AN EXPERIMENTAL INVESTIGATION INTO
 THE ROUTES TO BYPASS TRANSITION




                               Author

           ...
The substance of this thesis is the original work of the author and due ref-
               erence and acknowledgement has...
ABSTRACT

    This report documents an experimental investigation into the bypass route to transition
for zero-pressure gr...
Acknowledgments

First and foremost I would like to thank my supervisor Dr. Edmond Walsh for always being

there to answer...
Contents


Acknowledgments                                                                              iv

List of Tables...
2.1.8   Pressure and Temperature Readings . . . . . . . . . . . . . . . . . 25

         2.1.9   Boundary Layer Measuremen...
4.7.1   Deviation from Blasius . . . . . . . . . . . . . . . . . . . . . . . . 78

         4.7.2   Measure of Increased E...
List of Tables

 2.1   Geometric description of turbulence grids and range of turbulence charac-

       teristics availab...
List of Figures

 1.1   Smoke flow visualisation of streamwise streaks, streamwise wiggles and

       turbulent spots (Mat...
2.9   Illustration of experimental arrangement (not to scale). ∆L is the plate

      leading edge distance downstream of ...
2.18 Evolution of Λx and λx downstream of turbulent grids.         , PP1 grid;   ,

      PP2 grid; , SMR grid. Open symbo...
4.2   Comparison of transition onset Reynolds numbers against the transition

      onset correlations of Mayle (1991) and...
4.10 Energy spectra for the same conditions as figure 4.9. ×, y/δ = 0.3; +,

     y/δ = 0.46; ◦, y/δ = 0.8; ——, y/δ = 3.4. ...
4.19 Wavelet maps and fluctuation velocity traces at T u = 3.1% and 6% in the

      near-wall and boundary layer edge regi...
5.13 Basic sketch illustrating the interaction between negative jets and freestream

     vortices as the boundary layer a...
5.21 Variation in penetration depth (PD) as a function of y/δ (non-dimensional

      representation of PD), %T u and Rex ...
6.7   Wall-normal disturbance profiles of the low- and high-speed streaks corre-

      sponding to the same results presen...
6.13 Maximum positive and negative fluctuation velocities presented in terms

     of a disturbance intensity about the mea...
6.20 Zoomed in view for fluctuating voltage trace of all four simultaneously

     sampled probes at Reθ = 410 and T u = 1....
6.28 Fluctuating voltage traces for all four simultaneously sampled probes at

     Reθ = 450 and T u = 1.3%. The hotwire ...
6.35 (a) Simple illustration for the argument put forward against the concept that

     a turbulent spot may develop at t...
Nomenclature


A        Hotwire calibration constant                      [-]
a        Overheat ratio                     ...
Rex       Reynolds number based on streamwise distance        [-]
Reθ       Reynolds number based on momentum thickness   ...
δ            Boundary layer thickness                      [m]
δ1           Boundary layer displacement thickness         ...
Chapter 1

Introduction


1.1 Research Motivation

The transition from a laminar to a turbulent state is one of the most u...
of FST downstream of turbulence generating grids. Bypass transition occurs above some

level of freestream turbulence (FST...
region occupied over 20% of the total suction surface length. According to Bowles (2000)

it has been estimated that 20% s...
1.2 Laminar to Turbulent Transition: The Bypass Route

1.2.1 Experimental Investigations

The first experimental work into ...
Figure 1.1: Smoke flow visualisation of streamwise streaks, streamwise wiggles and turbu-
lent spots (Matsubara and Alfreds...
1976; Westin et al., 1998; Chong and Zhong, 2005). The major advantage of artificially

generating the turbulent structures...
The increase in wall shear stress can be attributed to the enhanced mixing caused by the

FST that has been shown experime...
theory failed to capture all of the physics involved in the transition process when com-

pared to experiment. This led to...
of long-lived, elongated streaks of alternating high and low streamwise speed”.

   Wundrow and Goldstein (2001) commented...
better represented the interaction between the freestream disturbances and the underlying

boundary layer.

   It has also...
DNS is illustrated in figure 1.2 where the low-speed streaks lift up towards the boundary

layer edge (denoted by δ). In DN...
δ
                                  Inflectional-
               FST                type wave


           U
             ...
of the peak locations of the maximum positive and negative fluctuation velocities in the

streamwise direction. They demons...
design and characterisation of the new flat plate and turbulence grids used in the investi-

gation of bypass transition. I...
number in order to accurately predict the increased energy dissipation rates to within 10%

of those measured.

   Chapter...
are observed and these results indicate the wall-normal size of the leading edge overhang.



1.4 Objectives

The main obj...
Chapter 2


Experimental Techniques, Facility and

Freestream Turbulence Characteristics


2.1 Introduction to Measurement...
• The spatial resolution of the probes is reasonably good with the possibility of mea-

      suring scales of the order 1...
R1     R2           Servo amplifier
                                                                   E



              ...
Rw = R20 [1 + α20 (Tw − T20 )].                              (2.1)


In the current investigation the overheat temperature...
16
                                     y=7.07x − 0.83

                                     R2 = 0.99




               ...
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Thesis Domhnaill Hernonfinal

