UNIVERSITY OF MINNESOTA. This is to certify that I have examined this bound copy of a doctoral thesis by. Bharathram Ganapathisubramani

Transcription

1 UNIVERSITY OF MINNESOTA This is to certify that I have examined this bound copy of a doctoral thesis by Bharathram Ganapathisubramani and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made. Ellen K Longmire Name of Faculty Adviser Signature of Faculty Adviser Date GRADUATE SCHOOL

2 Investigation of turbulent boundary layer structure using stereoscopic particle image velocimetry A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Bharathram Ganapathisubramani IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Ellen K Longmire, Adviser October 2004

3 c Bharathram Ganapathisubramani October 2004

4 Acknowledgments First and foremost, I express my sincere gratitude to my advisers Dr. Ellen Longmire and Dr. Ivan Marusic for giving me the opportunity to conduct this research. The guidance and support they provided during the course of this work is deeply appreciated. Their professionalism and wisdom has steered my professional development over the past five years and my growth as a scientist. I consider myself very fortunate to have been able to work with them. I am really indebted to William Hambleton, Dr. Nicholas Hutchins and Pramod Subbareddy for all their help and support without which this work would not have been possible. I will never forget the discussions over coffee and lunch breaks or the fruitless nights spent to get the Stereo PIV system working in the wind tunnel facility. I would also like to thank and acknowledge: The department of Aerospace Engineering and Mechanics for providing me the opportunity to attend graduate school in the University of Minnesota. The Graduate school of University of Minnesota and Doctoral dissertation fellowship program for the financial assistance. Arnauld Loyer for all the help with the stereoscopic PIV measurements in the jet flow facility. Dave Hultman, Steve Nunally and the workshop staff for their expertise in the machine shop and the willingness to help. The rest of my final examining committee, Dr. Paul Strykowski and Dr. Graham Candler for their helpful critique and counsel. My past and present colleagues, Daniel Khalitov, Ilija Milosevic, Zulfaa M Kassim, Gary Kunkel, Weston Heuer, Tikeshwar Naik, Stamatios Pothos, Aizaz Bhuiyan, David Kubat and Shirin Salber for their friendship. They were always willing to lend any support in the time of need. Ross Wagnild & Rayna DeMaster for all their help during summer 2004 and Josh Sharpe-Stirewalt & Sordy Muor for their help during summer i

5 All my friends for the fun, games and camaraderie over the course of my stay here in Minnesota. In particular, special thanks to Venu, Chari, Ramnath, Rahul and Bhramaji for all those late-night board-game sessions. Finally, I would like to thank my entire family, specially my parents, grandparents and my sister for their love and support over the course of my life and their patience over the last five years. I am eternally indebted to Priti for her unwavering friendship, support and love. This work was supported by the National Science Foundation under grants ACI , CTS and CTS and by the David and Lucile Packard Foundation. ii

6 This dissertation is dedicated to my grandparents and their unending love, support and inspiration. iii

7 Abstract The focus of this study is on understanding the dynamics of a zero-pressure-gradient turbulent boundary layer over a flat plate. The primary objective is to relate production of turbulence to vortex structures and study the cause and effect relationship between vortex structures and Reynolds shear stress. Stereoscopic particle image velocimetry was employed to obtain detailed measurements of the fluid flow in a wind tunnel. The vector fields in the logarithmic layer reveal signatures of vortex packets similar to those found by Adrian and co-workers in their PIV experiments. Groups of legs of hairpin vortices appear to be coherently arranged along the streamwise direction. These regions also generate substantial Reynolds shear stress ( uw), sometimes as high as 40Uτ 2. An objective feature extraction algorithm was developed to automate the identification and characterization of these packets of hairpin vortices. Hairpin packets contained anywhere between 2-10 hairpin vortices and sometimes were found to span across the entire vector field (> 2δ). Identified packets contribute close to 30% of the total Reynolds shear stress while occupying less than 5% of the total area in the log layer. Beyond the log layer, the spatial organization into packets breaks down. Instead, large individual vortex cores and spanwise strips of positive and negative wall-normal velocity are observed. Dual-plane PIV experiments were performed at two wall-normal locations to obtain all components of the velocity gradient tensor. The availability of the complete gradient tensor aid improved identification of vortex cores, determination of their orientation and their relationship to turbulence production. Inclination angles of vortex cores were computed using statistical tools (two-point correlations, joint p.d.f.) as well as instantaneous fields. The results indicate that most vortex cores are inclined in the downstream direction, however a small percentage of the cores are inclined backwards. The ratio of the number of forward to backward leaning cores decreases away from the wall; however the number density of backward leaning cores remains relatively a constant. A hypothetical model to represent the structure of the boundary layer is proposed that includes forward-leaning and backward-leaning vortex cores. iv

8 Contents 1 Introduction Motivation Previous work Visualization and Measurement: Instantaneous Structure Statistical Analysis of Measurements: Average Structure Objectives and Approach Experimental methods and facilities Wind tunnel facility Boundary layer characteristics Constant experimental conditions Trip wire Zero-Pressure-Gradient (ZPG) Conditions Wall shear stress - Skin friction Integral parameters Wake parameters Particle Image Velocimetry v

9 2.3.1 Flow tracers Cross correlation Stereoscopic setup Translation method Rotation or Angular displacement method Stereo calibration Lasers and laser timing Laser sheet optics Cameras and timing PIV Processing Interrogation Recursive interrogation Validation and Three dimensional field computation Uncertainty analysis of Stereo PIV in wind tunnel Dual-plane PIV: velocity gradient measurements Principle Experimental setup Timing configuration Laser sheet characteristics: determination of sheet separation Post Processing Uncertainty in computed gradients Results and discussion: stereoscopic PIV Mean and R.M.S statistics vi

10 3.2 Statistical analysis Instantaneous fields Logarithmic region Outer region Hairpin packets: Statistics Hairpin packets: Reynolds shear stress contributions Conclusions Results and Discussion: dual-plane PIV Mean and R.M.S statistics Instantaneous results Relationship between Hairpin-type vortices and turbulence production Hairpin structural features Statistical results Instantaneous results Possible interpretation Conclusions Summary and recommendations for future work 116 A Velocity interpolation for dual-plane data 119 B Velocity gradient computations 121 B.1 In-plane velocity gradients B.2 Out-of-plane gradients vii

11 C Uncertainty analysis for velocity gradients 123 D Two-point correlations 126 E Swirl strength calculations 128 E.1 Three-dimensional swirl strength E.2 Two-dimensional swirl strength F Feature extraction: Hairpin packets 132 G Feature Extraction: Vortex angles 135 viii

12 Nomenclature Primary flow parameters ν ρ U U V W x y z Coefficient of Kinematic viscosity Density of air Freestream streamwise velocity component Streamwise velocity component Spanwise velocity component Wall-normal velocity component Streamwise direction Spanwise direction Wall-normal direction Derived flow parameters x y z δ Π τ w = µ U z Streamwise separation Spanwise separation Wall-normal separation Displacement thickness Coles wake parameter Wall shear stress ix

13 θ C C 1 C p U τ = τw ρ S = U Uτ u v w uw U x U y U z V x V y V z W x W y W z ω x ω y ω z σ u Momentum thickness Bernoulli s constant Coles wake parameter Pressure coefficient Skin friction velocity Mixed scaling Fluctuating streamwise velocity component Fluctuating spanwise velocity component Fluctuating wall-normal velocity component Reynolds shear stress Streamwise velocity gradient in the streamwise direction Streamwise velocity gradient in the spanwise direction Streamwise velocity gradient in the wall-normal direction Spanwise velocity gradient in the streamwise direction Spanwise velocity gradient in the spanwise direction Spanwise velocity gradient in the wall-normal direction Wall-normal velocity gradient in the streamwise direction Wall-normal velocity gradient in the spanwise direction Wall-normal velocity gradient in the wall-normal direction Streamwise vorticity component Spanwise vorticity component Wall-normal vorticity component Root mean square of the fluctuating streamwise velocity component x

14 σ v σ w Root mean square of the fluctuating streamwise velocity component Root mean square of the fluctuating wall-normal velocity component σ ωx Root mean square of the fluctuating streamwise vorticity component σ ωy Root mean square of the fluctuating spanwise vorticity component σ ωz Root mean square of the fluctuating wall-normal vorticity component δu δv δw λ 2D λ 3D Ξ P i = uw U z R AB l y Uncertainty in streamwise velocity component Uncertainty in spanwise velocity component Uncertainty in wall-normal velocity component Swirl strength computed using 2-D velocity gradient tensor Swirl strength computed using complete velocity gradient tensor Streamwise zero-crossing parameter of any component Turbulence production Cross-correlation coefficient between two variables A and B Spanwise length-scale computed from two-point correlations Superscripts + Indicates viscous scaling. Also referred to inner- or wall-scaling. The variables are non-dimensionalized using U τ and ν. Camera/Experiment related parameters α t X Y U p Stereoscopic camera angle Time separation between the two laser pulses Horizontal direction in the PIV images Vertical direction in the PIV images Horizontal pixel displacement in any PIV result xi

15 V p dx left dx right dy left dy right dx f dy f dz f Vertical pixel displacement in any PIV result Horizontal pixel displacement in the left camera Horizontal pixel displacement in the right camera Vertical pixel displacement in the left camera Vertical pixel displacement in the right camera Actual fluid displacement in the x direction Actual fluid displacement in the y direction Actual fluid displacement in the z direction Miscellaneous d g Separation between points 1 and 2 d s In-plane grid spacing = x = y q 1 A specific velocity component at point 1 q 2 A specific velocity component at point 2 K Any velocity gradient δd g δk δk ip δk op Uncertainty in separation d g Uncertainty in any velocity gradient (K) Uncertainty in any in-plane velocity gradient Uncertainty in any out-of-plane velocity gradient δq 1 Uncertainty in any velocity component q 1 δq 2 Uncertainty in any velocity component q 2 ɛ xy ɛ xz Angle made by the line joining the origin and the highest negative covariance peak in ω x ω y plane with positive y axis Angle made by the line joining the origin and the highest negative covariance peak in ω x ω z plane with positive x axis xii

16 ɛ yz γ xy γ xz γ yz θ e Angle made by the line joining the origin and the highest negative covariance peak in ω y ω z plane with positive y axis Angle made by the line joining the origin and the highest positive covariance peak in ω x ω y plane with positive y axis Angle made by the line joining the origin and the highest positive covariance peak in ω x ω z plane with positive x axis Angle made by the line joining the origin and the highest positive covariance peak in ω x ω z plane with positive y axis Angle made by the vorticity vector with the streamwise-spanwise plane θ i = θ xz The angles in 1st and 2nd quadrants and 3rd and 4th quadrants of θ xz are grouped together θ xy θ xz θ yz Angle made by the projection of the vorticity vector in streamwisespanwise plane with streamwise direction Angle made by the projection of the vorticity vector in streamwisewall-normal plane with streamwise direction Angle made by the projection of the vorticity vector in spanwise-wallnormal plane with spanwise direction xiii

17 List of Tables 1.1 Wall-normal extent of various regimes in a turbulent boundary layer Quadrant splitting of Reynolds shear stress Turbulent boundary layer parameters Laser sheet thickness and pulse separation streamwise-spanwise plane measurement parameters Interrogation window parameters and offsets Vector field validation parameters Measurement volume and resolution parameters Mean and standard deviation of the artificial PIV object measurements Dual-plane experiments. Timing parameters for the experiments at z + = Dual-plane experiments. Timing parameters for the experiments at z/δ = Dual-plane experiments. Laser sheet thickness and pulse separation Dual-plane experiment: Measurement parameters Dual-plane experiments: Interrogation window parameters and offsets Dual-plane experiments: Vector field validation parameters xiv

18 2.14 Dual-plane experiments: Measurement volume and resolution parameters Uncertainties in velocity gradients at z + = 110. K is any gradient, σ K is the r.m.s of K, δk is the uncertainty in the gradient with value σ K and δk is the uncertainty in the gradient with value 2σ K Ensemble-averaged flow statistics. σ ωz is the r.m.s (root mean square) of the fluctuating wall normal vorticity Ensemble averaged flow mean and r.m.s statistics from dual-plane datasets. σ ω + x, σ ω + y, σ ω + z are the r.m.s (root mean square) of the fluctuating vorticity components Definitions of the angles computed form the covariance plots. Angles corresponding to two highest magnitude (positive and negative) covariance values are calculated Definitions of the projection angles in the three planes C.1 Uncertainties in velocity gradients. K is any gradient and δk is the uncertainty in the gradient. Please note that uncertainty in W depends only on uncertainties in U V z and, as continuity equation is x y used to compute to W z xv

19 List of Figures 1.1 Hairpin vortex as proposed by Theodorsen (1952). Figure reproduced from Hutchins (2003) Example of inclined hairpin structures forming regular features with interface inclined at approximately 20 to the surface (a) schematic and (b) photograph taken at Re θ = Flow is from right to left. Reproduced from Head & Bandyopadhyay (1981) Vortices identified in the simulated channel flow. Figure reproduced from Zhou et al. (1999) Four hairpin vortex signatures aligned in the streamwise direction at Re τ = 355. Velocity vectors are viewed in a frame of reference moving at U c = 0.8U. From Adrian et al. (2000b) (a) Uniform momentum zones (i), (ii) and (iii), with vortices A, B and C convecting slower than D, (b) conceptual scenario of nested packets of hairpins growing up from the wall. Reproduced from Adrian et al. (2000b) Wind tunnel facility Zero pressure gradient - C p as a function of station X, where the origin is the location of measurements Law of the wall and the data fitted to it. The circles and asterisks show the velocity profiles for the stereoscopic and dual-plane experiments respectively xvi

20 2.4 (a) Rack of Laskin nozzles for seeding the wind tunnel, (b) Closer look into a single Laskin nozzle, (c) Design drawings of a nozzle Cross correlation images and correlation function Uncertainty in classical PIV. x is the actual displacement of the particle, x is the projected displacement on the laser sheet and z is the out of plane displacement that is lost in PIV. X is the displacement recorded on the image plane (a) Translation method, (b) Angular rotation method Scheimpflug condition Calibration target (a) Top view shows the calibration markers in two planes, (b) Side view shows the mirror slots that is used for laser sheet alignment (a) Calibration target, (b) Identified Markers (a) Equation grid lines, (b) Camera fields of view and common area in which the final vector field is computed is the middle square Laser sheet optics including spherical and cylindrical lenses Stereoscopic arrangement Timing diagram of the laser and camera synchronization Adaptive interrogation technique. All interrogation boxes for the window computation are offset by V o. The four windows within any given window are offset by 0.5V in the first frame and 0.5V in the second frame, where V is the displacement computed in the window (a) Perspective view (b) side view of the experimental setup. H1 and H2 are linear polarization filters oriented to allow passage of horizontally polarized light. V1 and V2 are linear polarization filters that allow passage of vertically polarized light xvii

21 2.17 Top view of laser setup. V and H designate the direction of vertical and horizontal polarization respectively Timing diagram. The BNC box at the top is the master of the timing circuit. The timing of the TSI boxes can be changed by altering the integer n. In this experiment n was set at (a) Wall-normal vorticity (ω z + ) and (b) uw + at z + = (a) R uu correlation at z + = 92, (b) R uu correlation at z/δ = 0.5. Contour levels for R uu range from -0.1 to 1.0 with spacing of 0.1. Zero contours are not shown Dotted-line shows the difference between p.d.f. of positive and negative u of a certain streamwise length. x + is the streamwise distance over which the quantity remained above or below zero. The bin width chosen was 40 wall-units. The solid line shows a second order fit to the dotted-line R vv & R ww correlations at z + = 92 and z/δ = 0.5 The contour levels range from 0.1 to 1 with increments of (a) R uw correlation at z + = 92, (b) R uw correlation at z/δ = 0.5. The contour levels for R uw range from 0.05 to -0.5 with spacing of Zero contours are not shown p.d.f of positive and negative uw events. The p.d.f. ordinate is in log scale. x + is the streamwise length over which the quantity remains above or below zero R uu, R uw and R ww correlations. For z + = 92, (a) Streamwise direction at y + = 0 and (b) Spanwise direction at x + = 0. For z/δ = 0.5, (c) Streamwise direction at y + = 0 and (d) Spanwise direction at x + = Hairpin packets identified using the feature extraction algorithm. At z + = 92, for clarity only every second vector resolved is shown, (a) ω z +, (b) uw +. The flow is from left to right and the vectors shown have U subtracted xviii

22 3.9 Hairpin packets identified using the feature extraction algorithm. At z + = 150, every third vector resolved is shown, (c) ω z +, (d) uw +. The flow is from left to right and the vectors shown have U subtracted At z + = 198 (z/δ = 0.2), for clarity only every second vector resolved is shown, (a) ω z +, (b)w +. The flow is from left to right and the vectors shown have U subtracted At z + = 530 (z/δ = 0.5), for clarity only every second vector resolved is shown, (a) ω z +, (b)w +. The flow is from left to right and the vectors shown have U subtracted Length and width statistics of packets found by feature extraction algorithm. Total number of patches identified (N p ) at z + = 92, 150 and 198 are 1940, 3040 and 380 respectively (a) Typical uw + signature showing different events at z + = 92, (b) % contribution to uw +. Measuring field of view length for z + = 150 was twice that used for z + = Fluctuating streamwise velocity (u + ) contours at (a) z + = 110; U + 1 = 16.04, (b) z + = 130;U + 2 = Various quantities at z + = 110 (a) ω + z, (b) ω + x. The dark line in the plots is the envelope of a low-speed region identified as a hairpin packet by the algorithm described in appendix F Various quantities at z + = 110 (a) λ + 2D and (b) λ+ 3D. The dark line in the plots is the envelope of a low speed region identified as a hairpin packet by the algorithm described in appendix F Various quantities at z/δ = 0.5 (a) λ + 3D and (b) λ+ 2D. A box is drawn around the region identified as a spanwise hairpin head Various quantities at z/δ = 0.5 (a) ω + x, (b)ω + z, (c) ω + y and (d) w +. A box is drawn around the region identified as a spanwise hairpin head. 90 xix

23 4.6 Various quantities at z + = 110. (a) uw +, (b) Instantaneous production, P i = uw + U + z +. The dark line in the plots is the envelope of a low-speed region identified as a hairpin packet by the algorithm described in appendix F Two-point correlations of uw + ( U + / z + ) with (a) λ + 2D and (b)λ+ 3D. Contour levels are marked in the figures (a) JPDFs and (b) Covariance integrands of (ω x, ω y ) at z + = 110 and z/δ = 0.53, left to right. The contour increment for (a) is 10 and the first level shown is equal to the increment. The contour increment for (b) is ±0.01 and the first level shown is equal to the increment. Negative contours are shown with dotted lines and the zero contour is not shown (a) JPDFs and (b) Covariance integrands of (ω y, ω z ) at z + = 110 and z/δ = 0.53, left to right. The contour increment for (a) is 10 and the first level shown is equal to the increment. The contour increment for (b) is ±0.01 and the first level shown is equal to the increment. Negative contours are shown with dotted lines and the zero contour is not shown (a) JPDFs and (b) Covariance integrands of (ω x, ω z ) at z + = 110 and z/δ = 0.53, left to right. The contour increment for (a) is 10 and the first level shown is equal to the increment. The contour increment for (b) is ±0.01 and the first level shown is equal to the increment. Negative contours are shown with dotted lines and the zero contour is not shown Two-point auto correlations of λ 3D at z + = 110 and z/δ = The contour levels are 0.1 to 1.0 at increments of Two-point auto correlations of λ 3D separated into four quadrants according to ω x and ω z. (a)z + = 110 (b) z/δ = The contour increments are 0.1 and the outermost contour is equal to the increment.101 xx

24 4.13 Hypothetical model of an average vortex in a turbulent boundary layer constructed based on two-point correlations of swirl strength and vorticity JPDF and covariances A schematic of the wall-wake model reproduced from Perry & Marusic (1995). The structures that exist in the log region (Type-A) extend out to δ. Additional structures (Type-B) of the form shown are added in the outer wake region p.d.f. of inclination angle (θ e ) at z + = p.d.f. of (a) θ e and (b) Joint p.d.f. of inclination angle and λ 2D /λ 3D at z + = 110 (top) and z/δ = 0.53 (bottom), the dotted line is the function λ 2D /λ 3D = sin θ e p.d.f. of (a) θ xy (b) θ yz and (c) θ xz at z + = 110 and z/δ = Eddy inclination angle (θ i ) (a) p.d.f. and (b) Absolute number density A schematic representation of the structure of a boundary layer in the logarithmic region. (a) Perspective view and (b) Plan view A schematic representation of the structure of a boundary layer in the outer wake region. (a) Perspective view and (b) Plan view A.1 Two cases of grid interpolation. (a) The four nearest neighbors are equidistant, (b) Two of the four points are closer. Grid 1 is shown with solid lines and grid 2 is shown with dashed lines D.1 The number of samples in a correlation map as a function of separation for a 2-D array of size The contours shown range from 0 to with a spacing of G.1 Performance of region growing algorithm. (a) λ 3D (b) Identified regions.136 G.2 Effect of swirl strength threshold on Eddy inclination angle. (a) Probability distribution (b) Number density xxi

