest of the terms represent the convective acceleration in the pressure equation. It is apparent that the magnitude of the traveling velocity Q, and the amplitude of the fluctuation velocity Ui , determine the relative importance of the unsteady and convective inertia terms in the above equation. If small amplitude of oscillation and negligible variation of the amplitude in the downstream direction are assumed, then second order terms may be neglected. In that case eqn 4-4 reduces to - I * =  dJL+ V * L = fctfjeW-*'® - i ^ U ^ *1 ® . 4-5 p dx dt dx Q The terms on the right hand side are the forces that emanate from the pressure gradient. When inserted into eqn. 4-2, they interact with the rest of the boundary layer terms enhancing the phase lead or causing a phase lag in the profile. The reaction of the boundary layer to the external fluctuating disturbance is dependent on the relative significance of the traveling wave velocity and the mean free stream velocity. For large values of Q the second term on the right hand side of eqn. 4-5 vanishes. With Q equal to the mean free stream velocity, all the terms cancel out resulting in a constant pressure case where there is no effect from the external Chapter 4 Experimental Results and Discussion 90 fluctuations. But, when the traveling wave velocity is less than the mean free stream velocity, the convective term becomes significant tending to increased inertia. Let us consider a traveling wave velocity Q less than the mean free stream velocity Uo . In all cases of the experiments, a value approximately 0.8 times the free stream velocity was observed. The convective term in the pressure gradient is then dominant compared to a purely time dependent (stationary wave) perturbation. The effect of the convective term is therefore to add to the inertia term of the momentum equation in the boundary layer forcing the fluid to lag in phase. This analysis then explains the behavior of the experimental results with regard to phase shift. Increasing the frequency also leads to an increased phase lag as illustrated by the analysis. Due to the presence of additional stresses, it is therefore reasonable to obtain a phase lag in a turbulent boundary layer exposed to a traveling wave type external disturbance. 4.3.4 Amplitude Ratio Across the Boundary Layer In this section results of amplitude ratio measurements across the boundary layer are presented. All amplitudes in the boundary layer are referenced to the free stream fluctuation amplitude at each measurement location. These ratios indicate whether the fluctuation is amplified or attenuated as the wakes interact with the boundary layer. Figs. 4.15a and 4.15b show results for blade spacing S = 0.1 m, and Figs. 4.16a and 4.16b give results for S = 0.2 m. Figs. 4.15a and 4.16a illustrate the relative magnitudes of the fluctuation amplitudes for a particular disturbance frequency. Each figure shows amplitude ratios at the indicated measurement locations along the flat plate. For instance, across the bulk of the boundary layer, the 20 Hz disturbance frequency resulted in attenuation of amplitudes irrespective of downstream location. In close proximity to the wall, however, amplitude overshoots were recorded at all locations. Chapter 4 Experimental Results and Discussion 91 This phenomenon is another indication of how the inner and outer regions of the boundary layer respond differently to an externally imposed disturbance fluctuation. The damping of the fluctuation amplitude appears to be a function of the external oscillating frequency. Attenuation is stronger for the lower frequency shown. For instance, at 20 Hz, the minimum amplitude ratio ranged between approximately 0.55 to 0.7, while at 40 Hz, the amplitude ratios in most of the external region remained almost constant at about 0.9. But, in the inner layer there is a significant variation of amplitude amplification for the various disturbance frequencies . Fluctuation overshoots increased with increasing frequency. As in the case of the outer region, there is no clear trend of the amplitude overshoots" dependence on the respective reduced frequencies. The influence of the traveling wave type wake disturbance is indeed felt all across the boundary layer. As the structures in the inner and outer layers are different, their respective responses appear to be different. It seems that the interaction of the random fluctuation and the periodic fluctuation also contribute either to the undershoots or the overshoots of the amplitude ratios. 4.4 Wall Shear Stress Under Traveling Periodic Disturbances One of the objectives in this study was to investigate unsteady wall shear stress on a flat plate. As in the cases of unsteady boundary layer velocity measurements, the boundary layer was perturbed by an external organized fluctuation of the traveling wave type. It was the response of the surface shear stress that was sought as a result of wake-boundary layer interactions. One of the techniques employed to obtain the wall shear stress was glue-on probe measurement of the skin friction. The technique has been explained in Chapter 3. Several longitudinal locations were selected at which wall shear stress measurements were taken. The first stage in this series of experimentation Chapter 4 Experimental Results and Discussion 92 (after calibration) was to take wall shear stress measurements for steady turbulent flow. Such measurements were utilized for the purpose of comparison. One of the purposes of steady measurements was to validate the results of the probe by comparing results with empirical skin friction equations for steady turbulent boundary layers as presented in the literature [37,55]. The other purpose was for comparison with the results obtained when the boundary layer is disturbed by periodic fluctuations. In addition to the hot element measurements, other methods were applied to estimate the skin friction. One method provided an estimate of the skin friction coefficient using the velocity data. This technique utilizes the relationship of the velocity gradient with the shear stress evaluated at the wall. Yet another technique employed was the Clauser plot, which makes use of the logarithmic region of the boundary layer velocity profile to estimate skin friction coefficient. The cases investigated are similar to those for velocity measurements, as shown in Table 4 . 1 . This time, however, the number of downstream stations at which wall shear stress measurements were taken increased as shown in Table 4.3. The table shows the Reynolds numbers and the reduced frequencies for each case considered. Like the previous cases (velocity measurements), the reduced frequencies were determined from the free stream velocity, the downstream location, and the fluctuation frequency. 4.4.1 Steady Turbulent Boundary Layer Wall Shear Stress In this section the experimental results of steady skin friction coefficient measurements along the flat plate are presented. No organized fluctuation was imposed in this case, i.e., the wake generating rotor was not in motion. Fig. 4.17 presents skin friction coefficient as a function of Reynolds number based on the downstream distance, as Chapter 4 Experimental Results and Discussion 93 estimated by the various techniques. In this figure, skin friction coefficient obtained by measurements is compared with empirical equations (extrapolated) for turbulent boundary layers as given by White [55]. The results obtained by measurement is within the bounds of the empirical results. The gradual decrease of skin friction with increasing Reynolds number has been reproduced. In particular, the experimental results closely followed the 1/5 , h power equation, as can be observed on the figure Table 4.3 Experimental condi t ions for Wall Shear Stress Measurement. X[m] S=0.1m S=0.2m Re x10 5 20 Hz 30 Hz 40 Hz 10 Hz 15 Hz 20 Hz 0.07 0.144 0.47 0.7 0.93 0.23 0.35 0.47 0.1 0.205 0.67 1.0 1.33 0.33 0.5 0.67 0.15 0.308 1.0 1.5 1.99 0.5 0.75 1.0 0.2 0.411 1.33 2.0 2.67 0.67 1.0 1.33 0.3 0.616 2.0 3.0 4.0 0.99 1.5 2.0 0.4 0.822 2.67 4.0 5.33 1.33 2.0 2.67 0.5 1.03 3.33 5.0 6.66 1.66 2.5 3.33 0.6 1.23 4.0 6.0 7.99 1.99 3.0 4.0 0.7 1.44 4.67 7.0 9.33 2.33 3.5 4.67 4.4.2 Unsteady Wall Shear Stress Results by Glue-on Probe Measurement Whether the unsteady wake-boundary layer interaction has any bearing on the wall shear stress was determined by measuring instantaneous values of the wall shear stress Chapter 4 Experimental Results and Discussion 94 at several locations along the flat plate. In these cases, the wake generating rotor was in operation producing disturbance over a range of frequencies. At each station ensemble-averaged wall shear stress values were obtained from the glue-on probe for several cycles. Typical time histories of the ensemble-averaged wall shear stress under a periodic traveling wave disturbance are given in Fig. 4.18a. The figures depict skin friction coefficients at some longitudinal stations at the indicated frequencies. The effect of the perturbations is stronger at the first few X-stations than at the far downstream stations. The effect of the disturbance was to produce step-like responses at the wall. This phenomenon is clear for the 30 and 40 Hz frequencies. Fig. 4.18b also gives a sample time history of the random fluctuation of skin friction at X = 0.1 m for the same frequencies. It is apparent that the wall shear stress is responsive to the external stimulus. Fluctuations appear to anticipate the disturbance oscillations. These oscillatory responses diminish further downstream as illustrated in Fig. 4.18c. Although the ensemble average results are not oscillatory in the fashion of the perturbations, there is an increase in the magnitude of the wall shear stress whenever there is a wake disturbance. This may be best shown by making comparisons between the skin friction coefficients in steady turbulent boundary layer flows with the average skin friction values of the unsteady flows at various frequencies. Fig. 4.19 depicts plots of time-averaged skin friction coefficients as a function of Reynolds number for cases 1, 2, 3 and 4 where the blade spacing S = 0.1 m. Here comparisons are made among results with different frequencies and the steady flow result. A note should be made here that all wall shear stress values were normalized with the corresponding local free stream velocities. It is apparent that the wall shear stresses in the unsteady turbulent Chapter 4 Experimental Results and Discussion 95 boundary layer flow cases ( with a few exceptions) were higher in magnitude than the corresponding steady flow values . A closer observation of the steady and unsteady skin friction results reveal some marked differences in the response of the wall shear stress. While the steady flow result indicates decay in the downstream direction, the unsteady skin