University of Southern California Dissertations and Theses

Boundary layer and separation control on wings at low Reynolds numbers

(USC Thesis Other)

Boundary layer and separation control on wings at low Reynolds numbers

PDF

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content BOUNDARY LAYER AND SEPARATION CONTROL ON WINGS AT LOW REYNOLDS NUMBERS by Shanling Yang A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements of the Degree DOCTOR OF PHILOSOPHY (AEROSPACE & MECHANICAL ENGINEERING) 18 December 2013 Copyright 2013 Shanling Yang Dedication To my Father ii Acknowledgments I would like to rst and foremost thank my Abba Father for the abundant blessings, favor, provision, and encouragement that He has given me throughout this journey. Though per- haps not obvious or logical to others, it is truly by His hand that I have come to this point. Throughout this journey I have witnessed uncountable instances of His grace and unmerited favor, and He deserves the glory. I would also like to thank my mother, father, and brother, who have been extremely supportive and encouraging. I especially thank my father, who did not get the chance to witness my research failures and successes, but he has always been an inspiration and role model who I constantly look up to. My mother and brother have never ceased to support me during this journey, and I thank them for investing their time to read my papers, inquire about my presentations, and even make sure that I stay on schedule. I thank my family for being such a strong and loving foundation. I also cannot leave out thanking my family of brothers and sisters who have lifted me in constant prayer and encouragement. Many of them patiently stood by my side and bore my burdens with me when I felt completely inadequate to continue my work. I thank them for praying me through weak moments and for rejoicing with me during triumphs, even when they did not fully understand the technical aspects of my work. I know that their prayers have helped me combat the mental and emotional struggles I faced during this process. I would like to thank my advisor, Dr. Spedding, for his support and nurturing over these past few years. I have grown immensely as a researcher under Dr. Spedding's advisory, though I know that I have only uncovered a little portion of the much larger realm of research. It has been a pleasure to work with Dr. Spedding, and I thank him for being patient with me through each technical issue, serendipitous discovery, television taping, manuscript revision, iii and presentation. I also thank Dr. Spedding for the wonderful opportunity in 2012 to present my work at Lund University and witness the Lunds' extraordinary wind tunnel research. On a similar note, I would like to thank Dr. Eliasson for giving me the opportunity to present my work at KTH Royal Institute of Technology and meet exceptional researchers. It has also been a pleasure to work with Dr. Redekopp and Eric Lin for a portion of my research. They both provided an immense amount of help for the non-experimental analysis of my work, and I especially owe many thanks to Eric Lin for the time and eort he put aside in his own schedule to help me with the numerical analysis. I also would like to thank Ewald Schuster and Rodney Yates for all the machining work they have helped me with throughout the last few years. Without either of them, I would not have gotten past numerous mechanical conundrums. I thank both of them for taking time out of their schedules to assist me and for being so patient with me. Many thanks go to the rest of the USC faculty, sta, and fellow classmates who have contributed to my journey. I could not have come this far without the relationships I have developed with these outstanding mentors, colleagues, and friends. iv v Table of Contents Dedication ii Acknowledgments iv Abstract vii Preface x Chapter 1 Introduction 1 1.1 Aim and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Chapter 2 Fundamental of Aerodynamics 4 2.1 Laminar Boundary Layer and Separated Shear Layer Flows . . . . . 4 2.2 Flow Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Transition to Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Stability of Viscous Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Chapter 3 Separation Control by Acoustic Excitation 13 3.1 Acoustic Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Optimum Excitation Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Forcing Tones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.5 Angle of Attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.6 Sound Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.7 Tunnel Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.8 Forcing Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.9 Vortex Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Chapter 4 Mathematical Modeling of Flow and Sound 27 4.1 Sound and Fluid Flow Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Sound and Tollmien-Schlichting Instability Interaction . . . . . . . . 28 Chapter 5 Methods 32 5.1 Wing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 vi 5.2 Wind Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.3 Force Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.4 Particle Imaging Velocimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.5 Acoustic Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.6 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Chapter 6 Summary of Papers 40 6.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.5 Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Chapter 7 Paper I 46 Chapter 8 Paper II 74 Chapter 9 Paper III 106 Chapter 10 Paper IV 144 Chapter 11 Paper V 174 Chapter 12 Concluding Remarks 194 References 195 Appendix A Derivation of Sound and Fluid Flow Equation 200 Abstract In the transitional chord-based Reynolds number regime for aeronautics, 10 4 Re 10 5 , uid ow over a surface is prone to separation followed by possible reattachment and tran- sition to turbulence. The amplication of disturbances in the boundary layer promotes transition to turbulence, so boundary layer and separation control methods are especially favorable in this transitional Re regime. The use of sound to control ow separation at transitional and moderateRe for various smooth airfoils has been experimentally studied in the literature. Optimum excitation frequencies are reported to match the frequency or sub harmonics of the naturally occurring instabilities in the shear layer, and correlations between optimum frequencies for external acoustic forcing and tunnel anti-resonances have been ob- served. However, reported optimum frequency values based on the Strouhal number scaling St=Re 1=2 are not in complete agreement among the dierent reported studies. Little atten- tion has been given to distinguish the eects of standing waves from traveling sound waves. Mathematical and experimental studies of sound and boundary layer instability interactions have also yielded mixed results, suggesting that there still lacks a full understanding about the mechanism by which acoustic waves aect boundary layer ows. Results on boundary layer and separation control through acoustic excitation at low Re numbers are reported. The Eppler 387 prole is specically chosen because of its pre- stall hysteresis and bi-stable state behavior in the transitional Re regime, which is a result of ow separation and reattachment. External acoustic forcing on the wing yields large improvements (more than 70%) in lift-to-drag ratio and ow reattachment at forcing fre- quencies that correlate with the measured anti-resonances in the wind tunnel. The optimum St=Re 1=2 range for Re = 60,000 matches the proposed optimum range in the literature, but there is less agreement forRe = 40,000, which suggests that correct St scaling has not been determined. The correlation of aerodynamic improvements to wind tunnel resonances im- plies that external acoustic forcing is facility-dependent, which inhibits practical application. Therefore, internal acoustic excitation for the same wing prole is also pursued. vii Internal acoustic forcing is designed to be accomplished by embedding small speakers inside a custom-designed wing that contains many internal cavities and small holes in the suction surface. However, initial testing of this semi-porous wing model shows that the presence of the small holes in the suction surface completely transforms the aerodynamic performance by changing the mean chordwise separation location and causing an originally separated, low-lift state ow to reattach into a high-lift state. The aerodynamic improve- ments are not caused by the geometry of the small holes themselves, but rather by Helmholtz resonance that occurs in the cavities, which generate tones that closely match the intrinsic ow instabilities. Essentially, opening and closing holes in the suction surface of a wing, perhaps by digital control, can be used as a means of passive separation control. Given the similarity of wing-embedded pressure tap systems to Helmholtz resonators, particular attention must be given to the setup of pressure taps in wings in order to avoid acoustic resonance eects. Local acoustic forcing is achieved through the activation of internally embedded speak- ers in combination with thin diaphragms placed across the holes in the suction surface to eliminate Helmholtz resonance eects. Activating various speakers in dierent spanwise and chordwise distributions successfully controls local ow separation on the wing atRe = 40,000 and 60,000. The changes in aerodynamic performance dier from those observed through external acoustic forcing, indicating that internal acoustic forcing is facility-independent. Combining the eect of Helmholtz resonance and the eect of pure internal acoustic forcing yields a completely dierent set of performance improvements. Since the internal acoustic forcing studies in the literature did not separate these two eects, there is reason to question the validity of the true nominal performance of the wings in previously reported internal acoustic studies. Stability analysis is performed on experimental velocity proles by means of a numerical Orr-Sommerfeld solver, which extracts the initially least stable frequencies in the boundary layer using parallel and 2-d ow assumptions. Velocity proles of the E387 wing are chosen at a condition where acoustic excitation at various chordwise locations and frequencies promotes viii the originally separated, low-lift state ow into a reattached, high-lift state. Preliminary stability analysis of the ow at dierent chordwise stations for the wing in its nominal state (without acoustic excitation) indicates that the ow is initially stable. The least stable frequencies are found to be equal to, and sub harmonics of, the preferential acoustic forcing frequencies determined in experiments. However, potentially improper and oversimplied ow assumptions are most likely sources of inaccuracy since the Orr-Sommerfeld equation is not generally used for separated ows or for boundary layers that grow signicantly over the chord length. The reported numerical results serve as a basis for further validation. The initial hope to control separation by acoustic excitation was realized. While both external and internal acoustic forcing successfully mitigate ow separation and improve wing performance, internal acoustic forcing is the more practical active control method for low Re ying devices. Further investigations would entail investigating the eect of embedded sound sources on the stability characteristics of wings and small-scale ying devices. ix Preface This thesis considers the boundary layer and separation control methods of nite wings at low Reynolds numbers and is based on the following papers: I Yang, S.L., Spedding, G.R., \Spanwise Variation in Wing Circulation and Drag Mea- surement of Wings at Moderate Reynolds Number," Journal of Aircraft, Vol. 50, No. 3, 2013, pp. 791-797. II Yang, S.L., Spedding, G.R., \Separation Control by External Acoustic Excitation on a Finite Wing at Low Reynolds Numbers," American Institute of Aeronautics and Astro- nautics, Vol. 51, No. 6, 2013, pp. 1506-1515. III Yang, S.L., Spedding, G.R., \Passive Separation Control by Acoustic Resonance," Ex- periments in Fluids, Vol. 10, No. 54, 2013, pp. 1-16. IV Yang, S.L., Spedding, G.R., \Local Acoustic Forcing of a Wing at Low Reynolds Num- bers," American Institute of Aeronautics and Astronautics. 2013. In review. V Yang, S.L., \Stability Analysis of Experimental Velocity Proles Using an Orr-Sommerfeld Solver," 2013. x Chapter 1 Introduction For much time now, small-scale ight dynamics has no longer pertained only to birds, bats, and insects with the steadily increasing number of small-scale aerial vehicles that operate in the same Reynolds number regime. The chord-based Reynolds number, Re =Uc= (where U is the ight speed, c the chord, and is the kinematic viscosity), balances inertial and viscous forces and can be classied into dierent regimes. In aeronautics, the lowRe regime is approximatelyRe 10 5 (Fig. 1.1). The lowRe range can be further sub-divided: Re< 10 4 (ultra-lowRe), which classies most insect ight, and 10 4 Re 10 5 , which classies most bird ight. A growing number of micro aerial vehicles operate in the particular low Re sub-regime between 10 4 and 10 5 . At these Re, uid ow is prone to laminar boundary layer separation with possible transition to turbulent reattachment, thereby either favorably or adversely aecting wing and aircraft performance. This sub-regime can consequently be termed the transitional Re regime, where viscous eects cannot be ignored. In this regime, uid ow is extremely sensitive to small disturbances, such as environmental noise, turbulence levels, vibrations, surface irregularities, among others. Experiments on airfoils and wings at these low Re show discrepancies among dierent facilities, indicating the diculty in accurately characterizing the behavior of airfoils and wings at low Re. The acute sensitivity of air- foils and wings in the transitional Re regime has triggered the investigation of transition mechanisms in the boundary and shear layers and methods for separation control. 1 Figure 1.1: Re regime for ying objects, from [33]. 1.1 Aim and Objectives The tendency of ow separation over airfoils at low Re has prompted the investigation of separation control methods. One particular method that has only been moderately studied is the use of acoustic excitation. The reported investigations on external and internal acoustic excitation found in the literature describe improvements to airfoil performance with the presence of select acoustic tones; however, most of the acoustic forcing studies were done on airfoils and wings at stall or post-stall angles of attack,. Very few studies examined airfoils that experienced ow separation at low, pre-stall . Correlations between the most eective forcing frequencies and the most unstable fre- quencies in the separated shear layer or free wake have been reported. Studies have also shown that the most eective forcing frequencies match the anti-resonances of the tunnel test section. There seem to be several possible sources of mechanisms by which acoustic excitation can change the ow, including a frequency matching of the acoustic wave and a naturally occurring instability, and the presence of maximum velocity uctuations at the wing surface. 2 In general, the reported range of most eective frequencies is much larger (and even an order of magnitude dierent) for internal acoustic forcing than for external acoustic forcing, and the reported ranges dier slightly for various airfoils, Re, and . The discrepancies suggest the possibility of dierent mechanisms by which acoustic excitation changes the ow around a wing and prompts the need to clearly distinguish between standing waves (most likely to occur for external acoustic forcing) from traveling waves (most likely to occur for internal forcing). It is quite possible that completely dierent parameters need to be considered for the two types of acoustic forcing if each involves a dierent type of wave as the ow-changing mechanism. All of the internal acoustic forcing tests found in the literature used wing models that had uncovered spanwise ducts. This design limits the localization of acoustic forcing to spanwise strips at a single, or very few, chordwise locations, and there are no studies on the spatial distribution of various forcing frequencies and sound pressure levels. Furthermore, the sound ducts in these wings present the potential eects of acoustic resonance, which is not mentioned in any of the reported studies. It is possible that the eects of acoustic resonance have been overlooked in the studies on internal acoustic forcing of wings reported in literature. The aim of the current work is to experimentally investigate the laminar separation and reattachment process on a nite wing in the transitional Re regime under the eect of acoustic excitation. In particular, the sensitivity to small disturbances and the practicality of separation control by manipulation of boundary layer instabilities are investigated. The explored methods for separation control are external acoustic excitation, internal acoustic excitation, and acoustic resonance. 3 Chapter 2 Fundamentals of Aerodynamics 2.1 Laminar Boundary Layer and Separated Shear Layer Flows Fluid ow is governed by the in uence of inertia, viscosity, and external forces. The Navier- Stokes equations for incompressible ow balances these elements by mass conservation and momentum equations, given by r u = 0; (2.1) @u @t + (u r)u = 1 rp +r 2 u +f ext (2.2) where the velocity vector u =hu;v;wi, is the uid density, p is the pressure, is the kinematic viscosity of the uid, and f ext is the external force vector. In terms of the x- component of the velocity, and assuming steady, two-dimensional ow and no external forces, Eq. (2.2) can be expressed as u @u @x +v @u @y = 1 @p @x + @ 2 u @x 2 + @ 2 u @y 2 : (2.3) 4 The left-hand side of Eq. (2.3) is the net inertial force, the rst term on the right-hand side is the net pressure force, and the last term on the right-hand side is the net viscous force. At lowRe, viscous forces dominate the inertial forces, especially in the laminar boundary layer where the ow is still attached (Fig. 2.1). Adverse pressure gradients ( @p @x > 0) are most likely to occur when the boundary layer is still laminar, making the ow over a surface susceptible to separation, after which begins the separated shear layer, which consists of a laminar shear layer and turbulent shear layer, as well as a laminar-turbulent transition shear layer. For the ow to remain attached to the surface, the ow must have sucient energy to overcome the adverse pressure gradient, the viscous dissipation along the ow path, and the consequent energy loss due to changes in momentum. When the energy is insucient, the ow separates from the surface (Fig. 2.1a). The ow is also prone to transition from laminar to turbulent and may reattach (Fig. 2.1b) to form a turbulent boundary layer that can result in a laminar separation bubble. 2.2 Flow Instabilities Each stage of a developing ow contains dominating instabilities. Flow instabilities can be classied as local and global, and further into convective and absolute [24, 5]. If localized disturbances spread upstream and downstream and contaminate the entire ow, the velocity prole is considered locally absolutely unstable, whereas if disturbances are swept away from the source, the velocity prole is considered locally convectively unstable. Global instabilities can arise from feedback from either upstream-propagating vorticity (instability) waves or irrotational global pressure feedback. While the basic ow may be convectively unstable, instantaneous velocity proles of the disturbed basic ow can be absolutely unstable with respect to secondary disturbances, which can occur if the scale of the secondary instability is much smaller than that of that of the primary disturbance [30]. 5 Figure 2.1: Boundary layer characteristics for separated ow (a) and reattached ow (b). Dierent dominating instabilities are found in corresponding parts of a developing bound- ary layer. Figure 2.2 shows the development of the boundary layer over a at plate. Im- mediately aft of the leading edge, the boundary layer is still attached and laminar, as indi- cated by Stage 1. Slightly farther downstream, the laminar boundary layer is dominated by small-amplitude, viscous instabilities, commonly referred to as Tollmien-Schlichting (T-S) instabilities (Stage 2). When amplied, T-S waves can grow into larger, three-dimensional instabilities. Experimental results on boundary layer instability and the onset of turbulence on at plates [29] showed that initially small-amplitude and nominally two-dimensional wave 6 Figure 2.2: Laminar-turbulent transition in the boundary layer on a at plate, from [55]. disturbances became three-dimensional when the amplitude was increased with the formation of regions of peaks and valleys with short duration \spikes." At large enough amplitude of the T-S waves, the ow no longer remains attached to the surface, whereby the boundary layer becomes a separated shear layer, dominated by secondary instabilities, referred to as separated shear layer, or sometimes Kelvin-Helmholtz (K-H) instabilities. K-H waves are inviscid and cause the shear layer to roll up, and they have been shown to be responsible for shear layer and separation bubble unsteadiness [32]. Studies also identied large-scale vortex shedding as the primary cause of low Re number separation bubble reattachment and unsteadiness, whereas the role of small-scale turbulence was only secondary. In the separated shear layer, vortex merging can occur and contribute to the sub harmonic growth of periodic disturbances, although the growth is less obvious and pronounced as Re increases or angle of attack, , decreases (and as the height of the separated region decreases). Merged roll-up vortices have been observed [59] to shed at the rst sub harmonic of the natural frequency in the separated shear layer. Separated shear layer instabilities would begin at Stage 3 in Fig. 2.2. Experimental results on a at plate from [29] showed development of these secondary waves in the form of 7 -shaped instabilities upon large enough amplication of T-S waves. Secondary instabilities can continue to grow (Stage 4) into turbulent spots (Stage 5), which can then initiate the transition to fully turbulent boundary layer ow [46]. Transition has been shown to occur quickly after the roll-up and agglomeration of vorticity in both experiments [54, 35] and sim- ulations [56]. While T-S instabilities have been observed to initiate transition in separation bubbles, the rate of transition caused by T-S instabilities is lower than that caused by K-H instabilities, which results in a longer transition length. When the ow extends beyond the end of the surface (for nite bodies), dominating instabilities are wake prole, or free wake, instabilities. 2.3 Transition to Turbulence Transition to turbulence occurs when perturbations enter into and in uence the ow, and upon large enough amplication, break down nonlinearly. The physical mechanisms that prompt transition depend on the specic type of ow and the type of environmental distur- bances. Transition in boundary layer ows can be classied into two main types [25]. The rst type involves boundary layer instabilities (initially described by linear stability theo- ries), amplication, and interaction of various instability modes, which ultimately lead to the breakdown of laminar ow. In this rst type of transition, the environmental disturbances are small. The second type, often referred to as bypass transition, involves the direct nonlin- ear laminar ow breakdown under the in uence of external disturbances. Bypass transition occurs when high levels of environmental perturbations, such as free stream disturbances and surface roughness, are present. The rst type of transition to turbulence can be separated into three main aspects: receptivity, linear stability, and nonlinear breakdown (Fig. 2.3) [25]. In the rst stage of transition where the localRe is low, T-S waves are generated through receptivity. Receptivity is a process by which a disturbance, such as sound or vorticity, enters the boundary layer and establishes its signature in the resulting ow [43]. Receptivity involves the generation, 8 rather than the evolution, of instability waves in the boundary layer. Instability waves are generated when energy from long wavelength external disturbances are transferred to shorter wavelength T-S waves through local or global ow changes [28, 13, 12, 6]. The wavelengths of the naturally occurring disturbances are usually longer than those of the instability waves, so a wavelength conversion mechanism is required to transfer energy from the longer free stream disturbance to the shorter instability wave [27]. The slow (viscous), linear growth of these disturbances occurs in the second stage of transition: the linear stability region. In this stage, T-S instability waves propagate down the boundary layer and are either amplied if the ow is unstable, or attenuated. Sometimes the disturbance might even decay for a considerable distance before being amplied [13]. This stage is described by linear hydrodynamic stability theory, which can be used to describe two- and three-dimensional ows and disturbances. After the linear stability region, the disturbances continue to grow and become nonlinear, entering the third stage of transition: nonlinear breakdown. This stage occurs when the uid ow enters a phase of nonlinear breakdown, randomization, and a nal transition into a turbulent state. In the breakdown phase, the ow is transformed from a deterministic, regular, and generally two-dimensional laminar ow into a stochastic yet ordered, three-dimensional ow [25]. Studies on nonlinear breakdown include resonant phenomena that occur in the transition process as well as the detection and description of coherent structures in the transitional boundary layer [25]. Figure 2.3: Characteristics of a boundary layer transition to turbulence, from [25]. 9 Transition to turbulence is also driven by the formation and the resulting development of vortical structures in the separated shear layer. During the initial stage of transition, small-amplitude disturbances centered at a fundamental frequency have been observed to experience exponential growth in the separated shear layer [58, 54, 8]. The nal stage of transition, which results in turbulence, is associated with nonlinear interactions between these disturbances, and coherent structures have been shown to form during this stage of transition [59, 38, 54, 31]. This sub harmonic merging of roll-up vortices in the separated shear layer is followed by a rapid breakdown of the vortices [59]. Even though shear layer roll-up vortices break down during transition, they may interact with wake vortex shedding if the transition region extends into the near wake [59]. 2.4 Stability of Viscous Flows The viscous growth of disturbances in a oweld (the linear region in Fig. 2.3) is governed by linear stability theory. Stability analysis does not predict turbulence, which is experimentally observed, and there is no theory of transition; however, empirical predictions of transition can be made based on the spatial amplication rates of linearized stability theory [55]. Fluid motion with components (u;v;w;p) can be decomposed into a basic ow consisting of mean components (U;V;W;P ) and a superimposed perturbation motion consisting of uctuating components (u 0 ;v 0 ;w 0 ;p 0 ). For a two-dimensional incompressible ow with a two-dimensional disturbance assumption, the mean velocity components can be assumed as follows: U = U(y), V = W = 0. Boundary layer ow can be regarded approximately as a parallel ow since the dependence ofU ony is much greater than the dependence ofU onx. The pressure P (x;y) should be assumed to be dependent on x, as the pressure gradient @P @x can strongly aect the ow. For parallel ow at a surface, appropriate boundary conditions require that u 0 and v 0 vanish at the walls to satisfy the no-slip condition. 10 The equations for continuity and momentum for an incompressible, two-dimensional ow can be written in terms of the vorticity equation , given by @ @t + (ur)r 2 = 0; (2.4) where u = (u;v), is the kinematic viscosity, and is the vorticity. Introducing a stream- function, , and using the relations u = @ @y ; v = @ @x (2.5) and =r 2 ; (2.6) the vorticity equation in Eq. (2.4) can be written in terms of the streamfunction: @ @t + y @ @x x @ @y r 2 r 2 = 0: (2.7) To analyze the initial development of small perturbations of a uid ow, it is assumed that the mean ow is parallel and represented byU(y), nonlinear eects can be ignored, the disturbances can be represented by traveling waves, and the maximum amplied disturbances are two-dimensional. Using parallel ow assumption, the streamfunction can be written as (x;y;t) = ~ (x;y;t) + Z 1 1 U(y)dy; (2.8) where ~ (x;y;t) is the perturbation streamfunction, and R 1 1 U(y)dy is the mean ow con- tribution. ~ (x;y;t) can be represented as a traveling wave of the form ~ (x;y;t) =(y)e i(kx!t) ; (2.9) The two-dimensional ow assumption omits the vortex stretching term. 11 where (y) is the amplitude function, k is the wave number and ! is the frequency. The phase speed, c, is related to k and ! by the relation c = ! k : (2.10) Linearizing the vorticity equation, Eq. (2.7), in terms of ~ and substituting in Eq. (2.9) yields the Orr-Sommerfeld equation: (U(y)c) 00 (y)k 2 (y) U 00 (y)(y) = 1 ikRe 0000 (y) 2k 2 00 (y) +k 4 (y) : (2.11) The left-hand side of Eq. (2.11) comes from the inertial terms while the right-hand side comes from the friction terms of the equations of motion. The Orr-Sommerfeld equation poses an eigenvalue problem for a given Re and mean ow, U(y). For boundary layers, the boundary condition at the surface boundary (or \wall") there is a no-slip condition, and perturbation velocities vanish at the boundary. Hence, y =y wall = 0; (y wall ) = 0; (y wall ) 0 = 0: (2.12) Stability analysis can be achieved using the Orr-Sommerfeld equation. The common types of stability analysis are temporal, spatial, and combined stability, where temporal stability corresponds to absolutely unstable ows, and spatial stability corresponds to convectively unstable ows. For boundary layer ows, it is more common that disturbances develop in space rather than in time. Therefore, in the linear region of transition (described in Section 2.3) where T-S instabilities are either amplied or attenuated, spatial stability can be assumed, whereby! i = 0 andk =k r +ik i . The termk i is the spatial growth rate, and the minimum value of k i (or maximumk i ) at a given Re represents the most unstable initial growth of the ow instability. 12 Chapter 3 Separation Control By Acoustic Excitation Manipulation of the instability waves in the boundary layer can be advantageous. Decreasing the instability wave amplitudes may delay or avoid transition from laminar to turbulent boundary layer ow, while increasing the amplitudes may trigger earlier separation, leaving sucient time and downstream surface for the ow to reattach. The amplitudes of boundary layer instability waves can be aected by changing either the generation or propagation of the waves [26]. In the traditional view of boundary layer ow control, the origin of the inviscid in ectional instability is associated with the K-H instability in the separated shear layer. However, in a modied view [7], the origin of the inviscid in ectional instability is an extension of the instability caused by the adverse pressure gradient in the region upstream of separation. In this modied view, there is seemingly no direct connection between T-S and K-H instabilities, although each is independently connected to the upstream convectively unstable in ectional instability [18]. Since ows in the transitional and lowRe regimes are prone to separation, it is benecial to nd ecient methods of boundary layer and separation control. In general, separation control falls under three categories: body shape design, passive control, and active control. Body shape design involves fabricating the body surface shape to be well streamlined in order to maintain a higher energy level along the ow path so that ow separation does not occur. This method can delay, but will not always prevent, separation, and it is suitable for limited ow conditions. Passive control involves using passive devices such as vortex generators 13 or fences that are mounted to the body; these devices are not controlled by an external energy source but energize the ow by enhancing or accelerating ow transition. Active control involves introducing an external energy source to supplement the boundary layer energy. Active control methods include acoustic excitation, vibrating wires and aps, and steady and unsteady blowing/bleeding techniques (injection and removal of mass into and out of the boundary layer) by various synthetic jets devices. Active control devices require additional power sources but can be implemented without changing the original body shape. 3.1 Acoustic Excitation Traditionally, boundary layer instabilities have not been problematic in aeronautical appli- cations at higher Re. However, an increasing number of engineering interests, including un- manned aerial vehicles, micro air vehicles, wind turbines, sailplanes, rotor blades, hydrofoils, and high-altitude jet fans, operate in the low and transitional Re regimes. Consequently, boundary layer instabilities become a concern, and separation control becomes an impor- tant design parameter. Of the various methods to control separation, the use of acoustic excitation has been studied throughout the literature due to experimental and theoretical observations of ow receptivity to acoustic disturbances. In general, acoustic forcing can be classied as external or internal, where external forcing is achieved by an outside sound source emitting tones in the vicinity of a wing, and internal forcing is achieved by emitting sound from within the wing. Possibly the earliest investi- gation of the in uence of external sound on aerodynamic performance was experiments by Schubauer and Skramstad on boundary layer transition of a at plate [47]. In their exper- iments, regularly oscillating velocity uctuations corresponded to sound production in the wind tunnel, and sound at particular frequencies and amplitudes changed the boundary layer transition process. It was concluded that small disturbances by themselves do not produce transition, but small disturbances may grow according to stability theory and when large enough can cause turbulent ow. 14 Acoustic forcing of symmetric and mildly cambered airfoils at low and moderateRe using single frequency tones has been shown to eectively change wing performance. Improvements include increasing lift at particular angles of attack [14, 62, 2, 1], tripping the ow from low- to high-lift states [14], diminishing the size of the hysteresis loop in lift-drag curves [14], reducing the tendency toward ow separation over an airfoil [62, 2], as well as changing the behavior of the laminar separation bubble and turbulent boundary layer [63]. However, the eects of external acoustic excitation have been shown [62, 14] to be strongly correlated with wind tunnel test section resonances. The results from [14] are peculiar, since the tested airfoil (Eppler 61) is one of the many airfoils reported in literature [48, 49, 34, 51] that experiences bi-stable separation behavior at low and transitional Re in its nominal, unexcited state. 3.2 Optimum Excitation Frequency Literature results on acoustic excitation of airfoils and wings at low and moderate Re and pre- and post-stall show dependence of optimum (or preferential) excitation frequencies on Re, , and the dominating instabilities in the ow. The optimum excitation frequency, denoted as f e *, has been found to increase with increasing Re or increasing [63], while the range of eective excitation frequencies has been found to increase with increasing Re or decreasing [1, 57]. In some cases, the range of f e * increased when the forcing occurred at higher amplitudes (sound pressure levels) [4]. Tests from both external and internal acoustic forcing show that the values of f e * cor- respond to the most amplied instabilities that dominate the separated region. For pre- and immediately post-stall , K-H instabilities dominate the separated shear region, so f e * correspond to these separated shear layer instability frequencies,f s [63, 57, 21, 61]. For large post-stall , the dominating instabilities are due to free wake vortices, so f e * correspond to the free wake vortex shedding frequencies, f w [23, 41, 19, 4, 20]. The free wake frequencies are reported to be an order of magnitude lower than the separated shear layer frequencies, f w 0.1f s [19]. 15 Increased pressure uctuations on the surface of a NACA 65(l)-213 airfoil atRe 350,000 [1] were observed at optimum forcing frequencies. The increases in uctuating pressure yield increases in uctuating velocity on the wing surface, which has been described in [1] to add vortical activity near the surface and excite the instability waves in the separated shear layer. The shedding and excitation frequencies relate to Strouhal number, St, given by St = fc U (3.1) where f is the shedding or excitation frequency, c is the chord length, and U is the free stream velocity. In external acoustic forcing tests [63, 57], laminar separation was observed to be most eectively reduced when the parameterSt=Re 1=2 was between 0.02 and 0.03, and correspondingly, whenf e * =f s . However, some external acoustic forcing results at moderate Re [1] showed similar improvements in reducing separation at St=Re 1=2 that was an order of magnitude lower than the 0.020.03 range. Results from internal acoustic forcing tests [19, 4, 21] report a larger optimum St=Re 1=2 range of 0.001 0.02. Since the laminar boundary layer thickness over a at plate is = 5:2x p Re x (3.2) where x is the streamwise distance from the leading edge and Re x the Reynolds number at the location x, the St=Re 1=2 scaling is directly related to the growth of the boundary layer thickness,. St from Eq. (3.1) uses the chord as the length scale and is associated with the shedding instabilities. If the scaling is instead the size of the wake behind the body (more suitable for airfoils at high , which act like blu bodies [44]), then the projected height of the airfoil,c sin, is used as the length scale, as shown in Fig. 3.1. A Strouhal number that takes into account can be given by St = fc sin U : (3.3) 16 If the scaling is associated with the size of the separated region, then the height of the separated region (or the laminar separation bubble) is used as the length scale, as shown by s in Fig. 3.1. A corresponding separation Strouhal number is then St s = f s U e (3.4) where s is the momentum thickness of the separated region, and U e is the edge velocity of the boundary layer [39]. An optimum range of St s is reported to be between 0.008 and 0.016 [39]. Figure 3.1: Dierent length scales for dierent Strouhal number scalings. 3.3 Forcing Tones Most of the previous studies on external acoustic excitation used continuous monotones for the source of acoustic forcing. There are some claims [45] that continuous acoustic forcing is problematic if one wishes to measure T-S waves, because Stokes waves generated by the continuous free stream acoustic uctuations are superimposed on the T-S waves. However, the dominant transition process aected by acoustic excitation has been claimed [14] to be in the separated shear layer rather than in the attached boundary layer where the T-S waves exist. If sound waves and T-S do not interact eciently, then the K-H instabilities, rather than the T-S instabilities, should be given more attention in regards to exciting boundary layer ow. Earlier tests [47, 3] showed that single frequency sound is much more likely to cause boundary layer transition than a continuous sound spectrum, which implies that 17 in uencing the transition process requires the particular matching of the acoustic forcing frequency to that of the naturally occurring instability. 