EFFECT OF BOUNDARY LAYER THICKNESS AND PASSIVE VORTEX GENERATORS ON THE WAKE OF A BLUNT TRAILING EDGE PROFILED BODY. Wenyi Zhao

Transcription

1 EFFECT OF BOUNDARY LAYER THICKNESS AND PASSIVE VORTEX GENERATORS ON THE WAKE OF A BLUNT TRAILING EDGE PROFILED BODY by Wenyi Zhao A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Institute for Aerospace Studies University of Toronto c Copyright by Wenyi Zhao (27)

2 Abstract Effect of boundary layer thickness and passive vortex generators on the wake of a blunt trailing edge profiled body Wenyi Zhao Master of Applied Science Institute for Aerospace Studies University of Toronto 27 This thesis studies the effect of turbulent boundary layer thickness on the wake dynamics of a blunt trailing edge profiled body, and investigates a flow control technique by applying passive vortex generators with different size and spacing. The boundary layer thickness was varied by changing the model chord to thickness ratio from 3 to 5 and by changing the Reynolds number from, to 35,. Consequently, the displacement thickness in this study ranges from.77d to.39d, which results in a decrease of Strouhal number from.22 to.68, an increase in formation length from.863d to.48d, and an increase in the average wavelength of secondary instability from 2.2d to 2.33d. The effectiveness of the passive flow control was found to have little dependence on Reynolds number, and the most back pressure recovery of 3% and formation length elongation of 8% were achieved with the smallest passive vortex generators spaced at 2.4d, which is close to the wavelength of the secondary instability. ii

3 Acknowledgments I would like to express my special appreciation and thanks to my supervisor Professor Philippe Lavoie for the continuous support of my thesis study, for his patience, motivation, enthusiasm, and immense knowledge. I would also like to gratefully acknowledge the financial support from University of Toronto Connaught Fund and Natural Sciences and Engineering Research Council of Canada (NSERC). I am deeply grateful to Heather Clark, Ross Cruikshank and Zachery Berger, who made enormous contribution to setting up the experiments and processing the data. Finally, I must thank everyone in the FCET group for sharing their experience and enthusiasm. Discussions with them have been insightful. iii

4 Contents Acknowledgments Table of Contents Nomenclature List of Tables List of Figures iii iv vi ix x Introduction. Blunt Trailing Edge Airfoils Wake Characteristics and Direct Flow Control Secondary Wake Instability D Wake Control Thesis Objectives Experimental Setup and Analytical Approach 4 2. Experimental Model Blunt Trailing Edge Model with Adjustable Length Passive Vortex Generators Blunt Trailing Edge Model with Vortex Generators Measurement Techniques Pressure Tubes Hot-wire Anemometry iv

5 2.2.3 Particle Image Velocimetry Analysis of PIV Measurements Proper Orthogonal Decomposition Wavelength Identification of Secondary Instability Effect of Boundary Layer Thickness on Wake Dynamics 3 3. Boundary Layer Properties Formation Length and Vortex Shedding Frequency Wavelength of Secondary Instability Summary and Discussion Control Effectiveness of Passive Vortex Generators 4 4. Characteristics of Passive Vortex Generators Effect of Reynolds Number Effect of Size and Spacing Interference Between Vortex Generators Conclusions and Recommendations 66 References 69 v

6 Nomenclature Explanation Acronym NSERC FCET UTIAS DLBA DSMA PIV POD Natural Sciences and Engineering Research Council of Canada Flow Control and Experimental Turbulence University of Toronto Institute for Aerospace Studies Douglas Long Beach Airfoil Douglas Santa Monica Airfoil Particle Image Velocimetry Proper Orthogonal Decomposition Symbol a a i A A p b c C + C D C D C pb d d dt vortex generator length POD time-coefficients area peak to peak amplitude spanwise distance between the two velocity local minima chord length intercept constant drag coefficient fluctuating component of drag back pressure coefficient trailing edge thickness effective trailing edge thickness PIV time interval vi

7 f f c F h H l L L f L r f m nw n P OD N N t p P b P g P s P t R Re Re d St St d t thr t n t T E vortex shedding frequency focal length force vortex generator height Shape factor vortex generator trailing edge spacing number of model sections formation length formation length for reference case separation vector in auto-correlation number of wavelengths detected number of POD modes utilized total length of signal number of time steps pressure back pressure voltage gain to pressure conversion factor static pressure total pressures correlation coefficient Reynolds number Reynolds number based on trailing edge thickness Strouhal number Strouhal number based on trailing edge thickness airfoil thickness signal to noise threshold time step airfoil trailing edge thickness vii

8 u u + u u u p u rms U v v rms y + α δ δ V θ κ λ λ i λ z ν ρ τ w streamwise velocity wall-normalized streamwise velocity friction velocity streamwise velocity fluctuation velocity from pitot tube root mean square of streamwise velocity fluctuation freestream velocity wall-normal velocity fluctuation root mean square of wall-normal velocity fluctuation wall-normalized y distance phase angle boundary layer thickness displacement thickness differential voltage momentum thickness von Kármán constant spanwise characteristic length of control POD eigenvalues spanwise characteristic wavelength dynamic viscosity fluid density wall shear stress viii

9 List of Tables 2. Testing cases for various boundary layer thickness Dimensions of testing vortex generators Testing cases for vortex generators of different size and spacing Summary of measured quantities at the boundary layer and in the wake for blunt trailing edge model with various length at Re d =,, 2,, and 35, Estimate of Displacement Thickness (from Plane ) at Different Reynolds Numbers Comparison of maximum vortex generators at difference spacing Comparison of minimum vortex generators at difference spacing ix

10 List of Figures. Airfoils with blunt trailing edge modifications studies by Standish and Van Dam (23) with constant thickness to chord ratio t/c of 35% and trailing edge thickness-to-chord-ratio, t T E /c of %, 5%, and % Figure from Naghib-Lahouti et al. (22): von Kármán vortex shedding after a blunt trailing edge profiled body at Re = Figure from Williamson (996): A measure of the length of a mean recirculating region in (a) behind a body is given by the formation length L f, which is (usually) defined by the distance downstream from the cylinder axis to a point where the rms velocity fluctuations are maximized on the wake center line, as sketched in (b) Examples of two-dimensional flow control Figure from Bearman (965): Base pressure coefficient against inverse of vortex formation position., Re =.45 5 ;, Re = Figure from Williamson (996): secondary instabilities in the wake after a circular cylinder at (a) Re = 2 and (b) Re = Figure from Naghib-Lahouti et al. (24): Variation of the wavelength of secondary instability λ z with Reynolds number Examples of 3-D forcing devices and trailing edge modifications from literature, (a) helical strakes (b) sinusous axis (c) rectangular segmentation (d) M-shape segmentation (e) wavy trailing edge (f) wavy leading edge (g) small tabs (h) plasma actuators Picture and schematic of wind tunnel facility from Hearst (25) x

11 2.2 Schematic of full length blunt trailding edge model with endplates Wind tunnel picture in front of the model at full length Schemetic of model with 2 sections in the middle, showing the coordinate system Features of individual vortex generator Assembled trailing edge model with minimum vortex generators spaced at 2.4d Experimental setup for hot-wire measurements Experimental setup for PIV Top view diagram of laser beam path Side view diagram of laser beam path Illustration of PIV measurement planes Variation of cumulative energy with the number of POD modes used in reconstruction Streamwise component of reconstucted instantaneous fluctuating velocity field with 5 POD modes at c/d = 3 and Re d =, Example of streamwise velocity variation across the span on the indentified trace Comparison of biased and unbiased correlation as a function of distance shifted Boundary layer profiles with different model length at Re d =, and x/d = Skewness of the boundary layers with different model length at Re d =, and x/d = Boundary layer profiles in wall-normalized units with different model length at Re d =, and x/d = Shape factor of the boundary layers with different model length at x/d = Variation of velocity variance in streamwise (x) direction at Re d =, and y/d = with different model length. The graphs have been offset to facilitate visualization Variation of formation length with boundary layer displacement thickness at different Reynodls number xi

12 3.7 Power spectral density of streamwise velocity, measured at x/d =.5 and y/d =.8. The graphs have been offset to facilitate visualization Variation of vortex shedding frequency in terms of Strouhal number with boundary layer displacement thickness at different Reynodls number Average wavelength computed with different threshold and number of POD modes for model length of 3d at Re d =,. Labels indicate threshold values and the number of valid wavelengths detected Variation of secondary instability wavelength with boundary layer displacement thickness at different Reynolds number Strouhal number normalized by d at different boundary layer thickness Formation length normalized by d at different Reynolds number from Naghib- Lahouti et al. (24) Formation length normalized by d at different boundary layer thickness from current study Wavelength of secondary instability normalized by d at different Reynolds number from Naghib-Lahouti et al. (24) Wavelength of secondary instability normalized by d at different boundary layer thickness from current study Streamwise mean velocity field in vertical plane, 2, 3 and horizontal plane at y/d =.5 when maximum vortex generators are applied with spacing of 4.8d at Re = 35, Streamwise mean velocity field in vertical plane at Re = 35, when no control is applied Percent energy of the first 5 POD modes in different vertical planes with maximum vortex generators spaced at 4.8d, at Re = 35, Phase plots in different vertical planes based on coefficents of POD modes and 2 with maximum vortex generators spaced at 4.8d, at Re = 35, Variation of turbulence intensity along x at y = without vortex generator (reference case) at Re d = 35, xii

13 4.6 Variation of streamwise mean velocity along x at y = with maximum vortex generators spaced at 4.8d in comparison to the reference case (with no VG), at Re d = 35, Comparison of formation length with minimum vortex generators of different spacing at different Reynolds number. Horizontal lines represent base cases with no vortex generator Comparison of back pressure coefficient with minimum vortex generators of different spacing at different Reynolds number. Horizontal lines represent base cases with no vortex generator Comparison of normalized change in formation length with minimum vortex generators of different spacing at different Reynolds number Relation of normalized change in formation length and reduction in base pressure with minimum vortex generators of different spacing at different Reynolds number Comparison of formation length in vertical Plane with vortex generators of different size and spacing at Re d = 35, Comparison of back pressure coefficient with vortex generators of different size and spacing Comparison of total drag reduction in vertical Plane with vortex generators of different size and spacing at Re d = 35, Comparison of drag due to velocity fluctuation with vortex generators of different size and spacing Percent energy of the first 5 POD modes in vertical Plane with different vortex generators Eigenvalues of the first 5 POD modes in vertical Plane with different vortex generators Streamwise components of the fisrt two POD modes in Plane for the reference case with no vortex generator at Re = 35, Streamwise components of the fisrt two POD modes in Plane when maximum vortex generators of 2.4d spacing are applied at Re = 35, xiii

14 4.9 Phase plots based on POD modes and 2 in Plane (if not otherwise indicated) with different vortex generators. x-axis label a / 2λ and y-axis label a 2 / 2λ 2 omitted for visual clarity Phase-averaged velocity field in Plane with minimum vortex generators of different spacing at Re d = 35,. Background color indicates one-thousandth of vorticity Phase-averaged velocity field in Plane when no control is applied at Re d = 35,. Background color indicates one-thousandth of vorticity Spanwise variation of streamwise mean velocity induced by minimum and maximum vortex generators spaced at 4.8d, at.4d downstream of trailing edge Spanwise variation of streamwise mean velocity induced by minimum and maximum vortex generators spaced at 4.8d, at 4.4d downstream of trailing edge Profiles of streamwise velocity in different vertical planes at.4d downstream of the trailing edge with maximum vortex generators spaced at 4.8d Velocity difference between vertical Plane 3 and Plane at.4d downstream of the trailing edge with maximum vortex generators spaced at 4.8d Spanwise variation of streamwise velocity at x =.4d downstream of the trailing edge and near y =.5d, with maximum vortex generators spaced at 4.8d Spanwise variation of streamwise mean velocity induced by minimum and maximum vortex generators spaced at 2.4d, at.4d downstream of trailing edge Spanwise variation of streamwise mean velocity induced by minimum and maximum vortex generators spaced at 2.4d, at 4.4d downstream of trailing edge Streamwise mean velocity field in the horizontal plane with minimum vortex generators spaced at 2.4d Streamwise mean velocity field in the horizontal plane with maximum vortex generators spaced at 2.4d Front view illustration of the vortices generated by minimum and maximum vortex generators spaced at 2.4d xiv