friction appears to behave differently in the regions near the leading edge of the flat plate and at locations further downstream. All the frequencies tested exhibited an increasing skin friction coefficient in the near leading edge region. After reaching maximum values at about X = 0.15 m, then all cases show decay further down along the flat plate. The rate of decay is found to be higher for the higher frequencies. The behavior of the skin friction response near the leading edge looks like a response in a transitional flow regime. The response of the wall shear stress is, therefore, found to be highly frequency sensitive as is evident from the figure. Skin friction clearly increases with increasing frequency in this set of experiments. For instance, at x = 0.3 m (Re x = 0.6X105), the friction coefficient is greater by approximately 10%, 20%, and 78% for 20 Hz, 30 Hz and 40 Hz disturbance frequencies respectively. Fig. 4.20 also shows results obtained from a different set of experiments where the airfoil spacing was 0.2 m. Of course this change in spacing resulted in a different set of disturbance frequencies as indicated in Table 4 .1 . This set includes cases 5, 6 and 7. The trend of the skin friction coefficients is similar to the previous set of experiments, but the values here are even higher than the corresponding cases in the first set when compared to the steady flow result. However, the respective local free stream velocities were lower than the previous cases. Chapter 4 Experimental Results and Discussion 96 To interpret the various results, it is important to investigate the responses of the boundary layers very close to the wall, and the mechanisms by which they are driven. The viscous phenomenon and turbulent stress generating process may shed some light on what causes the wall shear stress to behave in a particular way. In general, it is known that shear stresses are dependent on the prevailing velocity gradients across the boundary layer and on the Reynolds stress terms resulting from the mixing and momentum exchanges throughout the layer. The presence of a periodically fluctuating disturbance, and its nonlinear interaction with the mean flow and random fluctuations, subjects the boundary layer to additional forces on top of the steady turbulent boundary layer case. Similar to the way in which random fluctuations lead to Reynolds stresses, periodic fluctuations also impart extra stresses to the boundary layer flow. The different forces can be identified experimentally as shown by Hussain and Reynolds [24] through the technique of triple decomposition. Such an analysis was also adopted by Telionis [7], by forming the governing differential equations similar to the steady turbulent boundary layer equations. For example, the mean momentum equation in the classical form is given as; du du 1 dp dud -— - <3 , — „ . _ u— + v — = - — - + v — — ( u ' v ' ) - — (uv) 4-6. dx dy p dx Qy z dy dy and the equation for the oscillating flow becomes; ~ ~ ~ ~ o ~ du ~ du du ~du du I dp d u d — — — + u — + u — + v — + v — = -—-— + v — = - + — (uv - uv) dt dx dx dy dy p dx Qy l dy - — (<M'V'> - i77) 4-7. dy It can be seen in these equations that Reynolds stress terms are obtained from the periodic and the random fluctuations. Therefore, an increase in the wall shear stress Chapter 4 Experimental Results and Discussion 97 appears to be significant due to the coupling and nonlinear interactions of the organized fluctuation with the boundary layer. In investigating the unsteady velocity profiles, it has been shown that there exists a departure from the profiles of steady turbulent boundary layer profiles. The departure was in the form of fuller velocity profiles for wake disturbed flows, characterized by increased velocity gradients at the wall. Increased velocity gradients in turn suggest higher stresses on the surface of the flat plate. Although it is difficult to quantify exactly what the contributions of the random and periodic fluctuations are separately, overall evaluation of the wall shear stress shows growth with the presence of organized fluctuations. One of the features observed in Figs. 4.19 and 4.20 is the transition-like behavior of the wall shear stress at the first three measurement locations. Although the basic boundary layer was turbulent prior to encountering a wake disturbance, the results indicate the intermittent nature of the boundary layer in the vicinity of the flat plate leading edge when the unsteady wakes are present. It seems appropriate to look into the kinematics of the flow just ahead of the leading edge. Looking from a stationary frame of reference ( hot wire, glue-on probe), the absolute flow speed fluctuates in magnitude and in direction as well. It has been found that the mean free stream velocity with the blades rotating was less than the steady flow free stream velocity near the leading edge. This was because of the blockage effect of the airfoils and the ensuing wake formation behind the blades. The free stream flow, however, recovered its speed in the downstream direction. It is, therefore, possible that the leading edge region was under intermittent flow conditions leading to transition-like skin friction coefficients. Chapter 4 Experimental Results and Discussion 98 4.4.3 Evaluation of Wall Shear Stress from Velocity Curve Fits One of the techniques available to approximate the wall shear stress in a boundary layer is making use of careful velocity measurements across the profile. In this procedure it is necessary to make sure that the velocity measurements close to the wall are as accurate as possible. Usually it is very difficult to produce accurate velocity measurements from hot wire probes in close proximity to the wall due to various limiting situations. In what follows, the ensemble-averaged velocity data for cases 1 to 4 have been used to approximate the wall shear stress at the wall. First the boundary layer velocity data at each measurement location were fit with ninth order polynomials for all unsteady turbulent boundary layer profiles as in Fig. 4.21a. For steady velocity profiles sixth order polynomials were used. Then the best fit curve equations for each velocity profile were formulated in terms of the transverse scale. By evaluating the velocity derivative of each equation at the wall, it was possible to evaluate the respective wall shear stresses for each case. Validity of the technique for steady flow cases has been confirmed as Fig. 4.21 presents a comparison between stress measurement and velocity curve fit for steady flow, in which agreement is reasonable. Results obtained through this technique as compared to the results of glue-on probe measurements for unsteady flows are illustrated in Fig. 4.22. The figure depicts skin friction coefficient in terms of the reduced frequencies shown in Table 4.3. Making use of the reduced frequency did not result in a collapse of the skin friction results. There are significant discrepancies between results obtained by wall shear stress measurements and those obtained from velocity curve fit data. The variation appears to be large for the lowest frequency in this set of Chapter 4 Experimental Results and Discussion 99 experiments. The gap, however, gets closer as the frequency was increased. The transition-like behavior was also not predicted by the velocity curve fit results. In addition, all the results rather overestimated the wall shear stress values irrespective of frequency and downstream measurement station. Nevertheless, certain trends of the wall shear stress in response to external periodic fluctuations have been reproduced. The increase of the skin friction when wakes interact with the boundary layer seems to be reproduced. Moreover, sensitivity of the skin friction to frequency is also in agreement with glue-on probe measurement results and the decay of skin friction in the downstream direction is replicated. A certain amount of discrepancy is expected from these two different results. Along with the errors associated with each probe, the difficulty of measuring velocity accurately very close to the wall enhances the error when a hot wire probe is employed. In this case the uncertainties may be in the order of 10% to 15%. Despite the variations in the results from the above techniques, the overall trends show a reasonable degree of agreement. 4.4.4 Estimation of Skin Friction Coefficient Using Clauser Plots Another technique that has been used to estimate the skin friction coefficients in turbulent boundary layer flows is the method introduced by Clauser [46]. Clauser plots make use of the boundary layer profile when velocity is plotted in terms of a dimensionless coordinate comprising the transverse scale, kinematic viscosity and the free stream velocity at several constant skin friction coefficients. The method entirely depends on the logarithmic region of the boundary layer. When a measured velocity profile is superposed on the graph of constant skin friction coefficients, a friction Chapter 4 Experimental Results and Discussion 100 coefficient for the measured velocity profile can be selected by matching. Details of the procedure are given in Appendix B. The basic assumption in employing this method is the existence of a logarithmic region in the boundary layer. This technique is valuable in approximating skin friction coefficients for steady turbulent boundary layers. Whether this technique is applicable to unsteady turbulent boundary layers has been investigated in this section. If the method holds true, it can be an easier way of approximating unsteady wall shear stress under periodic wake disturbance. Therefore, the experimental results of velocity measurement at different disturbance frequencies have been used to generate Clauser plots. Velocity profiles at various phases in a single cycle were superposed on the Clauser plots. First the validity of the experimental results for steady turbulent boundary layers was ensured. Fig. 4.23 illustrates Clauser piots for the steady flow case at four downstream measurement stations. The skin friction coefficients for each location can be read with reasonable accuracy. The decreasing trend of the friction coefficient is shown. Values comparable to glue-on probe skin friction measurements (with errors in the order of 10%) are obtained. Steady flow results are, therefore, believed to be reproduced in these plots and the validity of measured results is confirmed. Application of the same procedure to unsteady turbulent boundary layers gives the graphs depicted in Figs. 4.24 to 4.26 for the different wake generation frequencies. Each figure contains Clauser plots at the four velocity measurement locations along the flat plate. As the superposed phase velocity profiles indicate in the respective cases, it appears that matching values can be read for some locations better than for others. For instance, the profiles do not match at X = 0.1 m for all frequencies shown. Reading at X = 0.3 m and X = 0.5 m are relatively reasonable. Although the fluctuation in profiles Chapter 4 Experimental Results and Discussion 101 are noticeable at each location, it seems difficult to identify with certainty the fluctuating skin friction coefficients in any one case. Review of all the results furnished by this technique leads to the following conclusion regarding the use of Clauser plots for unsteady turbulent boundary layers. Even though there are occasions where the logarithmic region matches with specified C f values, a clear trend did not emerge in the process. It is, therefore, not advisable to apply this technique alone to evaluate skin friction coefficients in periodically disturbed unsteady turbulent boundary layers. 