  1. 1. AN EXPERIMENTAL INVESTIGATION INTO THE ROUTES TO BYPASS TRANSITION Author Domhnaill Hernon FOR THE DEGREE OF DOCTOR OF PHILOSOPHY COLLEGE OF ENGINEERING, UNIVERSITY OF LIMERICK, IRELAND Supervisor Dr. Edmond Walsh Submitted to the University of Limerick, November 2007
  2. 2. The substance of this thesis is the original work of the author and due ref- erence and acknowledgement has been made, when necessary, to the work of others. No part of this thesis has previously been accepted for any degree nor has it been submitted for any other award. Domhnaill Hernon Candidate The substance of this thesis is the original work of the author and due ref- erence and acknowledgement has been made, when necessary, to the work of others. No part of this thesis has previously been accepted for any degree nor has it been submitted for any other award. Dr. Edmond Walsh Principal Adviser This thesis was defended on the 5/10/2007. Examination Committee Chairman Dr. John Jarvis Univeristy of Limerick External Examiner Prof. Terrence W. Simon University of Minnesota Internal Examiner Dr. David Newport Univeristy of Limerick ii
  3. 3. ABSTRACT This report documents an experimental investigation into the bypass route to transition for zero-pressure gradient laminar boundary layer flow when under the influence of ele- vated freestream turbulence (FST). A complete understanding of how disturbances from the freestream enter the boundary layer and their subsequent evolution towards turbulent flow has been the source of intense investigation for some time, both experimentally and numerically. However, the breakdown process is still not fully understood and it is the main objective of this investigation to give some experimental insight into the pre-transitional flow characteristics that initiate the transition from a laminar to a turbulent state. What follows is a summary of the main results of this doctoral thesis. Through thermal anemometry a measure of the change in receptivity of the laminar boundary layer with increased FST level was achieved by examining the wall-normal pro- files of the skewness distribution. The location of peak negative skewness was used as a measure of the penetration depth of disturbances into the boundary layer and was shown to demonstrate the same parameter dependence as theory, i.e., the penetration depth scaled with the Reynolds number, the frequency of the largest freestream eddies and the shear stress. Furthermore, a relationship is presented that predicts the wall-normal location in the boundary layer where the most intense negative spike activity is observed and this location is considered to represent the most probable wall-normal location of turbulent spot produc- tion. These relationships represent the first attempt at predicting the wall normal location of breakdown. Taking mean statistics is not sufficient in fully explaining the routes to bypass tran- sition. By examining profiles of the maximum positive and negative of the streamwise fluctuation velocity it was demonstrated that the peak disturbance magnitudes of the low- and high-speed streaks far exceed those presented in RMS terms, i.e., the Klebanoff mode disturbance profiles. In this investigation, at transition onset, the peak disturbance associ- ated with the negative fluctuation velocity was approximately 40% of the freestream ve- locity for all turbulence intensities considered, and this was always greater than the peak positive value. Furthermore, the location of the peak negative disturbance moved towards the boundary layer edge and the location of the peak positive disturbance moved towards the wall as the location of transition onset approached. Due to the higher disturbance am- plitudes associated with the peak negative fluctuation velocities it can be inferred that the breakdown to turbulence most likely occurs on the low-speed regions that are lifted to the upper portion of the boundary layer. Simultaneous measurements between a hotwire probe traversed throughout the boundary layer thickness and hotfilm probes placed at the wall demonstrated that high-frequency structures with negative fluctuation velocity are observed away from the wall before any near-wall turbulent spots are formed and an estimate of the leading edge overhang size is gained. These results lend further support to the boundary layer edge breakdown concept under naturally occurring disturbance conditions. A num- ber of relationships are presented that allow for accurate prediction of the new parameters measured. iii
  4. 4. Acknowledgments First and foremost I would like to thank my supervisor Dr. Edmond Walsh for always being there to answer my questions and to engage in conversation on all aspects of engineering. I would like also to especially thank my girlfriend Lisa for all of her love and support and for always being there for me. My parents have always been a major influence in my life and it is fair to say that without their encouragement and support I would not have had the confidence to do a PhD. I would like to thank all of the lads that joined SRI at the same time as myself for their friendship and for the good fun we had. Thank you Colin, Dave, John (big thanks to John for letting me crash at his place) Mike, and Magali. Thanks also to my house mates Ronan and Lisa for the good company over the last number of years. I would especially like to thank Cian for his help with the SED programme, Dave for his help with Matlab and Kevin for his help with Adobe illustrator. I would also like to thank the technical staff at the University. Pat Kelly, Paddy O’Donnell, Ken Harris and especially John Cunningham and Paddy O’Regan. Dr. Philip Griffin gave me his Wavelet code that was donated to the group by Prof. Ralph Volino and for this I must thank both of them. I must also thank Dr. Marc Hodes and Dr. Alan Lyons who were instrumental in getting me a position with Bell Labs Ireland and also for allowing me the time to complete my thesis. I wish to give thanks to Prof. Donald McEligot for all of his engaging comments and for sharing his vast experience with me throughout our manuscript draft process. Also thanks to his wife Julie for taking the time to help with our correspondence over the last few years. I would like to thank Science Foundation Ireland (SFI) and the H.T Hallowell Jr Grad- uate Scholarship for their financial support over the three years. iv
  5. 5. Contents Acknowledgments iv List of Tables viii List of Figures xxi Nomenclature xxii 1 Introduction 1 1.1 Research Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Laminar to Turbulent Transition: The Bypass Route . . . . . . . . . . . . . 4 1.2.1 Experimental Investigations . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Theoretical Investigations . . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Numerical Investigations . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 Experimental Techniques, Facility and Freestream Turbulence Characteristics 17 2.1 Introduction to Measurement Techniques . . . . . . . . . . . . . . . . . . 17 2.1.1 Hotfilm and Hotwire Anemometry . . . . . . . . . . . . . . . . . . 17 2.1.2 Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.3 Hotwire Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.1.4 CTA System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.5 Hotwire Traverse Mechanism . . . . . . . . . . . . . . . . . . . . 23 2.1.6 Hotfilm Probe Set-up . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.7 Hotwire Frequency Response . . . . . . . . . . . . . . . . . . . . 25 v
  6. 6. 2.1.8 Pressure and Temperature Readings . . . . . . . . . . . . . . . . . 25 2.1.9 Boundary Layer Measurement Procedure . . . . . . . . . . . . . . 26 2.1.10 Flow Visualisation Techniques . . . . . . . . . . . . . . . . . . . . 26 2.2 Experimental Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.1 Wind Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.2 Flat Plate Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.3 Characterisation of the Flat Plate . . . . . . . . . . . . . . . . . . . 31 2.3 Freestream Turbulence Characteristics . . . . . . . . . . . . . . . . . . . . 35 3 Data Analysis Techniques 45 3.1 Statistical Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Near-wall Measurement Correction . . . . . . . . . . . . . . . . . . . . . . 47 3.3 Intermittency Estimation Procedure . . . . . . . . . . . . . . . . . . . . . 47 3.4 Time Derivative Detection of High-Frequency Events and Detection of Tur- bulent Spot Leading Edge Location . . . . . . . . . . . . . . . . . . . . . 48 3.5 Removal of High-Frequency Events . . . . . . . . . . . . . . . . . . . . . 49 3.6 Calculating the Dissipation Coefficient . . . . . . . . . . . . . . . . . . . . 51 3.7 Wavelet Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.8 Signal Level Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.9 Determination of the Maximum Positive and Negative of the Streamwise Fluctuating Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.10 Uncertainty Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4 Characteristics of Pre-transitional Flow up to the Point of Transition Onset 61 4.1 Definition of Transition Onset . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2 Non-dimensional Velocity Profiles . . . . . . . . . . . . . . . . . . . . . . 66 4.3 The Klebanoff Mode Disturbance Profiles . . . . . . . . . . . . . . . . . . 67 4.4 Disturbance Energy Growth . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.5 Linear Versus Nonlinear Receptivity Process . . . . . . . . . . . . . . . . . 71 4.6 Velocity Traces and Energy Spectra . . . . . . . . . . . . . . . . . . . . . 74 4.7 Enhanced Energy Dissipation Rates due to Elevated FST . . . . . . . . . . 78 vi
  7. 7. 4.7.1 Deviation from Blasius . . . . . . . . . . . . . . . . . . . . . . . . 78 4.7.2 Measure of Increased Energy Dissipation Rates . . . . . . . . . . . 79 4.8 The Change in Flow Structure from the Wall to the Freestream During the Early Stages of Turbulent Spot Development . . . . . . . . . . . . . . . . . 82 5 The Route to Bypass Transition: Part 1-The Receptivity of the Boundary Layer to Freestream Disturbances 88 5.1 Normalised Skewness Profiles and Signal Level Probability . . . . . . . . . 88 5.2 On the Significance of the Skewness and PDF Results . . . . . . . . . . . . 96 5.3 Parameter Dependence for Penetration Depth (PD) Measure Using Peak Skewness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6 The Route to Bypass Transition: Part 2-Disturbance Growth and Breakdown 110 6.1 Instantaneous Velocity Fluctuation Profiles . . . . . . . . . . . . . . . . . . 110 6.1.1 Maximum Positive and Negative Fluctuations Scaled with the Freestream Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.1.2 Maximum Positive and Negative Fluctuations Scaled with the Lo- cal Mean Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.2 A New Method To Predict the Location of Transition Onset Based on Local Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.3 Simultaneous Hotwire and Hotfilm Measurements . . . . . . . . . . . . . . 129 7 Conclusions and Recommendations for Future Work 147 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . 151 Bibliography 154 A MATLAB Code 163 B Example of Uncertainty Estimation 170 C Published Work 171 vii
  8. 8. List of Tables 2.1 Geometric description of turbulence grids and range of turbulence charac- teristics available. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.1 Tabulated results for each measured test condition presented in figure 4.2, where the leading edge distance downstream of the turbulence grid is ∆L. . 65 4.2 Comparison between the current measurements and the proposed transition onset models of Brandt et al. (2004). . . . . . . . . . . . . . . . . . . . . 73 5.1 Summary of the average locations for the peak negative skewness values. . 96 6.1 Summary of the peak locations for the high-speed streaks (hs) and the low- speed streaks (ls) when scaled with U∞ and Umean . . . . . . . . . . . . . . 125 viii