15 4.32 Comparison of spanwise variation of streamwise mean velocity profiles.4d downstream of trailing edge induced by maximum vortex generators at different spacing Comparison of spanwise variation of streamwise mean velocity profiles.4d downstream of trailing edge induced by minimum vortex generators at different spacing Comparison of measured and linearly superimposed spanwise variation of streamwise mean velocity with maximum vortex generators spaced at 2.4d. The superposition is the sum of two the velocity variation profiles caused the same generators spaced at 4.8d, with one shifted horizontally by 2.4d, and freestream velocity is subtracted from the sum Velocity field for different vortex generator configureations in the vertical planes. Background color represnets the velocity magnitude Velocity field in the horizontal plane for the case of maximum vortex generators spaced at 2.4d. Background color represnets the streamwise velocity Velocity field in the horizontal plane for the case of maximum vortex generators spaced at 4.8d. Background color represnets the streamwise velocity xv

16 Chapter Introduction Blunt trailing edge airfoils have many industrial applications thanks to their aerodynamic and structural benefits. However, the vortex shedding after the blunt trailing edge causes unwanted by-products such as form drag and tonal noise. With the knowledge of primary and secondary wake characteristics, various 2-D and 3-D flow control devices have been developed to mitigate the von Kármán vortex street. This thesis focuses on 3-D flow control because of its proven efficiency and advantages. More specifically, the wake dynamics after a blunt trailing edge profiled body is investigated under different turbulent flow conditions with and without 3-D passive flow control. The first part of the thesis is inspired by previous studies on secondary instabilities after a blunt trailing edge profiled body, and is devoted to study the effect of boundary layer thickness on wake characteristics including vortex formation length, vortex shedding frequency, and the wavelength of secondary instability. On the premises of the first, the second section intends to determine an optimal configuration of vane-type vortex generators evenly spaced on both upper and lower surfaces of a blunt trailing edge model as a method of distributed forcing in order to suppress the primary vortex shedding. The geometric features of a single vortex generator was determined based on previous studies on vane-type vortex generators in the application of boundary layer separation control. The effect of its size and spacing on the flow control in the wake is studied at different Reynolds number.

17 . Blunt Trailing Edge Airfoils Improvements on transonic and supercritical airfoil design have led to airfoils with large leading edge radius, upper surface with low curvature, and large amount of after chamber, which results in a blunt trailing edge (Henne, 99). As early as in the mid-9s, Chapman (955) theoretically determined and experimentally verified that, at Mach numbers of.5 and 2., a properly designed blunt trailing edge airfoil produces lower drag and higher liftcurve slope than a conventional design. Yoo (2) found that, at transonic Mach number and near the design lift coefficient, blunt trailing edge modification to the supercritical airfoil DLBA 86 improves its aerodynamic performance in various aspects including maximum lift coefficient, lift slope, lift-to-drag ratio, maximum lift-to-drag ratio, etc. Blunt trailing edge airfoils can also be found in other industrial applications such as truncated wind turbine blade for both aerodynamic and structural benefits. Standish and Van Dam (23) showed that the blunt trailing edge modification to a thick airfoil shown in Figure. results in a reduction in the adverse pressure gradient on the suction side, and an increase in the lift-curve slope and maximum lift coefficient. Besides, for a fixed airfoil maximum thickness, blunt trailing edge design increases both sectional area and sectional moment of inertia, which meets the structural demands of high load wind turbine blades (Jackson et al., 25). On the other hand, for either wind turbine blades or supercritical airfoils under subcritical conditions, blunt trailing edge modification with sharp corners induces flow separation, and as a result, vortex shedding and larger wakes illustrated in Figure.2 develop downstream of the Figure.: Airfoils with blunt trailing edge modifications studies by Standish and Van Dam (23) with constant thickness to chord ratio t/c of 35% and trailing edge thickness-to-chord-ratio, t T E /c of %, 5%, and %. Figure.2: Figure from Naghib-Lahouti et al. (22): von Kármán vortex shedding after a blunt trailing edge profiled body at Re = 55. 2

18 trailing edge (Thompson and Lotz, 22). The vortex shedding in the wake can significantly increase form drag, may cause structural damage, and is associated with unwanted tonal acoustic noise (Barone and Berg, 29). To alleviate these side effects, wake dynamics has been extensively studied and a number of flow control techniques have been developed to mitigate the wake downstream of bluff bodies..2 Wake Characteristics and Direct Flow Control The interaction of the separated shear layers after a bluff body forms periodic shedding of primary spanwise vortices, known as the von Kármán vortex sheet, first recorded by Kármán (9). Since then, significant research has been conducted regarding the vortex structures and wake characteristics of bluff bodies. For a circular cylinder, according to Hopf bifurcation instability, the transition from a steady wake flow field to laminar two-dimensional vortex shedding occurs at a Reynolds number Re of (Dušek et al., 994). The shedding frequency has been commonly expressed in terms of the non-dimensional Strouhal number St(d) = fd U, where f is the vortex shedding frequency, d is the characteristic length, and U is the freestream velocity (Rayleigh, 95). Another important wake parameter is the formation length L f defined as the streamwise distance at which the root-mean-square velocity fluctuations are maximized (Bearman, 965). Williamson (996) suggested that this distance also corresponds to the length of the mean recirculation region in the wake shown in Figure.3, where the pressure and shear stresses are in equilibrium on the boundary (Sychev, 982). To alleviate the vortex shedding after a bluff body, direct flow control techniques were developed. These techniques, such as splitter plates and others shown in Figure.4, have geometric features that are uniform across the span, and hence are referred to as 2-D forcing. Roshko (955) installed splitter plates after a cylinder and achieved drag reduction by eliminating a pressure valley that occurs without the splitter plate about one diameter downstream of the base, which is in the region where the vortices form. He also found that 3

19 Figure.3: Figure from Williamson (996): A measure of the length of a mean recirculating region in (a) behind a body is given by the formation length L f, which is (usually) defined by the distance downstream from the cylinder axis to a point where the rms velocity fluctuations are maximized on the wake center line, as sketched in (b). an increase in formation length is associated with a decrease in the level of maximum velocity fluctuation and a decrease in the base suction, expressed in terms of base suction coefficient C pb = P b P s (/2)ρU 2 = P b P s P t P s, where ρ is the fluid density, P b, P s, and P t are the back, static, and total pressures, respectively. Bearman (965) implemented splitter plates after a blunt trailing edge and successfully disturbed the interaction between the separated shear layers and hence delayed the vortex shedding. In his experiment, it was also discovered that, with splitter plates of various lengths, the formation length is closely inversely proportional to the base suction coefficient as shown in Figure.5. By placing a smaller cylinder behind the bluff body, Thiria et al. (29) reduced the turbulent drag of a 2-D bluff body with a blunt trailing edge by 7.5%. They showed that the recirculation length and the pressure inside were decreased by the presence of the control cylinder. Martín-Alcántara et al. (24) numerically studied single-cavity and multi-cavity modifications of a D-shaped body across the blunt trailing edge height, and observed that, in a given range of cavity height, the average drag decreased monotonically to an asymptote in both cases. Furthermore, with the multi-cavity control, considerable reduction in flow randomness was reached. 4

20 Figure.4: Examples of two-dimensional flow control. Figure.5: Figure from Bearman (965): Base pressure coefficient against inverse of vortex formation position., Re =.45 5 ;, Re =

21 .3 Secondary Wake Instability Many two dimensional control techniques are directly applied in the wake and require large changes in the geometry of airfoils, which is undesired in industrial applications. The exploration of 3-D flow control techniques with little aerodynamic or structural compromise has been inspired by the observation of secondary instabilities in the wake. Roshko (954) first reported the existence of spanwise periodic structure of the vortex street with cylinders at Re slightly below 5. Since then, structures of secondary wake instabilities and particularly the spanwise characteristic wavelength λ z have been carefully examined by researchers for various geometries at different flow conditions. Under some flow conditions, primary vortices in a wake are connected by pairs of streamwise vortices with characteristic spacing λ z. This phenomenon is commonly referred to as the secondary instability in the wake. The review from Williamson (996) summarized the three dimensional wake structure after a circular cylinder at different Reynolds number. The secondary (streamwise) vortices appear at a Reynolds number of 94 as a mode-a instability with λ z between 3d and 5d (Figure.6 a), where d is the diameter of the cylinder, and the sense of rotation of these vortices alternate every half shedding cycle. At 23 < Re < 26, mode-b instability with λ z around d dominates (Figure.6 b) with the same sense of rotation over multiple shedding periods. Figure.6: Figure from Williamson (996): secondary instabilities in the wake after a circular cylinder at (a) Re = 2 and (b) Re = 27. 6

22 Robichaux et al. (999) studied the secondary wake dynamics for a square cylinder and reported that mode-a occurs at a Reynolds number of around 62 with λ z of 5.22d, and mode- B occurs at Re 9 with λ z =.2d. They also observed a third distinct mode, named mode- S, the onset of which is at Re 2 with λ z of 2.4d-2.8d, in which the sense of rotation of the streamwise vortices alternate every full shedding cycle.dobre and Hangan (24) conducted an experimental study on the wake of a square cylinder at a Reynolds number of 22,. They observed a wake flow topology similar to mode-a for circular cylinders but with a different wavelength of 2.4d. As for blunt trailing edge profiled bodies, Ryan et al. (25) numerically studied the effect of aspect ratio (chord over thickness) from 2.5 to 7.5 on the secondary instability in a Reynolds number range of 25 to 7 based on model thickness. They found that the mode-a instability dominates when the aspect ratio is less than 7.5 with a spanwise wavelength of 3.5d to 3.9d. Mode-B dominates for larger aspect ratio with λ z = 2.2d, where the prime indicates the difference in spanwise wavelength from the mode-b associated with the cylinder. For aspect ratios of 7.5, 2.5, and 7.5, they observed mode-s with a wavelength of approximately d, which is similar to that of mode-b for a circular cylinder wake although the spatio-temporal symmetry is different. They also found that mode-b and mode-s are both more dominant than mode-a at aspect ratio of 7.5. Doddipatla (2) studied various wake structure after a blunt trailing edge with an aspect ratio of 2.5 at Reynolds number from 25 up to 46,. Despite the difference in spatiotemporal symmetry, they concluded that the overall spanwise spacing of the streamwsie vortices is robust through different flow regimes and ranges from.2d to 2.8d. At Reynolds number of 25 to 2,3, where the flow is transitional, along with other modes of instabilities, they observed small-scale streamwise vortices with an average spanwise spacing of.d, which are not associated with severe distortion of the Kármán vortices. Naghib-Lahouti et al. (22) studied the same geometry and found an average spanwise wavelength that varies from 2.d to 2.5d for mode-b as the Reynolds number is increased from 55 to 2,5. Figure.7 shows the further investigation of the model by Naghib- Lahouti et al. (24). They summarized the wavelength of the secondary instability found in previous studies with the same geometry and reported that λ z is relatively constant when the 7

23 Figure.7: Figure from Naghib-Lahouti et al. (24): Variation of the wavelength of secondary instability λ z with Reynolds number. Reynolds number ranges from 5, to 5,, which includes a region where the boundary layer upstream transitions from laminar to turbulent between Re 8, and,..4 3-D Wake Control Researchers started to experiment with 3-D flow control techniques to reduce the vortex shedding after a bluff body as early as in the fifties. Scruton and Walshe (957) wound three strakes of sharp-edged rectangular section as three helices around the surface of a cylinder, with a pitch of 5d to break the wake separation line (Figure.8a). Later, Woodgate and Maybrey (959) determined the optimum pitch for the three strakes was 5d. More recently, Owen and Szewczyk (2) studied a sinuous circular cylinder with a wavelength of 7.5d (Figure.8b) over a wide range of Reynolds number up to a maximum value of,, and observed a suppression of vortex shedding and drag reductions of up to 47%. They also stated that the modification of the separation line appears to be the most efficient when 8