3.4 Hysteresis At transitionalRe, many smooth airfoils [48, 49, 34, 51] exhibit unconventionally shaped lift- drag curves, where there appear to be two performance envelopes: a low- and high-lift state. Many of these \bi-stable state" airfoils also experience hysteresis between the two states, so that a given state depends on the time history or previous state. External acoustic excitation tests on an Eppler 61 airfoil at Re = 25,000 and 60,000, which experienced bi-stable states and pre-stall hysteresis [14], showed changes in lift and drag hysteresis with as well as with f e . Excitation promoted transition from low- to high-lift states at lower wing incidence, , and the high-lift state was maintained longer when was increased. There was also a reduction in the size of the hysteresis loop. Hysteresis was also noticed with the addition and removal of, or change in, acoustic forcing. At certain , the removal of the acoustic forcing caused the ow to revert back to its original state, while at other , even with the removal of the forcing, the ow remained in the high-lift state, indicating that there exists a certain region in the hysteresis loop where the high-lift state is unstable and will tend to revert back to the original low-lift state under insucient excitation [14]. Similar behavior occurred for a symmetric NACA 0025 airfoil at Re = 57k, 100k, and 150k, whereby once ow reattachment occurred, either f e could be altered slightly from the optimum value f e *, or the SPL could be lowered, and the boundary layer would remain attached and the ow would remain in the high-lift state [58]. Acoustic forcing is therefore capable of promoting hysteresis that depends on various factors, including , f e , and SPL. 18 3.5 Angle of Attack There are various reported results regarding the range of for which acoustic forcing is benecial, which is seemingly determined by the unexcited performance of a given airfoil. External acoustic excitation on a E61 airfoil atRe = 25,000 had little eect at high and low but the largest eect at inside the hysteresis loop ( 6 11 ) [14], as shown in Fig. 3.2. Figure 3.2: Lift coecient for the E61 airfoil at Re = 25,000 without excitation and excited at f e = 140Hz (St=Re 1=2 = 0.025), replotted from [14]. For a NACA 65(l)0-213 atRe = 350,000 and the GA(W)-1 atRe = 600,000 and 800,000, 16 appeared to be aected the most by external acoustic forcing [2, 1]. For a NACA 63 3 -018 airfoil at Re = 300,000 with internal acoustic excitation located at x=c = 0.0125 (near the leading edge), immediately post-stall values experienced the most benet. Just prior to stall, lift could either be improved or diminished depending on the selected forcing frequency [19]. On the other hand, tests on a LRN(1)-1007 airfoil at Re = 75,000 showed 19 that lift increased for all up to 38 with external acoustic excitation [61], whose results are replotted in Fig. 3.3. The E61 airfoil was tested at a lower Re and demonstrated pre- stall hysteresis and bi-states while the NACA 65(l)0-213, NACA 63 3 -018, GA(W)-1, and LRN(1)-1007 airfoils were tested at higher Re and did not experience pre-stall hysteresis or bi-stable states in their nominal, unexcited states, further indicating the Re dependence of bi-stable states. It is likely that the range of in which airfoils benet from acoustic excitation depends on the unexcited performance of the given airfoil and Re. Figure 3.3: Lift coecient for the LRN(1)-1007 airfoil at Re = 75,000 without excitation and excited at f e = 342Hz (St=Re 1=2 = 0.017), replotted from [61]. 20 3.6 Sound Amplitude The benecial eects of external acoustic forcing have been shown to increase with increasing SPL. In the external acoustic forcing tests of a NACA 65(l)-213 airfoil at Re 350,000 [1], not only didC Lmax increase with increasing SPL, but the value of at C L;max also increased, as seen in Fig. 3.4. The magnitude of uctuating pressures measured on the surface of the wing increased with increasing SPL [1], which was also observed for excitation at optimum forcing frequencies. In the same experiments, increased freestream velocities as well as larger chords required higher SPL values in order to obtain the same increase inC L , implying that eective SPL depends on Re. While results from internal acoustic testing also agreed that higher SPL values generate higher C L , some studies showed that lift improvement actually diminished after a certain amplitude level [4], suggesting that there are thresholds for eective SPL values. Figure 3.4: Lift coecient for the NACA 65(l)-213 airfoil atRe = 350,000 for unexcited and excited ow at f e = 665Hz (St=Re 1=2 = 0:007) with SPL = 139dB and 147dB, replotted from [1]. 21 3.7 Tunnel Resonance The eects of external acoustic excitation have been found to couple with and depend on tunnel resonances. The resonances in wind tunnel test sections dier among facilities de- pending on tunnel geometry. The fundamental cross resonances in a rectangular cross section can be calculated by f mn = a 0 2 " 1 U 0 a 0 2 #1 2 m H 2 + n W 2 1 2 ; (3.5) where m and n are normal and spanwise modes, respectively, H and W are the height and width of the test section, respectively, a 0 is the speed of sound, and U 0 is the freestream velocity [11]. For non-rectangular and more complicated tunnel cross sections, the funda- mental cross-resonances are less easy to calculate and most likely need to be obtained by empirical means. In external acoustic forcing studies on a LRN(1)-1007 airfoil [62], the cross-resonances in the wind tunnel test section induced large transverse velocity uctuations near the air- foil. These cross-resonances had the greatest eect in changing the airfoil performance and determined the optimum frequencies at which airfoil performance was improved. The eect of individual frequencies on the ow around an E61 airfoil was shown to scale with the acoustic response of the wind tunnel [14]. At frequencies corresponding to tunnel resonances (maxima in sound level), the ow required a higher gain to trip, whereas at the frequencies corresponding to tunnel anti-resonance (minima in sound level), the ow only required a small gain to trip. In a resonant duct, maxima in sound pressure level correspond to minima in induced uctuating velocity levels and vice versa. In acoustic wave elds, the gradient of pressure is maximum where the pressure uctuations cross zero and is consequently where the max- imum uctuations in velocity (90 out of phase with the pressure) are found. In a closed structure, the standing wave nodes that correspond to pressure uctuations equal to zero 22 (anti-resonances) will have minimum root-mean-square values. The nodes where the pres- sure uctuates between maximum and minimum but where the pressure gradient is close to zero will have maximum r.m.s. values. The results from [14] and [62] suggest that maximum improvement in wing performance occurs at maximum levels of uctuating velocity, which occur at minimum SPL (tunnel anti-resonances). There are further claims [14] that the mechanism by which acoustic disturbances aect the ow is through velocity uctuations rather than acoustic pressure uctuations. 3.8 Forcing Location A parameter specic to internal acoustic excitation is the location of the acoustic forcing. The common internal acoustic forcing setup used in the studies found in literature entails placing a speaker inside the wing with spanwise ducts and slots so that the sound travels through the entire span of the wing. Tests on a symmetric airfoil at = 0 and Re = 35,000 [22] showed that acoustic forcing applied ahead of the chordwise separation point, denoted asc s , was found to be more eective in increasing lift than forcing applied aft of c s , where higher SPL was required for the same eectiveness. Tests on a NACA 63 3 -018 airfoil at Re = 300,000 [19] showed that when the forcing location, which can be denoted as c e , was approximately equal to c s , the performance of the wing was most aected by acoustic forcing, especially in the post-stalled region. Separation occurred very near the leading edge of the airfoil (c s =c 0.01), and so forcing nearest to c s yielded the most benet, as seen in Fig. 3.5. While the eect of forcing was found to deteriorate as c e moved farther aft from c s , the values of f e * were reported not to be a function of c e [19]. These studies concluded that the nature of the local excitation control is due to hydro-dynamical disturbances rather than acoustics: if the nature of the control were due to acoustics, the forcing should not be sensitive to slot location because the acoustic wave length is much longer than the length scale of the wing model. 23 Figure 3.5: Lift coecient for the NACA 63 3 -018 airfoil at Re = 300,000 [19] for acoustic forcing at c e =c = 0.0125, 0.0625 and 0.1375. f e = 110Hz (St=Re 1=2 = 0.004). Internal acoustic forcing emitted near c s over the top of a symmetric airfoil (c e =c 0.15) at Re = 35,000 and at stalled angles of attack ( = 15 , 20 ) was found to eectively alter the separated region so that stall was delayed [23, 22]. When excitation was forced near c s , f e * was found to be equal to the separated shear instability frequency, f s , or a sub harmonic off s . As described in [23], the coupling (frequency matching) between the injected sound and separated shear layer instabilities causes the shear layer to resonate through its harmonics and induce additional vortical motions, and the presence of a pressure gradient further enhances receptivity of the ow to sound. Separation that otherwise occurs at the leading edge in the immediate post-stalled region can be reattached when f e =f s and when c e = c s [19]. Trailing edge acoustic forcing also alters the ow over a wing. Studies on internal acoustic forcing emitted near the trailing edge of an airfoil at = 0 and Re = 35,000 [22] showed that acoustic forcing near the trailing edge successfully controlled trailing edge separation and substantially altered the near wake development. While eective forcing frequencies were in the range 1/4f w to 2f w , the optimum forcing frequency was equal to the vortex 24 shedding frequency: f e * = f w . Acoustic disturbances generated pressure oscillations in the separation region, which modied the aft pressure recovery region and therefore reduced trailing edge separation [22]. Flow visualization showed that trailing edge acoustic forcing at certain forcing frequencies made the wake develop earlier and spread out farther. However, far downstream of the airfoil, the vortex shedding frequency remained constant and was unaected by acoustic forcing, which suggests that the trailing edge acoustic forcing only aects near wake ow. The high amplitude, low frequency acoustic forcing through a slot on the airfoil surface can be compared to the control of boundary layer separation by periodic blowing and suction [22]. At low frequency and high amplitude forcing, the acoustic forcing is composed of two halves of a cycle: one half is analogous to the suction of low momentum ow out of the boundary layer, and the other half of the cycle is analogous to the injection of high momentum ow into the boundary layer. 3.9 Vortex Dynamics Experimental studies on acoustic excitation of airfoils and wings include observations and measurements of vortical activity in the separated shear layer. Tests on a NACA 0025 air- foil at Re = 55k, 100k, and 150k [59] showed that disturbances centered at a fundamental frequency were amplied in the separated shear layer. The initial growth of these distur- bances was followed by the generation and growth of harmonics and a sub harmonic of the fundamental frequency, resulting from the nonlinear interactions between the disturbances [8]. The amplication of ow disturbances in the separated shear layer was shown [59] to cause the shear layer roll-up at a fundamental frequency (the frequency of the most ampli- ed disturbances). The shear layer transition process is associated with the decay of roll-up vortices [59, 35], and the sub harmonic growth of the disturbances in the shear layer is likely due to vortex merging [59, 37, 35]. Breakdown to turbulence has been observed to take place at the most upstream location where the separated shear layer reattaches [37]. Secondary and tertiary reverse- ow zones 25 occurred in the reattachment region, which were induced by vortices that were shed from the transition region and generated by the roll-up of two-dimensional vorticity within the shear layer [59]. In some instances, these vortices were shed at a sub harmonic of the fundamental separated shear layer frequency, indicating the presence of a vortex-pairing process. In the wake of the NACA 0025 airfoil [59], alternating vortices were shed, resulting in a similar pattern to a Karman vortex street. The wake vortices were less coherent and the vortex pattern less organized when a laminar separation bubble formed (when the ow reattached). 26 Chapter 4 Mathematical Modeling of Flow and Sound 4.1 Sound and Fluid Flow Interaction The interaction of sound waves and uid ow can be mathematically modeled by using the fact that sound perturbs the surrounding uid. For simplicity, assuming ow in the x-direction only, the continuity equation is @ @t + @ @x (u) = 0 (4.1) and the Navier-Stokes equation (Eq. (2.3)) without external forces is @u @t +u @u @x = 1 @p @x + @ 2 u @x 2 : (4.2) An initially unperturbed state at rest is assumed, as well as small perturbations in density, pressure, and velocity from the acoustic wave: = 0 + 0 , p =p 0 +p 0 , and u =u 0 . Taking the divergence of Eq. (4.1), and taking the time derivative of Eq. (4.2) and combining it with Eq. (4.1) (derivation in Appendix A) yields @ 2 u 0 @t 2 c 2 @ 2 u 0 @x 2 = @ @t @ 2 u 0 @x 2 : (4.3) Eq. (4.3) describes a modied wave equation where the temporal and spatial changes in velocity perturbations, u 0 , are related to both speed of sound, c, and viscosity, . The 27 viscous term involving the third partial of u 0 indicates that temporal changes in particle perturbation convection must also be considered in addition to the standard terms in a pure wave equation (when the right-hand-side of Eq. (4.3) equals zero). Pressure and velocity perturbations can also be related by the relation (derivation in Appendix A) 1 c 2 @ 2 p 0 @t 2 @ 2 p 0 @x 2 = 0 @ 3 u 0 @x 3 : (4.4) The real part of the particle frequency, !, is related to viscosity (derivation in Appendix A) by the relation ! 2 r k 2 =c 2 1 2 2 k 2 (4.5) where k is the wave number. For viscosity-dominant ows in the low and transitional Re regimes, especially in the laminar boundary layer, Eq. (4.5) can be important for relat- ing viscosity to frequencies associated with boundary layer instabilities. This example only assumes ow in one direction; including more dimensions in the ow eld to better pre- dict three-dimensional ow inevitably involves additional higher-order partial terms of the velocity perturbations in all three directions, (u 0 ;v 0 ;w 0 ). 4.2 Sound and Tollmien-Schlichting Instability Inter- action Experimental and theoretical work on exciting the boundary layer with acoustic waves have led to various conclusions regarding the interaction of sound and T-S instability waves. Experimental work [50] on the generation of T-S waves by sound showed that a T-S wave was generated with the same frequency as the sound wave but with a much smaller wavelength. Mathematical modeling [53] the ow eld as the sum of a sound wave and T-S wave agreed with the experimental data [50], leading to the conclusion that there is no interaction between 28 the two waves. The only relationship between the waves is in the setting up of initial conditions of the T-S waves at or before the leading edge of the surface. Numerical analysis on the interaction of acoustic and T-S waves over a at plate at 120,000Re 380,000 [40] assumed two dierent energy-feeding mechanisms: a continuous (plane) wave that fed energy to the T-S wave along the whole boundary layer, and a sound wave that interacted with the boundary layer only in localized regions. For the rst case (continuous wave interaction), and using the notation in [40], the total disturbance can be represented as u =A cos( i x!t) +B cos(!t) (4.6) where A is the T-S wave amplitude, B the sound wave amplitude, i the instability wave number, and! the instability frequency (with the assumption that both the sound wave and T-S wave have the same frequency). The Fourier amplitude of the total disturbance at some ! is C = A 2 +B 2 + 2AB cos( i x) 1=2 (4.7) and whenBA (the sound wave amplitude is much larger than that of the T-S instability), Eq. (4.7) can be expanded to C =A cos( i x) +B +O A 2 (4.8) which assumes that A and B are independent or at most weakly dependent. If this is the case, then the total disturbance amplitude will oscillate spatially with a wavelength equal to the T-S wavelength, the magnitude of the spatial mean will be associated with the sound wave, and the magnitude of the envelope about the mean will be associated with the T-S wave [40]. Earlier experiments [50] showed that the Fourier amplitude of the total disturbance oscillated spatially with the T-S wavelength. Comparisons of spatial amplitude variations were made between a T-S wave in the presence of sound wave and a pure T-S 29 wave, and results [40] showed that the T-S wave was generated by the boundary condition at the upstream boundary and propagated independently of the sound wave. For the second case (localized wave interaction), the parameter s = !x U (where ! is the disturbance frequency, x is streamwise location, and U the free stream velocity) was used to classify the Orr-Sommereld equation into two regimes: sO(1) and s>O(1). It was found that the Orr-Sommereld equation was valid only fors>O(1), in which case the T-S wave and sound wave were found to be independent. For 0sO(1), where the boundary layer and sound wave would interact near the leading edge, linear solutions for the T-S wave and Stokes-layer were found to be mathematically independent. It was therefore concluded that on a sharp, at plate, a sound wave can generate a T-S wave but not interact with it over the rest of the boundary layer [40]. Numerical results further suggested that localized disturbances may generate larger amplitude T-S waves than a plane wave disturbance. Under normal circumstances, the nature and physical characteristics of sound and T-S waves are so dierent that they will not interact. However, there can still be coupling between the two waves, and for this to occur, necessary conditions are either matching the frequencies or matching the phase velocities of the incident sound wave and excited instability wave [52]. Since the phase velocity is inversely proportional to the wave number (! =kc), a matching of wave numbers,k, will also result in a coupling between a sound wave and T-S wave. Using the notation in [52], the pressure associated with the T-S wave can be expressed as p i (x;y;t) =f (y) e i( i x!t) (4.9) where f(y) is the instability amplitude distribution across the boundary layer, i the insta- bility wave number, and! the instability frequency. The pressure associated with the sound wave can be expressed as p s (x;y;t) =Bg (y) e i(sx!t) (4.10) 30 where s is sound wave number (in thex-direction) andB the amplitude of the sound wave, and the function g(y) is amplitude distribution of the sound wave across the shear layer. Normally, s > i so that there is no interaction; however, if B is allowed to vary inx, then p s can be expressed then as p s (x;y;t) =B (x)g (y) e i(sx!t) = Z 1 1 ~ B (k)g (y) e i[(k+s)x!t] dk (4.11) where ~ B(k) is the Fourier transform of the spatially varying amplitude B(x). Then, the acoustic wave is a superposition of many wave components with a new wave number (k+ s ). The wave number spectrum is now continuous, and with rapid enough amplitude variation, the wave number spectrum of the sound wave could be broad enough to overlap the constant wave number of the instability wave. Then, there would be coupling and interaction between the two waves. There is hence a seemingly possible interaction of an acoustic wave with a T-S wave when the acoustic wave has a rapidly varying amplitude or when there is rapid spatial change in the mean ow velocity prole. 31 Chapter 5 Methods 5.1 Wing Models All tests were performed on wings with an Eppler 387 prole (inset of Fig. 5.1). The baseline model is a solid wing, CNC-machined from an aluminum block, with AR = 5.8 (span b = 52.7cm and chord c = 9cm). The model used for the internal acoustic forcing tests is a two-part aluminum wing with AR = 6 (b = 54cm and c = 9cm), which is custom designed to consist of a base and a lid that t together with a tongue-and-groove connection. The wing was manufactured by electrical discharge machining wire cutting (wire EDM), which is a thermal mass-reducing process that uses a constantly moving wire to remove material by rapid, controlled, and repetitive spark discharges. The base structure of the wing contains cavities and channels into which small speakers and wire connections are embedded. The lid is 1-mm thick and contains 180 0.5-mm diameter holes arranged in six spanwise arrays with 30 holes each. The six spanwise arrays are located at streamwise locations x=c = c 0 = 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6. The base of the wing contains 180 cavities that are aligned with the holes in the lid; these cavities are connected to each other through spanwise channels for wiring, and ultimately to an exit port aft of the quarter chord mount point. 32 (a) (b) Figure 5.1: (a) E387 wing consisting of a base with 180 speaker cavities and a lid with 0.5- mm diameter holes, and (b) prole view of the lid of the wing. Six spanwise rows of holes are located at x=c = 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6. 5.2 Wind Tunnel Experiments were performed in a closed-loop wind tunnel with an octagonal test section of wall to wall width 1.37m and 5.7m length and area contraction ratio of 7 to 1. The empty test-section turbulence level is 0.025% for spectral frequencies between 2Hzf 200Hz in the velocity range 5m/sU 26m/s, and the velocity at any point in a given cross section deviates by no more than 0.5% from the mean velocity for that cross section [60]. The (x,y, z) coordinate system is as follows: x is the streamwise direction,y is the spanwise direction, and z is the normal direction, with the origin at the leading edge and midspan (Fig. 5.2). 33 Figure 5.2: Wind tunnel setup. (x, y, z) are streamwise, spanwise, and normal directions. Origin is at leading edge and midspan. 5.3 Force Balance All wing models are mounted vertically on a sting that extends through the oor of the wind tunnel, which connects to a custom force balance (described in detail in [60] and [36]) placed below the wind tunnel oor. The cruciform-shaped force balance contains four strain gauges on each arm. Two of the arms are connected to a plate above, and the other two arms are connected to a plate below, so that the entire assembly resembles a sandwich structure with the cruciform containing the strain gauges as the middle layer. The strain gauges are parallel-plate-structures (PPS), which have been shown to yield a lower total de ection for the same strain than a sensor based on a uniform beam at the same gauge location [42]. 34 Figure 5.3: Force balance schematic, from [60]. Inset: parallel-plate-structure strain gauge setup. The force balance is capable of measuring lift, drag, and pitching moment. Measurements are averaged over at least 8000 samples at a sampling rate of 1000 Hz, and standard deviations are also calculated. Before data acquisition, static calibrations are performed from 0 to 360 mN in 4 mN steps at dierent moment arms, and the force balance measurement has an expected uncertainty of 0.1 mN, conrmed through calibration. The expected friction coecient on a at plat can be given by C f = 1:328 p Re x (5.1) and the friction drag by F f =C f q 1 S (5.2) 35 where q 1 is the freestream dynamic pressure. For a at plate the same size as the E387 wing at zero degrees of incidence and Re = 40k, the expected friction drag is 11 mN. In force balance measurements, is varied from10 to 20 in steps of 1 , and for some of the acoustic studies, is varied in steps of 0.1 inside the hysteresis loop. At least three tests are performed for both increasing and decreasing and lift, drag, and moment results are averaged. 5.4 Particle Imaging Velocimetry A Continuum Surelite II dual-head Nd:YAG laser generates pulse pairs separated by exposure times, t = 100300s. The two coaxial laser beams are converted into sheets of slowly varying thickness through a series of convergent-cylindrical-cylindrical lenses. The laser sheets are oriented in the xz-plane across the tunnel, illuminating single chordwise span stations on the wing, or downstream of it (Fig. 5.2), and can be moved in streamwise and spanwise directions. A Colt 4 smoke machine generates 1m paran-based particles. For the external acoustic excitation study, a Kodak ES 1.0 CCD array camera with 1008 x 1018 pixels and 85-mm focal length lens, placed above the wind tunnel, traverses in the spanwise direction in concert with, and at constant distance from, the scanned laser sheet and acquires images. For the acoustic resonance and internal acoustic excitation studies, a higher quality Imager Pro X 2M (1600 x 1200 x 14-bit) camera that is tted with a 85-mm focal length lens and an adjustable 70210-mm focal length lens is used to acquire images. PIV images are used to obtain qualitative ow visualization as well as quantitative measurements of velocity elds and proles, and spanwise vorticity, ! y . PIV processing uses a variant of the custom CIV algorithms described in [10] and [9]. A smoothed spline interpolated cross-correlation function is directly t with the equivalent splined auto-correlation functions from the same data. The vectors are passed through an automated rejection criterion, and obviously incorrect vectors are manually removed, after which the raw displacement vector eld is reinterpolated back onto a rectangular grid with 36 the same smoothing spline function [24]. The spline coecients are dierentiated analytically to obtain velocity gradient data. The uncertainty, which does not depend on velocity mag- nitude, is in fractions of a pixel and correlates with 0.5 5% infu;wg and approximately 10% in ! y . 5.5 Acoustic Forcing External acoustic forcing is accomplished using a SolidDrive SD1sm speaker, which is at- tached to the outside of the wind tunnel test section upstream of the wing model (Fig. 5.2). The SolidDrive SD1sm speaker, which has a frequency response of 60Hz15kHz, uses high-powered neodymium magnets and dual symmetrically opposed motors to convert audio signals into vibrations, which are transferred into solid surfaces upon direct contact. Placing the speaker on the outer wall of the wind tunnel test section converts the entire test section into an acoustic chamber. The vibrations from the speaker are negligible to the structure integrity of the wind tunnel test section. Sine waves from a waveform generator are ampli- ed by an adjustable gain Pyle Pro PCA1 2 x 15 W stereo power amplier with a frequency response of 20Hz40kHz3dB and 0.3% total harmonic distortion. The frequency and peak-to-peak voltage amplitude of the sine wave are changed directly from the waveform generator. Internal acoustic forcing is achieved with Knowles Acoustics Wide Band FK Series (WBFK-30095-000) speakers measuring 6.50mm in length, 2.75mm in width, and 1.95mm in height. The WBFK speaker frequency response is 400Hz1000Hz 3dB. Four EX1200-3608 eight-channel digital-to-analog-converters send sine waves of adjustable frequencies and am- plitudes to the WBFK speakers. The WBFK speakers are amplied with a Kramer VA-16XL balanced adjustable gain stereo audio amplier that has a frequency range of 20Hz40kHz. A 4944 1/4" B&K pressure eld microphone, which has a pressure-eld response of2 dB between 16Hz70kHz, is used to obtain acoustic measurements for the wind tunnel resonance study. A 4954-B 1/4" B&K free eld microphone, which has a free-eld response 37 of3 dB between 9Hz100kHz, is used to obtain all other acoustic measurements. Both microphones are calibrated using a B&K 4231 Acoustic Calibrator. 5.6 Stability Analysis Stability analysis on the experimental velocity proles is achieved by a numerical Orr- Sommerfeld solver based on the solver described in [17] and used in [16] and [15]. The solver extracts the initial instability properties, not the completely developed properties. The general process is outlined here, and a detailed description is found in Paper V. The Orr-Sommerfeld equation (refer to Section 2 for derivation) is given by (U(z)c) 00 (z)k 2 (z) U 00 (z)(z) = 1 ikRe 0000 (z) 2k 2 00 (z) +k 4 (z) : (5.3) Far from the boundary, derivatives in the velocity eld are small, whereby the term in Eq. (5.3) containing U 00 (z) can be neglected as z ! 1. Then, the asymptotic form of the eigenfunction (z) can be given by (z) =Ae kz +Be kz +Ce z +De z : (5.4) The exponential solutions in k are homogeneous (inviscid) solutions, and the exponential solutions in are the particular (viscous) solutions, where is given by = k 2 + (ikRe) (Uc) 1=2 : (5.5) Far above the boundary (z! +1), only the decaying solutions exist, so A = C = 0. At the boundary (or \wall"), the boundary conditions are 38 z =z wall = 0; (z wall ) = 0; (z wall ) 0 = 0: (5.6) A shooting method is implemented to solve the two-point boundary valued problem. The ordinary dierential equation is integrated downwards from an initial point z = +z i far above the boundary, as well as upwards from the boundary atz = 0. The fourth-order o.d.e. is simplied into a set of rst order o.d.e.'s, yielding four solution vectors, two at z = z i and two at z = 0. At both limits in z, an inviscid (homogeneous) solution i and a viscous (particular) solution v exist. The solution vectors are integrated towards a matching point, z =z m , above the boundary, and a fourth order Runge-Kutta integration scheme is used for marching the solutions towards the matching point. At the matching point, the Wronskian of the eigenvectors must vanish with proper choices of c andk. If this convergence criterion is not met, c is iterated on until the Wronskian meets the criterion. The solutions are for a spatial stability analysis, so the instability frequency, ! =kc, is real. The velocity proles at dierent chordwise locations for the E387 wing at Re = 60,000 and = 9 , which come from PIV results, are used in the Orr-Sommerfeld solver to obtain the initial most amplied instabilities in the boundary layer. The frequency associated with the value ofk =k r + ik i that contains min(k i ) atk r > 0 is the most unstable frequency for a given prole. The most unstable frequencies are solved for at various chordwise locations and compared to the experimentally-obtained values for preferential acoustic forcing frequencies. 39 Chapter 6 Summary of Papers 6.1 Paper I This paper considers the variation in drag and circulation across the span of an E387 wing atRe = 30k. Local two-dimensional drag coecients are measured at 35 span locations and six using non-intrusive particle imaging velocimetry. The momentum defect method is applied far enough downstream where the downstream pressure is equal to the undisturbed pressure and where turbulent ow is negligible. Variations in the spanwise prole drag, c d (y), are related to local ow variations on the wing itself, including the location of the separation point and instantaneous and time-averaged spanwise vorticity. The greatest drag variation occurs near the wing tip, which is expected due to the three-dimensionality of the ow. In the mid portion of the wing where the ow is expected to be nominally two- dimensional, the variation in drag is found to correlate with the spanwise vorticity and separation point location such that an increase in local drag is associated with an increase in local spanwise vorticity and a forward movement of the separation point. Integrated values of drag are compared with direct force balance data, and the distinction between prole and induced drag components is realized, along with estimations for inviscid and viscous span eciency factors, e i and e v , respectively. The magnitude of the measured drag variations are compared to the magnitude of measured drag discrepancies for the same airfoil reported 40 in the literature. Comparisons suggest that spanwise drag variation is a possible - although not the sole - contributor to the measured inconsistencies. 6.2 Paper II This paper details the study of external acoustic excitation on a E387 wing atRe = 40k and 60k. A magnetically driven speaker that converts acoustic signals into vibrations is placed on the outside of the wind tunnel test section so that when the speaker is turned on, the entire test section turns into an acoustic chamber. Total lift and drag forces on the wing are mea- sured directly by the force balance, and ow visualization and corresponding measurements of ow velocity and spanwise vorticity are obtained from PIV methods. Optimum forcing frequencies (the frequencies that yield the largest improvements in aerodynamic eciency, L=D) are shared between the two testRe, although the range of eective frequencies is much greater at the higherRe. Both forcing frequency and sound pressure level (sound amplitude) are varied, and correlations between improvements in L=D to both parameters are found. The E387 wing has a nominal behavior that includes pre-stall hysteresis and abrupt switch- ing between stable states which results from sudden ow reattachment and the appearance of a large separation bubble. Control of these dynamics is achieved using external acoustic forcing at select excitation frequencies and sound pressure levels. The global ow around the wing is eectively modied such that large, stable vortical structures appear in the sep- arated shear layer. Correlation between the eects of acoustic excitation and wind tunnel resonance shows that the anti-resonances in an enclosed chamber correspond to the largest improvement in wing performance. The resulting optimum frequencies, when normalized to Strouhal numbers, correlate with values reported in the literature, suggesting that a correct Re scaling has not yet been determined. 41 6.3 Paper III This paper describes the serendipitous discovery of the presence of open holes in the suction surface of a E387 wing as a means of passive separation control. A custom designed and electrical-discharge-machined wing with a E387 prole contains 180 0.5-mm diameter holes in the suction surface, designed for the emanation of sound from speakers embedded inside the wing. Initial testing of this wing without speakers but with open holes shows completely dierent lift and drag behavior than from the known behavior of the same prole wing. Blocking air passage through the holes - either by lling them in from the top of the wing or by placing diaphragms underneath the holes - yields lift and drag results that match those from an otherwise `solid' E387 wing. It is realized that the presence of small holes has a transformative eect on the aerodynamics by acoustic resonance that occurs in the backing cavities underneath the holes, as changing the cavity volume changes the calculated and measured instability frequencies immediately above the open holes. Opening dierent spanwise rows of holes at various chord locations changes the mean chordwise separation line location. PIV measurements show that the local ow at a given span station on the wing is aected by the presence of a nearby open or closed hole, implying the ability to locally and passively control ow separation. The wing with open holes (`perforated wing') is compared to a woodwind instrument, and practical consequences for passive ow control strategies are discussed together with potential problems in measurements through pressure taps in such ow regimes. 6.4 Paper IV This paper is the main paper of this thesis and details the successful local control of ow separation by local internal acoustic forcing. Small speakers are embedded inside a custom designed and fabricated E387 wing. Function generators with adjustable frequency and amplitude and variable gain stereo audio ampliers were used to send sine wave signals to each individual speaker. Force balance and PIV measurements yield lift and drag forces as 42 well as ow visualization of local ow separation. For a given spanwise row of speakers, a range of optimum forcing frequencies is determined. The investigation of the distribution of activated speakers shows that improvement in L=D correlates with spatial distribution, spacing, density, and amplitude of sound sources. The localization of separation control is apparent from the raw PIV images, which show dierences in the separation line and the presence of vortical structures depending on the activation of a local speaker. The eect internal acoustic excitation and the eect of acoustic resonance from open holes are combined and show signicantly dierent results than those from the individual eects. Previously reported studies found in the literature on internal acoustic forcing are questioned as those studies do not distinguish the eects of acoustic resonance from pure internal acoustic forcing. From the current results on local separation control by internal acoustic forcing, implications for control and stabilization of small aircraft are considered. 6.5 Paper V This technical report describes the Orr-Sommerfeld numerical solver used for extracting the initial ow instability properties of experimental velocity proles and the results thereof. A spatial stability problem using parallel and 2-dimensional ow assumptions is solved, and the preliminary results indicate that the ow in the boundary layer is initially bounded and stable. The least stable frequencies, which are associated with the minimum spatial growth rates,k i , can be determined for proles at dierent chordwise locations. Comparison of the numerical results to experimental results from local internal acoustic forcing show that the preferential excitation frequency, f e , matches the initial least stable frequency at the natural separation point, and that aft of the separation point, f e is a harmonic of the least stable frequency. Conclusions are less obvious for locations forward of the separation point. These preliminary results must be taken only as a basis for further verication due to potentially improper ow assumptions and an oversimplication of the problem. More studies can be done to vary dierent parameters in the solver, obtain better interpolations 43 of the experimental data, and even compare experimental velocity proles to Falkner-Skan proles. 