  9. 9. List of Figures 1.1 Smoke flow visualisation of streamwise streaks, streamwise wiggles and turbulent spots (Matsubara and Alfredsson, 2001). . . . . . . . . . . . . . . 5 1.2 Sketch of the bypass transition process adapted from Zaki and Durbin (2006). 12 2.1 Sketches of constant temperature anemometer probes (DantecDynamics, 2006). (a) Hotwire probe. (b) Hotfilm probe. . . . . . . . . . . . . . . . . . 18 2.2 Simple schematic of Wheatstone bridge used to maintain probes in constant temperature mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Hotwire calibration using King’s law. , Measured data points. ——, Trend line fit to data points. The equation of the trend line gives the co- efficients in King’s law with A = −0.83 and B = 7.075 and n = 0.45. . . . 21 2.4 Picture of the hot wire traverse mechanism (O’Donnell, 2000). . . . . . . . 23 2.5 Illustration of the hotfilm array. (a) Zoomed in picture of the hotfilm el- ements courtesy of Kevin Nolan. (b) Schematic of Senflex 9101 hotfilm elements at 180◦ orientation compared to (a). . . . . . . . . . . . . . . . . 24 2.6 Frequency response of the hotwire probe (Lomas, 1986). . . . . . . . . . . 25 2.7 Photograph of experimental facility. (a) (1) Controller unit; (2) AN-1005 CTA system; (3) Data acquisition computer; (4) DSPACE stepper motor controller; (5) Test section; (6) Diffuser; (7) Contraction. (b) (8) Flat plate; (9) Hotwire and stepper motor support; (10) Stepper motor; (11) Hotwire support; (12) Turbulence grid. . . . . . . . . . . . . . . . . . . . . . . . . 28 2.8 Sketch of the three flat plate designs (not to scale). (a) Roach and Brierley (1990). (b) Jonas et al. (1999). (c) Becker et al. (2002). . . . . . . . . . . . 30 ix
  10. 10. 2.9 Illustration of experimental arrangement (not to scale). ∆L is the plate leading edge distance downstream of the grids. . . . . . . . . . . . . . . . 31 2.10 Flow visualisation demonstrating the flow characteristics on the flat plate test surface with the flap angles set to zero, maximum (approximately +40 degrees) and -15 degrees using the oil and powder mixture at Rex = 1, 500, 000. The left hand figure is a full view of the plate with the flap at zero degrees and the pictures at the right hand side are of zoomed in regions of the plate with the flap at different settings. . . . . . . . . . . . . . . . . . 32 2.11 Flow visualisation demonstrating the flow characteristics on the flat plate test surface with various flap angles using the shear sensitive liquid crystal mixture at Rex = 1, 500, 000. (a) zero degrees. (b) maximum (approxi- mately +40 degrees). (c) -15 degrees. . . . . . . . . . . . . . . . . . . . . 33 2.12 (a) Measured velocity profiles compared to theoretical Blasius velocity pro- file. ◦, (Reθ , T u) =(178, 0.2%); , (Reθ , T u)= (303, 0.2%); , (Reθ , T u)= (351, 0.2%); +, (Reθ , T u)= (410, 1.3%); , (Reθ , T u)= (442, 1.3%); ——, Blasius profile. (b) The velocity distribution over the plate. . . . . . . . . . 35 2.13 Sketch of PP turbulence grid. . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.14 Turbulence decay for all three grids. , PP1, Red = 6000; , PP1, Red = 2600; ◦, PP2, Red = 760; , PP2, Red = 1310, +, SMR, Red = 100, , SMR, Red = 430. ——, Eqn. 2.5; - - -, Eqn 2.6. . . . . . . . . . . . . . . . 38 2.15 Examples of the non-dimensional autocorrelation function R(T). , Xgrid = 0.456 mm, Red = 6000. ; Xgrid =0.456 mm, Red =2600; ; Xgrid =0.456 mm, Red = 2600. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.16 Integral and dissipation time scale estimation from the autocorrelation func- tion, (R(T)) at T u = 1.3%. ——, R(T). . . . . . . . . . . . . . . . . . . . . 40 2.17 Integral length scale evolution. , PP1, Red = 6000; ∗, PP1, Red = 2600; ◦, PP2, Red = 760; , PP2, Red = 1310, +, SMR, Red = 100, , SMR, Red = 430; ×, estimation of Λx using Eqn. 2.10 for the SMR grid. ——, Eqn. 2.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 x
  11. 11. 2.18 Evolution of Λx and λx downstream of turbulent grids. , PP1 grid; , PP2 grid; , SMR grid. Open symbols are λx . . . . . . . . . . . . . . . . . 42 2.19 Non-dimensional freestream energy spectra compared to the von Karman one-dimensional isotropic approximation, Eqn. 2.10 (Hinze, 1975). , T u = 1.3%; •, T u = 4.2%. , T u = 6%. . . . . . . . . . . . . . . . . . . 43 3.1 Dual slope method for intermittency detection within the boundary layer, Reθ = 577. +, y/δ=0.3, γ = 0.1%. . . . . . . . . . . . . . . . . . . . . . . 48 3.2 (a) Voltage trace of hotwire placed near boundary layer edge (y/δ = 0.86) at T u = 1.3%. (b) Time derivative of the hotwire trace squared (du/dt)2 . . 49 3.3 Comparison between the magnitude of two turbulent spots at low- and high-turbulence intensity. Complete trace. ——, T u = 1.3%, Reθ = 577, y/δ = 0.3. ——, T u = 6%, Reθ = 165, y/δ = 0.14. . . . . . . . . . . . . 50 3.4 Full trace where the arrow signifies the the portion of the trace to be anal- ysed. Test conditions are T u = 1.3% and Reθ = 577 . . . . . . . . . . . . 50 3.5 Typical velocity profile in wall coordinates compared to the linear law of the wall. , Reθ =94, %T ule =7; ——, u+ =y + . (b) ——, Local rate of energy dissipation, , calculated for square symbols case in (a). . . . . . . 52 3.6 Example of the Mexican hat wavelet transform. . . . . . . . . . . . . . . . 53 3.7 Demonstration of the advantages of wavelet over FFT analysis. . . . . . . . 55 3.8 Comparison between the energy spectrum and the time-average of the wavel- et map, y/δ = 0.3, Reθ = 577 and %T u = 1.3. ——, Fourier spectrum. ◦, time-averaged wavelet spectrum. . . . . . . . . . . . . . . . . . . . . . . . 56 3.9 Sample illustration showing the calculation of the 5th and 95th statistical distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.10 Example of umax and umin determination. . . . . . . . . . . . . . . . . . . 58 4.1 Hotfilm detection of turbulent spot, U∞ =17 m/s and T u = 1.3%. Inlay caption is zoomed in view of the turbulent spot. . . . . . . . . . . . . . . . 62 xi
  12. 12. 4.2 Comparison of transition onset Reynolds numbers against the transition onset correlations of Mayle (1991) and Fransson et al. (2005). , Reθ = 442 − 577, T u = 1.3%; , Reθ = 186 − 209, T u = 3.1%; •, Reθ = 154 − 167, T u = 4.2%; , Reθ = 131 − 166, T u = 6%; +, Reθ = 123 − 139, T u = 7%. ——, Mayle (1991) correlation, Reθ = 400T u−5/8 . - - -, Fransson et al. (2005) correlation, Reθ = 745/T u, at γ=0.1. . . . . . . . . 64 4.3 Plot illustrating the relationship between urms,max and U τ at transition on- set for the same Reθ values presented in figure 4.2. , T u = 1.3%; , T u = 3.1%; •, T u = 4.2%; , T u = 6%; +, T u = 7%. . . . . . . . . . . . 66 4.4 Non-dimensional velocity profiles over the complete range of %T u com- pared to the linear law of the wall. ——, U + =Y + . - - - -, Blasius velocity profile. , Reθ = 303, T u = 1.3%; , Reθ = 186, T u = 3.1%; •, Reθ = 181, T u = 4.2%; , Reθ = 131, T u = 6%; +, Reθ = 139, T u = 7%. 67 4.5 Growth of disturbances through the laminar boundary layers investigated at transition onset. , (Reθ , Xle , T u)= (577, 0.556m, 1.3%); , ((Reθ , Xle , T u)=(209, 0.195m, 3.1%); , (Reθ , Xle , T u)=(167, 0.272m, 4.2%); •, (Reθ , Xle , T u)=(166, 0.255m, 6%); , (Reθ , Xle , T u)=(139, 0.325m, 7%). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.6 (a) Plot illustrating how urms,max varies in linear proportion to Re0.5 . (b) x 0.5 urms,max /U∞ scaled with Rex . (c) urms,max /U∞ %T u scaled with Re0.5 . x , T u = 1.3%; , T u = 3.1%; •, T u = 4.2%; , T u = 6%; +, T u = 7%. . 69 4.7 Streamwise energy growth of streaks as a function of Rex for various FST levels. , T u = 1.3% (Rex has been scaled down by a factor of 10 for visual purposes); , T u = 3.1%; •, T u = 4.2%; , T u = 6%; +, T u = 7%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.8 Energy growth on streamwise streaks. ——, Trend line fit to experimental data, G ∝ %T u2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.9 Velocity traces from the near-wall to the freestream region at T u = 1.3% and Reθ = 577 (near transition onset). . . . . . . . . . . . . . . . . . . . . 74 xii
  13. 13. 4.10 Energy spectra for the same conditions as figure 4.9. ×, y/δ = 0.3; +, y/δ = 0.46; ◦, y/δ = 0.8; ——, y/δ = 3.4. . . . . . . . . . . . . . . . . . 75 4.11 Velocity traces from the near-wall to the boundary layer edge region at T u = 3.1% and Reθ = 186 (near transition onset). (a) Select velocity traces. (b) Close-up view of double negative spike at y/δ = 0.93. (c) Close-up view of multiple negative spike at y/δ = 0.93. . . . . . . . . . . . 75 4.12 (a) Energy spectra for the same conditions as figure 4.11 at T u = 3.1%. +, y/δ = 0.13; ◦, y/δ = 0.34; ×, y/δ = 0.93; ——, y/δ = 1.2. (b) Reθ = 106. , y/δ = 1; - - -, y/δ = 1.2. . . . . . . . . . . . . . . . . . . . 76 4.13 Velocity traces from the near-wall to the boundary layer edge region at T u = 6% and Reθ = 131 (near transition onset). . . . . . . . . . . . . . . 76 4.14 Energy spectra for the same conditions as figure 4.13. ×, y/δ = 0.13; +, y/δ = 0.28; ◦, y/δ = 0.95; ——, y/δ = 1.2. . . . . . . . . . . . . . . . . . 77 4.15 Deviation in laminar velocity profiles due to increased FST compared to the Blasius velocity profile. ◦, (Reθ , Xle , T u) =(178, 0.255m, 0.2%); , (Reθ , Xle , T u)= (442, 0.455m, 1.3%); , ((Reθ , Xle , T u)=(209, 0.195m, 3.1%); , (Reθ , Xle , T u)=(167, 0.272m, 4.2%); •, (Reθ , Xle , T u)=(166, 0.255m, 6%); , (Reθ , Xle , T u)=(139, 0.325m, 7%); ——, Blasius profile. 79 4.16 Increase in laminar dissipation coefficient, Cd , with increase in turbulence intensity at leading edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.17 Variation in Cd for the measured test conditions in the current investigation compared to Eqn. (3.11), with variation in %T u and Reθ . , T u = 1.3%; , T u = 3.1%; •, T u = 4.2%; , T u = 6%; +, T u = 7%; ——, Expo- nential trendline given by Eqn. (4.3); - - - -, ±10%, . . . . . . . . . . . . . 81 4.18 Development of high-frequency structures at various stages through the boundary layer thickness at Reθ = 577 (transition onset) and T u = 1.3%. The upper plots are the wavelet maps and the lower plots are the corre- sponding fluctuation velocity/voltage traces. (a) Wall. (b) y/δ = 0.3. (c) y/δ = 0.41. (d) y/δ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 xiii
  14. 14. 4.19 Wavelet maps and fluctuation velocity traces at T u = 3.1% and 6% in the near-wall and boundary layer edge regions. (a) T u = 3.1%, Reθ = 186, y/δ = 0.13. (b) T u = 3.1%, Reθ = 186, y/δ = 1.07. (c) T u = 6%, Reθ = 165, y/δ = 0.14. (d) T u = 6%, Reθ = 165, y/δ = 0.77. . . . . . . . 86 5.1 Skewness distribution at T u = 1.3%. , Reθ = 352; , Reθ = 392; , Reθ = 577. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2 Skewness distribution at T u = 3.1%. , Reθ = 106; , Reθ = 176; , Reθ = 186. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.3 Skewness distribution at T u = 4.2%. ◦, Reθ = 76; , Reθ = 107; , Reθ = 115; , Reθ = 181. . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.4 Skewness distribution at T u = 6%. , Reθ =83; , Reθ =107; , Reθ =131. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.5 Skewness distribution at T u = 7%. , Reθ =66; , Reθ =93; , Reθ =123. . 93 5.6 Probability distribution functions (PDF) for Reθ = 577 and T u = 1.3%. ♦, y/δ = 0.3; +, y/δ = 0.5; ◦, y/δ = 1. . . . . . . . . . . . . . . . . . . . 94 5.7 Probability distribution functions (PDF) for Reθ = 186 and T u = 3.1%. ♦, y/δ = 0.11; ×, y/δ = 0.48; ◦, y/δ = 0.93. . . . . . . . . . . . . . . . . 94 5.8 Probability distribution functions (PDF) for Reθ = 182 and T u = 4.2%. ♦, y/δ = 0.17; ×, y/δ = 0.51; ◦, y/δ = 0.97. . . . . . . . . . . . . . . . . 95 5.9 Probability distribution functions (PDF) for Reθ = 131 and T u = 6%. ♦, y/δ = 0.1; ×, y/δ = 0.51; ◦, y/δ = 0.95. . . . . . . . . . . . . . . . . . . 95 5.10 Probability distribution functions (PDF) for Reθ = 123 and T u = 7%. ♦, y/δ = 0.17; ×, y/δ = 0.5; ◦, y/δ = 0.93. . . . . . . . . . . . . . . . . . . 96 5.11 Variation in shear from the wall to the boundary layer edge. , T u = 1.3% at Reθ = 577. , T u = 3.1% at Reθ = 186. , T u = 6% at Reθ = 131. All measurements near transition onset. . . . . . . . . . . . . . . . . . . . 97 5.12 Variation in shear from the wall to the boundary layer edge at approxi- mately the same Reθ . , T u = 1.3% at Reθ = 148. , T u = 3.1% at Reθ = 176. , T u = 6% at Reθ = 131. . . . . . . . . . . . . . . . . . . . 97 xiv