24 - Figure.8: Examples of 3-D forcing devices and trailing edge modifications from literature, (a) helical strakes (b) sinusous axis (c) rectangular segmentation (d) M-shape segmentation (e) wavy trailing edge (f) wavy leading edge (g) small tabs (h) plasma actuators. its wavelength is close to that of mode-a instability, and such conclusion holds even when the flow regime is not dominant by mode-a instability. Kim and Choi (25) extended the idea of sinuous axis on cylinders to active flow control. They numerically applied timeinvariant blowing and suction with a sinusoidal profile across the span of a circular cylinder and annihilated the primary vortex street and thus significantly reduced the mean drag and the fluctuations of lift and drag. The primary vortex shedding completely disappeared with the actuation of 5d wavelength at Re =, and was weakened at Re = 39 when the actuation wavelength was adjusted to πd. As for blunt trailing edge geometries, Tanner (972) suggested that to reduce drag it was most important to have a break in the separation line. He implemented various forms of segmented trailing edge on blunt trailing edge wings including the ones shown in Figure.8c 9

25 and.8d, and achieved drag reductions of up to 64%. The study on segmented trailing edges has been carried on by researchers including Rodriguez (99), Petrusma and Gai (994) and Deshpande and Sharma (22). Bearman and Tombazis (993) observed vortex dislocations after a blunt trailing edge occurring randomly in space and time. These are large scale three-dimensional features in the wake of nominally two-dimensional bluff bodies and are associated with local changes in vortex shedding frequency. They then implemented a spanwise wavy trailing edge (Figure.8e) after a blunt trailing edge section to control the locations of vortex splitting, which resulted in a reduction in the wake associated drag and an increase in formation length. The observations of vortex dislocations were further investigated from a numerical perspective by Najjar and Balachandar (998), who examined the wake after a normal flat plate and found that the flow gradually varies between two different regimes. In the regime where the streamwise vorticity is higher which indicates stronger secondary instability, the shear layer extends farther downstream, the von Kármán vortex street is less coherent, and the drag is lower. This suggests that three-dimensional flow control techniques, which encourage secondary instability by fixing vortex dislocations across the span, can alleviate the primary vortex and reduce drag in the wake. Tombazis and Bearman (997) implemented spanwise wavy trailing edges with different wavelength from 3.5d to 5.6d after a half ellipse shaped body. The most recovery in base pressure was achieved with the steepest trailing edge modification (peak-to-peak wave height of.5d and wavelength of 3.5d). However the attenuation of the peak shedding frequency was not obtained which suggests that the primary vortex shedding persisted. Bearman and Owen (998) investigated similar spanwise wavy modifications on thin plates normal to the flow and on the front side of rectangular cross-section bodies (Figure.8f) at Reynolds number of 4,. Drag reductions of up to 3% were achieved and the primary vortex shedding disappeared completely when the ratios of peak-to-peak wave height over wavelength (wave steepness factor) is greater than.9. Dobre et al. (26) extended the study on a square cylinder with wavy leading edge at high Reynolds number to a lower wavelength of 2.4d, in accordance to the turbulent mode-a wavelength of 2.4d observed by Dobre and Hangan (24). They achieved complete mitigation of the primary vortex shedding and significant

26 reduction in base drag with a wave steepness factor of.97. Darekar and Sherwin (2) numerically studied a square cylinder with sinusoidal axis of different wavelength and wave height in a laminar flow regime at Reynolds number of. Depending on the perturbations, the flow was characterized into five regimes. From mild to large geometric waviness, they observed deformation of the von Kármán vortices, alternation between dominantly two-dimensional and mildly three-dimensional, formation of streamwise vortices similar to mode-a instability, suppression of von Kármán vortex street with small unsteadiness due to hairpin vortices, and complete suppression of the primary vortex shedding with significant drag reduction. They also showed that the smallest wave steepness to suppress the primary vortex shedding occurs when the wavelength of waviness is close to the wavelength associated with mode-a instability (around 5.5d). Doddipatla (2) compared the effect of wavy trailing edge modifications with wavelengths of 2.4d and 5.6d on a flat plate with an elliptical leading edge at Reynolds numbers between 24, and 46,. The maximum base drag reduction was achieved at λ = 2.4d, and with the wavy trailing edge, the strength of the streamwise vortices were enhanced, which reduced the strength of the von Kármán vortex shedding. Besides trailing edge modifications, evenly spaced small devices are also means of 3-D forcing in the control of wake dynamics. Park et al. (26) installed non-staggered tabs (Figure.8g) with optimal spanwise spacing of.667d on the blunt trailing edge of a twodimensional bluff body which resulted in an 33% increase in the base pressure. Naghib- Lahouti et al. (25) implemented plasma actuators (Figure.8h) on both sides and near the trailing edge of a blunt trailing edge profiled body with a spanwise spacing of 2.4d. With a momentum coefficient (the ratio of momentum induced by the actuator and momentum in the freestream) of 2.45%, they attenuated the primary vortex street completely, and achieved 94% increase in formation length and more than 4% recovery in base pressure..5 Thesis Objectives In the study by Naghib-Lahouti et al. (24), λ z is unaffected as Reynolds number increases from 5, to 5, while the boundary transitions from laminar to turbulent and its

27 thickness changes with Reynolds number. Inspired by the fact that λ z does not change with boundary layer state, the first part of this thesis intends to verify whether the wavelength of the secondary instability after a blunt trailing edge profiled body stays unchanged when the boundary layer thickness varies by adjusting the model aspect ratio up to 5. Besides λ z, this thesis also examines other features in the wake including formation length L f and back pressure coefficient C pb. This may extend the previous studies on wake dynamics after a blunt trailing edge to flow conditions that are more representative in commercial applications. Ryan et al. (25) studied the effect of aspect ratio on the secondary instability after a blunt trailing edge geometry. However the study was conducted in the low Reynolds number region and the range of aspect ratios was only up to 7.5. A DSMA-523 transonic airfoil with blunt trailing edge has an aspect ratio of 96 UIUC (26), which is much larger than 7.5. With large aspect ratios for which the boundary layer is turbulent, the key difference in the near wake flow properties caused by models of different aspect ratios is the boundary layer thickness, which is the key property being varied in the first part of the thesis. 3-D flow control techniques are often associated with less geometric modification than 2-D ones, and have been shown to be similarly or more effective in alleviating the side effects of the primary vortex shedding after a bluff body. Studies have found that 3-D control approach is most effective when the spanwise periodicity of these devices is equal to the wavelength λ z of the dominant secondary wake instability. As explained by Choi et al. (28), when forcing is applied at a spanwise spacing close to λ z, the forced streamwise vortices disorganize and can eventually breakdown the spanwise structure of the von Kármán vortices. This implies that if the wavelength of secondary instability after a blunt trailing edge persists over a wide range of Reynolds number (Naghib-Lahouti et al., 24) and potentially over a wide range of other flow conditions, a flow control device with the same spanwise characteristic length can be effective under those various conditions as well. In the second part of this thesis, evenly spaced vane-type vortex generators are examined as a 3-D wake control device. This thesis aims to determine an optimal size and spacing of the vortex generator configuration to suppress the primary vortex shedding and to reduce the wake associated drag. Meanwhile it intends to find out, with these passive devices at the optimal configuration, how much formation length elongation, back pressure recovery, 2

28 and drag reduction can be achieved. It is also a focus to verify, with the passive vortex generators, whether a spacing close to the wavelength found in the first part of the thesis and in literature would result in the most effective control in the wake. 3

29 Chapter 2 Experimental Setup and Analytical Approach The experiments were performed in a closed-loop subsonic wind tunnel (Figure 2.) with a 5 m by.2 m by.8 m test section in the FCET lab at UTIAS. The wind tunnel produces freestream velocities up to 4 m/s and.5% turbulence intensity at U = m/s (Hearst, 25). In the first part of the experiment, different boundary layer thicknesses were produced under the same freestream velocity by varying the model length. The wake after the blunt trailing edge was characterized by hot-wire anemometry and particle image velocimetry (PIV) techniques. In the flow control experiments, a vortex generator design was implemented in different sizes and at different spanwise spacing. Their respective influence on the wake was examined by pressure tubes and PIV in horizontal and vertical planes. Figure 2.: Picture and schematic of wind tunnel facility from Hearst (25). 4

30 2. Experimental Model 2.. Blunt Trailing Edge Model with Adjustable Length Figure 2.2 shows the schematic of the experimental model for the first part of this thesis. It is bounded by acrylic endplates from d upstream of the model leading edge to 2d downstream of the trailing edge. The model has a constant thickness d of.254 m and span of 3.3d. It consists of a semi-elliptical leading edge with length of 3.5d, a rectangular trailing edge with length of 2.5d, and one to six flat sections in the middle. Each flat section is 24d in length, which allows 6 evenly spaced total chord to thickness ratios from 3d to 5d. All tested cases are summarized in Table 2.. To excite an early development of turbulent boundary layer, two narrow rough surfaces are placed on both sides at the end of the leading edge section, shown in Figure 2.3. As the model was being assembled, the leveling of the model surface was kept within ± o with respect to the bottom surface of the wind tunnel. For all experiments, the coordinate system is defined as shown in Figure 2.4, with the origin at the trailing edge in streamwise (x) direction and at the mid point in both wallnormal (y) and spanwise (z) directions. Figure 2.2: Schematic of full length blunt trailding edge model with endplates. 5

31 Table 2.: Testing cases for various boundary layer thickness Case Re d c/d Case Re d c/d Case Re d c/d, 3 7 2, , 3 2, , , 54 3, , , 78 4, 2 2, , 2 5, 26 2, , 26 6, 5 2 2, , 5 Figure 2.3: Wind tunnel picture in front of the model at full length. Figure 2.4: Schemetic of model with 2 sections in the middle, showing the coordinate system Passive Vortex Generators To encourage three-dimensionality in the wake after a blunt trailing edge, two arrays of evenly spaced vane-type vortex generators were installed on both sides of the blunt trailing edge model as a distributed forcing device, which introduces streamwise vortices at the trailing edge and interacts with the vortex shedding directly. Although not commonly seen as a distributed forcing device, passive vortex generators have been extensively studied and widely implemented for the purpose of boundary layer reattachment. Taylor (947) first introduced these devices, which consist of a row of small plates with height close to the boundary layer thickness, and found that momentum was transferred from the outer flow to the near wall region. Yao et al. (22) showed that the vortex structure formed by this type 6

32 of vortex generator remains fairly concentrated up to 2h downstream of the trailing edge of the vanes, where h is the vortex generator height. Shim et al. (24) performed cross-flow measurements after vortex generators with differently-shaped vanes in a flat plate turbulent boundary layer. In their measurements, it was shown that the flow is swept upwards in the pressure side region and downward in the suction side region by a vortex around the vortex generator vane, resulting in greater streamwise mean velocity on the suction side. They also demonstrated that, in comparison with trapezoidal and rectangular generators, the vortex of a triangular generator was formed at a lower position and laterally closer to the trailing edge of the vanes. Lin et al. (99) applied small submerged vortex generators (when h is smaller than the boundary layer thickness δ) for turbulent flow separation control, in which reattachment distance was reduced and pressure recovery was increased. By comparing the devices with different height, they concluded that the vortex generator with h/δ.8 provided the largest pressure recovery but was associated with the largest device-drag penalty. Meanwhile, at smaller device height, the one with h/δ. performed as well as the larger one with h/δ.4. They also suggested the vortex generators be placed no more than 2δ upstream of the baseline separation position. Lin (22) reviewed the studies on low-profile vortex generators to control boundary layer separation, and concluded that, when flow-separation locations are fixed and when the generators are placed within h upstream of the baseline separation, the low-profile vortex generators with. h/δ.5 can be as effective as conventional ones with height greater or equal to the boundary layer thickness. In the study to optimize triangular passive vortex generators for the control of a decelerating boundary layer by Godard and Stanislas (26), they showed that a counter-rotating vortex generator is more effective than a co-rotating one, the optimum skew angle is 8, and the length to height ratio (a/h) is not very sensitive in the range tested but recommended to be at least 2. This thesis intends to test a particular vortex generator design with three different sizing. The design implements the optimal geometric features suggested by the previous studied. It is a pair of counter-rotating triangular vortex generators with a skew angle of 8 o as shown in Figure 2.5. Its length to height (a/h) and trailing edge spacing to height (l/h) ratios are 2.5 and 2.25, respectively. The difference in size of the three vortex generators is summarized 7