25 G.3 Effect of filter width (number of points) on Eddy inclination angle. (a) Probability distribution (b) Number density xxii

26 Chapter 1 Introduction 1.1 Motivation A turbulent flow is three-dimensional and chaotic. The presence of swirling vortex structures (eddies) of various sizes, strength and orientation in the flow makes it complicated and difficult to understand. The presence of turbulence can be observed in the arbitrary motion of a plastic bag in the wind or in the flow around a space shuttle. Most fluid flows in engineering applications are turbulent. Turbulence can aid applications like combustion in engines but it is a deterrent in other applications like oil transport in pipelines. One of the most basic types of turbulent flows is the turbulent boundary layer. At a wall, the fluid velocity must match the velocity of the wall (i.e.zero, for stationary surfaces, this is called the no-slip condition). A boundary layer is the zone over which the average fluid velocity decreases from a freestream value to zero. The presence of a wall takes away fluid momentum, causing drag, a force that opposes the motion in the freestream direction. Various studies over the past century of turbulent boundary layers reveal velocity fluctuations in which slow moving fluid close to the wall is ejected outward and fast moving fluid farther away from the wall is swept inward towards the wall. This phenomenon is the dominant mechanism for the generation and sustenance of turbulence. Generation of turbulence increases drag. This additional drag, also known as the skin friction drag, leads to heavy losses and inefficiency in various applications. For example, the skin friction drag constitutes up to 50% of total drag in aircrafts, 70% in ships and over 90% in gas and oil pipelines. 1

27 It is believed that swirling eddies of various scales are responsible for the mechanism that generates and sustains turbulence. Therefore, it is of primary importance to characterize the eddy structure, in particular the manner in which eddies evolve, regenerate, sustain turbulence and lead to increased skin friction drag. The goal of this study is to understand the physics and basic dynamics that govern these eddies, and hence develop better control and prediction strategies. For example, if we can understand the nature of eddies and predict the location of the regions that generate turbulence, a simple control strategy involving destruction of these regions would help in reducing drag. This study would also be useful in aiding applications like suppression of flow separation and efficient combustion, where turbulence is necessary to achieve the desired results. 1.2 Previous work The literature review is classified into two categories, namely Visualization and Measurements: Instantaneous Structure where the instantaneous variation in velocity over a spatial region is examined, and Statistical Analysis of Measurements: Average Structure where characteristics of large datasets are isolated and analyzed using ensemble averaging techniques. Before discussing the relevant literature, the co-ordinate system used in the detailed review of the literature, in the experimental results and description is as follows, The streamwise direction is along the x axis, the spanwise is y and the wall-normal direction is along the z axis. (U, u), (V, v) and (W, w) are the instantaneous and fluctuating velocity components in the streamwise, spanwise and wall-normal directions respectively Visualization and Measurement: Instantaneous Structure The near wall region in turbulent boundary layers is an area where the viscous contribution dominates the stresses. This layer extends only to very small distance from the wall (usually, z + < 30, where z + = zu τ /ν. z is the wall-normal location, ν is the kinematic viscosity and U τ = τ w /ρ is the skin friction velocity. τ w is the wall shear stress and ρ is the density of the fluid). The early work on structure identification in boundary layers, first reported in the 1950s, goes back to flow visualizations performed in the near-wall region. Elongated streaky structures were observed by Hama 2

28 Layer Range Near-wall region z + < 30 Logarithmic region z + > 70 and z/δ < 0.2 Wake region z/δ > 0.2 Table 1.1: Wall-normal extent of various regimes in a turbulent boundary layer. et al. (1957). A series of studies followed that concentrated largely on this near-wall region (for example, Kline et al and Corino & Brodkey 1967) and described regions of sweeps (u > 0, w < 0, where fast fluid away from the wall is pushed towards the wall, also known as Q4 events, see table 1.2) and ejections (u < 0, w > 0, where slow moving fluid close to the wall is pushed upwards, Q2 events) which were considered the major source of turbulence generation and sustenance. Some researchers (Heist, Hanratty & Na 2000; Lyons, Hanratty & McLaughlin 1989) have proposed a model based on the existence of long quasi-streamwise vortices. Jimenez & Pinnelli (1999) performed numerical experiments on modified turbulent channels and used them to differentiate between several possible regeneration cycles for the turbulent fluctuations in wall-bounded flows. They presented a cycle, which is local to the near-wall region and independent of the outer flow. This cycle involved the formation of streamwise low and high velocity streaks from advection of the mean profile by streamwise vortices, and the generation of vortices from the instability of the streaks. They also found that interrupting these processes led to laminarization, displaying their importance to sustaining turbulence. Schoppa & Hussain (2002) performed direct numerical simulations (DNS) in a channel flow and analyzed the data using linear perturbation theory. Their analysis revealed that the transient growth of perturbations on low-speed streaks led to the formation of streamwise vortices. They concluded that generation of streamwise vortices from low-speed streaks possibly plays a dominant role in sustaining turbulence. The region outside the near-wall regime, z + > 70 and z/δ < 0.2 (where δ is the boundary layer thickness) is called the logarithmic layer. This is a region where the turbulence-related stresses dominate the viscous stress. It is also believed that this region of a boundary layer plays a major role in generation and sustenance of turbulence. Studies also show that the mean velocity profiles of different boundary layers (under different conditions) seem to collapse to a single universal logarithmic curve in this region (hence the name). As a result of this finding, it is believed that the dynamics of generation of turbulence 3

29 u² head z y z y x x u² neck / shoulders Q4 Q2 legs Q4 u z Figure 1.1: Hairpin vortex as proposed by Theodorsen (1952). Figure reproduced from Hutchins (2003). could be universal and similar for different types of boundary layers. Therefore it is of primary importance to study this region of the boundary layer. The region beyond the log-layer is called the wake region, and its structure depends on the flow conditions. Details of the extent of various regions in a boundary layer are given in Table 1.1. In our study, the focus is on the dynamics in the log region and beyond. Most early work in the field did not provide a model to explain the various experimental findings. A physical explanation for the existence of sweeps and ejections beyond the nearwall region was sought. In a landmark paper, Theodorsen (1952) proposed the first structural model to represent a turbulent boundary layer. Theodorsen proposed the existence of a hairpin-shaped vortical structure (hairpin or horseshoe vortex, as shown in figure 1.1) responsible for turbulence transport. In the model, the vortices were formed near the wall in low speed streaks and grew outwards with heads inclined downstream at 45. The spanwise dimensions of the vortex were proportional to the distance of the head from the wall. This vortex model was proposed to be the primary structure responsible for turbulence production and dissipation. Perhaps the first experimental evidence of hairpins was found by Offen & Kline (1975). 4

30 Figure 1.2: Example of inclined hairpin structures forming regular features with interface inclined at approximately 20 to the surface (a) schematic and (b) photograph taken at Re θ = Flow is from right to left. Reproduced from Head & Bandyopadhyay (1981). The authors used dye injections from various wall-normal locations to study the effect of low speed streaks near the wall on the dynamics in the outer region. As a result of these experiments, they were able to suggest that: (i) lift-up is associated with a disturbance caused at the wall by a mean motion towards the wall and (ii) such disturbances originate in the logarithmic region and are generated by the interactions of an earlier burst from upstream. Hence, a well-defined regenerative cycle was detected. Ejection leads to sweep leads to ejection and so forth. The authors invoked a model in which hairpin-shaped structures are inclined at an angle with the wall to explain the ejections and sweeps. The induced velocities on either side of the neck and leg and the upstream region of the transverse head of a hairpintype vortex are used to explain the sweep-type motions. Ejection is explained as a local separation caused by the adverse pressure gradient due to the passage of a transverse vortex further out in the boundary layer. Such events were postulated as strong contributors to the Reynolds shear stress within the boundary layer and consequently to surface drag. Offen & Kline also touched on vortex pairings as a possible mechanism for the growth of larger coherent structures in the wake region of a turbulent boundary layer. The presence of inclined hairpin vortices in the log region and wake region of turbulent boundary layers has also been evidenced in visualization experiments. Bandyopadhyay (1980) and Head & Bandyopadhyay (1981) performed visualizations in smoke-filled zero-pressure-gradient boundary layers, illuminated with a laser sheet inclined at angles of 45 and 135 with the principal flow axis. Head and Bandyopadhyay concluded that hairpin-type vortices are a major constituent of the 5

31 turbulent boundary layer for Reynolds numbers in the range 500 < Re θ < (Re θ = U θ/ν, where U is the free stream velocity and θ is the momentum thickness). These hairpin structures seemed to be inclined consistently at a characteristic angle to the wall of 45. It was also found that the vortex structure seemed to increase in aspect ratio (length to height) at higher Reynolds numbers. The fact that these hairpins extended from the wall to the edge of the boundary layer implies a general elongation of hairpin structures with higher Reynolds number. The characteristic shape of the hairpins progressed from loops at very low Re θ, through horseshoes at moderate Re θ, to elongated hairpin type structures at high Reynolds numbers. Head & Bandyopadhyay also proposed that the basic hairpin vortices could themselves be organized into larger coherent groups with a characteristic growth angle of about degrees as shown in figure 1.2. At very low Reynolds number it was noted that the hairpin scales are similar to those of the large-scale features like bulges that are observed to characterize the interface at the edge of a turbulent boundary layer. However, at the highest Reynolds numbers, the hairpins were highly elongated and narrowed, with cross-sectional dimensions much smaller than the larger scales. It was shown that at these Reynolds numbers, hairpins seemed to organize together to form the larger structures. Additional evidence of coherent groups was found in theoretical studies by Smith et al. (1991) and in an experimental investigation by Acarlar & Smith (1987b). These studies showed that, when a single hairpin vortex of sufficient circulation is inserted into a laminar boundary layer, it could spawn a series of smaller hairpin vortices, and that these vortices would travel together in a group with a common convection velocity as shown in figure 1.3. Zhou et al. (1999) simulated the evolution of a hairpin vortex in a low-reynolds number channel flow to investigate further the coherent groups of hairpin vortices (Re τ = U τ h/ν = 180, where U τ is the skin friction velocity and h is the channel half-width). A typical vortex associated with a strong Reynolds shear stress event present in a low speed region was extracted from a DNS database. The initial three-dimensional vortex was extracted by two-point velocity correlation of the velocity field (by linear stochastic estimation) conditionally averaged on a strong Reynolds shear stress event. The result was a symmetrical counter-rotating vortex pair. The precise nature of the extracted vortex depended strongly on the strength and height of the event. This vortex pair was introduced into a clean DNS channel flow (no background turbulence, but with a turbulent mean velocity profile). The introduced vortex pair rapidly 6

32 Figure 1.3: Vortices identified in the simulated channel flow. Figure reproduced from Zhou et al. (1999). forms into a hairpin vortex, with the small initial bridge forming the hairpin head, and two additional counter-rotating vortices projecting horizontally downstream. As the simulation proceeded, the primary vortex pair rapidly grew into an Ω-type vortex (which was termed the primary hairpin vortex - PHV). This is similar to the development found by Acarlar & Smith (1987a,b). The shape of the vortex (Ω shaped) might be a symptom of the very low Reynolds number of the flow as Head & Bandyopadhyay found horseshoe shaped loops at higher Reynolds numbers. As the time was marched forward, additional hairpins formed upstream of the primary vortex (termed secondary and tertiary hairpins, or SHV and THV in Figure 1.3). This again is similar to the findings of experimental visualizations performed by Acarlar & Smith (1987b). There were also signs of formation of additional streamwise vortices beneath lifted portions of the legs. Hence, very rapidly the primary hairpin spawned subsidiary structures to form the coherent group of vortices similar to the arrangement noted by Head & Bandyopadhyay (1981). The formation of the coherent group was found to depend strongly on the strength and location of the initial vortex. It was found that weaker vortices, whilst developing 7

33 Figure 1.4: Four hairpin vortex signatures aligned in the streamwise direction at Re τ = 355. Velocity vectors are viewed in a frame of reference moving at U c = 0.8U. From Adrian et al. (2000b). a primary hairpin vortex, could not spawn the additional secondary and tertiary vortices. One other question raised was the forced symmetry introduced due to the primary vortex. Robinson (1991) showed asymmetric hairpins were an integral part of a turbulent boundary layer (at low Reynolds number). Further tests conducted by Zhou et al. (1999) with an initial asymmetry produced more promising results. This addition of asymmetry reduced streamwise spacing between and increased the rate of spawning of additional vortices. An important conclusion of this study was that, under suitable conditions, an initial hairpin-type vortex can evolve rapidly into a collective group of several hairpins aligned at a mean angle to the wall of 45 and convecting at almost identical velocities as observed by Head & Bandyopadhyay (1981). Perhaps, the first quantitative visual evidence of coherent groups of hairpin vortices was found by Adrian, Meinhart & Tomkins (2000b). The authors performed planar particle image velocimetry (PIV) measurements in streamwise-wall-normal planes (x z planes) in a zero pressure gradient turbulent boundary layer across the Reynolds number range, 355 < Re τ < The authors noted compact regions of vorticity associated with circular streamlines located above and downstream of strong Q2 activity. They also found that the line along the center of strong Q2 activity and the center of the associated vorticity core was roughly inclined at 45 to the wall. The 8

34 Figure 1.5: (a) Uniform momentum zones (i), (ii) and (iii), with vortices A, B and C convecting slower than D, (b) conceptual scenario of nested packets of hairpins growing up from the wall. Reproduced from Adrian et al. (2000b). 9

35 vorticity cores were identified as numerous hairpin vortex heads in the logarithmic and outer wake layers. They also observed many instances where groups of 5-10 vortex heads covering a spatial streamwise distance of 2δ convected with a uniform velocity as seen in figure 1.4 (also noted by Head & Bandyopadhyay 1981). They also noted that the presence of these vortex heads explained the multiple ejections and sweeps found in previous hot-wire measurements (see Bogard & Tiederman 1986, Tardu 1995). The authors termed such a group of hairpin vortices a hairpin packet. The existence of vortices organized within packets can also help explain the long tails in streamwise velocity autocorrelations within the boundary layer observed by various researchers (see Brown & Thomas 1977, Townsend 1976 and Kovasznay, Kibens & Blackwelder 1970). Adrian et al. (2000b) also proposed the existence of a nested arrangement of hairpin packets across the boundary layer based on the existence of layers of uniform momentum zones as seen in figure 1.5. The figure shows regions labeled (i), (ii) and (iii) as three different uniform momentum zones, all comprising groups of hairpins. They postulated that packets of larger hairpin vortices in the outer wake region would skip over packets of smaller hairpin vortices in the log region. The streamwise separation between packets in the outer wake region would be greater than the separation in the log region. An alternate view on the structure of a turbulent boundary layer uses the concept of a superburst (Na et al. 2001). In this model large scale Reynolds-stress-producing events (such as those documented by Kovasznay et al. 1970) are visualized as plumes of low-momentum fluid ejected outward from the wall (see Nagakawa, Na & Hanratty 2003 and Nagakawa & Hanratty 2001). However, as discussed above, Adrian et al. (2000b) have shown that these large uniform-momentum regions, classified as superbursts can occur within hairpin packets Statistical Analysis of Measurements: Average Structure Examples of statistical techniques developed to study the turbulent flow structure include probability density function analysis, two point correlations, variable interval time average (VITA), conditional sampling and averaging, proper orthogonal decomposition (Lumley, Holmes & Berkooz 1996) and linear stochastic estimation (Adrian & Moin 1988). These techniques can all be applied to any experimental (PIV or 10

36 hot-wire) or computational dataset in varying capacities to isolate interesting flow features. The flow visualization studies discussed above were paralleled by more detailed flow measurements made using hot-wire anemometry which sought to investigate further, and expand upon the visually observed flow phenomena. Various researchers have performed extensive two-point correlation studies based on hot-wire data using Taylor s hypothesis. Favre, Gaviglio & Dumas (1957) acquired data at two wall-normal locations simultaneously in a turbulent boundary layer, and computed the transverse space-time double correlation (spanwise in space, streamwise in time). The measurements suggested that the eddy structure had an oblique pattern where the outer parts (farther from the wall) were further downstream than those closer to the wall. Kovasznay et al. (1970) constructed contour plots of space-time correlation of the three velocity components from fixed-probe data at a wall-normal location of z/δ = 0.5. They concluded that the individual bulges or bursts are strongly three-dimensional. They also found that these events are elongated in the streamwise direction and comparatively compact in the spanwise direction. Tritton (1967) found a decreasing trend in the streamwise length with wall-normal distance. Krogstad & Antonia (1994) found long tails (extending beyond 2δ) in the streamwise velocity auto-correlation in the streamwise direction of a turbulent boundary layer. The spanwise length-scales were smaller (less than 0.5δ). However, the spanwise velocity correlations were short (less than 0.5δ) in both streamwise and spanwise directions. Based on simulation data of turbulent channel flow, Moin & Kim (1985) concluded that the two-point correlations strongly support a flow model with hairpin vortices inclined at 45 to the wall in a channel flow. The authors also found that the size of vortices increased with wall-normal location. Wallace, Eckelmann & Brodkey (1972) used two-component measurements of u and w, and calculated Reynolds stresses. The authors conditionally analyzed the Reynolds stress contributions on the basis of the four possible combinations of u and w as shown in Table 1.2, finding that more that 100% of the net value of Reynolds stresses can be accounted for by the ejection and sweep events. At z + = 15 they found that the quadrant 2 and 4 events were each responsible for approximately 70% of the net Reynolds stress value. This was slightly different from Corino & Brodkey (1967) who had found that ejections accounted for 70% of the total Reynolds stress, and had assumed that sweeps would account for the remainder. Also it was noted that 11

37 Quadrant Sign of u Sign of w Sign of uw Type of motion Outward interaction (away from the wall) Ejections inward interaction (towards the wall) Sweep Table 1.2: Quadrant splitting of Reynolds shear stress. at z + = 15 sweeps and ejections were equal contributors to Reynolds shear stress. Below this height, sweeps dominated and above this height, ejections played a major role. Willmarth & Lu (1972), attempted more complicated measurements, again basically looking at Reynolds stress contributions. In terms of quadrant contributions to Reynolds stress, there was some disagreement with the results of Wallace et al. (1972). Again, it was concluded that ejections and sweeps are major contributors to Reynolds stress. However, Lu & Willmarth (1973) found that the second quadrant contribution is 85% larger than the fourth quadrant, as opposed to the 13% difference found by Wallace et al. (1972) at a similar height. This the authors attribute to the differences in Reynolds number between the two studies and the flow geometry used (Wallace et al used a channel flow). It is also possible that the hot-wires used in the higher Reynolds number experiments of Willmarth and Lu had a much larger (non-dimensional) probe length. Willmarth and Lu also made use of a detection probe in the near-wall viscous layer (at z + = 15) measuring streamwise velocity. When this measured value of u(z + = 15) rose above or below a given threshold, a cross-wire further from the wall (nominally z + = 30) was sampled. These measurements allowed them to conclude that a Q4 event at z + = 30 essentially accompanied the velocity at z + = 15 becoming high (and increasing), and that Q2 events occurred when u at (z + = 15) was low. The phase lag between the behavior at z + = 30 and z + = 15 indicates that large contributions to Reynolds stress comes from a forward leaning structure. This pattern is consistent with the convection of a hairpin-type vortex past the measuring station. They also made an interesting measurement where they correlated the streamwise vorticity (ω x ) with streamwise velocity fluctuation u. It was found that when u (z + = 15; y = 0) passed below a negative threshold, the streamwise vorticity ω x was positive at y > 0 12