  15. 15. 5.13 Basic sketch illustrating the interaction between negative jets and freestream vortices as the boundary layer advances towards transition onset. . . . . . . 100 5.14 Variation in the maximum and minimum of the instantaneous velocities from the wall to the boundary layer edge for T u = 1.3% ——, Average ve- locity; , Minimum instantaneous velocities; ♦, Maximum instantaneous velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.15 Variation in penetration depth (PD) measured from the boundary layer edge with %T u according to table 5.1. ——, trend line fit to data points, P D ∝ 0.0005%T u − 0.0005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.16 Variation of penetration depth (PD) measured from the boundary layer edge over the range of test conditions. ♦, T u = 1.3%; , T u = 3.1%; ◦, T u = 4.2%; , T u = 6%; +, T u = 7%. ——, Trend line fit to experimental data P D ∝ (ωRex )−0.47 . - - -, P D ∝ (ωRex )−0.133 by Jacobs and Durbin (1998). 104 5.17 Variation of penetration depth (PD) measured from the boundary layer edge over the range of test conditions. ♦, T u = 1.3%; , T u = 3.1%; ◦, T u = 4.2%; , T u = 6%; +, T u = 7%. ——, Trend line fit to experimental data P D ∝ (ωRex τw )−0.32 . - - -, P D ∝ (ωRex τ )−0.33 by Jacobs and Durbin (1998), where τ was replaced with the current experimentally measured τw values to allow for comparison. . . . . . . . . . . . . . . . . . . . . . . . . 105 5.18 Variation of penetration depth (PD) measured from the boundary layer edge over the range of test conditions. ♦, T u = 1.3%; , T u = 3.1%; ◦, T u = 4.2%; , T u = 6%; +, T u = 7%. ——, Trend line fit to experimental data √ P D ∝ (ωρUe Rex )−0.31 . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2 5.19 Variation in penetration depth (PD) as a function of y/δ (non-dimensional representation of PD), %T u and Rex . ——, trend line fit to data points, 0.378 y/δ = 164.3/%T uRex . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.20 Variation in penetration depth (PD) as a function of y/δ (non-dimensional representation of PD), %T u and Reθ . ——, trend line fit to data points, y/δ = 95/%T uRe0.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 θ xv
  16. 16. 5.21 Variation in penetration depth (PD) as a function of y/δ (non-dimensional representation of PD), %T u and Rex at transition onset. ——, trend line fit to data points, y/δ = 200/%T uRe0.4 . Note that when these results are x plotted in terms of Reθ the correlation becomes y/δ = 128/%T uRe0.73 . . . 109 θ 6.1 Maximum positive and negative fluctuation velocities presented in terms of a local disturbance intensity (%T uL ) at T u = 1.3%. (a) Reθ = 352. (b) Reθ = 392. (c) Reθ = 577 (transition onset) with high-frequency events removed. (d) Reθ = 577 without high-frequency events removed. , maximum negative value; ♦, maximum positive value; ×, RMS value. . 111 6.2 Statistical distribution of the 5th and 95th , 1st , 99th , 0.1st and 99.9th , and the 0.01st and 99.99th levels for T u = 1.3% and Reθ = 352. , 0.01th ; ♦, 99.99th ; ——, 0.1st ; - - -, 99.9th ; ——, 1st ; - - -, 99th ; ——, 5th ; - - -, 95th . . 113 6.3 Maximum positive and negative fluctuation velocities presented in terms of a local disturbance intensity (%T uL ) at T u = 3.1%. (a) Reθ = 106. (b) Reθ = 176. (c) Reθ = 186 (transition onset). , maximum negative value; ♦, maximum positive value; ×, RMS value. . . . . . . . . . . . . . . . . . 114 6.4 Maximum positive and negative fluctuation velocities presented in terms of a local disturbance intensity (%T uL ) at T u = 4.2%. (a) Reθ = 76. (b) Reθ = 107. (c) Reθ = 115. (d) Reθ = 181 (transition onset). , maximum negative value; ♦, maximum positive value; ×, RMS value. . . . . . . . . . 115 6.5 Maximum positive and negative fluctuation velocities presented in terms of a local disturbance intensity (%T uL ) at T u = 6%. (a) Reθ = 83. (b) Reθ = 107. (c) Reθ = 131 (transition onset). , maximum negative value; ♦, maximum positive value; ×, RMS value. . . . . . . . . . . . . . . . . . 115 6.6 Maximum positive and negative fluctuation velocities presented in terms of a local disturbance intensity (%T uL ) at T u = 7%. (a) Reθ = 66. (b) Reθ = 93. (c) Reθ = 123 (transition onset). , maximum negative value; ♦, maximum positive value; ×, RMS value. . . . . . . . . . . . . . . . . . 116 xvi
  17. 17. 6.7 Wall-normal disturbance profiles of the low- and high-speed streaks corre- sponding to the same results presented in figures 6.1, 6.3 and 6.5 where the same symbols apply. (a) T u = 1.3%. (b) T u = 3.1%. (c) T u = 6%. . . 117 6.8 Variation in wall-normal location of the peak minimum of the streamwise fluctuation velocities. (a) Results of Brandt et al. (2004) where the different line styles represent difference freestream integral length scales at constant T u = 4.7%. (b) Current results presented as a comparison to DNS. , T u = 1.3% where the Reynolds numbers have been reduced by a factor of 10 to allow for a clear comparison; , T u = 3.1%; •, T u = 4.2%; , T u = 6%; +, T u = 7%, where where δ1 is the displacement thickness . . . 118 6.9 Maximum positive and negative fluctuation velocities presented in terms of a disturbance intensity about the mean (% u/Umean ) at T u = 1.3%. (a) Reθ = 352. (b) Reθ = 392. (c) Reθ = 577 with high-frequency events removed. (d) Reθ = 577 without high-frequency events removed. , maximum negative value; ♦, maximum positive value. . . . . . . . . . . 121 6.10 Maximum positive and negative fluctuation velocities presented in terms of a disturbance intensity about the mean (% u/Umean ) at T u = 3.1%. (a) Reθ = 106. (b) Reθ = 176. (c) Reθ = 186 (transition onset). , maximum negative value; ♦, maximum positive value. . . . . . . . . . . . . . . . . . 121 6.11 Maximum positive and negative fluctuation velocities presented in terms of a disturbance intensity about the mean (% u/Umean ) at T u = 4.2%. (a) Reθ = 76. (b) Reθ = 107. (c) Reθ = 181 (transition onset). , maximum negative value; ♦, maximum positive value. . . . . . . . . . . . . . . . . . 122 6.12 Maximum positive and negative fluctuation velocities presented in terms of a disturbance intensity about the mean (% u/Umean ) at T u = 6%. (a) Reθ = 83. (b) Reθ = 107. (c) Reθ = 131 (transition onset). , maximum negative value; ♦, maximum positive value. . . . . . . . . . . . . . . . . . 122 xvii
  18. 18. 6.13 Maximum positive and negative fluctuation velocities presented in terms of a disturbance intensity about the mean (% u/Umean ) at T u = 7%. (a) Reθ = 66. (b) Reθ = 93. (c) Reθ = 123 (transition onset). , maximum negative value; ♦, maximum positive value. . . . . . . . . . . . . . . . . . 123 6.14 Evolution of the maximum negative and positive of the streamwise fluctua- tion velocities over the range of FST levels investigated. (a) Maximum neg- ative values. (b) Maximum positive values. , T u = 1.3%; , T u = 3.1%; •, T u = 4.2%; , T u = 6%; +, T u = 7%. . . . . . . . . . . . . . . . . . . 126 6.15 Trendline fit to maximum negative fluctuation values when scaled with 0.5 %T u and Rex . , T u = 1.3%; , T u = 3.1%; •, T u = 4.2%; , Tu = 6%; +, T u = 7%. ——, Trend line fit to experimental data, %T uL(max,neg) /%T u = 0.066(Rex )0.44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.16 Trendline fit to maximum positive fluctuation values when scaled with 0.5 %T u and Rex . , T u = 1.3%; , T u = 3.1%; •, T u = 4.2%; , Tu = 6%; +, T u = 7%. ——, Trend line fit to experimental data, %T uL(max,pos) /%T u = 0.082(Rex )0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.17 Comparison between the transition onset prediction correlations of Mayle (1991), Fransson et al. (2005) and the current results. ——, Correlation presented in the current investigation Reθ = (370/%T u)1.17 ; · · ·, Fransson et al. (2005) correlation Reθ = 745/%T u; - - - , Mayle (1991) correlation Reθ = 400%T u−5/8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.18 Non-dimensionalised skewness distribution at transition onset. Reθ = 410 and T u = 1.3%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.19 Fluctuating voltage trace for all four simultaneously sampled probes at Reθ = 410 and T u = 1.3%. The hotwire probe is at y/δ = 1.26. The hotwire probe is labelled ”wire” and the location of the film probes are given by their distance downstream of the plate leading edge. ——, xle = 221mm; ——, xle = 241.3mm; ——, W ire xle = 250mm; ——, xle = 261.6mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 xviii
  19. 19. 6.20 Zoomed in view for fluctuating voltage trace of all four simultaneously sampled probes at Reθ = 410 and T u = 1.3%. The SN probe is at y/δ = 1.26. (a) Fluctuating voltage. (b) Corresponding (du/dt)2 . Same legend as figure 6.19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.21 Fluctuating voltage trace for all four simultaneously sampled probes at Reθ = 410 and T u = 1.3%. The hotwire probe is at y/δ = 1.07. Same legend as figure 6.19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.22 Fluctuating voltage trace for all four simultaneously sampled probes at Reθ = 410 and T u = 1.3%. The hotwire probe is at y/δ = 1.07. (a) fluctuating voltage. (b) corresponding (du/dt)2 . Same legend as figure 6.19. 133 6.23 Fluctuating voltage trace for all four simultaneously sampled probes at Reθ = 410 and T u = 1.3%. The hotwire probe is at y/δ = 1.07. (a) fluctuating voltage. (b) corresponding (du/dt)2 . (c) fluctuating voltage. (d) corresponding (du/dt)2 . Same legend as figure 6.19. . . . . . . . . . . 133 6.24 Fluctuating voltage trace for all four simultaneously sampled probes at Reθ = 410 and T u = 1.3%. The hotwire probe is at y/δ = 0.9 (close to peak skewness). Same legend as figure 6.19. . . . . . . . . . . . . . . . 134 6.25 Fluctuating voltage trace for all four simultaneously sampled probes at Reθ = 410 and T u = 1.3%. The hotwire probe is at y/δ = 0.9. (a) Fluctuating voltage. (b) Corresponding (du/dt)2 . (c) Fluctuating voltage. (d) Corresponding (du/dt)2 . Same legend as figure 6.19. . . . . . . . . . . 135 6.26 Fluctuating voltage trace for all four simultaneously sampled probes at Reθ = 410 and T u = 1.3%. The hotwire probe is at y/δ = 0.9. (a) Fluctuating voltage. (b) Corresponding (du/dt)2 . Same legend as figure 6.19.136 6.27 Fluctuating voltage trace for all four simultaneously sampled probes at Reθ = 450 and T u = 1.3%. The hotwire probe is at y/δ = 0.9 (close to peak skewness). ——, xle = 221mm; ——, xle = 241.3mm; ——, W ire xle = 247.7mm; ——, xle = 261.6mm. . . . . . . . . . . . . . . . . 137 xix
  20. 20. 6.28 Fluctuating voltage traces for all four simultaneously sampled probes at Reθ = 450 and T u = 1.3%. The hotwire probe is at y/δ = 0.9. Same legend as figure 6.27. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.29 Fluctuating voltage trace for all four simultaneously sampled probes at Reθ = 450 and T u = 1.3%. The hotwire probe is at y/δ = 0.6. Same legend as figure 6.27. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.30 Fluctuating voltage trace for all four simultaneously sampled probes at Reθ = 450 and T u = 1.3%. The hotwire probe is at y/δ = 0.41. Same legend as figure 6.27. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.31 Fluctuating voltage trace for all four simultaneously sampled probes at Reθ = 158 and T u = 4.2%. The hotwire probe is at y/δ = 0.8. ——, xle = 231.2mm; ——, xle = 246.4mm; ——, W ire xle = 259.1mm; ——, xle = 272mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.32 Fluctuating voltage trace for all four simultaneously sampled probes at Reθ = 158 and T u = 4.2%. The hotwire probe is at y/δ = 0.8. (a) Fluctuating voltage. (b) Corresponding (du/dt)2 without wire. (c) Fluctu- ating voltage. (d) Corresponding (du/dt)2 without wire. Same legend as figure 6.31. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.33 Fluctuating voltage trace for all four simultaneously sampled probes at Reθ = 123 and T u = 7%. The hotwire probe is at y/δ = 0.58. ——, xle = 231.2mm; ——, xle = 241.3mm; ——, W ire xle = 254mm; ——, xle = 264.2mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.34 Fluctuating voltage trace for all four simultaneously sampled probes at Reθ = 123 and T u = 7%. The hotwire probe is at y/δ = 0.58. (a) Fluctuating voltage. (b) Corresponding (du/dt)2 . (c) Fluctuating voltage. (d) Corresponding (du/dt)2 . Same legend as figure 6.33. . . . . . . . . . . 142 xx