4.5 Concluding Remarks In this chapter, experimental results of turbulent boundary layers subject to traveling wave type external fluctuations have been presented. The primary variables of measurement were instantaneous velocity and wall shear stress. From the respective data, various boundary layer behaviors were investigated. All the unsteady velocity profiles obtained in this study provide the following points about turbulent boundary layers under the disturbance of traveling waves. First, the profiles fluctuate significantly within a single period of disturbance for all frequencies considered. A simple assumption of the equivalency of steady velocity profiles obtained from time averaging, and unsteady turbulent velocity profiles subject to traveling wave disturbances, is not, therefore, warranted. Second, the decay of oscillations is gradual in the downstream direction. Therefore, the influence and interaction of the wakes with the boundary layer is sustained longer as some other studies on wakes suggest [59]. The slow decay of an upstream generated wake and its interaction with the boundary layer has significance in multi-stage flow regimes of turbomachines. Chapter 4 Experimental Results and Discussion 102 The response of the boundary layer lags that of the free stream for all cases considered. The phase lags increased in magnitude towards the wall. Amplitude ratios indicated sensitivity of the boundary layer to traveling perturbations. Overshoots and undershoots suggest the fluctuating nature of the flow across the boundary layer. Wall shear stress results show sensitivity of the wall resistance to external fluctuation frequencies. Within the bounds of the experiment, skin friction coefficients increased with disturbance frequency. Resistance was high in all unsteady turbulent boundary layer measurements compared to steady boundary layer results. Therefore, using steady flow approximations to predict unsteady flow resistance would underestimate the losses associated with unsteadiness. It is important to take into account variations of boundary layer behaviors due to unsteadiness. Chapter 4 Experimental Results and Discussion 103 i i i in i i i i i E E E E d o d o n II n II X X X X I I I I 00 CO en/n II Chapter 4 Experimental Results and Discussion 104 LL Chapter 4 Experimental Results and Discussion 105 Chapter 4 Experimental Results and Discussion 106 Chapter 4 Experimental Results and Discussion 108 Chapter 4 Experimental Results and Discussion 109 1 • * 1 1 • 1 ' 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time (sec) Fig. 4.4c Periodic Velocity Fluctuation: X - 0.1 m, S = 0.2 m, f - 1 0 Hz E o Z> cz g to o _2 u_ o g OL ' 1 ' 1 1 1 ' 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time (sec) F ig.4.4d Periodic Velocity Fluctuation: X - 0.3 m, S • 0.2 m, f - 1 0 Hz Chapter 4 Experimental Results and Discussion 1 1 2 Fig. 4.5d Velocity Profiles at Various Phases in a Single Cycle: Unsteady Flow U e - 3.0 m/s, U r = 2.0 m/s, S - 0.2 m, f - 1 0 Hz Chapter 4 Experimental Results and Discussion 114 Fig. 4.5e Velocity Profiles at Various Phases in a Single Cycle: Unsteady Flow U e = 3.0 m/s, U r - 3.0 m/s, S = 0.2 m, f - 15 Hz Chapter 4 Experimental Results and Discussion 115 Fig. 4.5f Velocity Profiles at Various Phases in a Single Cycle: Unsteady Flow U e - 3 . 0 m/s, U r - 4 . 0 m/s, S = 0 .2m, f = 20 Hz Chapter 4 Experimental Results and Discussion 116 Fig. 4.6b Comparisons of Average Turbulent Boundary Layer Velocity Profiles for Steady and Unsteady Flows With Different Disturbance Frequencies Chapter 4 Experimental Results and Discussion 118 Chapter 4 Experimental Results and Discussion 119 Chapter 4 Experimental Results and Discussion 120 Chapter 4 Experimental Results and Discussion 121 Chapter 4 Experimental Results and Discussion 122 Chapter 4 Experimental Results and Discussion 124 Chapter 4 Experimental Results and Discussion 125 Chapter 4 Experimental Results and Discussion 126 Chapter 4 Experimental Results and Discussion 127 N N N X X X 2 o o II »»-II T 1 V i i } I lO E CM o O g<J)e|6uvesBMd o s >0 Hz 10 Hz: 0 Hz II **— n g — \ q J© CO E d O II o 8 o 8 g $ 9 | 6 u V 6 S B l | ' d o 8 ~ N N N X X X P. o o II II T 1 V ll \ \ • lO 8 t i l l ( ^ e i B u v e s e i i d o D o X 8° ^ g4>e|6uvesBL|'d c o •a CO 35 E (0 CD ± J CO 8 CD E o cr £ u. o> c CO QL CD « 5 <3> C '£> . c 2r o o CD T3 C c © t_ >» T3 CO CD CO c => CD J Z -*—' M— o its ! c W CD 13 x: 0. & O) LL Chapter 4 Experimental Results and Discussion 128 Chapter 4 Experimental Results and Discussion 129 100 •e-o O) c < (0 100 200 -300 -400 -500 • w • . f = 1 0 H z • I - f = 1 - f = 2 5 Hz-!0 H z : • 0.0 0.2 0.4 0.6 0.8 1.0 i ) X = 0.1m y / o 1.2 1.4 1.6 1.8 100 -400 -500 r i i i I i i i i i i i i i i i i i i i i i i i i i I i i f = 1 0 H z f = 1 5 H z f = 20 Hz 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 ii) X ° 0 . 3 m y / o Fig. 4 .14a Phase Shifts of the Unsteady Turbulent Boundary Layer Velocity Wi th Varying Wake Passing Frequency at Some Downstream Stations Chapter 4 Experimental Results and Discussion 130 Chapter 4 Experimental Results and Discussion t - co rnr-o o o o It U II u X X X X V / \ / 1 7 to N X u> cvi o in o env/nv u> d 3 ° — E E E E t - CO l O h -o o o o ii ii n n X X X X «o N d O CM o 9nv/nv u> d 3 to d 3 N O CO env/nv 0s-d Chapter 4 Experimental Results and Discussion 132 Chapter 4 Experimental Results and Discussion 133 Chapter 4 Experimental Results and Discussion 1 3.0 2.5 2.0 <D 1.5 < I 1 0 0.5 f = 10 Hz f = 15 Hz f = 20 Hz tt • • * . - -• 0.0 0.2 0.4 i) X = 0.1 m 0.6 0.8 1.0 1.2 1.4 1.6 1.8 y / 8 3.0 2.5 2.0 0.0 i i I i i i • I T T r i I i i i i f = 1 0 H z f = 1 5 H z f = 20 Hz 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 i i )X = 0 .3m y / 5 Fig. 4.16b Amplitude Ratio of the Periodic Fluctuation Across the Turbulent Boundary Layer: Same Station, Varying Frequency, S - 0.2 m Chapter 4 Experimental Results and Discussion 135 Chapter 4 Experimental Results and Discussion 136 Experimental Results and Discussion 137 Chapter 4 Experimental Results and Discussion 138 'O jo uojiBnpny tuopuBy 'O jo uopnpnu Luopuey E o b jo uoiiBnpny Luopustj Chapter 4 Experimental Results and Discussion 140 Chapter 4 Experimental Results and Discussion 142 Chapter 4 Experimental Results and Discussion 143 Chapter 4 Experimental Results and Discussion 144 Chapter 4 Experimental Results and Discussion Chapter 4 Experimental Results and Discussion 146 Chapter 4 Experimental Results and Discussion Chapter 5 C O M P U T A T I O N A L RESULTS A N D DISCUSSION 5.1 Introduct ion Computational fluid dynamics (CFD) has emerged as a viable tool in solving the governing equations of motion for boundary layer flows, as presented by Cebeci et al. [58], Fletcher [68] and others [56, 57, 64, 65, 66]. It complements experimental and theoretical fluid dynamics by providing an alternative technique of modeling real flows. Currently, well developed computer programs are available in the market for solving various flow problems. Since the governing equations for flows of practical interest are often complicated, an exact solution is not usually available. Computational techniques replace the governing partial differential equations with systems of algebraic equations thereby facilitating the use of computers to obtain numerical solutions. Some of the techniques include finite difference, finite element and finite volume methods, where the algebraic equations link together values of the dependent variables at adjacent grid points. For these methods, a grid of discrete points is distributed throughout the computational domain in the directions of time and space coordinates. Consequently, the process of converting the continuous governing equations to a system of algebraic equations depends on discretization. One application of CFD is the solution of boundary layer development on solid surfaces. Cebeci and Bradshaw [58] present the finite-difference methods as the most flexible, practical and efficient tools for the solution of the boundary layer equations. The required number of grid points, for accurate solution, depends on the geometric Chapter 5 Computational Results and Discussion 149 complexity and severity of the gradients of the dependent variables. In addition, due to the nonlinearity of the governing equations, the computational solution usually proceeds iteratively. Hence, it is necessary for the solution of each dependent variable at each grid point to be sequentially corrected using the discretized equations. In this chapter, Computational results of a turbulent boundary layer subjected to a sinusoidal external disturbance of traveling wave nature will be presented. Calculations of turbulent boundary layer development on a flat plate under the influence of a periodic free stream velocity have been performed. The numerical procedure of Cebeci and Carr [10] has been adopted here. Evans [57] modified the code to accommodate both standing and traveling wave free stream conditions for laminar boundary layers. Fletcher et al. [56] also used a similar procedure to solve turbulent boundary layers under the influence of standing wave free stream fluctuations. The purpose of the computational scheme was not to present stand alone results for turbulent boundary layers, rather it has the limited scope of comparing some of the experimental results presented in Chapter 4. For some parameters, (amplitude, frequency, and traveling wave velocity) values obtained during the experiments were used as inputs. Furthermore there is an implied assumption in using these experimental data in the computational scheme in that the somewhat jagged free stream periodic fluctuations, shown in Fig. 4.4, are assumed to be sinusoidal. The respective amplitudes, frequencies and traveling wave velocities were then applied to the solution procedure. This assumption of free stream disturbance equivalency may be subject to some errors when comparisons of results obtained by experiments and computation are carried out. In what follows, the calculation method and formulation of the governing equations will be discussed briefly. Results of the computation, such as the velocity profile, the wall Chapter 5 Computational Results and Discussion 150 shear stress, the shear stress across the boundary layer, and some comparisons with experimental results are subsequently presented. 