44 Paper I 45 Chapter 7 Spanwise Variation in Wing Circulation and Drag Measurement of Wings at Low Reynolds Numbers Yang, S. L. and Spedding, G. R. Aerospace and Mechanical Engineering Department University of Southern California Los Angeles, California 90089-1191 Journal of Aircraft, Vol. 50, No. 3, 2013, pp. 791-797. The measurement and prediction of aerodynamic performance of airfoils and wings at chord Reynolds numbers below 10 5 are both dicult and increasingly important in ap- plication to small-scale aircraft. Not only are the aerodynamics strongly aected by the dynamics of the unstable laminar boundary layer, but the ow is decreasingly likely to be two-dimensional as Re decreases. The spanwise variation of the nominally-two-dimensional ow along a two-dimensional geometry is often held to be responsible for the large varia- tions in measured prole drag coecient, c d at this scale. Here, local two-dimensional drag coecients are measured along a nite wing using non-intrusive PIV methods. Variations in c d (y) can be related to local ow variations on the wing itself. Integrated values can then be compared with direct force balance data, and the dynamical signicance of spanwise variability will be re-evaluated. 46 Nomenclature AR = aspect ratio b = wing half span (m) c = chord (m) c d = sectional prole drag coecient c d;PIV = sectional prole drag coecient obtained from PIV measurements C d;0 = minimum total drag coecient on an innite wing C D = total drag coecient on a nite wing C D;PIV +FB = total drag coecient on a nite wing from force balance and PIV C D;i = induced drag coecient on a nite wing C D;i;FB = induced drag coecient on a nite wing calculated from force balance measurements C D;0 = minimum total drag coecient on a nite wing C f = laminar skin friction coecient C L = total lift coecient on a nite wing c s = separation line location (m) D' = drag per unit span (N/m) e i = inviscid span eciency e v = viscous span eciency f = body force (N) l = vertical transect in the wake region of the wing n = normal vector to surface p = pressure (N/m 2 ) 47 p 0 = free stream pressure (N/m 2 ) q = dynamic pressure (N/m 2 ) Re = Reynolds number S = control surface around the wing U = mean velocity vector (m/s) U = mean component of velocity in x (m/s) U 0 = free stream velocity (m/s) x, y, z = coordinates are streamwise, spanwise, and normal directions = angle of attack (deg) = boundary layer thickness (m) = standard deviation = momentum integral (m) ! = vorticity vectory (rad/s) ! y = spanwise component of vorticity (rad/s) = normalized vorticity h*i = root-mean-square value I. Introduction STANDARD airfoil performance data do not often extend beneath Re 100,000, and when they do there are large discrepancies between studies of airfoils under ostensibly the same conditions [1, 2]. In the range 30,000 Re 100,000 in particular, there is heightened sensitivity to small variations in geometry and operating conditions. Gross performance parameters depend strongly on initial laminar boundary layer stability and separation, tran- sition to turbulence of the separated shear layer, and possible subsequent reattachment in some time-averaged sense. Though practical wings are nite in span, those of moderate aspect ratio share many of the characteristics that have been demonstrated for 2D airfoil sections [3], as the central part of the wing sees little in uence from the tip vortices. One 48 of the well-known characteristics that has not been measured for the nite wing however, is the possible variation in sectional drag coecient as measured from wake surveys [1, 4]. At these Re, wake surveys and calculating drag components pose particular problems (described below), and it is not known whether the variations will be large, or whether they could be responsible for discrepancies or variations in force measurement. The purpose of this paper is to carefully document the spanwise variation in measured sectional drag coecient in ex- periments that combine optical ow measurement techniques with direct force balance data from custom instrumentation with adequate resolution of the small forces involved at these Re. We may then determine whether quasi two-dimensional analysis is sucient (or where it is sucient) and then compare the variations with the previously-reported variations on two-dimensional geometry. Ultimately the ndings will extrapolate out to small UAVs with xed wing design. A. Drag Variation at Transitional Reynolds Numbers While much literature exists for the Eppler 387 [4], an airfoil commonly used on sailplanes and gliders, the agreement of measured lift and drag coecients among dierent facilities deteriorates as Re decreases, as seen in Figure 7.1. At Re = 60k, c d (y) is large, and while turbulence levels, acoustic noise, model accuracy, and physical vibrations may contribute to these measured drag discrepancies, it has also been suggested that spanwise drag variation is responsible [4, 5]. This is partly because measurements of relatively small drag forces are usually made not by direct measurement on the wing but by integrating velocity/pressure information from scanning or arrayed sets of pitot tubes in the wake, when sampling of spatially inhomogeneous data can be incomplete. Moreover, precisely at these low Re the validity of the method has been questioned due to the upstream in uence of wake rakes on the ow [6]. The spanwise drag variation,c d (y), of an E374 wing section has been measured at various angles of attack,, and downstream locations,x=c, forRe = 200,000 (Figure 7.2) [1]. Figure 7.2 shows that while variation is most pronounced at the farthest downstream locations, all stations vary across the span, and the variation is spread across the entire span. Total 49 Figure 7.1: Drag polars for the 2-D Eppler 387 at various Re from dierent facilities replotted from [7]. drag estimates would have to come from a number of span stations and one may imagine cases where it could be wrongly-estimated if measurement stations coincided with peaks or troughs. Most routine c d measurements are taken at some x=c (= 2.25 for Selig) where pressure gradients are small and the local ow is mostly parallel to the tube array. Indeed, for drag calculations made from pitot-static pressure measurements, a proper downstream wake survey location has been shown to be a function of the drag formulation equations themselves [8]. Steady-state equations have been shown to be applicable for survey regions suciency far downstream so there is negligible variation in static pressure [8, 9], but this is where the spanwise variation in Figure 7.2 is most pronounced. Finally, the larger variations inc d in Figure 7.1 are forRe< 100,000, and it is not clear how to extrapolate the results at 50 Figure 7.2: c d (y) of an E374 wing section at Re = 200,000, =6:4 atx=c = 1.0 (trailing edge) (solid circles), x=c = 1.7 (squares), and x=c = 2.23 (diamonds) re-plotted from [1]. Re = 200,000 in Figure 7.2 to this regime, where apparently the ow properties are much less predictable. B. Drag Calculations The momentum equations for a viscous ow in Einstein notation are @~ u i @t +~ u j @~ u i @x j = @ ~ P @x i +f i + @ 2 ~ u i @x j @x j (7.1) where ~ u i is the total instantaneous velocity comprised of mean, ~ U i , and uctuating, u i , parts ( ~ u i = U i +u i ), p i is the total pressure also comprised of mean and uctuating parts ( ~ p i = P i +p i ), is the density of the uid, is the dynamic viscosity, f i is the total body force per unit mass, and i;j = (x;y;z). For our wing model system, x is streamwise, y is spanwise, and z is normal to the chord,x, and span,b, when the wing is at zero, Eq. (7.1) can alternatively be written by decomposing into mean and uctuating velocity components to yield the Reynolds Averaged Navier-Stokes (RANS) equation: @U i @t +U j @U i @x j = @P @x i +f i + @ 2 U i @x j @x j @u i u j x i (7.2) where is the Reynolds stress tensor. In classical aeronautics applications, the ow around a xed wing in steady motion is assumed to be steady and inviscid. Furthermore, if a region of ow is surveyed far from the body, then the pressure there can be assumed to be equal 51 to the constant, undisturbed free stream pressure, p 0 , and turbulent motion of the ow is negligible. With these additional constraints, the time derivative term on the left side of Eq. (7.2), the pressure term, and the last two terms on the right side drop out, leaving U j @U i @x j =f i (7.3) Eq. (7.3) relates the body force in any direction with the mean momentum ux in that direction. It is then convenient to express this relationship in integral form so that forces in one region can be related to uxes through an enclosing control volume with surface area S and corresponding normal vector n, so Z S U i U j n j dS = Z S f i dS (7.4) The component of the force in the streamwise direction, f x , can therefore be calculated from the change in momentum ux between upstream and downstream surfaces, and when the ow and body geometry are uniform in one direction, such as the span, the drag force per unit span, D 0 , can be evaluated from just two line integrals: D 0 = Z l 1 1 U 2 1 dz + Z l 2 2 U 2 2 dz (7.5) where 1 andU 1 are the density and velocity of the uid upstream of the body at a vertical transect l 1 , and 2 and U 2 are the density and velocity downstream of the wing at l 2 , which contains all the wake region, W. IfU 1 =U 0 and = constant, the total positive drag on the body with span b can be written as D =U 2 0 b (7.6) where the momentum integral, depends only on the variation of mean velocity components over W : 52 = Z W U 2 U 0 U 2 U 0 2 ! dz (7.7) The section prole drag coecient c d is then c d = 2 c (7.8) The drag formulation of Eqs. (7.4) (7.8) diers slightly from other well-known methods in the literature [8, 10, 11, 12]. The total drag formulation in [11] includes a total pressure loss term and allows for non-zero cross plane velocity components, while Eqs. (7.4) (7.8) are more restrictive, assuming that the wake is surveyed far enough downstream so the downstream pressure is equal to the undisturbed pressure and turbulent ow is negligible (so average cross plane velocity components go to zero). Betz's equation for prole drag [10] takes the survey location only in the wake of the body by introducing a ctitious velocity component that is non-zero only in the region of the viscous wake while the current method does not use a ctitious velocity but rather takes the survey location far downstream. The equivalent equation for prole drag in [12] uses the total and static pressures measured close behind the body whereas the current method only considers velocities measured at two locations (upstream and downstream of the body). The total drag on a nite wing is commonly described as the sum of two components: C d =c d +C D;i (7.9) c d is the prole drag coecient in Eq. (7.9), which is a function of, and can be expressed c d =c d;0 +c d () (7.10) where c d;0 is the minimum drag coecient for a 2-d wing section. C D;i is the induced drag coecient, 53 C D;i = C 2 L e i AR (7.11) where C L is the lift coecient of a nite wing with aspect ratio AR, e i 1 is the inviscid span eciency factor, which accounts for departures from the ideal elliptic spanwise load distribution. It can be convenient and reasonable to write C D as a quadratic function of C L , and then a viscous span eciency factor, e v , which can be obtained through the slope approximation of the C D -C L 2 curve [3], can be used to write the total drag coecient as C D =C D;0 + C 2 L e v AR (7.12) where C D;0 is the minimum drag value from the C L -C D polar. The slope values from the C D - C L 2 curves dier between two-dimensional airfoils and nite wings, as shown in [3], so careful distinction between the two conditions must be made. Either of these two drag decomposition methods (using either Eqs. (7.9) & (7.11), or Eq. (7.12) alone) can be used to estimate drag components that are essentially inviscid (induced drag due to downwash behind a lifting wing) and viscous (prole drag from skin friction and boundary layer separation) in origin. Such a separation is simple at high Re, but perhaps less easy to disentangle at moderate to low Re, when the behavior of the viscous boundary layer is so in uential [13]. C. Objectives This paper provides the rst direct check on the spanwise variation of local c d measure- ments on a smooth airfoil of moderate thickness for Re < 100,000. When and if variation is found, the associated instantaneous and time-averaged velocity elds on the wing can be checked for conditions that may cause the observed uctuations. Direct association of the force coecients with the relevant ow eld is still quite rare in aeronautics practice but is important in regimes with such a rich variety of important ow behavior. The local drag measurements come from PIV-derived velocity elds, and the total inferred and integrated drag on the wing can be compared with direct force balance measurements. 54 II. Materials and Methods A. Experimental Setup Experiments were performed in a closed-loop wind tunnel with octagonal test section of wall-to-wall width 1.37 m, and 5.7 m in the streamwise direction. The empty test-section turbulence level is 0.025% for spectral frequencies between 2 Hz f 200 Hz in the velocity range 5 m/s U 26 m/s. Flow uniformity measurements showed no more than 0.5% velocity deviation from the mean velocity for a given cross section [14]. The wing was CNC-machined from a solid aluminum block with AR = 5.8 (span b = 52.7 cm and chord c = 9 cm) with an Eppler 387 airfoil section, as shown in Figure 7.3 inset. All measurements were made at Re = 30,000, which is deliberately set to be in a region of C L (C D ) space where abrupt switching between stable states can occur. Previous experimentation [15] on the same wing and same conditions shows that the ow separates before the trailing edge, so small variations in trailing edge thickness were not a concern. Particle Image Velocimetry (PIV) was used to estimate velocity components (u, w) in the two-dimensional plane (x, z) (Figure 7.3). A dual-head Continuum Nd-Yag laser was used to generate coplanar sheets in the smoke-seeded ow in the test section. Paran-based particles were generated with a Colt 4 smoke machine. The laser sheets were oriented in the xz-plane across the tunnel, illuminating single chord-wise span stations on the wing, or downstream of it. A Kodak ES 1.0 CCD camera with 1008 x 1018 pixels and an 85-mm focal length lens, placed above the wind tunnel, was traversed in the spanwise direction in concert with the scanned laser sheet and acquired images. The time between laser pulses was set to a nominal 260 s. Images were taken at 35 span stations spaced 1 cm apart, at three streamwise locations (Figure 7.3), and at six angles of attack, = 0, 2, 4, 6, 8, 10 . B. Spanwise Vorticity Measures Persistent features in the time-averaged wake proles can be traced upstream to the generating conditions on the wing, where the separation line location and spanwise vorticity 55 Figure 7.3: Three streamwise locations and three (of 35) spanwise stations at which PIV images were taken. The coordinate system origin is the leading edge at mid span. The trailing edge is x=c = 1.0. Inset: E387 prole. magnitudes in the separation region can be related to the wake structure. Contrasting regions of interest were studied for the ! y (x,z) measurements at two . For = 0 , where the majority of the ow across the top surface of the wing is still attached, the regions of interest were the fore- and aft-attached regions, denoted by \a" and \b" in Figure 7.4a. Region(a) encompassed the front (windward) half of the airfoil, following the boundary points along the top surface of the airfoil and extending to a height that enclosed the entire boundary layer prole. Region(b) covered the back (leeward) half of the airfoil with the same height as Region(a). Statistics from these two regions were collected separately. When the separation line has moved forward by a signicant fraction of the chord, such as at = 8 , the regions of interest were the attached region and the separated region, \a" 56 and \s" in Figure 7.4b. The attached region was the same as Region(a) for = 0 . The separated region was dened by a triangle from mid-chord to the trailing edge (Figure 7.4b). Figure 7.4: Regions of interest for spanwise vorticity elds. Top: = 0 fore (a) and aft (b). Bottom: = 8 attached (a) and separated (s). For all PIV images at a given span station and region of interest, i(a, b, s), the root- mean-square of the instantaneous spanwise component of vorticity! y values were calculated over that region to give a single r.m.s. vorticity value,h! y i i : h! y i i = v u u t 1 N N X k=1 (! y (fx;zg2i)) 2 k (7.13) where N is the total number of images. h! y i i was then normalized by the chord and mean velocity and is denoted h i i = h! y i i c U (7.14) The location of the separation point itself can be measured directly and independently from raw particle images. At values of where separation occurs on the front half of the wing (0.0 x=c 0.4) the separation line is visible as a thin dark line. In this line uid 57 has come directly from the boundary layer where fewer tracer particles (introduced in the exterior ow) have penetrated. c s is the chordwise location of this separation line. C. Force Balance Lift and drag forces were measured with a custom cruciform-shaped force balance (de- scribed in [14, 15]), placed below the wind tunnel oor. The force balance was capable of measuring lift, drag, and pitching moment. Measurements were averaged over 9000 samples at a sample rate of 1000 Hz. Careful calibration procedures were performed each day before data acquisition; static calibrations were performed from 0 to 360 mN in 4 mN steps at dierent moment arms. The force balance measurement has an expected uncertainty of 0.1 mN. The expected friction drag on a at plate of the same size as the E387 wing at zero degrees of incidence is approximately 11 mN. In force balance measurements, was varied from -10 to 20 in steps of 1 . Three tests were performed for both increasing and decreasing , and results were averaged. The force balance measures total drag (which will be labeled C D;FB ) as well as lift, C L . The prole drag values at the six dierent are obtained from the PIV measurements of momentum wake defect in the midsection of the wing (-0.4 y/b 0.4); this prole drag component will be labeled, c d;PIV . The span eciency (Eqs. (7.11), (7.12)) for the E387 wing is initially unknown, so C D;i;FB can be estimated by subtracting c d;PIV from C D;FB at each . Since C L is known at each , a least squares t for C D;i;FB values can be used to solve for e i and e v from Eqs. (7.11) and (7.12). The total drag achieved by adding c d;PIV and C i;FB with the calculated eciency values will be labeled C D;PIV +FB and is necessarily equal to C D;FB . The uncertainties in C D;i;FB derive primarily from the standard deviation in C L from force balance measurements, and the uncertainties in c d;PIV are obtained by methods explained in the following section. 58 III. Results A. Spanwise Drag Variation at Moderate Reynolds Numbers Correct estimates of c d from Eq. (7.10) require that converged time-averaged proles exist and that contributions from dp dx are negligible. This condition occurs at some distance downstream, estimated to be x=c > 3 for similar conditions [16]. Mean prole integrals R W U 2 (z)dz, where W is a vertical line in the wake of the wing where the wake defect exists, converged to within 13% after 120-210 image pairs, depending on. Satisfactory convergence of (x) can be claimed after x=c = 2.75, and all subsequent data use streamwise averages over x=c2 [3.0, 3.4]. The results of integrating Eq. (7.7) to obtain(y) at the six dierent at the downstream locationx=c = 3.4 yieldc d (y) for the E387 wing at Re = 30,000, shown in Figure 7.5. There is increasing variation near the wingtip (y=b =1.0) with increasing , which is mostly a consequence of the non-zero mean out-of-plane momentum ux. There is also measurable variation over the mid portion of the wing between -0.4 y=b 0.4 where the variation withy in c d is higher than the measurement uncertainty. c d (y 0 ) increases as increases, but there is no obvious variation in the absolute magnitude of c d with in the wing center. Since the wingtip eects are incidental to the main focus here, they are not analyzed further and the resulting focus will be on the mid portion of the wing. The momentum thickness, , was obtained by several methods, including averaging dif- ferent numbers of image pairs to obtain U (z), applying dierent interpolation methods to acquire the boundaries for the momentum defect regions, and using dierent integration methods to calculate from a given mean prole The uncertainty c d is the maximum dif- ference inc d values obtained using these various methods. The greatest relative variation in c d in the midsection of the wingspan (-0.4y=b 0.4) is 27% at = 0 . By denition, in this procedure, any measured variation must come from systematic and repeatable variations in prole amplitude and width. 59 Figure 7.5: c d (y) at = 0, 2, 4, 6, 8, 10 . The symbol size is chosen to match the size of the measurement uncertainty. 60 If there is variation in average values of and c d due to time-averaged variation in ow eld, then it should be possible to trace such variations upstream. c d (y) at = 0 and = 8 are compared at two dierent downstream locations, x=c = 3.4 and 2.0 (Figure 3.6). Figure 7.6: c d (y) at = 0 and 8 at x=c = 3.4 (solid circles) and x=c = 2.0 (squares). The symbol size matches the measurement uncertainty. Arrows denote data points in subsequent sections. At = 0 the pattern of above-threshold spanwise c d (y) variation matches for the two x=c locations. The dierence between the two sets of data is a slight oset (the calculated drag values are slightly higher further downstream). At = 8 the correlation is less obvious, but there are correlated variations, signicantly above noise, that are coherent in x. If the correlations in the wake are coherent in x, then it may be possible to trace their origin back to conditions on the wing. B. Spanwise Vorticity and Separation Point Location Variation At = 0 , instantaneous and time-averaged ! y look similar as the ow is steady and laminar separation occurs shortly before the trailing edge (Figure 7.7 a, b). At = 8 , ow separation is earlier and the separated shear layer has become unstable, generating coherent structures that impinge upon the downstream portion of the suction surface. ! y and ! y are not the same (Figure 7.7 c, d). The earlier separation of the boundary layer is associated with increased turbulent levels in the separated region and reduced aerodynamic performance. At 61 x 10mm, Re x 3300 and the boundary layer thickness = 5:2x p Rex 0.9mm. The grid resolution is 1.5mm, so the laminar boundary layer at the wall is not resolved. The boundary layer vorticity and possible presence of small separation and reattachment regions upstream of the trailing edge separation are therefore not accessible to this experiment. However, the statistics of the larger separated region can be used as indicators of variation in the separation point and conditions behind it. Figure 7.7: a) ! y (x;z) and b) ! y (x;z) at = 0 ; c) ! y (x;z) and d) ! y (x;z) at = 8 . Arrows are uctuating velocity vectors. At = 0 , vey=b stations in the mid portion of the wing (-0.5y=b 0.5) were chosen wherec d (y) varied similarly atx=c = 3.4 andx=c = 2.0 (indicated by arrows in Figure 7.6). h i a;b are shown at the dierent span stations in Figure 7.8. h i a varies with the resolved outer boundary layer vorticity over the attached part of the airfoil, and is therefore a measure of the strength of the bound vorticity on the wing. h i a may be expected to vary with lift 62 coecient but may not be sensitive to changes in drag. When ow separation is mild,h i b , which averages all spanwise vorticity over the aft surface, has a lower magnitude (the mean boundary layer has thickened) but still has very similar variation. Neitherh i a norh i b varies signicantly across the span or in phase withc d (y). Apparent variations in phase with variation in c d (y) do not rise above the measurement uncertainty. Flow separation does not occur until close to the trailing edge at = 0 , so there is no separated region along the top of the wing as in the case of higher . At = 8 sixy=b stations were also chosen wherec d (y) varied similarly atx=c = 3.4 and x=c = 2.0 (indicated by arrows in Figure 7.6). h i a;s and c s =c are shown at the dierent span stations in Figure 7.9. h i s varies as c d , showing, unsurprisingly, that high local c d is associated with high turbulence levels over the trailing half chord. There is no clear correlation ofh i s withc d . c s varies inversely withh i s so high turbulence in the separation region is associated with earlier separation and higher local c d . C. Estimation of Spanwise Eciency Factors In Figure 7.10, C D;FB is compared with values calculated from the wake measurements that use C D;i and the least squares t on e i (Eq. (7.11)) to match the sum of C D;i and C D;PIV +FB . The best t yielded e i = 0.83. This estimate uses the wake-measured variation of c d () to estimate the prole drag. An alternative is to use the force balance data to calculate the constant value of C D;0 (also sometimes known as the prole drag), and then the implicit variation of c d with C L 2 is included in the value of the viscous span eciency, e v , which can again be estimated by least-squares regression of C D;PIV +FB on . The t is satisfactory when e v = 0.3 (Figure 7.11) [3]. This estimation by least-squares regression essentially yields values of e v and e i that would otherwise be obtained through a linear t and slope calculation of the C D -C L 2 curve. The estimated slope at low is 0.14, compared with 0.09 for a dierent E387 wing at the sameRe in [3], where it was noted that such values 63 Figure 7.8: At = 0 , spanwise c d (y) (top, x=c = 3.15 in black, x=c = 1.75 in gray),h i b (middle), andh i a (bottom). have limited signicance when the lift-drag polars themselves have shapes very dierent from model assumptions. IV. Discussion There is measurable variation in c d (y) on the E387 wing at Re = 30,000 over all the tested (0 10 ). The large dierences in measuredc d (y) between the mid-span and 64 Figure 7.9: At = 8 , spanwise c d (y) (top, x=c = 3.15 in black, x=c = 1.75 in gray),h i s (middle), andc s =c (bottom). The uncertainty inh i i is the maximum dierence among the values ofh i i obtained from averaging over dierent number of image pairs. the wing tip (up to 98% at = 10 for 0.0 y/b 0.9), are related to the non-negligible momentum ux in out-of-plane directions, and are related directly to the lift on the nite wing. However, spanwise c d (y) variation also occurs in the mid portion of the wing (up to 27% variation at = 0 for 0.0 y/b 0.2), where the local ow may otherwise be considered to be close to two dimensional. 65 Figure 7.10: Combined force balance and PIV drag results. C D;FB (gray line + circles), c d;PIV (dashed line + diamonds), C D;PIV +FB using e i = 0.83 (solid line + squares). The c d (y) variation trends are preserved at dierent x=c locations, and tracing the ow back to on-wing conditions shows that, at = 8 ,where signicant ow separation occurs, h i s variation is directly proportional to c d (y) variation. This was not the case forh i a , or at = 0 where separation does not occur until near the trailing edge. The fore-aft move- ment of the separation point location showed that an increase inh i s corresponds to earlier separation, suggesting that at high , the location of the separation line c s is not uniform and two dimensional. The location of separation aects the size of the separated region above the airfoil, as measured by the magnitude of the spanwise vorticity in the separated region, and correlates with the wake momentum decit, and hence the local sectional drag, and its measurement. 66 Figure 7.11: C D;FB (gray line + circles), C D;0 (dotted line), C D;PIV +FB using e v = 0.3 (dashed line + triangles). The results of the c d (y) variation study at Re = 30,000 are quantied by the same normalization procedure as described in Section IB. and compared with literature results [1, 4] in Figure 7.12. While the values of drag variation are associated with two dierent Re and two dierent (but similar) airfoils, both show a local maximum at 6 deg, which is where the separation location is highly sensitive to small disturbances. The lower Re data show that the relative variation increases for small . The variation of c d with y is much higher in current experiments, but that is because Re is lower. The variation magnitude is shown as a function of Re in Figure 7.13. Clearly decreases as Re increases, but the increase in /c d with decreasing Re is consistent in both facilities. Finally, for the current study at Re = 30,000 is compared with c d among dierent facilities atRe = 60,000 in Figure 7.14. For 0 10 , atRe = 30,000 is less than c d 67 Figure 7.12: at Re = 200,000 for the E374 from [1] (squares) and Re = 30,000 for the E387 from the current study (solid circles). Figure 7.13: /c d at = 0 (solid circles) and 5 (gray squares) at various Re from Langley [4] and Dryden. The value at = 5 from Dryden is an average of the values at = 4 (bottom white square) and 6 (top white square). atRe = 60,000. Figure 7.13 shows that drag variation increases as Re decreases, so at Re = 60,000 will be lower still than at Re = 30,000. The observed c d (y) variation is much less 68 than the drag variation from literature at Re = 60,000 and therefore c d (y) variation is not the main cause of the discrepancies in measured c d among dierent facilities. Figure 7.14: Drag variation from dierent facilities (squares) and spanwisec d (y) from current study (solid circles). The separation of drag into its dierent components is achievable atRe = 30,000 ife i and e v are determined empirically. Since only the-dependent prole dragc d is obtained from the PIV-based measurements, a second method (in this case, direct force balance measurements) must be used to complete the drag measurement/calculation, which involves two unknowns, e and C D;i . At Re = 30,000, the drag measurements imply values of e i and e v of 0.83 and 0.3, respectively, which are very low compared with the usual high Re default values close to 1, but in agreement with previous ndings at moderate Re [3]. The existence of persistent spanwise variation in wake defect magnitude, and hence c d , along the span, supports the argument that a single parameter description of the departure from ideal uniform conditions may not be a very good re ection of the detailed ow eld on the wing. 69 V. Conclusion At transitional Re ows, particularly in the sub-regime 30,000Re 70,000, the drag values reported by various facilities for smooth airfoils including the E387 dier signicantly, and it has been suggested that spanwise drag variation is one possible cause of these dispari- ties. Airfoil and wing performance, especially at low Re, is extremely sensitive to separation location [17], and variations in separation location do indeed correlate with variations in local measured sectional drag coecients. However, the magnitude of these variations in the nominally two dimensional center section of the wing reported here is small, and cannot account for the dierences in reported results among dierent facilities. 70 Paper I References [1] Guglielmo, James J., and Selig, M. S., \Spanwise Variations in Prole Drag for Airfoils at Low Reynolds Numbers," Journal of Aircraft, Vol. 33, No. 4, 1996, pp. 699-707. [2] Simons, M., Model Aircraft Aerodynamics, 4th Ed., Special Interest Model Books, Poole, 1999. [3] Spedding, G. R. and McArthur, J., \Span Eciencies of Wings at Low Reynolds Num- bers," Journal of Aircraft, Vol. 47, No. 1, 2010, pp. 120-128. [4] McGhee, R. J., Walker, B. S., and Millard, B. F., Experimental Results for the Eppler 387 Airfoil at Low Reynolds Numbers in the Langley Low-Turbulence Pressure Tunnel, NASA TM-4062, 1988, pp. 25-26. [5] Mueller, T. J., \Aerodynamic Measurements at Low Reynolds Numbers for Fixed Wing Micro-Air Vehicles," presented at the \Development and Operation of UAVs for Military and Civil Applications" course held at the von Karman Institute for Fluid Dynamics, Belgium, September 13-17, 1999. [6] Lowson, M. V., \Aerodynamics of Aerofoils at Low Reynolds Numbers," Bristol UAV Conference 1999. [7] Selig, M. S., Guglielmo, J. J., Broeren, A. P., and Giguere, P., \Summary of Low-Speed Airfoil Data," Vol. 1, Soar Tech Publications, Virginia, 1995, pp. 19-20. [8] Takahashi, T. T., \On the decomposition of drag from wake survey measurements," AIAA Paper 97-0717, Jan. 1997. [9] Taylor, G. I., \The Determination of Drag by the Pitot Traverse Method," British ARC R&M 1808, 1937. [10] Betz, A., \A Method for Direct Determination of a Wing Section Drag," NACA Tech- nical Memorandum No. 337, 1925. [11] Bollay, W., \Determination of Prole Drag from Measurements in the Wake of a Body," Journal of Aeronautical Sciences, Vol. 5, No. 6, 1938, pp. 245-249. 71 [12] Jones, B. M., \Measurement of Prole Drag by the Pitot-Traverse Method," British ARC R&M 1688, 1936. [13] Brune, G. W. \Quantitative Low-Speed Wake Surveys." Journal of Aircraft, Vol. 31, No. 2, 1994, pp. 249-255. [14] Zabat, M., Farascaroli, S., Browand, F., Nestlerode, M., and Baez, J., \Drag Mea- surements on a Platoon of Vehicles," Research Reports, California Partners for Advanced Transit and Highways (PATH), Institute of Transportation Studies, UC Berkeley, 1994, (doi: 10.1146/annurev. .15.010183.001255). [15] McArthur, John. \Aerodynamics of Wings at Low Reynolds Numbers," Ph.D. Disser- tation, Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA, 2007. [16] Spedding, G. R. and Hedenstrom, A., \PIV-Based Investigations of Animal Flight," Experiments in Fluids, Vol. 46, 2009, pp. 749-763. [17] Lissaman, P. B. S., \Low Reynolds Number Airfoils," Annual Review of Fluid Mechan- ics, Vol. 15, 1983, pp. 223-239. 72 Paper II 73 Chapter 8 Separation Control by External Acoustic Excitation on a Finite Wing at Low Reynolds Numbers Yang, S. L. and Spedding, G. R. Aerospace and Mechanical Engineering Department University of Southern California Los Angeles, California 90089-1191 American Institute of Aeronautics and Astronautics, Vol. 51, No. 6, 2013, pp. 1506-1515. At Reynolds numbers approaching those of micro-air vehicles (both engineered and nat- ural), the Eppler 387 airfoil (in common with many other smooth proles) can have multiple lift and drag states at a single wing incidence angle. Pre-stall hysteresis and abrupt switch- ing between stable states result from sudden ow reattachment and the appearance of a large separation bubble. Here, we show that control of the dynamics can be achieved using external acoustic forcing. Separation control, hysteresis elimination, and more than 70% increase in lift:drag ratio are obtained at certain excitation frequencies and sound pressure levels. The global ow around the wing is eectively modied, and large, stable vortical structures appear in the separated shear layer. Correlation between the eects of acoustic excitation and wind tunnel resonance shows that the anti-resonances in an enclosed chamber 74 correspond to the largest improvement in wing performance. Implications for control and stabilization of small aircraft inside and out of enclosed boxes are considered. Nomenclature AR = aspect ratio b = wing half span (m) c = chord (m) C D = total drag coecient on a nite wing C L = total lift coecient on a nite wing f e = excitation frequency (Hz) f e * = optimum excitation frequency (Hz) f e o = uneasily excitable frequency (Hz) f s = separated shear layer instability shedding frequency L/D = lift-to-drag ratio SPL = sound pressure level (dB) St = Strouhal number St = angle of attack based Strouhal number St s = separation Strouhal number Re = chord-based Reynolds number U = free stream velocity (m/s) U e = edge velocity of boundary layer (m/s) U a = advection speed (m/s) = angle of attack (deg) 0 = hysteresis loop-preceding angle of attack (deg) = percent change in parameter (%) 75 I. Introduction A growing number of micro aerial vehicles have been in development, production, and use for multiple applications. The ight regime in which many of these miniature aircraft systems operate is where the chord-based Reynolds number, Re, lies between 10 4 10 5 , which is considered to be a low Re regime in aeronautics. Here, complex ow characteristics can either favorably or adversely aect wing performance. Two main approaches can be taken in this design space: either avoid it altogether, or manipulate and force the ow toward favorable conditions that maximize wing performance. In this study, the latter approach is taken using active separation control through acoustic excitation. At low Re, adverse pressure gradients are most likely to occur when the boundary layer is still laminar, making the ow over an airfoil susceptible to separation. When the ow has sucient energy to overcome the combined eects of adverse pressure gradient, viscous dissipation, and change in momentum, the ow remains attached. Conversely, when the ow has insucient energy, the ow separates from the wing surface, then often transitions from a laminar to turbulent state, and may then reattach as a turbulent boundary layer. In such a case, the separated region forward of the reattachment point will be termed a laminar separation bubble (LSB). The performance of the Eppler 387, a high performance sail plane airfoil usually used at Re > 200,000, has been shown to be strongly aected by the presence of a laminar separation bubble at lower Re [1]. In the regime 30,000 Re 80,000 the E387 has complex ow characteristics where the lift-drag curves show pre-stall hysteresis and abrupt jumps between what appear to be multiple performance envelopes due to ow separation and reattachment, as seen in Figure 8.1. Numerous laminar airfoils also experience such behavior in similar Re regimes [2, 3]. These airfoils can have more than one lift or drag state at a single angle of attack. In Figure 8.1, there appear to be two sets of curves to which the C L (C D ) polar may be attracted. Small perturbations can lead to a transition from one to another of what we shall term bi-stable state. Previous work [4] has shown that state-switching corresponds to 76 the presence or absence of reattachment and that the process is close to two-dimensional or spanwise uniform [5]. Figure 8.1: Bi-stable states in C L C D polars for the E387 wing [4]. Since ow separation and reattachment both strongly aect wing performance, separation control is of clear practical signicance. Active separation control involves introducing an external energy source to supplement that of the boundary layer, and common methods include external and internal acoustic excitation, vibrating wires and aps, blowing, bleeding, and synthetic jets. The basis for ecient energy-based mechanisms to induce separation control is boundary layer receptivity, when a particular disturbance such as an acoustic pressure wave or vortex structure can interact with the boundary layer and establish its signature in the resulting disturbed ow. When the initial disturbances are suciently 77 large, they can grow nonlinearly and result in turbulent ow. When they are small, they can still excite disturbances in the boundary layer, such as Tollmien-Schlichting (T-S) waves [6]. When the boundary layer does separate, the detached shear-layer is susceptible to Kelvin- Helmholtz (K-H) mode instabilities. The unstable waves grow and their roll-up into coherent structures and transition to turbulence are associated with a high degree of unsteadiness and facilitation of the reattachment process as high momentum uid from the external ow is swept into the region close to the airfoil surface [7]. Since the possible ow reattachment is critical to the selection of bi-stable state alternatives, proposed ow control strategies should be targeted at both wall-bounded and free shear layer modes. Several studies have focused on external acoustic excitation as a means to modify the ow and control separation around a wing at various Re and in various ow states. External forcing at single frequency tones has been shown to eectively change wing performance in the range 25,000 Re 800,000 by increasing lift at particular angles of attack [2, 8, 9, 10], tripping the ow from low- to high-lift states [2], diminishing the size of the hysteresis loop in lift-drag curves [2], reducing the tendency toward ow separation [8, 9], and changing the basic behavior of the laminar separation bubble and turbulent boundary layer [11]. The eects of external acoustic excitation have been shown in [8] and [2] to be strongly correlated with wind tunnel test section resonances. Previous literature results on acoustic excitation at low and moderate Re and pre- and post-stall show the dependence of optimum excitation frequencies on Re,, and the dom- inant intrinsic instabilities in the ow. The optimum excitation frequency has been found to increase with increasing Re or increasing [11], while the range of eective excitation frequencies has been found to increase with increasing Re or decreasing [11, 13]. It has also been suggested that the optimum excitation frequencies correspond to the most amplied instabilities in the separated region. For pre- and immediately post-stall , K-H instabili- ties dominate the separated shear region, and so the optimum excitation frequencies have been reported to correspond to these shear layer instability frequencies [10, 12, 13, 14, 15]. 