  21. 21. 6.35 (a) Simple illustration for the argument put forward against the concept that a turbulent spot may develop at the wall, where xU∞ is the unknown veloc- ity that the upper portion of the spot must travel in order for the near-wall breakdown hypothesis to be correct. (b) Boundary layer edge breakdown scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 xxi
  22. 22. Nomenclature A Hotwire calibration constant [-] a Overheat ratio [-] B Hotwire calibration constant [-] b Wavelet constant [-] Cd Dissipation coefficient [-] Cf Skin friction coefficient [-] c Wavelet conversion from scale to frequency [-] d Grid bar diameter [m] d1 Energy converted to heat by viscous dissipation [Wm2 k−1 ] E Disturbance energy [-] E Energy spectral density [m2 s−1 ] E Expectation operator [-] Ec Temperature corrected voltage [volts] Es Temperature corrected measured voltage [volts] E0 Zero voltage [volts] f Frequency [Hz] fc Frequency cut-off [Hz] G Disturbance energy growth [-] g(t) Time series [-] h Maximum height of frequency response curve [-] k(y) Wall-normal wavenumber [m−1 ] M Turbulence grid mesh length [m] n Power law exponent in hotwire calibration [-] R(T ) Autocorrelation function [-] Rw Wire resistance [ohms] R20 Wire resistance at room temperature [ohms] Red Reynolds number based on hotwire diameter [-] xxii
  23. 23. Rex Reynolds number based on streamwise distance [-] Reθ Reynolds number based on momentum thickness [-] T Temperature [K] T Time delay [s] Tu Turbulence intensity [-] T uL Local turbulence intensity [-] Tr Reference temperature [K] Ts Sensor temperature [K] Tw Temperature of wire [K] T20 Ambient temperature [K] U Local, instantaneous streamwise velocity [m/s] U Mean streamwise velocity (Umean ) [m/s] Uτ Skin friction velocity [m/s] U+ Non-dimensional streamwise velocity [-] u Fluctuating streamwise velocity [m/s] V Mean tranverse velocity [m/s] Vs Measured mean voltage [volts] vs Measured fluctuating voltage [volts] w(t) Wavelet function [-] w(f, t) Wavelet coefficient [-] x Streamwise distance [m] Y+ Non-dimensional wall-normal velocity [-] y Wall-normal distance [m] Greek Symbols α20 Thermal coefficient of resistance at ambient [K−1 ] β Spanwise wavenumber [m−1 ] γ Intermittency [-] ∆L Distance between turbulence grid and leading edge [m] ∆t Time difference [s] xxiii
  24. 24. δ Boundary layer thickness [m] δ1 Boundary layer displacement thickness [m] Local rate of energy dissipation [m2 s−3 ] Rate of energy dissipation per unit area [ms−3 ] η Blasius parameter [-] θ Momentum thickness [m] Λ Integral length scale [m] λ Dissipation length scale [m] µ Dynamic viscosity [Nsm−2 ] µ3 Third central moment about the mean [-] ν Kinematic viscosity [m2 s−1 ] ρ Density [kg.m−3 ] Σ Summation sign [-] σ Standard deviation [-] τ Shear stress [Nm−2 ] τ Time constant [s] τw Wall shear stress [Nm−2 ] Φ Incompressible viscous dissipation function [Jm−3 s−1 ] ω Frequency of the largest freestream eddies [Hz] Subscripts le Leading edge conditions max Maximum quantity mean Time-averaged quantity min Minimum quantity neg Negative quantity pos Positive quantity RM S Root mean square ∞ Freestream parameter xxiv
  25. 25. Chapter 1 Introduction 1.1 Research Motivation The transition from a laminar to a turbulent state is one of the most ubiquitous features of fluid flow. Traditionally, the term laminar flow suggested that the flow is comprised of laminae or layers of fluid which glide over one another with relative ease and little mixing. At some stage in the laminar boundary layer development the flow becomes unstable and transitions to a turbulent state. Given long enough all flow types will transition and this fact gave rise to the statement by Moin and Kim (1997) that in nature ”turbulence is the rule rather than the exception”. There are generally three main types of boundary layer transition scenario depending on the level of ambient disturbance; one is known as the orderly route to transition where Tollmien-Schlichting (TS) instabilities develop and the flow eventually breaks down into patches of turbulence called turbulent spots. This mode of transition is well understood (Tollmien, 1929; Schlichting, 1933) and occurs when the level of freestream disturbance is very low. In the second mode of transition the TS instabilities are bypassed, hence the term bypass transition is used to describe this process (Morkovin, 1969), and turbulent spots develop almost spontaneously. The third type of transition scenario occurs in regions of highly adverse pressure gradient such as in a gas turbine cascade (Van Treuren et al., 2002) where the flow breaks down into turbulent motion at or just downstream of the separation zone. This investigation deals with the bypass mode of transition induced by elevated levels 1
  26. 26. of FST downstream of turbulence generating grids. Bypass transition occurs above some level of freestream turbulence (FST), usually above 1% of the freestream velocity. The bypass transition process is studied extensively due its industrial applications by virtue of the fact that it occurs in regions where elevated levels of FST are present such as in the downstream wake passage of a gas turbine blade. Although the transition process has been studied for some time (the most relevant early example is Reynolds (1883)), until recently the mechanisms involved in the bypass route to transition remained elusive both experimentally and numerically. Recently, the full Navier- Stokes equations have been solved using direct numerical simulations (DNS) thereby giv- ing much sought after insight into the complete transition process. However, experimental support of such complex results is necessary and challenging. The complexity of the transition process can be highlighted by considering the various factors that affect the destabilisation of laminar flow to a turbulent state in real life flow processes, such as: pressure gradient, surface curvature, surface roughness, heat transfer effects, disturbances in the freestream and the spectrum of the freestream flow. The impor- tance of these parameters is recognised in this investigation; however, the objective of this study is to focus on the fundamental fluid dynamics leading to the generation of turbulent spots in zero pressure gradient (ZPG) flow subject to varying levels of FST. The study of each individual factor named above is outside the scope of the current investigation. A noteworthy point is that the results herein are fundamental and have the potential to be extended to include these factors in future work. It is not only necessary to study the transition process due to its ubiquitous nature, it is also necessary to understand it due to its many and varied industrial applications. It is well known that laminar to turbulent transition can have adverse effects on wall shear stresses and heat transfer rates in fluid flow processes, with the gas turbine engine being the most widely quoted. Wang and Zhou (1998) and Brandt (2003) state that as much as 50-80% of the surface of a typical turbine blade can be covered by transitional flow and this will have severe consequences on the overall losses and thereby increasing the operating cost of the engine drastically. Walsh and Davies (2005) measured the transition region in realistic Mach number conditions over a turbine blade profile and showed that the transition 2
  27. 27. region occupied over 20% of the total suction surface length. According to Bowles (2000) it has been estimated that 20% savings could be attained if the majority of flow on a gas turbine blade was kept laminar. Bowles (2000) also comments on the reduced life span of the turbine blades due to the increased heat transfer properties involved in the transition process. The breakdown process on a gas turbine is significantly more complicated than that on a flat plate. One major source of this complexity is due to the fact that the highly turbulent wake (T u = 20% according to Mayle (1991)) from upstream rows impinges on the leading edge of the blade and causes the transition process to be unsteady. This type of FST is more complicated than grid generated turbulence which can be considered statistically steady under controlled conditions (Lewalle et al., 1997). Furthermore, in a gas turbine engine the flow also has to contend with pressure gradients, streamwise curvature, surface roughness and very large heat transfer rates with temperatures high enough to melt the blades. The main goal of this investigation is to gain further insight into the physical attributes of pre-transitional flow with the prospect of this increased understanding improving the modelling of the flow. Specifically, this thesis examines the laminar region up to the point of breakdown to turbulent motion. The breakdown from a laminar to a turbulent state contains three main stages: 1) the receptivity phase, i.e., how disturbances from the freestream enter into the boundary layer, 2) the disturbance amplification phase, i.e., the development of the disturbances in the boundary layer up to the point of breakdown to turbulent motion and 3) the breakdown of these disturbances into turbulent motion. Recently the similarity between ZPG transition and pressure gradient transition phe- nomena has been demonstrated (Maslowe and Spiteri, 2001; Zaki and Durbin, 2006) thereby further illustrating the importance of completely understanding the transition process in ZPG flow. However, the transition process on a flat plate is still not fully understood and the latest analytical and numerical results are still lacking experimental support. An overview of the literature describing the current understanding of the stages of laminar to turbulent breakdown in a ZPG flow follows. This section is separated into experimental, theoreti- cal and numerical investigations. Following this, Section 1.3 will give an overview of the thesis and finally Section 1.4 will outline the main objectives of the work. 3
  28. 28. 1.2 Laminar to Turbulent Transition: The Bypass Route 1.2.1 Experimental Investigations The first experimental work into the transition process on a flat plate was by Dryden (1937). The main observation from this work was that the streamwise fluctuations in the boundary layer were of low frequency and were several times larger in amplitude than the fluctuations found in the freestream. Following from this, experimental work into the bypass transition process has demonstrated that under the influence of elevated FST the laminar boundary layer develops high- and low-speed fluctuations relative to the mean streamwise velocity that are extended in the streamwise direction and are termed streaky structures (Matsubara and Alfredsson, 2001). Experiments have shown that the maximum of the root mean square (RMS) streamwise disturbance is located approximately in the middle of the boundary layer and at transition onset the maximum streamwise disturbance is approximately 10% of the freestream velocity (Roach and Brierley, 2000; Matsubara and Alfredsson, 2001). As stated by Matsubara and Alfredsson (2001), as a matter of comparison, a TS wave has its amplitude maximum located in the near-wall region and breaks down at amplitudes of approximately 1% of the freestream velocity. An illustration of the streaky structures in a laminar boundary layer is shown in fig- ure 1.1(a) where the low-speed streaks (regions of smoke) and the high-speed streaks (re- gions without smoke) are clearly visible. These streaky structures were first discovered by Klebanoff (1971) where he referred to them as ”breathing modes” due to the fact that their occurrence seemed to correspond to a thickening and thinning of the boundary layer. The time-averaged representations of the streaky structures are termed Klebanoff modes, due to Kendall (1985). Kendall (1985) demonstrated that the low-frequency fluctuations associ- ated with the streaky structures in the flow grow proportionally with the laminar boundary layer thickness (δ). After this linear growth secondary instabilities on the streaks develop leading to ex- ponential growth signifying the breakdown to turbulence. Under certain conditions the streaky structures have been shown to develop a streamwise waviness which eventually breaks down into a turbulent spot, as shown in the flow visualizations of Matsubara and 4