5.2 Computat ional Scheme 5.2.1 Equations of Motion For a two-dimensional, incompressible, unsteady turbulent boundary layer, the governing differential equations of phase averaged velocities are expressed as: ^ + ^ = 0 5-1 dx dy du du du dUP dUp d • + M —— + V —— = — + Ue + 5-2 x Su (v+em) — dt dx dy dt dx dy\ The above equations cannot be solved without additional closure equations for the turbulence stress terms. With a turbulence model of the eddy viscosity concept and the pertinent boundary conditions, the governing equations can then be solved numerically. Two separate algebraic equations are formulated for the eddy viscosity. One of the equations treats the inner part of the profile while the other equation expresses the eddy viscosity in the outer region of the boundary layer. Despite its simplicity as compared to other schemes that utilize higher order turbulence closure models, the eddy viscosity formulation has been found useful in engineering applications. The boundary layer equations, and the boundary conditions are expressed in terms of the transformed variables; X = X, t = t, r\=y[U0/vxf12 . Defining a dimensionless stream function f(x,t,x\) =vp I JvxUQ , the transformed governing equation and the boundary conditions become Chapter 5 Computational Results and Discussion 151 dx dx UQ dt 5-3 where / dUD UQ dx u, dU„ dU„ dx dt m and the boundary conditions transform to, at n = 0, / = / = 0 TI ->rioo, By introducing new dependent variables u(x,Ti,t) and V(x,Ti,t) to equation 5-3, where f = u and u' = V, the algebraic equations in the flow domain can be solved. For further details reference should be made to the paper by Cebeci and Carr [10]. 5.2.2 Numerical Formulation All calculations are based on the finite-difference method developed by Cebeci and Smith [1] and later modified by Evans [57] to accommodate traveling wave free stream disturbances. The numerical formulation chosen was the Keller and Cebeci [58] "box" method. One advantage of the "Keller box" method is that non-uniform grid spacings can be used across the boundary layer, in the streamwise direction, and the time dimension. This method uses the basic idea of writing the governing system of equations in the form of a first-order system. The resulting first-order equations are formulated by centered-difference derivatives and averages at the midpoints of net rectangles and net segments formed by the descritization process. As a result several implicit and nonlinear algebraic Chapter 5 Computational Results and Discussion 152 equations are generated throughout the calculation domain. For solutions, Newton's method is employed. A typical sample grid spacing and a cell are given in Fig. 5.1. ^n-1 P4 j-1/2 kn p1 P2 n-1 "n-1/2 'n Fig. 5.1 Net Rectangle for Difference Approximation [58]. In applying this method to the wake-boundary layer interaction flow domain, the free stream velocity was represented by a sinusoidally fluctuating velocity as expressed in equation 2.2 and repeated below as: Ue{x,t) = Uo + Af/esinco(/ - x IQ) = Uo[l + AUe lUo since (t - x / Q) 2-2 or Ue = Uo + c V ' w ( ' ~ * / G ) The mean free stream velocity, the fluctuation amplitude, the fluctuation frequency and the traveling wave velocity were all taken from the experimental results. The turbulent boundary layer on the flat plate was then discretized in the time, longitudinal and transverse dimensions. The period for any cycle was determined from the experiments and equal time steps were used during computation. The streamwise dimension had Chapter 5 Computational Results and Discussion 153 unequal grid spacings equivalent to the measurement stations as specified for wall shear stress measurements. The flow was specified to be turbulent starting from the tripping location (X = 0.02 m). The grid spacing across the boundary layer was non-uniform and was generated by a geometric progression where the ratio of any two adjacent intervals is a constant. The code was solved for turbulent flows with a maximum number of nodes of 61 across the boundary layer. At each downstream location, the solution marches in the time direction; then the governing equations are solved for each specified time station. The linearized forms of the equations were solved iteratively until a prescribed convergence criterion was met. The code uses the wall shear stress parameter f w " as the basis for it convergence. Calculation terminated when — — - < 0.1. 5/w + 2 / w For turbulent boundary layers subjected to traveling organized fluctuations, the solutions were sensitive to the time and transverse spacings. For instance, the solution diverges if the time step is on the order of 1 millisecond and when the change in vertical spacing very close to the wall is less than 0.02. Therefore, the smallest time and vertical space scales that could be resolved were limited to the order of the above figures. 5.2.3 Boundary & Initial Conditions In computing unsteady turbulent boundary layers, careful specification of the boundary conditions is required. Conditions upstream, at the wall and in the free stream need to reflect the features of the actual flow. The condition in the free stream reflects the traveling nature of the disturbance. The traveling speed of the disturbance for all cases of frequencies investigated was about 0.8 times the local free stream velocity. The velocity in the boundary layer approaches the Chapter 5 Computational Results and Discussion 154 external fluctuating velocity in the free stream which was modeled as in equation 2.2. It is to be recalled that the derivatives of the free stream velocity with respect to time and longitudinal space were used to represent the pressure fluctuation impressed on the boundary layer. In addition, all quantities with the exception of the wall shear stress vanish at the flat plate. With proper specification of initial conditions at some initial upstream surface the problem is completely formulated. The conditions required in this case are: t  = tQ and X ^ XQ ; u = u0(x,x\) X  = X0 and t2-t0 ; u = u0{x\,i) These conditions provide velocity distributions everywhere at some initial time, and for subsequent time, conditions are given on some upstream station. When the conditions are applied to equation 5-3, initial velocity fields can be obtained. For instance, for the initial conditions at t = to and x > x 0 steady state flow was assumed. The following table indicates the conditions applied to the computational scheme. These values were either conditions imposed during the experiments, or results obtained and used as inputs to the calculation. 5.3 Traveling Wave Free Stream Oscil lat ion Results 5.3.1 Turbulent Boundary Layer Velocity Profiles One of the outputs from the computation process was the velocity profile across the turbulent boundary layer. The calculations furnish values of the dimensionless velocity f with respect to the similarity variable n or the coordinate y. The boundary layer was Chapter 5 Computational Results and Discussion 155 Table 5-1 Experimental Condit ions and Results Applied to the Computat ion: U°° = 3.0 m/s Rotor Velo. U r [ m/s ] A.F.Spacing S [ m ] Frequency: f [ H z ] co [rad/s ] Period: T [s] Amplitude: B [ % ] 2.0 0.1 20 125.66 0.05 ~ 10 3.0 0.1 30 188.49 0.033 ~ 10 4.0 0.1 40 251.33 0.025 - 10 assumed turbulent starting from 0.02 m from the leading edge, corresponding to the experimental condition. For steady flow, calculation in the x-y plane was sufficient for there was no variation in the time dimension. Fig. 5.2 illustrates results of the steady turbulent boundary layer profiles obtained at several longitudinal locations on the flat plate. The profiles show a slight increase in fullness in the stream direction for the inner region. Fig. 5.2a shows comparisons of calculated and experimental results of the steady velocity profiles. The results are in good agreement except at x = 0.1 m where the calculated values overpredict the experimental values. The discrepancy at this station reflects that the experimental velocity profile was not fully turbulent. Steady flow turbulent boundary layer velocity calculations may be used as test cases to ensure whether the results are in agreement with known semi-empirical formulae or experimental results. The concern, however, of this study was the response of the turbulent boundary layer to a free stream traveling oscillatory perturbation. Therefore, the next stage was to perform computations of boundary layer characteristics at several phases in a single period. Typical phase-averaged unsteady turbulent Chapter 5 Computational Results and Discussion 156 boundary layer velocity profiles as a consequence of the external fluctuating stimuli are shown for various phase angles in Fig. 5.3. These profiles are for case 2, where the frequency of oscillation was 20 Hz. The figures provide some interesting features of the profiles at each station. The outstanding feature of the profiles is the unsteadiness generated throughout the boundary layer due to the interactions of the free stream oscillatory disturbance with the steady turbulent boundary layer. It is apparent that the velocity profiles vary significantly from phase to phase. The response of the boundary layer is more accentuated in the outer region than in the inner region. For instance, looking at the profiles at X = 0.3 m, the fluctuation amplitude reaches up to 20% of the mean velocity in the external region of the boundary layer. The profile fluctuations are, however, stronger in the internal region here than was the case in the experimental results. In general, the profiles decrease in fullness as the velocity defect goes towards its maximum and gets fuller as the flow accelerates and the defect decreases. Some of the profiles appear to revert to transition-like behavior during some of the phases in a cycle. Variations of profiles in a cycle are indicators of the nature of the forces acting across the boundary layers. Contrary to the assumption of a constant pressure gradient across the boundary layers for steady turbulent flows, the pressure gradient in periodically disturbed boundary layers fluctuates in a single period. It is this oscillatory pressure gradient that forces the velocity profiles to fluctuate in the boundary layers. The oscillatory nature of the boundary layer profiles is generally in agreement with the profiles obtained experimentally, but there seems to be a slight difference in the magnitude of the fluctuations. In both cases the unsteady flow velocity profiles are seen to be quite different from steady turbulent boundary layer profiles. Comparison of the average Chapter 5 Computational Results and Discussion 157 profiles also shows significant variation from the time-mean of steady flow velocity profiles. Therefore, one should be cautious in approximating unsteady boundary layer behavior with steady velocity profiles when such values are used for design of turbomachinery blades and stages. Figs. 5.3a - 5.3c compare calculated and measured phase-averaged velocity profiles at the indicated frequencies. Although the calculated average velocity profiles underestimate the experimental results, particularly at downstream stations for f = 20 Hz, the agreement improves as the frequency increased. Comparison of calculated and measured velocity phase shifts are shown in Fig. 5.3d. Both calculation and experiment show a similar qualitative trend although quantitative agreements are poor. As opposed to the interaction of standing-wave perturbations with the boundary layer, which generates a velocity phase advance (phase lead), traveling-wave type interactions with the boundary layer impose a negative velocity phase shift (phase lag) as indicated in the figure. The calculation procedure has successfully captured this important feature of traveling-wave type unsteady boundary layer. 