78 For large post-stall , the dominating instabilities are due to free wake vortices, when the optimum frequencies correspond to the vortex shedding frequencies [13, 17]. The optimum excitation frequencies can be related to the Strouhal number, St, given by St = fc U (8.1) where f is the shedding or excitation frequency, c is the chord length, and U is the free stream velocity. Laminar separation was observed to be most eectively reduced when the parameter St=Re 1=2 was between 0.02 and 0.03 based on the excitation frequency [11, 12]. Recalling that the boundary layer thickness over a at plate,/Re 1=2 , theSt=Re 1=2 scaling is directly related to the growth of the boundary layer thickness, . St from Eq. (8.1) uses the chord length as the length scale and is associated with the shedding instabilities. If the scaling is instead the size of the wake behind the body (more suitable for airfoils at high , which act like blu bodies [20]), then the projected height of the airfoil is used as the length scale, and a Strouhal number that takes into account can be given by St = fc sin() U : (8.2) If the scaling is associated with the recirculation time of the separated region, then the height of the separated region (or of the laminar separation bubble) is used as the length scale, and a separation Strouhal number is St s = f s U e (8.3) where s is the momentum thickness of the separated region, and U e is the edge velocity of the boundary layer [18]. An optimum range of St s is reported to be between 0.008 and 0.016 [18]. This paper provides a study on the eects of external acoustic excitation on the forces and ow elds of an E387 wing in a Re regime where pre-stall hysteresis and abrupt switching 79 of bi-stable states occur. At low Re, the aerodynamic performance (C L , C D ) of the E387 and many other smooth airfoils is notoriously sensitive to small changes in environmental and/or boundary conditions, and this study reports the rst of a series of experiments to unambiguously establish the basic ow conditions associated with the force variation, with a view to exploiting this sensitivity for control. If successful, then internal acoustic forcing can be examined for the same wing, and signicant variations in lift and drag could in principle be generated with no moving parts on the wing. II. Materials and Methods A. Experimental Setup Experiments were performed in a closed-loop wind tunnel with octagonal test section of wall-to-wall width 1.37 m, and 5.7 m in the streamwise direction. The empty test-section turbulence level is 0.025% for spectral frequencies between 2Hzf 200Hz in the velocity range 5m/s U 26m/s. Flow uniformity measurements showed no more than 0.5% velocity deviation from the mean velocity for a given cross section [19]. The wing was CNC- machined from a solid aluminum block with AR = 5.8 (span b = 52.7 cm and chord c = 9 cm) with an Eppler 387 airfoil section. The (x;y;z) coordinate system is as follows: x is the streamwise direction,y is the spanwise direction, andz is the normal direction, with the origin at the leading edge and midspan (Figure 8.2). External acoustic forcing was accomplished using a SolidDrive SD1sm speaker, which was attached to the outside of the wind tunnel test section upstream of the wing model. The SD1sm has a usable frequency response range of 60Hz15kHz and uses neodymium magnets and dual symmetrically opposed motors to convert audio signals into vibrations, which are transferred into solid surfaces upon direct contact. Placing the speaker on the outer wall of the wind tunnel test section converts the entire test section into an acoustic chamber. The vibrations from the speaker did not impact the structure of the wind tunnel. Sine waves from a waveform generator were amplied by an adjustable gain Pyle Pro PCA1 80 2 x 15 W stereo power amplier with a frequency response of 20Hz 40kHz3dB and 0.3% total harmonic distortion. The frequency and peak-to-peak voltage amplitude of the sine wave were changed directly from the waveform generator. A 4944 1/4" B&K pressure eld microphone, which has a pressure-eld response of2dB between 16Hz f 70kHz, was used to obtain acoustic measurements for the wind tunnel resonance study. A 4954-B B&K free eld microphone, which has a free-eld response of3 dB between 9Hzf 100kHz, was used to obtain all other acoustic measurements. Both microphones were calibrated using a B&K 4231 Acoustic Calibrator. B. Force Balance Lift and drag forces were measured with a custom cruciform-shaped force balance de- scribed in [19] and [16], placed below the wind tunnel oor. The force balance was capable of measuring lift, drag, and pitching moment. Measurements were averaged over 8000 sam- ples at a sampling rate of 1000 Hz. Careful calibration procedures were performed each day before data acquisition and static calibrations were performed from 0 to 360 mN in 4 mN steps at dierent moment arms. The electromechanical force balance measurement has an expected uncertainty of 0.1 mN. The expected friction coecient on a at plat can be given by C f = 1:328 p Re x (8.4) and the friction drag by F f =C f q 1 S (8.5) where q 1 is the dynamic pressure. For a at plate the same size as the E387 wing at zero degrees of incidence, the expected friction drag would be 11 mN. In force balance measurements, was varied from -10 to 20 then back down to -10 in steps of 1 outside of the hysteresis loop region and in steps of 0.1 in the hysteresis loop region, and for each Re at least three tests were performed and results were averaged. 81 C. Particle Imaging Velocimetry Particle Image Velocimetry (PIV) was used to estimate velocity components (u;w) in the two-dimensional plane (x;z) (Figure 8.2). A Continuum Surelite II dual-head Nd:YAG laser was used to generate pulse pairs separated by exposure times, t = 100300s. The two coaxial laser beams were converted to sheets of slowly varying thickness through a series of convergent-cylindrical-cylindrical lenses. The ow was seeded with 1m smoke particles from a Colt 4 smoke generator and imaged onto a Kodak ES 1.0 1008 x 1018 dual frame CCD array camera. PIV processing used a variant of the customised CIV algorithms described in [22] and [23]. A smoothed spline interpolated cross-correlation function was directly t with the equivalent splined auto-correlation functions from the same data. Obviously incorrect vec- tors that passed by an automated rejection criterion were manually removed and the raw displacement vector eld was reinterpolated back onto a complete rectangular grid with the same smoothing spline function [24]. The spline coecients are dierentiated analytically to yield velocity gradient data. The uncertainty does not depend on velocity magnitude but is xed in fractions of a pixel, but when rescaled to conditions reported here, we may expect uncertainties of 0.55% infu;wg and about 10% in gradient-based quantities, such as the spanwise vorticity ! y = @w @x @u @z ; which is displayed on a discrete colorbar whose step size is set to the measurement uncer- tainty. D. Acoustic Excitation at Constant Amplitude and Constant SPL The eects of dierent excitation frequencies, f e , on lift and drag forces at Re = 40k and 60k were examined. At each Re, a value of 0 immediately preceding the hysteresis loop was chosen. For Re = 40k, 0 = 10 , and for Re = 60k, 0 = 8 . For the acoustic study at constant amplitude,f e from the waveform generator was varied while keeping both the waveform generator peak-to-peak voltage amplitude and the power amplier volume 82 Figure 8.2: Wind tunnel setup. (x, y, z) are streamwise, spanwise, and normal directions. Origin is at leading edge and midspan. constant. Consequently, the SPL at a given location in the wind tunnel was not constant for this portion of the study. For the acoustic study at constant SPL, the power amplier was kept at a constant volume setting while the peak-to-peak voltage amplitude levels from the waveform generator were varied at each f e to yield a constant SPL measured at the wing leading edge and midspan. The excitation frequencies that produced maximum improvements in aerodynamic performance are considered to be optimum excitation, or easily excitable, frequencies and denoted as f e *, which is not meant to denote global optimum values (but could in fact be local optimum values). The frequencies that made the least improvements are considered to be uneasily excitable frequencies and denoted as f o e . After f e * values were determined, 83 the SPL was varied by changing the waveform generator peak-to-peak voltage. In both the constant amplitude and constant SPL studies, the force balance measured lift and drag forces, and PIV yielded ow eld characteristics. In general, the speaker was kept on while changing the frequency during the frequency sweep. However, at and around the optimum frequencies, the speaker was turned o (allowing the ow to return to its nominal state) and then turned back on. This was done to ensure that there was no hysteresis occurring. E. Wind Tunnel Resonance Since acoustic amplitudes in a closed box vary greatly in space, spatial maps of the test section response were measured. The B&K 4944 pressure eld microphone was placed inside the empty wind tunnel test section without ow and traversed in 2 cm steps in (x;y;z) to form three planes that would intersect the wing if it were in place. The planes traversed by the microphone were the yz-plane at quarter chord (x=c = 0.25), xz-plane at midspan (y=c = 0.0), and xy-plane at leading edge (z=c = 0.0) (Figure 8.3). The power amplier volume and waveform generator peak-to-peak voltage level were kept constant, and four excitation frequencies were used: two values of f e * and two values of f o e . SPL values were averaged over 15,000 samples at a sampling rate of 2500 Hz. Figure 8.3: Wind tunnel resonance measurement planes: a) yz-plane, b) xz-plane, c) xy- plane. 84 III. Results A. Eppler 387 Performance at Low Reynolds Numbers Figure 8.4 shows the abrupt increases in lift and decreases in drag for the E387 at par- ticular pre-stall values, which decrease as Re increases. Counter-clockwise hysteresis also occurs so that the high-lift state is preserved longer as the wing incidence is decreased. Changes between separated ow and reattached ow conditions over the suction surface of the wing cause the jumps between bi-stable states, as observed from PIV ow eld data (not shown here). More than an 85% dierence in L=D can occur over a 0.1 change in (atRe = 40k, the dierence in L=D is 87% between 12.3 and 12.4 , Figure 8.4b). B. Acoustic Excitation at Constant Amplitude At each, Re = 40k and 60k, distinct increases in L=D can be observed at particular excitation frequencies, as indicated by Figure 8.5, which plots the percent change in L=D with varying acoustic excitation frequencies. Maxima in (L=D) occur at f e * = 525Hz and 660Hz for both Re = 40k and 60k, but at 800Hz there is no improvement for Re = 40k. The range of f e * is larger for higher Re, which agrees with the observations in [11, 13]. However, for the same excitation conditions, the L=D improvement is greater at Re = 40k where (L=D) reaches 74% for f e = 525Hz. The maximum (L=D) atRe = 60k is only 56% atf e = 800Hz. The correspondingC L (C D ) and L=D curves at the indicated f e * values are shown in Figure 8.6. When the ow is excited at f e *, hysteresis is largely eliminated for both Re = 40k and 60k. Excitation at f e * removes most, but not all, of the drop in L=D() at moderate , and the magnitude of the improvement varies with . The higher Re case shows the closest achievement of a at, high (L=D) over a broad range of (Figure 8.6d). The original PIV images with densely-seeded ow show dark lines that are the path lines of uid originating in the relatively particle-poor boundary layer. 85 Figure 8.4: C L (C D ) curves (a) and L=D() curves (b) for the E387 at Re = 30k (circles), 40k (triangles), 50k (squares), and 60k (diamonds). Error bars are the standard deviation from multiple tests. 86 Figure 8.5: (L=D) for dierentf e at constant amplitude at = 10 at Re = 40k (top) and at = 8 at Re = 60k (bottom) with f e * values indicated. 87 Figure 8.6: C L (C D ) forRe = 40k (a) andRe = 60k (b) andL=D forRe = 40k (c) andRe = 60k (d) for unexcited ow (dotted gray line + circles), f e = 525Hz (black line + triangles), f e = 660Hz (gray line + squares), and f e = 800Hz (black line + diamonds). 88 Figure 8.7 compares the unforced (left column) and forced (right column) ows for = 8 at Re = 60k. The unforced ow separates at a well-dened location about c/4 from the leading edge. In the forced ow the path line lifts slightly and late, and then marks a series of dark spots located above the airfoil surface. There is no obvious sign of a large- scale detachment. The time-average spanwise vorticity shows that in fact separation has occurred close to the leading edge but that the ow then reattaches to form, in the mean, a recirculation zone that is large in both x and z. The recirculation zone attaches stably to the suction surface. It can be termed a laminar separation bubble. Note that this bubble is much larger and occupies a dierent chordwise location than the well studied laminar separation bubble that appears on the SD7003 airfoil [21] and which, by contrast, has almost no dynamical signicance. Figure 8.7: Separation and spanwise vorticity for ow over E387 at = 8 and Re = 60k without forcing (left) and with forcing at f e = 800Hz (right). Vorticity elds are superim- posed on a uctuating velocity vector eld with unity scaling. 89 C. Acoustic Excitation at Constant SPL The eects off e on (L=D) at constant SPL at Re = 60k and 40k together with the SPL variation withf e in an empty wind tunnel measured at x = 0, y = 0, and z = 0 are shown in Figure 8.8. Figure 8.8c shows that the tunnel acoustic pressure is not uniform as a function of frequency, and measurements taken in the normal position of the wing show variations in SPL of up to 30dB. These variations are due to constructive and destructive interference of primary and re ected waves in the tunnel test section which has no special acoustic treatment of the walls. In acoustic wave elds, the gradient of pressure,j5pj, is maximum where the pressure uctuations cross the zero line, and this consequently is where the maximum induced particle velocity is found, 90 out of phase with the pressure uctuations. When the acoustic wave eld is dominated by standing waves (ultimately caused by the container geometry), nodes that correspond to zero crossings will have the lowest r.m.s. values. The high r.m.s. values, by contrast, occur where the pressure uctuates between maximum and minimum but where the pressure gradient is close to zero. The anti-resonance regions in f e are where (L=D) is highest. The ow is most easily switched to its highL=D state when the acoustic wave induced velocity eld has its highest amplitude. Note that the sensitivities in Figure 8.8a, b are adjusted for constant amplitude SPL. The resulting performance curves from constant SPL excitation over the range of at the three most excitable frequencies of Figure 8.8b are shown in Figure 8.9. Hysteresis is again eliminated, although the original dips inL=D (gray curves in Figure 8.9) are not completely eliminated. Not all experience the same magnitude of L=D improvement, similar to the results from acoustic excitation at constant amplitude. Of the three values of f e * (415Hz, 520Hz, and 675Hz), the lower two produce better overall lift-drag curves. Qualitative results of the ow eld and corresponding spanwise vorticity eld at two values of f e * (415Hz and 520Hz) and two values of f o e (445Hz and 550Hz) are shown in Figure 8.10 and Figure 8.11, respectively. The particle images for the two f o e (Figure 8.10 b, d) have the same dark separation line as seen before the wing changes to the high-lift, low-drag state in normal, unexcited 90 Figure 8.8: Eect of f e on (L=D) at Re = 60k (a) and 40k (b) and corresponding wind tunnel SPL response measured at (x = 0, y = 0, and z = 0) (c). conditions. At the two f e * (Figure 8.10 a, c), the dark line, previously detached from the airfoil surface, has moved closer to the surface, and the previously-noted vortical structures can be seen close to the surface starting at x/c 0.3, most notably at 520Hz (Figure 8.10 c). These vortical structures move along the suction surface of the wing from leading edge 91 Figure 8.9: Eect of f e on C L and C D (a) and L=D (b) at Re = 40k for unexcited ow (dotted gray line + circles), f e = 415Hz (black line + triangles), f e = 520Hz (gray line + squares), and f e = 675Hz (black line + diamonds) at constant SPL = 75.5dB 92 Figure 8.10: Raw PIV images for ow at Re = 40k with acoustic excitation atf e * (a, c) and f o e (b, d) (SPL = 75.5dB). Figure 8.11: Spanwise vorticity elds for ow at Re = 40k with acoustic excitation at f e * (a, c) andf o e (b, d) (SPL = 75.5dB). Fluctuating velocity vectors are scaled by a factor of 4. 93 to trailing edge as observed through a time series of acquired images. The spanwise vorticity elds (Figure 8.11), obtained from the raw PIV images (Figure 8.10), reveal that exciting the ow at f o e has no eect on the ow, which remains separated over the aft half of the airfoil, but excitation at f e * produces a region of circulation over the front half of the wing, corresponding to a reattached ow state. D. SPL Dependence The results of varying SPL on wing performance for a value of f e * (520Hz) and f o e (550Hz) are shown in Figure 8.12. Figure 8.12 shows that varying the forcing amplitude at f e * changes the magnitude of (L=D) and the range of over which the change is seen. Changes inL=D can be obtained by forcing atf o e but require a much higher amplitude. For ow at Re = 40k and = 10 , a 77%L=D improvement is achieved with an SPL = 77.8dB at f e * = 520Hz, but the same improvement requires a much higher SPL of 91.8dB at f o e = 550Hz. A 14dB change in SPL is a 0.1mPa change in pressure. Figure 8.12: L=D() for acoustic excitation at f e * = 520Hz (a) and f o e = 550Hz (b). Varying SPL can lead to quite smooth variations in (L=D), and Figure 8.13 shows that SPL can be used as a control parameter for (L=D) that has its own hysteresis loop, not in , as in Figures 8.4, 8.6, 8.9, and 8.12, but in SPL. In Fig. 8.13, the excitation is held 94 constant at f e * = 520Hz. The separated and reattached ow states are indistinguishable from those achieved with varying . Figure 8.13: Hysteresis of L=D and SPL at Re = 40k and = 10 . f e is held constant at 520Hz. Spanwise vorticity superimposed on a uctuating velocity eld is shown for the four indicated points on the hysteresis loop. Fluctuating velocity vectors have unity scaling. E. Wind Tunnel Resonance Figures 8.14 8.16 show the spatial variation in measured SPL for constant amplitude forcing of the speaker/tunnel wall. The two f e * cases are shown in a and b and the two f o e 95 Figure 8.14: Wind tunnel resonance in the yz-plane (normal to the chord and the mean ow). cases are shown in c and d in each case. Figure 8.14 shows the SPL response in the yz-plane, Figure 8.15 the xz-plane, and Figure 8.16 the yz-plane. The SPL varies signicantly (by 15dB) over length scales that are comparable to the span (b 6c) in thefy, zg plane normal to the wing and across the free stream (Figure 8.14). The acoustic source is at the tunnel wall in z + and direct re ections will come from the opposite wall in z . The corresponding distribution infx, zg (Figure 8.15) is more uniform, with only minor variation in x, where there are no direct re ectors. There are also smaller variations in z, which suggests that the large variations infy, zg of Figure 8.14 come also 96 Figure 8.15: Wind tunnel resonance in the xz-plane (parallel with the chord and the free stream). from re ections in y. The wind tunnel test section is octagonal and so this is expected. In Figure 8.16 thefx, yg plane lying coplanar with the wing chord at = 0 has amplitude variations in y that are similar to those of Figure 8.14 and rather small variations in x. In reverse order, a region of SPL min occurs at y = 0 over all x in Figure 8.16 for f e *. In Figure 8.15 this trough is at z = 0, uniform in x. In Figure 8.14, the minimum is sharp at z = 0, y = 0. In this particular wing/facility geometry, spatial minima in SPL occur on the wing at mid span at this particular frequency. This SPL min is associated with a maximum eciency of ow modication. The obverse is also true: spatial maxima occur at 97 Figure 8.16: Wind tunnel resonance in the xy-plane (parallel with the chord and with the span). the wing center (and ow measurement point) for the frequencies f o e that are least eective in disturbing the ow. IV. Discussion Previous literature results suggest that the values off e * correspond to the most amplied instabilities in the separated shear layer (K-H instabilities) for pre- and immediately post- stall. If the values off e * are used in the calculation of St in Eq. (8.1) and then normalized byRe 1=2 , the optimum range ofSt=Re 1=2 for Re = 60k is approximately 0.015St=Re 1=2 98 0.035, and for Re = 40k the optimum range is 0.025St=Re 1=2 0.045, as shown in Figure 8.17. The reported optimum range ofSt=Re 1=2 between 0.02 and 0.03 [11, 12] coincides more with the higher Re results here. The fact that results for both Re do not overlap inSt=Re 1=2 suggests that the correct scaling has not been identied. Figure 8.17: (L=D) as a function of St and St=Re 1=2 . A shear layer frequency, f s , can be obtained given a mean advection speed, U a , and the spatial separation of the vortical structures in the free shear layer, x s (observable from instantaneous spanwise vorticity elds): f s = U a x s (8.6) 99 where x s is the average separation between two adjacent vortical structures in the x-direction, and U a , is the time-averaged streamwise velocity at the location of the vortical structures. For the case of Re = 40,000, using the average distance between the centers of the distinct vortices for x s yields f s = 445 125Hz, where the uncertainty comes from using dierent adjacent vortices and the uncertainty in location of the vortex centers. While this range of f s encompasses the observed f e * = 520Hz, the large uncertainty in f s suggests that vortical structures in the shear layer are not shed regularly when the ow is separated. Although the vortical structures can be detected from vorticity elds, none can be clearly seen in the raw PIV images. In contrast, when the ow is forced atf e *, distinct structures are evident in the raw PIV images, like Figure 8.10c. Ifx s is the spatial separation between the dark patches, andU a is calculated from the time-averaged velocity eld at thefx, zg location of the corresponding structures, then another shedding frequency can be calculated from Eq. (8.6). For the case of forcing at f e * = 520Hz, the average f s is equal to 1110 30Hz. This value of f s is a second harmonic of 550Hz 30Hz, which equals the observedf e * = 520Hz. The uncertainty in this shedding frequency also comes only from using dierent adjacent vortices and the uncertainty in location of the vortex centers. The noticeably smaller uncertainty for f s for reattached ow implies that the shedding is much steadier than for the separated case. The agreement of estimated f s with the observed f e * suggests that forcing at intrinsic most-amplied frequencies of the free shear layer could be the most eective way to control the ow. However, if this were the case, the preferred St, or even range of St=Re 1=2 , would not vary with Re, but they do. Moreover, the proposed physical mechanism based on a resonance with f s entirely ignores the tunnel resonance dependence. It is most likely that the variations in eective acoustic forcing, with spatial location and with frequency, are coexisting with preferred modes in the natural (unenclosed) system. Since the full wind tunnel/wing system is neither general nor simple, there may be limited benet in disentangling the various contributors whose relative in uence is likely measured by continuous amplitude variation, rather than having either one be completely responsible. 100 The quite subtle amplitude and frequency sensitivities and their dependence on the fa- cility could explain much of the well known variation between facilities in aerodynamic performance of smooth airfoils. The test section size and shape will determine a response map for a range of frequencies for some given acoustic source. The geometry of that response map relative to the physical wing in the tunnel will strongly aect the frequency response and sensitivity of the system. When acoustic sources include not only external noise but also the wind tunnel fan and motor assembly, it is small wonder that observations vary. Of more practical interest will be responses that are not strong functions of re ected acoustic waves, and this could be arranged outside of a tunnel in free ight or in a specialized open section, anechoic tunnel. Since we would like to pursue the possibility of localized forcing from sources inside the wing, it is possible that alternative experiments could succeed where re ections from low amplitude, local forcing are not strongly in uenced by re ection. V. Conclusion The distinct jumps between bi-stable states and pre-stall hysteresis of the E387 wing, particularly in the Re regime 40,000Re 60,000, can be either provoked or eliminated by acoustic excitation at optimum excitation frequencies, yielding more than a 70% increase in L=D. Forcing at these optimum frequencies also completely changes the global ow over the wing by reattaching the formerly separated ow and forming a laminar separation bubble. Improvement by acoustic excitation is a function of f e and SPL, and in this experiment, optimum excitation values f e * correlate with wind tunnel anti-resonances. While f e * and associatedSt* are not inconsistent with previous literature results, the correct Re scaling is not apparent. The documented dependence on test section geometry and acoustic resonance could explain many of the previous discrepancies in the literature at similarRe, corroborating and extending the original observations also found in the literature. Further experiments using local on-wing forcing may help to distinguish the dierent eects in an open ow. 101 Paper II References [1] McGhee, R. J., Walker, B. S., and Millard, B. F., Experimental Results for the Eppler 387 Airfoil at Low Reynolds Numbers in the Langley Low-Turbulence Pressure Tunnel, NASA TM-4062, 1988, pp. 25-26. [2] Grundy T. M., Keefe G.P. and Lowson M.V., \Eects of Acoustic Disturbances on Low Re Aerofoil Flows." In Fixed and Flapping Wing Aerodynamics for Micro Air Vehicle Applications, Vol. 195, pp. 91112. Reston, Virginia: American Institute of Aeronautics & Astronautics, 2001. [3] Simons, M., Model Aircraft Aerodynamics, 4th Ed., Special Interest Model Books, Poole, 1999. [4] Spedding, G. R. and McArthur, J., \Span Eciencies of Wings at Low Reynolds Num- bers," Journal of Aircraft, Vol. 47, No. 1, 2010, pp. 120-128. [5] Yang, S. L. and Spedding, G. R., \Spanwise Variation in Wing Circulation and Drag Measurement of Wings at Moderate Reynolds Number," Journal of Aircraft, Vol. 50, No. 3, 2013, pp. 791-797. [6] Reshotko, E. Boundary-Layer Stability and Transition. Ann. Rev. Fluid Mech. 8, 1976, pp. 311-349. [7] Lin, J. C. M., Pauley, L. L., \Low-Reynolds Number Separation on an Airfoil," American Institute of Aeronautics and Astronautics, Vol. 34, No. 8, 1996. [8] Zaman, K. B. M. Q., Bar-Sever, A., \Eect of Acoustic Excitation on the Flow Over a Low-Re Airfoil," Journal of Fluid Mechanics, Vol. 182, 1987, pp. 127148. [9] Ahuja, K.K., Whipkey, R.R., and Jones, G.S., \Control of Turbulent Boundary Layer Flows by Sound," AIAA Paper No.1983-0726, 1983. [10] Ahuja, K. K., Burrin, R. H., 1984, \Control of Flow Separation by Sound," AIAA Paper 84-2298, Oct. 1984. 102 [11] Zaman, K.B.M.Q., McKinzie, D.J., \Control of Laminar Separation Over Airfoils by Acoustic Excitation," American Institute of Aeronautics and Astronautics, Vol. 29, No. 29, 1991. [12] Yarusevych, S., Kawall, J. G., Sullivan, P. E., 2002, \In uence of Acoustic Excitation on Airfoil Performance at Low Reynolds Numbers," 23rd International Council of the Aeronautical Sciences Congress, 813 September, Toronto, Ontario, Canada. [13] Hsiao, F. B., Jih, J. J., and Shyu, R. N., \The Eect of Acoustics on Flow Passing a High-AOA Airfoil," Journal of Sound and Vibration, Vol. 199, No. 2, 1997, pp. 177-188. [14] Zaman, K.B.M.Q., \Eect of Acoustic Excitation on Stalled Flows Over an Airfoil," American Institute of Aeronautics and Astronautics, Vol. 30, No. 6, 1992, pp. 1492-1499. [15] Nishioka, M., Asai, M., Yoshida, S., \Control of Flow Separation by Acoustic Exci- tation," American Institute of Aeronautics and Astronautics, Vol. 28, No. 11, 1990, pp. 1909-1915. [16] McArthur, John. \Aerodynamics of Wings at Low Reynolds Numbers," Ph.D. Disser- tation, Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA, 2007. [17] Huang, L.S., Maestrello, L., Bryant, T. D., \Separation Control Over an Airfoil at High Angles of Attack by Sound Emanating From the Surface," AIAA Paper No. 87-1261, 1987. [18] McAulie, B. R., Yaras, M. I., \Transition Mechanisms in Separation Bubbles Under Low- and Elevated-Freestream Turbulence," Journal of Turbomachinery, Vol. 132, No. 1, 2010. [19] Zabat, M., Farascaroli, S., Browand, F., Nestlerode, M., Baez, J., \Drag Measurements on a Platoon of Vehicles," Research Reports, California Partners for Advanced Tran- sit and Highways (PATH), Institute of Transportation Studies, UC Berkeley, 1994. (doi: 10.1146/annurev. .15.010183.001255) [20] Roshko, A., \On the Drag and Shedding Frequency of Two-Dimensional Blu Bodies," NACA Report TN 3169, 1954, pp.1-29. [21] Radespiel, R., Windte, J., and Scholz, U. \Numerical and Experimental Flow Analysis of Moving Airfoils with Laminar Separation Bubbles," American Institute of Aeronautics and Astronautics, Vol. 45, No. 6, 2007, pp. 13461356. [22] Fincham, A. M. and Spedding, G. R., \Low cost, high resolution DPIV for measurement of turbulent uid ow," Experiments in Fluids, Vol. 23, 1997, pp. 449-462. [23] Fincham, A., and Delerce, G., \Advanced optimization of correlation imaging velocime- try algorithms," Experiments in Fluids, Vol. 45, No. 29, 2000, pp. 13-22. 103 [24] Spedding, G.R. and Rignot, E.J.M., \Performance Analysis and Application of Grid Interpolation Techniques for Fluid Flows," Experiments in Fluids, 15 (1993), 417-430. 104 Paper III 105 Chapter 9 Passive Separation Control by Acoustic Resonance Yang, S. L. and Spedding, G. R. Aerospace and Mechanical Engineering Department University of Southern California Los Angeles, California 90089-1191 Experiments in Fluids, Vol. 10, No. 54, pp. 1-16, 2013. At transitional Reynolds numbers, the laminar boundary layer separation and possible reattachment on a smooth airfoil, or wing section, is extremely sensitive to small variations in geometry or in the uid environment. We report here on the results of a pilot study that unexpectedly adds to this list of sensitivities. The presence of small holes in the suction sur- face of an Eppler 387 wing has a transformative eect upon the aerodynamics, by changing the mean chordwise separation line location. These changes are not simply a consequence of the presence of the small cavities, which by themselves have no eect. Acoustic resonance in the backing cavities generates tones that interact with intrinsic ow instabilities. Prac- tical consequences for passive ow control strategies are discussed together with potential problems in measurements through pressure taps in such ow regimes. 106 I. Introduction An emerging generation of practical Micro-Air Vehicles (MAVs) operates at ight speeds and characteristic length scales that brings them into an especially challenging ight regime, where abrupt changes in ight performance can result from very small, and often uncon- trolled, changes in the geometry and/or environmental conditions. For example [1], [2], and [3] show data from a number of facilities with factors of two variation in sectional drag coe- cient (c d , withc l the corresponding sectional lift coecient) at moderate, pre-stall, geometric angles of attack, . Experimental diculties are exacerbated because the relatively small forces (of mN or less) are dicult to measure directly on standard force balance equipment, so scanning arrays of pitot tubes are used to measure the streamwise momentum defect to estimate the prole drag, and it is possible that variations in local sectional properties across the span could increase the measurement uncertainty greatly. [4] showed that at a chord-based Reynolds number, Re 8 10 4 (Re = Uc=, where U is the ight speed, c is the wing chord, and is the kinematic viscosity), there were measurable variations in c d across the span of an E387 wing, but that they were not of sucient magnitude to account for disparities in the literature. The E387 airfoil is a well-studied airfoil, originally designed for sailplanes at moderate Re ( 2 10 5 ). The E387 has been referred to as the low Re calibration standard [5], since at low Re (< 1 10 5 ), it experiences laminar separation without reattachment, laminar separation with turbulent reattachment, and turbulent trailing edge separation. Although these phenomena that occur at lowRe do not make the E387 a strong candidate for practical MAV design, it can be used as a testbed for the study of transitional phenomena where small variations can have a large dynamical eect. Performance data for the same E387 wing used in [4] that included a lower range of Re values than customary in typical wind tunnel studies (Re ranged from 1 8 10 4 ) [6], [7] showed that at moderate = 4 8 the ow can be in either one of two states. In one state, a lower envelope of c l (c d ) curves marks the performance characteristics where trailing edge 107 separation gradually moves forward on the wing, and c d increases rapidly for > 4 , while c l rises rather slowly to maximum values of about 0.8. This low-lift state will be referred to as SI. In a second state, the initial separation point has moved close to the leading edge, but the ow reattaches before the trailing edge. This high-lift state will be referred to as SII. Consequently, at any given c d , c l has a value about 40% higher than in SI. The lower-lift envelope of SI is occupied by all points at Re 2 10 4 , and the upper envelope of SII is characteristic of polars for Re 8 10 4 . In between, and at interim 4 10 , the ow can separate either close to the leading edge, or further aft, and the global aerodynamic force coecients can match with either state SI or SII. The state switch SI{SII has hysteresis so the ow at any depends on the time history, or previous state. The peculiar (when contrasted with the usual, simple C-shape curves at higherRe) shape of the lift-drag polar of the E387 at intermediate Re is not actually restricted to this single prole shape, but is quite characteristic of a class of airfoils that have smooth rounded leading edges with some minimum prole thickness. 74 of 94 airfoils in [1], [2] and [3] at Re< 10 5 have this polar shape, with non-unique values of c l over some range of c d , and 18 of 31 proles compiled by [8] have it too. The polar shape is due to the abrupt forward movement of the separation line, with an accompanying reduction in c d and increase in c l [11]. Absent any other evidence at these Re, it is reasonable to assume that the same dynamics occur in all these airfoil/wing systems. The acute sensitivity of an airfoil conguration at intermediate Re (E61 at Re = [25, 35, 50, 60]10 3 ) was noted by [9], who also showed a dependence on the background acoustic environment. It was further demonstrated that the most eective forcing frequencies coincided with resonant modes in the tunnel test section, and therefore that sensitivity to ambient acoustic noise will be facility-dependent. The same result was shown in [7], where control of the SI{SII transition and its hysteresis loop around transitional could be achieved at resonant modes in the tunnel. The spatial distribution of the sound pressure levels was not uniform but local minima in SPL at the wing location were associated with the most eective forcing frequencies. 