38 and negative at y < 0. Such anti-symmetric vorticity patterns on either spanwise side of what was previously determined to be an ejection characteristic, led them to conclude that streamwise vorticity is associated with the ejection near the wall. Blackwelder & Kaplan (1976) took a new look at the structure of turbulent boundary layers using the VITA (Variable Interval Time Averaging) technique. This form of conditional analysis built on the preceding results which showed that the bursting process is associated with large streamwise velocity fluctuations. A short window was moved through the signal, over which the local variance was calculated. When this local variance exceeded a given percentage of the long-term value, a detection was activated, and the surrounding signal was added to the ensemble average. Using rakes of probes, aligned in the wall-normal or spanwise direction, Blackwelder and Kaplan were able to analyze the spatial correlation of the burst event. Streamwise correlations revealed that the convection velocity near the wall was greater than the local mean. Blackwelder & Kaplan also used an x-wire located above their detection probe and thus were able to map the conditional wall-normal velocity and the Reynolds stress associated with a VITA detection. They showed that a VITA detection was accompanied by a positive w prior to detection and a negative w after detection. Thus, a Q2 event is expected to be followed by a Q4 Reynolds stress event. Bogard & Tiederman (1986) conducted an evaluation of the various detection algorithms by comparing them with visually observed ejections. A burst event was judged to be occurring when dye-marked sub-layer fluid was observed to move away from the wall. Not surprisingly, the quadrant technique had the best correlation with the visually observed events (in looking for dye moving away from then wall corresponded with Q2 events). However, it was found that VITA could also detect these events provided that the threshold and the window length were reliably chosen. The authors also demonstrated that the trailing edge of an ejection is characterized by negative u and positive w, followed closely in time by positive u and negative v, in other words, a conditional Q2 event followed by Q4. Johansson & Alfredsson (1982) used a spatial counterpart of the VITA technique (VISA) with a DNS database to study the three-dimensionality of sweeps and ejections. Since the database provided the instantaneous three-dimensional flow field surrounding a detection point, a unique snapshot of the detected structure could be constructed. It was found that instantaneous shear layers were prevalent and persis- 13

39 tent features of the near-wall region. They were found to be highly correlated over extended streamwise distances, and important contributors to turbulence production (via Q2 and Q4 events). It was also noted that the VISA ensemble averaged structure bore a strong resemblance to instantaneous structure of a Q2 or Q4 event. Despite the obvious pitfalls of conditional analysis like spatial or temporal filtering as a result of averaging, the ensemble averaged VITA detected structure was demonstrated to be representative of the instantaneous case. Johansson et al. (1983) also highlighted that the ensemble averaging imposes a spanwise symmetry on the flow picture on either side of detection. Instantaneously they demonstrated that this symmetry does not exist. This can be seen in various instantaneous flow fields in their paper, where the vortex organization and the slow and fast moving regions are not arranged symmetrically, even though correlations and other ensemble averages are symmetric. Townsend (1976) showed that a pair of roller eddies inclined at an angle of 30 is consistent with the two-point correlation measurements performed by Grant (1958). He also claimed that attached double-roller eddies are the dominant structures in the turbulent boundary layer. Perry & Chong (1982) formulated a model based on Townsend s attached eddy hypothesis, featuring Λ shaped vortices of different scales inclined at a fixed angle. Later, Perry & Marusic (1995) refined this model to compute second order statistics and found good quantitative agreement with experimental results. However, the agreement in velocity statistics using representative (average) Λ shaped vortices does not necessarily imply that individual instantaneous hairpin vortices are always symmetric. The existence of hairpin structures of various shapes and sizes including asymmetric hairpins in the region above the buffer layer were documented by Robinson (1991) in his analysis of the early low Reynolds number (Re τ = 300) DNS dataset generated by Spalart (1988). Statistical evidence for the presence of hairpin packets is found in Christensen & Adrian (2001). The authors used linear stochastic estimation on PIV data to compute the conditionally averaged velocity field associated with swirling motion. The PIV data was in x z planes, from turbulent channel flow at Re τ = 547 and They found that the conditional structure consisted of a series of hairpin vortex heads along a line inclined at with the wall. This result is consistent with the same group s observations relating to hairpin packets. Marusic (2001) refined a previous model based on the attached eddy hypothesis to include vortex packets and found that packets of eddies could better match the turbulence statistics in the logarithmic 14

40 region, in particular the long tails found in streamwise velocity fluctuation autocorrelations. More recently, Tomkins & Adrian (2003) studied the spanwise growth of structures in a turbulent boundary layer. They obtained PIV measurements in x y planes from the buffer layer to the top of the log layer at Re τ = 426 and They found that the dominant motions are large-scale low-speed regions elongated in the streamwise direction. These regions are bounded by an arrangement of vortices organized in the streamwise direction. The authors also presented statistical evidence based on spectral analysis for spanwise growth, showing that the spanwise length-scale varies linearly with distance from the wall revealing a self-similar growth of an average spanwise structure. However, examination of instantaneous velocity fields indicated no obvious growth of individual structures. They proposed that this scale growth occurs by merging of vortex packets on an eddy-by-eddy basis through a vortex reconnection mechanism. 1.3 Objectives and Approach At the onset of this study, the state of knowledge suggested that turbulent boundary layers consist streamwise vortices in the near-wall region and forward-leaning hairpinshaped vortices in the log region and beyond. Various mechanisms have been proposed for their formation and existence. Physical models based on these hairpin-shaped vortices explain the generation of Reynolds shear stress through sweeps and ejections. Generation of Reynolds shear stress is believed to be a primary source for sustenance of turbulence and increased surface drag. Previous experimental and theoretical studies also showed that hairpin vortices could travel together in coherent groups called hairpin packets. The characteristics of hairpin packets are not thoroughly investigated or quantified, however. Many questions remain unanswered regarding the physical aspects of hairpin packets, geometric nature of individual hairpin eddies and their relative contributions to Reynolds shear stress. The primary objective of this study is to answer some of the questions in order to understand the structure of a turbulent boundary layer in the logarithmic region and beyond. Answers to some or all of the following questions are sought: 15

41 Do hairpin packets exist? What is the typical arrangement, symmetry and orientation of hairpin vortices in a packet? What is the geometric structure of a typical hairpin vortex? What is the contribution of the Reynolds shear stress generated by these packets to the overall drag? Are there any other drag related Reynolds shear stress producing mechanisms? Stereoscopic particle image velocimetry (SPIV), an optical technique that measures all three velocity components of a plane of flow, is employed to obtain the velocities, fluctuations, gradients and stresses in streamwise-spanwise (x y) planes of a zero pressure gradient turbulent boundary layer to study its structure. Data are acquired at various wall-normal locations across the boundary layer with emphasis on the logarithmic region. This enables comparison of the structure of the eddies across the boundary layer. Dual plane Stereo PIV is employed also to acquire data in two planes separated by a small distance simultaneously. This enables us to obtain the complete velocity gradient tensor. With the help of the complete gradient tensor, in addition to the complete Reynolds stress tensor, the orientation of the eddies, their convective acceleration and their contribution to turbulence production can be investigated. This can lend further insight into the dynamics of the structures in a turbulent boundary layer. 16

42 Chapter 2 Experimental methods and facilities 2.1 Wind tunnel facility The experiments in this study were performed in an open-return suction-type boundary layer wind tunnel (manufactured by ELD Inc., Minnesota) shown in figure 2.1. The tunnel is run by a fan that has variable-pitch blades and is powered by a 40 HP motor. The range of speeds in the test section varies from 2 to 50 m/s. The entrance to the wind tunnel is m, after which is a hexagonal honeycomb cell flow straightener. After the honeycomb, the flow passes through three graduated, stainless steel screens. The open area ratio of the screens is approximately 60%. After the screens, the flow enters the three-dimensional 9:1 contraction. After the contraction is the working section. The working section is 1.2 m wide, 4.7 m long, and nominally 0.3 m high. The tunnel floor is comprised of four 1 thick removable aluminum plates that are extremely smooth. At the rear of the test section, the final bottom plate has a glass panel to facilitate PIV measurements. The side walls of the test section are comprised of clear Plexiglas. The final section of the side walls is equipped with glass windows for optical clarity. The speed of the tunnel is controlled manually, both with the pitch of the fan blades, and with a Toshiba H3 variable torque adjustable speed drive. During the experiments, the pitch of the fan blades was held constant, and the frequency of the Toshiba controller was adjusted to change the wind tunnel speed. The roof of the tunnel is adjustable, which allows the user to set the stream- 17

43 18 Figure 2.1: Wind tunnel facility.

44 wise pressure gradient. For these experiments, a nominally zero-pressure-gradient boundary layer was desired, and the height of the ceiling was adjusted accordingly. All measurements were made at a location 3.3 m downstream of a trip-wire spanning the width of the tunnel at the beginning of the test section. The flow in the test section is assumed two-dimensional with freestream turbulence intensity with respect to the mean of 0.1%. 2.2 Boundary layer characteristics For the present study, the wind tunnel is set to produce a canonical nominally zeropressure-gradient (ZPG) turbulent boundary layer Constant experimental conditions Since the experiments were conducted over a relatively large period of time it was necessary to determine a way in which to establish a consistent set of conditions under which all experiments were run. To do this, the Reynolds number per meter (U /ν) was fixed from the first experiment, and an attempt was made to hold it constant throughout the entire experimental process. This was done by setting the freestream velocity such that the desired U /ν was achieved. The variation of U /ν was kept to ±0.1% at the start of all experiments. To account for temperature drift with the operation of the tunnel, the tunnel was allowed to warm up at least several hours prior to each experiment Trip wire The diameter of the trip wire was chosen to ensure a fully turbulent boundary layer at the test position that was neither over stimulated (trip wire diameter too large) nor close to or encountering transition (trip-wire diameter too small). For the experiments used in this study, the diameter of the trip wire was chosen as 1.5 mm. Erm & Joubert (1991) studied the effects of the size of trip-wires on the development of turbulent boundary layers on smooth surfaces. They found that, for a correctly stimulated boundary layer with a freestream velocity of 10 ms 1, the diameter of the trip wire should be 1.2 mm which gives a Reynolds number based on the diameter of the wire of Re d = 770. At a freestream velocity of approximately 9 ms 1, the size of the trip wire used in this study yields Re d 850. This is considered to be acceptable and 19

45 C p Station location: x (mm) Figure 2.2: Zero pressure gradient - C p as a function of station X, where the origin is the location of measurements. agrees with the previous studies of Li (1989) which suggests 600 < Re d < 900, and Marusic (1991) which used a 0.4 mm trip wire for a flow having a freestream velocity of 30 ms Zero-Pressure-Gradient (ZPG) Conditions The wind tunnel is designed such that the pressure gradient of the flow can be changed by adjusting the top surface of the tunnel. The top surface of the tunnel is constructed with separate panels and these panels are adjustable in four different locations along the length. The panels can be moved up or down depending on the required pressure gradient. To ensure ZPG, the top surface must be oriented such that the free stream velocity (U ) remains a constant along the length of the tunnel in spite of the growing boundary layer that would otherwise tend to increase the speed of the free-stream. Holding U a constant is the same as setting the streamwise pressure gradient to zero. This can be proved by applying Bernoulli s equation along a streamline. Bernoulli s equation is applicable in this case because the freestream of the flow is independent of the viscous effects near the wall. Let p be the pressure at any point, U be the free-stream velocity, and ρ the density of the fluid. As the flow does not go through 20

46 any incline, the gravitational potential is a constant. p ρu 2 = C, (2.1) dp dx + ρu du dx = 0 (2.2) dp dx = ρu du dx (2.3) For the streamwise pressure gradient (dp/dx) to be zero along the boundary layer, either U = 0, which is not practical, or the streamwise velocity gradient du /dx has to be zero. The streamwise velocity gradient can be made zero by maintaining U a constant along the length of the tunnel. This is achieved by maintaining the pressure coefficient (C p ) nominally close to zero along the length of the tunnel. The streamwise freestream pressure distribution in terms of the pressure coefficient (C p ) is given by, C p = p p ( ) 2 ref 1 ρu 2 ref 2 = 1 U (2.4) U ref where U ref is the reference freestream velocity at the inlet of the test section and p ref is the reference pressure at the inlet. Differentiating equation 2.4 with respect to streamwise variable x also yields equation 2.3. Practically, ZPG is set by a trial-and-error method where the freestream velocity is measured at different stations along the length of the tunnel using a pitot tube and a pressure transducer. The walls are adjusted so that U remains a constant and the value of C p is maintained within ±0.01 at all stations. The value of C p at various stations along the length of the test section is shown in figure 2.2. The data shows that the boundary layer is subject to a very weak adverse pressure gradient. Near the measuring station, the value of C p is which is very minimal. For the purpose of this study, this boundary layer flow can be considered to be subjected to nominally zero-pressure-gradient conditions Wall shear stress - Skin friction The friction velocity (U τ = τ w /ρ, where τ w is the shear stress at the wall and ρ is the density of air) of the boundary layer was measured using the Clauser chart method (Clauser (1954)). Clauser chart method assumes and relies on the existence 21

47 u + =(1/κ)log(z + ) + A U Logarithmic region z + Figure 2.3: Law of the wall and the data fitted to it. The circles and asterisks show the velocity profiles for the stereoscopic and dual-plane experiments respectively. of a law of the wall. The logarithmic law of the wall, U = 1 [ ] U τ κ ln zuτ + A (2.5) ν is rewritten incorporating the freestream velocity at z = δ, U U τ = 1 [ ] κ ln δuτ + A (2.6) ν giving, U U = U τ κu ln [ ] zu ν + U [ ] τ Uτ ln + U τ A (2.7) κu U U To obtain the skin friction velocity, the mean velocity profile (U/U ) is plotted as a function of ln[zu /ν]. The data in the logarithmic region is least square fitted to equation 2.7 with universal constants κ = 0.42 and A = 5.0. This fit yields a value for U /U τ from which U τ is determined. The skin friction velocity thus obtained is used to calculate U +. 22

48 The mean velocity profile was measured using a pitot tube as well as a boundary layer type hot-wire probe. The details on the working and calibration of the hot-wire probes can be found in Kunkel (2003). The law of the wall in equation 2.5 is shown in figure 2.3, and the data fits the law in the log region that reigns in the range z + > 50 and z < 0.2δ Integral parameters The integral parameters to characterize the turbulent boundary layer are given in table 2.1. The boundary layer thickness was calculated following the integral method in Perry & Li (1990) as, where δ is the displacement thickness, and, δ = C 1 = δ = δ U C 1 U τ (2.8) 0 The momentum thickness is given by, θ = 0 0 U U (1 UU ) dz (2.9) ( U U U τ ) d z δ. (2.10) (1 UU ) dz. (2.11) Wake parameters Based on an extensive survey of existing data, Coles (1956) extended the logarithmic law of the wall in equation 2.5 to be applicable across the entire boundary layer by incorporating a wake function, U = 1 [ ] U τ κ ln zuτ + A + Πκ [ ] ν W. (2.12) zδc Here Π is the wake strength, which depends on the streamwise development of the boundary layer (for example, the pressure gradient), W is the wake function, and δ c is Coles boundary layer thickness which is slightly less than the boundary layer thick- 23

49 Parameter Symbol Stereo PIV Dual-plane experiments experiments Freestream velocity (ms 1 ) U Displacement thickness (mm) δ Momentum thickness (mm) θ Shape factor δ /θ Coles wake parameter Π C Skin friction velocity (ms 1 ) U τ Mixed scaling S = U /U τ Boundary layer thickness (mm) δ Reynolds number based Re θ on momentum thickness Reynolds number based on Re τ skin friction velocity Table 2.1: Turbulent boundary layer parameters ness according to the integral formulation in equation 2.8. Using this formulation, Π is determined from the maximum deviation of the mean velocity data from the law of the wall profile (figure 2.3) which is, ( ) U = 2Π C U τ κ. (2.13) However, this formulation does not have a zero slope at the edge of the boundary layer (z = δ). To account for this, Lewkowicz (1982) proposed a modified universal wake function, using the absolute boundary layer thickness, δ. Furthermore, Jones et al. (2001) found that the use of this formulation resulted in a poorer fit with experimental data. To remedy this, Jones et al. (2001) use a modified form of logarithmic law of the wall and law of the wake. Using this function, the flow parameter Π is again found from the maximum deviation of the law of the wall profile, ( ) U = U τ 288Π 3 κ(12π + 1) 2 (2.14) PIV measurements were made in streamwise-spanwise planes of a turbulent boundary layer at a single Reynolds number at four different wall-normal locations. The boundary layer parameters are given in table 2.1. The tunnel characteristics drifted 24

50 a small amount in the time between the single-plane stereoscopic PIV experiments and the dual-plane experiments. Table 2.1 lists the boundary layer characteristics for both sets of experiments. 2.3 Particle Image Velocimetry Particle Image Velocimetry (PIV) is a member of a broader class of velocity measuring techniques that measure the motion of small, marked regions of a fluid by observing the locations of the markers at two or more times. PIV is an instantaneous, nonintrusive full field flow measuring technique, based on optical recording. This method returns to the fundamental definition of velocity and estimates the local velocity, u from : u( x, t) = x( x,t), where x is the displacement of a marker, located at x at t time t, over a short time interval t. The experimental set-up of a PIV system typically consists of several subsystems. In most applications, tracer particles have to be added to the flow. These particles have to be illuminated in a plane of the flow at least twice within a short time interval. The light scattered by the particles has to be recorded either on a single frame or on a sequence of frames. The displacement of the particle images between the light pulses has to be determined through evaluation of the PIV recordings. The process of velocity measurement by PIV can be divided into the following stages: 1. Seeding the flow with small, passive tracer particles which follow the motion of the fluid; Seed particles are suspended in the fluid to trace the motion of the fluid and provide a recording signal. 2. Illumination of the measurement zone with a two-dimensional pulsing light sheet; A thin slice of the flow field is illuminated by a light-sheet; the illuminated seeding scatters the light. 3. Image capture, using either a photographic camera, a video camera or a CCD camera; The first pulse of the laser captures images of the initial position of seeding particles onto the first frame of the camera. The camera frame is advanced, and the second frame is exposed to the light scattered by the particles from the second pulse of laser light. There are thus two camera images: the 25

51 first showing the initial positions of the seeding particles and the second showing their final positions due to the movement of the flow field. 4. Analysis of the image by dividing it into a number of small interrogation areas and calculating one velocity vector for each interrogation area. 5. Post-processing of the resulting vector map to remove systematic errors, noise and erroneous vectors Flow tracers The particle image velocimetry technique functions on the presumption that small particles can follow the flow. These particles should also scatter the laser light and illuminate the cameras positioned to capture images. The size of the particle that should be used depends on the smallest scale of the flow. It has been found that certain non-toxic liquids like Glycerin-water mixture, Olive oil etc can be atomized and be used a tracer particles in air flows. Glycerin water mixture cannot be used in large scale experiments like a wind tunnel facility used in this study because of the difficulty in generating large amounts of droplets continuously. Olive oil is the most commonly used liquid since oil droplets are edible and not toxic. Therefore olive oil was chosen as the source of flow tracers. The mean size of the particles generally depends on the type of liquids being atomized, but it is also dependent on the conditions under which the oil is atomized. Hence we used Laskin nozzles for generating droplets. The aerosol generator consists of eight closed cylindrical containers with two air inlets and one aerosol outlet (see figure 2.4(a)). One of the air supply pipe- mounted at the top- dips into olive oil inside the container; the pipe is closed at its lower end (see figure 2.4(b)). Four Laskin nozzles, 1 mm in diameter, are equally spaced along the circumference on each pipe. A horizontal circular impactor plate is placed inside the container, so that a small gap of about 2 mm is formed between the plate and the inner wall of the container. The second air inlet and the aerosol outlet are connected directly to the top. A detailed drawing of the nozzle design is shown in figure 2.4(c). Compressed air is applied to the Laskin nozzles which creates air bubbles within the liquid. Due to the shear stress induced by the tiny sonic jets, small droplets are generated and carried inside the bubbles towards the oil surface. Big particles are 26

52 (a) Impactor plate Laskin nozzles (b) (c) Figure 2.4: (a) Rack of Laskin nozzles for seeding the wind tunnel, (b) Closer look into a single Laskin nozzle, (c) Design drawings of a nozzle. 27

53 retained by the impactor plate; small particles escape through the gap and reach the aerosol outlet. The number of particles can be controlled by the pressure on the inlet of the nozzle and by increasing the number of containers. The olive oil droplets generated using the Laskin nozzles have a size of about 1 µm. The important question is if these particles are small enough to capture all the important features of the flow. Considering the motion equation for the particle can be reduced to the acceleration term and the Stokes drag force, the time response of the particle is given by, τ P = ρ P d 2 P 18µ (2.15) where ρ P and d P are the density and the diameter of the particle, and µ is the viscosity of the ambient fluid. Considering our particles, with a mean diameter of 1µm, and a fluid density of about 0.91Kgm 3 (water = 1000 Kgm 3 ) this gives a time response τ P 2.80 µs; i.e. f P Hz (where f P is the frequency response of the particle). It is primarily important that there are no features of the flow that carry significant energy beyond this frequency. A pre-multiplied energy spectrum generated using the velocity data obtained with a hotwire anemometer at the wall normal location of z + = 100 clearly indicated that there is very minimal or virtually no energy present in frequencies greater than f P. This shows that the particles can follow the fluid in its small scale motions, and more precisely are able to give information about the turbulent motion Cross correlation A cross correlation algorithm was used to evaluate the pixel displacements for PIV. The acquired data consists of series of two images (one for each laser pulse) separated by a small time. The seed particles in the flow move a small distance in this time as shown in figure 2.5. Each pixel in the image has an intensity (I) associated with it. If a seed particle is present in an area, that area would look brighter in the image and hence have a higher intensity. The algorithm divides the image into small interrogation areas (of user defined size) and for each of these zones the correlation function is computed to identify an average displacement. This task is performed by superposing the interrogation area in frame 1 onto the corresponding area in frame 2 28