5.3.2 Computational Skin Friction Coefficient Results During the calculation process, skin friction coefficients were determined from the wall shear stress parameter / w " . In fact the code calculates the shear stress across the whole boundary layer at each r] location. The wall shear stress was then evaluated at the surface of the flat plate. In a similar fashion to the calculation of the velocity profiles, the wall shear stress was first computed for steady turbulent boundary layer flows. Calculations of unsteady turbulent boundary layer wall friction followed, after the code was ascertained to be valid Chapter 5 Computational Results and Discussion 158 for steady flows. The calculation domain covered stations from X = 0.1 m to X = 0.7 m in the flow direction. The calculated steady flow skin friction coefficients were in a close agreement with empirical equations presented by White [55]. The result closely follows the 1/5th power equation. A comparison of the calculated skin friction coefficients with experimental results is shown in Fig. 5.4. Glue-on probe skin friction measurement results and those found from velocity curve fit data are superposed on the computed results. The agreement is reasonable, and it is therefore concluded that the computational scheme is valid to predict wall shear stress for steady turbulent boundary layer flows. With the exception of laminar boundary layers [9], skin friction calculations for unsteady turbulent boundary layers under traveling wave disturbances are very scarce. Hence, the computational scheme was expanded to include turbulent boundary layers in such free stream perturbing conditions. The state of wall shear stress under these circumstances is of enormous practical importance. To examine the unsteady wall shear stress responses, calculations were performed for the disturbance frequencies shown in Table 5 -1 . Fig. 5.5 depicts calculated results of average skin friction coefficients in terms of the Reynolds number based on the longitudinal axis X. In the same figure the calculated steady flow skin friction is also included for comparison. All the unsteady skin friction coefficients are larger than the steady case. For instance, at Reynolds number of 2x10 4 the skin friction coefficient is twice the steady value for a frequency of 40 Hz. Variations are strong near the leading edge, but appear to converge in the downstream direction. The results thus strongly suggest the frequency dependence of the response of the unsteady wall shear stress to traveling free stream fluctuations. Within the bounds of the Chapter 5 Computational Results and Discussion 159 frequencies tested, the higher the frequency, the higher the response of the wall shear stress. In addition, all the computed skin friction coefficients exhibit the usual trend of decreasing with the increase of Reynolds number. The rate of decay, however, appears to be higher near the leading edge compared to the downstream positions where the rate decreases. The mechanism of increasing skin friction coefficient with frequency supports the description given in Chapter 4. The results obtained by calculation are similar in most features to the experimental data, and detailed comparisons will be given in the next section. Variation of the skin friction coefficient can also be seen by observing its trace in time. Fig. 5.6 presents the time history of the skin friction coefficients at several positions on the flat plate for different frequencies. The curves at the specified frequencies indicate the fluctuations of c y in a single period at the locations specified on the flat plate. The results emphasize two aspects of the wall shear stress variation in response to the free stream fluctuations. The first aspect is the fluctuating nature of the wall shear stress in one cycle. The fluctuations in skin-friction coefficient are strong at X = 0.1 m for all frequencies considered, but appear to flatten in the downstream direction. The fluctuating amplitude at the first calculation station is higher than at the downstream stations where the disturbance has dissipated some of its energy due to mixing. The other feature of the results is the decreasing trend of the magnitude of the wall shear stress in the streamwise direction. This is the expected trend as was found experimentally for steady turbulent boundary layers. From the results, one can presume that with a sinusoidal external disturbance, the wall shear stress responds sinusoidally and the amplitude of the wall shear fluctuation decreases in the flow direction. Chapter 5 Computational Results and Discussion 160 5.3.3 Comparisons of Calculated and Experimental Results of Skin Friction Coefficients In this section, comparisons of unsteady skin friction coefficients obtained by calculations, by measurement, and by velocity curve fits will be presented and discussed. Figs. 5.7 to 5.9 Present averages of the unsteady skin friction coefficients in terms of the reduced frequency w for different frequencies. For free stream oscillation frequency of 20 Hz, the results are not in good agreement with each other. The transition-like trend indicated by measurement was not reproduced by the curve fit ( f rom velocity data) or by the calculations. Both methods overestimated the values obtained by measurement. The Computational results lie between the curve fit and measured results. The calculated results, however, converge to the measured results in the downstream direction. The discrepancies among the various methods appear to narrow down with increasing frequency of free stream oscillation. Figs. 5.8 and 5.9 exhibit this phenomenon for the 30 and 40 Hz frequencies respectively. Although the calculated and velocity curve fit results still overestimate the measured results, the gap closes as the frequency is increased. However, compared to steady boundary layer results, all the periodically unsteady turbulent boundary layers result in significant increases in wall friction. Therefore, in evaluating the wall shear stress for unsteady turbulent boundary layers, one should not solely rely on steady flow results. Concerning the discrepancy between direct measurement and computational results, one responsible factor may lie in the assumption of the free stream fluctuation. When the computational scheme was employed it was assumed that the free stream oscillation was sinusoidal. The jagged nature of the experimental periodic fluctuations, as shown in Chapter 5 Computational Results and Discussion 161 Fig. 4.4, were therefore inaccurately modeled by sinusoidal periodic fluctuations in the computational scheme as expressed by equation 2.2. Moreover, the angular fluctuation in the absolute velocity vector of the rotor exit, that is, before the flow interacts with the boundary layer on the flat plate in the experiments, was not considered in the calculation model. Therefore, directional fluctuations were not incorporated in the numerical model. It has been also noted that random turbulent fluctuations existed within the wakes of the airfoils. No random turbulent fluctuation was included in the free stream fluctuating disturbance model in the calculations. Although it is believed that the major physical phenomena in traveling wave type unsteady turbulent boundary layers were predicted by the calculations, more refined modeling of the free stream periodic fluctuations corresponding to the experimental results may be required to close the gap between calculation and experiment. The greatest differences in skin friction coefficients are seen between wall shear stress measurement and those obtained from velocity curve fits. Skin friction coefficients obtained by curve fits overestimated the measurement results more than the calculated results. One source of error for the velocity curve fit data could be the difficulty of taking measurements of velocities very close to the wall. In fact the velocities were extrapolated to the wall whereby the wall shear stress was derived from the velocity gradients. Errors may also be associated with the curve fits to the measured data. All these factors are believed to contribute to the deviation among the results. Plotting Cf against the reduced frequency did not collapse all the data onto a single curve. An attempt has been made to plot Cy in a logarithmic scale in order to bring the plots closer to each other. If the errors are accounted for, this procedure appears to Chapter 5 Computational Results and Discussion 162 provide a better functional relationship between logcy and w. If that is the case, the reduced frequency can be considered as a similarity parameter for unsteady turbulent boundary layer skin friction coefficient determination. The impact of the traveling wave velocity variation on the response of the boundary layer profile and the wall shear stress was not investigated. In all the experiments and the calculations, a single traveling wave velocity ( 0.8 U e ) was employed. To explore to what extent it can influence the boundary layer characteristics requires more detailed investigations. Patel [12] has presented significant influences of the traveling wave velocity for the frequencies he considered ( f < 12 Hz ). Another aspect of the free stream disturbance is the standing wave type oscillation. This case needs to be investigated as well to see if the boundary layer behaves differently from the traveling wave case. Preliminary calculations indicate substantial differences in the responses of the boundary layer to the two types of oscillatory perturbations. Results indicate variations in magnitude as well as in phase. 