108 In continuation of the acoustic forcing tests reported in [7], the next step in testing the response to acoustic forcing was to embed arrays of small speakers inside the wing prole, and to do that, arrays of small holes were drilled in a wing lid that enclosed cavities to house the speakers. In this short note, we report on how the presence of the holes themselves profoundly changes the properties of the wing, locking the ow onto the high lift state SII. II. Materials and Methods A. Wing Model The aluminum wing had an Eppler 387 prole section (Fig. 9.1) with aspect ratio of 6 (span of 54 cm and chord of 9 cm). The wing was custom designed to consist of a base and a lid, which t together with a tongue-and-groove connection. The wing, which originally started as a solid piece of aluminum, was manufactured by electrical discharge machining wire cutting (wire EDM), which is a thermal mass-reducing process that uses a constantly moving wire to remove material by rapid, controlled, and repetitive spark discharges. The removed particles are ushed with a dielectric uid, which also regulates the discharge and keeps the wire and metal cool. The tolerance on the wire EDM is0.05 mm. The lid of the wing was 1 mm thick and contained 180 0.5 mm diameter holes arranged in six spanwise arrays with 30 holes each. The six spanwise arrays were located at streamwise locations x=c = 0:1; 0:2; 0:3; 0:4; 0:5; 0:6. The base of the wing contained 180 cavities that were aligned with the holes in the lid; these cavities were connected to each other through spanwise channels for wiring, and ultimately to an exit port aft of the quarter chord mount point. B. Experimental Setup Experiments were performed in the low turbulence Dryden wind tunnel at USC, where the empty test-section turbulence level is 0.025% for spectral frequencies between 2f 200 Hz in the velocity range 5U 26 m/s. Measurements on ow uniformity yielded no more than 0.5% velocity deviation from the mean for a given cross section [10]. The wing assembly was mounted vertically to a sting at one wingtip, at the quarter-chord point (Fig. 9.2). The 109 (a) (b) Figure 9.1: (a) E387 wing with 180 0.5-mm diameter holes in the 1-mm thick lid, and (b) prole view of the lid of the wing. ow is known to be sensitive to small disturbances (e.g., [3, 30]) and placing the mount point at one tip connes this particular disturbance to a region that at moderate is dominated by the induced ow of the tip vortex [4], and is therefore not closely associated with determining separation points on the central part of the wing, where most measurements are focused. As noted in the gure, coordinate axesfx;y;zg run streamwise, spanwise and vertically, respectively, with origin at the midspan leading edge at = 0 . Time-averaged lift and drag forces were measured with a custom force balance with measurement uncertainty of 0.1 mN, described in [10], [11], [6]. C. Particle Imaging Velocimetry Particle Image Velocimetry (PIV) was used for ow visualization and estimation of veloc- ity componentsfu;wg infx;zg. A Continuum Surelite II dual-head Nd:YAG laser generated 110 Figure 9.2: Wind tunnel setup. (x;y;z) are streamwise, spanwise, and normal directions. Origin is at leading edge and midspan. pulse laser pairs separated by exposure times, t = 100 300s. A series of convergent- cylindrical-cylindrical lenses converted the two laser beams into slowly-varying thickness laser sheets. The ow was seeded with 1m smoke particles from a Colt 4 smoke generator and captured by an Imager Pro X 2M (1648 x 1214 x 14-bit) camera. PIV processing was based o of the customized CIV algorithms described in [12] and [13] in which a smoothed spline interpolated cross-correlation function was directly t with the equivalent splined auto-correlation functions from the same data. Obviously wrong vectors that passed by an automated rejection criterion were manually removed and the raw displacement vector eld was reinterpolated back onto a complete rectangular grid with the same smoothing spline function [14]. The spline coecients are dierentiated analytically to generate velocity gradient data. The uncertainty is in fractions of a pixel, and when re-scaled 111 to the test conditions reported here, expected uncertainties are 0.5{5% infu;wg and 10% in gradient-based quantities, such as the spanwise vorticity: ! y = @w @x @u @z : D. The Imperfect Test Environment No physical experiment conforms perfectly to its nominal conguration. Here we docu- ment some of the departures from ideal test conditions. Comparative tests indicate that none of these are critical in in uencing the basic phenomena described here, but ows at transi- tionalRe are known to be sensitive to a number of sometimes poorly- or partially-controlled environmental variables, and it is useful to at least know what some are. Background Acoustic Environment This paper concerns the sensitivity of wing aerodynamics to acoustic perturbations that occur in a background non-zero acoustic ambient. No special attempts were made to modify the tunnel geometry to tailor the background acoustics, either from obvious sources (such as the downstream fan) or from re ections at test section walls. The tunnel is not an anechoic chamber. Acoustic power spectra were measured with a shielded 1/4" B&K microphone oriented normal to the mean ow direction. Each power spectrum is an average of at least 10 individual spectra, each taken from 10,000 samples and sampled at 10,000 Hz. The acoustic power spectra with and without ow in the empty test section were measured at the equivalent midspan, quarter chord, and top surface of the wing (with no wing present), and are shown in Fig. 9.3. A constant peak at 300 Hz, with no wind on, measures the background environmental noise. Though striking by itself, its amplitude is small when compared with peaks that appear when the wind is on. These include an increase in the 300 Hz component and another at about 360 Hz. These will come from the fan noise combined with self-noise of the microphone head. Though shielded, zero self-noise cannot be obtained. The background acoustic spectrum is subtracted from all subsequent spectra. 112 Figure 9.3: Acoustic power spectra in an empty wind tunnel with ow at U = 6.7m/s (Re = 40k), and without ow. Note the change in axis scales. Wing Tip De ections The maximum absolute de ection amplitude, z 0 , was measured at the free wing tip (y=b = 1:0) from image sequences taken from the camera mounted on top of the wind tunnel. z 0 () was measured for Re = 40k and 60k for the wing with all holes covered (the solid wing), and results are shown in Fig. 9.4. At Re = 40k before and after the low- to high-lift transition, the wing tip de ection z 0 =c 0:11%. At Re = 60k, a dierence in wing tip de ection is observed when SI-SII transition occurs. In the low-lift (SI) state, z 0 =c 0:2%, and after transition to the high-lift (SII) state, z 0 =c 0:07%. The dierence is likely due 113 to the increased in uence of unsteady aerodynamic forces, which are not resolved in this experiment, but are implied by the variations in spanwise vorticity, to be shown later. Figure 9.4: Wing tip de ections normalized by the chord (z 0 =c) atRe = 40k and 60k for the wing with all holes closed (ie, solid wing). Surface Roughness Surface roughness of sucient height can act as a boundary layer trip, and wings with dierent roughness can act as though they operate at dierent Re. The nontrivial eect in ow separation and transition at these Re has been well documented [15]. The surface roughness near the wing tip (y=b0:99) was measured at dierent points along the chord by a Ambios Technology XP Stylus Proler with a vertical resolution of 1:5 10 9 m. For a given chordwise 5-mm segment, multiple scans were made where each data set consisted of 37,300 points. A 5th-order polynomial curve was t to each data set and removed to obtain the relative, small-scale surface roughness. At x=c 0:3, where the wing is visibly and tangibly the smoothest, the maximum measured roughness height, h r was 6.3 microns, At x=c 0:9, where the wing is the roughest, the maximum measured h r 15 microns. 114 The roughness may be compared with the likely boundary layer thickness,, which for a at plate at zero incidence is, = 5 r x U ; (9.1) wherex is the distance from the leading edge. Atx=c = 0:3, = 1:3 mm, andh r = = 510 3 . At x=c = 0:9, = 2:1 mm, and h r = = 0:01. Thus, the surface roughness is small compared with a boundary layer thickness. We will be considering the eect of small cavities of 0.5 mm diameter, which could also be argued to be acting as roughness elements. However the roughness is also a small fraction of the cavity diameter, 1:3% and 3% at the smoothest and roughest points on the wing, respectively. Since the hole geometry is at least two orders of magnitude larger than the surface roughness, its eect can be clearly distinguished from the small-scale distributed manufacturing roughness. Subsequent tests described later will isolate the geometric surface and volumetric eects. III. Results A. Open Holes on the Suction Surface Figure 9.5 compares the aerodynamic performance at Re = 40k and 60k of the baseline solid wing and a second wing with the same geometry except for the presence of the arrays of 0.5 mm diameter holes. AtRe = 40k, the baseline wing has the characteristic jump from SI to SII at = 12 (Fig. 9.5a, b). This transition is marked by an increase in C L from 0.7 to values above 1, and a reduction in drag coecient by about 10%. AtRe = 60k, the jump from SI to SII occurs at = 9 (Fig. 9.5c, d), with similar C L increase and drag reduction. The wing with holes (also referred to as the perforated wing) has no such transition and is in the upper lift state, SII, at all . Consequently, before the SI-SII transition for the baseline wing (0 12 at Re = 40k and 0 9 at Re = 60k), the perforated 115 wing has a higher eciency, as measured by L=D, than the baseline case. The performance improvement is entirely passive, with no active energy input. Figure 9.5: The eect of open holes on the aerodynamic performance of an E387 wing at Re = 40k (a, b) and 60k (c, d). B. Surface Geometry and Cavity Flows A comparison was made between two cases where the holes were covered dierently. In one case, the holes were lled and sealed with modeling clay, and in the other case, the holes were covered and sealed at the underside of the lid so that each hole became a cavity with aspect ratiow=h = 1=2. The resulting forces on the wing for the two cases show that a wing 116 Figure 9.6: Small surface cavities have no eect on wing performance (Re = 40k). populated with small, sealed cavities performs just as though the cavities were absent, which is the same as the baseline wing (Fig. 9.6). Any eect of the unsealed cavities on the ow depends upon the presence of the backing cavities, and the holes in the surface themselves do not act like roughness elements (unsurprising given their size, as previously noted), and their geometry alone does not appear to generate secondary uid ows that then aect the boundary layer separation and/or reattachment. C. Chord Location The SI{SII transition is triggered by a forward chordwise movement of the mean sepa- ration line. The early separation leaves sucient time, in fractions of (c=U), so that tran- sition to turbulence and reattachment can occur. Chordwise local disturbances could ef- fectively promote this transition, and this concept was tested by leaving open individual spanwise rows of holes at a givenx=c location, while closing all others. For a row with open holes, 15 holes spaced 3.2 cm apart (y=b = 0:12) were left open across most of the span 117 (0:83y=b 0:83). TheC L (C D ) andL=D curves for the wing with selected rows of open holes located at x o =x=c = 0:1; 0:2; 0:3; 0:4 are shown in Fig. 9.7 (Re = 40k) and Fig. 9.8 (Re = 60k). For bothRe cases, single rows of open holes at specic chord locations allow the envelope of C L (C D ) and (L=D)() to be populated with results that are intermediate between the extremes where all holes are open or all closed. As the row location moves closer to the leading edge, the performance curves are pulled toward the SII state, as generated with all holes open. When x o = 0:1, there is no abrupt jump in either L or D and the curves lie close to SII over all. AtRe = 40k, the curves once again show sharp jumps between states when x o 0:2, and for x o 0:3, the performance curves are much closer to, though still measurably improved upon, the original solid baseline wing. At Re = 60k, the curves show the sharp jumps when x o 0:3. D. Open and Closed Exit Port The original design of the wing included two exit ports at the right wing tip which were meant to be used for electrical connections. The results shown in Figs. 9.59.8 were for the wing with open exit ports. Figure 9.9 shows theL=D curves for the wing with dierent rows of open holes with closed exit ports at Re = 40k and 60k. At the lower Re (Fig. 9.9a, b), the family of curves crossing from SI to SII is restricted to 7 in the pre-stall regime. If the eect of the holes is to trigger separation (and then re-attachment) towards the leading edge, then they are less eective when the large cavity volume is closed o. This phenomenon does not occur, however, at the higher Re, where the curves are nearly the same for both open and closed exit ports (9.9c, d). While the eect of opening and closing the exit port aects the performance at the lower Re, this variation was not pursued in further detail. The remaining results were obtained with open exit ports. A possible reason for the performance improvements from the open holes is acoustic resonance in the chambers that back them. If the resonant acoustic modes are in the ap- propriate frequency range, then intrinsic ow instabilities could be amplied as the acoustic waves impinge upon the boundary layer or separated shear layer. 118 Figure 9.7: Single rows of holes at varying x=c allow the envelope of performance curves to be populated between SI and SII at Re = 40k. 119 Figure 9.8: Similar variation in performance curves between SI and SII occur for dierent rows of open holes at Re = 60k. 120 Figure 9.9: L=D for dierent spanwise rows of open holes with open and closed exit ports at Re = 40k (a, b) and Re = 60k (c, d). E. Acoustic Measurements In this experimental setup, it is not straightforward to insert a measurement probe into the ow while leaving the sensitive SI{SII transition unchanged. In tests on a same-sized 121 E387 wing under acoustic excitation at similarRe and pre-SI-SII transition [7], the place- ment of a microphone near the top surface of the wing prevented the ow from reattaching when it otherwise normally would, thus keeping the ow in SI. Since the characteristics of SI-SII transition can be monitored here, one can search for arrangements where it is not aected by the microphone presence. As an example check on possible feedback from the presence of the microphone to the wing, power spectra were taken with the microphone placed at dierent distances from the wing surface (in the z-direction) aty=b = 0 andx=c = 0:1. Examples measured above the wing with open holes at x=c = 0:1 and closed exit port are shown in Fig. 9.10. By monitoring the SI-SII state through jumps in the overall lift force, an adequate distance between the tip of the microphone nose cone and the wing surface was determined to be z=c = 0:15, and all subsequent power spectra were obtained with the microphone placed at that location. Figure 9.10: Power spectra at Re = 40k and = 10 for the wing with open holes at x=c = 0:1 and open exit port measured at various distances, z=c, from the top surface. 122 The power spectra at Re = 40k are shown in Fig. 9.11 for no wing (a), wing with open holes at x=c = 0:1 and closed exit port (b), wing with open holes and open exit port (c), and holes covered (d). For cases (b)(d), = 10 . When the holes are closed (Fig. 9.11d) and when holes are open at x=c = 0:1 with closed exit ports (Fig. 9.11b), the wing is in SI (corresponding to Fig. 9.9b), and the power spectra have a peak at f = 600 Hz for these two cases. This value of 600 Hz is later shown to match an experimentally measured vortex passage frequency above the wing surface. When holes are open atx=c = 0:1 with open exit ports (Fig. 9.11c), the wing is in SII (corresponding to Fig. 9.9a), and the power spectrum has no peak at f = 600 Hz. A natural frequency of 600 Hz occurs over the wing in SI, but once the wing is in SII, it is no longer measurable in the external ow eld. Similarly, the 400 Hz peak, which is highest when the wing is in low-lift SI, is reduced in amplitude when the ow is controlled to state SII, and it could mark a harmonic of a ow instability frequency that is suppressed in the presence of control. F. Cavity Volume The preceding results suggest that the aerodynamic improvements in Fig. 9.5 could be caused by pressure uctuations within the cavities and corresponding velocity uctuations at the orices due to acoustic resonance. If this is so, then varying the cavity volume/geometry should aect the resonant frequencies in predictable ways. In the original wing design, the cavities were not uid-dynamically isolated from each other, as shown by the schematic in Fig. 9.12, but this base topology could easily be changed. The performance of the wing at Re = 40k was analyzed when the holes in the lid were left open but the cavities were isolated from each other so that each hole communicated only with its own local chamber, or reservoir. Figure 9.13 shows that the isolated cavity model behaves similarly as the original perforated wing, being in SII at positive. However, in the range 0 12 , L=D values for the wing with isolated cavities are slightly lower than 123 Figure 9.11: Power spectra at Re = 40k. (a) No wing; (b) wing with closed holes; (c) wing with open holes (x=c = 0:1) and open exit port; (d) wing with open holes (x=c = 0:1) and closed exit port. For (b)(d), = 10 . 124 Figure 9.12: Schematic of cavities interconnected by a channel. Each cavity was designed to house speakers with an adjacent channel for wiring. Figure 9.13: C L (C D ) (left) and L=D (right) for a E387 wing at Re = 40k with closed holes, open holes, and open holes with isolated cavities. for the wing with connected cavities. Topologically, the isolated cavity model is the same as the AR = 1=2 sealed holes. The dierence lies only in the volume of the cavity. For a single row of open holes located at x=c = 0:1, lift and drag forces were compared for two dierent internal cavity volumes. Part of the volume directly below the holes was 125 reduced by approximately 24%. The resulting C L (C D ) and L=D curves at Re = 40k are shown in Fig. 9.14. The reduced cavity volume case yields a slightly lower L=D curve between4 12 . A decrease in performance in the same range was also observed at Re = 60k. Figure 9.14: C L (C D ) (left) andL=D (right) for a E387 wing with 15 open holes atx=c = 0:1 and two dierent internal cavity volumes. G. Spanwise Variation The in uence of spanwise uniformity of the open holes was examined. In Figs. 9.5 - 9.14, open holes spanned from0:83 y=b 0:83. At x=c = 0:1, only nine holes centered at midspan (0:48 y=b 0:48) were left open while all those closer to the wing tips were closed. Partial coverage with open holes at midspan (Fig. 9.15) generates lift-drag polars and (L=D)() curves that are intermediate between those of fully closed and fully open holes. The intermediate curves still show sharp SI{SII transitions. The intermediate result suggests that the integrated force on the whole wing can be controlled by spanwise variation of local conditions. Figure 9.16 demonstrates that this concept is true. The gure shows example time-averaged spanwise vorticity elds from ve spanwise planes under the same conditions. The span sections are aty=b =f0.0, -0.15, -0.30, 126 -0.44, -0.59g. The rst location (station #1) is at midspan, directly at an open hole location. Here, and in the next two neighboring locations, a large leading edge separation bubble is followed by reattachment shortly after mid-chord. The reattached ow corresponds locally to state SII. The last two stations aty=b =f0:44;0:59g lie either side of an open hole, but both show a short separation bubble that does not reattach. The eective separation point is much closer to the leading edge and the wake is much wider. This ow state corresponds to SI. The ow at station #4 is controlled not only by the most proximate hole state, but also by the ow in adjoining station #5. Raw PIV images also provide qualitative information about the ow behavior by the presence of a dark separation line, caused by the presence of a shear layer where the uid particle velocity is zero. In Fig. 9.16, raw images are shown at y=b =0:15 (station #2), where the dark separation line lies close to the suction surface, corresponding to a locally attached (SII) ow, as well as aty=b =0:59 (station #5), where the separation line is farther from the surface, corresponding to a locally separated (SI) ow. Figure 9.15: C L (C D ) (left) and L=D (right) for a E387 wing at Re = 40k with mid portion holes open (0:48y=b 0:48) at x=c = 0:1. 127 Figure 9.16: Spanwise vorticity elds, ! z (x;z) are averaged over 20 independent samples to give a map whose strongest features are steady, but an indication of the unsteady structure remains. This is superimposed on similarly time-averaged uctuating velocity vector elds at spanwise sections y=b = 0:0;0:15;0:30;0:44;0:59 for the E387 at Re = 40k and = 11 with open holes at x=c = 0:1 and 0:48y=b 0:48. Every third vector is plotted. Raw PIV images at y=b = 0:0;0:15 show dark separation lines that distinguish between locally attached and locally separated ow. 128 IV. Discussion On a wing that is characterized by abrupt jumps in lift and drag coecient between what we have termed SI, where the ow separates at some point between mid-chord and the trailing edge, and SII, where separation close to the leading edge is followed by reattachment, the dynamics are very sensitive to a number of dierent perturbations. In a study originally aimed at acoustic forcing of the ow, it was found, quite by accident, that the presence of the holes themselves in the wing suction surface was sucient to aect the global behavior. When holes are present, the ow state switches from SI to high-lift SII, with no other control input necessary. The mechanism is entirely passive, and a control strategy might simply involve sliding lids open and shut to modify local ow characteristics. The eect on the wing is approximately local, so that chordwise strips can aect local sectional c l and c d . Since each local ow state is either in SI or SII, one may think of the wing as a device whose lift and drag coecients can be manipulated under digital control. Local states can be on or o, and asymmetries across the span will lead to rolling moments, while symmetrically (about midspan) actuated hole opening can yield pre-determined total lift and drag from the envelope of possibilities. A. Cavity Flows The performance characteristics of the perforated wing could be due to some form of cavity ow. In general, cavity ow can be categorized into three main types, as detailed in [16]: (a) uid-dynamic, where oscillations come from the instability of the cavity shear layer and are enhanced through a feedback mechanism, (b) uid-resonant, where oscillations are strongly coupled with resonant (standing) wave eects, and (c) uid-elastic, where os- cillations are linked to solid boundary motion. It is also possible to have combinations of dierent types of cavity ow. In purely uid-dynamic cavity ow, the feedback mechanisms that enhance the oscilla- tions are driven by the presence of the cavity downstream edge [16]. This type of cavity ow includes the ow over a cavity covered by a perforated plate, whose uses include acoustic 129 lining for sound attenuation [17]. In studies on the shear layer oscillations along a perforated plate backed by a cavity [18], the predominant frequency varied with impingement length and in ow velocity. Without the perforated plate, the shear layer was separated with an in ection point, while with a perforated plate, the shear layer was bounded with no in ection point. It was also suggested in [19] that unied, large-scale motion occurs through the plate perforations and induces jet ow at the downstream end of the perforated plate. The long- wavelength instabilities along a perforated plate of length L can be expressed as a Strouhal number,fL=U, wheref is a shedding or oscillation frequency. The reported values offL=U are on the order of 0.5-0.6 [20]. If the chord of the wing is used as an eective plate length, the calculated fL=U 8 is an order of magnitude larger. In classical cavity ows, the cavity length must be several times larger than a boundary layer thickness at the upstream lip in order for amplication to occur of the instabilities in the cavity shear layer. Estimates of from Eq. (9.1) ranged from about 1-2 mm along the chord, which are always larger than the streamwise cavity length of 0.5 mm. Moreover, in the current tests, a cavity of nite depth (equal to twice its diameter), by itself, has no eect on the performance of the wing. Changes in ow occur only when the holes are connected to a larger chamber and when the chamber volume is varied, suggesting the presence of acoustic resonance eects. B. Helmholtz Resonance Fluid-resonant cavities include Helmholtz resonators, which are distinguished by the very large ratio of cavity volume to cavity orice area [16]. In general, a Helmholtz resonator is a device in which a volume of compressible uid is enclosed by rigid boundaries with a single small opening and can be modeled by a second-order mass-spring system where the uid in the orice has an eective mass and the compressibility of the uid in the chamber is the stiness [21]. The resonator has a natural frequency, and when the instabilities in the ow match the natural Helmholtz frequency, ow-excited resonance occurs. In such cases, small pressure disturbances can produce large velocity uctuations at the orice and large pressure uctuations inside the resonator [21]. 130 The Helmholtz resonant frequency is f H = a 2 r V ; (9.2) wherea is the speed of sound, is a quantity that represents the resistance of uid passage through the orice, and V is the cavity volume [22]. For a circular orice, the resistance is = r 2 h + 1:697r ; (9.3) where r is the orice radius, and h is the orice thickness (neck height) [22]. For multiple orices in a single cavity, the resistance terms can be summed, yielding a resonant frequency f H = a 2 r 1 + 2 +::: V : (9.4) Equations 9.3 and 9.4 were used to calculate the Helmholtz resonant frequency for a single spanwise row of 15 open holes with interconnected cavities (Fig. 9.14). The combined cavity and channel volume is approximately 14200 mm 3 . Solving forf H yieldsf H = 650 3 Hz, where the uncertainty comes from the uncertainty in cavity and channel volume mea- surements. An instability frequency can be estimated from the properties of the vortical structures above the wing, f i = U a x s ; (9.5) where U a is the streamwise advection speed, and x s is the average separation between vor- tices, as observed in PIV data (e.g. Fig 9.17). At Re = 40k and = 10 for the case of 15 open holes at x=c = 0:1, the calculated instability frequency is f i = 630 90 Hz, where the uncertainty comes from using dierent pairs of neighboring vortical structures and the uncertainty in locating the vorticity centers. The calculatedf i is equal to the calculatedf H . For the reduced volume case where the volume is approximately 10800 mm 3 (Fig. 9.14), the 131 calculated Helmholtz resonance is f H = 750 3 Hz. The calculated instability frequency is f i = 730 50 Hz, which again equals f H . Figure 9.17: Instantaneous spanwise vorticity elds over the aft portion of the wing (begin- ning at x=c = 0.5) at Re = 40k and = 10 with 15 open holes at x=c = 0:1. The time between successive vorticity elds is 0.1 s. The calculation of f H in Eq. 9.2 is independent of ow speed. For a given resonator volume and orice radius, the same resonant frequencies should be generated for varying ow speeds. This was conrmed by calculating f i at Re = 60k and = 8 for the case of 15 open holes at x=c = 0:1. For the original volume, f i = 650 90 Hz, and for the reduced volume, f i = 710 50 Hz, which are equal to the values of f i at Re = 40k for the two dierent volumes, respectively. 132 When the number of open holes is doubled from 15 to 30 (while keeping all other param- eters the same), the calculated Helmholtz resonance is f H = 925 3 Hz. At Re = 60k and = 8 , the calculated instability frequency is f i = 830 100 Hz. The large variation could be due to the fact that the interconnected cavities might not truly act as a single large cavity, but rather a series of small cavities. In the example given in [22], when two resonators are connected together in series, one of the resonators can experience a slightly dierent resonant frequency depending on the geometry of the connecting channel between the two resonators. In the same example, if one of the cavities is open to the atmosphere, then the resulting resonant frequency can be as much as a factor of 2.4 dierent than a simple resonator [22]. For the current wing, opening and closing the exit ports yielded dierent results, which were especially prominent at the lowerRe. The ratio of the calculatedf i for open and closed exit ports at Re = 40k is f i;open =f i;closed 2. Since the power spectra in Fig. 9.11 were measured at x=c = 0:1, and separation occurs at x=c = 0:2; f = 600 Hz most likely matches a readily amplied frequency that grows rst in the still-attached boundary layer, and then subsequently after separation. The peak in power spectra at f = 600 Hz disappears when the ow is reattached and the wing is in SII. 600 Hz is close to the calculated f H and equal to f i , suggesting that a matching of the Helmholtz resonance with the naturally occurring instability frequency in the boundary layer promotes ow reattachment and SI{SII transition. Once this frequency matching occurs, the reattached SII shear layer contains structures like those in Fig. 9.17 that propagate with the initial SI separated boundary layer frequency, which is equal to the Helmholtz resonant frequency. For airfoil shapes, such as the E387, that exhibit distinct jumps between the SI and SII states at transitionalRe, the same Helmholtz resonant frequencies in perforated wings can be expected over the entireRe range, and should yield similar results as this study atRe = 40k and 60k. Key parameters that change the resonant frequency are orice radius, orice neck height, and cavity volume. Additional parameters such as the number of orices and 133 connecting channel geometry need to be considered when the resonator is more complicated, as in the case of the current perforated wing. C. Similarities to Woodwind Instruments Helmholtz resonance is the primary means for how many woodwind instruments work, such as the ocarina or a transverse ute. The distinct frequencies of woodwind instruments depend on various parameters, including instrument size and geometry, selection of open and closed nger holes, nger hole size and spacing, type of mouthpiece, and even the angle at which the air ows over the mouthpiece. There exists an extensive amount of details on the physics and design of woodwind instruments (ie, [23], [24], [25], [26]), and the main sound production mechanism that governs woodwind instruments similarly governs the perforated wing. The semi-porous wing in the current study contains numerous combinations of open and closed holes along dierent spanwise and chordwise arrays, similar to the nger holes on a woodwind instrument. The wing is much like an ocarina when the exit port is closed, and a transverse ute when the exit port is open. However, the wing has no single mouth- piece or embouchure, but rather spanwise rows of embouchures that can be blown across simultaneously. It should also be noted that f H in Eq. 9.2 is independent of cavity geometry, which explains why ocarinas can be made into so many dierent shapes, as long as the internal volume is controlled. The orice geometry of Helmholtz resonators was shown in [27] not to be a signicant factor in the resulting resonant frequencies. In the current study, the intricate interior design of the wing makes the wing a more complicated resonator that may require dierent mathematical formulations to obtain the precise resonant frequencies for various combinations of open and closed holes, angle of attack, etc. In [7], a solid E387 wing atRe = 40k was acoustically excited by an external sound source. At excitation frequencies from the sound source that matched tunnel anti-resonances (415, 520, and 660 Hz) the wing experienced an almost 80% increase inL=D. One of the optimum 134 excitation frequency values (660 Hz) matches the calculated f H for the perforated wing. The calculated values of f i , also obtained from the PIV data, are the second harmonics of the excitation frequencies, f e . Results from [7] indicate that the ow over the wing is altered when external acoustic excitation induces maximum uctuating velocities on the wing (at tunnel anti-resonances). Minimum pressure and maximum velocity also occur at the embouchure of a ute, where the air ow occurs [28]. The current study suggests that the open holes act similarly to embouchures for Helmholtz resonators that amplify the natural frequency of the uid instabilities through traveling acoustic waves. That contrasts with external acoustic excitation, where the most amplied frequency is driven by the matching of acoustic frequencies with standing waves set up by tunnel resonance. Here, the traveling acoustic waves from the Helmholtz resonators induce pressure uctuations at the orices that propagate into the separated shear layer immediately above. The perforated wing shares some characteristics with woodwind instruments and may be thought of as a type of multi-embouchure woodwind instrument that uses a combination of cavity resonance and pipe ow to alter the nominal characteristics of the boundary and separated shear layers. D. Pressure Tap Measurements There are certain practical implications of these results. One is that if airfoils and wings are instrumented with pressure taps in this transitional regime, then their chamber volumes must be carefully selected to remove resonant frequencies from possible interactions with the intrinsic ow instabilities, or those induced at the orice opening. A typical pressure tap is connected by a tube to a pressure transducer, which contains a cavity, as shown in the schematic in Fig. 9.18. In a multiple-pressure tap setup, the connecting tubes are typically connected to a scanning valve which then connects to the transducer. The pressure tubes can vary in length depending on the placement of the pressure taps in the wing model, but essentially a pressure tap setup closely resembles a Helmholtz resonator. Calculations off H from Eqs. (9.2) and (9.3) can be made for dierent tube lengths (neck heights), h, assuming constant orice/tube radius, or constant cavity volume. Assuming an orice radius (equal to tube radius) of 0.5 mm, and setting the cavity volume to those for 135 Figure 9.18: A typical pressure tap geometry (schematic adapted from [31]) has the essential components of a Helmholtz resonator. three models of a Validyne pressure transducer (DP15, DP303, and Type 45-14), yields the curves in Fig. 9.19a. Alternatively, setting the cavity volume to that of the Type 45-14, while varying the orice diameter (tube radius) gives the curves in Fig. 9.19b. f H calculated here gave resonant frequencies between 600 950 Hz. In the acoustic studies reported in [7], excitable frequencies that cause SI-SII transition at Re = 60k range between 400 1000 Hz. Figure 9.19 suggests that using pressure transducers with smaller cavity volumes and/or having smaller orice/tube radii may be problematic for tube lengths less than about 14 cm. This could happen when the pressure transducers are located directly inside the wing, which results in shorter tube lengths. Longer pressure tubes (greater than 20 cm) seem to be safer in order to avoid resonance eects from pressure taps, although there still must be a balance to avoid attenuation of the pressure response. Given the similar conguration of a pressure tap system to a Helmholtz resonator, the presence of pressure taps may modify the ow over a wing, preventing SI{SII transition. The ow will also then be non-uniform and forced at spanwise scales dictated by the pressure tap spacing. For example, data on the E387 airfoil in the sameRe regime [30] included sectional pressure coecients, obtained by 129 0.5 mm diameter pressure orices on the wing surface. The data show signicant spanwise variation inc d atRe = 100; 000, and the lowest values of c d were measured at the location of the pressure taps. These ndings parallel those from the 136 Figure 9.19: Relationship between Helmholtz frequency, f H , and tube length (neck height), h for constant orice/tube radius and variable cavity volume (a), and for constant cavity volume and variable orice/tube radius (b). current study, which show higherC L and lowerC D for congurations with open holes (Figs. 9.5, 9.7, 9.8, 9.9, 9.15). The local sectional ow state (SI or SII), local wing circulation, and section drag have been shown to be correlated for the E387 wing in thisRe regime [4]. Here, it is clear that the presence of open holes induces spanwise variation in wing circulation and hence section drag coecient. Measured lift and drag coecients for the E387 vary widely throughout the literature, especially at lowerRe. The relative dierence in measuredc d for the E387 airfoil atRe = 60k among dierent facilities [1] can be expressed as a fraction of the maximum c d , C d;lit = c d;max c d;min c d;max : (9.6) 137 The result is given in Fig. 9.20. The data include experiments in which some of the models had pressure orices and some did not. The variation in C D for the E387 wing from the current study at Re = 60k with all open holes and all closed holes is shown as C D;USC = jC D;open C D;closed j C D;closed ; (9.7) and is also plotted in the same gure. The values of C d;lit are higher than the values of C D;USC , but they are not orders of magnitude apart, unlike a similar measure derived from the natural spanwise variation tests in [4]. The disruptive eect of holes in pressure tap measurements upon the global ow properties could be a signicant factor in the observed variations in the technical literature. We may note also that the same results here show that small geometric cavities by themselves do not aect wing performance, so, for example, surface mounted MEMS probes that have small cavities should not be problematic. The large changes caused by small geometries promoting resonance at the orices suggest that very slightly porous wings may also operate permanently in the high lift SII state. Little attention has yet been given to the eects of porosity in feathered wings of birds, for example, which could be important. However, small geometric cavities by themselves will not aect wing performance, implying that several types of surface mounted MEMS probes should not be problematic for this particular ow and airfoil. V. Conclusion In the initial stages of testing the response of an E387 wing to acoustic excitation, the presence of small holes in the suction surface of the wing was found to signicantly change the overall aerodynamic performance. This discovery led to an independent study of the eects of open holes on the forces and local ow dynamics of the E387 wing at the transitional Re = 40k where ow separation and reattachment determine whether the wing is in a low SI or high SII lift state, respectively. Switching from SI to SII can be promoted by forcing through acoustic resonance of the small chambers when their resonant modes are close to the most unstable modes in the original (hole-free) ow. The large eect of the 138 Figure 9.20: C d;lit for the E387 airfoil at Re = 60k among dierent facilities [1], and C D;USC between open and closed holes for the E387 wing at Re = 60k. small holes suggests that some caution is required in interpreting and designing pressure tap measurements in this transitional Re range. In principle, the passive eects of the holes+chambers ought to be replaceable through equivalent, local forcing through small, embedded sources. Either one could be used for local, digital control of forces and moments on the wing, but with active acoustic forcing through loudspeakers, the frequency is an independent control parameter, not depending on the cavity geometry, and this possibility will be investigated in the future. 