33 in Table 2.2. The maximum vortex generator is twice as large as the minimum one, and the medium one is of the intermediate size. For all cases, the vertical edges of the vanes are aligned with the edge of the square base, such that the back edge of each vortex generator is.5d upstream the trailing edge of the model. Figure 2.5: Features of individual vortex generator. Table 2.2: Dimensions of testing vortex generators. min VG med VG max VG l.9d.35d.8d a d.5d 2d h.4d.6d.8d 2..3 Blunt Trailing Edge Model with Vortex Generators In the study of passive flow control, the blunt trailing edge model with 2 middle sections is utilized (Figure 2.4), and the 2.5d long trailing edge section is replaced by a 4.5d long one, 8

34 Figure 2.6: Assembled trailing edge model with minimum vortex generators spaced at 2.4d. which is adaptive to various passive flow control devices shown in Figure 2.6. This results in a total model length of 56d and a boundary layer of approximately.75d thick at Re d = 35, (Section 3.). The trailing edge consists of three parts: the base with slots, the spacers flush mounted on both top and bottom sides of the base, and the individual vortex generators which fit within the spacers. The base of the trailing edge has a flat trailing edge of d thick, on which 3 pressure tubes are flush mounted. The pressure tubes are placed along the horizontal centerline, one at the middle of the span and the other two, to each side, 7d away from the central one. There are three sets of spacers:.8d, 2.4d, and 3.6d. The 2.4d and 3.6d ones are designed to accommodate vortex generators with base size of 2d by 2d. Due to geometric limit, the slot size is forced down to.2d by.2d on the.8d spacer, and only the minimum vortex generators can be built on top of such small base size. The 2.4d spacers are also utilized to test the cases with 4.8d spacing by alternating the mounting of individual vortex generators and flat plates. All tested cases are summarized in Table

35 Table 2.3: Testing cases for vortex generators of different size and spacing Case Re d size spacing Case Re d size spacing Case Re d size spacing, min.8d 2, min.8d 2 35, min.8d 2, min 2.4d 2 2, min 2.4d 22 35, min 2.4d 3, min 3.6d 3 2, min 3.6d 23 35, min 3.6d 4, min 4.8d 4 2, min 4.8d 24 35, min 4.8d 5, med 2.4d 5 2, med 2.4d 25 35, med 2.4d 6, med 3.6d 6 2, med 3.6d 26 35, med 3.6d 7, med 4.8d 7 2, med 4.8d 27 35, med 4.8d 8, max 2.4d 8 2, max 2.4d 28 35, max 2.4d 9, max 3.6d 9 2, max 3.6d 29 35, max 3.6d, max 4.8d 2 2, max 4.8d 3 35, max 4.8d 2.2 Measurement Techniques 2.2. Pressure Tubes To provide an estimation of the freestream velocity for wind tunnel control, a pitotstatic tube was mounted near the front of the test section, approximately 8d upstream of the model trailing edge. In the first part of the experiment, a second pitot tube was positioned 6d downstream and 5d below the trailing edge of the model to measure the local freestream velocity near the wake region, and to determine the reference velocities for hot-wire calibration. Each pitot tube was connected to a MKS Baratron Torr pressure transducer, and the velocity data were computed from pressure measurements by u p = 2(P t P s ) ρ = 2 V P g, ρ where V is the acquired differential voltage data and is converted to the pressure difference by the voltage gain to pressure conversion factor P g of Pa/V. In the second part of the experiment, the front pitot-static tube was moved closer to the model and positioned about d upstream of the model leading edge. The pressure tubes from the trailing edge of the model were averaged and connected to one end of the pressure transducer via three T-shaped connectors, and the transducer measures the pressure difference between the back pressure and the static pressure from the front pitot-static tube 2

36 ( V 2 ). The back pressure coefficient is then calculated by C pb = P b P s P t P s = V 2 V, where V is the measurement from the front pitot-static tube Hot-wire Anemometry Hot-wire anemometry was employed to obtain time-resolved measurements of the boundary layer and in the wake. A 5 µm diameter copper-plated tungsten wire with a sensing length of.5 mm soldered on a Dantec Type 55H probe connected to a Dantec 56C constant temperature anemometer was employed for the hot-wire measurements. The anemometry was operated at an overheat ratio of approximately.6. Each data point was sampled at 2 khz and acquired by a National Instruments SCC-68 Data Acquisition system after a.5 khz low-pass filter. The hot-wire was calibrated against the velocity measured with a pitotstatic tube, and the raw hot-wire voltage data were converted to velocity measurements using King s law (Jørgensen, 2). The temperature in the test section, measured by an Omega T-type thermocouple, was controlled within 3 C variation for each dataset, and temperature correction was applied using the methodology of Hultmark and Smits (2). Figure 2.7: Experimental setup for hot-wire measurements. 2

37 The experimental setup is shown in Figure 2.7. A single hot-wire, oriented to measure the streamwise velocity, was mounted on a 3 directional traverse system. Airfoil shaped leading and trailing edges were attached to the z-direction traverse to reduce its disturbance to the flow. At the beginning and end of each set of measurements for a particular model length, the hot-wire was aligned with the pitot tube in both x and y directions and calibrations were performed to account for hot-wire drift. Hot-wire measurements were performed in the vertical (xy) plane at z = to obtain boundary layer profiles, formation length, and Strouhal number with different model length at Re d =,, 2,, and 35,. The hot-wire was traversed from y/d = to y/d = 2.5 for boundary layer measurements at x/d = -.5. Streamwise hot-wire measurements were performed at x/d =. to 2.35 and y/d =, in order to determine the formation length, which corresponds to the position of maximum velocity variance in the x direction along the centerline. To determine the Strouhal number, velocity spectra were measured at x/d =.5 and y/d =.8. The sampling time for each data point was 35 s, 5 s, and 2 s for boundary layer profiles, streamwise variance, and spectra measurements, respectively Particle Image Velocimetry Particle Image Velocimetry (PIV) was used in both parts of the study to obtain quantitative measurement of the near wake flow. Figure 2.8 shows the experimental setup for PIV measurements in the horizontal plane. Seeding particles were generated by a LaVision aerosol generator using di-ethylhexyl sebacate oil. A thin laser beam was generated by a Quantel EverGreen Laser with output energy of 2 mj per pulse, redirected by 3 mirrors and passed through a cylindrical lens with a focal length of f c = -25 mm and a spherical lens with f c = 5 mm, forming a thin laser sheet. Figures 2.9 and 2. show the top and side view diagrams of the laser beam path through the cylindrical and spherical lenses. The two lenses were spaced by 475 mm to achieve a columnated sheet. The sheet was aligned with the upper surface of the model (y/d =.5) and illuminated particles for x/d from to 5. A scoms LaVision CCD camera with a AF Micro-Nikkor 6 mm f/2.8d lens was utilized to take snapshots of illuminated particles in a field of view from z/d -4 to 4, which is right after the thinnest portion of the laser sheet where it starts to diverge. This field of view was 22

38 chosen to avoid too many particles moving out of the laser sheet between the pulses, which were separated by dt. The thickness of the laser sheet in alignment mode was measured by a ruler to be approximately.2 mm to.7 mm from z/d = 4 to -4. Figure 2. shows the measurement planes for the second part of the experiment. The horizontal plane was located at the same vertical location of y/d =.5 and the field of view was from z/d 7 to z/d 2. Vertical Plane was in the middle of two neighboring vortex generators, Plane 3 was located at the center of a vortex generator, and Plane 2 was in the middle of Plane and Plane 3, where it was close to the edge of a vortex generator. All vertical planes were relative to the vortex generators, and therefore their locations in the z coordinate changed as the positioning of the vortex generators changed. Measurements in vertical Plane were taken for all cases, measurements in Plane 2 were taken for cases 4, 7,, 4, 7, 2, 24, 27, and 3, and measurements in Plane 3 were taken for cases 8,, 8, 2, 28, and 3 (Table 2.3). To illuminate the vertical planes, the same lenses, placed at a lower position, were used to generate a horizontal sheet and then a flat mirror was used to Figure 2.9: Top view diagram of laser beam path. Figure 2.: Side view diagram of laser beam path. Figure 2.8: Experimental setup for PIV. 23

39 redirect the laser sheet vertically. The position of the vertical plane can hence be adjusted by moving the flat mirror along the z direction. For all experiments, independent pairs of snapshots were taken with a recording rate of approximately Hz. The time interval dt between each pair of frames was 8 µs, 4 µs, and 2 µs for Re d =,, 2,, and 35,, respectively. Velocity vectors were obtained by cross-correlating the interrogation region in one frame to the search region in the second frame, using pixel interrogation windows with 5% overlap, resulting in a vector spacing of around.5d. Using the built-in function of DaVis software version 8..5, which incorporates the method of correlation statistics, the uncertainties in the horizontal plane measurements were estimated to be less than 4% of the freestream velocity (Wieneke and Prevost, 24). Figure 2.: Illustration of PIV measurement planes. 24

40 2.3 Analysis of PIV Measurements 2.3. Proper Orthogonal Decomposition Proper orthogonal decomposition (POD), is a common tool in approximating low-dimensional descriptions of turbulent fluid flows (Holmes, 22). POD has been implemented in both sections of this thesis to filter out small-scaled turbulent noise, to extract dominant flow structures, and to analysis flow field from an energetic perspective. This section provides a brief review of the POD method as described by Tropea et al. (27). Suppose {u = u(x, t n ), x R 3, t n R + } is a set of snapshots obtained by PIV measurements at N t different time steps t n over a spatial domain Ω(x Ω). The objective of POD is to extract from this set of random vector fields a coherent structure which is defined as the deterministic function which is best correlated on average with the realizations u(x, t n ) (Lumley, 967). This is equivalent to finding an L 2 space of functions Φ, on which the mean-squared projection of u(x, t n ) is maximized in a time-average sense. Since the magnitude of Φ is insignificant and can be removed, symbolically this problem becomes finding Φ such that, < (u, Φ) 2 > < (u, Ψ) 2 > = max, (Φ, Φ) = Φ 2 =, Φ 2 Ψ L 2 (Ω) Ψ 2 where <. > denotes an average over time, (.,.) and. denote the L 2 inner product and norm over Ω, respectively. For Φ in real space, (u, Φ) = Ω u(x) Φ(x)dx, u = (u, u). It has been mathematically shown that the above maximization problem is equivalent to solving the eigenvalue problem R(x, x )Φ(x )dx = λφ(x). Ω The two-point spatial correlation tensor R(x, x ) is estimated under stationarity and ergod- 25

41 icity assumptions as R(x, x ) = N t u(x, t i ) u(x, t i ), N t i= where denotes the dyadic product. Since the number of spatial points in the data sets is much greater than the number of snapshots N t and the snapshots are linearly independent, the method of snapshot POD is implemented to drastically reduce the computational effort (Sirovich, 987). When we assume that Φ has a special form in terms of the original data: N t Φ(x) = a(t j )u(x, t j ), and substitute Φ into the above two equations, we obtain j= N t i= [ Nt N j= t ( Ω ] N t u(x, t j ) u(x, t i )dx )a(t j ) u(x, t i ) = λ a(t j )u(x, t j ). j= A sufficient condition for this equation to hold is N t N j= t ( Ω ) u(x, t j ) u(x, t i )dx a(t j ) = λa(t i ), which can be rewritten as the eigenvalue problem CV = λv, where C ij = N t u(x, t i ) u(x, t j )dx and Ω V = [a(t ), a(t 2 ),..., a(t Nt )] T. Since C is a nonnegative Hermitian matrix, it has a complete set of eigenvalues and orthogonal eigenvectors λ (n) and V (n) for n =,...,N t, with λ () λ (2)... λ (Nt). The magnitudes of the time coefficients a (n) (t i ) and POD eigenfunctions Φ (n) (x) can be imposed 26

42 by N t (V (n), V (m) ) = N t N t i= Φ (n) (x) = N t λ (n) a (n) (t i )a (m) (t i ) = λ (n) δ nm, N t j= a (n) (t j )u(x, t j ), and such that they satisfy the common properties as for the classical POD: u(x, t) = N P OD n= a (n) (t)φ (n) (x), Φ (n) (x) Φ (m) (x)dx = δ nm, and Ω T T a (n) (t)a (m) (t)dt = λ (n) δ nm. When u(x, t) is the fluctuating velocity field, as suggested by Cordier and Bergmann (28), a measure of the integrated turbulent kinetic energy is provided by κ Ω = ( u, u ) Ω N P OD = n= λ (n) Wavelength Identification of Secondary Instability The wavelength of the secondary instability is identified from the velocity field captured with PIV on the horizontal measurement plane at y/d =.5. Each snapshot is reconstructed with a number of POD modes such that the high-order modes with low energy contribution are filtered out. Figure 2.2 shows the percentage of cumulative energy in the reconstructed velocity field as a function of the number of POD modes for cases and 8 from Table 2.. For a given number of POD modes, the cumulative energy is slightly lower with the later case in which the boundary layer is thicker and the flow is more turbulent. However, this difference is small and 7% of the cumulative energy can be preserved in a velocity field reconstructed with about 5 modes for both cases. Therefore, 5 POD modes are used in reconstructing the velocity fields for all tested cases. 27

43 Figure 2.3 shows an example of reconstructed instantaneous fluctuating velocity field with 5 POD modes at c/d = 3 and Re d =,. To estimate the wavelength length of the secondary instability across the span, a smoothed trace, shown by the dotted line, is identified along the local minimum in the streamwise velocity as a function of spanwise location. Figure 2.4 shows the variation of streamwise velocity across the span on the identified trace, and the spanwise wavelength is estimated using the autocorrelation of the velocity trace. Two types of autocorrelation coefficients can be computed: biased and unbiased. The formulas in discrete form are given below: R biased (m) = R unbiased (m) = N m N n= (X n+m X)(X n X), and V ar(x) N m N-m n= (X n+m X)(X n X), V ar(x) where X and X are the original and shifted signals, respectively, N is the total number of data points, m is the separation vector. Since the spanwise coordinates are evenly spaced, m can be converted to spanwise wavelength by a multiple of the spatial resolution. % energy Re d =,; c/d = 3 Re d = 35,; c/d = 5 x / d npod Figure 2.2: Variation of cumulative energy with the number of POD modes used in reconstruction z / d Figure 2.3: Streamwise component of reconstucted instantaneous fluctuating velocity field with 5 POD modes at c/d = 3 and Re d =,. Figure 2.5 shows the biased and unbiased autocorrelations computed from the signal in Figure 2.4. The distance shifted z is considered as a valid wavelength if the corresponding 28

44 autocorrelation value reaches a local maximum and is above a certain threshold. The threshold is an indicator of signal to noise level and local maximum values below the threshold are considered as noise. In the biased case, the covariance of original and shifted signals is normalized by the total signal length N regardless of the length of overlapping signal (N m) used in the computation. Therefore, the magnitude of the biased autocorrelation decays as the shifted distance increases. This definition of autocorrelation biases towards the smaller shifted distance where more data points are involved in the product summation. As a result, some physical wavelength such as the one at z/d 2.2 in the example case may not be detected with slightly larger threshold. On the other hand, the unbiased autocorrelation has large variation at large shifted distance. When the shifted distance is large, the unbiased autocorrelation is computed from an averaged quantity over a limited portion of the signal. Therefore, with the unbiased autocorrelation, the wavelength detected at large shifted distance, such as the one at z/d 6.3, may be unclear in the velocity signal and can be caused by noise alone. As a trade off, only the first half of the unbiased autocorrelation values are used in finding the spanwise wavelength (half length of autocorrelation). Section 3.3 provides further justification of the choices of the number of POD modes utilized (n P OD ), the signal to noise threshold (thr), biased vs. unbiased, and half vs. full length of autocorrelation. u (m/s) Correlation unbiased biased threshold z / d δ z / d Figure 2.4: Example of streamwise velocity variation across the span on the indentified trace. Figure 2.5: Comparison of biased and unbiased correlation as a function of distance shifted. 29

45 Chapter 3 Effect of Boundary Layer Thickness on Wake Dynamics In this chapter, the effect of boundary layer thickness on wake dynamics is studied. The boundary layer thickness is varied by changing the model length and by testing at different Reynolds numbers. Meanwhile, the boundary layer is maintained to be turbulent for all cases, which is verified by examining boundary layer properties such as log-scaled profile and boundary layer shape factor. The variation of wake dynamics as a result of different boundary layer thickness is studied by observing the change in the characteristic quantities in the wake including Strouhal number, wake formation length, and the wavelength of secondary instability. These wake characteristics are extracted from the time-resolved velocity measurements from hot-wire anemometry and from the horizontal plane velocity fields recorded via PIV. 3

46 3. Boundary Layer Properties Boundary layer profiles with different model length at Re d =,, and x/d =.5 are shown in Figure 3., where L = to 6 stands for the number of model sections between the leading edge and trailing edge. The shape of the boundary layer becomes less convex as the model length increases. At the smallest model length, as y increases, u increases, exceeds the freestream velocity, and then decelerates back. This is owing to the fact that the flow was accelerated at the leading edge of the model, resulting in an overshoot of velocity near the surface of the model. When the model length is short, this overshoot is still evident in the boundary layer profiles since there is not enough distance for it to be smoothed out. Due to the overshoot and the variation in the boundary layer shape, it is difficult to define the boundary layer thickness by the conventional method of finding the y-location where the velocity reaches 99% of freestream velocity. Mariotti and Buresti (23) defined boundary layer thickness by the point where the magnitude of skewness is maximum, and the same definition is applied in this thesis. Figure 3.2 plots the skewness measurements along the boundary layers at different model lengths for Re d =, and x/d = -.5. It is illustrated that the minimum point in skewness is easily distinguishable and unique for each case. 3 Re =, x =.5d 3 Re =, x =.5d y / d L = L = 2 L = 3 L = 4 L = 5 L = 6 y / d L = L = 2 L = 3 L = 4 L = 5 L = u / u fs Figure 3.: Boundary layer profiles with different model length at Re d =, and x/d = skewness Figure 3.2: Skewness of the boundary layers with different model length at Re d =, and x/d = -.5. Figure 3.3 shows the boundary layer profiles with different model length at Re d =, 3

47 in wall-normalized units u + and y + defined by u + = u u, y+ = u y ν, and τw u = ρ, where ν is the dynamic viscosity, u is the friction velocity, and τ w is the wall shear stress. Since τ w could not be measured directly with the experimental setup, it was estimated based on the measured boundary layer profiles. By definition, ( ) u τ w = µ y where µ is the dynamic viscosity. Therefore, a initial guess of τ w was determined by fitting a straight line to the boundary layer profile in the near-wall (linear) region. The value of τ w was then adjusted by iteratively fitting to the log region of the boundary layer according to the law of wall:, y= u + = κ log y+ + C +, where κ is the von Kármán constant of.4 and C + is the intercept of 5., both of which are in the range of common values suggested in the literature (Coles, 956; Buschmann and Gad-el Hak, 23; Guo et al., 25). The wall-normalized boundary layer profiles all collapse together in the log region between y + to 3. At low y +, u + at the shortest model length is relatively higher than the ones at longer model length, which is due to the velocity overshoot as observed in Figure 3.. This phenomenon also affected the shape factor shown in Figure 3.4. Shape factor is defined as the ratio of displacement thickness δ over momentum thickness θ, calculate from δ = ( u(y) ) dy, and θ = U ( u(y) u(y) ) dy, U U respectively. The shape factor H is slightly below.4 and slowly decreases as the Reynolds number increases for all model length. This is in agreement with the results from Naghib- Lahouti et al. (24) and indicates the boundary layer is turbulent for all tested cases (Schlichting, 979). 32

48 25 2 Re =, x =.5d.32.3 u+ 5 5 L = L = 2 L = 3 L = 4 L = 5 L = y+ Figure 3.3: Boundary layer profiles in wallnormalized units with different model length at Re d =, and x/d = -.5. H = (δ /θ) Re d =, Re d = 2, Re = 35, d # of sections Figure 3.4: Shape factor of the boundary layers with different model length at x/d = Formation Length and Vortex Shedding Frequency The formation length L f by definition is the streamwise distance along the wake centerline at which the velocity fluctuations are maximum (Bearman, 965). Figure 3.5 shows the variation of the streamwise velocity variance at Re d =, with different model length. Due to the noise in the data, it was found that the maximum of the velocity variance measurements is not necessarily the physical maximum. A second order polynomial function was fitted around the measured maximum to find a approximated maximum shown by the circles in Figure 3.5. Figure 3.6 shows the variation of formation length with boundary layer displacement thickness at different Reynolds number. For all Reynolds number, the formation length increases as the displacement thickness increases. This result is consistent with the ones of other researchers such as Rowe et al. (2) who attributed this observation to the longer time (distance) it takes for vorticity from thicker boundary layers to be carried across the wake when the boundary layers are effectively farther apart. Additional evidence of this phenomenon is presented in the vortex shedding frequency and boundary layer thickness relation. Figure 3.7 shows the power spectrum density of single-point velocity measurements in the form of Strouhal number St d with different model length at Re d =,. The highest peaks in the spectra correspond to the vortex shedding frequencies, and the secondary peaks are their harmonics. The variation of Strouhal number with displacement thickness at different Reynolds numbers is shown in Figure 3.8. It can be 33

49 easily observed that the Strouhal number decreases as the boundary layer thickness increases for all Reynolds number. Physically, with the boundary layers spaced farther apart, the vorticity travels from one side of the model to the other in a longer time, and hence sheds at a lower frequency. From both Figure 3.6 and Figure 3.8, the data at different Reynolds numbers appear to collapse to a single trend, which indicates that the variations of formation length and Strouhal number are accounted for by the difference in boundary layer thickness and are not sensitive to changes in freestream velocity. Variance (m 2 /s 2 ) Re =, y = x / d L = L = 2 L = 3 L = 4 L = 5 L = 6 Figure 3.5: Variation of velocity variance in streamwise (x) direction at Re d =, and y/d = with different model length. The graphs have been offset to facilitate visualization. L f / d Re d =, Re d = 2, Re d = 3, δ* / d Figure 3.6: Variation of formation length with boundary layer displacement thickness at different Reynodls number. PSD (m 2 s 2 Hz ) Re =, x =.5d, y =.8d L = L = 2 L = 3 L = 4 L = 5 L = 6 St d Re d =, Re d = 2, Re d = 3, St(d) Figure 3.7: Power spectral density of streamwise velocity, measured at x/d =.5 and y/d =.8. The graphs have been offset to facilitate visualization δ*/d Figure 3.8: Variation of vortex shedding frequency in terms of Strouhal number with boundary layer displacement thickness at different Reynodls number. 34

50 3.3 Wavelength of Secondary Instability For each model length and Reynolds number, the wavelength of the secondary instability is estimated by averaging the collection of the wavelengths identified using the method described in Section Figure 3.9 demonstrates, with c/d = 3 and Re =,, how the resultant average wavelength is affected by the choices of n P OD, thr, unbiased or biased autocorrelation, and half or full length of autocorrelation taken into account. Shown in the legends, nw, the number of wavelengths detected when n P OD = 5, decreases as the threshold value increases thr =., nw = 67 thr =.5, nw = 827 thr =.2, nw = 566 thr =.25, nw = 367 thr =.3, nw = 223 thr =.4, nw = 56 thr =.5, nw = λ z / d.5 thr =., nw = 226 thr =.5, nw = 42 thr =.2, nw = 862 thr =.25, nw = 682 thr =.3, nw = 57 thr =.4, nw = 244 thr =.5, nw = 86 λ z / d npod 2 3 npod (a) unbiased, half (b) biased, half thr =., nw = 283 thr =.5, nw = 247 thr =.2, nw = 228 thr =.25, nw = 79 thr =.3, nw = 486 thr =.4, nw = 938 thr =.5, nw = thr =., nw = 682 thr =.5, nw = 4 thr =.2, nw = 72 thr =.25, nw = 48 thr =.3, nw = 233 thr =.4, nw = 56 thr =.5, nw = λ z / d 5.5 λ z / d npod 2 3 npod (c) unbiased, full (d) biased, full Figure 3.9: Average wavelength computed with different threshold and number of POD modes for model length of 3d at Re d =,. Labels indicate threshold values and the number of valid wavelengths detected. 35