  29. 29. Figure 1.1: Smoke flow visualisation of streamwise streaks, streamwise wiggles and turbu- lent spots (Matsubara and Alfredsson, 2001). Alfredsson (2001) (figure 1.1). The upper parts of figure 1.1(a) and (c) show the stream- wise waviness that Matsubara and Alfredsson (2001) refer to as streamwise ”wiggles”. The wiggles signify the exponential growth on the streak and eventually the flow breaks down into a turbulent spot as illustrated in figure 1.1(b) and (d). The flow process here is that the streak disturbance magnitude reaches a sufficiently high level for secondary instabil- ities on the streaks to develop (the wiggle referred to previously) which will provoke the breakdown to turbulence. It will be demonstrated in this investigation that the negative component of the fluc- tuating velocity is critical in the transition process. A number of experimental studies have demonstrated the evolution of structures with associated negative fluctuation veloc- ity from the near-wall region towards the boundary layer edge (see e.g., Wygnanski et al., 1976; Kuan and Wang, 1990; Blair, 1992; Kendall, 1998; Westin et al., 1998; Chong and Zhong, 2005). Well upstream of transition onset Blair (1992) observed large scale struc- tures with associated negative fluctuation velocities near the boundary layer edge region; Blair termed these structures negative spikes. Approaching the point of transition onset the negative spikes developed high-frequency components and Blair commented that the demarcation between multiple negative spikes and a turbulent structure became ‘blurred’. Artificially generated turbulent spots have been investigated extensively (Wygnanski et al., 5
  30. 30. 1976; Westin et al., 1998; Chong and Zhong, 2005). The major advantage of artificially generating the turbulent structures is that it allows for an ensemble-averaged representation of the spots as the same disturbance is generated every time. Chong and Zhong (2005) showed that high-frequency negative fluctuations develop near the boundary layer edge be- fore any near-wall positive fluctuations were observed; however, they did not discuss the connection between their results and DNS and the question still remains as to the similarity between artificially and naturally occurring turbulent spots. The importance of elevated freestream turbulence intensity at the leading edge (T u) in the bypass transition process has been further demonstrated in the work of Fransson et al. (2002). They demonstrated that the initial disturbance energy in the boundary layer is proportional to T u2 and the transition Reynolds number is inversely proportional to T u2 . Recent investigations such as Roach and Brierley (2000), Jonas et al. (2000) and Frans- son et al. (2005) have discussed the importance of accurate determination of the FST length scales and energy spectra in the laminar boundary layer receptivity process. By virtue of the fact that the length scales have an effect on the receptivity process it can be expected that they too will affect the transition process, a fact demonstrated by Jonas et al. (2000). Through some of the above referenced investigations into the bypass transition process came the observation that under the influence of increased FST a laminar velocity profile deviates substantially from the theoretical velocity profiles of Blasius and Pohlhausen. This characteristic has been observed by many experimental investigators such as Fasihfar and Johnson (1992), Roach and Brierley (2000), Matsubara and Alfredsson (2001) and Hernon and Walsh (2007). Although this deviation in velocity gradient has been noted previously, the associated increase in energy dissipation (where the energy dissipation rate is equivalent to the entropy generation rate) due to enhanced viscous shear rates has not been quantified. This departure in velocity profile compared to the well established theories of Blasius and Pohlhausen has a two fold effect on the velocity gradients of the flow; 1) a reduction in the velocity gradient in the outer layer region occurs, increasing the boundary layer thickness compared to theory; 2) an increase in the velocity gradient in the near-wall region develops causing an increase in the wall shear stress with an attendant increase in the rate of energy dissipation. 6
  31. 31. The increase in wall shear stress can be attributed to the enhanced mixing caused by the FST that has been shown experimentally by Volino and Simon (2000) and numerically by Jacobs and Durbin (2001) to penetrate into the boundary layer. The effect of increasing the near-wall velocity gradient has significantly more influence on the energy dissipation rate as the majority of energy dissipation takes place in the near-wall region. Wavelet techniques have been used to great advantage in fluid dynamic processes, espe- cially in the phenomenon of transition to turbulence which is a complicated and stochastic process. The advantage of wavelet over fast Fourier transform (FFT) analysis is that the wavelet provides both sequential and spectral information on the flow. Wavelets allow for a more in depth view of the time-dependent scales found in the complex process that is laminar to turbulent transition under the influence of elevated FST. Lewalle et al. (1997) used a number of different wavelet techniques as a means of gaining more information on the various time scales present in the transition process through ensemble and condi- tional statistics. They found that there was a lack of energy-dominant time scales inside the turbulent spots. They identified transport dominant scales within the spots which were responsible for the largest part of the Reynolds stress. Kaspersen and Krogstad (2001) used continuous wavelet techniques to detect turbulent spots and burst phenomena and Volino (2002) used wavelet analysis to define the variation in turbulent scales throughout the tran- sition process. 1.2.2 Theoretical Investigations Theoretical studies into the transition process have been beneficial in elucidated some of the transition mechanisms and a brief review of the most important work relating to the experimental results presented in this investigation are given here. According to Andersson et al. (2001) the classical starting point for a theoretical de- scription of the transition process involves linear stability theory. Using this method expo- nentially growing solutions to the linearised Navier-Stokes equations are sought to signify the breakdown to turbulence. If these solutions are not found then the flow is considered to be stable. These authors state that many flow types may undergo transition for Reynolds numbers well below the critical value predicted by linear stability theory. Therefore, the 7
  32. 32. theory failed to capture all of the physics involved in the transition process when com- pared to experiment. This led to the development of further theories that could predict the algebraic/transient growth on the streaks before the evolution of the exponential behavior. According to Berlin and Henningson (1999) a re-examination into the Orr-Sommerfeld and Squire operators showed that the eigenfunctions are nonorthogonal. This implies that disturbances other than exponentially growing instabilities can grow in the boundary layer and this theory is referred to as nonmodal, transient or algebraic growth. The theories on transient growth have been developed to accurately predict the features of the streaks before they breakdown to turbulent motion. These theories show that the growth on the streaks is linear along the length of the plate and scales with the boundary layer thickness upstream of the exponential growth that signifies the breakdown to turbulence. According to Andersson et al. (2001), Wundrow and Goldstein (2001) and Zaki and Durbin (2005) the secondary instability on the streak can be caused by the presence of inflection type instabilities in the laminar flow velocity profile. It has been demonstrated theoretically that the streaks are caused by streamwise vor- tices at the leading edge of the plate (Landahl, 1975, 1980; Andersson et al., 1999, 2001; Luchini, 2000). One of the first sound theoretical attempts at explaining the streak growth phenomenon was proposed by Landahl (1975, 1980) and termed the linear lift-up effect. According to Andersson et al. (2001) this theory argued that vortices aligned in the stream- wise direction advect the mean velocity gradient towards and away from the wall, generat- ing spanwise inhomogeneities, i.e, the streaks. In this theory a wall-normal displacement of fluid (due to the freestream vortices) will initially retain its horizontal momentum and therefore generate the low- and high-speed regions within the flow. Andersson et al. (1999) and Luchini (2000) have since predicted that the energy growth of the optimal disturbances in the freestream is proportional to the distance from the lead- ing edge. These theories on optimal disturbances also accurately predict the wall-normal shape of the time-averaged streamwise fluctuations and show that streamwise-orientated disturbances in the freestream are the cause of maximum growth of the high- and low- speed streaky structures in the boundary layer. It was stated by Andersson et al. (2001) that ”the streamwise vortices leave an almost permanent scar in the boundary layer in the form 8
  33. 33. of long-lived, elongated streaks of alternating high and low streamwise speed”. Wundrow and Goldstein (2001) commented on the significance of looking at maximum disturbance magnitudes compared to RMS values when considering what disturbances are critical in causing the transition to turbulence. Their investigation also illustrated the de- velopment of negative spike structures and the evolution of an inflectional velocity profile near the boundary layer edge as the point of transition onset approached. Furthermore, in the majority of studies on the transition process the disturbances that amplify to generate turbulent spots are presented in time-averaged form (e.g., the so-called Klebanoff mode) where the most prominent disturbances that lead to transition remain recondite due to the averaging process. In order to more fully understand the bypass transition process it is necessary to under- stand how freestream disturbances (such as freestream vortices, freestream acoustic waves and most importantly freestream turbulence in the context of this investigation) penetrate and interact with the underlying boundary layer. This phase in the flow’s development is referred to as the receptivity phase. According to Reshotko (1994), receptivity describes the signature in the boundary layer of some externally imposed disturbance. More recently the study of receptivity has been examined in a new light mainly due to the recent devel- opments in the theory of shear sheltering (see e.g., Jacobs and Durbin, 1998; Hunt and Durbin, 1999; Maslowe and Spiteri, 2001; Zaki and Durbin, 2005, 2006). Jacobs and Durbin (1998) solved the model problem of a two-dimensional Orr- Sommerfeld disturbance for a Blasius boundary layer profile and determined that the pene- tration depth (PD) of disturbances (defined as the point where the disturbance eigenfunction (φ) dropped and remained below 0.01 which was defined by the location in the boundary layer where the freestream function lost its oscillatory nature) from the freestream into the boundary layer is given by P D ∝ (1/ωR)−0.133 , where ω and R are the frequency of the largest freestream disturbances and the Reynolds number of the flow. This dependence was also demonstrated in Maslowe and Spiteri (2001) for varying pressure gradients. Zaki and Durbin (2005) stated that the measure of PD by Jacobs and Durbin (1998) was rather sub- jective as they restricted attention to a single wall-normal wavenumber. Zaki and Durbin (2005) further enhanced the concept of PD by introducing a coupling coefficient which 9
  34. 34. better represented the interaction between the freestream disturbances and the underlying boundary layer. It has also been demonstrated that streamwise streaks can be generated by the interac- tion of oblique waves in the freestream which combine to form streamwise vortices which in turn force the generation of streaks in the boundary layer (Schmid and Henningson, 1992; Berlin and Henningson, 1999; Brandt et al., 2002, 2004). This transition scenario is initiated with the introduction of two oblique waves with small amplitude. The two oblique waves can be characterised by (ω ± β), where ω is their angular frequency and ±β is their spanwise wavenumbers. This type of investigation has opened up a new type of study where different mechanisms for triggering or delaying transition are being developed (Fransson et al., 2002). There are generally two different receptivity mechanisms proposed in the literature (Brandt et al., 2004): 1) a linear mechanism for streak generation which is due to the dif- fusion and/or propagation of freestream streamwise vortices into the boundary layer. If the linear process is present it has been shown that the growth of the streaks is caused directly √ by the freestream vortices at the leading edge and is proportional to Rex . 2) a nonlinear mechanism involving oblique modes in the freestream was proposed by Berlin and Hen- ningson (1999) and Brandt et al. (2002). As stated in Brandt et al. (2004) this nonlinear mechanism is a two step process. Firstly, the nonlinear generation of streamwise vorticity occurs due to the interaction between the oblique modes which penetrate the underlying boundary layer. Secondly, the streaks are then formed due to the lift-up effect. This non- linear mechanism acts along the length of the plate above the boundary layer and causes the continuous forcing from the freestream on the underlying boundary layer. 1.2.3 Numerical Investigations DNS studies have been a major influence in elucidating the routes to bypass transition (see e.g., Wu et al., 1999; Jacobs and Durbin, 2001; Brandt et al., 2004; Zaki and Durbin, 2005, 2006). These studies have also illustrated that the low-speed components of the flow lift away from the wall towards the boundary layer edge due to the upward motion of the vor- tical structures found in the freestream region. The breakdown process put forward by 10
  35. 35. DNS is illustrated in figure 1.2 where the low-speed streaks lift up towards the boundary layer edge (denoted by δ). In DNS the low-speed regions are termed negative or backward jets due to the fact that their associated instantaneous velocity (U) is significantly less than the local mean value (U ), therefore such structures have negative fluctuation velocity (u). The low-speed streaks (negative jet structures) can be lifted up to the height of the mean boundary layer thickness. When the low-speed streaks reach the outer region of the bound- ary layer they are subject to inflection point instability and a form of Kelvin-Helmholtz type instability develops (Zaki and Durbin, 2005) which is signified by rolling of the shear layer. At this stage the flow is locally unstable to the high-frequency disturbances con- tained in the freestream and the disturbance at the boundary layer edge region intensifies until eventually the lifted low-speed streak (negative jet) breaks down into a turbulent spot in the outer region of the boundary layer. As stated previously, Zaki and Durbin (2005) developed a coupling coefficient which gave a measure of the propensity of a certain mode (where mode refers to the frequency content and hence the size of freestream disturbance) to penetrate and continue oscillating within the boundary layer. In that paper, using DNS, Zaki and Durbin (2005) demonstrated that with two weakly penetrating modes at the inlet the boundary layer was weakly perturbed; with two strongly penetrating modes at the in- let, Klebanoff modes were produced in the boundary layer; with one strongly penetrating low-frequency mode and one weakly penetrating high-frequency mode the complete tran- sition process was simulated. This work further illustrates the necessity for the interaction between low- and high-frequency modes in order to initiate the transition process. This route to bypass transition differs significantly from those presented before the application of DNS where it was believed that turbulent structures were generated, almost spontaneously, in the near-wall region. One postulate put forward (Johnson and Dris, 2000) was that transition occurs due to a local near-wall flow separation which develops when the instantaneous velocity drops below 50% of the mean value. Furthermore, before the development of DNS, such was the contention that the flow broke down in the near-wall region that some experiments only examined the flow in this region (Lewalle et al., 1997; Johnson and Dris, 2000). Andersson et al. (2001) used DNS to follow the nonlinear evolution of boundary layer 11
  36. 36. δ Inflectional- FST type wave U Lifted streaks Turbulent spot Laminar Transitional Turbulent Figure 1.2: Sketch of the bypass transition process adapted from Zaki and Durbin (2006). streaks. They showed that the streak critical amplitude via the sinuous mode is approxi- mately 26% of the freestream velocity. This is the critical amplitude for the onset of sec- ondary instability but not necessarily for the breakdown of the flow which develops when an inflectional profile is present. The critical amplitude of the streaks when undergoing the varicose mode of breakdown was shown be approximately 37% of the freestream ve- locity; therefore, the sinuous mode is far less stable than the varicose mode. Brandt et al. (2004) demonstrated that either sinuous or varicose secondary instabilities on the streaks may be the precursors to turbulent spot formation and they also investigated the receptivity of the boundary layer to changes in the freestream integral length scale (Λ). They demon- strated that, by varying the energy spectrum of the freestream disturbances, the transition location moved to lower Reynolds numbers when the integral length scale of the FST was increased. It is a demanding experimental task to vary T u while keeping the same scales of turbulence, or vice versa, and it is in this regard that DNS becomes a sensible option. In that study they also found two distinct receptivity mechanisms depending on the energy content of the external disturbance: 1) if low-frequency modes diffuse into the boundary layer at the leading edge the streaks are generated in the normal way due to streamwise vorticity through the linear lift-up effect and 2) if the freestream disturbances are mainly located above the boundary layer a non-linear process is needed to generate the streaks within the boundary layer. The DNS of Brandt et al. (2004) also illustrated the evolution 12
  37. 37. of the peak locations of the maximum positive and negative fluctuation velocities in the streamwise direction. They demonstrated that the peak of the negative value shifted to- wards the boundary layer edge and the peak of the positive value shifted towards the wall as the point of transition onset approached. Their study further demonstrates the impor- tance of examining maximum disturbance levels associated with the low- and high-speed streaks when compared to time-averaged results. Their study also demonstrated that it is on the low-speed components of the flow that are located away from the wall where the flow first breaks down. Although it may be said that DNS has been the most influential technique used to inves- tigate the breakdown to turbulence since the invention of the hotwire probe, the disadvan- tage of DNS is that it is computationally very expensive. For example, the study by Zaki and Durbin (2006) used 26 million grid points and took several months to solve. Currently, DNS studies neglect the leading edge region due to the increased numerical cost; however, it is well known that the leading edge region is critical in the receptivity process and there- fore also in the transition process (Hammerton and Kerschen, 1996; Westin et al., 1998; Kendall, 1998; Saric et al., 2002; Fransson, 2004, discuss the importance of the leading edge in the receptivity process). Therefore, experimental techniques are still a necessary approach to solving many of the complex flow problems, especially one as complex as the transition process. 1.3 Thesis Overview The main aim of this work is to provide new insights into the bypass transition process over a range of turbulence intensities for ZPG flow. In order to do this the subsonic wind tunnel at the University of Limerick is utilised and a range of experimental and data analysis techniques are implemented to gain further insight into the flows development. Chapter 2 details the experimental techniques, facility and freestream turbulence char- acteristics. The experimental techniques section describes the operation of the constant temperature probes and the hotwire system as well as the other measurement techniques used throughout the investigation. The section on the experimental facility includes the 13
  38. 38. design and characterisation of the new flat plate and turbulence grids used in the investi- gation of bypass transition. It is demonstrated that the flow characteristics on the flat plate and downstream of the turbulence generating grids all compare favourably to those in the literature giving confidence in the design of the test facility and also in the various data analysis techniques used. Chapter 3 presents the advanced data analysis techniques used throughout the current investigation. The data analysis techniques are explained fully and this will allow for a better understanding of the interpretation of the measurements throughout the main body of the thesis. This section will include details on the hotwire calibration, the intermittency detector function, the estimation of the energy dissipation in the flow, the determination of the signal level probability and the techniques used to gain further insight into the spectral content of the flow through the fast Fourier transforms and wavelet analysis. The determi- nation of the maximum positive and negative of the streamwise fluctuation velocities are also presented. This chapter will be concluded with an account of the various uncertainties involved in the measurements and data analysis. Chapter 4 presents some important characteristics of pre-transitional flow up to the emergence of turbulent spots near the wall. This chapter discusses the velocity profiles, the Klebanoff mode disturbance profiles and the shear-sheltering phenomena is also illustrated through the velocity traces and energy spectra. Results are presented on the scaling of the peak disturbance associated with the Klebanoff mode over a range of turbulence intensities and Reynolds numbers. The question of whether the receptivity mechanism is linear or non-linear in grid generated turbulence experiments is addressed and comparisons to recent numerical simulations are made. Another important aspect of the flow is the change in structure of the high-frequency turbulent spots as the flow is traversed from the wall to the freestream. This chapter provides an important introduction to some of the flow features and associated terminology that will form the basis of discussion in the results chapters that follow. Chapter 4 also quantifies and provides a new correlation that accounts for the increased energy dissipation rates in a laminar boundary layer when under the influence of elevated FST. The correlation presented has the advantage that it needs only information on the FST level at the leading edge of the plate and also the momentum thickness Reynolds 14