139 Paper III References [1] Selig, M. S., Guglielmo, J. J., Proeren, A. P., Giguere, P., \Summary of Low-Speed Airfoil Data Vol. 1," pp. 19-21. SoarTech Publications, Virginia Beach, VA (1995) [2] Selig, M. S., Lyon C.A., Giguere, P., Ninham, C.P. and Guglielmo, J.J., \Summary of Low-Speed Airfoil Data Vol. 2," SoarTech Publications, Virginia Beach, VA (1996) [3] Lyon, C. A., Broeren, A. P., Giguere, P., Gopalarathnam, A. and Selig, M. S., Summary of Low-Speed Airfoil Data Vol. 3, Soartech Publication, Poole (1997) [4] Yang, S. L., and Spedding, G. R., \Spanwise Variation in Wing Circulation and Drag Measurement of Wings at Moderate Reynolds Number," J. Aircraft, 50, 3, pp. 791-797 (2013) [5] Somers, D.M. and Maughmer, M.D., \Experimental Results for the E 387 Airfoil at Low Reynolds Numbers in the Penn State Low-Speed, Low-Turbulence Wind Tunnel," AHS Specialists Conference on Aeromechanics (2008) [6] Spedding, G. R., McArthur, J., \Span Eciencies of Wings at Low Reynolds Number," J. Aircraft, 47, pp. 120-128 (2010) [7] Yang, S. L., and Spedding, G. R., \Separation Control by External Acoustic Excitation on a Finite Wing at Low Reynolds Numbers," American Institute of Aeronautics and Astronautics, 51, 6, pp. 1506-1515 (2013) [8] Simons, M., Model Aircraft Aerodynamics, 4th Ed., Special Interest Model Books, Poole, (1999) [9] Grundy T. M., Keefe G.P. and Lowson M.V., \Eects of Acoustic Disturbances on Low Re Aerofoil Flows." In Fixed and Flapping Wing Aerodynamics for Micro Air Vehicle Applications. Vol. 195, pp. 91112. Reston, Virginia: American Institute of Aeronautics & Astronautics, (2001) [10] Zabat, M., Farascaroli, S., Browand, F., Nestlerode, M., Baez, J., \Drag Measurements on a Platoon of Vehicles," Research Reports, California Partners for Advanced Transit and Highways (PATH), Institute of Transportation Studies, UC Berkeley (1994) 140 [11] McArthur, John. \Aerodynamics of Wings at Low Reynolds Numbers," Ph.D. Disser- tation, Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA (2007) [12] Fincham, A. M. and Spedding, G. R. \Low Cost, High Resolution DPIV for Measure- ment of Turbulent Fluid Flow," Experiments in Fluids, 23, pp. 449-462 (1997) [13] Fincham, A. M. and Spedding, G. R. \Advanced Optimization of Correlation Imaging Velocimetry Algorithms," Experiments in Fluids, 29, pp. 13-22 (2000) [14] Spedding, G.R. and Rignot, E.J.M., \Performance analysis and application of grid interpolation techniques for uid ows," Experiments in Fluids, 15, pp. 417-430 (1993) [15] Lissaman, P.B.S., \Low-Reynolds-Number Airfoils," Annual Review of Fluid Mechanics, 15, pp. 223-239 (1983) [16] Rockwell, D. and Naudascher, E., \Review - Self-Sustaining Oscillations on Flow Past Cavities," Journal of Fluids Engineering, 100, pp. 152-165 (1978) [17] Guess, A.W., \Calculation of Perforated Plate Liner Parameters From Specied Acous- tic Resistance and Reactance," Journal of Sound and Vibration, 40, pp. 119-137 (1975) [18] Celik, E. and Sever, A.C. and Rockwell, D., \Shear Layer Oscillation Along a Perforated Surface: a Self-Excited Large-Scale Instability," American Institute of Aeronautics and Astronautics, 14, 12, pp. 4444-4447 (2002) [19] Celik, E. and Sever, A.C. and Rockwell, D., \Self-Sustained Oscillations Past Perforated and Slotted Plates: Eect of Plate Thickness," American Institute of Aeronautics and Astronautics, 43, 8, pp. 1850-1853 (2005) [20] Ekmekci, A. and Rockwell, D., \Self-Sustained Oscillations of Shear Flow Past a Slotted Plate Coupled with Cavity Resonance," Journal of Fluids and Structures, 17, pp. 1237- 1245 (2002) [21] Morris, S. C., \Shear-Layer Instabilities: Particle Image Velocimetry Measurements and Implications for Acoustics," Annual Review of Fluid Mechanics, 43, pp. 529-550 (2011) [22] Strutt, J. W. S., \On the Theory of Resonance," Scientic Papers, 1, pp. 77-118 (1964) [23] Fletcher, N.H. and Rossing, T.D., \The Physics of Musical Instruments," 2nd Ed., (1998) [24] Benade, A.H. and French, J.W., \Analysis of the Flute Head Joint," Journal of the Acoustical Society of America, 37, 4, pp. 679-691 (1965) [25] Benade, A.H., \Fundamentals of Musical Acoustics," (1967) 141 [26] Coltman, J.W., \Sounding Mechanism of the Flute and Organ Pipe," Journal of the Acoustical Society of America, 44, 4, pp. 983-992 (1968) [27] Chanaud, R.C., \Eects of Geometry on the Resonance Frequency of Helmholtz Res- onators," Journal of Sound and Vibration, 178, 3, pp. 337-348 (1994) [28] Dickens, P. and France, R. and Smith, J. and Wolfe, J., \Clarinet Acoustics Introducing a Compendium of Impedance and Sound Spectra," Acoustics Australia, 35, 1, pp. 17-24 (2007) [29] Cohen, K. and Aradag, S. and Siegel, S. and Seidel, J. and McLaughlin, T., \A Method- ology Based on Experimental Investigation of a DBD-Plasma ActuatedCylinder Wake for Flow Control," in Low Reynolds Number Aerodynamics and Transition, (2012) [30] McGhee, R. J., Walker, B. S., Millard, B. F., Experimental Results for the Eppler 387 Airfoil at Low Reynolds Numbers in the Langley Low-Turbulence Pressure Tunnel. NASA TM-4062, pp. 25-26 (1988) [31] Freeman, L.A., Carpenter, M.C., Rosenberry, D.O., Rousseau, J.P., Unger, R., McLean, J.S., \Use of submersible pressure transducers in water-resources investigations," in Techniques of Water Resources investigations, 8-A3, U.S. Geological Survey, (2004) 142 Paper IV 143 Chapter 10 Local Acoustic Forcing on a Finite Wing at Low Reynolds Numbers Yang, S. L. and Spedding, G. R. Aerospace and Mechanical Engineering Department University of Southern California Los Angeles, California 90089-1191 American Institute of Aeronautics and Astronautics, 2013. In review. At transitional Reynolds numbers (10 4 10 5 ), many smooth airfoils experience laminar ow separation and possible turbulent reattachment, where the occurrence of either state is strongly in uenced by small changes in the surrounding environment. The Eppler 387 airfoil is one of many airfoils that can have multiple lift and drag states at a single wing incidence angle. Pre-stall hysteresis and abrupt switching between stable states occur due to sudden ow reattachment and the appearance of a separation bubble close to the leading edge. Here, we demonstrate control of the ow dynamics by localized acoustic excitation through small speakers embedded beneath the suction surface. The ow can be controlled not only through variations in acoustic power and frequency, but also through spatial variations in forcing location. Implications for control and stabilization of small aircraft are considered. 144 Nomenclature AR Aspect ratio b Wing semi-span (m) c Chord (m) c 0 Normalized chordwise coordinate (m) C D Total drag coecient on a nite wing C L Total lift coecient on a nite wing c s Separation line location (m) f e Excitation frequency (Hz) f s Separated shear layer instability shedding frequency (Hz) L=D Lift-to-drag ratio SPL Sound pressure level (dB) St Strouhal number Re Reynolds number u;v;wVelocity components in (x;y;z) (m/s) U 0 Free stream velocity (m/s) x;y;zCoordinates in streamwise, spanwise, and normal directions Angle of attack (deg) ! y Spanwise component of vorticity (rad/s) Superscript Preferential value I. Introduction A growing number of micro aerial vehicles operate in a particular ight regime where the chord-based Reynolds number,Re =U o c= (whereU o is the ight speed,c the chord, and is the kinematic viscosity), lies between 10 4 and 10 5 . In this regime, laminar boundary layer 145 separation and then possible turbulent reattachment can either favorably or adversely aect wing performance. The Eppler 387 airfoil, along with many other smooth airfoils, can be in either one of two states: a low-lift state (SI in Fig. 10.1) where separation occurs prior to the trailing edge without reattachment, or a high-lift state (SII) where initial separation is followed by the formation of a laminar separation bubble and ow reattachment. A number of intrinsic unstable modes in the laminar separation bubble problem have been identied [1, 2, 3], from Tollmien-Schlichting waves in the still-attached, laminar boundary layer to Kelvin- Helmholtz instabilities in the later separated shear layer, together with possible mixed modes in between. Various of these modes are known to be susceptible to acoustic perturbation, and we propose to exploit this to test the possibility of active control between the SI and SII ow states. Figure 10.1: C L (C D ) andL=D curves show bi-stable states for the E387 wing at variousRe. A. Separation Control by Acoustic Excitation Previous tests of acoustic excitation in the Dryden Wind Tunnel involved acoustically exciting a E387 wing by a speaker that was placed on the outside of the wind tunnel test section, turning the entire test section into a resonating chamber [4]. Acoustic forcing of the ow around an E387 wing at Re = 40k and 60k at certain excitation frequencies, f e , increased lift at certain angles of attack, tripped the ow from low- to high-lift state (SI 146 - SII), eliminated pre-stall hysteresis, and promoted ow reattachment. In an enclosed chamber, minima in the rms acoustic power correspond to minimum uctuating pressure and maximum velocity uctuations, and external acoustic excitation at values of f e that correlated with test section anti-resonances yielded the largest improvement in L=D. It was also shown that a harmonic of one of these optimum excitation frequencies matched the shear layer instability frequency, f s , suggesting that the ow over the wing is altered when acoustic excitation matches a naturally occurring instability in the shear layer. Internal acoustic excitation of airfoils and wings has been shown to increase lift and delay and/or prevent separation, sensitive to both excitation frequency and sound pressure level (SPL) [6, 7, 8, 9, 10]. The most common internal acoustic forcing experiment has a speaker inside the wing with spanwise slots so that sound travels through the entire span of the wing, exiting at the open slot(s), and it has been shown [7, 10] that acoustic forcing is most eective when applied at a point,x 0 =c =c 0 , close to or before the chordwise separation point, c s , so that c 0 c s . The eect of acoustic forcing was found to deteriorate as c 0 moved farther from c s , and forcing aft of c s required considerably higher SPL to achieve the same reduction in L=D. However, the optimum values of forcing frequency, f e , were reported to be independent of c 0 [7]. When excitation was forced near c s , f e was found to be equal to the separated shear instability frequency, f s , or a sub harmonic [9, 7]. Tests from both external and internal acoustic forcing show that the values of f e cor- respond to the dominant natural instabilities in the separated region. A non-dimensional Strouhal number, St, can be written St = f e c U o : (10.1) If the main frequency selection depends on lengthscales in the viscous boundary layer or in the separated shear layer, then a modied Strouhal number,St =St=Re 1=2 may be relevant. However, studies of internal acoustic forcing at both transitional and moderate Re show a 147 large range of St from 0:001 0:04, with subranges specic to particular wing [6, 7, 8], and generally-applicable scaling laws may be elusive. B. Objectives This paper reports on the eects of internal acoustic excitation on the forces and ow elds of an E387 wing in a Reynolds number regime where pre-stall hysteresis and abrupt switching between bi-stable states occur. This study is the continuation of a series of experiments to acoustically excite the boundary layer instabilities and control ow separation of wings at transitional Re. As an extension of the external acoustic forcing tests, the experiments reported here aim to eliminate the role of global standing waves that inevitably occur with facility-dependent resonances by truly localizing the acoustic forcing. Applications of this research to small-scale aircraft could lead to energy-ecient separation control as well as overall aerodynamic improvement with no moving parts. II. Materials and Methods A. Wind Tunnel and Instrumentation Experiments were performed in the Dryden wind tunnel at USC. Lift and drag forces were measured with a custom force balance (described in detail in previous experiments [11, 12, 13]) placed below the wind tunnel oor. Particle Imaging Velocimetry (PIV) was used to estimate velocity components (u;w) in the two-dimensional plane (x;z) with the same setup used by Yang et al. [4] (Fig. 10.2) but with an improved resolution Imager Pro X 2M (1648 x 1214 x 14-bit) camera. B. Acoustic Equipment The internal sound sources were Knowles Acoustics Wide Band FK Series (WBFK) speakers with dimensions 6:50 2:75 1:95 mm, with a frequency response of 400 Hz- 1000 Hz3 dB. EX1200-3608 16-bit DACs were used to generate sine waves with adjustable 148 Figure 10.2: Wind tunnel setup. (x;y;z) are streamwise, spanwise, and normal directions. Origin is at leading edge and midspan. frequency and amplitude, and these were amplied with Kramer VA-16XL variable gain stereo audio ampliers with a3 dB frequency range of 20 Hz 40 kHz. C. Wing Model The wing model had an Eppler 387 prole (inset of Fig. 10.3a) with an aspect ratio AR = 6 (span b = 54 cm and chord c = 9 cm). The model was a two-part aluminum wing composed of a base and a lid, as shown in Figure 10.3. The base of the wing contained cavities and channels into which speakers and wires were embedded. The lid, which contained 0.5 mm diameter holes for sound emission, slid over and locked into the base by a tongue-and- groove connection. The model had a total of 180 speaker cavities arranged in six rows, each with 30 cavities, as noted in Fig. 10.3b. The individual holes in the wing suction surface were 149 (a) (b) Figure 10.3: (a) E387 wing consisting of a base with 180 speaker cavities and a lid with 180 0.5 mm diameter holes, and (b) prole view of the lid of the wing. Six spanwise rows of holes are located at x=c =f0:1; 0:2; 0:3; 0:4; 0:5; 0:6g. From Yang et al. [5]. either covered with tissue paper diaphragms from the underside of the lid, leaving cavities of width-to-depth ratio of 1/2, or lled in with modeling clay. III. Results A. Baseline Performance of the E387 Wing Cavities with a width-to-height ratio of 1/2 do not aect the basic performance of the wing [5], and the perforated wing with holes covered from the bottom of the lid performed the same as a solid wing of the same size, used in prior external acoustic forcing experiments [4]. The standard behavior of the wing at these transitional Re includes the sudden jump from the low-lift (SI) state to high-lift (SII) state at a pre-stall angle of attack, as seen in Fig. 10.4. At Re = 40k, with external acoustic forcing, the current wing had similar values 150 Figure 10.4: C L (C D ) and (L=D) curves for the E387 wing at Re = 40k (black) Re = 60k (gray) without excitation. of f e * as the solid E387 wing, the two most eective frequencies being 420 Hz and 520 Hz (St = 0.028 and 0.035, respectively). B. Excitation at Dierent Reynolds Numbers In initial tests, 15 speakers were activated (phase-synchronized in time) over0:83 y=b 0:83 and spaced (y=b) = 0:12 apart at the chordwise excitation locationc 0 = 0:1. Lift and drag forces were measured for a series of dierent excitation frequencies, f e , at a pre-SII angle of attack, 0 , forRe = 40k and 60k (as noted in Fig. 10.4). The changes inL=D from internal forcing at c 0 = 0:1, shown in Fig. 10.5, show that at Re = 40k, a single optimum excitation value occurs at 500 Hz (St = 0:034), for (L=D) 97%. On the other hand, at Re = 60k, a broad range off e exists between 200 Hzf e 1500 Hz (0:007St 0:055), for (L=D) 57%. L=D at f e = 500 Hz is about the same, regardless of Re. L=D() at Re = 40k and 60k are shown in Fig. 10.6. Improvements in L and D occur within a small 151 Figure 10.5: L=D for the E387 wing at Re = 40k and = 11 (black), and Re = 60k and = 9 (gray), resulting from excitation at dierent frequencies and corresponding parameter St . Acoustic excitation occurs at the same amplitude settings. Figure 10.6: L=D for the E387 wing at a)Re = 40k with and without excitation atf e = 500 Hz (St = 0:034), and b)Re = 60k with and without excitation atf e = 800 Hz (St = 0:029). 152 range of low, pre-stall where the bi-stable states exist (5 11 forRe = 40k and 2 10 for Re = 60k). C. Spanwise Distribution Until now, internal acoustic forcing was always applied uniformly across the span [6, 7, 8, 9, 10], though it is well-known that three-dimensional instabilities can rapidly promote transition to turbulence in otherwise nominally two-dimensional structures, such as mixing layers [14, 15, 16, 17]. The possible role of spanwise forcing variation in in uencing three- dimensional transition was examined atRe = 60k and 0 = 9 . Speakers located atc 0 = 0:1 and spaced evenly were forced at f e = 800 Hz (St = 0:029). The dierent span fractions of activated speakers were 0%; 12%; 24%; 36%; 48%, and 83% of the total span, centered around midspan. Figure 10.7 shows that between the extreme congurations (one speaker aty=b = 0 and 15 speakers between0:83y=b 0:83), a family of curves exists and that a broader spanwise coverage produces improved aerodynamic performance. Note that two eects are combined here: an increase in spanwise coverage and also simply a larger number of speakers, and hence a higher acoustic power input to the ow. Figure 10.8 expresses the nearly linear relation between increase in (L=D) and number of speakers. One can also vary the number of speakers in a xed span fraction, and Fig. 10.9 shows the variation in L=D as a function of number of speakers, or speaker density over0:36 y=b 0:36. Increasing the number of speakers (with decreased spacing between them) results in a higher (L=D). Figure 10.10 once again shows a correlation of increasing wing performance with increasing number of speakers, but this correlation is not as linear as for the varying spanwise distribution with constant spacing (Fig. 10.8), so while (L=D) varies with the spanwise distribution of sound sources, it depends also on their density. Although the conguration yielding the greatest performance improvement would be a continuous line source (where the limit of (y=b) goes to zero), the optimum eciency in terms of excitation energy vs. propulsion energy requirement would be some intermediate value that exploits the nite area of a nominally local excitation source. It is not yet clear whether the optimum spanwise spacing that would be found corresponds to a natural three-dimensional mode in 153 Figure 10.7: L=D for the E387 wing at Re = 60k for dierent spanwise distributions of activated speakers. Forcing is at f e = 800 Hz (St = 0:029). Figure 10.8: (L=D) for the E387 wing at Re = 60k and = 9 for varying number of speakers spaced evenly apart, forced at f e = 800 Hz (St = 0:029). the separated, or bubble shear layer, or whether it simply re ects the nite area in uenced by the acoustic waves as they propagate from the small source through the shear layer. Experiments to verify this might involve deliberate three-dimensional geometric forcing of 154 the ow at one wavelength and varying the spanwise wavelength of acoustic excitation sources around that. Figure 10.9: L=D for the E387 wing atRe = 60k with activated speakers atc 0 = 0:1 between 0:36y=b 0:36 with dierent spacings. f e = 850 Hz (St = 0:031). Figure 10.10: (L=D) for the E387 wing at Re = 60k and = 9 for varying number of speakers and xed span coverage, forced at f e = 850 Hz (St = 0:031). 155 D. Amplitude Variation External acoustic excitation of the same E387 wing [4] and reports on internal acoustic excitation of others [6, 7, 8, 9, 10] showed that (L=D) varies predictably with acoustic forcing amplitude. Similar to Fig. 10.7, the number of activated speakers at c 0 = 0:1 was varied, but the number of diaphragms was kept constant. Though the chambers beneath each speaker in the wing could be mechanically sealed, it could not be ensured that they were acoustically isolated, so inactive diaphragms could still act as passive acoustic sources. Fig. 10.11 can be compared with (Fig. 10.7) where the inactive speaker locations had no diaphragm. The entire baseline (L=D) is raised, and small, local forcing can have a pronounced eect on global L=D. For example, the eectiveness of single speaker forcing when surrounded by passive diaphragms is much greater; at = 8 ; L=D 9, a result that is only achieved with a row of 9 speakers in Fig. 10.7. Thus, additional diaphragms act as additional sound sources in the wing surface. Figure 10.11: L=D for the E387 wing atRe = 60k for dierent numbers of activated speakers beneath a constant number of diaphragms. f e = 800 Hz (St = 0:029). The black circles on the wing schematic indicate activated speakers and white circles inactivated speakers. 156 A separate amplitude test maintained a constant speaker and diaphragm conguration and varied the amplitude directly from the sound amplier. Five speakers centered around midspan at c 0 = 0:1 were forced at f e = 800 Hz (St = 0:029). The speaker and diaphragm conguration is shown in Fig. 10.12. Figure 10.12 shows that (L=D) is a simple function of forcing amplitude, with measured acoustic power at the shear layer A 3 <A 2 <A 1 . Figure 10.12: Eect of varying amplitude for ve activated speakers and 15 diaphragms covering0:83y=b 0:83 at c 0 = 0:1 and forced at f e = 800 Hz (St = 0:029). E. Localized Excitation While maintaining the same diaphragm conguration, activating nine speakers between 0:48 y=b 0:48 (curve (b) in Fig. 10.13) yields the same result within measurement uncertainty as activating ve speakers within the same span, where the spacing between speakers is doubled (curve (c) in Fig. 10.13) and electrical power input is almost halved. 157 This economy is not unexpected because extra diaphragms were found to act as additional sound sources, albeit at reduced amplitude. However, when the holes over the inactivated speakers were blocked, resulting in only ve total diaphragms (one for each speaker), the L=D improvements were signicantly lower (curve (d) in Fig. 10.13). Curve (d) lies between the curve associated with the nominal, unexcited state (curve (a)) and the dome-shaped L=D curves associated with the high-lift state (curves (b) and (c)). Figure 10.13: L=D for the E387 wing atRe = 60k with activated speakers between0:83 y=b 0:83 at c 0 = 0:1. Excitation with 15 diaphragms is at 800 Hz (St = 0:029) and with 5 diaphragms is at 900 Hz (St = 0:033). Raw particle images can be used to mark a dark separation line as particle-poor uid from the boundary layer is released into the otherwise uniform exterior distribution. These images can be used to directly infer ow conditions on the wing. Figure 10.14 shows the raw PIV images for the four cases labeled a-d in Fig. 10.13 at = 9 . Activated speakers 158 are indicated by solid black circles, inactivated speakers with diaphragms over them are indicated by white circles, and solid wing sections (ie, where holes are lled in) have no special marker. Arrows show the streamwise position where the dark separation lines can no longer be readily distinguished from the background. At a span station over an activated speaker (Fig. 10.14b-d), the dark separation line ends at x=c 0:5 followed by dark patches that mark distinct vortical structures, which undersampled time-series show have a very regular passage frequency. Over a solid portion of the wing, the end of the separation line is farther downstream (x=c 0:6) and is not followed by distinct vortical structures (Fig. 10.14a-d). When no speakers are activated, as in Fig. 10.14a, the visible separation line also ends at the farther downstream location x=c 0:6 across both solid wing sections and diaphragms. The alternating pattern of separation line ending points is the same in Fig. 10.14b and c, further indicating that diaphragms over inactivated speakers still act as sound sources. It is plausible that acoustic excitation amplies a naturally-occurring instability in the originally separated shear layer (Fig. 10.15a), returning high-speed uid close to the surface. The forced vortical structures then move down the airfoil chord and result in a ow which is re-attached, in the time-averaged sense (Fig. 10.15b). The small dierences in separation line stability and persistence of Fig. 10.14 are not easily resolved in PIV measurement, where scales smaller than a correlation box size are not observable, but the overall spanwise vorticity distributions for forced and unexcited ows are similar to previous measurements in globally-forced experiments [4]. Fig. 10.16a shows that at = 9 , the natural state is SI, where the ow separates close to the half-chord, and does not re-attach. The gradual forward movement of the separation line from close to the trailing edge at low accounts for the increase in drag and rather small lift increments as increases. In Fig. 10.16b, the ow is acoustically forced atf e = 800 Hz, and has switched to SII. A region of separated ow close to the leading edge is followed by reattachment so the global ow sees a wing with high eective camber. This is what leads to the higher lift and lower drag in SII. 159 (a) (b) (c) (d) Figure 10.14: Raw PIV images for dierent span stations across the wing at Re = 60k and = 9 . Speakers are located at c 0 = 0:1, and f e = 900 Hz (St = 0:033). (a) Diaphragms with inactivated speakers (no forcing), (b) diaphragms with activated speakers, (c) diaphragms with and without activated speakers, and (d) diaphragms with activated speakers with double the spacing of (b). Arrows indicate the downstream locations where the separation line vanishes. F. Chordwise Location The detailed instability mechanisms behind both forced and unforced ows described here are not necessarily simple to specify and not necessarily general to all cases. Possible receptive 160 Figure 10.15: Schematic of the separation line (a) when the ow is unexcited in the SI state and (b) when the ow is excited in the SII state, where separation occurs slightly earlier and the separation line vanishes earlier as vortical structures form. sites to nite amplitude disturbances include, from upstream to trailing edge, Tollmien- Schlichting waves in the attached laminar boundary layer, in ectional mean proles in the bubble shear layer, Kelvin-Helmholtz-type instabilities in the separated shear layer, and wake prole instabilities aft of the trailing edge (TE). Moreover, there can be acoustically- propagated feedback from the TE back to any one of the upstream modes, altering the incoming ow state [18, 1, 19, 2]. More detailed investigation of the natural and forced mean ow proles will follow later, but here initial evidence on the eect of chordwise-local acoustic forcing is shown. Six rows of ve speakers over0:48 y=b 0:48 spaced (y=b) = 0:24 apart were excited. The six rows of speakers were located atc 0 =f0:1; 0:2; 0:3; 0:4; 0:5; 0:6g. An example of this speaker conguration (at c 0 = 0:1) is shown in the wing schematic associated with curve (d) in Fig. 10.13. The holes over inactivated speakers were blocked so that the diaphragms corresponded only to activated speakers. At = 9 , each row of speakers was individually forced over a sweep of frequencies, and resulting (L=D) is shown in Fig. 10.17. The largest range of eective f e occurs at c 0 = 0:1, where (L=D) 50%. For rows of speakers at c 0 > 0:1, the range of eective f e is much narrower. The range of f e where L=D > 1 2 L=D max can be denoted as R, and is plotted for dierent c 0 in Fig. 10.18a. R decreases with increasingc 0 untilc 0 = 0:5, whereR is approximately the same between 0:4 c 0 0:6. In general, the magnitude of the maximum (L=D) decreases with downstream 161 Figure 10.16: ! y (x;z) superimposed on uctuating velocity vector elds for the wing at y=b = 0 (a) with no excitation and (b) with excitation at f e = 800 Hz. Re = 60k and = 9 . Vectors are scaled arbitrarily to 6 times the displacement in experiment. row location, as shown in Fig. 10.18b. (L=D) is constant for c 0 = 0:4 and 0.5 but then slightly increases at c 0 = 0:6. So far, measurements show that acoustic sources located close together, across the entire span, and nearest to the leading edge of the wing cause the highest increase in (L=D). A subsequent study was done to determine if the number of acoustic sources alone was a major factor in eective excitation. Measurements of (L=D) were taken atc 0 = 0:1 for two other speaker congurations where a constant number of speakers was maintained, but the spacing between speakers was varied. Figure 10.19 shows the eect of varying the speaker spacing, (y=b), on (L=D) for the wing at Re = 60k and = 9 at select frequencies. For almost all f e values, a simple correlation exists for (L=D) vs: (y=b). For a given power input (a constant number of activated speakers), a larger spacing between acoustic sources yields 162 Figure 10.17: (L=D) for various f e for ve speakers spaced (y=b) = 0:24 apart between 0:83y=b 0:83 at dierent c 0 locations. Re = 60k, and = 9 . 163 Figure 10.18: (a)R, the range off e whereL=D> 1 2 L=D max , and (b) (L=D) max at dierent x=c for ve speakers spaced (y=b) = 0:24 apart between0:83y=b 0:83 at Re = 60k, and = 9 . Figure 10.19: (L=D) at select frequencies for dierent speaker spacing, (y=b), at Re = 60k and = 9 . Five speakers are located at c 0 = 0:1. 164 Figure 10.20: L=D curves at Re = 40k for inactivated speakers with diaphragms and open holes at c 0 = 0.1. a larger improvement in wing performance, implying that the number of acoustic sources alone does not in uence the ow, but rather the distribution of these acoustic sources. G. Combined Eect of Holes and Internal Forcing Open holes in the suction surface of the wing drive the ow to the high-lift state through passive resonance [5]. Covering the holes with thin diaphragms removes this eect, as shown in Fig. 10.20. Individually, the eect of open holes and the eect of internal acoustic forcing both improve wing performance. At Re = 40k, neither of these methods by themselves produces dome-shaped L=D curves, such as those at Re = 60k (ie, Fig. 10.6b). In the region of interest (5 12 ), a local minimum in L=D occurs at at = 7 , as seen in Fig. 10.20. At this , a row of 15 speakers at c 0 = 0:1 was forced at varying f e , and the corresponding (L=D) are shown in Fig. 10.21. In the case of pure internal acoustic excitation at Re = 40k and = 9 (Fig. 10.5), a single peak in L=D occurs at f e 500 Hz 165 Figure 10.21: (L=D) at various f e for the wing with open holes at c 0 = 0:1 at Re = 40k and = 7 . (St 0:034). However, in the case of both open holes and internal acoustic excitation, the range of eective f e is much wider (f e = 100Hz 500 Hz or St = 0:007 0:034). When the speakers were forced atf e = 200 Hz without diaphragms in the lid of the wing, the combined eect of active acoustic forcing plus open holes gives the highestL=D increase, and the resulting dome-shaped L=D curve is shown in Fig. 10.22. In the same gure, L=D associated with diaphragms and no forcing, open holes and no forcing, and pure internal acoustic forcing at c 0 = 0:1 are also plotted. The performance of the wing with open holes only and with pure internal acoustic forcing are nearly the same, supporting the idea that forcing through a passive Helmholtz resonator mechanism, and an equivalent active acoustic source are equivalent. The L=D curves for these two cases lie between the two extremes of closed holes without forcing and open holes with forcing. At Re = 40k, pure internal 166 Figure 10.22: Combined eect of open holes and acoustic forcing at Re = 40k. acoustic excitation does not yield a completely high-lift state, but removing the diaphragms allows the high-lift state to be achieved. IV. Discussion A. Localized Separation Control Besides separating the phenomena associated with pure internal acoustic forcing from those associated with acoustic resonance due to open holes, the results reported here provide a study of spatially-localized acoustic forcing, which was not achievable with the wing-speaker arrangements used previously. Here, local spanwise ow separation is evidently dierent in locations where there are speakers, diaphragms or just a solid surface. The presence of a local acoustic source changes the ow separation quasi-locally, and the amplitude and symmetry of variations in L=D can be selected by appropriate selection of forcing pattern geometry. Most previous internal acoustic forcing studies were for higher Re (> 2 10 5 ) [6, 7, 8, 9, 10], and none used wing proles having bi-stable state behavior like the E387 airfoil. However, the bi-stable state is actually common for many smooth airfoils for Re< 10 5 , and during cruise conditions (at low, pre-stall), these airfoils could naturally experience abrupt 167 changes in aerodynamic eciency. This is why airfoil selection is so critical at moderateRe, and this paper shows how the potentially catastrophic, abrupt changes in bothL andD can be controlled, locally on the wing. Ultimately the purpose of the local acoustic forcing is to selectively amplify intrinsic ow instabilities so as to eciently exert a strong in uence on the ow through a small control amplitude, measured in acoustic power, or electrical drain on a battery. We are also interested in using the local ow control to understand further which instability mechanisms are most receptive and under which conditions. This paper presents the acoustic forcing results only and further work needs to be done to link this more closely with the various possible instability modes and amplication mechanisms. B. Facility Independence The internal acoustic excitation results reported here dier from earlier reported external acoustic excitation results on the same wing [4]. The most prominent dierences in wing performance can be seen in theL=D vs. f e graphs. AtRe = 60k, the (L=D)(f e ) curve from external forcing is discontinuous, showing particular preferentialf e values that increaseL=D, while the equivalent curve from internal forcing is continuous, showing that all values of f e within a given range will increase L=D. Results from the external forcing study determined a dependence of optimumf e on wind tunnel resonances, which explains why only a selection of f e values improve wing L=D. There is no such dependence for internal acoustic forcing, so this eciency enhancement and separation control technique can, in principle, be applied to standard ying devices in open ight. C. Spanwise Slots All previous studies of internal acoustic excitation have used a single speaker lying be- neath an uncovered spanwise channel or slit in the wing [6, 7, 8, 9, 10]. Some of the wing models [7] not only contained uncovered spanwise slits but also additional pressure taps along the suction surface. In light of the study by Yang et al. on Helmholtz resonance from open holes [5], there is reason to question the validity of the true nominal performance of 168 Figure 10.23: (L=D) of the E387 wing (a) with external acoustic forcing at = 8 (replot- ted from Yang et al. [4]) and (b) with internal acoustic forcing from 15 speakers at c 0 = 0:1 at = 9 . any wing with open cavities regardless ofRe, since the resonance mechanism is independent of ow speed. The modications in wing performance previously observed in the literature may have originated, not from pure internal acoustic excitation, but from a combination of acoustic resonance and internal forcing eects, as observed here (e.g. Fig. 10.22). Although most previous studies were for airfoils and wings at large, post-stall, when Helmholtz resonance would be unimportant, at lower , it is quite likely that a combined resonance and internal forcing phenomenon would cause the observed changes in wing performance. V. Conclusion There is a particular practical range of ight Reynolds number, 10 4 Re 10 5 , where, at pre-stall , the ow over the suction surface of an airfoil or nite wing can suddenly 169 switch states that we call SI and SII. In SI the laminar boundary layer simply separates at some point before the trailing edge. In SII, separation is closer to the leading edge, and reattachment occurs following a laminar separation bubble. SII is associated with much higher L=D than SI. Here we show that local acoustic forcing can be used to selectively control local ow separation, and combinations of active and passive acoustic forcing allow access to an envelope of L=D() curves. Special care must be taken to separate out these two contributors, which have often been combined in the literature. Local acoustic forcing appears to be eective in modifying the ow but the mechanism is not yet clear. Future work on calculating unstable modes of both attached and separated ow proles will help, and further practical steps include measuring the roll moments on a wing under asymmetric internal acoustic forcing. With proper spacing, frequency, and amplitude selection, small embedded speakers could replace movable control surfaces for small-scale ying devices. 170 Paper IV References [1] Marxen, O. and Rist, U., \Mean ow deformation in a laminar separation bubble: sepa- ration and stability characteristics," Journal of Fluid Mechanics, 660, pp. 37-54 (2010) [2] Marxen, O. and Henningson, D.S., \The eect of small amplitude convective disturbances on the size and bursting of a laminar separation bubble," Journal of Fluid Mechanics, 671, pp. 1-33 (2011) [3] Marxen, O. and Lang, M. and Rist, U., \Discrete linear local eigenmodes in a separating laminar boundary layer," Journal of Fluid Mechanics, 711, pp. 1-26 (2012) [4] Yang, S. L., and Spedding, G. R., \Separation Control by External Acoustic Excitation on a Finite Wing at Low Reynolds Numbers," American Institute of Aeronautics and Astronautics, 51, 6, pp. 1506-1515 (2013) [5] Yang, S. L., and Spedding, G. R., \Passive Separation Control by Acoustic Resonance," Experiments in Fluids, 10, 54, pp. 1-16 (2013) [6] Hsiao, F. B., Jih, J. J., and Shyu, R. N., \The Eect of Acoustics on Flow Passing a High-AOA Airfoil," Journal of Sound and Vibration, 199, 2, pp. 177-188 (1997). [7] Hsiao, F. and Liu, C. and Shyu, R., \Control of Wall-Separated Flow by Internal Acoustic Excitation," American Institute of Aeronautics and Astronautics, 28, 8, pp. 1440-1486 (1989). [8] Chang, R. C., Hsiao, F. B., Shyu, R. N., \Eect of Acoustics on Flow Passing a High- AOA Airfoil," Journal of Sound and Vibration, 199, 2, pp. 177-188 (1997). [9] Huang, L.S., Maestrello, L., Bryant, T. D., \Separation Control Over an Airfoil at High Angles of Attack by Sound Emanating From the Surface," AIAA Paper No. 87-1261, (1987). [10] Huang, L.S. and Bryant, T. D. and Maestrello, L., \The Eect of Acoustic Forcing on Trailing Edge Separation and Near Wake Development of an Airfoil," American Institute of Aeronautics and Astronautics, 1st National Fluid Dynamics Congress, (1988). 171 [11] Zabat, M., Farascaroli, S., Browand, F., Nestlerode, M., Baez, J., \Drag Measurements on a Platoon of Vehicles," Research Reports, California Partners for Advanced Transit and Highways (PATH), Institute of Transportation Studies, UC Berkeley (1994) [12] McArthur, John. \Aerodynamics of Wings at Low Reynolds Numbers," Ph.D. Disser- tation, Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA (2007) [13] Spedding, G. R. and McArthur, J., \Span Eciencies of Wings at Low Reynolds Num- bers," Journal of Aircraft, Vol. 47, No. 1, 2010, pp. 120-128. [14] Gaster, M. and Grant, T., \An experimental investigation of the formation and devel- opment of a wave packet in a laminar boundary layer," Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 347, pp. 253-269 (1975). [15] Schlichting, H., \Boundary-Layer Theory," 539, (1968). [16] Lin, J. C. M., Pauley, L. L., \Low-Reynolds Number Separation on an Airfoil," Ameri- can Institute of Aeronautics and Astronautics, 34, 8, (1996). [17] McAulie, B. R. and Yaras, M. I., \Transition Mechanisms in Separation Bubbles Under Low and Elevated Free Stream Turbulence," Proceedings of the ASME Turbo Expo 2007, (2007). [18] Diwan, S. S. and Ramesh, O. N., \On the Origin of the In ectional Instability of a Laminar Separation Bubble," Journal of Fluid Mechanics, 629, pp. 263-298 (2009). [19] Jones, L.E. and Sandberg, R.D. and Sandham, N.D., \Stability and receptivity charac- teristics of a laminar separation bubble on an aerofoil," Journal of Fluid Mechanics, 648, pp. 257-296 (2010). 