54 Figure 2.5: Cross correlation images and correlation function. and computing the integral shown below. R I1 I 2 ( X, Y ) = I 1 (X, Y )I 2 (X + X, Y + Y )dxdy (2.16) where I 1 is the intensity at a point (X, Y ) of the first image and I 2 is the intensity at a point (X + X, Y + Y ) of the second image. ( X, Y ) are the displacements of the particles in the area. The goal of the algorithm is to extract the ( X, Y ) displacement for maximum correlation R. A Gaussian curve is fitted to the peak and its two neighbors to find a sub-pixel resolved correlation peak. The displacement corresponding to the Gaussian peak of the correlation function is taken to be the final displacement for that interrogation spot Stereoscopic setup 1 In spite of all its advantages, the classical PIV method is only capable of recording the projection of the velocity vector in the plane of the light sheet. The out-of-plane velocity component is lost (figure 2.6)- the vectors are two-dimensional projections on the object plane of the full three-dimensional vectors- while the in-plane components are affected by an unrecoverable error due to the perspective transformation. One method capable of recovering the complete set of velocity components involves 1 Parts of this section appear in Ganapathisubramani et al. (2002), published in Journal of Turbulence, 03, #17 (2002) 29

55 Object plane x x z d o d i Image plane X Figure 2.6: Uncertainty in classical PIV. x is the actual displacement of the particle, x is the projected displacement on the laser sheet and z is the out of plane displacement that is lost in PIV. X is the displacement recorded on the image plane. making an additional PIV recording from a different viewing axis using a second camera. This can be generally referred to as stereoscopic PIV recording. Stereoscopic imaging eliminates the shortcomings mentioned above. The working of stereoscopic PIV utilizes a basic concept from geometry: if the perceived length and width of any three dimensional object is recorded from two different perspectives, it is possible to reconstruct the actual three dimensions of the object using the recorded information. Simultaneous views from two off-axis directions provide sufficient information to extract the out-of-plane component as well as to correct for the errors in the in-plane components. Stereoscopic PIV allows the measurement of three components of velocity in a plane. It only differs from standard 2D PIV by the use of two imaging systems rather than one. It uses two cameras to view a flow field from two perspectives so that the out of plane velocity component can be measured; each camera can 30

56 Object plane Object plane Lens plane Image plane Lens plane Camera #1 Camera #2 Camera #1 Camera #2 Image plane (a) (b) Figure 2.7: (a) Translation method, (b) Angular rotation method. be considered as a standard 2D-PIV measurement set-up. There are two basic configurations in stereoscopic PIV systems. The translation method uses two cameras whose axes are oriented parallel to each other and orthogonal to the light sheet. In the translation method, both cameras view a common area encompassing the symmetry line of the two-camera system, but from different off-axis directions. The second configuration, called the angular-displacement method, uses two cameras whose axes are not parallel, but are rotated inward so that they intersect at the midpoint of the domain to be recorded. These methods are schematically depicted in figures 2.7(a) and 2.7(b) Translation method This method involves off-axis translation of the image plane from the optical axis, parallel to the object plane. This means that the images have no geometric distortion due to the perspective view, and the two views are thus easier for correspondence. The primary drawback of this method is that it is limited to small angles (due to lens vignetting), which means that the out-of-plane measurement is also limited in accuracy Rotation or Angular displacement method In this study, the angular-displacement Stereo PIV method was used, in which the cameras are rotated inwards such that their axes intersect at the midpoint of the domain to be recorded as shown in figure 2.7(b). Since the object plane is not parallel 31

57 Object Plane Camera lens plane Image plane Lens axis Figure 2.8: Scheimpflug condition to the lens plane, there is geometric distortion due to the perspective view. Therefore it becomes more difficult to obtain particle images that are well focused across the image plane. To overcome this problem, the Scheimpflug condition that requires the object, the lens and the image planes to intersect at a common line, is enforced (see figure 2.8). This arrangement, however, introduces a strong perspective distortion and the magnification factor varies across the image plane. These issues can be overcome by calibrating the configuration Stereo calibration The calibration system used, includes a black calibration target with white markers as shown in figure 2.9(a). The target has two planes of marker points such that alternate markers are in different planes. The planes are separated by 1 mm in depth. The target is placed such that the two planes of markers are on either side of the center of the laser sheet. A fiducial point in the center (See figure 2.10(a)) acts as a reference or the origin. The location of this point is an input, and the coordinates of other marker points are computed relative to the fiducial mark. This two-plane target setup allows the computation of the calibration coefficients without traversing the target in the out-of-plane direction. However, the out-of-plane component of the velocity is only first order accurate. The side of the target is fitted with a plate that has a mirror aligned with the centerline between the two planes of the target as shown in figure 2.9(b). The laser 32

58 Figure 2.9: Calibration target (a) Top view shows the calibration markers in two planes, (b) Side view shows the mirror slots that is used for laser sheet alignment. (a) (b) Figure 2.10: (a) Calibration target, (b) Identified Markers. 33

59 sheet is aligned with the target using this mirror such that the sheet illuminates a measurement plane in-between the two planes of calibration. The calibration plate is placed on machined blocks of various heights that elevate the target, parallel with and above the bottom wall of the tunnel to achieve the desired wall-normal location. The laser sheet strikes the mirror slot and reflects back. The optical path of the reflected sheet is made co-planar with the incident sheet to ensure alignment of the laser sheet with the calibration target. Sufficient illumination is provided such that the cameras can clearly focus and image the target. The calibration image analysis identifies the calibration markers in the image and locates the centroid of each. Figure 2.10(b) shows the identified calibration markers (for the left camera) by drawing a box around them, the fiducial mark is highlighted with a diamond. The calibration marker points are sorted into a grid, with each marker point having a row and column grid index. This grid structure, with the fiducial mark used as the base point, is used to match the calibration points recorded on the image to target locations in the object plane (which is the plane of the laser sheet). The result of this image analysis is a calibration points file with a record of the (X, Y ) image pixel location and the target marker (x, y, z) location in the object plane given for each calibration marker. This points file is used to generate the mapping functions shown in equations X left = f 1 (x, y, z) (2.17) Y right = f 2 (x, y, z) (2.18) Y left = f 3 (x, y, z) (2.19) Y right = f 4 (x, y, z), (2.20) where f is the generated mapping function. The mapping functions are also used to find the local magnification for each 3D vector in the left and the right images. The local magnification is found by computing the derivatives of equations with respect to the three spatial directions. The derivatives of the mapping functions represent the gradient of particle displacement (in pixels) in the image plane to the particle displacement (in mm) in the object plane. These gradients show the pixel displacements in the X or Y direction in the image plane that is caused by particle motion in the x, y or z directions in the object plane. Each point in the object plane has a unique set of displacement calibration factors. To locate the velocity vectors in the object plane (plane of the laser sheet), a three-dimensional vector grid 34

60 (a) (b) Figure 2.11: (a) Equation grid lines, (b) Camera fields of view and common area in which the final vector field is computed is the middle square. is created to define the locations where the three-dimensional velocity is desired. After calibration, images are acquired, and an in-plane cross-correlation algorithm computes the relevant pixel displacements for each camera. transformation equations is used to obtain the fluid displacements: X left = dx f ( dx left dx Y left = dx f ( dy left dx X right = dx f ( dx right dx Y right = dx f ( dy right dx ) + dy f( dx left dy The following set of ) + dz f( dx left ) (2.21) dz ) + dy f( dy left dy ) + dz f( dy left dz ) (2.22) ) + dy f( dx right dy ) + dy f( dy right dy ) + dz f ( dx right ) (2.23) dz ) + dz f ( dy right ) (2.24) dz where ( X left, Y left ) and ( X right, Y right ) are the pixel displacements from crosscorrelation based on each camera view, and (dx f, dy f, dz f ) are the unknown fluid displacements in the laser-sheet plane. The coefficients within the parentheses are the displacement gradients obtained by computing the derivatives of mapping functions generated by calibration. This system of four equations with three variables is solved using a least squares error method. The in-plane 2D pixel displacements for the two cameras are computed using the previously-mentioned cross correlation algorithm. 35

61 Experiment z + sheet t name thickness (mm) µs tunnel20apr tunnel6jul tunnel17may tunnel14jun tunnel1feb Table 2.2: Laser sheet thickness and pulse separation. Figure 2.11(a) shows the plot of the equation grid lines which are the grid lines along which the mapping functions are generated. Figure 2.11(b) shows the field of view with respect to the left and the right cameras. The two outer boxes in the figure represent the image area as viewed by the left and the right cameras. The smaller box embedded within, represents the common field of view of the two cameras, and it is this region in which the three components of the velocity are reconstructed. A uniform grid is prescribed in this overlap region and the vectors are computed at each node of the grid Lasers and laser timing A pair of Nd:YAG lasers (Bigsky, CFR 200 series) were used as illumination sources. These lasers emit a pulse energy of 120 mj/pulse with a pulse width of 7 ns, at a repetition rate of 10 Hz. It is important to excite the laser flash lamp at this frequency to maintain the consistency of the laser beam profile. However, the laser Q-switch can be operated at sub harmonics of 10 Hz ( 5, 3.33, 2.5 Hz etc). The timing and the pulse separation of the laser pulses and the cameras are controlled by a TSI synchronizer box, which in turn is controlled by Insight (5.0) software. The optimal timing between the laser pulses was determined by a trial and error method. For large time separations, the seeding particles exit the plane of the laser sheet, and as a result the signal to noise ratio in cross correlation algorithm is lower and fails to perform efficiently. But, at the same time t should not be too small such that the pixel displacements are very small. This will result in limited resolution of the velocity in terms of pixel displacement leading to a vector field with large uncertainties. By using a trial and error method, the value of t was fixed at an optimal value that achieves good dynamic resolution for the velocity but at the same 36

62 Figure 2.12: Laser sheet optics including spherical and cylindrical lenses. time has an acceptable signal to noise ratio in the cross correlation computations. The values of t used in various experiments are given in table Laser sheet optics Figure 2.12 reveals the optical components used to convert the circular laser beam into a planar laser sheet that illuminates the measurement region. The laser beam is focused using a spherical lens of focal length 1.0 m and subsequently passed through a cylindrical lens with a focal length of -30 mm that spreads the beam into a sheet. The laser sheet thus formed illuminates a streamwise-spanwise plane parallel to the bottom wall of the wind tunnel. The thickness of the laser sheet reaches a minimum at the focal length of the spherical lens before it diverges. The sheet thickness values for the various experiments performed at different wall-normal locations are listed in table 2.2. The laser sheet is aligned parallel to the bottom wall, at the desired wall-normal distance using the alignment mirror that is a part of the calibration target. The system is designed such that the laser sheet location can be adjusted vertically by traversing the complete sheet optics setup as shown in figure

63 Figure 2.13: Stereoscopic arrangement Cameras and timing Light scattered by the olive oil droplets is captured by a pair of CCD cameras. A schematic of the camera orientation and the laser sheet alignment is shown in figure Two different sets of cameras were used to acquire data at different wall-normal locations. A pair of Kodak Megaplus ES-1.0 cameras ( pixel resolution, 8 bit gray-scale) were used for data acquisition at z + = 92 and z/δ = 0.2&0.5. A pair of higher resolution TSI powerview cameras ( pixel resolution, 12 bit gray-scale) cameras were used to acquire data at z + = 100&150. Further details on the cameras used in different experiments are given in table 2.3. Nikon Micro Nikkor 60 mm f/2.8 lenses were used with the cameras. The cameras were aligned in a plane parallel with the x y flow plane and inclined at angles of α = 20 with the z axis as shown in figure This stereoscopic angle is critical for the accuracy of the out-of-plane component. Lawson & Wu (1997) found that an optimum performance is achieved in the range 20 < α < 30. However, there are other practical limitations like light sheet intensity and particle light scatter that contribute to the determination of the stereoscopic angle. The timing of the lasers and the cameras is as shown in the diagram in figure The pulse repetition rate specifies the separation between two successive laser pulse sequences, it is also known as the sampling rate as it is the rate at which the sequence of data is acquired. Its maximum value is fixed by the ability of the camera to transfer the images acquired to the frame grabber/hard disk/ram. The Kodak Megaplus and TSI Powerview cameras used in this experiment have a readout time of 33 ms per frame. The total time that should elapse before the next capture is 66 ms (two frames readout time) plus the exposure time of the first frame. 38

64 Experiment z + z # of Cameras field δ name realizations (N t ) used of view tunnel20apr Megaplus 1.2δ 1.2δ tunnel6jul Powerview 2.4δ 2.4δ tunnel17may Powerview 2.4δ 2.4δ tunnel14jun Megaplus 1.2δ 1.2δ tunnel1feb Megaplus 1.2δ 1.2δ Table 2.3: streamwise-spanwise plane measurement parameters. Maximum Pulse Repetition rate Synchronizer TTL camera Trigger Laser Pulses Pulse Delay δ t Camera Exposure Image 1 Image 2 Camera digital Image output Image readout 33 ms 1 Image 2 readout 33 ms Figure 2.14: Timing diagram of the laser and camera synchronization. 39

65 This exposure time was fixed at 255 µs. The sampling rate of the system is a function of the laser operating frequency and the data transfer capability of the computer. The PCI bus (to which the frame grabber cards are connected) is capable of transferring at a maximum rate of 100 MB/sec. Hence there is a limitation on the rate of data acquisition. It was found that the TSI powerview cameras could continuously stream data at only 3.33Hz while the Kodak Megaplus can acquire and transfer data at a much faster rate ( 10Hz at best, due to the limitation to the lasers which can operate only at 10 Hz). Pulse delay (ms) specifies the delay time from the start of a sequence to the first laser pulse. This delay must be optimized such that the first laser pulse is fired within the first exposure and the second laser pulse is fired during the second exposure. This is important for two frames to be unique. If the pulse delay is too small, both lasers will fire during the first exposure and if it is too large, the lasers will fire during the second exposure. The value of pulse delay was optimized and fixed at 0.25 ms. 2.4 PIV Processing Interrogation The two images from the cross correlation CCD cameras are interrogated using a twoframe cross-correlation algorithm with a discrete window offset. The interrogation is carried out using the Insight (5.0) software. The images are first interrogated with a coarse window pixels in size and zero window offset. The mean displacement is calculated, and the window in the second frame is offset by this mean value for all further calculations to increase the quality of the correlation peak Recursive interrogation All final vector fields were computed using the adaptive central-difference technique outlined in Wereley & Meinhart (2001) using Insight 5.0 software. In this algorithm, a coarse vector field with resolution pixels is computed and validated initially (the second window is offset using the information from the preliminary window calculation). The same image is interrogated again with a window. At this stage, the interrogation box in frame 1 is offset upstream, and the box in frame 2 is offset downstream by half the displacement computed in the first pass as shown in figure This technique was found to be more accurate and reduced pixel-locking 40

66 frame 2 Vo Vo 64 x64 window frame 1 32 x 32 window second frame offset by Vo for all interrogation areas frame 2 V/2 V/2 frame 1 16 x 16 window First frame offset by V/2 Second frame offset by V/2 V New displacement V Vn Figure 2.15: Adaptive interrogation technique. All interrogation boxes for the window computation are offset by V o. The four windows within any given window are offset by 0.5V in the first frame and 0.5V in the second frame, where V is the displacement computed in the window. and other bias errors. All images from both the left and right cameras were analyzed using this procedure. The resolution of the vector field, type of interrogation, window size and offsets for all the experiments is given in table Validation and Three dimensional field computation The above procedure yields close to vectors per image set in the Megaplus cameras and vectors in the TSI powerview cameras. These vector fields are then validated to remove any spurious vectors. Usually the erroneous vectors are fewer than 1% of the total number of vectors computed. The computed vector field is first validated using the standard deviation since the flow is homogeneous in the plane. A standard deviation value of 4.0 is used to validate and remove the erroneous 41

67 Experiment z + Overlap First pass Second pass Offset name pixel pixel pixel pixel pixels tunnel20apr % tunnel6jul % tunnel17may % tunnel14jun % tunnel1feb % Table 2.4: Interrogation window parameters and offsets Experiment z + Validation Standard Interpolation neighborhood name engine deviation engine size tunnel20apr 92 Gaussian 4.0 local mean 3 3 tunnel6jul 92 Gaussian 4.0 local mean 3 3 tunnel17may 150 Gaussian 4.0 local mean 3 3 tunnel14jun 198 Gaussian 4.0 local mean 3 3 tunnel1feb 530 Gaussian 4.0 local mean 3 3 Table 2.5: Vector field validation parameters Experiment z + x + y + z + # of name vectors tunnel20apr tunnel6jul tunnel17may tunnel14jun tunnel1feb Table 2.6: Measurement volume and resolution parameters 42

68 vectors in both U p and V p pixel displacements. The holes are interpolated using the local mean of a 3 3 neighborhood. The validation and interpolation parameters of the various datasets are given in table 2.5. The validated vector fields from the left and right cameras in liaison with the calibration coefficients are used to compute all three velocity components as outlined in section The final vector field resolution is given in table 2.6. Velocity gradients were computed using the final vector field as shown in appendix B. Wall-normal vorticity component (ω z ) was calculated using the in-plane velocity gradients Uncertainty analysis of Stereo PIV in wind tunnel There are various sources of uncertainty in any experiment. These uncertainties are broadly classified in the following paragraphs. System disturbance Effects System disturbance effects are induced by the presence of flow tracers in the flow thereby altering the flow conditions. It might also be that the tracers are not following the flow in the first place. But, detailed studies have shown that tracers of certain size and properties indeed follow the flow and do not affect the flow itself. Hence we should be able to find seeding that will not have a large relaxation time, the particle must react to the flow and follow the flow path. Most widely used flow tracer for gas flows is olive oil droplet with a mean diameter of 1 µm, which was found to be a very good flow tracer. The time response of the olive oil particles is found to be of the order of 10 6 secs (see section 2.3.1). System Sensor interaction This is negligible in a PIV analysis, as the response of the sensor is instantaneous and there is generally no information that is lost. The errors associated with the system-sensor interactions are usually neglected. Calibration errors This plays an important role in the stereo PIV technique. Calibration corresponds to more than two thirds of the experiment and one must be really cautious during this process. One must ensure that the two cameras are looking at exactly the same point 43

69 in the target. This can be checked by performing the calibration and overlapping the images from the two cameras and checking if the dots lie on top of one another. There is an error in curve fitting to obtain the calibration coefficients. It must be ensured that the experimental setup is not disturbed by any means between calibration and image acquisition. Calibration is necessary every time the experiment is performed. There is also the error in the final curve fitting to obtain the fluid displacements, and this gives rise to residual pixels. Observational errors These errors are associated with the calibration process. First, we need to align the laser sheet and the calibration target and this purely depends on our ability to observe the reflection of the laser sheet from the mirror aligned with the target and noticing if the sheet retraces its path. Secondly, the focusing of the image is purely dependent on an individual s observational skills. The velocity after uncertainty calculations is written as, f actual = f measured β + δf (2.25) β = Velocity bias (2.26) δf = B 2 o + B P 2 1, (2.27) where f is any velocity component, B o is the zeroth order bias error which is the system bias, B 1 is the first order bias arising due to the error in the least square fit and the Gaussian peak fit. P 1 is the first order precision error (also called Precision limit) is a statistical error and is given by, P 1 = tσ f N where t is a constant obtained from t tables and its value is 1.96 for a large number of samples at 95% confidence level. σ f is the variance of f and N is the number of samples. P 1 is associated with the randomness of the data that cannot be predicted and appears only at readout. Quantifying these errors is not a simple task. The bias errors can be quantified using solid body translations. This is done by taking a solid body and translating it with known displacements and then using Stereo PIV to compute these displacements. 44

70 Bjorkquist (1998) found that the errors associated with the x, y and z components was -0.28%, -0.18% and -0.26% respectively for 1 mm translations using a solid body. To test the performance and accuracy of the stereoscopic PIV system in the wind tunnel facility, an experiment of solid body translations was performed similar to the one performed by Bjorkquist (1998). The TSI Powerview (2k 2k) cameras were placed in stereoscopic arrangement as shown in figure All parameters like focal length of the lenses, field of view and camera angles were the same as in the experiments. The solid body measurements involved moving a sheet of paper with aluminum filings glued randomly over its area. This paper with filings was glued to a Newport XYZ traverse with micrometer adjustments. The aluminum filings paper was used to simulate a high number density of random size seed particles usually seen in a PIV experiment. The object was illuminated with a constant halogen light source. The images acquired were of very good quality and the vector field computed did not require any validation to remove spurious vectors. The artificial PIV object plane was traversed mm (this was done because the traverse was based on FPS units, and 40 divisions on the traverse corresponded to mm movement) in all three directions. This was repeated 10 times to obtain a large sample space for statistical convergence. The images were processed with TSI Insight (5.0) using two frame cross correlation, identical to the algorithm used in processing vector fields in the experimental datasets. The three-dimensional vector field covered an area of 78 mm 80 mm. The statistics from the vector fields are shown in table 2.7. The absolute errors in the displacements are 0.26%, 0.43% and 1.02% in the streamwise, spanwise and wall-normal directions respectively. The r.m.s errors are 0.23%, 0.71% and 1.01% relative to mm displacement in the streamwise, spanwise and wall-normal directions. The r.m.s error, is a system error which includes all possible PIV measurement errors including error in pixel displacement, calibration error, traversing error and the least squares fit error used to compute actual displacement from pixel displacement. This includes all system bias errors shown in equation 2.27 (B 0 and B 1 ) and represents the total uncertainty in the displacements. This result is in qualitative agreement with the uncertainty analysis performed by Prasad & Adrian (1992) for a translation-type stereoscopic arrangement. The authors found relative r.m.s errors of 0.2% in the in-plane components and 0.8% in the out-of-plane 45

71 Mean Standard Traverse (mm) deviation (mm) mm streamwise (x) measured dx measured dy measured dz mm streamwise (x) measured dx measured dy measured dz mm spanwise (y) measured dx measured dy measured dz mm spanwise (y) measured dx measured dy measured dz mm wall-normal (z) measured dx measured dy measured dz mm wall-normal (z) measured dx measured dy measured dz Table 2.7: Mean and standard deviation of the artificial PIV object measurements. 46

72 component. The first order precision error (P 1 ) is extremely small and is negligible since a large number of samples are included in the computation (N = ). It can be noted from the r.m.s values that the maximum uncertainty is in the outof-plane motion. However, this case of greater than 1 mm motion is an extreme one. Usually, the laser sheet thickness is nominally 0.5 mm, and the out-of-plane motion is restricted to within that distance to obtain a high quality vector field. 2.5 Dual-plane PIV: velocity gradient measurements Dual plane PIV is a natural extension of the PIV method in which velocity components are measured in two planes simultaneously. The primary motivation of performing multiple plane PIV is to study the flow field in two different two dimensional planes simultaneously and understand the instantaneous structure of the flow. In turbulent boundary layers, there are three primary applications for multiple plane PIV, to measure acceleration based on PIV vector fields recorded with a time delay (See Christensen 2001), for correlation measurements based on two dimensional spatially separated planes as performed by Kahler (2004), and finally, to perform velocity gradient measurements using two 2-D planes of data, spatially separated by a small distance. Various researchers have used multiple hot-wire probes to measure multiple components of the gradient tensor simultaneously including the nine sensor vorticity probe developed by Balint, Wallace & Vukoslavcevic (1991). Similar point gradient measurements have been made using Laser Doppler Anemometry (LDA)(see e.g.ötügeny, Su & Papadopoulosz 1998). However, the hot-wire and LDA gradient measurements are typically limited to a point in space, and Taylor s hypothesis must be applied to make inferences about spatial flow structures. Standard PIV has been used in various studies to obtain four in-plane components of the velocity gradient tensor over a plane instantaneously. This enabled calculation of one vorticity component (out-of-plane 47

73 vorticity) over a planar field. Stereoscopic PIV and scanning PIV give the out-ofplane velocity component, and hence two more components of the velocity gradient tensor can be determined. However, the in-plane components of the vorticity vector remain unresolved with this method. Holographic particle image velocimetry (HPIV) can provide the complete velocity gradient tensor over a volume (Meng & Hussain 1995 and Zhang et al. 1997). In the in-line and off-axis holographic PIV techniques (for example, see Scherer & Bernal 1997 and Barnhart et al. 1994) acquired holograms are reconstructed numerically and analyzed to provide a fully resolved 3-D velocity field over a volume. Any velocity gradient can then be determined by differentiation. Another method capable of resolving the complete gradient tensor is Dual-plane stereoscopic PIV (DSPIV). This technique, which enables measurement of the complete velocity gradient tensor over a plane, has been employed previously by a number of research groups for various purposes. Kahler (2004) used a polarization-based dual-plane technique to compute correlations between planes with various separations in a turbulent boundary layer. Two independent stereo PIV systems measured three velocity components in two parallel planes, with the two pairs of light sheets having orthogonal polarization. The property of polarization of scattered light was used to image light from the two sheet pairs onto two independent camera pairs. The working principle of this method was based on the concept that light scattered by small particles (diameter < 10µm) possesses the same polarization properties as the incident light. Polarization filters were placed in front of the camera lenses to allow passage of light with specific polarization. Hu et al. (2001) used the same technique to investigate large-scale features in a lobed jet. Mullin & Dahm (2004) employed a frequency-based DSPIV technique to measure the velocity gradient tensor in a turbulent shear flow. Their technique is similar to the above in principle, except that two independent stereo PIV systems measured velocity components in differentially-spaced planes illuminated by pairs of 532 nm and 635 nm laser light sheets. This method thus relied on color filters on the camera lenses to isolate one plane from another. In the present study, a three-camera polarization based dual-plane technique is used to measure the full velocity gradient tensor. The continuity equation is employed in combination with the PIV data to determine the appropriate quantities. The overall objective of this study is to use the three dimensional velocity gradient data to study the geometric structure of vortices and relate them to turbulence production. 48

74 2.5.1 Principle A two system approach is used in this study to compute spatial velocity gradients from two independent velocity vector fields separated by a small distance. Two independent PIV systems are necessary to acquire data in the spatially-separated planes. An independent PIV system includes two laser heads, one or two CCD cameras (two for stereoscopic measurements) and a timing circuit to synchronize the lasers and the cameras. The basic principle of this technique involves making a separate PIV measurement with each PIV system, such that the planes of data from the two systems are separated by a nominal distance ( 1 mm). However, since the measurements are made simultaneously and since the depth of field of camera lenses are greater than 1 mm, the particle images viewed by the cameras of system 1 will also be imaged by the cameras on system 2 and vice-versa. This effect would introduce strong correlated noise into the image pairs. This would render the concept of velocity data in spatially separated planes to compute gradients useless. Therefore, this effect must be removed to accurately measure velocities and its gradients. The solution is to identify a method in which the PIV images for system 1 are independent of those for system 2. There are multiple ways to achieve the desired solution. One is to color code the seed particles. This is achieved by using two different colored light sources for the two light sheets. For example, pairs of Nd:YAG 532 nm (green) lasers and Nd:YAG pumped 635 nm (red) dye lasers can be used in liaison with optical color filters on the cameras to isolate the particle images in two planes as done by Mullin & Dahm (2004). The second method is to make use of the fact that small seed particles maintain the properties of incident light. The scattered light from particles of 10 µm diameter or less retains the polarization of the incident light. This enables coding of particles using the polarization of incident light (Christensen 2001, Raffel et al. 1998, Hu et al. 2001). Hence, the laser pulses for system 1 would be horizontally polarized and the pulses for system 2 would be vertically polarized. With the polarization of the incident light fixed, the cameras of the two systems are fitted with linear polarizing filters with appropriate orientation (horizontally polarized light let through for system 1 and vertically polarized light through for system 2) to separate one plane of particle images from the other. 49

75 2.5.2 Experimental setup Dual-plane PIV experiments were performed in the streamwise-spanwise planes of a turbulent boundary layer. The primary goal for performing these experiments was to resolve the complete velocity gradient tensor which could be used in computing the complete vorticity vector and other quantities like 3D swirl strength (see E) and instantaneous Reynolds shear stress production. Two independent PIV systems capture data simultaneously in neighboring streamwise spanwise planes separated by a small distance as shown in figure System 1 is a stereoscopic system used to provide all three velocity components over a plane illuminated by sheet 1, and system 2 is a single-camera planar PIV system. The single-camera system measures the streamwise-spanwise velocity components in the higher plane illuminated by sheet 2. Simultaneous measurements are performed utilizing the polarization property of the laser light sheets to isolate one plane to one set of cameras (see e.g. Raffel et al. 1998, Hu et al. 2001, Christensen & Adrian 2002). PIV data were captured in the log layer and in the outer wake region of a turbulent boundary layer. System 1 includes two Kodak Megaplus 1k 1k pixel resolution cameras equipped with Nikon 105mm lenses. The stereoscopic angle for the cameras is fixed at 25. The lenses are fitted with linear polarizers (two in series) oriented to allow the passage of horizontally polarized light only. Light sheets are generated by a pair of Spectra- Physics PIV-400 series Nd:YAG lasers (350 mj/pulse) that are horizontally polarized. The laser beams are split using the beamsplitter (CVI-laser) as shown in figure One half (that is horizontally polarized) is used to illuminate the planes for system 1. The other half is passed through a half-wave plate (CVI laser) to rotate the polarization by 90 and is used to illuminate the planes for system 2. System 2 includes one TSI Powerview camera with 2k 2k pixel resolution and a Nikon 50mm lens. The magnification in the single camera was matched (approximately) with the stereo pair by moving the camera away from the object plane. The linear polarizers in this case are oriented to allow the passage of vertically polarized light only. Each laser sheet pair is aligned independently to illuminate a specific wall-normal location. Note that the linear polarizers for the stereoscopic cameras are not parallel to each other. The first polarizer is oriented along a plane parallel to the light sheet and the second polarizer is fitted to the camera lens and its axis is rotated at the stereoscopic angle. This configuration worked the best in eliminating the scattered light from the 50

76 (a) (b) Figure 2.16: (a) Perspective view (b) side view of the experimental setup. H1 and H2 are linear polarization filters oriented to allow passage of horizontally polarized light. V1 and V2 are linear polarization filters that allow passage of vertically polarized light. 51

77 To sheet 1 To sheet 2 Spectra Physics Nd:YAG lasers H H 50/50 non polarizing beam splitter 1/2 wave plate H V V mirror Figure 2.17: Top view of laser setup. V and H designate the direction of vertical and horizontal polarization respectively. second plane. This configuration was arrived at, by a trial and error method and the physical reasoning for this is not completely clear. Preliminary dual-plane experiments were performed with a two pairs of Nd:YAG 120 mj lasers and three TSI powerview 2k 2k cameras. Details of these experiments are given in Ganapathisubramani et al. (2004c) and Ganapathisubramani et al. (2004b) Timing configuration The timing and triggering of the entire system is controlled by a master timing box (BNC-500A). This box has 4 channels but only the first two channels (M1 and M2) are used. The channel M1 triggers the laser timing box continuously at 15 Hz as shown in figure The laser timing box (BNC-500C) in turn triggers the two heads of the lasers at 15 Hz. Channel M2 of the master timing box has the capability of providing one trigger pulse for every n pulses of M1. That is, if channel M1 is running at 15 Hz, M2 is capable of triggering at 15/n Hz. In this experiment, n was set at 15, that is, the trigger frequency of M2 is set at 1 Hz. The timing and triggering of each camera system is controlled using a TSI synchronizer box. The timing of the cameras works exactly based on the timing diagram shown in figure In order to ensure simultaneous capture of images, the trigger from channel M2, which is in synchronization with the laser timing box is provided to both synchronizer boxes for image capture at a frequency of 1 Hz. Externally triggering the TSI synchronizer box with channel M2 ensures that the laser timing 52

78 f/n Hz (where n is any integer) M1 M2 BNC master timing box E T1 T2 T3 T4 BNC laser timing box f = 15 Hz TSI box 1 TSI box 2 Ext Cam Cam Ext Cameras F1 Q1 F2 Q2 Spectra Physics Nd:YAG lasers Cameras Figure 2.18: Timing diagram. The BNC box at the top is the master of the timing circuit. The timing of the TSI boxes can be changed by altering the integer n. In this experiment n was set at 15. Signal frequency (Hz) function delay (µs) relative to M1 M1 15 To laser 0 BNC box 1 timing box M2 1 To TSI 0 BNC box 1 boxes T1 15 Flash lamp 105 BNC box 2 of laser 1 Latency in TSI box T2 15 Q-switch BNC box 2 of laser 2 Latency + Q-SW T3 15 Flash lamp t (= 150) BNC box 2 of laser 2 Latency + PIV T4 15 Q-switch t(= 150) BNC box 2 of laser 2 Latency + PIV + Q-SW Table 2.8: Dual-plane experiments. Timing parameters for the experiments at z + =

79 Signal frequency (Hz) function delay (µs) relative to M1 M1 15 To laser 0 BNC box 1 timing box M2 1 To TSI 0 BNC box 1 boxes T1 15 Flash lamp 125 BNC box 2 of laser 1 Latency in TSI box T2 15 Q-switch BNC box 2 of laser 2 latency + Q-SW T3 15 Flash lamp t (= 120) BNC box 2 of laser 2 latency + PIV T4 15 Q-switch t(= 120) BNC box 2 of laser 2 latency + PIV + Q-SW Table 2.9: Dual-plane experiments. Timing parameters for the experiments at z/δ = and image capture are on synchronization. The frequency of image acquisition is low (1 Hz) to provide ample time interval between successive captures. The latency of the TSI synchronizer boxes (time taken to start image acquisition on receiving an external trigger) is a critical issue that needs to be addressed. In order to synchronize the firing of the lasers with image acquisition, the time delays in channels T1, T2, T3 and T4 of the laser timing box have to be altered to account for the latency. Details of the respective delays included in the four channels are given in Tables 2.8 and 2.9. It was found that the latency in the TSI synchronizer boxes depends on t (PIV delay), however the reason for this variable latency is unknown. The data from the two systems are streamed continuously to the RAM of two Dell Precision workstations and saved at regular intervals to disks Laser sheet characteristics: determination of sheet separation The thickness of the laser sheet and the separation between the sheets of Systems 1 and 2 were determined by performing burn tests. A laser alignment paper was placed in the measurement region (these burn tests were performed at multiple spanwise locations), and the laser sheets were exposed to it. The laser sheets leave a permanent mark on the paper. The paper was then scanned into a TIFF image using a 54

80 Experiment z + t sheet thickness separation separation name µs 1, 2 (mm) (mm) wall-units dp , dp , Table 2.10: Dual-plane experiments. Laser sheet thickness and pulse separation. Experiment z + z # of field δ name realizations (N t ) of view dp plane δ 1.1δ plane δ 1.1δ dp plane δ 1.1δ plane δ 1.1δ Table 2.11: Dual-plane experiment: Measurement parameters. high-quality scanner. The resulting TIFF image was then used in liaison with the appropriate magnification factor to obtain the sheet thickness and separation values. Details of the thickness and separation for the dual plane experiments are given in Table The uncertainty in determining the laser sheet thickness and separation was estimated to be 0.1 mm (100µm) Post Processing Vector fields for the 1k 1k cameras are computed using a recursive Nyquist algorithm described in using Insight 6.1 on a pixel window. The resulting vector Experiment z + Overlap First pass Second pass Offset name pixel pixel pixel pixel pixels dp plane % plane % dp plane % plane % Table 2.12: Dual-plane experiments: Interrogation window parameters and offsets. 55

81 Experiment z + Validation Standard Interpolation neighborhood name engine deviation engine size dp plane Gaussian 4.0 local mean 3 3 plane Gaussian 4.0 local mean 3 3 dp plane Gaussian 4.0 local mean 3 3 plane Gaussian 4.0 local mean 3 3 Table 2.13: Dual-plane experiments: Vector field validation parameters. Experiment z + x + y + z + # of name vectors dp plane plane dp plane plane Table 2.14: Dual-plane experiments: Measurement volume and resolution parameters. fields are validated using the same validation schemes described in section The vectors from each camera in the stereo plane are then combined using suitable magnification factors to compute all three velocity components (see 2.4.3). The vector fields for the single 2k 2k camera are computed using the same recursive algorithm, but on a pixel window. This was chosen to match the spatial resolution of the 2k 2k camera with the 1k 1k stereoscopic camera pair. The vector field from the single camera is resampled and mapped to the grid of the stereo measurement using the inverse distance interpolation method (see Appendix A). Details of vector computation, validation and spatial resolution are given in Tables 2.11, 2.12, 2.13 and The single camera vector field from the upper plane in liaison with the stereoscopic data from the lower plane is then used to compute all velocity gradients in the lower plane. A second order central difference method is used to compute all possible in-plane gradients while a first order forward difference is used to compute the wall-normal gradients of the streamwise and spanwise velocities. Finally, the continuity equation is used to recover the wall-normal gradient of the wall-normal velocity (see Appendix B for details). With the out-of-plane gradients provided by 56

82 K σ K (s 1 ) δk (s 1 ) % δk σ K δk (s 1 ) % δk 2σ K U x U y U z V x V y V z W x W y W z Table 2.15: Uncertainties in velocity gradients at z + = 110. K is any gradient, σ K is the r.m.s of K, δk is the uncertainty in the gradient with value σ K and δk is the uncertainty in the gradient with value 2σ K. the dual-plane data, we are able to compute the complete swirl strength and vorticity vector. The three vorticity components are directly calculated using the computed gradients Uncertainty in computed gradients A complete uncertainty analysis was performed to quantify the validity of the computed gradients. The uncertainty in any gradient is dependent on the differencing scheme used to compute it. A simple error propagation analysis was used to compute the uncertainties in the gradients at a single point as described by Kline & McClintock (1953). Complete details of the error propagation scheme are given in appendix C. Table 2.15 lists the computed absolute uncertainty in all 9 gradients and values normalized by each r.m.s (root mean square) of the gradients. Typically, in the velocity fields presented, we are interested in gradients that are twice the r.m.s. values. Hence, the relative uncertainties in larger gradients are smaller. For in-plane velocity gradients, the primary contributions to the uncertainty come from the pixel resolution and stereoscopic reconstruction. The uncertainty in sheet separation makes a significant contribution to the uncertainty in the gradient in the wall-normal direction. The absolute uncertainty in the gradient increases with uncertainty in sheet separation. δk and δk (column 3 and 5, table 2.15) are the absolute uncertainties computed for 57

83 the r.m.s and twice the r.m.s values of the gradients respectively and δk represents typical uncertainties in the measurements. It is clear from the table that δk for all in-plane gradients is exactly equal δk. However, δk for the out-of-plane gradients of U and V is marginally higher than the corresponding value of δk. The uncertainty in sheet separation causes this difference. Improving the accuracy in determination of sheet separation is thus an obvious goal for any future studies. The results in Table 2.15 reflect the findings of Westerweel (1994), where the author found that gradients can be computed with 10-20% accuracy, provided the velocity data is within 1-2% accuracy. The uncertainty in the wall-normal gradients cannot be improved by using the dualplane stereoscopic PIV technique (4 cameras). Accuracy of wall-normal gradients depends only on accuracy of the velocity component and determination of sheet separation. Therefore, three-component velocity field information in two differentiallyseparated planes is redundant, however it could be used to validate the technique by comparing the computed gradients against continuity (See Mullin & Dahm 2004). 58

84 Chapter 3 Results and discussion: stereoscopic PIV Instantaneous and statistical results from the single plane stereoscopic PIV datasets are presented in this chapter. Various statistical quantities were computed at different wall-normal locations and are compared. Instantaneous vector fields are presented to provide a comprehensive view of the structure of the turbulent boundary layer. 3.1 Mean and R.M.S statistics The values of U computed from the PIV data at each wall-normal location agreed well with the hot-wire measurements. Table 3.1 shows a summary of some ensemble averaged turbulence statistics. The values shown compare well with previous data in the literature (Balint, Wallace & Vukoslavcevic 1991;Naguib & Wark 1992;Adrian et al. 2000b). Please note that all quantities presented in this chapter are normalized using U τ and ν and are denoted with a superscript Statistical analysis 1 Figures 3.1(a) and 3.1(b) show color contours of instantaneous vorticity (ω + z ) and Reynolds shear stress ( uw + ) at z + = 92. The vectors include U U and V. The 1 Parts of this section appear in Ganapathisubramani et al. (2004a), accepted for publication in Journal of Fluid Mechanics. 59

85 y x + (a) y x + (b) Figure 3.1: (a) Wall-normal vorticity (ω + z ) and (b) uw + at z + =

86 z + U + u 2+ v 2+ w 2+ uw + + σ ωz Table 3.1: Ensemble-averaged flow statistics. σ ωz the fluctuating wall normal vorticity. is the r.m.s (root mean square) of ω + z plot clearly reveals a pattern of cores of opposite sign of ω + z organized along the streamwise direction that are indicative of hairpin legs intersecting the measurement plane. Examination of the vectors in the figures reveals that a region of uniform momentum is found sandwiched between the cores of positive and negative vorticity. The Reynolds shear stress plot contains values as large as 20U 2 τ in the zone of uniform momentum between the vortex cores. The instantaneous velocity realizations indicate the presence of hairpin structures and packets producing considerable Reynolds shear stress as a common and recurrent feature in the log layer. Before delving into analyzing the instantaneous velocity signatures, it is useful to study the behavior of the various flow parameters statistically. Two-point correlations of the fluctuating velocity components were computed using 750 independent velocity fields at each of the four wall-normal locations, z + = 92 and 150 in the log region and z/δ = 0.2 and 0.5 (z + = 198 and 530 respectively) in the wake region. The computations were made according to the procedure outlined in Appendix D. Note that, for all streamwise-spanwise correlation plots, the flow is from left to right. A positive x corresponds to correlation (R AB ) of variable B at a downstream location from variable A and a negative x characterizes a location of B upstream of A. A similar sign convention applies to the spanwise direction. Figures 3.2(a) and 3.2(b) show the two-point auto-correlation R uu at z + = 92 and z/δ = 0.5 respectively. For z + = 92, it is immediately obvious that there is significant spatial coherence in the streamwise direction such that the correlation coefficient extends further than 1500 wall units (1.5δ). The equivalent plot at z/δ = 0.5 (z + = 530) is much shorter, extending a distance of only 0.6δ. These locations are chosen to show 61

87 y + 0 y x (a) x (b) Figure 3.2: (a) R uu correlation at z + = 92, (b) R uu correlation at z/δ = 0.5. Contour levels for R uu range from -0.1 to 1.0 with spacing of 0.1. Zero contours are not shown. representative views of both the log region and wake region of the turbulent boundary layer. Additional correlation plots from intermediate planes suggest that the extent of the streamwise coherence drops off significantly beyond the log layer. This reduced streamwise coherence is clearly evident from the vector fields beginning at z/δ = 0.2 (z + = 198), which typically contain evidence of larger individual vortex cores, upwash and downwash regions, and possibly cross sections of individual hairpin heads (see Longmire, Ganapathisubramani & Marusic 2001 and section 3.3.2). The R uu measurements at z/δ = 0.5 appear very similar to the space-time correlation results of Kovasznay et al. (1970) obtained at a similar height (z/δ = 0.45, z + = 580) in a boundary layer with similar Re θ and Re τ (this was the closest wall-normal location in the previous study). At z + = 92, the spanwise width of the positive correlation region is about 400 wall units, with a change in sign occurring at y + = 200. At z/δ = 0.5, however, the positive correlation is wider with the sign change shifted to y + = 400. This location is again consistent with the results of Kovasznay et al. (1970). The spanwise results demonstrate that typical flow structures increase in width with increasing wall-normal distance. At z + = 92, the long negative correlation outboard of the positive correlation region supports the notion of adjacent low and high speed zones extending in the streamwise direction. This result is entirely consistent with a hairpin vortex model, where the legs of an individual hairpin vortex become increasingly streamwise and elongated near the wall. The long streamwise coherence at z + = 92 could be caused either by long zones of negative u as observed 62

88 Ξ(u) x + Figure 3.3: Dotted-line shows the difference between p.d.f. of positive and negative u of a certain streamwise length. x + is the streamwise distance over which the quantity remained above or below zero. The bin width chosen was 40 wall-units. The solid line shows a second order fit to the dotted-line. in the packet structures or by long zones of positive u which are sometimes observed on either side of the packets. In order to investigate the relative contributions, lines of vectors in the streamwise direction were examined for the streamwise length between zero crossings. A zero crossing is defined as the location at which u changes sign. The probability density is computed for positive and negative u (streamwise velocity fluctuations) versus streamwise length (length is the distance over which the quantity remains above zero or below zero). This is similar to the zero crossing method defined by Kailasnath & Sreenivasan (1993). Separate probability density functions are generated for lengths of positive and negative u between zero crossings. The normalized difference between these functions is then defined as, Ξ(u) = pdf(u > 0) pdf(u < 0) 100, (3.1) pdf(u > 0) + pdf(u < 0) and the result at z + = 92 is plotted in Figure 3.3. A negative value in the ordinate of the plot indicates that strips of negative u occur more frequently than strips of positive u. Note that the streamwise length plotted ( x + ) is limited by the field of 63

89 R vv (z + = 92) R vv (z/δ = 0.5) R ww (z + = 92) R ww (z/δ = 0.5) y x + Figure 3.4: R vv & R ww correlations at z + = 92 and z/δ = 0.5 The contour levels range from 0.1 to 1 with increments of 0.1. view. The plot shows that shorter negative and positive u excursions ( x + < 1400) occur with approximately equal probability. However, for excursions longer than 1500 wall units, those with negative u occur significantly more frequently than those with positive u. This result suggests that the long positive streamwise correlations in R uu (see figure 3.2(a)) are dominated by long low-speed regions such as those noted to occur within vortex packets. To clarify this point a second order polynomial was fitted to the raw data. This second order curve (shown as solid line in figure 3.3) clearly reveals the trend discussed above. The primary advantage of stereo PIV data is the availability of all three velocity components. Hence, it is possible to compute all possible velocity auto- and crosscorrelations over the streamwise-spanwise plane. Figure 3.4 shows the R vv and R ww correlations at the two representative wall-normal locations respectively. Both correlations are compact in both streamwise and spanwise directions indicating that the spanwise and wall-normal velocity fluctuations are localized and do not possess an extended streamwise or spanwise coherence across the boundary layer. There is a small increase in the spanwise extent of both R vv and R ww correlations away from the wall as seen in figure 3.4. This increase is attributed to the increase in the geometric size of a representative average vortex structure. Also, note that the spanwise extent of R vv is larger than R ww in both locations. These results are in qualitative agreement with the findings of Krogstad & Antonia (1994) who performed experiments using X-wire arrays. Figures 3.5(a) and 3.5(b) show the R uw correlation (streamwise-wall-normal) at z + = 92 and z/δ = 0.5 respectively. As with R uu, the R uw correlation extends over a long streamwise distance at z + = 92, but the correlation values are negative 64

90 y + 0 y x (a) x (b) Figure 3.5: (a) R uw correlation at z + = 92, (b) R uw correlation at z/δ = 0.5. The contour levels for R uw range from 0.05 to -0.5 with spacing of Zero contours are not shown. for y + = 0. The maximum magnitude of R uw is about Given the result that low-speed events are primarily responsible for long correlations, the negative values of R uw in figure 3.5 suggest that slow moving fluid is associated with upwash (flow away from the wall) over long streamwise distances. Further away from the wall at z/δ = 0.5, the R uw correlation decreases in length and increases in width similar to the behavior of R uu. Again, at this location, the size and shape of the spatial correlation is similar to that found by Kovasznay et al. (1970) from space-time correlations. The maximum magnitude of R uw at this position is which is close to the correlation values found in previous hot-wire experiments by Tritton (1967). One other point worth observing is the sign change of the R uw correlation in the spanwise direction. The presence of the sign change suggests that slow moving fluid is correlated with flow towards the wall (downwash) at a distance of about 200 wall units on either side. This observation is in accordance with the inclined hairpin vortex theory( Theodorsen 1952; Perry & Chong 1982; Perry & Marusic 1995). A zero crossing analysis was also performed on the uw product at z + = 92. Similar to the case of u, separate probability density functions were determined for the length of streamwise strips between zero crossings for both uw < 0 and uw > 0. The resulting pdf s are plotted in figure 3.6. These distributions are narrower than those based on u generated for figure 3.3, which can be expected based on the correlation plots. The distribution of uw < 0 is broader than that of uw > 0 indicating that uw < 0 is associated with slightly longer 65

91 10-2 uw > 0 uw < 0 p.d.f ( uw ) x + Figure 3.6: p.d.f of positive and negative uw events. The p.d.f. ordinate is in log scale. x + is the streamwise length over which the quantity remains above or below zero. structures. This result might be explained by the argument that uw < 0 events derive naturally from angled eddies generating upwash and downwash while uw > 0 events are more random. This is consistent with all existing literature on Reynolds shear stress quadrant analysis (see Lu & Willmarth 1973). It was found that the Q2 and Q4 events always dominate the Q1 and Q3 events throughout the boundary layer which explains uw < 0. When uw < 0 was broken down into separate zero crossing length distributions for Q2 (u < 0) and Q4 (u > 0), no significant difference was observable (and therefore the separate distributions are not shown in the plot). This is also in agreement with the results of Willmarth & Lu (1972) who show that Q2 and Q4 contributions are of similar strength. In Figure 3.7(a), the R ww correlation at z + = 92 is compared with R uu and R uw results along the centerline in the streamwise direction. First, note that the R ww correlation drops very sharply when x suggesting that the streamwise coherence in w is typically quite short as would occur for ejections associated with individual hairpins. On the other hand, the peak in R uw drops off more slowly, and the correlation has significant magnitude over the entire range plotted and beyond. (The normalized correlation magnitude for x + = 2000 is , or 11% of the peak value). The R uu correlation extends beyond the stream- 66

92 R uu R ww R uw R uu R ww R uw x (a) y (b) R uu R ww R uw R uu R ww R uw x (c) y (d) Figure 3.7: R uu, R uw and R ww correlations. For z + = 92, (a) Streamwise direction at y + = 0 and (b) Spanwise direction at x + = 0. For z/δ = 0.5, (c) Streamwise direction at y + = 0 and (d) Spanwise direction at x + = 0. 67

93 wise range examined, clearly indicating presence of streamwise regions longer than 2δ. Taken together, these results suggest that slow moving fluid is correlated with discrete zones of upwash over an extended streamwise distance both upstream and downstream, again as would be expected for packets of hairpin vortices moving with similar convection speeds. Note that the R uw correlation is slightly asymmetric in the streamwise direction such that the correlation has a longer tail for positive x + than for negative x +. This can be explained by the following argument. Typically, vortex packet structures have fairly uniform streamwise velocity over their streamwise extent (see Ganapathisubramani et al. 2003). On the other hand, the results of Adrian et al. (2000b) suggested that the hairpins at the downstream end of the packet tend to be larger than those at the upstream end leading to stronger positive values of w at the downstream end. The resulting R uw values for large positive x + should thus be larger than for large negative x +. The spanwise extent of the structures is shown in figure 3.7(b) in which R uu, R uw and R ww are compared in the y direction at x = 0. The R uu correlation in the spanwise direction drops from 1 to a negative value at y The spanwise R uu also indicates a general trend where the value drops below zero on either side of the peak indicating the occurrence of alternating high-speed and low-speed regions in the spanwise direction. The R uw correlation changes sign from negative to positive at around the same spanwise displacement as R uu suggesting that the Reynolds shear stress production has a very similar spanwise scale to the streamwise velocity. Figure 3.7(b) also shows that the R ww correlation is narrower in the spanwise direction than R uu, and that its negative lobe is weaker suggesting in general that coherence in w is narrower than in u. In any case, the presence of negative lobes in R ww suggests that upwash locations are frequently bounded by downwash as would occur in hairpin vortices where an ejection between angled legs would be accompanied by inward motion beneath the legs. The streamwise and spanwise extent of the correlations further from the wall (z/δ = 0.5) are depicted in figures 3.7(c) and 3.7(d). It is evident from these two figures that the R uu correlations are shorter in the streamwise direction and broader in the spanwise direction than at z + = 92 indicating a reduced streamwise and increased spanwise spatial coherence. Tomkins & Adrian (2003) found that an eddy conditioned on slow moving zones in the streamwise-spanwise plane bore a striking resemblance to a hairpin vortex signature. Similar to the present results, they also found that the spanwise width of the structure 68

94 and the separation between the legs increased away from the wall which is analogous to the increase in the spanwise extent of the R uu correlations. It is worth mentioning that the present results do not agree with some of the trends found recently by Christensen (2001) who performed spatial correlations of streamwise wall-normal PIV data taken in a turbulent channel flow with very similar Re τ to the present study (Re τ = 1141). Christensen reported R uu, R ww, and R uw values with no spanwise displacement at various wall-normal positions including z + = 103 (z/h = 0.09, where h is the channel half-width). Within the Christensen data, R uu and R uw were elongated in the streamwise direction compared with R ww, as in the present data. Christensen s fields covered x values up to 0.72h representing a shorter streamwise range than the present data. Nevertheless, direct comparison of like contour levels shows that the channel data exhibit longer correlations at z + = 103 (z/h = 0.09) than the present boundary layer data at z + = 92 (z/δ = 0.09). In addition, the length of the channel correlations increased with wall normal distance in contrast to the length of the present boundary layer correlations which decreased from the log to the wake region. Perhaps, differences in boundary layer and channel results are not surprising, given that fully-developed channel flow includes additional structures initiated on the opposite wall that can influence the measurement regions. This additional influence from the opposite wall in low Reynolds number flows was documented by various researchers. Wei & Willmarth (1989) suggested that opposite near-wall regions constantly exchange vorticity and Reynolds shear stress while Antonia et al. (1992) investigated this phenomenon (in experiment and simulation) by introducing temperature increases on one wall and detecting it on the opposite wall. They found that fluid from the slightly heated wall region can reach the near wall-region of the opposite wall. The penetration depth and frequency of occurrence of thermal excursions decreases with Reynolds number. In a channel flow at a fairly low Reynolds number, therefore, no outer region exists within which the turbulence generated by the wall-shearing dies out (as occurs in an isolated boundary layer). If a contour level of R uu (say 0.5) is chosen to define a representative streamwise length scale (l x ), the present results reveal an increase in l x through the log region (between z + = 92 and 150), but then a decrease in l x in the outer wake region (at z/δ = 0.2 and 0.5). Further work to complement the present results would clarify this increasing trend through the log region. However, Christensen s channel flow data clearly show increasing streamwise length scales throughout the log region. It should 69

95 be noted that the entire outer region in a channel flow is effectively a log region (as evidenced in the mean velocity profiles in Christensen 2001, page-61) and there is no intermittency as one would find in the outer region of a boundary layer. 3.3 Instantaneous fields 2 Although the two-point correlation studies reveal the existence of large-scale coherent features which could be hairpin packets, the instantaneous structure of coherence cannot be isolated using correlations or other averaging techniques. Hence a feature extraction was developed to isolate the hairpin packets present in the flow. The algorithm was designed such that, it searched for regions of strong positive and negative wall-normal vorticity (ω z ). Further, it looked for points of strong instantaneous Reynolds shear stress ( uw) between the region of positive and negative vorticity. It then used the identified points as seed points for a region growing algorithm to isolate a region of low momentum (or slow moving fluid) enveloped by cores of vortices that produce strong Reynolds shear stress. Such a region identified by the algorithm was labeled a hairpin packet and its primary features, such as length, width, and contribution to total Reynolds shear stress were recorded. Details of the feature extraction algorithm are outlined in Appendix F. The contour/vector plots shown in figures 3.8 and 3.9 are from individual realizations, but the patterns are representative of those found in many vector fields Logarithmic region Examples of packets found by the feature extraction algorithm at z + = 92 and 150 are shown in figures 3.8 and 3.9. Plots of instantaneous vorticity (ω + z ) and Reynolds shear stress ( uw + ) at z + + = 92 are shown in figures 3.8(a) and 3.8(b). The ω z plot clearly reveals a pattern of cores of opposite sign of ω + z organized along the streamwise direction that are indicative of hairpin legs intersecting the measurement plane. Contours of λ + 2D (it is the measure of the frequency of swirl and is derived from 2-D in-plane velocity gradient tensor) were also studied and confirmed the presence of swirling motion corresponding to the centers of blobs of vorticity. For detailed mathematical definition of swirl please see Appendix E. The region outlined in black 2 Parts of this section appear in Ganapathisubramani et al. (2003), published in Journal of Fluid Mechanics, 478, pp (2003) 70

96 (A) is the zone of uniform momentum identified by the extraction algorithm. The presence of this zone sandwiched between the cores of positive and negative vorticity solidifies the claim that the hairpin vortices convect with similar speeds as part of a group. The Reynolds shear stress plot contains values as large as 20U 2 τ in the zone of uniform momentum between the vortex cores. Further, the negative direction of the vector arrows in the patch in combination with positive uw + values indicates the presence of a Q2 event where slow moving fluid is ejected away from the wall. The vorticity plot reveals at least five pairs of vortex cores aligned in the streamwise direction, and the uw + plot shows five distinct regions of strong uw + just upstream of a pair. These regions are likely caused by hairpins inclined at an angle to the wall that eject fluid upward between their legs. Finally, note that the packet extends all the way across the field ( 1.2δ). Additional plots of uw + (not shown) revealed values as high as 40Uτ 2 within packets compared with the mean value of only 0.91Uτ 2. Plots of complete fields depicting vorticity, swirl strength and Reynolds shear stress within and surrounding packets can be seen in Longmire, Ganapathisubramani & Marusic (2001). Usually, the hairpin vortices observed have two legs. The legs may be aligned or somewhat offset in streamwise location. Sometimes, the hairpins appear to have single legs or have significant asymmetries in the strength of each leg (see the single negative vortex core to the left of region marked (A) at x + = 50 and y + = 280). Although Robinson (1991) saw mostly single-legged hairpins in DNS datasets at low Reynolds numbers, Tomkins & Adrian (2003) in their PIV measurements of x y planes at higher Reynolds numbers found both symmetric and asymmetric structures. Figures 3.9(a) and 3.9(b) are larger views with plots of ω + z and uw + at z + = 150. The total field length is twice as long as in 3.8(a) and 3.8(b). The region marked (B) is a zone of uniform momentum identified by the feature extraction algorithm. The contours of ω z + and λ + 2D (not shown) along with the vectors again show regions of swirling motion bounding the patch. The Reynolds shear stress shows discrete zones of high magnitude along the entire length of (B). The zones of strong uw + always seems to occur between pairs of swirls on opposite sides of the patch. In this patch, the maximum values of uw + are smaller than in figure 3.8(b), but they are still significantly higher than the global mean ( uw + = 0.78). The length of the patch is 1.9δ. Many shorter patches like (C) were also identified at both z + = 92 and 150. These patches usually contained either one or two vortex core pairs and a small area con- 71

97 (a) (b) Figure 3.8: Hairpin packets identified using the feature extraction algorithm. At z + = 92, for clarity only every second vector resolved is shown, (a) ω z +, (b) uw +. The flow is from left to right and the vectors shown have U subtracted. 72

98 (a) (b) Figure 3.9: Hairpin packets identified using the feature extraction algorithm. At z + = 150, every third vector resolved is shown, (c) ω z +, (d) uw +. The flow is from left to right and the vectors shown have U subtracted. 73

99 tributing to the Reynolds shear stress. There are also even smaller patches like the one just above patch (B). On further investigation of the λ + 2D plot, it is clear that this is a single hairpin vortex encompassing a small area of significant uw +. However, it is unclear whether this or other smaller patches travel with nearby longer patches. Sometimes, patches containing 2-3 hairpins were closely followed by an additional short patch at a streamwise distance of wall units. The region marked (D) exhibits Q4 characteristics. From the direction of the vector arrows it is quite clear that u is greater than zero and uw + is positive, so w must be less than zero. These regions frequently appeared adjacent to packets identified by the algorithm at both z + = 92 and 150. The inclination of hairpin vortices with respect to the wall is such that the region outside of the legs/necks should induce Q4 events. These Q4 regions often carry a significant amount of Reynolds shear stress, and the maximum local values are comparable to the maximum values observed in a Q2 type event ( uw + 40). The patches identified by the algorithm are typically aligned in the streamwise direction but sometimes inclined at small angles such that they have a spanwise component. The series of vortex cores are at times offset in the spanwise direction by wall units so that the zones of uniform momentum are oriented at an angle of to the streamwise direction Outer region The data acquired further from the wall (z + = 198 and 530) revealed a variety of coherent structures, but no patterns of long low-speed streaks, elongated regions of high Reynolds shear stress or organization of vorticity in the form of a train of hairpins. Thus, the results indicate that the legs of most hairpins do not extend as far as z/δ = 0.2 (z + = 198). However, it is possible that the largest hairpin in a given packet crosses this measurement plane such that only individual structures are evident. Typical structures observed included individual vortex cores, large regions of upwash with scales approaching 300 wall units, and spanwise strips of wall-normal velocity with alternating signs possibly indicative of hairpin heads. Contour plots of wall-normal vorticity (ω z ) and velocity fluctuations (w) at z/δ = 0.2 and z/δ = 0.5 are shown in figures 3.10 and The regions marked (E) and (F) in figure 3.10 reveal large wall-normal vortex cores and the region marked (G) is an area of strong upwash 74

100 E 800 y G F x + (a) E 800 y G F x + (b) Figure 3.10: At z + = 198 (z/δ = 0.2), for clarity only every second vector resolved is shown, (a) ω + z, (b)w +. The flow is from left to right and the vectors shown have U subtracted. 75

101 y H I x + (a) y H I x + (b) Figure 3.11: At z + = 530 (z/δ = 0.5), for clarity only every second vector resolved is shown, (a) ω + z, (b)w +. The flow is from left to right and the vectors shown have U subtracted. 76

102 and downwash just outside the log region (z/δ = 0.2). Figure 3.11 (outer wake region, z/δ = 0.5) also reveals a large region of upwash (marked (H)) and spanwise strips of positive and negative wall-normal velocity fluctuations (marked (I)) that could indicate spanwise heads. A closer look at the region marked (H) indicates that the large region of upwash contains four possible hairpin core pairs in a typical hairpin packet arrangement. However, this arrangement extends only about 500 wall units in the streamwise direction. This feature is extremely rare and is not a dominant occurrence in the outer wake region. However, this feature could be explained by the nested hierarchy of hairpin packets proposed by Adrian et al. (2000b) (See figure 1.5(b)). The reduced spatial coherence in the streamwise direction at the edge of the logarithmic layer and beyond is consistent with the results obtained by Adrian et al. (2000b) in their PIV measurements of streamwise-wall-normal planes. This behavior is also consistent with the attached eddy modeling study by Marusic (2001), in which the kinematic state of the boundary layer was modeled by an ensemble of eddies of varying height. The results of that study suggested that packets with long streamwise coherence did not need to extend beyond the log layer in order to obtain the proper correlation statistics. In this region, only uncorrelated eddies were required Hairpin packets: Statistics Figures 3.12(a) and 3.12(b) show the length and width distributions of the patches identified by the feature extraction algorithm in all vector fields at z + = 92, 150 and 198. The ordinate is the number of patches found in a certain bin divided by the total area searched in all of the vector fields. The length distribution clearly illustrates the lack of patches at z + = 198 relative to the two planes in the logarithmic layer. The peak of this distribution in all three planes, which occurs near 100 wall units, is caused by the identification of single vortex cores. The long tail of the distribution at z + = 150 (where measurement planes had the largest dimension) reveals a significant number of packets with length on the order of 2δ. This result is supported by the findings of Tomkins & Adrian (2003) whose PIV datasets revealed zones of uniform momentum stretching as long as 2.5δ. Much larger measurement planes (difficult to achieve with the available illumination energy) would be required to determine the maximum length of the patches. The width distributions in figure 3.12(b) peak from 77

103 30 70 σ L x z + = 92 z + = 150 z + = 198 σ W x z + = 92 z + = 150 z + = Length (viscous units) (a) Width (viscous units) (b) Figure 3.12: Length and width statistics of packets found by feature extraction algorithm. Total number of patches identified (N p ) at z + = 92, 150 and 198 are 1940, 3040 and 380 respectively wall units, and do not exhibit long tails. The shape of these distributions indicates that the patches are typically narrow in the spanwise direction. The length and width results are consistent with two-point correlations and autocorrelations of the streamwise velocity found in the literature (e.g. Townsend, 1976, pg. 112) which indicate that the zones of uniform momentum extend to a length of order 3δ in the streamwise direction. Although the width distribution found by the feature detection algorithm appears accurate, the length distribution might change with a more sophisticated algorithm. In many cases, individual short patches were found separated by distances of about 100 wall units in the streamwise direction, and these patches were not merged by the algorithm. From examination of individual fields, we suspect that these short patches in reality could be part of longer patches with length of order δ. Thus, with improved identification techniques, the length distribution curve would most likely become flatter such that the peak near 100 viscous units would decrease in value while the number of patches with greater length would increase Hairpin packets: Reynolds shear stress contributions In figure 3.13(a), values of uw have been extracted along streamwise lines of fields at z + = 92 for a typical instantaneous realization. The curve marked Q2 shows uw including the approximate centerline of an identified patch where the extent of the 78

104 10 0 Q4 event No event Q2 event uw z + = 92 z + = 150 uw Q4 Q2 % contribution to Patch found on Q2 event x Length (viscous units) (a) (b) Figure 3.13: (a) Typical uw + signature showing different events at z + = 92, (b) % contribution to uw +. Measuring field of view length for z + = 150 was twice that used for z + = 92. patch is marked. The values of uw are large through most of the patch and reach a maximum near 35U τ 2. This behavior can be compared with a line taken through a short sweep (marked Q4). The feature extraction algorithm does not pick out these regions because the combination of vorticity sought finds zones with negative u only. The Q4 line fluctuates fairly rapidly between positive and negative values of uw of significant magnitude. The maximum magnitude of 20U τ 2 is associated with a short negative peak. A line along a zone not including any packet or obvious strong event is also shown. In this case, the curve oscillates around zero at very small magnitudes. This behavior, in fact, occurs along many streamwise lines in the PIV plots, which is not surprising considering the shape of the probability density function of uw found in Willmarth & Lu (1972). The ratio of the total Reynolds shear stress from Q2 areas to that of Q4 areas was about 1.1. This is lower than the value of 1.35 reported by Lu & Willmarth (1973) who used conditional sampling of hot-wires. Nevertheless, this result is congruent with the hairpin vortex model where by virtue of the orientation of a hairpin vortex, Q2 ejections occur between the legs and Q4 sweeps on either side. The average length of strong individual bursts(q2) and sweeps(q4) (defined as a connected region of strong upwash or downwash where uw + is positive and greater than 2σ uw + ) were computed and found to be 0.04δ. These lengths are comparable to the results of Lu & Willmarth (1973). 79

105 Figure 3.13(b) shows the percentage contribution to the total Reynolds shear stress as a function of the packet length identified by the feature extraction algorithm. The percentage contribution is computed as the ratio of total uw associated with all the patches of a certain length to the global sum of uw in all vector fields. The plot indicates that packets with lengths greater than δ contribute significantly to the Reynolds shear stress relative to the smaller packets at both wall-normal positions shown. This clearly illustrates the importance of these long packets to turbulence production and transport of momentum in the logarithmic region. The contribution of all detected patches to the Reynolds shear stress was summed in order to evaluate the percentage contribution to the mean. The percentage contribution was computed as the ratio of uw associated with all the patches to the global sum of uw in all the vector fields. At z + = 92, the patches were found to contribute 27% of all Reynolds shear stress even though they occupied only 4% of the total area in the vector fields. At z + = 150, the contribution was 24%, and the patches occupied 4.5% of the total area. It is worth noting that the percentage contribution information is affected by the choice of the Reynolds shear stress seed threshold in the feature identification algorithm. For low thresholds (0.5σ + uw ), the contributions are as high as 55% from a total area of 15% due to the inclusion of all possible singular vortex cores with substantial Reynolds shear stress. However, above a certain value of the threshold, i.e. 1.5σ + uw, the contributions taper off towards the values quoted above. It is important to note that the patches frequently have regions of positive uw in addition to the dominant negative uw. Hence, if only the negative values of uw were included, the contribution would be even higher. The patches thus make a sizable contribution to the Reynolds shear stress considering the fact that they do not include certain Q2 events and all possible Q4 events. 3.4 Conclusions In this section, results of stereo PIV measurements in streamwise-spanwise planes of a zero pressure gradient turbulent boundary layer were presented. The velocity fields and the corresponding derived gradient quantities paint a good picture of the types of structures intersecting and crossing through different heights in the logarithmic layer and beyond. Two-point correlations were computed to understand the statistically significant features of the flow. Key conclusions from the statistical survey are as 80

106 follows, The streamwise-spanwise R uu and R uw correlations suggest that, in the log region, long narrow zones with uniform speed contain intermittent zones of upwash or downwash over their length as found previously by other researchers (Tritton 1967; Kovasznay et al. 1970; Bogard & Tiederman 1986; Tardu 1995) using point measurement techniques and Taylor s hypothesis. The data also seem to indicate an increase in the streamwise length scale through the log region (between z + = 92 and 150) and then a decreasing trend in the outer wake region (between z/δ = 0.2 and 0.5). These conclusions are based on only a few wall-normal positions, however, and therefore it would be useful to test the trend in the log region with future work. Investigation of the zero-crossing events indicates that long low speed zones are statistically more significant than high speed zones in the log region, and hence are the more significant contributors to the tail of R uu correlations. The fact that R uw is negative at y + = 0, suggests that these elongated low-speed zones are associated with regions of upwash. However, the short R ww correlations indicate that these regions of upwash are spatially compact. This picture is consistent with the idea that packets of hairpin vortices dominate the long streamwise coherence in the correlations in the log region. Based on the global statistical features, an algorithm for hairpin packet identification was presented that searched for zones of uniform momentum enveloped by cores of vorticity of opposite sign that contained seed points of significant Reynolds shear stress. Some of the key conclusions from analyzing the results of this algorithm and the resulting vector fields from the stereo PIV experiments are : Packets of hairpin vortices, sometimes extending over a length of 2δ, were identified in the logarithmic region of the boundary layer using an objective feature extraction algorithm. This result is consistent with the conclusions of Adrian et al. (2000b) based on their PIV datasets in x z planes. Here we used stereo PIV datasets in x y planes which enabled us to quantify the Reynolds shear stress contribution and spanwise orientation of these structures. 81

107 The packets carry a large percentage of the Reynolds shear stress which suggests that this packet structure is an integral part of the turbulence transport mechanism. The packets contribute more than 25% to uw even though they occupy less than 4% of the total area. The streamwise spatial coherence and organization of vortices is seen to break down beyond the log layer, consistent with the findings of Marusic (2001) who used the attached eddy hypothesis. The stereo PIV datasets at z = 0.2δ and 0.5δ reveal a wide range of structures but do not include long low speed streaks caused by organization of hairpin vortices. Instead, there were regions of upwash, adjacent spanwise strips of positive and negative wall-normal velocity oriented and large single vortex cores. 82

108 Chapter 4 Results and Discussion: dual-plane PIV Instantaneous and statistical results from the dual-plane PIV datasets are presented in this chapter. Various statistical quantities were computed at two wall-normal locations and are compared. Instantaneous velocity and vorticity fields are presented to provide a comprehensive view on the structure of the turbulent boundary layer. 4.1 Mean and R.M.S statistics The values of mean and r.m.s statistics of the velocity and vorticity components computed from the dual-plane PIV data at z + = 110 and z/δ = 0.53 are listed in Tables 4.1. The values shown compare well with previous experimental and DNS data in the literature (Balint, Wallace & Vukoslavcevic 1991;Naguib & Wark 1992). The validity of the dual-plane-stereo gradients in the log region can be judged by computing the wall-normal gradient of mean streamwise velocity ( U/ z) and comparing this value with the wall-normal gradient predicted by the log law: U = 1 [ ] U τ κ ln zuτ + A ν 83

109 Parameter z + = 110 z/δ = 0.53 U σ u σ v σ w uw σ ω + x σ ω + y σ ω + z Table 4.1: Ensemble averaged flow mean and r.m.s statistics from dual-plane datasets. σ + ω x, σ + ω y, σ + ω z are the r.m.s (root mean square) of the fluctuating vorticity components. rearranging the terms, U = U τ κ ln[z] + U [ ] τ κ ln Uτ + A ν = U τ ln[z] + A κ Differentiating the above equation with respect to z, U z = U τ κz where κ = 0.41 is the universal log-law constant. For wall-normal location of z + = 110, the log-law predicts the gradient to be m 1. The average value from an ensemble of 1200 images (with resolution of vectors) is m 1. The error in the mean value of the gradient is thus 2.8% which is well within the expected uncertainty for this first order difference quantity. 4.2 Instantaneous results In all vector plots presented, the flow is from left to right, and the local mean U is subtracted from the vectors to clearly illustrate the slow and fast moving zones. Figures 4.1(a) and 4.1(b) show the streamwise velocity fluctuation (u + ) contours from the lower (z + = 110) and the upper planes (z + = 130) respectively. It is clear that the two planes are very well correlated. The streamwise velocity signature consists of narrow and elongated low- and high-speed regions. The region marked with a box in 84

110 y x (a) y x (b) Figure 4.1: Fluctuating streamwise velocity (u + ) contours at (a) z + = 110; U + 1 = 16.04, (b) z + = 130;U + 2 =

111 the figure contains a hairpin packet identified by the feature extraction algorithm described in appendix F. In subsequent plots, we present a zoomed in view of the boxed area in order to focus on the details of the packet region. The wall-normal (ω + z ) and streamwise (ω + x ) components of vorticity of the boxed region are shown in figures 4.2(a) and (b). As seen from figure 4.1, this region is centered on a low-speed zone. The plot of ω z + shows that the top of the low-speed region is bounded by negative values and the bottom by positive values. Also, the vectors seem to indicate that these regions of vorticity contain swirling motions indicative of vortex cores. However, the vortices seen here are not merely wall-normal vortices. This point becomes clear from the ω + x plot. The regions of positive ω + z have predominantly positive ω + x, and regions of negative ω + z have negative ω + x. The data thus suggest the presence of vortices inclined at an angle with respect to the streamwise direction consistent with the notion of hairpin-type structures. Note, however, that inclined hairpins are not the only type of instantaneous structures observed. Examination of various vector fields in the two neighboring planes indicates evidence of some structures that are inclined at 90 degrees to the streamwise direction and others that are completely streamwise. To present more concrete evidence that regions of strong vorticity correspond with vortex cores, we examine plots of the instantaneous swirl strength for the same boxed region shown in figure 4.1. Figure 4.3(a) reveals the instantaneous two dimensional swirl strength (λ + 2D ) at z+ = 110. This plot in tandem with ω z + (see figure 4.2(a)) shows that 2-D swirl isolates regions that are swirling about an axis aligned with the wall-normal direction. A visual comparison between the locations of swirl in the lower and upper planes (not shown here) indicates a forward tilt to most structures as they are offset in the positive streamwise direction in the upper plane. (This trend is observable in many pairs of planes, in particular when they are viewed in rapid succession). However, if the inclination of vortices with respect to the wall is small, λ + 2D does not do a good job at identifying them. Figure 4.3(b) presents a plot of the full swirl strength (λ + 3D ) that was computed utilizing the complete velocity gradient tensor. The plot reveals that the 3-D swirl identifies not only the hairpin shaped vortices cutting across the plane but also additional regions that are not isolated by 2-D (wall normal) swirl. The magnitudes of λ + 3D are typically larger than λ+ 2D in the same location suggesting some deviation from the wall-normal direction. Note for example, the location at the downstream end of the packet that contains elongated regions of significant λ + 3D but not λ+ 2D. These regions could possibly coincide with streamwise legs of hairpin vortices that are cutting across the measurement volume 86

112 ω + z y (a) x + ω + x y (b) Figure 4.2: Various quantities at z + = 110 (a) ω + z, (b) ω + x. The dark line in the plots is the envelope of a low-speed region identified as a hairpin packet by the algorithm described in appendix F. x + 87

113 λ + 2D y (a) x + λ + 3D y (b) Figure 4.3: Various quantities at z + = 110 (a) λ + 2D and (b) λ+ 3D. The dark line in the plots is the envelope of a low speed region identified as a hairpin packet by the algorithm described in appendix F. x + 88

114 y + 0 A x + (a) y + 0 A x + (b) Figure 4.4: Various quantities at z/δ = 0.5 (a) λ + 3D and (b) λ+ 2D. A box is drawn around the region identified as a spanwise hairpin head. 89

115 y y x x (a) (b) y y x x (c) (d) Figure 4.5: Various quantities at z/δ = 0.5 (a) ω x +, (b)ω z +, (c) ω y + and (d) w +. A box is drawn around the region identified as a spanwise hairpin head. 90

116 and hence are not apparent in the 2-D swirl plot. The fact that λ + 3D identifies the spanwise heads of hairpin vortices becomes more obvious in the vector fields obtained in the outer region of the boundary layer. Figures 4.4(b) and 4.4(b) show contour plots of λ + 3D and λ+ 2D respectively. As discussed earlier in section 3.3.2, the outer region of the boundary layer does not contain significant signatures of hairpin packets, rather it contains large vortex cores, spanwise heads and large region of upwash and downwash. A spanwise hairpin head (H) is marked with a box in figure 4.4. It is clear from these two plots that λ + 3D clearly identifies this region while λ + 2D does not pick up the complete structure but isolates small patches (presumably due to measurement noise) along the structure. A zoomed in look at this boxed region is depicted in figure 4.5. Figures 4.5(a)-4.5(c) shows contours of ω + x, ω + z and ω + x respectively. The contours levels of ω + x and ω + z are relatively small clearly indicating rotational inactivity in the streamwise and wallnormal directions. However, ω y + plot shows a positive spanwise vorticity along the region of high λ + 3D revealing a spanwise vortex. Further, figure 4.5(d) indicates that the wall-normal velocity shows spanwise strips of positive and negative fluctuations on either streamwise side of the high swirl region. This is similar to previous observation in figure 3.11 clearly revealing spanwise heads of hairpin vortices Relationship between Hairpin-type vortices and turbulence production In order to investigate the relationship between Reynolds stress production and hairpin-type vortices in the log layer, a closer look into the boxed region in figure 4.2 is presented. Figure 4.6 reveals a zoomed in view of the region identified as a hairpin packet (outlined by the thick black line) by the algorithm described in appendix F. Figure 4.6(a) show contours of uw + in the region identified as a hairpin packet. The plot demonstrates that the hairpin packet is a low-speed region that contains significant Reynolds shear stress in spatially compact regions. Figure 4.6(b) shows instantaneous production, defined as P i = uw + ( U + / z + ) 1. Some locations of strong production lie close to swirling zones. Figure 4.6(b) shows high production 1 This is consistent with Brodkey et al. (1973) who provided an interpretation of production, based on instantaneous Reynolds shear stress and instantaneous wall-normal gradient, where the production almost exactly balances dissipation. Other definitions have also been proposed (see Bradshaw 1974). 91

117 u w y x + u w + ( U + / z + ) (a) y (b) x + Figure 4.6: Various quantities at z + = 110. (a) uw +, (b) Instantaneous production, P i = uw + U +. The dark line in the plots is the envelope of a low-speed region identified z + as a hairpin packet by the algorithm described in appendix F. 92

118 y x (a) y x + (b) Figure 4.7: Two-point correlations of uw + ( U + / z + ) with (a) λ + 2D and (b)λ+ 3D. Contour levels are marked in the figures. levels in the packets near the legs of hairpins (for example, the middle of the packet at x + = 0 has high production zones on either side of a swirl). Other examples (not shown here) reveal strong Reynolds stress production in regions that lie close to zones of significant λ + 3D (but, no λ+ 2D ) that could be heads or streamwise legs of hairpin structures. The representative relationship between the hairpin vortex and the Reynolds shear stress production can be further understood by computing the two-point correlations. Figures 4.7(a) and 4.7(b) reveal two-point correlations between P i and λ + 2D, and P i and λ + 3D respectively, computed according to the procedure outlined in appendix D. Figure 4.7(a) shows that swirl normal to the measurement plane occurs predominantly on either spanwise side of the location of production. This suggests that inclined vortex legs contribute to Reynolds stress production with the sites of production occurring on either spanwise side of the leg. This can also be seen instantaneously in figure 4.6, where production sites seem to occur in the vicinity of a vortex. Figure 4.7(b) corresponds with the presence of swirling activity in the neighborhood of production. This would include streamwise hairpin legs, angled necks and spanwise heads. This plot suggests that the majority of swirling zones occur in the downstream side of a production site. The lobe on the upstream side is presumably the contribution from spanwise heads, since turbulence production can be present on either 93

119 streamwise side of a spanwise head. However, the downstream extent of the correlation (> 100 wall-units, which is greater than the size of a nominal vortex neck and head) suggests that production is not necessarily influenced by only spanwise heads and angled necks. Therefore streamwise legs (similar to those in figure 4.3(b)) are also presumably a part of the dynamics. 4.3 Hairpin structural features The geometric structure of a typical vortex core in a turbulent boundary layer is much debated. There are various viewpoints on the shapes and sizes of vortices that may be present. However as discussed in chapter 1, there is significant evidence that in the logarithmic region and beyond, the boundary layer contains many hairpin-shaped vortices (with one or two legs). In the following sections, several analytical tools are used to determine the geometric structure of a typical vortex Statistical results An average or probable structure of vortex can be isolated using methods based on statistical analysis. In this section, the vortex angles and a qualitative look at the geometric structure of a vortex are studied using various statistical tools. Ong & Wallace (1998) performed a joint probability density analysis of various components of vorticity obtained using hot-wire measurements to study the dominant vortex orientation. This analysis was similar to the quadrant splitting analysis of Reynolds shear stress developed independently by Wallace et al. (1972) and Willmarth & Lu (1972). This analysis involved determining the joint probability density function (JPDF), P (a, b) of any two variables a and b, where ab = abp (a, b) da db This integral of the covariance integrand, abp (a, b) over a differential area dadb, represents the contribution of that particular simultaneous combination of sign and magnitude of a and b to the covariance ab. Wallace & Brodkey (1977) plotted contours of P (u, w) and uwp (u, w) (covariance) to study the dominant contributors to the 94

120 Angle Plane Definition γ xy ω x ω y Angle made by the line joining the origin and the highest positive covariance peak with positive y axis ɛ xy ω x ω y Angle made by the line joining the origin and the highest negative covariance peak with positive y axis γ yz ω y ω z Angle made by the line joining the origin and the highest positive covariance peak with positive y axis ɛ yz ω y ω z Angle made by the line joining the origin and the highest negative covariance peak with positive y axis γ xz ω x ω z Angle made by the line joining the origin and the highest positive covariance peak with positive y axis ɛ xz ω x ω z Angle made by the line joining the origin and the highest negative covariance peak with positive y axis Table 4.2: Definitions of the angles computed form the covariance plots. Angles corresponding to two highest magnitude (positive and negative) covariance values are calculated. Reynolds shear stress. Ong & Wallace (1998) used a similar analysis. However they used JPDF and covariance of various vorticity components to study the structure of a boundary layer. The authors used JPDF and covariance of (ω x, ω y ), (ω x, ω z ) and (ω y, ω z ) to determine a dominant structure, finding that vorticity filaments that contribute most to the covariance ω x ω z were inclined downstream with the streamwise direction, when projected to the x z plane. They also found that this angle decreases with increasing distance from the wall. The inclination angles in (x, y) and (y, z) planes with the y axis, computed using (ω x, ω y ) and (ω y, ω z ) covariances respectively indicated an increasing trend with distance from the wall. An identical analysis was performed using the dual-plane datasets at z + = 110 and z/δ = 0.53 to study the scale and orientation of vortex cores. The definitions of the angles computed from the covariance peaks is given in table 4.2. Figure 4.8 shows the JPDF and covariance of (ω x, ω y ) in the two planes. The covariance plots at z + = 110 reveal the dominance of the contributions from quadrant 1 (ω x > 0, ω y > 0) and quadrant 2 (ω x < 0, ω y > 0). The angles of inclination made with the positive y axis can be inferred from the locations of the peaks in these covariances as outlined in table 4.2. The angle γ xy (positive peak, quadrant 1) was determined to be 43 95

121 and ɛ xy (negative peak, quadrant 2) was 41 at z + = 110 and 47 and 45 at z/δ = This shows a slight increase in the angles away from the wall consistent with the results of Ong & Wallace (1998). This result in liaison with the results of Ong & Wallace (1998) where the angle increased from ±16 at z + = 30 to ±42 at z + = 89, suggests that vorticity filaments tend to align themselves close to ±45 with the spanwise direction. It is also important to note that the mean spanwise vorticity in both wall-normal locations is positive and hence stronger lobes in quadrants 1 and 2 can be expected. Figure 4.9 shows the JPDF and covariance of (ω y, ω z ) in the two planes. The covariance plots at z + = 110 reveal the dominance of the contributions from quadrant 1 (ω y > 0, ω z > 0) and quadrant 4 (ω y > 0, ω z < 0). The angles of inclination made with the positive y axis can be inferred from the location of the peaks in these covariances in 1 st and 4 th quadrants. The angles γ yz (quadrant 1) and ɛ yz (quadrant 4) were determined to be 41 and 38 at z + = 110 and 30 and 33 at z/δ = This reveals a decreasing trend in the angles which is in contrast to the findings of Ong & Wallace (1998) who found that the angle increased from ±27 at z + = 20 to ±37 at z + = 89. However, the notion that there are more spanwise heads in the outer region, supports the decreasing trend detected. Figure 4.10 shows the JPDF and covariance of (ω x, ω z ) in the two planes. The angled orientation of the JPDF contours reveals that positive ω z is correlated with positive ω x and negative ω z is correlated with negative ω x. This further reinforces the viewpoint that most vortices are inclined downstream. The covariance plots at z + = 110 also reveal the dominance of the contributions from quadrant 1 (ω x > 0, ω y > 0) and quadrant 3 (ω x < 0, ω z < 0). The angles of inclination made with the positive x axis are inferred from the peaks in covariances. The angles γ xz (quadrant 1) and ɛ xz (quadrant 3) at z + = 110 were computed as 39 and 143. The covariance at z/δ = 0.53 show that the forward-leaning structure is not as dominant in the outer region, as the contour levels in the 4 quadrants are comparable, however the values in quadrants 1 and 3 are marginally higher. The angles γ xz (quadrants 1) and ɛ xz (quadrant 3) are 31 and 150 respectively. This reveals a decreasing trend in the angle with wall-normal location that is consistent with the finding of Ong & Wallace (1998). Even though, only the angle values from the first and third quadrants are reported, the fact that there is considerable population density in the second and fourth quadrants indicates the presence of other types of structures in both the log and the outer region. An average vortex structure and size can be found by computing 96

122 ω + y ω + y ω + y ω + y ω + x ω + x (a) ω + y ω + y ω + y ω + y ω + x ω + x (b) Figure 4.8: (a) JPDFs and (b) Covariance integrands of (ω x, ω y ) at z + = 110 and z/δ = 0.53, left to right. The contour increment for (a) is 10 and the first level shown is equal to the increment. The contour increment for (b) is ±0.01 and the first level shown is equal to the increment. Negative contours are shown with dotted lines and the zero contour is not shown. 97

123 0.2 ω + y 0.2 ω + y ω + z 0 ω + z ω + y ω + y (a) 0.2 ω + y 0.2 ω + y ω + z 0 ω + z ω + y ω + y (b) Figure 4.9: (a) JPDFs and (b) Covariance integrands of (ω y, ω z ) at z + = 110 and z/δ = 0.53, left to right. The contour increment for (a) is 10 and the first level shown is equal to the increment. The contour increment for (b) is ±0.01 and the first level shown is equal to the increment. Negative contours are shown with dotted lines and the zero contour is not shown. 98

124 ω + z 0 ω + z ω + x ω + x (a) ω + z 0 ω + z ω + x ω + x (b) Figure 4.10: (a) JPDFs and (b) Covariance integrands of (ω x, ω z ) at z + = 110 and z/δ = 0.53, left to right. The contour increment for (a) is 10 and the first level shown is equal to the increment. The contour increment for (b) is ±0.01 and the first level shown is equal to the increment. Negative contours are shown with dotted lines and the zero contour is not shown. 99

125 y z + = x + y z/δ = x Figure 4.11: Two-point auto correlations of λ 3D at z + = 110 and z/δ = The contour levels are 0.1 to 1.0 at increments of 0.1. the two-point auto-correlations of swirl strength (λ 3D ). Figure 4.11 shows the auto correlation of λ 3D at z + = 110 and z/δ = The extent of the outer most contour levels can be interpreted as a representative length-scale of the vortex core. This figure does not show any discernible difference in the size of higher contour levels in the log and the outer region. However, the lower contour levels indicate that the size of a representative structure is bigger in the outer wake region than the log region which is consistent with the conclusions from instantaneous vector fields. These correlation plots include contributions from all possible structures that include forward- and backward-leaning eddies and spanwise head. Hence, arriving at a possible conclusion on the orientation of these individual eddies is not possible. The shape of the structure can be further studied by separating swirl strength λ 3D fields into the four separate fields based on the four quadrants in the (ω x,ω z ) plane to distinguish between forward-leaning and backward-leaning cores as shown below, λ 1 = λ 3D, for ω x > 0, ω z > 0 = 0, otherwise λ 2 = λ 3D, for ω x < 0, ω z > 0 = 0, otherwise λ 3 = λ 3D, for ω x < 0, ω z < 0 = 0, otherwise 100

126 0 z + = 110 For swirls, ω + < 0 z ω + < 0 x 0 z/δ = 0.53 For swirls, ω + < 0 z ω + < 0 x y + y For swirls, ω + > 0 z ω + > 0 x For swirls, ω + > 0 z ω + > 0 x x (a) x z + = 110 For swirls, ω + z < 0 ω + x > 0 0 z/δ = 0.53 For swirls, ω + z < 0 ω + x > 0 y + y For swirls, ω + z > 0 ω + x < For swirls, ω + z > 0 ω + x < x (b) x Figure 4.12: Two-point auto correlations of λ 3D separated into four quadrants according to ω x and ω z. (a)z + = 110 (b) z/δ = The contour increments are 0.1 and the outermost contour is equal to the increment. 101

127 λ 4 = λ 3D, for ω x > 0, ω z < 0 = 0, otherwise Two-point auto correlations for each of these separated swirl strength was computed following the technique outlined in Appendix D. The goal of this computation is to isolate a possible shape in the streamwise-spanwise plane of the vortex cores of various streamwise-wall-normal orientations. The shape of the contours of the auto-correlation of these separated swirl parameters at both wall-normal locations imply that both forward- and backward-leaning vortex cores are angled inwards towards each other with increasing wall-normal distance. This does not imply that there would not be any individual forward leaning or backward leaning structures of any other spanwise orientation. The correlation simply isolates an average shape of the structure. Another interesting note inferred from these correlation plots is that the forward-leaning structures seemingly are bigger (in core size) than the backward-leaning cores. A hypothetical model for an average vortex structure can be constructed based on the shapes of these contours and the angles computed from the covariance plots. Figure 4.13(a) shows a schematic representation of a forward-leaning Λ shaped eddy and its projection on the streamwise-spanwise plane. The projections in the first and third quadrants of the forward-leaning eddy are qualitatively similar to the contour shapes in figure 4.12(a). Figure 4.13(b) illustrates a backward-leaning Λ shaped structure and its projection onto the x y plane. The projections of the eddy are similar to the contour shapes in figure 4.12(b). Note that these correlations do not suggest that the negative leg of the hairpin-type vortex has to appear in the positive spanwise side of a positive leg. Therefore an alternate view based the wake eddy structure (Type-B) proposed by Perry & Marusic (1995) (figure 4.14) can also explain the shapes of the contour Instantaneous results The dual-plane data can also be used to compute the inclination angle of any given vortex structure by determining the orientation of the vorticity vector averaged over the region of the vortex core. It is important to distinguish this from an instantaneous vorticity vector angle at a point. The instantaneous vorticity field contains many small-scale fluctuations. By computing the average vorticity vector averaged over a 102

128 (a) (b) Figure 4.13: Hypothetical model of an average vortex in a turbulent boundary layer constructed based on two-point correlations of swirl strength and vorticity JPDF and covariances. Figure 4.14: A schematic of the wall-wake model reproduced from Perry & Marusic (1995). The structures that exist in the log region (Type-A) extend out to δ. Additional structures (Type-B) of the form shown are added in the outer wake region. 103

129 0.015 λ 3D (all) λ 3D (λ 2D > 0) p.d.f(θ e ) Figure 4.15: p.d.f. of inclination angle (θ e ) at z + = 110. θ region identified as a vortex core by the swirl strength λ + 3D, the small-scale variations are averaged out, leading, to determination of the orientation of the vortex core. This orientation can then be interpreted as the local inclination of that vortex. A region of vortex core is isolated using a region growing algorithm that locates connected regions of swirl greater than a specified threshold. The technique and other nuances of identifying a vortex core to compute the inclination angle are described in appendix G. Figure 4.15 reveals the probability density function (p.d.f.) of the inclination angle (θ e ) that vortex cores make with the x y plane at z + = 110 (This angle is also called the elevation angle). This distribution (square symbols) includes a wide range of structure angles at this wall-normal location. Note that many structures have small inclination angles. Further study, including the investigation of the azimuthal angle made by the projection of the vorticity vector onto the x y plane with the x axis, reveals that most λ + 3D regions with small inclinations are spanwise structures indicative of heads of smaller hairpin vortices or other in-plane oriented vortices. In order to obtain the inclination angles of cores that are not spanwise heads or streamwise legs, the average vorticity vector in isolated regions of λ + 3D that include λ + 2D (λ+ 2D > 0) was computed. This additional criterion filters out spanwise and 104

130 streamwise structures which do not contribute to λ + 2D. The resulting p.d.f., shown by circles in figure 4.15 does not have a peak at zero inclination angle. Hence, the use of λ + 2D is essential to study the orientation of vortices that are not spanwise heads or streamwise legs. In all further p.d.f. plots of vortex angles, spanwise heads and streamwise legs are not included in the p.d.f., unless otherwise specified. Figure 4.16(a) shows the comparison of the inclination angles (θ e ) at z + = 110 and z/δ = This p.d.f. yields peaks at ±38 for z + = 110 and ±33 for z/δ = 0.5. Note, however, that the peaks are broad, and a wide range of inclination angles is present in each location. This result suggests that the dominant inclination angle decreases with wall-normal distance. Figure 4.16(b) is a plot of the joint probability distribution of the ratio of 2-D swirl strength to 3-D swirl strength and the inclination angle (θ e ). It is worth noting that, mathematically λ 2D will always be less than or equal to λ 3D for any orientation. The distribution indicates a unique relationship between this ratio and the inclination angle of vortex structure with respect to the cutting plane (x y plane in this instance). Velocity fields induced around idealized hairpin vortices (with and without curvature) were computed using Biot-Savart calculations to calculate the ratio λ 2D /λ 3D as a function of the hairpin angle. The results from this computation suggest that the ratio of the two swirl strengths varies as sin θ e. The value of this ratio from the experiments follows this theoretical finding as seen in figure 4.16(b). Note that this plot does not reveal any information about the streamwise or spanwise oriented structures, since the ratio was computed only for cores where λ 2D is greater than zero. The fact that the distribution is dense in the angle range 20 < θ e < 50 at both wall-normal locations indicates that most vortex structures are inclined in that range of angles. However, on closer examination of the plot, it can be concluded that the range of angles at z + = 110 is larger than at z/δ = The projections of the vorticity vector in the x y, y z and x z planes can be used to compute the projection angles in the plane, similar to the covariance and JPDF covered in section However, the present calculation gives the angles of vortex cores isolated from instantaneous vector fields. The definitions of the various angles made by the vorticity projections in the three planes are given in table 4.3. Figure 4.17(a) shows the p.d.f. of θ xy at both wall-normal locations. This figure clearly reveals that a large number of structures have orientations in the range 0 < 105

131 z + = 110 z/δ = 0.53 p.d.f(θ e ) (a) θ Ratio θ e (b) Figure 4.16: p.d.f. of (a) θ e and (b) Joint p.d.f. of inclination angle and λ 2D /λ 3D at z + = 110 (top) and z/δ = 0.53 (bottom), the dotted line is the function λ 2D /λ 3D = sin θ e 106

132 0.006 z + = z/δ = z + = z/δ = 0.53 p.d.f(θ xy ) θ p.d.f(θ yz ) θ (a) (b) p.d.f(θ xz ) 0.01 z + = z/δ = θ (c) Figure 4.17: p.d.f. of (a) θ xy (b) θ yz and (c) θ xz at z + = 110 and z/δ =

133 Angle Plane Definition Range θ xy x y angle made by the projection 180 θ xy 180 with positive x axis θ yz y z angle made by the projection 180 θ yz 180 with positive y axis θ xz x z angle made by the projection 180 θ xz 180 with positive x axis θ i x z defined the same as θ xz, however it groups Q1 and Q3 vortices (forward-leaning) 0 θ i 90 Q2 and Q4 vortices (backward-leaning) 90 < θ i 180 Table 4.3: Definitions of the projection angles in the three planes. θ xy < 180 (1st and 2nd quadrants in the x y plane). This result is in accordance with the findings of the covariance analysis in section where the covariance peaks in the 1st and 2nd quadrants were higher than the 3rd and 4th quadrants. The p.d.f. s in both wall-normal locations have three modest modes. The first mode is in the range 0 < θ xy < 30 presumably due to positive legs of a Λ shaped eddies. However, it might include contributions from positive streamwise legs (or vortices at small inclination angle that is picked up by λ 2D ) whose angle would be close to 0. The second mode, which is in the range 75 < θ xy < 105 are the contributions from both positive and negative legs. This mode would also include contributions from spanwise heads (weak signatures picked by λ 2D ) and the inclination of such spanwise heads would be close to 90. Finally the third mode is in the range 155 < θ xy < 180. This would include contributions from negative legs of Λ shaped eddies and other negative streamwise legs at small inclination angles whose inclination would be close to 180. Regardless of the modes, the plot of θ xy shows relatively flat region for 0 < θ xy < 180 that suggests that instantaneous structures do not have a preferred x y orientation in the first and second quadrants. Figure 4.17(b) reveals the p.d.f. of θ yz at z + = 110 and z/δ = This plot (in both wall-normal locations) has peaks at ±45. This shows that most structures are inclined at ±45 in the y z plane. This result can be explained by using the Λ shaped eddy model shown in figure All positive legs of a Λ shaped vortex will have an inclination in the first quadrant (0 < θ yz < 90 ) while the negative legs will have inclination in the 4th quadrant ( 90 < θ yz < 0 ). Note that the curves are broad and there are a wide range of eddy structures at various angles. The shape 108

134 z + = 110 z/δ = z + = 110 z/δ = 0.53 p.d.f(θ i ) number(θ i ) θ θ (a) (b) Figure 4.18: Eddy inclination angle (θ i ) (a) p.d.f. and (b) Absolute number density. of the curve in the 1st and 4th quadrants in figure 4.17(b) are similar in both wallnormal locations suggesting an equal probability in the inclination. Also, the results compare well with the angles computed from the covariance plots. The inclination angle in the x z plane, θ xz is shown in figure 4.17(c). This plot reveals peaks in the first and third quadrants of the x z plane, that is analogous to the forward leaning positive and negative legs of a hairpin-type vortex. The angles in the first and third quadrants and the angles in the second and fourth quadrants can be combined to represent a single eddy inclination angle (θ i ) that varies from 0 θ i 180. This eddy inclination angle is computed on the assumption that the eddies are symmetric, and all forward leaning cores (both positive and negative legs as shown in figure 4.13) are accumulated into one group while all backward leaning cores are grouped together. The range of angles for the forward leaning cores is 0 θ i 90 while the backward leaning cores range as 90 < θ i 180. The resulting p.d.f. of the eddy inclination angle is given in figure 4.18(a) and it has peaks at angles of 46 at z + = 110 and 42 at z/δ = These results are comparable with a 45 hairpin inclination as suggested by various researchers over the past century. The decrease in θ i with wall-normal location is similar to the to results obtained from the covariance plots. The ratio of the area under the curve for 0 < θ i 90 and 90 < θ i 180 was computed to study the relative density of forward and backward leaning cores. This ratio was found to be 5.3 at z + = 110 and 2.4 at z/δ = This indicates that the number of forward-leaning cores is much larger in the log region 109

135 than in the wake region which is in congruence with the results from covariance plots in section and the finding of Ong & Wallace (1998). Figure 4.18(b) shows the absolute number density of θ i at the two wall-normal locations. Clearly, a larger number of total vortex cores were found in the log region than in the wake region. An interesting point to note from this figure is that the number density of the backward-leaning cores in the log region and the wake region remains relatively constant thereby suggesting a universality in the number of backward leaning cores. Also, the correlation plots in figure 4.12 suggested that an average backward-leaning core is smaller in size than an average forward-leaning core Possible interpretation A possible interpretation of the structure of the turbulent boundary layer based on the results in section and is outlined below. A schematic representation of the structure of a boundary layer in the log region is given in figure In the log region, the number of forward-leaning cores is much larger than backward-leaning cores as seen from figure 4.18(b). Since, the correlation plots in figures 4.12 indicated that backward leaning cores are smaller in size, figure 4.19 shows 4 small-scale backward-leaning cores. All the other cores shown are leaning forward. Forward-leaning cores are present in a range of scales as proposed by Perry & Marusic (1995) in the wall-wake model, however it can be hypothesized that the number of small-scale forward-leaning cores is to equal the number of backwardleaning cores. This is consistent with the presence of small-scale cores of all possible orientations which can be considered a relatively universal small-scale feature across the boundary layer (analogous to Kolmogorov hypothesis). Therefore their total number density, including both forward- and backward-leaning types remains the same across the layer. Figure 4.20 shows the schematic of a representative boundary layer in the outer wake region. This shows 4 small-scale forward- and 4 small-scale backward-leaning cores (number density is universal, same for log and wake region), and all the other larger cores are forward leaning. However, the total number of cores in the outer region is smaller and hence the ratio of number of forward leaning to the number of backward leaning cores is smaller. Note that this model does not incorporate of streamwise and spanwise oriented cores that could be possible legs or heads of these hairpin-type vortices. 110

136 (a) (b) Figure 4.19: A schematic representation of the structure of a boundary layer in the logarithmic region. (a) Perspective view and (b) Plan view. 111

137 (a) (b) Figure 4.20: A schematic representation of the structure of a boundary layer in the outer wake region. (a) Perspective view and (b) Plan view. 112