  39. 39. number in order to accurately predict the increased energy dissipation rates to within 10% of those measured. Chapter 5 describes the change in receptivity of pre-transitional flow when under the in- fluence of elevated FST by analysing the skewness function. The location of peak negative skewness is shown to penetrate further into the boundary layer with increased FST level. It was decided to use the location of peak negative skewness as a measure of the change in receptivity of the boundary layer due to the response in the peak skewness level to the change in forcing from the freestream. Once the location of peak skewness was shown to demonstrate the correct parameter dependence an equation is presented that predicts the wall-normal location of the most probable point of the breakdown to turbulent motion. The term most probable is used as this predicts the location where the most negative spike ac- tivity is observed. These results represent novel experimental findings that account for the change in receptivity of a laminar boundary layer when under the influence of elevated FST under naturally occurring disturbance conditions. The importance of the Klebanoff mode disturbance was discussed in Chapter 1 and in more detail in Chapter 4; however, due to the time averaging of the streaky structures in the flow, the Klebanoff mode gives little information in the way of describing which flow structures may propagate and grow to the point of breakdown. For this reason it was de- cided to analyse the time traces with respect to the maximum positive and negative of the streamwise fluctuating velocities and present these in terms of a local disturbance magni- tude when normalised with the freestream velocity. Chapter 6 presents such distributions over a range of Reynolds numbers and FST levels. The results give experimental support to the recent DNS results which demonstrated that the flow breaks down into turbulent motion near the boundary layer edge region on the low-speed streaks. Relationships are presented that accurately predict the maximum disturbances on the low- and high-speed streaks and these relationships also provide a new means of predicting the location of tran- sition onset based on local parameters. These results are further supported by simultaneous measurements between a hotwire probe that is traversed from the wall to the freestream and a hotfilm array at the wall. The results demonstrate that the high-frequency turbulent structures are produced near the boundary layer edge before any near-wall turbulent events 15
  40. 40. are observed and these results indicate the wall-normal size of the leading edge overhang. 1.4 Objectives The main objectives of the current investigation are to: 1) Design and characterise a new flat plate and turbulence grid facility which will allow for comparison against the wealth of experiments in the literature on ZPG flow when under the influence of elevated freestream turbulence. 2) The importance of elevated FST throughout the transition process was highlighted in the introduction. The elevated levels of FST impressed upon the underlying laminar boundary layer must have some measurable effect on the energy dissipation rates in the flow; how- ever, little information exists in the literature to account for this. To this end, the increased energy dissipation rates over a range of turbulence intensities need to be quantified and a correlation developed that accurately predicts this increase with minimum a priori knowl- edge thus allowing for simple implementation into numerical codes. 3) Finally, a major goal of this work is to provide experimental insights into the bypass transition to turbulence and give experimental support to the recent direct numerical simu- lations and theoretical investigations that have enlightened the flow’s development towards the breakdown to turbulence. Experimental support of such advanced theories and simula- tions has been achieved slowly due to the inherent difficulties in measuring such a stochas- tic process as the breakdown to turbulence. This objective contains two main aspects. Firstly, it is hoped to gain insight into the receptivity phase in which disturbances from the freestream penetrate into the laminar boundary layer and secondly, it is hoped to gain insight into the way streamwise disturbances evolve to the point of breakdown. Following these novel measurements it will be possible to provide new relationships that accurately predict the change in flow structure when under the influence of elevated FST. 16
  41. 41. Chapter 2 Experimental Techniques, Facility and Freestream Turbulence Characteristics 2.1 Introduction to Measurement Techniques 2.1.1 Hotfilm and Hotwire Anemometry For almost 100 years hotwire anemometry has been the main research tool in the field of fluid dynamics. One early example using hotwire anemometry which is cited extensively in the literature is Dryden (1937). It is fair to say that the majority of experimental studies into boundary layer phenomena owe their success to the hotwire or hotfilm probe (see fig- ure 2.1). It was not until relatively recently that new techniques have been put to good use, such as particle image velocimetry (PIV) and Laser Doppler Anemometry (LDA), however, when compared to the advantages of hotwire anemometry these optical techniques suffer greatly. The main advantages of hotwire anemometry are (Bruun, 1995): • A very broad range of velocities can be measured from 0.3m/s to supersonic speeds. • Standard hotwire probes have very high temporal resolution. This implies that these probes can measure fluctuations in the flow in the kHz range. 17
  42. 42. • The spatial resolution of the probes is reasonably good with the possibility of mea- suring scales of the order 1mm. • There are various probe types available, such as: single normal (SN), slant, x-wire, triple wire and 4 wire probes which can measure all three components of velocity and may also include temperature measurement. However, as the complexity of the probe increases the size also increases which implies the advantage of good spatial resolution is lost. • The output from the probe can provide instantaneous measurements as well as allow for the calculation of higher order moments which give insightful information into the flow characteristics. The two types of probe used in the current investigation are the SN hotwire probe and the hotfilm probe shown in figure 2.1. The hotwire probe is shown in figure 2.1(a) and this probe is usually mounted to some form of traverse system that allows the probe to move in set increments from the wall to the freestream. The single hotfilm probe is shown in figure 2.1(b) and this probe is typically attached to the wall. An array of hotfilms was also used in the current investigation and is detailed in section 2.1.6. (a) Ceramic probe support Stainless steel prongs Electrical connections for probe holder 5µm Tungsten wire (b) 16 0.1mm dia. Cu-wire Length 55mm 8 0.9 0.1 Figure 2.1: Sketches of constant temperature anemometer probes (DantecDynamics, 2006). (a) Hotwire probe. (b) Hotfilm probe. 18
  43. 43. R1 R2 Servo amplifier E R3 Bridge voltage Probe (Rw) Figure 2.2: Simple schematic of Wheatstone bridge used to maintain probes in constant temperature mode. 2.1.2 Principle of Operation The basic principle behind the operation of a hotwire probe is that there is a cooling effect of a flow on a heated body. A Constant Temperature Anemometer (CTA) system basically consists of a probe, an electronic feedback circuit which drives the probe and produces an output voltage that is proportional to the velocity, and a signal conditioning circuit which provides noise reduction and signal amplification. Shown in figure 2.2 is a simple rep- resentation of the Wheatstone bridge that is used to keep the hotwire probe in constant temperature mode. The bridge should be almost balanced at room temperature. The wire (represented by Rw in figure 2.2) is connected to one arm of the bridge and is heated by an electric current. The servo amplifier holds the bridge in balance by continuously control- ling the current to the sensor. This control of current keeps the sensor operating at constant resistance and therefore also at constant temperature. Point E in figure 2.2 represents the bridge voltage and is a measure of the heat transfer from the sensor which can be directly related to a velocity through a calibration procedure. Bruun (1995) states that the sensitivity of the hotfilm or wire probe to fluctuations in the flow is dependent on the overheat ratio at which the hotwire is operated; however, the probe can be destroyed if the overheat ratio is too high due to oxidation effects. The value at which the variable resistor on the CTA bridge is set to give the desired sensor overheat ratio may be evaluated using the following equation 19
  44. 44. Rw = R20 [1 + α20 (Tw − T20 )]. (2.1) In the current investigation the overheat temperature of the hotwire and hotfilm probes was 250◦ C and 110◦ C, respectively. 2.1.3 Hotwire Calibration Each hotwire probe needs careful calibration in order to be sure that the quantities measured have the desired accuracy. This is especially the case when taking near-wall measurements as the accuracy of the measurement decreases due to the reduction in velocity as the wall is approached. In the current investigation a single normal hotwire is used extensively and the following calibration procedure is outlined for this probe type. Bruun (1995) states that a hotwire probe should be calibrated in situ, i.e., in the environ- ment that the actual measurements will be taken. The flow in which the probe is calibrated must also be of very low turbulence intensity and the probe must be placed perpendicular to the flow. It was explained previously that the basic operation of a hotwire probe is that the increase in power needed to maintain the bridge at constant temperature can be related to the fluid velocity. Therefore, in order to relate the increased heat transfer from the sensor to the velocity of the flow the probe must be calibrated in a known velocity field. This is achieved by placing the hotwire probe in the wind tunnel test section in close proximity to a Pitot-static probe, but not close enough that any interference effects are present. Initially the voltage across the Wheatstone bridge is measured without any flow over the sensor. This is referred to as the zero voltage (E0 ). At each power setting on the wind tunnel the differential pressure from the Pitot-static probe, the temperature of the airflow in the tunnel and the output voltage of the hotwire probe are measured for a range of flow velocities. The calibration of the probe is carried out according to King’s law which provides for a linear relationship between the Wheatstone bridge output voltage and the fluid velocity of the flow and is given by 20
  45. 45. 16 y=7.07x − 0.83 R2 = 0.99 E2−E2 (Volts2) 12 0 8 c 4 0 0 0.5 1 1.5 2 2.5 Ren d Figure 2.3: Hotwire calibration using King’s law. , Measured data points. ——, Trend line fit to data points. The equation of the trend line gives the coefficients in King’s law with A = −0.83 and B = 7.075 and n = 0.45. Es − E0 = A + BRen . 2 2 d (2.2) Here, Es is the voltage sensed by the hotwire when there is a flow passing over the probe, A, B, Red are the calibration constants and the Reynolds number of the flow based on the diameter of the hotwire (d) and n is an exponent that is recommended to be in the range 0.4-0.5 (Bradshaw, 1971; Bruun, 1995). In the current investigation the hotwire calibration was typically obtained between test velocities of 0.4m/s-20m/s. Variation in fluid temperature above the temperature at which the probe was calibrated was compensated for by using the technique of Kavence and Oka (1973) given by 0.5 TS − T∞ Ec = Es . (2.3) TS − TR Shown in figure 2.3 is a typical calibration graph obtained during the current set of experiments. The calibration constants can be read directly from the graph by relating the equation of the trend line fit to the measured data points to King’s law given by Eqn. 2.2, 21

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