5.3.4 Calculated Shear Stress Profiles Across the Boundary Layer The calculation scheme also provides shear stress values across the boundary layer. Although no measurement of shear stress was taken across the boundary layer (except at the wall) in any of the experiments, the computational results give some indication of what to expect due to the external disturbances. Figs. 5.10 to 5.12 depict such calculated shear stress results at various phases in a period for the frequencies indicated. All cases show strong shear stress fluctuations at the wall. This is consistent with the fluctuations of the wall shear stress presented in the previous section. The values of the shear stress reach maxima at or very close to the wall, and then decay at a Chapter 5 Computational Results and Discussion 163 fast rate to about the n = 2.0 location and remain almost flat for the remainder of the profile. It appears that the shear stress is less frequency sensitive in the outer region than in the inner region. The inner region, therefore, responds readily to the external fluctuation. 5.4 Concluding Remarks A numerical procedure has been employed to predict some of the experimental results for unsteady turbulent boundary layers. Although the calculations were limited in scope, the results obtained are interesting and encouraging. The wall shear stress was found to respond strongly to the free stream disturbance frequency, and to grow with increasing frequency. All results confirm the high wall shear stress values observed in unsteady turbulent boundary layers compared to steady results obtained in the experiments. Unsteady turbulent velocity profiles vary significantly within a single period in response to the external fluctuations. The velocity fluctuations are strong in the external region of the profile. All cases indicate non-negligible fluctuations as a result of non-linear interactions which do not justify the implementation of steady turbulent boundary layer results in a design process that involves unsteady flows. It is believed that agreement of the experimental results and computations carried out are promising. However, more detailed calculations involving unsteady turbulent boundary layers subject to both standing and traveling wave free stream oscillations at various frequencies is recommended. Chapter 5 Computational Results and Discussion 164 Chapter 5 Computational Results and Discussion 165 Chapter 5 Computational Results and Discussion 168 Chapter 5 Computational Results and Discussion 170 Chapter 5 Computational Results and Discussion 171 Chapter 5 Computational Results and Discussion 173 Chapter 5 Computational Results and Discussion 174 *0 r juepiyeoo UOIJOUJ upis ^ Chapter 5 Computational Results and Discussion 175 Chapter 5 Computational Results and Discussion 176 Chapter 5 Computational Results and Discussion 177 Chapter 5 Computational Results and Discussion 178 Chapter 5 Computational Results and Discussion 179 1 8 0 Chapter 6 CONCLUSIONS A N D RECOMMENDATIONS 6.1 Summary A fundamental study of unsteady wake-boundary layer interaction has been conducted. Experimentally, traveling-wave disturbances were generated upstream of a flat plate by a rotating mechanism representing rotor blades in a turbomachine stage. Unsteadiness in either a real machine or the physical model is marked with periodic regularity resulting from the relative motion of the rotor and the stator. The boundary layer was tripped near the leading edge to generate a turbulent flow prior to wake interactions. However, the experimental results suggest that at the near-leading edge stations of the flat plate, the flow regime was transitional turbulent flow. In the computational procedure, the governing equations were formulated numerically and an eddy viscosity model was used to close the equations. The external disturbances were mathematically modeled with traveling sinusoidal oscillations. Subsequently, numerical solutions were generated and comparisons made with the experimental results. 6.2 Conclus ions 1. The experimental results show that the mean skin friction in periodically unsteady boundary layer flow due to external traveling oscillatory disturbances should not be estimated with steady flow skin friction results. For a free stream amplitude Au/Ue of order 0 .1 , frequency parameter w « 0.33 - 9.33 and range of Reynolds number Re x = (0.144 to 1.44)x105, skin friction values on a tripped flat plate at x = 0.02 m from the elliptical leading edge increased by up to twice relative to that of the undisturbed Chapter 6 Summary and Conclusions 181 the elliptical leading edge increased by up to twice relative to that of the undisturbed boundary layers. With the exception of near-leading edge stations, that show transitional behavior, skin friction decreased with increasing frequency parameter but values remained higher than that of the steady case. Computational results, obtained by making use of the experimental data inputs and a traveling-wave type external sinusoidal velocity, also confirm similar responses of the wall shear stress to organized external disturbances. Therefore, in approximating skin friction losses in a turbomachine stage flow, there is a clear need to account for losses due to periodic unsteadiness. 2. Within the bounds of the study (Au/Ue = 10% , m « 0.33 - 9.33 and Re x = (0.144 to 1.44)x10 5), of the methods used to determine the wall shear stress in unsteady boundary layers on the tripped flat plate subject to periodic traveling wave perturbations, the Clauser plot was found to be inconclusive; even average values could not be determined with certainty. Therefore, this method may not be used as a dependable means to predict wall shear stresses in periodic unsteady boundary layer flows. On the other hand, skin friction coefficients from glue-on probe measurement and computation generally agree in their trend along the flow direction. 3. Approximating periodic unsteady turbulent boundary layer velocities by time mean steady turbulent boundary layer velocities cannot be justified. For the external amplitude ratio of 0.1 and Reynolds number (0.144-1.44)x10 5, results suggest that unsteady boundary layer profiles on the tripped flat plate, whose elliptic leading edge was located at about 40 mm perpendicular to the plane of the trailing edges of linearly moving airfoils Chapter 6 Summary and Conclusions 182 shedding traveling wakes, undergo a series of profile fluctuations ( up to ±10% of the mean velocity in the outer boundary layer region) during a single period of perturbation. The mean of the unsteady ensemble averaged velocity profiles exhibited decreased shape factors ( H * 1.5 to 1.4) compared to those of the steady profiles (H * 1.7 to 1.5). The profiles suggest enhanced momentum exchanges across the boundary layer due to the unsteadiness. That the phase-averaged velocity profiles were not equivalent to the mean steady velocity profiles is in contrast to previous unsteady turbulent boundary layer studies of Karlsson [11] on standing-wave type perturbation and that of Patel [12] on traveling-wave type disturbance. However, the results qualitatively agree in the fullness of the perturbed boundary layer profile with the works of Evans and Yip [13], Telionis [25] and Evans and Holland [72]. 4. The random fluctuations in turbulent boundary layers are coupled with the periodic, traveling-wave wake disturbances. Although Patel [12] observed insensitivity of the random fluctuations to the periodic traveling fluctuations for the cases he considered ( f = 4 - 12 Hz, free stream fluctuations of about 0.11U e), the results in this work are qualitatively in agreement with the results of Houdeville et al. [19]. Their investigation, however, was on standing-wave type free stream perturbation. Turbulence levels increase when subjected to external periodic fluctuations. For the range of frequency parameter tested, random fluctuation intensities of order 3% - 1 0 % were recorded in the free stream for the unsteady cases as opposed to 0.7% for the steady cases. The higher the periodic disturbance frequency, the more intense the random fluctuations are. These external organized fluctuations with discrete Chapter 6 Summary and Conclusions 183 frequencies behave as sources of additional random fluctuations, especially in the outer region of the boundary layer, on top of the "natural" random fluctuations. Thus, care must be taken regarding unsteady wake interaction effects when modeling the Reynolds' stress terms in closing the equations of motion. A mechanism is required to take into account the impact of wake-boundary layer interactions on the random turbulent fluctuations when developing computational schemes. 5. Traveling-wave wake fluctuations having a wave speed less than the upstream velocity (Q/U e < 1.0) and amplitude ratio of 10% generate negative phase shifts across the boundary layer. For the range of frequency parameters investigated (w ra 0.33 - 9.33) experimentally, velocity phase lags of order one period prevail at the wall. Although phase lags increase (from zero in the free stream) towards the wall for any given frequency of disturbance, no clear trend has been observed with respect to the frequency parameter. Uncertainties close to the wall are in the range of 10% to 15%. Computational results also indicate phase lags in the boundary layer when subjected to the traveling sinusoidal perturbations. 6. The unsteady turbulent boundary layer calculation procedure, adapted from Cebeci and Carr [10], and modified to accommodate traveling-wave type free stream oscillations, was used for comparison with the experimental data. Calculations were limited to a free stream oscillation of about 0.1 U e and Re x = (0.144 to 1.44)x105 . The flow was considered turbulent starting from the trip-wire location on the flat plate. The computational scheme is sensitive to time and transverse grid spacings closest to the Chapter 6 Summary and Conclusions 184 wall. Calculated average velocity profiles compare reasonably with the experimental velocity profiles with a maximum margin of error of about ±12 %. Calculated skin friction coefficients over-predict experimental results over the periodic fluctuation frequencies tested (f = 20, 30, 40 Hz).The quantitative agreement between calculated and measured phase shifts is poor. Despite the discrepancies in phase shifts between experiment and calculations, both methods captured the negative phase shift aspect in response to traveling wave perturbations. The negative phase shift aspect of the boundary layer response to traveling wave disturbances for wave speed Q < Ue, is in agreement with earlier similar studies [12,13,72]. The discrepancies between calculated and experimental results are attributed to: a) modeling the wakes of the airfoils by a sinusoidal traveling-wave type free stream velocity; b) the varying angle of attack of the flow on the flat plate which was not accounted for in the calculations; c) the random turbulent fluctuations inside the individual wakes that were not considered in the calculations. In general, for the cases considered, the calculation procedure is promising. With proper modeling of the traveling-wave type fluctuations and the angle of attack on the flat plate, the calculation procedure may be further developed to more closely predict unsteady turbulent boundary layers subject to traveling-wave disturbances. 