51 λ z / d Re d =, Re d = 2, Re d = 35, δ / d Figure 3.: Variation of secondary instability wavelength with boundary layer displacement thickness at different Reynolds number. for all of the four cases. The average wavelength computed from the first half of unbiased autocorrelation (Figure 3.9a) converges well for a large range of signal to noise threshold from. to.3 when n P OD is greater than. Meanwhile, with a threshold between. and.3, the average wavelength stays relatively constant as n P OD increases from to. However, for the other cases (Figures 3.9b 3.9c and 3.9d), the average wavelength is sensitive to the choice of signal to noise threshold. The definition of an exact threshold from the experimental data can be biased, which makes the estimation of wavelength unreliable for these three cases. Therefore, the first half of the unbiased autocorrelation is utilized in the computation of the average wavelength to give robust estimations. Since the averaged wavelength is approximately unchanged as the threshold varies from. to.3 and the constraint for the number of POD modes is n P OD >, parameters of thr =.2 and n P OD = 5 (about 7% energy reconstruction) are applied for all cases of model length and Reynolds number. Figure 3. shows the variation of the average spanwise wavelength with boundary layer displacement thickness at Re =,, 2,, and 35,. The errorbars indicate the standard errors of the wavelengths computed by the above method. As the displacement thickness increases, the wavelength of the secondary instability increases slightly from 2. to 2.4. This range of wavelengths is in agreement with previous studies with similar geometries 36

52 (Ryan et al., 25; Doddipatla, 2; Naghib-Lahouti et al., 24). Similar to St d and L f, λ z has little dependence on Reynolds number, and its variation with Reynolds number can be accounted for in the change of boundary layer thickness. 3.4 Summary and Discussion Table 3. provides a summary of the boundary layer properties and the wake characteristic quantities for all tested model lengths and Reynold numbers. As the boundary layer thickness changes, the effective thickness of the trailing edge changes, and so do the wake quantities St d, L f, and λ z. In the literature where changes in Reynolds number result in variation of boundary layer thickness (Bull et al., 995; Doddipatla, 2; Naghib-Lahouti et al., 24), the changes in wake quantities due to the increase in boundary layer thickness is often accounted for when the quantities are normalized by the effective trailing edge thickness d = d + 2δ. δ in the current study varies from.77d to.39d, which results in a 42% increase in d from.54d to.638d. Table 3.: Summary of measured quantities at the boundary layer and in the wake for blunt trailing edge model with various length at Re d =,, 2,, and 35,. Case Re d c/d δ/d δ /d θ/d H St d L f /d λ z /d, ±.3 2, ±.3 3, ±.3 4, ±.4 5, ±.3 6, ±.4 7 2, ±.3 8 2, ±.3 9 2, ±.3 2, ±.3 2, ±.4 2 2, ± , ± , ± , ± , ± , ± , ±.4 37

53 .5.4 Re d =, Re d = 2, Re d = 35, St(d ) δ / d Figure 3.: Strouhal number normalized by d at different boundary layer thickness. Figure 3. shows the variation of Strouhal number normalized by d at different boundary layer thickness. The lowest St(d ) values (.233), which are at the smallest aspect ratio of 3 (δ/d.9), match closely with the values reported by Bull et al. (995) (St(d ) =.229, c/d = 2-52, δ/d.6-.6) and Naghib-Lahouti et al. (24) (St(d ) =.23, c/d = 2.5, δ/d.3). As boundary layer thickness increases, although St d decreases by 7% (Table 3.), St(d ) increases by 9% from.233 to.277. This suggests that the decrease in the vortex shedding frequency due to the increase in boundary layer thickness is over-compensated by the normalization of d. Figures 3.2 and 3.3 compare the d normalized formation lengths found in the current study to the ones reported in the previous studies (Doddipatla, 2; Naghib-Lahouti et al., 24). Similarly Figures 3.4 and 3.5 compare the d normalized spanwise wavelengths to the ones from the previous studies. The ranges indicated in red in Figures 3.2 and 3.4 match the range of Reynolds numbers investigated in the current study. The model aspect ratios in the studies by Doddipatla (2) and Naghib-Lahouti et al. (24) are both 2.5. Hence, L f /d and λ z /d measured at the smallest δ/d in the current study are at conditions (aspect ratio of 3) closest to ones in the previous studies. L f /d at δ/d.9 in Figure 3.3 is about 5% smaller than the ones in previous studies, and λ z /d at δ/d.9 in Figure 3.5 is about 5% smaller than the ones reported by Doddipatla (2) and Naghib-Lahouti 38

54 et al. (24). However, both L f /d and λ z /d would match the ones in previous studies if the trends in Figures 3.3 and 3.5 were extended to smaller boundary layer thickness. Naghib-Lahouti et al. (24) suggested that there is no significant change in L f /d and λ z /d for Reynolds number greater than, (.28 <= δ/d <=.3). However, decreases in L f /d and λ z /d were observed in the current study with a significantly wider range of Re d =, Re d = 2, Re d = 35,.2 L f / d δ / d Figure 3.2: Formation length normalized by d at different Reynolds number from Naghib- Lahouti et al. (24). Figure 3.3: Formation length normalized by d at different boundary layer thickness from current study Re d =, Re d = 2, Re d = 35, λ z / d δ / d Figure 3.4: Wavelength of secondary instability normalized by d at different Reynolds number from Naghib-Lahouti et al. (24). Figure 3.5: Wavelength of secondary instability normalized by d at different boundary layer thickness from current study. 39

55 boundary layer thickness (up to 2.5d). As boundary layer thickness increases, L f /d decreases from.78 to.64 by 8% and λ z /d decreases from.86 to.39 by 25%. Comparing to the quantities normalized by d (Table 3.), L f /d increases by 2% and λ z /d increases by %. Similar to the Strouhal number, the normalization of d does not make the variations in L f and λ z smaller. The wake quantities obtained from the current study agree with the ones from the previous studies when normalized by d. However, to account for the changes in the wake quantities, d is not a good scaling factor in this study where the boundary layers are much thicker than the ones previous investigated. 4

56 Chapter 4 Control Effectiveness of Passive Vortex Generators The purpose of this chapter is to optimize the effectiveness of flow control by varying the size and spacing of vortex generators with a particular design. The performance of each configuration is evaluated by how much formation length L f is elongated and how much back pressure, expressed in base suction coefficient C pb, is recovered. Features extracted from POD analysis such as modal energy distribution, phase plot, and mode shape are utilized to understand the change in the flow field dynamics. To understand the characteristics of the chosen vortex generator design, velocity fields have been measured in the horizontal plane at the model surface and in three vertical planes at different spanwise locations. To study the effect of size and spacing, the same vortex generator geometry is scaled to three different sizes referred as minimum, medium, and maximum. All of the three vortex generators have been studied at the spacing of 2.4d, 3.6d, and 4.8d, and the minimum ones have also been investigated at.8d spacing. To examine how control effectiveness varies with freestream velocity, the experiment is performed at three different Reynolds numbers:,, 2,, and 35,. When larger vortex generators are placed with small spacing, interference between neighbouring vortex generator pairs has been observed, and the characteristics of the interference is thereby investigated. 4

57 4. Characteristics of Passive Vortex Generators To analyze the characteristics of this particular design of passive vortex generator, the flow field around a single vortex generator has been measured in different planes. For this purpose, the velocity fields in the vertical planes, 2 and 3, and the horizontal plane at y/d =.5 (shown in Figure 2.) are measured when the maximum vortex generators are applied with λ/d = 4.8 at Re = 35,. Figure 4. plots the mean streamwise velocity field in these four measurement planes. Figure 4.2 shows the mean streamwise velocity field in a vertical plane under the same flow conditions without forcing. Comparing Figures 4. and 4.2, the recirculation region is elongated in all planes when control is applied. The velocity field in Plane is the closest to the velocity field without intervention of vortex generators. In Plane 2, located near the edge of a vortex generator vane, the shear layers are brought further apart due to the interaction between the flow field and each vane of the vortex generators. As the flow encounters the front side of a vortex generator vane, the locally decelerated flow results in the high pressure that brings momentum away from the wall and along the vane towards the freestream. On the back side of the vane, flow is transported towards the wall and across the vane due to the low pressure, resulting in a vortex around each vortex generator vane. In Plane 3, located at the center of a vortex generator, the streamwise velocity field after the blunt trailing edge approaches freestream velocity the fastest. This is due to the streamwise momentum brought by the spanwise components of the vortices generated by both vanes. POD analysis (as described in section 2.3.) was implemented to decompose the velocity fields into time-independent spatial modes and time coefficients. The spatial modes are ordered from the most to the least energetic based on their associated eigenvalues, and the primary vortex shedding structure can be isolated since it is described by the first two POD modes (Deane et al., 99). Figure 4.3 compares the energy distribution of the first five POD modes in different vertical planes in the case of maximum vortex generators spaced at 4.8d. When vortex generators are applied, the percentage of energy in the first two modes, which represents the primary vortex shedding, is lower than the reference case in all vertical planes, and the opposite is true for higher order modes. This indicates that, with these vortex 42

58 2.5 Vertical Plane 2.5 Vertical Plane y / d.5 y / d x / d x / d y / d Vertical Plane 3 x / d Horizontal Plane x / d z / d Figure 4.: Streamwise mean velocity field in vertical plane, 2, 3 and horizontal plane at y/d =.5 when maximum vortex generators are applied with spacing of 4.8d at Re = 35, Base Case y / d x / d 5 Figure 4.2: Streamwise mean velocity field in vertical plane at Re = 35, when no control is applied. 43

59 generators, a less portion of the total energy is contained in the vortex shedding, while more is distributed to higher order modes. The time-varying coefficients of the first two POD modes are further investigated by plotting their normalized values. Van Oudheusden et al. (25) suggested that if the von Kármán vortex shedding can be described as a purely periodic motion, the time coefficients of the first two POD modes will follow a circle with radius of in the normalized plane (a (t)/ 2λ vs. a 2 (t)/ 2λ 2 ), and change in the intensity of the vortex when shed leads to the deviation of radius from unity. Figure 4.4 compares the phase plots in different vertical planes with vortex generators spaced at 4.8d to the case when no vortex generator is applied. The percentage of snapshots that fall within a radius of ±.2, which represents how dominant and organized the primary vortex shedding is, is indicated for each case. The phase plot in Plane 2, which is close to the edge of the vortex generator, is slightly more spread out than the reference case while not much increase in the scatter is observed in the other two planes with vortex generators being applied. Figure 4.3 and 4.4 suggest that by applying this particular vortex generator geometry, although the percent energy in von Kármán vortices is decreased, the primary vortex street is not structurally disrupted or much distorted by streamwise (secondary) vortices. As reviewed in Section.2, L f is defined as the streamwise distance from the trailing edge where u rms, or turbulence intensity is maximized (method ), and it also corresponds to the length of the mean recirculation region, which ends at a point where the mean streamwise velocity crosses zero (method 2). In the literature, method is commonly used with both hot-wire and PIV measurements to determine formation length (Tombazis and Bearman, 997; Park et al., 26; Naghib-Lahouti et al., 22). However, in the current study, the peak in turbulence intensity along the wake centerline is unclear within a certain range due to noise in the measurements and the effect of vortex generators in some cases. Therefore, the formation length in the current study is determined by finding the point along the wake centerline where the mean streamwise velocity crosses zero. Figure 4.5 plots the turbulence intensity along the wake centerline for flow field constructed with different numbers of POD modes, when no vortex generator is applied and Reynolds number is 35,. The vertical dashed line corresponds to the location where the 44

60 .5 No VG: 72.%.5 4.8d max P: 74.5% POD Mode % Energy, Max VG s Spaced at 4.8d 34 Flat Middle of VGs 3 Center of VG Edge of VG a 2 / (2λ 2 ) /2.5.5 a 2 / (2λ 2 ) /2.5.5 % Energy a / (2λ ) /2 4.8d max P2: 65.8%.5.5 a / (2λ ) /2 4.8d max P3: 7.3% Mode Index a 2 / (2λ 2 ) /2.5.5 a 2 / (2λ 2 ) /2.5.5 Figure 4.3: Percent energy of the first 5 POD modes in different vertical planes with maximum vortex generators spaced at 4.8d, at Re = 35,..5 a / (2λ ) /2.5 a / (2λ ) /2 Figure 4.4: Phase plots in different vertical planes based on coefficents of POD modes and 2 with maximum vortex generators spaced at 4.8d, at Re = 35, streamwise velocity crosses zero. As the number of POD modes increases, the peak in the turbulent intensity shifts slightly towards the trailing edge. Meanwhile, the formation length found by the current method closely matches the peaks in turbulence intensity in the reconstructions with 2, 5, and 5 modes. This proves that the point along the centerline where the mean streamwise velocity changes sign is consistent with the location of maximum u rms. Figure 4.6 compares the variation of mean streamwise velocity along the wake centerline at Re d = 35, with maximum vortex generators spaced at 4.8d to the case when no vortex generator is applied. As shown, the point where the mean streamwise velocity changes from the upstream direction to downstream (where it crosses zero) is unique for each case, which makes the determination of formation length less ambiguous. In all of the three vertical planes, the formation length is longer when control is applied. It is the longest in Plane 3 located at the center of a vortex generator pair and the shortest in Plane located at the middle of two neighbouring vortex generators. Formation length in Plane 2 is slightly longer than the one in Plane. This pattern of how the control strategy extends the formation length across the span of the model agrees well with the results from Naghib-Lahouti et al. 45

61 (22). The observations in this section shows that Plane gives the most conservative estimate of the control effectiveness when the vortex generators are not placed too close to each other. Therefore, the formation length in Plane (which gives the shortest formation length for all cases tested) is compared for different cases as one method of evaluating the control effectiveness, and unless otherwise specified, results presented in the following sections were obtained from measurements in Plane Turbulence Intensity Along x, No VG, Re = 35, 3 modes 2 modes 5 modes 5 modes raw L f u / u fs x / d Figure 4.5: Variation of turbulence intensity along x at y = without vortex generator (reference case) at Re d = 35,. Figure 4.6: Variation of streamwise mean velocity along x at y = with maximum vortex generators spaced at 4.8d in comparison to the reference case (with no VG), at Re d = 35,. 4.2 Effect of Reynolds Number For each vortex generator configuration, measurements of the flow field have been taken at three different Reynolds numbers:,, 2,, and 35,. Figure 4.7 and 4.8 illustrate the effect of Reynolds number on formation length and base pressure coefficient, respectively. With the minimum vortex generators, the overall trend of how formation length or base pressure coefficient varies with the spacing has little dependence on Reynolds number. Figure 4.7 shows that the measurements at Re d = 2, coincide with the ones at Re d = 35,, and the formation length is slightly longer when the Reynolds number is 46

62 Re =, Re = 2, Re = 35,.2.22 Re =, Re = 2, Re = 35, L f / d λ / d C pb λ / d Figure 4.7: Comparison of formation length with minimum vortex generators of different spacing at different Reynolds number. Horizontal lines represent base cases with no vortex generator. Figure 4.8: Comparison of back pressure coefficient with minimum vortex generators of different spacing at different Reynolds number. Horizontal lines represent base cases with no vortex generator.,. This difference is explained by the longer formation length for the baseline case and the thicker boundary layer at Re d =,. When the change of formation length is normalized by d, where d = d+2δ and δ is the displacement thickness (listed in Table 4.), the measurements collapse for all Reynolds numbers as shown in Figure 4.9. It should be noted that the estimate of displacement thickness has a large uncertainty of about 2% due to a few limitations. Since all measurements have been taken in the wake, the velocity profile for the boundary layer at separation is approximated based on measurements obtained at.36d downstream of separation. Measurements across the wall-normal direction Table 4.: Estimate of Displacement Thickness (from Plane ) at Different Reynolds Numbers Re =, Re = 2, Re = 35, Reference.d.88d.8d.8d min.69d.47d.38d 2.4d min.55d.36d.25d 3.6d min.32d.3d.d 4.8d min.3d.8d.4d 2.4d med.92d.7d.57d 3.6d med.49d.24d.5d 4.8d med.42d.24d.d 2.4d max.244d.22d.22d 3.6d max.76d.52d.4d 4.8d max.52d.33d.26d 47

63 are taken every.4d in y, which limits the accuracy of the displacement thickness estimate. It should also be noted that for some cases, such as when the maximum vortex generators are applied at 2.4d spacing, the streamwise vortices generated by the control devices can directly affect the velocity profiles in the measurement plane due to small spacing of the vortex generators. Hence the locally estimated displacement thickness may deviate from the mean over the entire span. Figure 4.8 shows that the Reynolds number causes little difference in the measurements of the back pressure coefficient. As can be observed from Figure 4.8 and 4.9, C pb and (L f L r f )/d vary similarly with the spacing of vortex generators. As the two normalized quantities are plotted, shown in Figure 4., the increase in base pressure, C pb, and the increase in formation length, (L f L r f )/d, are closely correlated (consistent with the observations from Bearman, 965) and the trend is unaffected by Reynolds number. 4.3 Effect of Size and Spacing It was shown in the previous section that the effect of flow control has little dependence on Reynolds number, and therefore, later analysis is focused on the cases at Re = 35,. Figure 4. and 4.2 compare the formation length and back pressure coefficient with vortex generators of different sizes and spacings. For all cases, both formation length and back.6.5 Re =, Re = 2, Re = 35,.4 (L f L f r ) / d λ / d Figure 4.9: Comparison of normalized change in formation length with minimum vortex generators of different spacing at different Reynolds number. Figure 4.: Relation of normalized change in formation length and reduction in base pressure with minimum vortex generators of different spacing at different Reynolds number. 48

64 pressure coefficient are increased relative to the baseline case with the application of control. With minimum vortex generators, the formation length is the longest and the back pressure coefficient is the highest when they are placed with 2.4d spacing, matching the wavelength of the secondary instability found in Section 3.3. This agrees with the results from earlier works, which demonstrated that distributed forcing control is the most effective when the forcing wavelength matches the natural wavelength of the secondary instability (Kim and Choi (25), Darekar and Sherwin (2), Dobre et al. (26), Doddipatla (2), etc.). The performance of the case with minimum vortex generators spaced at.8d, which is smaller than but close to λ z, is only slightly degraded. This demonstrated that the control effects are relatively insensitive to spacing near the optimal one and hence the spacing of the forcing does not need to be exactly the same as the wavelength of the secondary instability. Comparing vortex generators of difference sizes but constant spacing of 4.8d, the control is slightly more effective as the size of vortex generators increases. However, as the spacing gets smaller, the streamwise vortices from the neighboring vortex generators start to interact with each other, which results in a less effective control. This effect is discuss in more details in the next section (4.4). Formation length and back pressure are two indicators of the control effectiveness on the reduction of drag due to the primary vortex shedding. The wake associated drag can be Formation Length min VG med VG max VG no VG Back Pressure Coefficient L f / d.2 C pb min VG med VG max VG no VG λ / d Figure 4.: Comparison of formation length in vertical Plane with vortex generators of different size and spacing at Re d = 35, λ / d Figure 4.2: Comparison of back pressure coefficient with vortex generators of different size and spacing. 49

65 estimated by the method proposed by Naghib-Lahouti et al. (25), which is based on formulations developed by Van Oudheusden et al. (27) and Bohl and Koochesfahani (29). The force based on planar PIV data can be estimated by the following equation (Van Oudheusden et al., 27): F i = ρ S ū i ū j n j ds ρ S u i u j n jds + µ S ( ūi + ū ) j n j ds x j x i S pn i ds, (4.) where the overbar denotes time-averaged quantities, S stands for the surface integral of the control volume, n is the unit vector normal to the boundary, and i and j are indices in tensor notation. According to Bohl and Koochesfahani (29), the last term in Equation 4., which is the mean pressure at the downstream boundary of the control volume, can be estimated by: p y = ρ v2 rms y. (4.2) Combining Equations 4. and 4.2, the drag coefficient can be estimated using the velocity measurements in the near-wake region (Bohl and Koochesfahani, 29): C D = 2 A W u U ( u ) dy + 2µ U U A 2 W ( u y + v ) dy + 2 x A W [ (vrms ) 2 ( urms ) ] 2 dy, U U (4.3) where the integral over W should be calculated outside of the recirculation zone, in the near-wake region. It is calculated in Plane at 4.9d downstream of the trailing edge for the current study. Figures 4.3 shows the percentage reduction of total drag with and without the vortex generators. In agreement with the trend in formation length and back pressure recovery, the maximum drag reduction of 2% was achieved with the minimum vortex generators spaced at 2.4d. The total drag was increased with the application of the other vortex generator configurations due to the drag introduced by the vortex generators themselves and the inefficiency in mitigating drag from the primary vortex shedding. component of drag ( C D ), represented by the last term of Equation 4.3. Figure 4.4 shows the fluctuating When control is applied, C D was reduced for all cases, and it was completely eliminated with the minimum 5

66 Figure 4.3: Comparison of total drag reduction in vertical Plane with vortex generators of different size and spacing at Re d = 35,. Figure 4.4: Comparison of drag due to velocity fluctuation with vortex generators of different size and spacing. vortex generators at.8d and 2.4d, with the medium vortex generators at 2.4d and 3.6d, and with the maximum vortex generators at 3.6d and 4.8d. Fluctuating drag was still significant when the minimum vortex generators were spaced at 3.6d and 4.8d, and when the medium vortex generators were spaced at 4.8d. This is potentially because the strength of the control required to stabilize the flow field was not reached by means of either enlarging the size or condensing the spacing. A more detailed investigation of the effect of the forcing on the flow field is performed with the assistance of POD analysis by comparing the modal energy content for different cases. Figure 4.5 and 4.6 plot the percentage of energy and eigenvalues of the first five POD modes for vortex generators with different sizes and spacings. In Figure 4.5, comparing to the reference case, the percent energy in the first two modes are lowered when control is implemented for most cases, while the contrary trend holds for the the remaining modes, such as mode 3-5 shown in the figure. This is an indication of energy in the primary vortex being brought to higher order vortex structures. With minimum and medium vortex generators spaced at 4.8d, little change in energy distribution is observed. However, in terms of absolute energy as represented by the eigenvalues in Figure 4.6, the energy is significantly lowered when control is applied for all cases. This means that the total energy of fluctuating velocity is lowered by applying any of the studied vortex generator configurations. A greater 5

67 35 3 % Energy of POD Mode min VG med VG max VG no VG 3 % Energy d: Bi layer 2.4d: Hollow 3.6d: Part filled 4.8d: Filled Mode Index Figure 4.5: Percent energy of the first 5 POD modes in vertical Plane with different vortex generators. Figure 4.6: Eigenvalues of the first 5 POD modes in vertical Plane with different vortex generators. percentage of the energy is transferred from the first two modes to higher order modes when the vortex generators are placed closer. This differs slightly from the order of control effectiveness evaluated based on the change in formation length and base pressure. One possible interpretation is that the change in the vortex shedding dynamics is not the only factor 52

68 that governs the change in formation length and pressure recovery. It is also noted that the mode shapes of the first two modes are slightly different from the reference case when there is interference between vortex generators (to be discussed in Section 4.4), which could result in a smaller eigenvalue corresponding to the mode. The difference in the mode shapes is most significant when the maximum vortex generators are spaced at 2.4d: Figures 4.7 and 4.8 plot the streamwise components of the first two POD modes in Plane with no mode of u Base Case mode 2 of u y/d y/d x/d x/d 25 Figure 4.7: Streamwise components of the fisrt two POD modes in Plane for the reference case with no vortex generator at Re = 35,. mode of u 2.4d max mode 2 of u y/d y/d x/d x/d 25 Figure 4.8: Streamwise components of the fisrt two POD modes in Plane when maximum vortex generators of 2.4d spacing are applied at Re = 35,. 53

69 vortex generator and with the maximum vortex generators spaced at 2.4d at Re d = 35,..8d min: 3.6% 2.4d min: 43.% 3.6d min: 73.3% 4.8d min: 79.9% Base Case: 72.% 2.4d med: 34.5% 3.6d med: 7.8% 4.8d med: 8.3% 2.4d max P3: 67.7% 2.4d max: 3.5% 3.6d max: 57.% 4.8d max: 74.5% Figure 4.9: Phase plots based on POD modes and 2 in Plane (if not otherwise indicated) with different vortex generators. x-axis label a / 2λ and y-axis label a 2 / 2λ 2 omitted for visual clarity. Figure 4.9 plots the normalized time coefficients of the first two POD modes in phase space for all cases at a Reynolds number of 35,. Comparing Figures 4.9 and 4.5, in cases where the first two POD modes contain a larger percentage of energy, it is more probable for a snapshot to fall within a circle of ±.2 radius. No significant change, or slight increase in the percentage of points between the concentric circles is observed for cases of minimum and medium vortex generators at 3.6d, and of all vortex generators at 4.8d, which implies that the structure of the primary vortices is not disrupted at these conditions. However, for these cases, the absolute energy in the first two modes has been decreased and both formation 54

70 length and base pressure coefficient have increase due to the control. A similar phenomenon was observed in earlier studies such as Naghib-Lahouti et al. (25) and Clark (25), and they suggested that the formation length is elongated by reinforcing the oscillatory mode and reducing the irregularities in the vortex shedding. The phase plots are noticeably more scattered with vortex generators spaced at.8d and 2.4d. However, even with the case of maximum vortex generators spaced at 2.4d, where the smallest percentage of data points fall within the range of radius, the structure of the primary vortex shedding persists in other vertical planes, such as Plane 3, where the points describing the shedding mode are not more spread out than those for the reference case. This means that the primary vortex shedding is partially suppressed while it is not completed eliminated by the streamwise vortices generated by the vortex generators. This can be confirmed by considering the phase-averaged velocity fields in Figures 4.2 and 4.2. For each case, the phase-averaged velocity field was computed.5.8d min.5 2.4d min.5.5 y / d y / d y / d x / d d min x / d y / d x / d d min x / d Figure 4.2: Phase-averaged velocity field in Plane with minimum vortex generators of different spacing at Re d = 35,. Background color indicates one-thousandth of vorticity. 55

71 .5 reference Figure 4.2: Phase-averaged velocity field in Plane when no control is applied at Re d = 35,. Background color indicates one-thousandth of vorticity. by taking an average over the snapshots with the same phase angle of π ± π. The phase 4 32 angle of the primary vortex shedding is determined by the normalized time coefficients of the first two POD modes: ( α(t) = tan a2 (t)/ ) 2λ 2 a (t)/. λ Observed from Figures 4.2 and 4.2, with the application of the minimum vortex generators, the strength of the primary vortex shedding is reduced, and the vortex formation is delayed. This effect is the most significant when the minimum vortex generators are spaced at 2.4d. However, the primary vortex shedding is still present even with the application of the most effective control configuration. The performance of vortex generators of different sizes is studied by examining their influence on the streamwise velocity in the shear layer (y/d =.5) at different downstream locations. Figure 4.22 compares the spanwise variation of streamwise mean velocity profiles at.4d downstream of the trailing edge induced by minimum and maximum vortex generators at 4.8d. The minimum vortex generators bring momentum from the outer flow to the shear layer in a concentrated manner, which increases the streamwise velocity by a large magnitude of about 25% over a small spanwise distance of around.6d. On the other hand, the maximum vortex generators increase the streamwise velocity by a slightly smaller magnitude ( 8%) but over a wider range of around 2.5d. The maximum ones also transfer momentum from their 56

72 sides to the middle in the case of 4.8d spacing, as shown by the dips (local minimum) along the green curve. Figure 4.23 compares the same velocity profiles but at a different downstream locations of 4.4d. It shows that the spanwise velocity variation dissipates faster with smaller vortex generators, as might be expected since larger vortex generators are expected to be more energetic. As shown earlier in Figure 4., the formation length is increased by 8% in the case of the minimum vortex generators spaced at 2.4d, while with the spacing of 4.8d, the increase in formation length is only 7%, 9%, and 3% for the minimum, medium, and maximum vortex generators, respectively. Even though the size is doubled for the maximum vortex generators in comparison to the minimum ones, the control effectiveness is not improved significantly as compared to changing the spacing of the forcing. Lin et al. (99) also found the effect of vortex generator height insignificant. He showed that vortex generators with h/δ. performed as well as the larger ones with h/δ.4 in the control of turbulent flow separation. Godard and Stanislas (26) evaluated the effect of vortex generator height on wall shear stress. He concluded that as the height of vortex generators increases from.2δ to.46δ, the wall shear stress doubled in the plane between two vortex generators while it barely changes in the plane of the vortex generator center. The effect of sizing being inconsequential to the change in formation length and pressure u (m/s) Spanwise Variation of Streamwise Velocity at x/d = d min 2 4.8d max z/d Figure 4.22: Spanwise variation of streamwise mean velocity induced by minimum and maximum vortex generators spaced at 4.8d, at.4d downstream of trailing edge. u (m/s) Spanwise Variation of Streamwise Velocity at x/d = d min 2 4.8d max z/d Figure 4.23: Spanwise variation of streamwise mean velocity induced by minimum and maximum vortex generators spaced at 4.8d, at 4.4d downstream of trailing edge. 57

73 recovery is partially due to the fact that the peak to peak velocity variation near the trailing edge induced by the maximum vortex generators is not much larger than the one caused by the minimum generators. One may argue that the peak to peak difference with the maximum generators may be larger in a slightly different horizontal plane since the larger generators effectively brings the boundary layer higher, as shown from the difference in displacement thickness (Table 4.). Figure 4.24 plots the velocity profiles in different vertical planes at the streamwise location of.4d, with the maximum vortex generators spaced at 4.8d. The Plane curve, located in the middle of two vortex generators, represents the spanwise mean velocity for area not directly affected by the vortex generators. Shown in Figure 4.22, the velocity at Plane 3, although slightly smaller than the spanwise maximum, is greater than the one at Plane and Plane 2. Figure 4.24 shows that this pattern continues to hold at any given y location above the horizontal measurement plane. The magnitude of streamwise velocity variation across the span can be represented by the difference between the measurement in Plane 3 and the one in Plane at a particular vertical position, and this difference is plotted against y in Figure As shown in the figure, the velocity difference in the boundary layer is maximum at y/d =.57, with a magnitude of 3. m/s. Comparing to Figure 4.22, this difference is smaller than the one caused by the minimum vortex generators in the measured horizontal plane. To further demonstrate that the velocity variation induced by the maximum vortex generators is not greater than the one caused by the minimum generators, the streamwise velocity values at y/d =.52 and.57 (the peak in Figure 4.24) from the vertical plane measurements at different spanwise locations are plotted in comparison with the horizontal plane measurements in Figure The points at y/d =.57 at different spanwise locations are all above the red curve representing the horizontal plane measurements, and the ones at y/d =.52 all lie beneath. Meanwhile, the velocity difference between Plane and Plane 3 (indicated by the dashed lines) has no noticeable change among the three slightly offset horizontal planes. This figure also suggests that the horizontal measurement plane is located between.52d and.57d, which is reasonable since the thickness of the laser sheet is around mm (.4d). When the vortex generators are spaced at 2.4d, the spanwise velocity perturbation demonstrates a different pattern. Figure 4.27 and 4.28 show the spanwise variation of streamwise 58

74 d max VG, x/d =.4 P: Middle of VGs P2: Edge of VG P3: Center of VG y/d u (m/s) Figure 4.24: Profiles of streamwise velocity in different vertical planes at.4d downstream of the trailing edge with maximum vortex generators spaced at 4.8d. Figure 4.25: Velocity difference between vertical Plane 3 and Plane at.4d downstream of the trailing edge with maximum vortex generators spaced at 4.8d d max VG, x/d =.4 y/d =.57 y/d =.52 horizontal plane measurements u (m/s) z/d Figure 4.26: Spanwise variation of streamwise velocity at x =.4d downstream of the trailing edge and near y =.5d, with maximum vortex generators spaced at 4.8d. velocity at.4d and 4.4d, respectively, for minimum and maximum vortex generators spaced at 2.4d. In the near field, the minimum vortex generators create much larger velocity perturbations across the span than the maximum ones, which is very different from the case of 4.8d. This difference is believed to be associated with the interference between neighboring vortex generators. This will be discussed further in section (4.4). The 2.4d case demonstrates a sim- 59

75 u (m/s) Spanwise Variation of Streamwise Velocity at x/d = d min 2.4d max z/d Figure 4.27: Spanwise variation of streamwise mean velocity induced by minimum and maximum vortex generators spaced at 2.4d, at.4d downstream of trailing edge. u (m/s) Spanwise Variation of Streamwise Velocity at x/d = d min 2 2.4d max z/d Figure 4.28: Spanwise variation of streamwise mean velocity induced by minimum and maximum vortex generators spaced at 2.4d, at 4.4d downstream of trailing edge. ilar behavior in the far field as the 4.8d case, where larger velocity variation persists further downstream with the larger vortex generators transferring more energy from the outer flow to the shear layer. 4.4 Interference Between Vortex Generators To study the decrease in spanwise variation of streamwise velocity caused by the maximum generators at 2.4d in more details, the streamwise mean velocity fields in the horizontal plane for the minimum and maximum vortex generators at 2.4d spacing are plotted in Figure 4.29 and 4.3. The drawings below the velocity contours are the scaled top view of the vortex generators. Region and in these two figures correspond to the local minima in Figure 4.27 for the minimum and maximum vortex generator cases, respectively, and region 2 in Figure 4.29 matches the local maximum in Figure 4.27 for the minimum vortex generator case. With the minimum vortex generators, momentum in the shear layer (at the horizontal measurement plane) is transferred from region to region 2 as the flow crosses the vortex generator vane, making region a local minimum. However, the small spacing between neighboring vanes of the large vortex generators prevents the transfer of momentum down from the shear layer due to the increase blockage, and instead, the momentum from the 6

76 outer flow is transferred to the inner side of the vanes, as illustrated by Figure 4.3. At the same time, when placed close to each other, the vanes of two neighbouring vortex generators effectively create a funnel such that the flow between the two neighbours is accelerated. Figure 4.32 and 4.33 compare the local streamwise velocity perturbations.4d downstream of the trailing edge, with the maximum and minimum vortex generators, respectively, at different spanwise spacing. Table 4.2 and 4.3 summarize the differences due to spacing quantitatively, where A p is the peak to peak amplitude and b is the spanwise distance between the two local minima caused by one vortex generator. Figure 4.34 illustrates the effect of interference graphically by comparing the measured and linearly superimposed spanwise variation of streamwise velocity with maximum vortex generators at spacing of 2.4d. The superposition is constructed by shifting and adding the spanwise velocity profiles resulted from the maximum vortex generators at 4.8d, with the local freestream velocity subtracted. It can be observed that the superimposed curve has larger amplitude than the measured one, which means that the interference is non-linear and that the amplitude of velocity variation caused by the maximum vortex generators at 2.4d would be larger if there were no interference. Figure 4.29: Streamwise mean velocity field in the horizontal plane with minimum vortex generators spaced at 2.4d. Figure 4.3: Streamwise mean velocity field in the horizontal plane with maximum vortex generators spaced at 2.4d. 6

77 Figure 4.3: Front view illustration of the vortices generated by minimum and maximum vortex generators spaced at 2.4d. Table 4.2: Comparison of maximum vortex generators at difference spacing λ/d A p (m/s) % b/d % % 2. -4% % % Table 4.3: Comparison of minimum vortex generators at difference spacing λ/d A p (m/s) % b/d % % %.2-4.8% %.2-4.8% With the maximum vortex generators at 3.6d, in comparison with those at 4.8d, the amplitude is increased slightly by 2.5% and the effective width b is decreased by 4%, indicating the onset of interference between neighbouring vortex generators. At the spacing of 2.4d, the interference becomes more severe, with 23% increase in amplitude and 9.5% decrease in effective width. For the cases with minimum vortex generators, no interference is observed at 3.6d spacing, while for the 2.4d case and the.8d case are similar to the case of maximum u (m/s) Spanwise Variation of Streamwise Velocity at x/d = d max 2 3.6d max 4.8d max z/d Figure 4.32: Comparison of spanwise variation of streamwise mean velocity profiles.4d downstream of trailing edge induced by maximum vortex generators at different spacing. u (m/s) Spanwise Variation of Streamwise Velocity at x/d = d min 2.4d min 3.6d min 4.8d min z/d Figure 4.33: Comparison of spanwise variation of streamwise mean velocity profiles.4d downstream of trailing edge induced by minimum vortex generators at different spacing. 62