172 Paper V 173 Chapter 11 Stability Analysis of Experimental Velocity Proles Using an Orr-Sommerfeld Solver Yang, S. L. Aerospace and Mechanical Engineering Department University of Southern California Los Angeles, California 90089-1191 Empirically-acquired velocity proles of separated ow and sound-induced reattached ow over a wing are examined for their initial instability properties. The most unstable frequencies at various streamwise (chordwise) stations across the wing surface are obtained by numerical analysis of the Orr-Sommerfeld equation. The velocity proles come from wind tunnel experiments on an airfoil that is particularly sensitive to small external disturbances which can promote an originally separated, low-lift state ow into a reattached, high-lift state. Numerical results of the initial instability developments are compared to experimental results for separation control by acoustic excitation as a further step in determining the mechanism by which sound aects boundary layer uid ow. 174 I. Introduction In aeronautics, when the chord-based Reynolds number Re = UC= (where U is the freestream velocity,C is the wing chord, and is the kinematic viscosity) lies in the particular regime 10 4 Re 10 5 , uid ow over a surface undergoes complex and complicated ow phenomena. In this Re regime, uid ow in a boundary layer is extremely sensitive to small environmental or geometry variations, causing uid ow to be prone to separation with possible turbulent reattachment. The sensitivity to small disturbances and changes is a result of the onset and amplication of instabilities in the boundary layer. A developing boundary layer contains several stages through which initially laminar ow can transition to a fully turbulent ow. Slightly downstream of the leading edge, small- amplitude, viscous instabilities, commonly referred to as Tollmien-Schlichting (T-S) insta- bilities dominate the still laminar boundary layer. When amplied, T-S waves can grow into larger, three-dimensional instabilities. When the T-S instabilities reach large enough amplitudes, the ow no longer remains attached to the surface, and the boundary layer becomes a separated shear layer dominated by inviscid separated shear layer instabilities. Also referred to as Kelvin-Helmholtz (K-H) instabilities, they cause the shear layer to roll up and have been shown in experiments to be responsible for shear layer and separation bubble unsteadiness [1]. Secondary instabilities can continue to grow into turbulent spots, which can then initiate the transition to fully turbulent boundary layer ow [2]. The linear transition to turbulence involves receptivity, linear stability, and nonlinear breakdown. In the receptivity stage, where the localRe is low, T-S instabilities are generated when a disturbance of longer wavelength (i.e., sound or vorticity) enters the boundary layer and disturbs the resulting ow [6]. Since receptivity involves the generation, rather than the evolution, of instability waves in the boundary layer, a wavelength conversion mechanism is required to transfer energy from the longer free stream disturbance to the shorter T-S instability [7]. The linear stability stage of transition involves the slow (viscous), linear growth of disturbances, where T-S waves propagate down the boundary layer and are either 175 amplied if the ow is unstable, or attenuated. Sometimes the disturbance might decay over a considerable distance before being amplied [8]. The present study is directed towards extracting properties of the initial growth of in- stabilities in the boundary layer of a wing at Re = 6 10 4 from experimentally-obtained velocity proles. Wind tunnel studies on separation control by acoustic excitation for the Eppler 387 airfoil at low Re [3, 5] showed that certain sound source distributions, forcing frequencies, and sound amplitudes can generate up to 70% increases in lift-to-drag ratio, L=D, by shifting the location of the separation point on the wing surface and prompting transition from an initially low-lift, separated state into a high-lift, reattached state. While results indicated that local separation control was achievable, the mechanism by which the acoustic disturbances were aecting the intrinsic ow over the wing was unclear. Attempts were made to experimentally measure the most amplied frequencies in the boundary layer over the wing surface, but the presence of a measurement probe in the shear layer aected the nominal performance of the wing, inhibiting ow reattachment and the expected transition from low- to high-lift states. Previous experiments in the literature also used measurement probes to obtain the most amplied instabilities, but there was little validation reported on the eects of the probes on the nominal performance of the wing models. The study reported here is a continuation of the experimental work in the Dryden wind tunnel on separation control of wings at low Re [3, 5, 4]. Here, details are reported on a numerical approach to solve for the initial growth of boundary layer instabilities of the E387 airfoil using empirical data from non-intrusive particle imaging velocimetry (PIV). The stability problem is a that of a boundary layer ow that is assumed to be parallel and 2-dimensional. An Orr-Sommerfeld solver based on the solver described in [17] and used in [11] and [10] solves the spatial stability problem for the initial behavior of the ow, and the amplied frequencies are extracted and compared to the optimum forcing frequencies observed from experimentation. 176 II. Empirical Velocity Proles Velocity proles U(z) for the E387 wing at Re = 60k and = 9 are obtained by non- intrusive PIV methods described in [3, 4, 5]. In this coordinate system, x is streamwise,y is spanwise, andz is normal to the wing surface, and the origin is dened to be at the leading edge and midspan of the wing (Fig. 11.1). Figure 11.1: Coordinate system is dened as: x is streamwise,y is spanwise, andz is normal to the wing. Flow eld images, which are taken with a 210-mm lens, have approximately 200 pixels/cm image resolution. The ow eld data are processed on a grid that yields 80 points per velocity prole from a customized algorithm described in [13] and [14]. The free stream velocity can be subtracted out of the total velocity proles to leave only the uctuating velocity,u. Figure 11.2 depicts the total and uctuating velocity proles,U(z) andu(z), respectively, at various chordwise stations for the wing atRe = 60k and wing incidence = 9 . Figure 11.2a shows velocity proles over the wing in its nominal state (without acoustic excitation). Points of in ection occur at x=C = x 0 0.4. For x 0 > 0.4, the ow is separated, and reverse ow occurs where @U @x < 0. Figure 11.2b shows velocity proles over the wing with internal forcing from ve speakers located at x 0 = 0.1, all forced in phase at f e = 900 Hz. Under internal acoustic forcing, in ection points occur much closer to the leading edge (x 0 0:1) where a laminar separation bubble begins to forms over the front half of the airfoil, and the ow is mostly attached over the back half. For each velocity prole, the second derivativeU 00 (z) is numerically solved for by a nite dierence scheme. The value of U 00 (z = 0) is not clear from the current PIV data due 177 Figure 11.2: Total and uctuating velocity proles, U and u, respectively, for the wing (a) without forcing and (b) with internal forcing from 5 speakers at f e = 900 Hz at x 0 = 0.1. Re = 60k and = 9 . Velocity values are arbitrarily scaled. to inadequate image resolution near the boundary. Therefore, a 1-d linear interpolation must be done on U 00 (z), and the value of U 00 (z = 0) is initially extrapolated to zero. It should be noted that the extrapolation to U 00 = 0 at z = 0 is a simple assumption; an accurate extrapolation requires more information about the ow at the boundary, which is later discussed. Figure 11.3 shows an example prole, U(z), and corresponding interpolated second derivative, U 00 (z), taken at midspan and x 0 = 0.1 for the unexcited wing at Re = 60,000 and = 9 . U(z) andU 00 (z) are normalized by the free stream velocity, U 1 , and the vertical distance z is normalized by the displacement thickness given by = Z 1 0 1 U U 1 dz: (11.1) 178 Figure 11.3: Empirical velocity proleU(z) (left) and second derivativeU 00 (z) with interpo- lation to U 00 (z = 0) = 0 (right) at x 0 = 0.1 for the unexcited wing at Re = 60,000, = 9 . The original prole U(y) is obtained from PIV data taken at midspan. III. Orr-Sommerfeld Solver The linear instability properties of the empirical velocity proles are determined by nu- merical solutions of the Orr-Sommerfeld equation [12], (U(z)c) 00 (z)k 2 (z) U 00 (z)(z) = 1 ikRe 0000 (z) 2k 2 00 (z) +k 4 (z) : (11.2) In Eq. (11.2), (z) is the amplitude function, k is the wave number, and c is the phase speed. k and c are related by the frequency, !: c = ! k : (11.3) The Reynolds number in Eq. (11.2) is based on from Eq. (11.4), such that Re = U : (11.4) 179 A. Numerical Integration of the Orr-Sommerfeld Equation The Orr-Sommerfeld numerical solver is based on the solver described in [9]. The Orr- Sommerfeld solver uses a shooting technique for the two-point valued boundary problem. The asymptotic boundary condition far from the wall (z! +1) can be obtained using the asymptotic nature of U(z). When the form of the solution is known, the Orr-Sommerfeld equation is cast in terms of a set of rst order dierential equations that are integrated towards a matching point above the wing surface. Here, a matching condition is satised with proper choices of k and c. The current study is a spatial stability analysis, whereby the wave number k is complex and the particle frequency ! is real. The spatial instability growth rate is given by k i , the imaginary part of the wave number. k i > 0 signies that the ow is initially contained and stable, while k i < 0 signies that the ow is initially unbounded and unstable, so the condition associated with the minimum value of k i (or maximumk i ) is the most unstable condition where the instability has the fastest initial growth rate. Note that this study extracts initial instability properties and not the fully developed properties. B. Boundary Conditions Far from the boundary, derivatives in the velocity eld are small, whereby the term in Eq. (11.2) containing U 00 (z) can be neglected as z!1. Then, the asymptotic form of the eigenfunction (z) can be found: (z) =Ae kz +Be kz +Ce z +De z : (11.5) The exponential solutions in k are homogeneous (inviscid) solutions, and the exponential solutions in are the particular (viscous) solutions, where is given by = k 2 + (ikRe) (Uc) 1=2 : (11.6) 180 Far above the boundary (z! +1), only the decaying solutions may exist, so thatA =C = 0. At the boundary (or \wall"), z =z wall = 0; (z wall ) = 0; (z wall ) 0 = 0: (11.7) C. A Shooting Method A shooting method is implemented to solve the two-point boundary valued problem. Integration of the ordinary dierential equation starts from an initial point z = +z i far above the boundary (ie, the surface of the airfoil), proceeding downward, as well as from the boundary at z = 0, proceeding upward. The fourth-order o.d.e. is simplied into a set of rst order o.d.e.'s d ~ dz =F ~ ; Re ; (11.8) where ~ = [ 0 00 000 ]: (11.9) The solution vectors at z =z i are ~ i = 2 6 6 6 6 6 6 4 e kz ke kz k 2 e kz k 3 e kz 3 7 7 7 7 7 7 5 and ~ v = 2 6 6 6 6 6 6 4 e z e z 2 e z 3 e z 3 7 7 7 7 7 7 5 ; (11.10) and the solutions at z = 0 are 181 ~ i = 2 6 6 6 6 6 6 4 0 0 1 1 3 7 7 7 7 7 7 5 and ~ v = 2 6 6 6 6 6 6 4 0 0 1 i 3 7 7 7 7 7 7 5 ; (11.11) where the subscript \i" corresponds to the inviscid (homogeneous) solution, and the subscript \v" corresponds to the viscous (particular) solution. The solution vectors are integrated towards the matching point,z =z m , above the bound- ary, and a fourth order Runge-Kutta integration scheme is used for marching the solutions towards the matching point. At the matching point, the Wronskian of the eigenvectors must vanish with proper choices ofc andk. The Wronskian is obtained from the four eigenvectors in Eqs. (11.10) and (11.11), W = + i + 0 i + 00 i + 000 i + v + 0 v + 00 v + 000 v i 0 i 00 i 000 i v 0 v 00 v 000 v ; (11.12) where the \+" corresponds to solutions marched downward from z = z i (Eq. (11.10)), and the \-" corresponds to solutions marched upward from z = 0 (Eq. (11.11)). When the absolute value of the Wronskian is small, for examplejWj< 110 7 , then the eigenvalue has been chosen correctly. If the eigenvalue has not been properly chosen, an iterative technique that expands a Taylor series for the Wronskian is employed, whereby W n+1 =W n + @W n @c j n (c n+1 c n ) +::: (11.13) wheren corresponds to the current estimate ofc, andn+1 corresponds to the next calculation based on the updated estimate of c. In order to converge on the eigenvalue, W n+1 = 0, so an updated estimate of c is found from the Taylor series expansion as 182 c n+1 =c n W n @W @c j n : (11.14) This iteration process continues until the convergence criterion for W is met. D. Iteration Process The iteration process for determining the eigenvalues is described here. A value of fre- quency, f, is rst provided, which is converted into ! by the relation ! = 2f. An initial guess is made for k = k r + ik i , which is taken from pre-denedfk r ;k i g pairs. These pairs are formed by the relations k r =r cos; k i =r sin; (11.15) where r is a radius and is an angle. 10 arbitrary values of r between 0.0011.0 and 10 arbitrary values of between 0 360 yield a total of 100fk r ;k i g pairs that are the initial guesses for k. Each guess for k corresponds to an initial \guess" for c from the relation in Eq. (11.3). For eachk, an iteration process is performed onc until the convergence criterion onW is met. For each!, 100 values ofk are computed. The value of interest is thefk r ;k i g pair containing min(k i ) for k r > 0. The general iteration process is as follows: Input f (in Hz) and compute ! = 2f Make rst guess for k =k 1 using (r 1 ; 1 ) from Eq. (11.15) { Calculate c 1 =!=k 1 ; compute W 1 { Make second guess for c =c 2 ; compute W 2 and @W @c j 2 { Estimate a new guess for c =c 3 { Iterate on c until the convergence criterion for W is met, at which point a nal value c =c f is found { Calculate k f;1 =!=c f , the nal k value for the rst guess 183 Make a new guess for k =k 2 using (r 1 ; 2 ) Iterate on c; calculate k f;2 Repeat process for all (r;) pairs to obtainfk f;1 ;k f;2 ;:::;k f;100 g Find min(k i ) for k r > 0 This process is then repeated for various values of f. E. Loss of Linear Independence As pointed out in [9], the loss of linear independence is common to o.d.e.'s which have exponential eigenvalue solutions with real parts that vary greatly. In the Orr-Sommerfeld equation, the inviscid eigenfunction i has exponential solutions away from the boundary with growth ratesk, which are of order O(1), while the viscous eigenfunction v has exponential solutions away from the boundary of order O(Re 1=2 ), such that Re( )Re(k): (11.16) The linear independence of the two eigenfunctions is lost as the slower growing solu- tion, i , becomes contaminated through truncation error by the faster growing solution, v . The contaminated growth occurs when explicit integration methods are used, so a pseudo-orthogonalization technique is implemented to maintain the uniqueness of the two eigenvectors. At each time step of the integration solver during the Runge-Kutta marching process, each i and v undergo the pseudo-orthogonalization process. The norms of the eigenvectors are given by jjN i jj = 00 i 000 i 000 v 00 v (11.17) and 184 jjN v jj = 000 v 00 v 00 i 000 i : (11.18) Each of the four components of the eigenvector undergo pseudo-orthogonalization such that ~ i = ~ i 000 i 000 v ~ v jjN i jj ; (11.19) and ~ v = ~ v 00 v 00 v ~ i jjN v jj ; (11.20) whereby the new eigenvectors have the form ~ i = ( i 0 i 1 0); (11.21) and ~ v = ( v 0 v 0 1): (11.22) The dot product of Eqs. (11.21) and (11.22) is close to zero so that the linear indepen- dence of the two eigenvectors is maintained at each time step of integration. The values of ~ then become the new values of ~ for the next integration step. IV. Results Proles for the unexcited wing located atx 0 =f0:1; 0:2; 0:3; 0:4g (the red-colored proles in Fig. 11.4) were used in the Orr-Sommerfeld solver. At each value of f, a nal value of k meeting the condition min(k i ) fork r > 0 was obtained. Figure 11.5 shows the spatial growth rate, k i , as a function of frequency, f, for proles at the dierent x locations. The rst observation to note is that k i > 0 in all cases, indicating that the ow is initially bounded and stable. The least stable frequencies can be determined by nding the minima in the 185 k i vs: f curves shown in Fig. 11.5. At x 0 = 0.1 and 0.2, the k i vs: f curves are quite at. Fewer points were generated for thex 0 = 0.1 prole due to the elimination of obviously incorrect, singular results. More points were generated for the x 0 = 0.2 prole, for which a local minimum in k i occurs around f = 500Hz, and even lower values of k i at f > 1000 Hz. At x 0 = 0.3, min(k i ) occurs between 200300 Hz, and at x 0 = 0.4, min(k i ) occurs around 200 Hz. Figure 11.4: Velocity proles U(z) for the wing without forcing at Re = 60k and = 9 . Red-colored proles are at x 0 =f0:1; 0:2; 0:3; 0:4g. Figure 11.5: Velocity proles U(z) for the wing without forcing at Re = 60k and = 9 . Red-colored proles are at x 0 =f0:1; 0:2; 0:3; 0:4g. 186 V. Discussion A. Initial Comparisons of Experimental and Numerical Results In the internal acoustic forcing experiments reported in [5], the largest range of eective forcing frequencies (200 Hz f 1500 Hz) occurred at x 0 = 0.1, as shown in Fig. 11.6, replotted from Yang et al. At x 0 = 0.2 and 0.3, a single value of eective forcing frequency atf = 300 Hz exists, followed by a range of eective frequencies (800 Hzf 1200 Hz). For x 0 > 0:4, the eective range of frequencies is narrower (900 Hz f 1100 Hz). Of particular interest are the empirical and numerical results at x 0 = 0.3. The single eective frequency at 300 Hz that was left unexplained in the internal forcing experiments is the frequency where the minimum k i occurs in the present numerical analysis. If this frequency is in fact the initially least stable frequency, then in the experiments the ow was excited into a high-lift state when the acoustic frequency matched the initial, naturally occurring, least stable frequency at x 0 = 0.3. It should also be noted that x 0 = 0.3 is the location of the natural separation location for the wing at the given Re and . Correlation of experimental and numerical results are less obvious at pre-separation x 0 locations, although aft of the natural separation point (x 0 = 0.4) it seems that the values off include a harmonic of the initially least stable frequency in the boundary layer. Previously reported studies on internal acoustic forcing claim that when the acoustic source is near the separation point, the values of f match the separated shear layer instabilities, f s , or sub harmonics of f s [19, 20]. The present analysis and comparison to experimental results suggests that the wing is excited into a high-lift state when there is a direct matching of f with the naturally-occurring, least stable frequency in the boundary layer at the natural separation location. Aft of the natural separation point, the ow is excited into a high-lift state when f matches a harmonic of the naturally occurring, least stable boundary layer frequency. 187 Figure 11.6: (L=D) for various f e from internal acoustic forcing at dierent x=c locations at Re = 60k, and = 9 , from Yang et al. [5]. 188 B. Oversimplied Problem The present comparisons are merely speculations as there are several possible sources of inaccuracy in particular, improper ow assumptions and oversimplication of the problem. First, a parallel and 2-dimensional ow was assumed for the experimental velocity proles when the actual ow is over a curved surface that has a growing boundary layer with increas- ing streamwise location. The wing model is also nite in span, so three-dimensional ow behavior, especially with external disturbances, is likely to occur. For each velocity prole, the second derivative U 00 (z) was extrapolated to zero. However, U 00 (z) is more than likely non-zero at the boundary. Since the image resolution is inadequate to resolve the velocity at the boundary, other means to obtain the actual velocity information at z = 0 are required. One method is to use the relationship between velocity and pressure from the momentum equation. For a two-dimensional ow without external forces, the momentum equation gives U @U @x +W @U @z = 1 @P @x + @ 2 U @x 2 + @ 2 U @z 2 : (11.23) At the boundary, z = 0, U = 0; W = 0; @ 2 U @x 2 = 0: (11.24) Then, the two terms on the left-hand side and the @ 2 U @x 2 term on the right-hand side of Eq. (11.23) go to zero, which leaves the relation @ 2 U @z 2 = 1 @P @x : (11.25) If the pressure at dierent x locations on the wing surface is known, then Eq. (11.25) can be used to solve for U 00 (z = 0), and a proper extrapolation can be done. No pressure data was taken in the Dryden wind tunnel experiments, nor could it be; however, pressure on the 189 E387 airfoil has been measured at the sameRe in other facilities (i.e., [15, 16, 17, 18]), which could be implemented into the interpolation procedure. C. Rened Test Parameters The current analysis only used 100 dierent (r;) pairs to solve for k at each f. Sev- eral computational runs at x 0 = 0.1 gave obviously incorrect, singular values, which were discarded. The incorrect values most likely occurred because even after 100 (r;) pairs, the proper valuesk were still not found. Increasing the number of (r;) pairs, although increas- ing computational time, would expand the locus offk r ;k i g values, most likely improving the accuracy of the nal k values. D. Falkner-Skan Prole Matching Another approach for solving the initial stability properties is using Falkner-Skan proles instead of the experimental velocity proles in the numerical solver. For example, a family of velocity proles with perturbation terms to independently vary the strength of back ow velocity and depth of reserved- ow region, as used in [11], is given by U() =f 0 (;)ae ( 0 )= 0 ; (11.26) where f(;) is determined by the Falkner-Skan equation f 000 +ff 00 +[1 (f 0 ) 2 ] = 0; (11.27) with boundary conditions f(0) = 0; f 0 (0) = 0; f 0 (1) = 1: (11.28) Equations (11.26)(11.28) produce a family of proles for various a, 0 , and values. The experimental velocity proles can be compared to the family of Falkner-Skan proles, and the 190 closest matching Falkner-Skan prole to the experimental prole can then be used in the Orr- Sommerfeld solver. An obvious problem with this method is that there are likely no closely matching proles simply due to the nature of empirical results, but if experimental proles do closely match Falkner-Skan proles, issues such as having inadequate ow information at the boundary would be eliminated. VI. Conclusion Stability analysis is performed on empirical velocity proles using a numerical Orr- Sommerfeld solver. The results of the spatial stability problem describe only the initial behavior of the boundary layer ow, rather than the fully developed ow behavior. Prelimi- nary results from the numerical analysis indicate that the ow is initially bounded and stable in the boundary layer. Comparisons of the numerical results to the experimental results from which the velocity proles are taken suggest that internal acoustic excitation at frequencies matching the initially least stable frequencies in the boundary layer trigger the ow from a separated, low-lift state into a reattached, high-lift state. These preliminary ndings are still questionable due to improper ow assumptions and an oversimplied problem. Further tests can be done to obtain more accurate second derivative interpolations of the velocity proles, vary the test parameters in the solver, or compare experimental proles with Falkner-Skan proles. The results reported here are from a preliminary attempt to numerically determine the least stable frequencies in the initial development of the boundary layer and only serve as a basis for further investigation and verication. Acknowledgments This report could not have been generated without the help of Dr. Larry Redekopp, Eric Lin, Tawan Tantikul, and Debbie Hammond. Special thanks go to Eric Lin for dedicating much appreciated time and eort to help develop and test the solver. 191 Paper V References [1] Lin, J. C. M., Pauley, L. L., \Low-Reynolds Number Separation on an Airfoil," American Institute of Aeronautics and Astronautics, 34, 8, (1996). [2] Schlichting, H., \Boundary-Layer Theory," 539, (1968). [3] Yang, S. L., and Spedding, G. R., \Separation Control by External Acoustic Excitation on a Finite Wing at Low Reynolds Numbers," American Institute of Aeronautics and Astronautics, 51, 6, pp. 1506-1515 (2013) [4] Yang, S. L., and Spedding, G. R., \Passive Separation Control by Acoustic Resonance," Experiments in Fluids, In revision. (2013) [5] Yang, S. L., and Spedding, G. R., \Local Acoustic Forcing on a Finite Wing at Low Reynolds Numbers," American Institute of Aeronautics and Astronautics, In review. (2013) [6] Reshotko, E., \Boundary-Layer Stability and Transition," Annual Review of Fluid Me- chanics, 8, pp.311-349 (1976) [7] Kerschen, E. J., \Boundary Layer Receptivity Theory," The University of Arizona (1993) [8] Goldstein, M. E. and Hultgren, L. S., \Boundary-Layer Receptivity to Long-Wave Dis- turbances," in Annual Review of Fluid Mechanics, 21, pp. 477-488 (1989). [9] Hammond, D. A., \Solving the Orr-Sommerfeld Equation Using a Pseudo- Orthogonalization Technique," University of Southern California (1996) [10] Hammond, D.A. and Redekopp, L.G., \Global Dynamics and Aerodynamic Flow Vec- toring of Wakes," Journal of Fluid Mechanics, 338, pp. 231-248 (1997) [11] Hammond, D.A. and Redekopp, L.G., \Local and Global Instability Properties of Sep- aration Bubbles," Journal of Fluid Mechanics, 17, 2, pp. 145-164 (1998) [12] White, F.M., Viscous Fluid Flow, 2 (1991) 192 [13] Fincham, A. M. and Spedding, G. R. \Low Cost, High Resolution DPIV for Measure- ment of Turbulent Fluid Flow," Experiments in Fluids, 23, pp. 449-462 (1997) [14] Fincham, A. M. and Spedding, G. R. \Advanced Optimization of Correlation Imaging Velocimetry Algorithms," Experiments in Fluids, 29, pp. 13-22 (2000) [15] McGhee, R. J. and Walker, B. S. and Millard, B. F., \Experimental Results for the Eppler 387 Airfoil at Low Reynolds Numbers in the Langley Low-Turbulence Pressure Tunnel," NASA, 4062, pp. 204 (1998) [16] Selig, M.S. and Donovan, J.F. and Fraser, D.B., \Airfoils at Low Speeds Data," Soartech8, H. A. Stokely, Virginia Beach, VA (1989) [17] Selig, M. S., Guglielmo, J. J., Proeren, A. P., Giguere, P., \Summary of Low-Speed Airfoil Data Vol. 1," pp. 19-21. SoarTech Publications, Virginia Beach, VA (1995) [18] Selig, M. S., Lyon C.A., Giguere, P., Ninham, C.P. and Guglielmo, J.J., \Summary of Low-Speed Airfoil Data Vol. 2," SoarTech Publications, Virginia Beach, VA (1996) [19] Hsiao, F. and Liu, C. and Shyu, R., \Control of Wall-Separated Flow by Internal Acous- tic Excitation," American Institute of Aeronautics and Astronautics, 28, 8, pp. 1440-1486 (1989). [20] Huang, L.S., Maestrello, L., Bryant, T. D., \Separation Control Over an Airfoil at High Angles of Attack by Sound Emanating From the Surface," AIAA Paper No. 87-1261, (1987) 193 Chapter 12 Concluding Remarks The aim and objectives of the current research were to experimentally investigate the laminar separation and reattachment process on a nite wing at low Reynolds numbers. In particular, acoustic excitation was chosen as the method for boundary layer and separation control. Both external and internal acoustic excitation were found to successfully control ow separation and improve the aerodynamic performance of an Eppler 387 wing. Acoustic resonance was further discovered to passively control ow separation. Numerical analysis of the instabilities of the experimental ow proles suggest that the mechanism by which sound interacts with boundary layer ow and improves wing performance is related to amplifying the least stable instabilities in the shear layer of the wing. Findings from the current research also force one to question the validity of many existing literature results on wing performance at low and moderate Reynolds numbers. A logical next step would be to investigate the eects of internal acoustic forcing on the stability characteristics of low Reynolds numbers wings and small-scale ying devices. 194 References [1] Ahuja, K., and Burrin, R. H. Control of ow separation by sound. In AIAA 9th Aeroacoustics Conference (1984). [2] Ahuja, K., Whipkey, R., and Jones, G. Control of turbulent boundary layer ows by sound. In AIAA 8th Aeroacoustics Conference (1983). [3] Boltz, F. W., Kenyon, G. C., and Allen, C. Q. The boundary layer transition characteristics of two bodies of revolution, a at plate, and an unswept wing in a low turbulence wind tunnel. Nasa, Ames Research Center, 1960. [4] Chang, R. C., H. F. B. S. R. N. Forcing level eects of internal acoustic excitation on the improvement of airfoil performance. American Institute of Aeronautics and Astronautics 298, 58 (1992), 823{829. [5] Collis, S. S., Joslin, R. D., Seifert, A., and Theofilis, V. Issues in active ow control: theory, control, simulation, and experiment. Progress in Aerospace Sciences 40 (2004), 237{289. [6] Crouch, J. D. Localized receptivity of boundary layers. Physics of Fluids 4, 7 (1992), 1408{1414. [7] Diwan, S. S., and Ramesh, O. N. On the origin of the in ectional instability of a laminar separation bubble. Journal of Fluid Mechanics 629 (2009), 263{298. [8] Dovgal, A., and Kozlov, V. On nonlinearity of transitional boundary-layer ows. Philosophical Transactions: Physical Sciences and Engineering 352, 1700 (1995), 473{ 482. [9] Fincham, A., and Delerce, G. Advanced optimization of correlation imaging ve- locimetry algorithms. Experiments in Fluids 29 (2000), 13{22. [10] Fincham, A. M., and Spedding, G. R. Low cost, high resolution dpiv for measure- ment of turbulent uid ow. Experiments in Fluids 23 (1997), 449{462. [11] Goldstein, M. Aeroacoustics. NASA SP. Scientic and Technical Information Oce, National Aeronautics and Space Administration, 1974. 195 [12] Goldstein, M. E., and Hultgren, L. S. The evolution of Tollmien-Schlichting wave near a leading edge part 2. Journal of Fluid Mechanics 129 (1983), 443{453. [13] Goldstein, M. E., and Hultgren, L. S. Boundary-layer receptivity to long-wave disturbances. In Annual Review of Fluid Mechanics, vol. 21. 1989, pp. 477{488. [14] Grundy, T. M., Keefe, G., and Lowson, M. Eects of acoustic disturbances on low re aerofoil ows. In Fixed and Flapping Wing Aerodynamics for Micro Air Vehicle Applications, T. J. Mueller, Ed. 2001, pp. 91{112. [15] Hammond, D., and Redekopp, L. Global dynamics and aerodynamic ow vectoring of wakes. Journal of Fluid Mechanics 338 (1997), 231{248. [16] Hammond, D., and Redekopp, L. Local and global instability properties of sepa- ration bubbles. Journal of Fluid Mechanics 17, 2 (1998), 145{164. [17] Hammond, D. A. Solving the orr-sommerfeld equation using a pseudo- orthogonalization technique. Tech. rep., University of Southern California, 1996. [18] Hong, G. Numerical investigation to forcing frequency and amplitude of synthetic jet actuators. American Institute of Aeronautics and Astronautics 50, 4 (2012), 788{796. [19] Hsiao, F., Liu, C., and Shyu, R. Control of wall-separated ow by internal acoustic excitation. American Institute of Aeronautics and Astronautics 28, 8 (1989), 1440{1486. [20] Hsiao, F., Shyu, R., and Chang, R. C. High angle-of-attack airfoil performance improvement by internal acoustic excitation. American Institute of Aeronautics and Astronautics 32, 3 (1994), 655{657. [21] Hsiao, F. B., Jih, J. J., and Shyu, R. N. The eect of acoustics on ow passing a high-aoa airfoil. Journal of Sound and Vibration 199, 2 (1977), 177{188. [22] Huang, L., Bryant, T. D., and Maestrello, L. The eect of acoustic forcing on trailing edge separation and near wake development of an airfoil. In 1st National Fluid Dynamics Congress (1988). [23] Huang, L., Maestrello, L., and Bryant, T. D. Separation control over an airfoil at high angles of attack by sound emanating from the surface. In AIAA 19th Fluid Dynamics, Plasma Cynamics and Lasers Conference (1987). [24] Huerre, P., and Monkewitz, P. A. Local and global instabilities in spatially developing ows. Annual Review of Fluid Mechanics 22 (1990), 473{537. [25] Kachanov, Y. Physical mechanisms of laminar-boundary-layer transition. Annual Review of Fluid Mechanics 26 (1994), 411{482. 196 [26] Kerschen, E. J. Boundary layer receptivity. In AIAA 12th Aeroacoustics Conference (1989). [27] Kerschen, E. J. Boundary layer receptivity theory. Tech. rep., The University of Arizona, 1993. [28] Kerschen, E. J., Choudhari, M., and Heinrich, R. A. Generation of boundary layer instability waves by acoustic and vortical free-stream disturbances. In Laminar- Turbulent Transition. 1990, pp. 477{488. [29] Klebanoff, P. S., Tidstrom, K. D., and Sargent, L. M. Wave mechanics of breakdown. Journal of Fluid Mechanics 12 (1962), 1{34. [30] Landahl, M. Wave mechanics of breakdown. Journal of Fluid Mechanics 56 (1972), 775{802. [31] Lang, M., Rist, U., and Wagner, S. Investigations on controlled transition devel- opment in a laminar separation bubble by means of lda and piv. Experiments in Fluids 30 (2004), 43{47. [32] Lin, J. C. M., and Pauley, L. L. Low-Reynolds number separation on an airfoil. American Institute of Aeronautics and Astronautics 34, 8 (1996), 1570{1577. [33] Lissaman, P. Low-Reynolds-number airfoils. Annual Review of Fluid Mechanics 15 (1983), 223{239. [34] Lyon, C. A., Broeren, A. P., Giguere, P., Gopalarathnam, A., and Selig, M. S. Summary of Low-Speed Airfoil Data, vol. 3. Soartech Publication, 1997. [35] Malkiel, E., and Mayle, R. E. Transition in a separation bubble. Journal of Turbomachinery 118 (1996), 752{759. [36] McArthur, J. Aerodynamics of Wings at Low Reynolds Numbers. PhD thesis, Uni- versity of Southern California, 2007. [37] McAuliffe, B. R., and Yaras, M. I. Separation bubble transition measurements on a low-re airfoil using particle image velocimetry. In Proceedings of the ASME Turbo Expo 2005 (2005), American Society of Mechanical Engineering. [38] McAuliffe, B. R., and Yaras, M. I. Transition mechanisms in separation bubbles under low and elevated free stream turbulence. In Proceedings of the ASME Turbo Expo 2007 (2007), American Society of Mechanical Engineering. [39] McAuliffe, B. R., and Yaras, M. I. Transition mechanisms in separation bub- bles under low- and elevated-freestream turbulence. Journal of Turbomachinery 132, 1 (2010). 197 [40] Murdock, J. W. The generation of a Tollmien-Schlichting wave. In Proceedings of the Royal Society of London (1980), vol. A 372, pp. 517{534. [41] Nishioka, M., Asai, M., and Yoshida, S. Control of ow separation by acoustic excitation. American Institute of Aeronautics and Astronautics 28, 11 (1990), 1909{ 1915. [42] Ono, K., and Hatamura, Y. A new design for 6-component force/torque sensors. In Mechanical Problems in Meausuring Force and Mass (1986), Springer Netherlands, pp. 39{48. [43] Reshotko, E. Boundary-layer stability and transition. Annual Review of Fluid Me- chanics 8 (1976), 311{349. [44] Roshko, A. On the drag and shedding frequency of two-dimensional blu bodies. Naca, 1954. [45] Saric,W.,White,E.B.,andReed,H.L. Boundary-layer receptivity to freestream disturbances and its role in transition. In 30th AIAA Fluid Dynamics Conference (1999). [46] Schlichting, H. Boundary-Layer Theory, vol. 539. McGraw-Hill, 1968. [47] Schubauer, G. B., and Skramstad, H. K. Laminar-boundary layer oscillations and transition on a at plate. NACA 909, National Bureau of Standards, 1948. [48] Selig, M. S., G. J. J. P. A. P. G. P. Summary of Low-Speed Airfoil Data, vol. 1. Soartech Publication, 1995. [49] Selig, M. S., C.A., L., Giguere, P., Ninham, C., and Guglielmo, J. Summary of Low-Speed Airfoil Data, vol. 2. Soartech Publication, 1996. [50] Shapiro, P. The in uence of sound upon laminar boundary layer instability. Tech. rep., Massachusetts Institute of Technology, 1977. [51] Simons, M. Aircraft Aerodynamics. Special Interest Model Books Ltd., 1999. [52] Tam, C. K. W. Excitation of instability waves by sound - a physical interpretation. Journal of Sound and Vibration 105, 1 (1986), 169{172. [53] Thomas, A., and Lekoudis, S. Sound and tollmien-schlichting waves in a blasius layer. Physics of Fluids 21, 11 (1987), 2112{2113. [54] Watmuff, J. Evolution of a wave packet into vortex loops in a laminar separation bubble. Journal of Fluid Mechanics 397 (1999), 119{169. [55] White, F. Viscous Fluid Flow, 2 ed. McGraw-Hill Professional Publishing, 1991. 198 [56] Yang, Z., and Voke, P. R. Large-eddy simulation of boundary-layer separation and transition at a change of surface curvature. Journal of Fluid Mechanics 439 (2001), 305{333. [57] Yarusevych, S., Kawall, J. G., and Sullivan, P. E. In uence of acoustic exci- tation on airfoil performance at low Reynolds numbers. In 23rd International Council of the Aeronautical Sciences Congress (2002). [58] Yarusevych, W., Kawall, J. G., and Sullivan, P. E. Airfoil performance at low reynolds numbers in the presence of periodic disturbances. Journal of Fluids Engineering 128 (2006), 587{595. [59] Yarusevych, W., Sullivan, P. E., and Kawall, J. G. On vortex shedding from an airfoil in low-Reynolds-number ows. Journal of Fluid Mechanics 632 (2009), 245{270. [60] Zabat, M., Farascaroli, S., Browand, F., Nestlerode, M., and Baez, J. Drag measurements on a platoon of vehicles. California partners for advanced tran- sit and highways (path), institute of transportation studies, University of California Berkeley, 1994. [61] Zaman, K. Eect of acoustic excitation on stalled ows over an airfoil. American Institute of Aeronautics and Astronautics 30, 6 (1992), 1492{1499. [62] Zaman, K., and Bar-Sever, A. Eect of acoustic excitation on the ow over a low-re airfoil. Journal of Fluid Mechanics 182 (1987), 127{148. [63] Zaman,K.,andMcKinzie,D. Control of laminar separation over airfoils by acoustic excitation. American Institute of Aeronautics and Astronautics 29, 7 (1991), 1075{1083. 199 Appendix A Derivation of Sound and Fluid Flow Equation For ow in the x-direction only, the mass conservation equation is @ @t + @ @x (u) = 0 (A.1) and the Navier-Stokes equation for incompressible ow without external forces is @u @t +u @u @x = 1 @p @x + @ 2 u @x 2 : (A.2) Assume an initially unperturbed state at rest (u 0 = 0) and small perturbations in density, pressure, and velocity: = 0 + 0 ( 0 0 ); p =p 0 +p 0 (p 0 p 0 ); u =u 0 : (A.3) Substituting the relations from Eq. (A.3) into Eqs. (A.1) and (A.2), neglecting all high-order 0 and u 0 terms, and using the relationship c 2 = p 0 0 (A.4) yields 200 @ 0 @t + 0 @u 0 @x = 0 (A.5) and 0 @u 0 @t +c 2 @ 0 @x = 0 @ 2 u 0 @x 2 : (A.6) Taking the divergence of Eq. (A.5), and applying @ 0 @x = 0 and @ 0 @t = 0, gives @ 2 0 @x@t + 0 @ 2 u 0 @x 2 = 0: (A.7) Taking the time derivative of Eq. (A.6), and applying @ 0 @x = 0 and @ 0 @t = 0, gives 0 @ 2 u 0 @t 2 +c 2 @ 2 0 @x@t = 0 @ @t @ 2 u 0 @x 2 : (A.8) Multiplying Eq. (A.7) by c 2 and subtracting it from Eq. (A.8) gives @ 2 u 0 @t 2 c 2 @ 2 u 0 @x 2 = @ @t @ 2 u 0 @x 2 : (A.9) The pressure perturbations, p 0 , can also be related to velocity perturbations, u 0 . The time derivative of Eq. (A.5) is @ 2 0 @t 2 + 0 @ @t @u 0 @x = 0; (A.10) and taking the divergence of Eq. (A.6) is 0 @ @t @u 0 @x +c 2 @ 2 0 @x 2 = 0 @ 3 u 0 @x 3 : (A.11) Subtracting Eq. (A.11) from Eq. (A.10) and relating pressure and density by Eq. (A.4) gives the relation 201 1 c 2 @ 2 p 0 @t 2 @ 2 p 0 @x 2 = 0 @ 3 u 0 @x 3 : (A.12) For a pure sound wave, the wave equation is @ 2 u 0 @t 2 c 2 @ 2 u 0 @x 2 = 0: (A.13) Take u 0 to be of the form u 0 ='exp [ikz i!t]: (A.14) It follows that @ 2 u 0 @t 2 = ! 2 u 0 ; (A.15) @ 2 u 0 @x 2 = k 2 u 0 ; (A.16) and @ @t @ 2 u 0 @x 2 = ik 2 ! u 0 : (A.17) Use the fact that ! =! r + i! i ; (A.18) where ! r and ! i are the real and imaginary parts of !, respectively. Substituting Eqs. (A.15)(A.17) into Eq. (A.9) and separating real and imaginary components yields ! i = 1 2 k 2 (A.19) 202 and ! 2 r ! 2 i =c 2 k 2 +! i k 2 : (A.20) Substituting Eq. (A.19) into Eq. (A.20) then gives ! 2 r k 2 =c 2 1 2 2 k 2 : (A.21) 203