6.3 Recommendat ions This study was of a fundamental nature in order to develop a better understanding of periodic unsteady turbulent boundary layers on turbomachinery blading. As the influence Chapter 6 Summary and Conclusions 185 of organized fluctuations on the development of boundary layers is of significant practical importance in turbomachinery stages, further detailed study of the unsteady wake-boundary layer interaction should be undertaken. It would be useful to conduct more fundamental experiments and make use of the computational technique used in this work to explore further the characteristics of the unsteady boundary layer. The study may be extended to three-dimensional unsteady boundary layer problems. The effect of traveling-wave speed also needs to be investigated. More study is required to determine to what extent discrete frequencies influence the boundary layer development. 186 R E F E R E N C E S [ I ] Cebeci, T. and Smith, A.M.O., "Analysis of Turbulent Boundary Layers," Academic Press, 1974, pp. 1-46. [2] Hinze, P.O., "Turbulence," McGraw-Hill Inc., 1975, 2nd Ed. [3] Rosenhead, L , "Laminar Boundary Layers," Dover Publications, Inc., 1963. [4] Lighthill, M.J., "The Response of Laminar Skin Friction and Heat Transfer to Fluctuations in the Stream Velocity," Proc. R. Soc. Lond., 1954, Vol. 224, pp. 1-23. [5] Hill, P.G. and Stenning, A.H., "Laminar Boundary Layers in Oscillatory Flow," Journal of Basic Engineering, 1960, Vol. 82, pp. 593-608. [6] Patel, M.H., "On Laminar Boundary Layers in Oscillatory Flow," Proc. R. Soc. Lond. , 1975, Vol. A347, pp. 99-123. [7] Telionis, D.P., " Unsteady Boundary Layers, Separated and Attached," Journal of Fluids Engineering, 1979, Vol. 101, pp. 29-43. [8] Greitzer, E.M., "An Introduction to the Unsteady Flow in Turbomachines; Thermodynamics and Fluid Mechanics of Turbomachinery," A.G.A.R.D. Special Course, 1985. [9] Evans, R.L, "Computation of Unsteady Laminar Boundary Layers Subject to Traveling Wave Free Stream Fluctuations," AIAA Journal, 1989, Vol. 27, pp. 1644-1646. [10] Cebeci, T. and Carr, L.W., "A Computer Program for Calculating Laminar and Turbulent Boundary Layers for Two-Dimensional Time-Dependent Flows," NASA TM-78470, 1978. [ I I ] Karlsson, S.K.F., "An Unsteady Turbulent Boundary Layer," Journal of Fluid Mechanics, 1959, Vol. 5, pp. 622-636. [12] Patel, M.H., "On Turbulent Boundary Layers in Oscillatory Flow," Proc. R. Soc. Lond. , 1977, Vol. A353, pp. 121-144. [13] Evans, R.L. and Yip, R.S.K., "An Experimental Investigation of Wake-Boundary Layer Interaction," Journal of Fluids and Structures, 1988, Vol. 2, pp. 313-322. [14] Carr, L.W., "A Review of Unsteady Turbulent Boundary Layer Experiments," NASA TM-81297, 1981. [15] Brembati, F., "An Investigation of an Unsteady Turbulent Boundary Layer," von Karman Institute of Fluid Dynamics P.R. 1975-17,1975. 187 [16] Cousteix, J. et al., "Structures and Development of a Turbulent Boundary Layer in an Oscillatory External Flow," ONERA TP 14,1977. [17] Evans, R.L, " Boundary Layer Development on an Axial-Flow Compressor Stator Blade," Journal of Eng. for Power, 1978, Vol. 2, pp. 287-293. [18] Mizushina, T. et al., "Structure of the Turbulence in Pulsating Pipe Flow," J. Chem. Eng. Japan, 8, 3, 1975. [19] Houdeville, R. et al., "Experimental Analysis of Average and Turbulent Boundary Layer," ONERA TP No. 30, 1976. [20] Lyrio, A.A., et a l . , " An Analytical Method for the Computation of Steady and Unsteady Turbulent Boundary Layer Flows Including the Transitory Stall Regime in Diffusers," Dept. of Mech. Eng., Stanford University, Rept. ME-PD23, 1981. [21] Simpson, R.L. et al., "Measurement of Unsteady Turbulent Boundary Layers With Pressure Gradients," SMU Rept. WT-6, Southern Methodist University, 1980. [22] Acarya, M. and Reynolds, W.C., "Measurements and Prediction of a Fully Developed Turbulent Channel Flow With Imposed Controlled Oscillations," Stanford University, Dept. Mech. Eng. Rept. TF-8,1975. [23] Ramaprian, B.R. and Tu, S.W., "An Experimental Study of Oscillatory Pipe Flow at Transitional Reynolds Numbers," J. Fluid Mech. 1980, Vol. 100, pp. 513-544. [24] Hussain, A.K.M.F. and Reynolds, W.C., "The Mechanics of an Organized Wave in Turbulent Shear Flow," J. Fluid Mech. 1970, Vol. 4 1 , pp. 241-258. [25] Telionis, D.P., "Unsteady Viscous Flows," Springer-Verlag, 1981. [26] Liu, X. and Rodi, W., "Experiments on Transitional Boundary layers With Wake-Induced Unsteadiness," J. Fluid Mech. 1991, Vol. 231, pp. 229-256. [27] Pfiel, H. et al., "Investigation of Laminar-Turbulent Transition of Boundary Layers Disturbed by Wakes," J. of Eng. for Power, 1983, Vol. 105, pp. 130-137. [28] Dong, Y. and Cumpsty, N.A., "Compressor Blade Boundary Layers," J. of Turbomachinery, Parts 1 & 2, 1990, Vol. 112, pp. 222-240. [29] Hodson, H.P., "Boundary Layer and Loss Measurements on the Rotor of an Axial-Flow Turbine," J. of Eng. for Gas Turbines and Power, 1984, Vol. 106, pp. 391-399. [30] Lakshminarayana, B. and Raj, R.," Characteristics of the Wake Behind a Cascade of Airfoils," J. Fluid Mech. 1973, Vol. 61 , pp. 707-730. 188 [31] Horlock, J.H., "Unsteady Flows in Turbomachines," The Institution of Engineers, Australia, 1968. [32] Evans, R.L, "Turbulence and Unsteadiness measurements Downstream of a Moving Blade Row," ASME paper No. 74-GT-73,1974, pp. 1-7. [33] Walker, G.J., "The Unsteady Nature of Boundary Layer Transition on an Axial-Flow Compressor Blade," ASME Paper No. 74-GT-135,1974. [34] Kerrebrock, J.L. and Mikolajczak, A.A., "Intra-Stator Transport of Rotor Wakes and its Effect on Compressor Performance," J. Eng. for Power, 1970, pp. 359-368. [35] Winters, G.J., "An Outline of the Techniques Available for the Measurement of Skin Friction in Turbulent Boundary Layers," Prog. Aerospace Sci. 1977, Vol. 18, pp. 1-57. [36] Haritonidis.T.H., "The Measurement of Wall Shear Stress," Advances in Fluid Mechanics measurements, 1989, vol. 45, pp. 231-261. [37] Schlicting, H., "Boundary Layer Theory," McGraw-Hill Inc. 7th Ed. 1979. [38] Coles, D., "The Law of the Wake in the Turbulent Boundary Layer," J. Fluid Mech. 1956, Vol. 1, pp. 191-226. [39] Ludweig, H., "Instrument for Measuring the Wall shear Stress of Turbulent Boundary Layers," NACATM 1284, 1950. [40] Liepmann, H.W. and Skinner, G.T., "Shearing Stress Measurements by Use of Heated Elements," NACA TN 3268,1954. [41] Bellhouse, B.J. and Schulz, D.L, "Determination of Mean and Dynamic Skin Friction, Separation and Transition in Low speed Flow With a thin Film Heated Element," J. Fluid Mech. 1966, Vol. 24, Part 2, pp. 379-400. [42] Menendez, A.N. and Ramaprian, B.R., "The Use of Flush-Mounted Hot-Film Gauges to Measure Skin Friction In Unsteady Boundary Layers," J. Fluid Mech. 1985, Vol. 161, pp. 139-159. [43] Bellhouse, B.J. and Schulz, D.L, "The Measurement of Fluctuating Skin Friction in Air With Heated Thin Film Gauges," J. Fluid Mech. 1968, Vol. 32, Part 4, pp. 675-680. [44] Holland, R.M., "The Effects of Periodic Wake Disturbances on Flat Plate Turbulent Boundary Layers," M.A.Sc. Thesis, UBC, 1993. [45] Hodson, H.P., " Measurements of Wake-Generated Unsteadiness in the Rotor passages of Axial-Flow Turbines," ASME Paper No. 84-GT-189,1984. [46] Clauser, F.H., "Turbulent Boundary layers in Adverse Pressure Gradient," J. Aero. Sci. 1954, pp. 91-108. 189 [47] Patel, V.C., "Calibration of The Preston Tube and Limitations on its Use in pressure Gradients," J. Fluid Mech. 1965, Vol. 23, Part 1, pp. 185-208. [48] Hodson,, H.P., "Unsteady Boundary Layers on Axial-Flow Turbine Rotor Blades," Ph.D. Dissertation, University of Cambridge, 1982. [49] Wills, J.A.B., "The Correction of Hot-Wire Readings for Proximity to a Solid Boundary," J.F.M, 1961, pp. 388-396. [50] Preston, J.H., "The Determination of Turbulent Skin Friction by Means of Pitot Tubes," J. Roy. Aero. Soc. 1954, Vol. 58, pp. 109-121. [51] Head, M.R. and Rechenberg, I., "The Preston Tube as a Means of Measuring Skin Friction," J. Fluid Mech. 1962, Vol. 14, pp. 1-17. [52] Ludweig, H. and Tillmann, W., "Investigations of the Wall Shear Stress in Turbulent Boundary Layers," NACA TM 1285, 1950. [53] Diller, T.E. and Telionis, D.P., "Time-Resolved Heat Transfer and Skin Friction Measurements in Unsteady Flow," Advances In Fluid Mechanics Measurements, 1989, Vo l .45, pp. 323-355. [54] Squire, L.C. et al., "The Interaction of a Wake With a Turbulent Boundary Layer," J. Roy. Aero. Soc. 1985, pp. 72-81. [55] White, F.M., "Viscous Fluid Flow," McGraw-Hill Inc. 2nd Edition, 1991. [56] Fletcher, R.H. et al., "Calculation of Unsteady turbulent Boundary layers," ASME Paper No. 87-GT-53,1987. [57] Evans, R.L, "Unsteady laminar Boundary Layers Subject to Standing Wave or Traveling Wave Free Stream Fluctuations," Cambridge University Engineering Department Report, CUED/A -Turbo/TR 124,1988. [58] Cebeci, T. and Bradshaw, P., "Momentum Transfer in Boundary Layers," McGraw-Hill Inc., 1977. [59] Wygnanski, I. et al., "On the Large Scale Structures in Two-Dimensional, Small Deficit, Turbulent Wakes," J. Fluid Mechanics 1986, Vol. 168, pp. 31-71. [60] Franke, G.F. and Henderson, R.E., "Unsteady Stator Response to Upstream Rotor Wakes," J. Aircraft, 1979, Vol. 17, No. 7, pp. 500-507. [61] Paxon, D.E. and Mayle R.E., "Laminar Boundary Layer Interaction With an Unsteady Passing Wake," J . of Turbomachinery 1991, Vol. 113, pp. 419-427. [62] Greenblatt, D. and Damelin, S.B., "Laminar Boundary Layers Subjected to High Frequency Traveling Wave Fluctuations," AIAA J., 1993, Vol. 31 , No. 5, pp. 957-959. 190 [63] Bario, F. et al., "An Experiment Concerning the Confluence of a Wake and a Boundary layer," J. Fluids Eng. 1982, Vol. 104, pp. 18-24. [64] Greenblatt, D., "The Effect of Wake-Passing and Free Stream Turbulence on Laminar Gas Turbine Blade Boundary layers," J. of Turbomachinery 1994, Vol. 116, pp. 384-391 . [65]Mankbadi, R.R., "A Study of Unsteady Rotor-Stator Interactions," J. of Turbomachinery 1989, Vol. 111, pp. 394-400. [66] Rodi, W. et al., "Computational Study of the Unsteady Flow Due to Wakes Passing Through a Channel," Turbulent Shear Flows 6, Springer-Verlag Berlin Heidelberg 1989, pp. 255-268. [67] Ackerman, J.D. et al., "Preston Tubes for the Measurement of wall Shear Stress," J. Fluids Eng. 1994, Vol. 116, pp. 645-652. [68] Buckingham, E., "On Physically Similar Systems: Illustrations of the Use of Dimensional Equations," Phys. Rev. 1914, Vol. 4, No. 4, pp. 345-376. [69] White, F. M., "Fluid Mechanics," 2nd Edition, 1986 [70] Evans, R.L., "Free Stream Turbulence Effects on the Turbulent Boundary Layer," 1974, ARC CP 1282, U.K. [71 ] Saravanamuttoo, H.I.H. et al. "Gas Turbine Theory", 3rd edition, 1985. [72] Holland, R.M. and Evans, R.L. , "The Effects of Periodic Wake Structures on Turbulent Boundary Layers", Journal of Fluid Structures, 10, pp. 269-280,1996. 191 APPENDIX A DIMENSIONAL A N A L Y S I S The velocity profile in an unsteady turbulent boundary layer flow depends on many variables. Some of the variables are density, viscosity, angle of attack, free stream velocity, rotor velocity, airfoil chord, spacing of airfoils, wall shear stress, distance from the leading edge, distance from the surface and time. The functional relationship may be expressed as: u = f(p,ii,a,uQO,ur,c,s,Tw,x,Y,t,...) A-1 The number of parameters need to be reduced in order to conduct a reasonable experiment. This can be accomplished by implementing the well-known technique of dimensional analysis [ 68, 69 ] . Appropriate dimensionless groups must be formulated and the relative importance of each evaluated so that a minimum number of dimensionless groups remain on which the experiment is based. Since in this particular case we have twelve variables, by selecting three primary dimensions as Mass (M), Length (L) and Time (T) and choosing three repeating variables, nine dimensionless groups would result from the analysis. Let the repeating variables be p, IL, and X. The other dimensionless groups will be formed by combining them in turn with the repeating variables as follows: 111 = paUoobXcUr = M U? -\a = M W a = 0, b = -1 ,c = 0 , a n d I I I = ^ -Uoo Flow coefficient n 2 = P auJx cv = M M LT = M°L°T° 192 a = -1, b = -1, c = -1, and II2 = Reynolds number II3 = paUocbXcS = ~ M~ a .T. 0t0t0 [L] C[L] = M UL UT .-. a = 0, b = 0 , c = -1, and TJ3 = — Blade solidity II4 = p a f /oo f o X c C ~ M~ a ~L L3J .T. lb [L] C[L] = M°L°T 0 a = 0, b = 0 , c = -1, and IT4 X Dimensionless length n 5 =p a c /oo f e x c y = ~ M~ a " L" L3J T. [L]C[L] = M°L°T 0 .-. a = 0, b = 0 , c = -1, and TJ5 = — X Transverse length Yl6 = PauJx ct = ~ M~ a L .T. [L] C[T] = M°L°T 0 .-. a = 0, b = 1 , c = -1, and TJ6 = Uoot Uoo X 10X Reduced frequency n 7 = p aU00 bX cu = M [lV .-. a = 0, b = -1 , c = 0, and TJ7 = Velocity profile 193 n 8 = p auj>x cTW = M M LT 2 = M°L°T° a = - 1 , b = -2 , c = 0, and IT8 = Friction coefficient II9 = p a t / o o f o X c a = ~ M~ a L L3J T_ [L]C=M°L°T° a = 0, b = 0 , c = 0, and TJ9 = a Angle of attack The velocity profile can therefore be written as a function of the other dimensionless groups as: U0 fl Ur UooX S C Y 10X T l / o o ' V 'x'X'X'Uoo 'pC/oo 2 ,a, . . . A-2 By setting some of the dimensionless groups constant or neglecting some of them based on their relative importance in the function and allowing the remaining variables to change one at a time, a set of experiments may be conducted from which the functional relationships are determined empirically. 194 APPENDIX B E S T I M A T I O N O F W A L L S H E A R S T R E S S COEFFICIENT BY C L A U S E R P L O T S The basic assumption in using the estimation technique of Clauser [46] plots rests on the concept that the velocity profile agrees with the law of the wall especially in the logarithmic region. It has been found that the log law of the wall can be expressed as Ux K B-1 where K and B are universal constants determined from various experiments. The shear velocity UT is defined as «r = 1 w V v ) ; T w is the wall shear stress in a flow. Define the coefficient of wall friction by 2 T Cf = —* ! r - ; Uoo is the mean free stream velocity. J pU£> Therefore, — = - c 7 T O2 C f and = p 2 J ^ -.1/2 -Uoo'Cf 2 1 B-2 Substitution of the above equation in the law of the wall equation is performed as follows; - I l n [ l / 2 t V 0 O2 C / ]1 / 2 K y(l I 2Uj-Cf)m + B, B-3 195 Uoo[Cf I 2 ] 1 / 2 K + ln B-4 IVoo K 2 J 1/2 -In V 2 , + B B-5 This equation can be simplified to a modified law of the wall equation as U-Ahx(yU° + c B-6 where A = — K (Cf\ V 2 , 1/2 , and K V EL 2 1/2 + B J A and C are determined by substituting different values of C f from which the relationship between u/LL and In(ylWv) is found graphically. If a velocity profile agrees with the law of the wall, there would be a linear region parallel to any of the lines with wall friction coefficient values. The wall shear stress is then estimated from the value of C f that coincides with the velocity profile in the logarithmic region. For the flat plate considered in the experiment, the values of K and B are 0.41 and 5.0 respectively [ 55]. The procedure to obtain the family of curves in the Clauser plots is to specify a friction coefficient value and evaluate A and C using the above equations. Given the kinematic viscosity of air at atmospheric pressure and room temperature (0.0000146 m2/s) and the free stream velocity of the fluid in the wind tunnel ( 3.0 m/s), a number of linear curves are drawn relating the coordinate represented by (ylWv) and the velocity ( u / l L ) . For instance one of the linear curves on the semi-logarithmic scale if Cf = 0.008 is 196 — - 0.154626 In f ^ - ^ - ] - 0.10964. t/oo I V J Therefore, if a velocity profile obtained through hot wire measurement is superposed on the same semi-logarithmic axes as the wall friction values, the region of overlap between the two is sought to determine the wall friction coefficient. The following figure illustrates constant Cf lines generated for Clauser plots. Fig. B-1 Generation of Constant Skin friction Lines for Clauser Plots 197 APPENDIX C CALIBRATION OF WALL SHEAR STRESS GAUGES Measurement of local skin friction by a hot-film is based on the concept of the rate of heat transfer from a solid surface to a moving fluid. If a flat plate is heated at a specific (hot-film) location, a thermal boundary layer develops, starting at the heated section, due to the sudden jump in the surface temperature at the film location. The basic assumption that relates heat transfer to shear stress is that the thermal boundary layer developing over the hot film lies entirely within the viscous sublayer. Consider a flat plate on which a viscous boundary layer grows. Let the surface temperature be raised at x = xi and kept constant for a short distance L. Fig. C.1 shows a schematic of the flat plate. y Fig. C.1 Schematics of Viscous and Thermal Boundary Layers In the case where the boundary layer is turbulent, it is assumed that the thermal boundary layer is still submerged within the viscous sublayer thereby justifying the neglect of turbulent diffusion in the sublayer. Therefore, neglecting viscous dissipation, the governing equation for the distribution of temperature T will be 198 where U and V are velocity components in the x and y directions, p, k and c p are the density, thermal conductivity and specific heat of the fluid respectively. If the inertia terms are neglected in the viscous sublayer, the above equation reduces to: a r k d 2T dt pcp dy' C-2 The equation of motion for the incompressible flow boundary layer on the flat plate becomes: du d 2u — « o — T C-3 dt dy 2 Liepmann and Skinner [40] deduced a relationship between the local skin friction T W and heat transfer rate Q w by dimensional analysis. Their method is adopted to obtain the wall shear stress-heat transfer relationship. The wall shear stress T w can be expressed in terms of the boundary layer thickness 5 as: x w u — where 5 J u — . C-4 5 V U By the same analogy, the heat transfer at the wall where the hot element is located can be expressed by: qw & — where the thermal boundary layer is 5 r °= I — — -—— C-5 5 r \pcpu$T) M(5 t) is velocity at the edge of the thermal boundary layer thickness. Since the region very close to the wall will be of primary importance for the heat transfer, u(5 7-) can be replaced by the first term of a series starting from the wall [40], « ( 5 r ) = du — l — + . 2 Thus, u(bT) * ^ i d r . C-6 Substituting equation C-6 in to C-5 gives 199 5 T* oc —n -P Cp  TW The thermal boundary layer is then expressed as: f , \V*, xl/3 I r — Yi i Kpcpj x - Xl Inserting equation C-8 in to the heat flux at the wall gives: qw 04 kAT Kkn , 1/3 xl/3 K x~ xi) C-7 C-8 C-9 For a hot element of length L, C-9 can be integrated to obtain the total heat transfer as: ( p c r A 1/3 kAT j C-10 For a very small element where L << x^ the wall shear stress can be assumed constant in the range of integration. In that case the integrated heat flux becomes: kATrwmL 2/3 C-11 Therefore, T w « G w Since the equations were obtained based on local considerations only, equation C-11 is valid for both laminar and turbulent boundary layers. If the quantities on the right hand side of equation C-11 are known, except for the wall shear stress, the heat transfer can be expressed as follows: — <* Cons tan * * T , 1 / 3 w If heat is supplied electrically, the voltage output E can be expressed in terms of the wall shear stress as: E 2 = A + Bxwm. C-12 200 A and B' are calibration constants. The wall shear stress gauge has been calibrated by using a Preston tube as a standard. The non-dimensional relationship between Preston tube reading and local skin friction can be represented [47,50,51 ] by: twD2 _ -4 P u2 _ / l ( A „ ™ 2 \ ApD C-13 where Ap is the pressure difference between the Preston tube reading and the static pressure at the wall, and D is the outer diameter of the Preston tube. Patel [47] produced empirical calibration equations covering a wide range of flow conditions and sizes of Preston tubes. These equations are currently used by many researchers [35,36,48,53] to calibrate wall shear stress gauges. Patel's calibration equations are: a) y* - + 0:037 where y* < 1.5; b) y* = 0.8287 - 0.1381** + 0.1437** 2 - 0 .006** 3 , 1.5 < y* < 3.5; c) JC* = y* + 21og(1.95y* + 4.10), 3.5 < y* < 5.3; * . A p D 2 * JWD2 x = l o g =-, y = l 0 g - = - _ 4 p u z 4 p u z Fig. C.2 shows the correlation of the measured pressure difference parameter by a Preston tube to the corresponding wall shear stress parameter obtained from Patel's calibration equations. The wall shear stress thus obtained was then correlated with the voltage difference across the shear stress gauge during the calibration process. This procedure led to the calibration curve given in Fig. 3.7. Considering the errors in the correlations given by Patel [47], errors in pressure measurement on the flat plate, and other possible experimental errors, the uncertainty in the wall shear stress was estimated to be on the order of 10%. 201 

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