DEVELOPMENT AND ERROR ANALYSIS OF A CONRAD PROBE FOR. Zhou Yuan

Transcription

1 DEVELOPMENT AND ERROR ANALYSIS OF A CONRAD PROBE FOR MEASUREMENTS OF D VELOCITY IN A LAMINAR BOUNDARY LAYER by Zhou Yuan A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Aerospace Science and Engineering University of Toronto Copyright c 1 by Zhou Yuan

2 Abstract Development and error analysis of a Conrad probe for measurements of D velocity in a laminar boundary layer Zhou Yuan Master of Applied Science Graduate Department of Aerospace Science and Engineering University of Toronto 1 The present study proposes to use a Conrad probe for transient growth study by measuring the two-dimensional velocity behind an array of roughness elements in the Blasius boundary layer. A look-up table approach is proposed to increase the accuracy of the data reduction process at low velocities, based on the results of the calibration performed in a round jet. A velocity correction method is proposed to minimize the errors due to high velocity shear and wall-proximity in the Blasius boundary layer by comparing Conrad probe results to previous hot-wire data. Measurements of the steamwise velocity perturbation obtained with the Conrad probe agree with previous studies. The measured spanwise perturbation confirms the transport process suggested by simulation. The results also show that the perturbation amplitude increases by increasing both the freestream velocity and roughness elements height. However, the mechanism for changing the perturbation amplitude of influence differs for these two parameters. ii

3 Acknowledgements It is a pleasure to thank all who made this thesis possible and who made my time of studying here an enjoyable experience. First of all, I would like to thank my supervisor Professor Philippe Lavoie for giving me the opportunity to work on this topic, for his patient guidance, encouragement and advice. Thank you for helping me through the difficulties I encountered as an international student. Thank you for the wonderful Christmas parties you hosted at your home every year. I would like to thank everyone in FCET group. Thanks to Ronald Hanson for his great help. Thank you for teaching me how to start my experiments at the beginning. Thank you for the help during the measurements and all through this project. Thanks to Heather Clark for spending so much time on reading through and editing the drafts of this thesis. Thank you for giving me the help on how to explain my work well. Thanks to Dr. Arash Naghib Lahouti for the advices on both the project and my thesis. Thanks to Rafik Chekiri for the help in my thesis writing. Thanks to Jason Hearst for kind helps when I started my study in an English speaking country for the first time. Thanks to Dr. John Murphy, Luke Osmokrovic, Leon Li and Nicole Houser for the suggestions. Thanks to our alumni, Dr. Leticia Gimeno for explaining the PIV technique to me, thanks to Jagannath Rajasekaran and Denis Palmeiro. It was really my pleasure to meet such a wonderful group of people. I would like to thank all the members in the RAC, for their advice and suggestions in the early stage of this project. Thanks to the second reviewer of my thesis, Professor C. A. Steeves, for his time and valuable comments. Thanks to Professor D.W. Zingg and Professor C.P.T. Groth for the courses at the first year, through which I gradually got used to the life as a student here. I would also like to thank all my friends at UTIAS, for their help and support. Last but not least, many thanks to my parents for their continual support and even several visits all the way from China. Thanks dad for sharing his experience as a researcher. Thanks mom for staying with me last Christmas and coming for my DMS. iii

4 Contents 1 Introduction Motivation Present study Background 4.1 Transient growth theory Viscous linear stability theory Transient growth mechanism Optimal perturbations theory Boundary layer experimental works Two-hole Conrad probe Experimental details Jet flow calibration Calibration apparatus Calibration range and procedure Wind tunnel experiment Experimental Apparatus Experimental conditions Experimental procedure Conrad probe data reduction Data reduction scheme Previous data reduction scheme Reynolds number effect on the flow field Proposed Re dependent scheme iv

5 4.1.4 Evaluation of LUT method Pressure probe measurement errors Pitot tube studies Conrad probe correction method Results and discussions Streamwise evolution Streamwise velocity perturbations Spanwise perturbation Effect of freestream velocity Effect of roughness elements height Conclusions and future work Conclusions Future work A Uncertainty analysis 73 A.1 Uncertainty on measured results A.1.1 Uncertainty on calculated results Bibliography 75 v

6 Nomenclature A i, B i LUT method coefficients, i = 1 for p 1, i = for p, page 31 C p Pressure recovery coefficient, page 9 C p Non-dimensional angular coefficient by Ulk (1), page 6 C pa Non-dimensional velocity coefficient by Ulk (1), page 6 D Outer diameter of Conrad probe tubes, page 14 D + Non-dimensional probe diameter defined by McKeon et al. (3), page 38 E v Disturbance kinetic energy, page 7 E in Disturbance kinetic energy at input location, page 8 E out Disturbance kinetic energy at output location, page 8 G Maximum growth factor, page 8 H Shape factor H = δ /θ, page 43 Re d Reynolds number based on probe geometry, page 7 Re h Reynolds number based on roughness elements height Re h = U(h)h/ν, page 3 T Ambient temperature in wind tunnel measurements, page U Time-averaged streamwise mean velocity, page 1 U Freestream velocity, page 5 V Time-averaged wall-normal direction velocity, page 1 W Time-averaged spanwise velocity, page 1 vi

7 U Streamwise velocity correction for boundary layer measurements, page 38 W Spanwise velocity correction for boundary layer measurements, page 45 y Position correction for boundary layer measurements, page 36 z Spanwise spacing between roughness elements, page 1 Φ LUT method mapping function for θ, page 3 Φ u Spanwise-wavenumber power spectrum of u, page 5 Φ w Spanwise-wavenumber power spectrum of w, page 55 α Streamwise wavenumber in wavelike solutions, page 6 α c Non-dimensional velocity gradient defined by McKeon et al. (3), page 37 β Spanwise wavenumber in wavelike solutions, page 6, non-dimensional spanwise wavenumber β = πδ/λ, page 5 δ Blasius similarity length scale δ = νx v /U, page 4 δ Boundary layer displacement thickness, page 5 η Non-dimensional wall-normal direction in boundary layer, η = y/δ, page 4 η y Wall-normal direction vorticity, page 6 κ κ = α + β, page 6 λ Spanwise-wavelength, page 11 U spanwise-averaged U, page 1 W Spanwise-averaged W, page 45 D Differentiation with respect to wall-normal direction, page 6 L os Orr-Sommerfeld operator, page 7 L sq Squire operator, page 7 ν Kinematic viscosity of air, page 5 vii

8 ω Frequency in wavelike solutions, page 6 ψ Tip angle of Conrad probe tubes, page 14 u in Velocity vector at input location, page 8 u out Velocity vector at output location, page 8 θ Flow yaw angle, page 14 θ Boundary layer momentum thickness, page 43 θ c Recorded θ in test, page 3 φ Flow pitch angle, page 14 a, a k, b k Third-order Fourier analysis coefficients, page 58 a i, b i, c i, d i, e i, f i LUT method coefficients, i = 1 for p 1, i = for p, page 31 d Inner diameter of Conrad probe tubes, page 14 d h Roughness elements diameter, page 1 e U Velocity error coefficient of data reduction methods, page 33 e θ Angular error coefficient of data reduction methods, page 33 h Roughness elements height, page 1 k Dimensional spanwise wavenumbers k = π/λ, page 5 p Perturbation pressure, page 6 p 1 Left pressure port reading of Conrad probe, page 14 p Right pressure port reading of Conrad probe, page 14 p a Atmospheric pressure, page 19 p d Dynamic pressure, page 19 p o Total pressure, page 9 p s Static pressure, page 19 viii

9 p avg Average of p 1 and p, p avg = (p 1 + p )/, page 6 u Streamwise perturbation velocity, page 5 u rms Spanwise standard deviation of u, page 43 v Wall-normal direction perturbation velocity, page 5 v rms Spanwise standard deviation of v, page 58 w Spanwise perturbation velocity, page 5 w rms Spanwise standard deviation of w, page 45 x Streamwise direction in boundary layer flow, page 5 x f Streamwise output location, page 8 x h Streamwise distance between leading edge and roughness elements, page 1 x v Streamwise location from virtual leading edge, page 4 y Wall-normal direction in boundary layer flow, page 5 y o Wall-normal direction initial location in boundary layer measurements, page 3 z Spanwise direction in boundary layer flow, page 5 ix

10 List of Tables 3.1 The specifications transducers used to measure p 1, p and p d Summary of experiment conditions and streamwise positions of the measurement planes x

11 List of Figures 1.1 Photograph of a Conrad probe taken under a microscope The streaky structures observed in boundary layers using smoke flow visualization (Matsubara & Alfredsson, 1) The initial perturbation velocity and the corresponding velocity at the output location calculated by the optimal theory (Andersson et al., 1999; Luchini, ). 9.3 Schematic of the roughness elements experimental setup (Fransson et al., 4) Roughness elements experiment results (Fransson et al., 4) Typical cross-wire (X) probe for simultaneous measurement of two velocity components, all dimensions in millimeters (figure from Butterfield, 1999) Three-view plot and notations of Conrad probe Design of Conrad probe (Bryer & Pankhurst, 1971) Schematic of the calibration setup for the Conrad probe Jet velocity related to blower control signal voltage Photograph of the Conrad probe Flowchart for the calibration process Schematic of the boundary layer model CCD camera picture of the Conrad probe position setting The curves fitted into polynomials for c pa = f(θ) and c p = g(θ) (figure from Ulk, 1) Angular coefficient C p under low velocities (a) The schematic of the high Reynolds number flow, and (b) of the low Reynolds number flow around the probe The viscous effect on the round, square-ended Pitot tubes (Tropea et al., 7) Comparison of the pressure port readings p 1 and p, with and without smoothing. 3 xi

12 4.6 The angular-averaged velocity error of LUT and Ulk s methods. Displayed as functions of flow speed The angular-averaged angular error of LUT and Ulk s methods. Displayed as functions of flow speed Effect of a Pitot probe on the streamline pattern: (top) in the shear flow and (bottom) in a uniform flow near the wall (McKeon et al., 3) A portion of a velocity profile (from MacMillan, 1956). The horizontal axis stands for the destines from the wall and the vertical axis stands for the velocity. The two curves on the figure are the measured profile (labeled as U) and the true profile (as U a ) respectively. The upper-left data point on the measured profile is the directly measured velocity. It can be corrected to either to the upper-right point on the true profile by a displacement correction or to the upper-right point by a velocity correction (a) Wall proximity correction for pitot tubes by Bryer & Pankhurst (1971), data from MacMillan (1956); (b) Total displacement correction y/d by MacMillan (1956). U τ D/ν is a factor of y/d, which indicated the measurement condition. U τ is the friction velocity defined by the relation: shear stress at wall = τ = ρuτ ; y is the distance from the wall; ν is the kinematic viscosity. y/d is solved as a function of U τ D/ν and y/d (which showed as the inverted D/y on the figure) Non-dimensionless spanwise-averaged streamwise velocity profiles U /U of hot-wire and Conrad probe measurements comparing to Blasius solution at flow condition of U = 5 m/s, x = 3, 4 and 495 mm from plate leading edge U/U calculated at η =.5-7 in.5 increment. For the results at the flow condition of U = 5 m/s, h = 1.5 mm, x = 65 mm gives the smallest D/δ value and x = 5 mm gives the largest D/δ values. The dashed lines are iso-η lines. The solid lines are the proportional fits between U/U and D/δ and only the data at U = 5 m/s, h = 1.5 mm, x = 5-65 mm are used for the fit The relative error of calculated x v after each iteration, U = 7.5 m/s, h = 1. mm and x = 3 mm from plate leading edge. The settings of the initial values are x v = 1,, 3, 4 and 5 mm respectively as shown in the legend.. 43 xii

13 4.14 Non-dimensional spanwise-averaged streamwise velocity profiles U /U measured with the Conrad probe, before and after correction, compared to that measured with the hot-wire Displacement thickness δ shown at the streamwise location from the virtual leading edge x v, measured for U = 5 m/s and h = 1.5 mm Shape factor H from the virtual leading edge x v at flow condition of U = 5 m/s and h = 1.5 mm. The dashed line corresponds to the Blasius solution of H = (a) Non-dimensional spanwise-averaged spanwise velocity profiles W /U see corrections for figure 4.14, and (b) spanwise standard divination of spanwise perturbation velocities w rms /U of Conrad probe measurements before and after correction Contour plots of the normalized streamwise perturbation velocity u/u measured by the Conrad probe at U = 5 m/s, h = 1.5 mm, (a) x = 3 mm, (b) x = 45 mm, and (c) x = 65 mm Contour plots of the normalized streamwise perturbation velocity u/u measured by the hot-wire probe at U = 5 m/s, h = 1.5 mm, (a) x = 3 mm, (b) x = 45 mm, and (c) x = 65 mm Comparison of Conrad probe and hot-wire u rms /U and w rms /U profiles at (a) x = 3 mm, (b) x = 45 mm and (c) x = 65 mm Ratio between the Conrad probe and the hot-wire u rms /U peak value at U = 5 m/s, h = 1.5 mm. When the ration u rms /U = 1, it means the Conrad probe and the hot-wire measuring the same peak value Spanwise-wavenumber power spectra Φ u /U and Φ w /U computed at the y location where u rms peaks. Measured at U = 5 m/s, h = 1.5 mm and (a) x = 3 mm, (b) x = 45 mm and (c) x = 65 mm Spanwise velocity profiles of streamwise perturbation u/u measured with the Conrad probe. Computed at the y location where u rms peaks, U = 5 m/s, h = 1.5 mm Disturbance energy contained in the full spectrum ΣΦ u (β)/u, fundamental mode Φ u (β 1 )/U and ΣΦ u (β)/u Φ u (β 1 )/U, measured with the Conrad probe. Computed at the y location where u rms peaks, U = 5 m/s, h = 1.5 mm. 54 xiii

14 5.8 Contour plots of the normalized spanwise perturbation velocity w/u measured by the Conrad probe, at U = 5 m/s, h = 1.5 mm, (a) x = 3 mm, (b) x = 45 mm, and (c) x = 65 mm Disturbance energy contained in the full spectrum, fundamental mode and the higher modes of the Conrad probe spanwise-wavenumber power spectra Φ w /U at the respective wall-normal locations of the maximum disturbance energy at U = 5 m/s, h = 1.5 mm Spanwise velocity profiles of W/U before and after smoothing. Measured with the Conrad probe, at U = 5 m/s, h = 1.5 mm and x = 45 mm Contour plots of the normalized wall-normal direction perturbation velocity v/u calculated by the Conrad probe data at U = 5 m/s, h = 1.5 mm, (a) x = 3 mm, (b) x = 45 mm, and (c) x = 65 mm Calculated v rms /U profiles at U = 5 m/s, h = 1.5 mm, x = 3, 45 and 65 mm Contour plots of the three-dimensional perturbation (a) u/u, (b) v/u and (c) w/u at x = 35 mm. Data from Lavoie et al. (8) Spanwise divination of the three-dimensional perturbation at x = 35 mm, data from Lavoie et al. (8) Contour plots of the normalized streamwise perturbation velocity u/u measured by the Conrad probe at x = 3 mm, h = 1. mm. From the top to the bottom are (a) U = 5 m/s, (b) U = 7.5 m/s, and (c) U = 9 m/s Contour plots of the normalized spanwise perturbation velocity w/u measured by the Conrad probe at x = 3 mm, h = 1. mm. From the top to the bottom are (a) U = 5 m/s, (b) U = 7.5 m/s, and (c) U = 9 m/s Comparison of Conrad probe spanwise standard divination profiles of (a) u rms /U and (b) w rms /U at x = 3 mm Contour plots of the normalized streamwise perturbation velocity u/u measured by the Conrad probe at U = 5 m/s, x = 3 mm, (a) h = 1.5 mm, and (b) h = 1. mm Contour plots of the normalized spanwise perturbation velocity w/u measured by the Conrad probe at U = 5 m/s, x = 3 mm, (a) h = 1.5 mm, and (b) h = 1. mm A.1 The uncertainty of (top) U and (bottom) θ solved by LUT xiv

15 Chapter 1 Introduction 1.1 Motivation The transition of laminar boundary layers to a turbulent state has important implications for many aeronautical applications, because a turbulent boundary layer produces much higher skin-friction drag than an equivalent laminar boundary layer. A popular approach to reduce the skin-friction drag is to delay the laminar to turbulence transition for as long as possible. In order to delay the process which leads to a breakdown of the laminar boundary layer, the transition process has to be understood first. However, the transition process is a very complex phenomenon. As reviewed by Saric et al. () and Reshotko (1), when weak initial disturbances are present, the initial growth can be described by classical stability theory. In this classical theory, transition is believed to be triggered by the exponential growth of twodimensional Tollmien-Schlichting (T-S) waves. Until the early 199 s, the predominant view of transition was centered on the T-S wave. When the background disturbance level is elevated, transition can occur at subcritical Reynolds numbers (of classical theory) due to the influence of three-dimensional perturbations (Saric et al., ). This phenomenon is known as transient growth, which does not follow the classical stability theory of T-S waves but still remains linear (Schlichting & Gersten, ). The transient growth process has attracted significant attention in recent years. Butler & Farrell (199, 1993) showed that nominally stable three-dimensional perturbations in a laminar boundary layer can experience large algebraic growth before decaying exponentially under the effect of viscosity. Later numerical analysis of optimal growth theory showed that the initial disturbance with the form of spanwise-periodic pairs of streamwise counter-rotating vortices can lead to the largest energy growth over a given streamwise distance (Andersson 1

16 CHAPTER 1. INTRODUCTION et al., 1999; Luchini, ). This initial disturbance creates streamwise high and low velocity streaks downstream, which was predicted by the earlier study of Landahl (198) and validated later by the experimental work of Matsubara & Alfredsson (1). In order to better understand the receptivity of the transient growth type disturbances and thus verify optimal growth theory, experimental studies were performed to investigate the streamwise velocity disturbance downstream of a spanwise array of roughness elements (White, ; White et al., 5; Fransson et al., 4). In each of these studies, a roughness array was used to generate transient growth disturbances in a Blasius boundary layer. Their work confirmed the existence of the flow structures predicted by numerical works, but the results also showed flow behavior that was sub-optimal to the flow field predicted numerically. Since the measurements were limited to the streamwise velocity component only, it has been difficult to quantitatively examine the details of the initial perturbation introduced by the roughness element array, thus making comparison to theory uncertain. The measurement of all components of the initial velocity perturbation presents many challenges, primarily due to the very high shear present in the boundary layer. This prevents the use of standard multi-sensor hotwire probes due to their unacceptable resolution in the wall-normal direction. The seeding and lighting problems in thin laminar boundary layer also prevents the use of laser-based diagnostic methods, which is discussed in more detail in section.3. Thus the focus of this thesis is to develop a multi-components velocity measurement technique without the problems that conventional technique possess when measuring velocity in a thin boundary layer. 1. Present study In the present work, a Conrad probe is used to measure the streamwise and spanwise velocity components simultaneously in the laminar boundary layer behind an array of roughness elements. As shown in figure 1.1, the Conrad probe is a two-hole pressure probe that measures two-dimensional velocity components. Previous studies (Brebner, 195; Bryer et al., 1958; Ulk, 1) showed that the Conrad probe is a compact, reliable, robust and economic instrument that is also easy to manufacture and calibrate. In the interest of measuring the present flow field, the Conrad probe was first calibrated in a jet flow and the results were used to develop a look-up table based data reduction scheme. Aiming at studying the multi-dimensional flow induced by roughness elements, the probe was then used in a wind tunnel experiment to measure the streamwise and spanwise velocity components in the thin laminar boundary layer. A velocity correction method is developed to correct the error due to shear and wall-proximity

17 CHAPTER 1. INTRODUCTION 3 Figure 1.1: Photograph of a Conrad probe taken under a microscope. There is a 1 mm ruler beneath the probe tip. effects by comparing Conrad probe data to previous hot-wire results by Hanson et al. (1, 1) which were taken under the same experimental conditions. In this thesis, the background of transient growth theory and the Conrad probe are reviewed in Chapter. Chapter 3 introduces the setup of the round jet calibration and the wind tunnel boundary layer experiment. Chapter 4 introduces the data reduction scheme proposed in the present study and the correction method for the laminar boundary layer flow. The experimental results are discussed in Chapter 5 and conclusions follow in Chapter 6.

18 Chapter Background This chapter provides the background to the present study. The introduction to the transient growth begins with a derivation of the theoretical solution in section.1. The optimal perturbation theory and its numerical applications are reviewed in section.. Section.3 summarizes the experimental works aiming at validating the numerical predictions and section.4 introduces the Conrad probe, which is the measurement instrument used in the present study..1 Transient growth theory The study of transient growth emanated from the studies on the three-dimensional disturbance amplification mechanism. Ellingsen & Palm (1975) showed that a finite disturbance without streamwise variation can lead to a linear increase in the streamwise velocity component in an inviscid, incompressible and nonstratified shear layer, even though the basic velocity does not possess any inflection point (which gives a stable flow according to the two-dimensional stability criterion). Theoretical analysis by Landahl (198) showed that all parallel inviscid shear flows are unstable to a wide class of three-dimensional disturbances. The result is independent of whether the flow is stable to the two-dimensional T-S wave (Schmid & Henningson, 1; Drazin & Reid, 4). The three-dimensional disturbance causes a localized three-dimensional up and down motion near the wall, which is called the lift-up mechanism (Landahl, 198). The lift-up motion creates the high and low speed streaks that are streamwise elongated in the boundary layer and the disturbance energy contained in the boundary layer grows algebraically in time according to the temporal inviscid formulation by Landahl (198). This theoretical prediction was supported by a later flow visualization by Matsubara & Alfredsson (1) as shown in figure.1. 4

19 CHAPTER. BACKGROUND 5 Figure.1: The streaky structures were observed in boundary layers using smoke flow visualization. The four figures are the flow fields subjected to different freestream turbulence intensities and freestream velocities. Flow direction is from left to right (figure from Matsubara & Alfredsson, 1)..1.1 Viscous linear stability theory Transient growth theory was summarized in the book by Schmid & Henningson (1) and the derivation is summarized in this literature review. The spatial coordinates x, y and z denote the streamwise, wall-normal and spanwise directions and u, v and w are the x-, y- and z-direction perturbation velocity components. The derivation begins with the linearized three-dimensional momentum and continuity equations for infinitesimal disturbances in the parallel flows, which are given as follows: u t + U u x + vu = p x + 1 Re u, (.1) v t + U v x w t + U w x u x + v y + w z = p y + 1 Re v, (.) = p z + 1 Re w, (.3) =, (.4) where U = U(y) is the streamwise parallel base flow and the prime ( ) denotes a y-derivative. The Reynolds number is given as Re = U δ /ν with the freestream velocity U, the boundary layer displacement thickness δ and the kinematic viscosity ν. Taking the divergence of the

20 CHAPTER. BACKGROUND 6 momentum equations (.1) - (.3), then using (.4), yields the equation for the perturbation pressure, viz. p = U v x. (.5) The pressure p can be removed from (.) by substituting (.5) to obtain the following: [( t + U ) U x x 1 ] Re 4 v =. (.6) Defining the y-direction vorticity η y = u/ x w/ x, the following equation is satisfied: [ t + U x 1 Re ] η y = U v z. (.7) Equations (.6) and (.7), together with the initial conditions and the boundary conditions given as v(x, y, z, t = ) = v (x, y, z), (.8) η y (x, y, z, t = ) = η (x, y, z), (.9) v = v = η y = at solid boundary and far field, (.1) describe an arbitrary three-dimensional disturbance. Equations (.8) and (.9) stand for the general expressions of the initial disturbance. The essence of this set of equations is the viscous linear incompressible three-dimensional boundary layer approximation of the N-S equations. Assume a set of wavelike solutions (α and β are streamwise and spanwise wave numbers, ω is the complex frequency) in the form, v(x, y, z, t) = ṽ(y)e i(αx+βz ωt), (.11) η y (x, y, z, t) = η y (y)e i(αx+βz ωt), (.1) the three-dimensional Orr-Sommerfeld and Squire equations can be solved by applying these solutions for (.6) and (.7), with the prescribed boundary conditions (.1): [( iω + iαu)(d κ ) iαu 1 Re (D κ ) ]ṽ =, (.13) [( iω + iαu) 1 Re (D κ )] η y = iβu ṽ, (.14) where κ = α + β and D stands for differentiation with respect to y-direction. Equation (.13) is the classical two-dimensional Orr-Sommerfeld equation for ṽ and (.14) is the Squire

21 CHAPTER. BACKGROUND 7 equation for normal vorticity η. These two equations can also be written into matrix form with the Orr-Sommerfeld and Squire operator L os and L sq shown as below, ( ) ( ) ( κ D ṽ Los iω + 1 η y iβu L sq ) ( ) ṽ =, (.15) η y L os = iαu(κ D ) + iαu + 1 Re (κ D ), (.16) L sq = iαu + 1 Re (κ D ). (.17) Equation (.15) describes the evolution of the three-dimensional wavelike perturbation and can explain transient growth theoretically. The classical T-S wave is a solution of the twodimensional Orr-Sommerfeld equation (.13). When the amplitude of the initial disturbance increases but remains small to satisfy the linearized governing equations, the distributed disturbance will start to show its three-dimensional characteristics and the Squire equation (.14) should be coupled with the Orr-Sommerfeld equation to give solutions of the three-dimensional problem (Schmid & Henningson, 1). The lift up mechanism is a result of the forcing of the Squire modes by the wall-normal velocity perturbations due to the coupling operator in the second term of (.15)..1. Transient growth mechanism The transient growth mechanism can be illustrated in the view of disturbance kinetic energy, which is expressed as (Schmid & Henningson, 1) E v = 1 u i u i dv = E(u), (.18) V where the choice of the volume V is dependent on the flow geometry. According to the twodimensional stability theory, the disturbance kinetic energy grows and decays when different forms of initial perturbations are introduced into the flow field. However, for certain forms of initial disturbances that should be stable by two-dimensional stability criteria, the kinetic energy experiences a linear growth before its final decay. This transient growth becomes important because the linear growth process may trigger an early transition in the flow field. The transient growth mechanism can be explained by the eigenmodes of the theoretical equations. The nonorthogonal modes are contained in the initial disturbance and each single mode decays spatially. However, since these modes decay at different rates, their superposition can cause a linear growth before the final decay (Schmid & Henningson, 1). This occurs in

22 CHAPTER. BACKGROUND 8 physical space when the disturbance energy gained by the redistribution of the flow field in a certain downstream range is larger than the loss of the energy associated with the decay of the initial disturbance. Numerical and experimental works have been performed to study the flow patterns that can induce transient growth, which are introduced in following sections.. Optimal perturbations theory Optimal perturbations theory is derived by solving the coupled Orr-Sommerfeld and Squire equations in the zero pressure gradient boundary layer flow (i.e. Blasius boundary layer) numerically. The basic idea of the optimal perturbations theory is to identify what type of initial condition can achieve a maximum disturbance kinetic energy gain at the output location by inputting different types of initial perturbation and solving the resulting flow field numerically. By choosing two streamwise in and out locations in the flow field, the velocity vectors u in and u out and the disturbance kinetic energies at these two locations are given as (Andersson et al., 1999) E in = E(u in ), E out = E(u out ). The maximum growth factor is defined as the ratio of the disturbance kinetic energies at these two locations (Andersson et al., 1999), G = max E out E in. (.19) The term optimal perturbation was first introduced by Butler & Farrell (199, 1993), to denote the shape of the input flow field leading to maximum amplification of the disturbance energy. The authors studied the temporal stability of parallel shear flows, including Blasius flow, and the results showed that nominally stable three-dimensional perturbations in a laminar boundary layer can experience large algebraic growth before decaying exponentially under the effect of viscosity. These studies showed that the optimal perturbation is in the form of longitudinal vortices that induced disturbances in the form of streamwise elongated streaks, which confirmed the lift-up amplification mechanism of Landahl (198). Andersson et al. (1999) studied the problem spatially in the Blasius boundary layer, as defined above. They chose the leading edge as the input location and the maximum gain G was calculated as a function of streamwise output location x f, spanwise wavenumber β and the Reynolds number Re. Their study suggested that the optimal initial perturbations are spanwise periodic pairs of counter-rotating vortices (figure.a) and this optimal initial perturbation produces streamwise

23 CHAPTER. BACKGROUND 9 elongated streaks of high and low speed fluid within the boundary layer at the output location (figure.b). Luchini () adopted a spatial backtracking optimization technique that is independent of Reynolds number. As shown in figure.(c), a similar result was obtained for the initial perturbation. It was noticed that when the condition Re was applied for Andersson et al. (1999), their results were equivalent those of (Luchini, ). Figure.: (a) Initial perturbation velocity vectors in the spanwise and wall-normal plane, u component is zero (adapted from Andersson et al., 1999); (b) contours of constant streamwise velocity representing the downstream response that results from the optimal disturbance, here the solid lines represent positive values and the dashed lines represent negative values (adapted from Andersson et al., 1999); (c) initial perturbation velocity profiles of v and w (figure from Luchini, ); (d) downstream response corresponding to the optimal perturbations at the output location, v and w components are zero at this point (figure from Andersson et al., 1999).

24 CHAPTER. BACKGROUND 1 The numerical prediction showed that streamwise counter-rotating vortex pairs can transfer high-velocity fluid down to the low-velocity region in the wall-normal direction and, similarly, moves the low-velocity fluid up to the high-velocity region. This initial perturbation creates the high and low-speed streamwise streaks that are often found in boundary layers and produces a disturbance kinetic energy growth through the redistribution of perturbation velocities..3 Boundary layer experimental works Experimental studies by White and co-workers (White, ; White & Ergin, 4; White et al., 5) and Fransson et al. (4) validated spatial optimal theory by analyzing results obtained behind a roughness element array over a flat plate (see for instance figure.3). In this type of measurement, Blasius boundary layer flow was produced over the flat plate and a spanwise array of roughness elements was used to generate transient growth disturbances at certain locations downstream of the leading edge. The roughness elements can generate counter-rotating streamwise vortex pairs behind them, which give a flow pattern similar to the optimal perturbation as discussed above. Moreover, this experimental setting can introduce an adjustable input perturbation that is comparable to the initial condition in the optimal theory. As shown in figure.3, different flow conditions can be achieved by changing the experimental setting of roughness element diameter d h, height h, spanwise distance z, streamwise distance from leading edge to roughness elements x h, and freestream velocity U. These conditions are corresponding to different Re, β, and x f in optimal perturbation theory. In other words, this experimental setting introduces the disturbance under controlled amplitude and specific wavenumbers. It also provides disturbance input at a single streamwise location; thus, the source of disturbance growth observed downstream of the array is unambiguous. During the measurements, the time-averaged streamwise velocity U is measured over y-z planes in the boundary layer at different locations downstream of the roughness array. To better study the behavior of the streamwise spatial perturbation velocity u, the spanwise mean profile U is subtracted from U at each measurement point. Single hot-wires were chosen as the measurement instrument to obtain enough resolution in the thin laminar boundary layer. Previous experiments by White (); White et al. (5); Fransson et al. (4) concluded that the general trends and qualitative behaviors of the flow are correctly captured by optimal theories. As shown in figure.4a, experimental studies confirmed the existence of streamwise high and low u regions in the flow field, which was predicted by Andersson et al. (1999) (as shown in figure.b). The streamwise perturbation velocity profile (figure.4b)

25 CHAPTER. BACKGROUND 11 Figure.3: Schematic of the roughness elements experimental setup. d in the figure is denoted as d h, k as h and x k as x h in present study (figure from Fransson et al., 4). also agrees with numerical results (in figure.d). However, the results also indicated that the measured flow field is sub-optimal to the predicted optimal perturbation mainly due to the limitations of the experimental geometry and the complexity of the real flow field. First, it is difficult to input disturbances or perform measurement at the leading edge, which is chosen as the in location in the numerical studies. Second, it is impossible to obtain perfect vortex pairs with a single spanwise wavenumber. The spectral analysis showed that the energy is contained in higher modes, which means the single wavenumber of β = πδ/ z (also wavelength λ = z) is not satisfied completely in the experiments. The sub-optimal behaviors indicated that the real flow field is even more complex than the predictions. Since previous measurements were limited to the streamwise velocity component, it is difficult to quantitatively examine the details of the initial perturbation introduced by the roughness element array, which makes the comparison to theory (figure.a) unrealistic. This suggests that in order to obtain further understanding of the transient growth phenomenon and to evaluate this sub-optimal behavior quantitatively, multiple-dimensional flow field measurements are necessary. Cross-wire (X) probe is commonly used in two-dimensional velocity measurements. However, as shown in figure.5, the spanwise resolution of the probe is 1.5 mm along spanwise based on the active length of the wire, which is unacceptable in laminar boundary layer measurements. The high shear and low mixing rate also make it difficult to reach sufficient seeding density levels required for laser-based diagnostic methods. Moreover, unwanted laser light reflections occur over the surface of the flat plate which makes the entire boundary layer show as the bright region. Due to the limitations of other multi-dimensional measurement techniques, the present study proposes to use a two-hole Conrad probe to measure the streamwise and spanwise velocity components in the flow field.

26 CHAPTER. BACKGROUND 1 (a) (b) Figure.4: (a) u contour at different downstream locations, positive and negative values are denoted as solid and dashed lines, respectively. The dots are U contour lines in.1u increment (figure from Fransson et al., 4). (b) Disturbance profiles in the wall-normal direction for different downstream locations at U = 7 m/s (figure from Fransson et al., 4). Figure.5: Typical cross-wire (X) probe for simultaneous measurement of two velocity components, all dimensions in millimeters (figure from Butterfield, 1999).

27 CHAPTER. BACKGROUND 13.4 Two-hole Conrad probe As shown in figure 1.1, the Conrad probe is a type of two-hole pressure probe that measures the flow velocity magnitude and the yaw angle relative to the probe (defined in figure.6). It was first introduced by Conrad in 195 and widely used for two-dimensional flow field measurement in the 195 s and 196 s (Brebner, 195; Black, 1953; Brebner & Wyatt, 1961; Horlock, 1957). The probe configuration and calibration methods were also studied from then on (Brebner, 195; Bryer et al., 1958; Bryer & Pankhurst, 1971). Later in the 198 s to 199 s, the Conrad probe was used in combination with other measurement techniques, e.g. hot-wire anemometer and flow visualization methods (Winkelmann, 1981; Ho, 199). Recently, the performance of this probe was compared to other types of pressure probes in boundary layer measurement during in-flight tests (Bender et al., 1; Ulk, 1). The coordinate system of the flow is defined relative to the probe orientation in figure.6. Early research showed that the Conrad probe can be used for two-dimensional flow measurement since the probe is sensitive to yaw angle but relatively insensitive to pitch angle. After the study by Conrad (195), Brebner (195) first used the name Conrad probe and used it to measure the flow field velocity, dynamic pressure, total head and yaw angle 5 mm behind the trailing edge of a 59 sweptback wing. The calibration curve and data reduction scheme for the Conrad probe were also discussed in the study. It was shown that for the yaw angle 1 θ 1, the non-dimensional pressure difference in the two tubes of the probe was proportional to θ and that the probe reading was the same for pitch angles of φ = 6 and φ = 1. The probe was also used in studies of swept wings (Black, 1953; Brebner & Wyatt, 1961) and an axial flow compressor (Horlock, 1957). Bryer et al. (1958) and Bryer & Pankhurst (1971) discussed the optimal configuration, calibration curve, and measurement method of the Conrad probe in detail and concluded that this probe is generally superior to other types of multi-hole probes for boundary layer measurements. Bryer et al. (1958) studied the influence of the tube inner and outer diameter d and D, and the tip angle ψ experimentally. The authors compared the performances of eight types of pressure probes, including four different types of Conrad probe, in a uniform stream and a turbulent boundary layer. The Conrad probe showed its advantage in obtaining accurate results when calculating displacement thickness and momentum thickness. By analyzing the experimental data, Bryer et al. (1958) suggested that a construction of ψ = 35 gave a better yaw sensitivity than a tip angle of 6, and a smaller outer diameter with d/d ratio less than.6 gave more accurate results in boundary layer measurement. They also proposed a simplified

28 CHAPTER. BACKGROUND 14 Figure.6: Three-view plot and notations of Conrad probe. The pitch angle φ is in x-y plane, the yaw angle θ is in x-z plane and the tip angle is shown as ψ. data reduction method for the Conrad probe and reported that the probe reading was sensitive to yaw but insensitive to pitch angle, similar to earlier studies. A later book by Bryer & Pankhurst (1971) summarized the study of Bryer et al. (1958) and suggested probe configurations with outer diameters D =.8, 1. or 1.5 mm, d/d =.6, and ψ = 35. These parameters were chosen based on wind tunnel experiments as well as the ease of manufacturing. As shown in figure.7, the tubes were connected to larger diameter tubing at D from the tip, then bent 9 at 4D for the boundary layer measurement. These lengths were chosen by balancing probe strength and avoiding flow blockage. An 8D tube was used for the probe pole so as to withstand aerodynamic loading during in calibration and wind tunnel experiments. In the 198 s to 199 s, the Conrad probe was used in combination with other measurement techniques. Winkelmann (1981) used hot-wire, split-film, and Conrad probes, as well as oil and smoke flow visualization, to study separated flow on a finite wing. Winkelmann reconstructed the surface flow and the three-dimensional wake of the wing. The flow visualization was also used in enhancing the Conrad probe measurement in a subsonic intake duct flow by Ho (199). Recently, the Conrad probe became a candidate for measuring the swept wing boundary layer in flight tests for its robustness and its fabrication simplicity. Bender et al. (1) compared the performance of a Conrad probe, total pressure probe and single-hole rotatable probe on flight tests of a 3 o swept wing laminar boundary layer. The streamwise and spanwise velocity components over the wing were measured. The study showed that the streamwise velocity acquired by the Conrad probe is comparable to the results obtained from total and

29 CHAPTER. BACKGROUND 15 Figure.7: Design of Conrad probe (figure from Bryer & Pankhurst, 1971). θ in the figure is denoted as ψ in this thesis. rotatable probes. However, the Conrad probe is less time consuming than the rotatable probe during the multi-dimensional velocity measurement. The thesis by Ulk (1) detailed the design, calibration, and performance evaluation of the Conrad probe, which was then applied on a subsonic in-flight measurement. After rigorous literature review, it was concluded that d/d =.6 and ψ = 45 o give the optimal performance for the Conrad probe. Two versions of the probe were manufactured: the.64 (1.656 mm) outer diameter prototype probe and the final D =. (.58 mm) one. A data reduction method different from Bryer & Pankhurst (1971) was proposed and a relative calibration curve was also introduced. Sensitivity, uncertainty and the performance within the flight test of a swept wing boundary layer was then discussed. The Conrad probe used in the present study is designed based on this literature review and the present boundary layer experimental setup is similar to the previous studies introduced in section.3. The streamwise and spanwise perturbation velocities obtained in present study are analyzed and discussed based on the transient growth concept and the optimal perturbation theory that are introduced in this chapter.

30 Chapter 3 Experimental details This chapter introduces the experimental setup and procedures used in the present study. The first section introduces the calibration apparatus. The second section describes the facility, the flow condition and the measurement procedure of the boundary layer wind tunnel experiment. 3.1 Jet flow calibration To use the Conrad probe in the boundary layer experiments, the probe was initially calibrated in a jet flow by exposing the probe to a flow with known velocity magnitude U and yaw angle θ. The pressure readings p 1 and p were recorded while varying U and θ within a certain range. As discussed in section 4.1, a data reduction scheme is proposed to relate p 1 and p to U and θ by building a function to relate the variables based on the calibration results. When the Conrad probe was used in the wind tunnel experiment, p 1 and p were recorded, and U and θ were determined by using the data reduction method. The coordinate system and the notations are as defined in figure Calibration apparatus The calibration system included a round jet driven by a blower, the Conrad probe with its rotatable stand, a total pressure tube measuring the reference velocity, three pressure transducers and the data acquisition system (figure 3.1). The calibrations were performed using the jet flow from a.6 mm diameter circular nozzle jet. The source of the jet was a METEK brushless DC blower that was driven by a 1 volt DC power supply. To increase flow uniformity, the flow from the blower was passed through porous elements before entering an axisymmetric contrac- 16

31 CHAPTER 3. EXPERIMENTAL DETAILS 17 tion. The blower rotational speed was controlled by an input voltage signal that was provide by a NI (National Instruments) USB-69 data acquisition (DAQ) board. To isolate the vibration of the blower, a flexible connection was used between the blower and the jet. Figure 3. shows the relationship between the jet velocity and input voltage to the blower as measured by a total pressure tube. Data set No.1 and No. were obtained in separate experiments. Over the range of 1 to 3 volts, the velocity changes linearly with input voltage from 1 to 19.5 m/s, which covered the velocity range required for the present calibration. Figure 3.1: Schematic of the calibration setup for the Conrad probe. jet velocity (m/s) data No.1 data No. Linear fit blower control voltage (volt) Figure 3.: Jet velocity related to blower control signal voltage. The velocity was measured at the center of the calibration jet exit plane. The Conrad probe used in the present work has been manufactured in-house based on the design outlined by Bryer & Pankhurst (1971) and Ulk (1). It consists of two stainless steel tubes soldered together (figure 3.3). Thinner tubes are preferred to achieve higher spatial

32 CHAPTER 3. EXPERIMENTAL DETAILS 18 resolution, however, the probe will take a longer time to respond to velocity changes if a smaller tube diameter is used. The outer and inner diameters of the tubes for the present Conrad probe are D =.5 mm and d =.3 mm, which were chosen to achieve a reasonable spatial resolution and settling time before sampling. As shown in figure 1.1, the tips of the probe tubes were cut by a disc sander to give a ψ = 45 o section relative to the probe axis. To avoid blockage due to the probe pole, the probe tubes were bent to an angle slightly larger than 9 o at roughly 4 mm from the tip. Both of the tubes were connected to thicker stainless tubes with an outer diameter of 1. mm and inner diameter of.6 mm after the bend. The thicker tubes were soldered together and covered by a D = 3 mm pole to facilitate mounting in the wind tunnel. Figure 3.3: Photograph of the Conrad probe. An aluminum probe stand was designed to hold the Conrad probe vertically during the calibration, to avoid the probe pole blocking the flow. By rotating the probe stand, different yaw angles were achieved for the Conrad probe while the probe tip remained at the center of the jet nozzle exit plane. The stand was driven by a stepper motor which has a resolution of steps per degree. The Motion Group (TMG) MMC-4S-SC 4-axis stepper motor controller was used to control the motor. A total pressure tube with an outer diameter of 1. mm and inner diameter of.6 mm was used to measure the reference velocity U at the outlet of the jet nozzle. It was placed beside the Conrad probe on a separated stand. The distance between the Conrad probe and the total pressure tube was chosen as 8 mm to limit the interaction between the two probes and also ensure the tips of them were both inside the jet potential core, in which region gives a uniform velocity. Three differential pressure transducers (as listed in table 3.1) were used to record the pressure from the total pressure probe and the two ports of the Conrad probe. PVC tubes were used to connect the pressure probes and the transducers. The transducer high pressure sides were

33 CHAPTER 3. EXPERIMENTAL DETAILS 19 connected to the Conrad probe ports and the total pressure tube, which sensed the stagnation pressures, and the low pressure sides were open to atmospheric pressure p a. Since the static pressure p s is equal to the atmospheric pressure p a in a free jet flow (Bruun et al., 1988), the differential pressure transducers measured the dynamic pressures from the total pressure tube (p d ) and the Conrad probe (p 1 and p ). The data acquisition system included a computer and the DAQ board that collected the output voltage signal of the pressure transducers. Matlab was used to interface the computer with the DAQ board and motor controller. Table 3.1: The specifications transducers used to measure p 1, p and p d. Type of the transducers Measurement range accuracy p 1 MKS Baratron 1AD 1 Torr ±.1% of reading p MKS Baratron 1AD 1 Torr ±.1% of reading p d MKS Baratron 5 1 Torr ±.3% of full scale 3.1. Calibration range and procedure The calibration was performed for velocities in the range of U = 1 1 m/s in.5 m/s increment from 1 5 m/s and 1. m/s increment from 6 1 m/s. The yaw angles range was from θ = 3 o to 3 o in 5 o increments. This calibration range covered the flow conditions that appeared in later boundary layer measurements. The velocity and yaw angle calibration points were used to fit polynomials to the data for better accuracy. Various sampling times were selected to ensure a 1% statistical convergence on velocity, ranging from 3 sec at low velocities to as short as sec for the high velocities. A separate test was performed to decide the sample time (more details are given in the appendix A). During the calibration, the pressure output from the Conrad probe and the total pressure tube were measured for different flow speeds and angles according to the procedure shown in figure 3.4. After adjusting the initial flow angle θ = o manually and measuring pressure transducer offset voltages at U = m/s, the test started with the lowest jet speed U = 1 m/s and the first flow angle θ = 3 o. After recording the data, the Conrad probe was rotated 5 degrees to obtain a flow angle of θ = 5 o and the process was repeated until θ = 3 o. Then the probe was turned back to θ = 3 o and a higher jet velocity was applied. The whole process was automated and took one hour. Since the calibration result is very stable for a given Conrad probe, the probe is calibrated once in the jet and does not require subsequent calibration.

34 CHAPTER 3. EXPERIMENTAL DETAILS Setting initial flow angle o = of the Conrad probe, measuring pressure transducer offset voltages. Setting the first calibration velocity U = 1 m/s Setting the first flow angle = -3 Increasing by 5 Sampling p1, p and pd Represent this as an equality. If U < 5 m/s, increase by.5 m/s; If U 5 m/s, increase by 1 m/s. = 3 U = 1 m/s Calibration finished Figure 3.4: Flowchart for the calibration process.

35 CHAPTER 3. EXPERIMENTAL DETAILS 1 3. Wind tunnel experiment The two-dimensional velocity measurements downstream of an array of roughness elements was performed in a laminar boundary layer in the wind tunnel. The definition of x- (streamwise), y- (wall-normal), and z- (spanwise) directions in the wind tunnel measurements are shown in figure 3.5. U, V, W are time-averaged velocities in these three directions, respectively. The streamwise and spanwise time-averaged velocity components (U and W ) were measured inside the Blasius boundary layer using the Conrad probe. Figure 3.5: Schematic of the boundary layer model Experimental Apparatus The experiments were carried out in the test section of the University of Toronto Institute for Aerospace Studies (UTIAS) closed-loop wind tunnel. The working section of the wind tunnel is 1. m wide,.8 m high and 5 m long. The turbulence intensity during the experiment was below.5% at freestream velocities up to 1 m/s. A flat plate was placed horizontally in the test section to generate a laminar boundary layer flow. The flat plane is.1 m long, 1. m width and was manufactured from a 1.7 mm thick cast aluminum plate with near mirror surface finish. The Blasius boundary layer flow has been established over the plate (Hanson et al., 11, 1) and streamwise steady streaks were generated by an array of cylindrical roughness elements mm downstream of the leading edge. The roughness elements are 5 mm in diameter with a spanwise spacing of z = mm between each elements. The roughness elements height was adjustable continuously from h = to 1.75 mm and was measured by a

36 CHAPTER 3. EXPERIMENTAL DETAILS laser displacement meter with an accuracy of ±.1 mm. The Conrad probe was mounted on a holder that allowed the Conrad probe pitch, yaw and roll angles to be independently adjusted. The holder was attached to a three-axis traversing system in the test section. The stepper motors used to drive the traversing system have a resolution of 1 mm per 8 steps in all three dimensions. An MMC - 4S Multiple Stepper Motor Controller from the Motion Group was used to control the stepper motors. The same differential pressure transducers used for the calibration (MKS Baratron 1AD 1 Torr and 1 Torr) were used in the wind tunnel experiment. The high pressure sides of the transducers were connected to the Conrad probe ports and the low pressure sides were connected to the static port of a Pitot-static probe that was placed in the freestream, since the static pressure p s inside a closed-loop wind tunnel is not equal to the atmosphere pressure p a. To minimize damping of pressure flactuations and decrease the time constant of the system, the transducers were placed on the roof of the wind tunnel, to achieve the shortest possible length of PVC tubing between the probe and the transducers. The transducer output signals were sampled by the DAQ board at a sampling rate of 1 Hz. Since the ambient temperature T was different from the room temperature in the tunnel measurements, an Omega Thermocouple (Type T) was used to measure the temperature inside the test section simultaneously with the pressure transducer signal recordings at each test point. The measured T inside the test section and the atmospheric pressure p a were used to calculate the fluid properties such as viscosity and density (p a was obtained from Toronto hourly observed weather released by Environment Canada at e.html and the distance from the laboratory to Environment Canada measurement station is less than 1 kilometer). 3.. Experimental conditions In the present study, roughness elements with z = mm were used in all measurements. The measurements were carried out in various combinations of roughness height h, freestream velocity U and downstream positions x, as summarized in table 3.. The streamwise evolution of the perturbation generated by the roughness elements was studied by measuring the streamwise and spanwise velocity components U and W at 9 x-locations with U = 5 m/s and h = 1. mm. The effect of the perturbation level was studied by measuring U and W at x = 3 mm with U = 5, 7.5 and 9 m/s for h = 1. and 1.5 mm. The experimental conditions of U and h were chosen to avoid laminar to turbulence transition by limiting the amplitude of the perturbation level. The downstream position of x = 3 mm was chosen because previous

37 CHAPTER 3. EXPERIMENTAL DETAILS 3 experiments showed that the perturbation was expected to be larger at this streamwise location (Hanson et al., 1). U and W velocity profiles were measured at 49 different heights from the plate at 3 evenly spaced spanwise locations covering a spanwise interval of z. The Reynolds number used in the present experiment is defined as Re h = U(h)h/ν, where U(h) is the streamwise velocity of the undisturbed boundary layer flow at the apex of the roughness element. Table 3.: Summary of experiment conditions and streamwise positions of the measurement planes. U (m/s) h (mm) x (mm) Re h , 3, 35, 4, 45, 495, 55, 6, Experimental procedure To minimize misalignment errors, the Conrad probe was adjusted before measurements in the Blasius flow without roughness elements. The optimal yaw and roll angles were achieved by obtaining the minimum difference between the readings of p 1 and p both inside the boundary layer and in the freestream. Since the probe is not sensitive to small pitch angles, φ = 1 o was used to make sure the probe tip touched the wall first when traversing downwards. This gave an advantage in setting the initial y o location by making sure that the probe was not pushed against the plate or bent at the y o position. Before measurements, the roughness elements were raised to the desired height and the initial y location of the probe tip geometric center was set to y o = D/ by letting the probe tip barely touch the wall. During the adjustment, a CCD camera was used to observe the zoomed-in image of the spacing between the probe tip and the plate surface (figure 3.6) and y o was set by controlling the y-direction traverse with an accuracy of.5 mm. Setting of initial wall locations was crucial in the present study because unlike previous hot-wire measurements (White & Ergin, 4), the geometric wall-normal locations of the Conrad probe measurements were found directly from the traverse reading. This is discussed further in section 4.. At each x location, the measurement started at the initial wall-normal location y o, right

38 CHAPTER 3. EXPERIMENTAL DETAILS 4 Figure 3.6: CCD camera picture of the Conrad probe position setting. The reflection of the probe on the boundary layer plate is also visible. behind one of the roughness elements at z = z, as defined in figure 3.5. The y-direction profile was performed at each spanwise location starting from the bottom of the boundary layer. The Conrad probe was traversed to a higher y location after sampling at each point and the process was repeated until the probe entered the freestream. A smaller spacing between data points was used at the bottom of the boundary layer and the spacing was gradually increased as the probe departed from the wall. The probe was moved to the next location along negative z-direction. The y-direction profile measurements were repeated until all the z-direction locations were scanned over z and the probe reached z = z. Different sampling times were used for the measurements at different y locations and was decided by an auxiliary experiment to ensure a 1% statistical convergence. A Matlab program was used to drive the Conrad probe and sample the data by controlling the traversing system and DAQ card. To reduce the streamwise misalignment error between the traversing system and the flat plate, the y-direction offset was measured between the spanwise location of z = z and z at each measurement plane. The y-offsets relative to z = z were solved for each spanwise measurement location by a linear interpolation. The initial location at each spanwise location was adjusted according to the offset values when the Conrad probe was sent back to the bottom of the boundary layer to measure the next profile. The spanwise misalignment was evaluated by adjusting the initial wall-normal location y o manually when the streamwise location was changed. To obtain the two-dimensional velocity field in the boundary layer, an additional boundary layer correction method is proposed in this study, as introduced in chapter 4.

39 Chapter 4 Conrad probe data reduction In order to obtain accurate results for the Conrad probe boundary layer measurements, a new data reduction scheme and a new form of the streamwise and the spanwise velocity correction method are proposed. In Section 4.1, the data reduction scheme proposed here is introduced. The method is developed especially for low-speed measurement based on free-jet calibration data such that low Re effects are considered. In Section 4., a velocity correction scheme is proposed for the boundary layer measurements, which accounts for the the error produced by the velocity gradient and wall proximity effects. According to this method, the Conrad probe results are corrected by comparing the boundary layer measurements data to the previous hotwire data obtained under the same experimental conditions by Hanson et al. (1). 4.1 Data reduction scheme Historically, a number of data reduction schemes have been suggested (Brebner, 195; Bryer et al., 1958; Bryer & Pankhurst, 1971; Ulk, 1) for the use of Conrad probe in the measurements of incompressible inviscid flow. In the current transient growth study, the Conrad probe is exposed to a flow field that has much lower Re compared to all other previous studies, such that viscous effects on the probe cannot be neglected. To account for these effects, a look-up table based data reduction scheme is proposed Previous data reduction scheme According to the different ways of solving the flow field, the previously developed Conrad probe data reduction methods are divided into two main groups (Tropea et al., 7): the 5

40 CHAPTER 4. CONRAD PROBE DATA REDUCTION 6 nulling methods used in early studies (Brebner, 195; Bryer et al., 1958; Bryer & Pankhurst, 1971), and the non-nulling methods proposed recently (Ulk, 1). Bryer & Pankhurst (1971) summarized the nulling methods of Conrad probe based on Brebner (195) and Bryer et al. (1958). When using this method, the angular orientation of the Conrad probe is altered about the yaw at each measurement point until both holes give the same pressure reading, which indicates the probe is aligned with the flow direction. The flow angle can then be determined and the velocity magnitude is obtained from the pressure readings at this particular angle with some simple velocity calibration of the probe. The non-nulling methods need a more complex calibration on both the flow velocity U and yaw angle θ. The methods then solve the velocity and angle using the functions obtained from the calibration data. Ulk (1) summarized previous non-nulling methods and proposed a method that relies on the non-dimensional angular coefficient C p and the velocity coefficient C pa defined as: where p avg C pa = p avg p d = f(θ), (4.1) C p = (p 1 p ) p avg = g(θ), (4.) = (p 1 + p )/. The functions f(θ) and g(θ) in (4.1) and (4.) are obtained by polynomial fits using the calibration data (see for instance figure 4.1). When reducing the data, the flow angle θ is solved by calculating (p 1 p )/p avg from recorded p 1 and p and inverting the function in (4.). Then p d = p avg /f(θ) is solved by substituting θ and p avg into (4.1). The nulling method needs no calibration or only simple calibration and offers the advantage of solving U and θ independently. However, it is very time consuming since the probe has to be rotated until p 1 = p at each measurement point. The accuracy of the method also depends on the rotational resolution and the probe manufacturing tolerance. The non-nulling method avoids rotating the probe during the measurements, but the probe has to be calibrated at more flow angles and more complex calibration functions are required.

41 CHAPTER 4. CONRAD PROBE DATA REDUCTION 7 Figure 4.1: The curves fitted into polynomials for c pa = f(θ) and c p = g(θ) (figure from Ulk, 1) Reynolds number effect on the flow field When a non-nulling method is used in the Conrad probe measurements, both the flow speed U and the yaw angle θ are solved by the calibration functions. When a flow field is independent of Reynolds number, the flow velocity and yaw angle can be solved independently. However, when the flow field is sensitive to Reynolds number, the effects of the velocity and yaw angle have to be considered together. Presently, most non-nulling data reduction schemes for multihole pressure probes, including the Conrad probes, are Reynolds number independent (Tropea et al., 7; Bryer & Pankhurst, 1971). They rely on the assumption that the probe angular and velocity sensitivities can be decoupled from the effect of Re d. This assumption has been shown to hold for velocities above 15 m/s (Bryer et al., 1958; Ulk, 1). Ulk (1) showed that the calibration coefficients are Reynolds number independent by analyzing the calibration data of the yaw angle 3 o θ 3 o in 6 o increments at velocities of 7.4, 38.7 and 47. m/s. The non-dimensional angular and velocity coefficients C p and

42 CHAPTER 4. CONRAD PROBE DATA REDUCTION 8 C pa were solved based on the calibration data and were compared for these three calibration velocities at each yaw angle. Figure 4.1 showed that the coefficients are not a function of flow velocity, which indicates that the probe sensitivity is independent of Reynolds number. The calibration was simplified because it was performed at only a few velocities while the calibration functions f(θ) and g(θ) were also simplified by fitting the averaged results of C p and C pa at the three calibration velocities such that they only depend on the yaw angle. However, the assumption that calibration coefficients C p and C pa are only a function of yaw angle does not hold at the low velocity between 1 1 m/s. As shown in figure 4., the angular coefficient C p and its uncertainty are solved using the calibration data. It is shown that the C p curves have larger slopes when the velocity is higher and the curves are more flat when the flow velocity is low. Thus result indicates that C p is also a function of calibration velocity. A physical explanation is given by figure 4.3. The flow stagnates directly on the probe tip at the higher velocity. However, at low velocity, the flow changed its direction due to the thick boundary layer of the probe, which causes the streamline to bend. The extent of the bend and thus the the angular character is dependent on the Reynolds number. 3 U = 1.5 m/s U = 3.5 m/s U = 6 m/s U = 1 m/s 1 C p θ Figure 4.: Angular coefficient under low velocities. C p is solved according to (4.), using current calibration data. Error bars are used to show uncertainty in data values. Previous studies on the blunt-nosed cylindrical Pitot tubes were summarized in Tropea et al. (7). The pressure recovery coefficient shows the difference between the measured

43 CHAPTER 4. CONRAD PROBE DATA REDUCTION 9 (a) (b) Figure 4.3: (a) The schematic of the high Reynolds number flow, and (b) of the low Reynolds number flow around the probe. value and the true velocity in the flow field, which was defined as C p = p o p s 1/ρU, (4.3) where p o is the total pressure, p s is the static pressure and U is the local flow velocity. Defining the probe Reynolds number Re d in terms of the inner diameter of the probe tubes, local velocity and the flow kinematic viscosity (Tropea et al., 7), figure 4.4 shows that the viscous effect on the probe is not negligible when Re d is low. When Re d is less than 1, C p gives a percentage difference of over 1% and it increases rapidly at very low Re d number. Tropea et al. (7) explained that in the low Reynolds number viscous flow, the viscosity may affect the flow around the probe tip such that the measured total pressure differs significantly from the true pressure. Though the design of the Conrad probe differs from the Pitot tube designs, the Conrad probe velocity result is also affected by the flow viscosity at low Re d. This effect should be considered in present study since Re d = 1 relates to a velocity approximately 5 m/s for the Conrad probe. It can be concluded that the viscosity around the probe alters the flow patten in the low Reynolds number flow and the corresponding change in the Conrad probe readings is related to the probe Reynolds number Re d. Thus the information for the velocity magnitude and flow angle are coupled, thus the Reynolds number independent assumption does not hold. In order to account for the correlation between the flow angle and speed in current low Re d measurements, a look-up table (LUT) based non-nulling data reduction method is proposed.

44 CHAPTER 4. CONRAD PROBE DATA REDUCTION 3 Figure 4.4: The viscous effect on the round, square-ended Pitot tubes (figure from Tropea et al., 7, p. ). The data in the figure are from early works: (A) Sherman (1953), (B) Hurd et al. (1953), (C) MacMillan (1954), (D) Barker (19). Re din in the figure is denoted as Re d Proposed Re dependent scheme Look-up table (LUT) method The proposed LUT method in this research is inspired by the data process scheme used with X-wire, which solves the flow field information based on a mapping obtained via a velocity and angle calibration (Lueptow et al., 1988; Burattini & Antonia, 5). The X-wire probe provides simultaneous measurements of two components of the velocity using two hot-wire sensors, which are usually oriented perpendicular to each other and at a 45 o angle to the mean flow. To produce the look-up table, Burattini & Antonia (5) performed the calibration at set velocities and flow angles, and the voltage pair (E 1, E ) from the sensors were measured. The voltage responses of the wires are then mapped into a look-up table of known velocities and angles. The look-up table is inverted when reducing the experimental data. In Burattini & Antonia (5), two-dimensional cubic spline interpolation was proposed to reduce the data between the calibration points. The LUT concept used in this study is associated to the pressure pair (p 1, p ) recorded in the calibration instead of the voltage pairs. This is related to the known velocity and flow angles as: U = Π(p 1, p ), (4.4) θ = Φ(p 1, p ). (4.5)