Abstract (if available)

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Aerodynamics at low Re: separation, reattachment, and control

PDF

Aerodynamics of wings at low Reynolds numbers: boundary layer separation and reattachment

PDF

Aerodynamic investigation of a novel aircraft configuration

PDF

Modeling and analysis of parallel and spatially-evolving wall-bounded shear flows

PDF

Biomimetics and bio-inspiration for moderate Reynolds number airfoils and aircraft

PDF

Wing selection and design for flapping flight at small scale

PDF

RANS simulations for flow-control study synthetic jet cavity on NACA0012 and NACA65(1)412 airfoils.

PDF

Numerical modeling of separated flows at moderate Reynolds numbers appropriate for turbine blades and unmanned aero vehicles

PDF

Numerical study of shock-wave/turbulent boundary layer interactions on flexible and rigid panels with wall-modeled large-eddy simulations

PDF

Numerical study of flow characteristics of controlled vortex induced vibrations in cylinders

Asset Metadata

Creator Yang, Shanling (author)

Core Title Boundary layer and separation control on wings at low Reynolds numbers

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Aerospace Engineering

Publication Date 10/11/2013

Defense Date 09/12/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag acoustic excitation,active separation control,Aerodynamics,boundary layers,external acoustic excitation,flow control,fluid dynamics,Helmholtz resonance,internal excitation,local acoustic forcing,low Reynolds numbers,OAI-PMH Harvest,passive separation control,separation control

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Spedding, Geoffrey R. (committee chair), Bickers, Gene (committee member), Eliasson, Veronica (committee member), Radovich, Charles (committee member), Redekopp, Larry G. (committee member)

Creator Email shanling.yang@gmail.com,shanling.yang@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-337675

Unique identifier UC11296172

Identifier etd-YangShanli-2096.pdf (filename),usctheses-c3-337675 (legacy record id)

Legacy Identifier etd-YangShanli-2096.pdf

Dmrecord 337675

Document Type Dissertation

Format application/pdf (imt)

Rights Yang, Shanling

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA