BOUNDARY LAYER AND SEPARATION

CONTROL ON WINGS AT LOW

REYNOLDS NUMBERS

by

Shanling Yang

A Dissertation Presented to theFACULTY OF THE USC GRADUATE SCHOOLUNIVERSITY OF SOUTHERN CALIFORNIA

In Partial Fulfillment of theRequirements 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 first 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 effort 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, staff, 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, 104 ≤ Re ≤ 105,

fluid flow over a surface is prone to separation followed by possible reattachment and tran-

sition to turbulence. The amplification 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 flow separation at

transitional and moderate Re 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/Re1/2 are not in complete agreement among the different reported studies. Little atten-

tion has been given to distinguish the effects 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 affect boundary layer flows.

Results on boundary layer and separation control through acoustic excitation at low

Re numbers are reported. The Eppler 387 profile is specifically chosen because of its pre-

stall hysteresis and bi-stable state behavior in the transitional Re regime, which is a result

of flow separation and reattachment. External acoustic forcing on the wing yields large

improvements (more than 70%) in lift-to-drag ratio and flow reattachment at forcing fre-

quencies that correlate with the measured anti-resonances in the wind tunnel. The optimum

St/Re1/2 range for Re = 60,000 matches the proposed optimum range in the literature, but

there is less agreement for Re = 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 profile 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 flow 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

flow 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 effects.

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 effects. Activating various speakers in different spanwise and

chordwise distributions successfully controls local flow separation on the wing at Re = 40,000

and 60,000. The changes in aerodynamic performance differ from those observed through

external acoustic forcing, indicating that internal acoustic forcing is facility-independent.

Combining the effect of Helmholtz resonance and the effect of pure internal acoustic forcing

yields a completely different set of performance improvements. Since the internal acoustic

forcing studies in the literature did not separate these two effects, 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 profiles by means of a numerical

Orr-Sommerfeld solver, which extracts the initially least stable frequencies in the boundary

layer using parallel and 2-d flow assumptions. Velocity profiles 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 flow into a reattached, high-lift state. Preliminary

stability analysis of the flow at different chordwise stations for the wing in its nominal state

(without acoustic excitation) indicates that the flow 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 oversimplified

flow assumptions are most likely sources of inaccuracy since the Orr-Sommerfeld equation

is not generally used for separated flows or for boundary layers that grow significantly 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 flow separation and improve wing

performance, internal acoustic forcing is the more practical active control method for low

Re flying devices. Further investigations would entail investigating the effect of embedded

sound sources on the stability characteristics of wings and small-scale flying devices.

ix

Preface

This thesis considers the boundary layer and separation control methods of finite 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 Profiles Using an Orr-Sommerfeld

Solver,” 2013.

x

Chapter 1

Introduction

For much time now, small-scale flight 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/ν (where

U is the flight speed, c the chord, and ν is the kinematic viscosity), balances inertial and

viscous forces and can be classified into different regimes. In aeronautics, the low Re regime is

approximately Re ≤ 105 (Fig. 1.1). The low Re range can be further sub-divided: Re < 104

(ultra-low Re), which classifies most insect flight, and 104 ≤ Re ≤ 105, which classifies most

bird flight.

A growing number of micro aerial vehicles operate in the particular low Re sub-regime

between 104 and 105. At these Re, fluid flow is prone to laminar boundary layer separation

with possible transition to turbulent reattachment, thereby either favorably or adversely

affecting wing and aircraft performance. This sub-regime can consequently be termed the

transitional Re regime, where viscous effects cannot be ignored. In this regime, fluid flow

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 different facilities, indicating the difficulty 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 flying objects, from [33].

1.1 Aim and Objectives

The tendency of flow 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, α. Very few studies examined airfoils

that experienced flow separation at low, pre-stall α.

Correlations between the most effective 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 effective 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 flow, including a frequency matching of the acoustic wave and

a naturally occurring instability, and the presence of maximum velocity fluctuations at the

wing surface.

2

In general, the reported range of most effective frequencies is much larger (and even an

order of magnitude different) for internal acoustic forcing than for external acoustic forcing,

and the reported ranges differ slightly for various airfoils, Re, and α. The discrepancies

suggest the possibility of different mechanisms by which acoustic excitation changes the

flow 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 different parameters need to be

considered for the two types of acoustic forcing if each involves a different type of wave as

the flow-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 effects of acoustic resonance, which

is not mentioned in any of the reported studies. It is possible that the effects 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 finite wing in the transitional Re regime under the effect 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 flow is governed by the influence of inertia, viscosity, and external forces. The Navier-

Stokes equations for incompressible flow balances these elements by mass conservation and

momentum equations, given by

∇ · u = 0, (2.1)

∂u

∂t+ (u ·∇)u = −1

ρ∇p+ ν∇2u + fext (2.2)

where the velocity vector u = 〈u, v, w〉, ρ is the fluid density, p is the pressure, ν is the

kinematic viscosity of the fluid, and fext is the external force vector. In terms of the x -

component of the velocity, and assuming steady, two-dimensional flow and no external forces,

Eq. (2.2) can be expressed as

u∂u

∂x+ v

∂u

∂y= −1

ρ

∂p

∂x+ ν

(∂2u

∂x2+∂2u

∂y2

). (2.3)

4

The left-hand side of Eq. (2.3) is the net inertial force, the first 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 low Re, viscous forces dominate the inertial forces, especially in the laminar boundary

layer where the flow 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 flow 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 flow to remain attached to the surface, the flow must have sufficient energy

to overcome the adverse pressure gradient, the viscous dissipation along the flow path, and

the consequent energy loss due to changes in momentum. When the energy is insufficient,

the flow separates from the surface (Fig. 2.1a). The flow 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 flow contains dominating instabilities. Flow instabilities can be

classified as local and global, and further into convective and absolute [24, 5]. If localized

disturbances spread upstream and downstream and contaminate the entire flow, the velocity

profile is considered locally absolutely unstable, whereas if disturbances are swept away from

the source, the velocity profile 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 flow may be convectively unstable,

instantaneous velocity profiles of the disturbed basic flow 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 flow (a) and reattached flow (b).

Different 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 flat 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 amplified, T-S waves can grow into larger, three-dimensional

instabilities. Experimental results on boundary layer instability and the onset of turbulence

on flat 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 flat 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 flow 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 identified 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, α, decreases (and as the height of the

separated region decreases). Merged roll-up vortices have been observed [59] to shed at the

first 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 flat plate from [29] showed development of these secondary waves in the form of

7

Λ-shaped instabilities upon large enough amplification 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 flow [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 flow extends beyond the

end of the surface (for finite bodies), dominating instabilities are wake profile, or free wake,

instabilities.

2.3 Transition to Turbulence

Transition to turbulence occurs when perturbations enter into and influence the flow, and

upon large enough amplification, break down nonlinearly. The physical mechanisms that

prompt transition depend on the specific type of flow and the type of environmental distur-

bances. Transition in boundary layer flows can be classified into two main types [25]. The

first type involves boundary layer instabilities (initially described by linear stability theo-

ries), amplification, and interaction of various instability modes, which ultimately lead to the

breakdown of laminar flow. In this first type of transition, the environmental disturbances

are small. The second type, often referred to as bypass transition, involves the direct nonlin-

ear laminar flow breakdown under the influence of external disturbances. Bypass transition

occurs when high levels of environmental perturbations, such as free stream disturbances

and surface roughness, are present.

The first type of transition to turbulence can be separated into three main aspects:

receptivity, linear stability, and nonlinear breakdown (Fig. 2.3) [25]. In the first stage of

transition where the local Re 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 flow [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 flow 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 amplified if the flow is unstable, or attenuated. Sometimes

the disturbance might even decay for a considerable distance before being amplified [13]. This

stage is described by linear hydrodynamic stability theory, which can be used to describe

two- and three-dimensional flows 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 fluid flow enters a phase of nonlinear

breakdown, randomization, and a final transition into a turbulent state. In the breakdown

phase, the flow is transformed from a deterministic, regular, and generally two-dimensional

laminar flow into a stochastic yet ordered, three-dimensional flow [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 final 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 flowfield (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 amplification rates of linearized stability theory [55].

Fluid motion with components (u, v, w, p) can be decomposed into a basic flow consisting

of mean components (U, V,W, P ) and a superimposed perturbation motion consisting of

fluctuating components (u′, v′, w′, p′). For a two-dimensional incompressible flow with a

two-dimensional disturbance assumption, the mean velocity components can be assumed as

follows: U = U(y), V = W = 0. Boundary layer flow can be regarded approximately as a

parallel flow since the dependence of U on y is much greater than the dependence of U on x.

The pressure P (x, y) should be assumed to be dependent on x, as the pressure gradient ∂P∂x

can strongly affect the flow. For parallel flow at a surface, appropriate boundary conditions

require that u′ and v′ vanish at the walls to satisfy the no-slip condition.

10

The equations for continuity and momentum for an incompressible, two-dimensional flow

can be written in terms of the vorticity equation∗, given by

[∂

∂t+ (u · ∇)− ν∇2

]Ω = 0, (2.4)

where u = (u, v), ν is the kinematic viscosity, and Ω is the vorticity. Introducing a stream-

function, Ψ, and using the relations

u = ∂Ψ∂y, v = −∂Ψ

∂x(2.5)

and

Ω = −∇2Ψ, (2.6)

the vorticity equation in Eq. (2.4) can be written in terms of the streamfunction:

[∂

∂t+ Ψy

∂

∂x−Ψx

∂

∂y− ν∇2

]∇2Ψ = 0. (2.7)

To analyze the initial development of small perturbations of a fluid flow, it is assumed

that the mean flow is parallel and represented by U(y), nonlinear effects can be ignored, the

disturbances can be represented by traveling waves, and the maximum amplified disturbances

are two-dimensional. Using parallel flow assumption, the streamfunction can be written as

Ψ(x, y, t) = ψ(x, y, t) +

∫ ∞−∞

U(y)dy, (2.8)

where ψ(x, y, t) is the perturbation streamfunction, and∫∞−∞ U(y)dy is the mean flow con-

tribution. ψ(x, y, t) can be represented as a traveling wave of the form

ψ(x, y, t) = φ(y)ei(kx−ωt), (2.9)

∗The two-dimensional flow assumption omits the vortex stretching term.

11

where φ(y) is the amplitude function, k is the wave number and ω is the frequency. The

phase speed, c, is related to k and ω by the relation

c =ω

k. (2.10)

Linearizing the vorticity equation, Eq. (2.7), in terms of ψ and substituting in Eq. (2.9)

yields the Orr-Sommerfeld equation:

(U(y)− c)(φ′′(y)− k2φ(y)

)− U ′′(y)φ(y) =

1

ikRe

(φ′′′′(y)− 2k2φ′′(y) + k4φ(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 flow, 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 = ywall = 0,

φ(ywall) = 0,

φ(ywall)′ = 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 flows, and spatial stability corresponds to convectively

unstable flows. For boundary layer flows, 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 amplified or attenuated, spatial stability can

be assumed, whereby ωi = 0 and k = kr+iki. The term ki is the spatial growth rate, and the

minimum value of ki (or maximum −ki) at a given Re represents the most unstable initial

growth of the flow 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 flow, while increasing the amplitudes may trigger earlier separation, leaving

sufficient time and downstream surface for the flow to reattach. The amplitudes of boundary

layer instability waves can be affected by changing either the generation or propagation of

the waves [26]. In the traditional view of boundary layer flow control, the origin of the

inviscid inflectional instability is associated with the K-H instability in the separated shear

layer. However, in a modified view [7], the origin of the inviscid inflectional instability is an

extension of the instability caused by the adverse pressure gradient in the region upstream of

separation. In this modified view, there is seemingly no direct connection between T-S and

K-H instabilities, although each is independently connected to the upstream convectively

unstable inflectional instability [18].

Since flows in the transitional and low Re regimes are prone to separation, it is beneficial

to find efficient 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 flow path so that flow separation does not occur.

This method can delay, but will not always prevent, separation, and it is suitable for limited

flow 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 flow by enhancing or accelerating flow transition. Active

control involves introducing an external energy source to supplement the boundary layer

energy. Active control methods include acoustic excitation, vibrating wires and flaps, 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 flow receptivity to acoustic disturbances.

In general, acoustic forcing can be classified 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 influence of external sound on aerodynamic performance was experiments by

Schubauer and Skramstad on boundary layer transition of a flat plate [47]. In their exper-

iments, regularly oscillating velocity fluctuations 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 flow.

14

Acoustic forcing of symmetric and mildly cambered airfoils at low and moderate Re using

single frequency tones has been shown to effectively change wing performance. Improvements

include increasing lift at particular angles of attack [14, 62, 2, 1], tripping the flow from low-

to high-lift states [14], diminishing the size of the hysteresis loop in lift-drag curves [14],

reducing the tendency toward flow separation over an airfoil [62, 2], as well as changing the

behavior of the laminar separation bubble and turbulent boundary layer [63]. However, the

effects 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 α 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 α show dependence of optimum (or preferential) excitation frequencies

on Re, α, and the dominating instabilities in the flow. The optimum excitation frequency,

denoted as fe*, has been found to increase with increasing Re or increasing α [63], while

the range of effective excitation frequencies has been found to increase with increasing Re

or decreasing α [1, 57]. In some cases, the range of fe* 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 fe* cor-

respond to the most amplified instabilities that dominate the separated region. For pre-

and immediately post-stall α, K-H instabilities dominate the separated shear region, so fe*

correspond to these separated shear layer instability frequencies, fs [63, 57, 21, 61]. For large

post-stall α, the dominating instabilities are due to free wake vortices, so fe* correspond to

the free wake vortex shedding frequencies, fw [23, 41, 19, 4, 20]. The free wake frequencies

are reported to be an order of magnitude lower than the separated shear layer frequencies,

fw ≈ 0.1fs [19].

15

Increased pressure fluctuations on the surface of a NACA 65(l)-213 airfoil at Re ≈ 350,000

[1] were observed at optimum forcing frequencies. The increases in fluctuating pressure yield

increases in fluctuating 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 effectively reduced when the parameter St/Re1/2 was between 0.02 and 0.03, and

correspondingly, when fe* = fs. However, some external acoustic forcing results at moderate

Re [1] showed similar improvements in reducing separation at St/Re1/2 that was an order

of magnitude lower than the 0.02−0.03 range. Results from internal acoustic forcing tests

[19, 4, 21] report a larger optimum St/Re1/2 range of 0.001 − 0.02.

Since the laminar boundary layer thickness over a flat plate is

δ =5.2x√Rex

(3.2)

where x is the streamwise distance from the leading edge and Rex the Reynolds number at

the location x, the St/Re1/2 scaling is directly related to the growth of the boundary layer

thickness, δ. 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 α, which act like bluff bodies [44]), then the projected height of

the airfoil, c sinα, is used as the length scale, as shown in Fig. 3.1. A Strouhal number that

takes α into account can be given by

Stα =fc sinα

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

θs in Fig. 3.1. A corresponding separation Strouhal number is then

Stθs =fθsUe

(3.4)

where θs is the momentum thickness of the separated region, and Ue is the edge velocity

of the boundary layer [39]. An optimum range of Stθs is reported to be between 0.008 and

0.016 [39].

Figure 3.1: Different length scales for different 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 fluctuations are superimposed on the T-S waves. However,

the dominant transition process affected 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 efficiently, then the K-H instabilities, rather

than the T-S instabilities, should be given more attention in regards to exciting boundary

layer flow. 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

influencing the transition process requires the particular matching of the acoustic forcing

frequency to that of the naturally occurring instability.

3.4 Hysteresis

At transitional Re, 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 α as well as with

fe. Excitation promoted transition from low- to high-lift states at lower wing incidence,

α, and the high-lift state was maintained longer when α 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 α, the removal of the acoustic

forcing caused the flow to revert back to its original state, while at other α, even with the

removal of the forcing, the flow 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 insufficient excitation [14]. Similar behavior

occurred for a symmetric NACA 0025 airfoil at Re = 57k, 100k, and 150k, whereby once

flow reattachment occurred, either fe could be altered slightly from the optimum value fe*,

or the SPL could be lowered, and the boundary layer would remain attached and the flow

would remain in the high-lift state [58]. Acoustic forcing is therefore capable of promoting

hysteresis that depends on various factors, including α, fe, and SPL.

18

3.5 Angle of Attack

There are various reported results regarding the range of α for which acoustic forcing is

beneficial, which is seemingly determined by the unexcited performance of a given airfoil.

External acoustic excitation on a E61 airfoil at Re = 25,000 had little effect at high and low

α but the largest effect at α inside the hysteresis loop (α ≈ 6 − 11) [14], as shown in Fig.

3.2.

Figure 3.2: Lift coefficient for the E61 airfoil at Re = 25,000 without excitation and excitedat fe = 140Hz (St/Re1/2 = 0.025), replotted from [14].

For a NACA 65(l)0-213 at Re = 350,000 and the GA(W)-1 at Re = 600,000 and 800,000,

α ≈ 16 appeared to be affected the most by external acoustic forcing [2, 1]. For a NACA

633-018 airfoil at Re = 300,000 with internal acoustic excitation located at x/c = 0.0125

(near the leading edge), immediately post-stall α values experienced the most benefit. 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 α 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 633-018, 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 α in which airfoils benefit from acoustic

excitation depends on the unexcited performance of the given airfoil and Re.

Figure 3.3: Lift coefficient for the LRN(1)-1007 airfoil at Re = 75,000 without excitationand excited at fe = 342Hz (St/Re1/2 = 0.017), replotted from [61].

20

3.6 Sound Amplitude

The beneficial effects 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 did CLmax increase with increasing SPL, but the value of α at CL,max also increased,

as seen in Fig. 3.4. The magnitude of fluctuating 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 in CL, implying that

effective SPL depends on Re. While results from internal acoustic testing also agreed that

higher SPL values generate higher CL, some studies showed that lift improvement actually

diminished after a certain amplitude level [4], suggesting that there are thresholds for effective

SPL values.

Figure 3.4: Lift coefficient for the NACA 65(l)-213 airfoil at Re = 350,000 for unexcited andexcited flow at fe = 665Hz (St/Re1/2 = 0.007) with SPL = 139dB and 147dB, replottedfrom [1].

21

3.7 Tunnel Resonance

The effects of external acoustic excitation have been found to couple with and depend on

tunnel resonances. The resonances in wind tunnel test sections differ among facilities de-

pending on tunnel geometry. The fundamental cross resonances in a rectangular cross section

can be calculated by

fmn =a0

2π

[1−

(U0

a0

)2] 1

2 [(mπH

)2

+(nπW

)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, a0 is the speed of sound, and U0 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 fluctuations near the air-

foil. These cross-resonances had the greatest effect in changing the airfoil performance and

determined the optimum frequencies at which airfoil performance was improved. The effect

of individual frequencies on the flow 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 flow required a higher gain to trip, whereas at the frequencies

corresponding to tunnel anti-resonance (minima in sound level), the flow only required a

small gain to trip.

In a resonant duct, maxima in sound pressure level correspond to minima in induced

fluctuating velocity levels and vice versa. In acoustic wave fields, the gradient of pressure

is maximum where the pressure fluctuations cross zero and is consequently where the max-

imum fluctuations in velocity (90 out of phase with the pressure) are found. In a closed

structure, the standing wave nodes that correspond to pressure fluctuations equal to zero

22

(anti-resonances) will have minimum root-mean-square values. The nodes where the pres-

sure fluctuates 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 fluctuating velocity, which

occur at minimum SPL (tunnel anti-resonances). There are further claims [14] that the

mechanism by which acoustic disturbances affect the flow is through velocity fluctuations

rather than acoustic pressure fluctuations.

3.8 Forcing Location

A parameter specific 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 α = 0 and Re =

35,000 [22] showed that acoustic forcing applied ahead of the chordwise separation point,

denoted as cs, was found to be more effective in increasing lift than forcing applied aft of cs,

where higher SPL was required for the same effectiveness. Tests on a NACA 633-018 airfoil

at Re = 300,000 [19] showed that when the forcing location, which can be denoted as ce,

was approximately equal to cs, the performance of the wing was most affected by acoustic

forcing, especially in the post-stalled region. Separation occurred very near the leading edge

of the airfoil (cs/c ≈ 0.01), and so forcing nearest to cs yielded the most benefit, as seen in

Fig. 3.5.

While the effect of forcing was found to deteriorate as ce moved farther aft from cs,

the values of fe* were reported not to be a function of ce [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 coefficient for the NACA 633-018 airfoil at Re = 300,000 [19] for acousticforcing at ce/c = 0.0125, 0.0625 and 0.1375. fe = 110Hz (St/Re1/2 = 0.004).

Internal acoustic forcing emitted near cs over the top of a symmetric airfoil (ce/c ≈

0.15) at Re = 35,000 and at stalled angles of attack (α = 15, 20) was found to effectively

alter the separated region so that stall was delayed [23, 22]. When excitation was forced

near cs, fe* was found to be equal to the separated shear instability frequency, fs, or a sub

harmonic of fs. 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 flow to sound. Separation that otherwise occurs at the

leading edge in the immediate post-stalled region can be reattached when fe = fs and when

ce = cs [19].

Trailing edge acoustic forcing also alters the flow over a wing. Studies on internal acoustic

forcing emitted near the trailing edge of an airfoil at α = 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 effective forcing frequencies

were in the range 1/4fw to 2fw, the optimum forcing frequency was equal to the vortex

24

shedding frequency: fe* = fw. Acoustic disturbances generated pressure oscillations in the

separation region, which modified 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

unaffected by acoustic forcing, which suggests that the trailing edge acoustic forcing only

affects near wake flow. 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

flow out of the boundary layer, and the other half of the cycle is analogous to the injection

of high momentum flow 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 amplified 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 amplification of flow 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-

fied 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-flow 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 flow reattached).

26

Chapter 4

Mathematical Modeling of Flow and Sound

4.1 Sound and Fluid Flow Interaction

The interaction of sound waves and fluid flow can be mathematically modeled by using

the fact that sound perturbs the surrounding fluid. For simplicity, assuming flow in the

x-direction only, the continuity equation is

∂ρ

∂t+

∂

∂x(ρu) = 0 (4.1)

and the Navier-Stokes equation (Eq. (2.3)) without external forces is

∂u

∂t+ u

∂u

∂x= −1

ρ

∂p

∂x+ ν

∂2u

∂x2. (4.2)

An initially unperturbed state at rest is assumed, as well as small perturbations in density,

pressure, and velocity from the acoustic wave: ρ = ρ0 + ρ′, p = p0 + p′, and u = u′. 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

∂2u′

∂t2− c2∂

2u′

∂x2= ν

∂

∂t

(∂2u′

∂x2

). (4.3)

Eq. (4.3) describes a modified wave equation where the temporal and spatial changes

in velocity perturbations, u′, are related to both speed of sound, c, and viscosity, ν. The

27

viscous term involving the third partial of u′ 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

c2

∂2p′

∂t2− ∂2p′

∂x2= −νρ0

∂3u′

∂x3. (4.4)

The real part of the particle frequency, ω, is related to viscosity (derivation in Appendix A)

by the relation

ω2r

k2= c2 − 1

2ν2k2 (4.5)

where k is the wave number. For viscosity-dominant flows 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 flow in one direction; including more dimensions in the flow field to better pre-

dict three-dimensional flow inevitably involves additional higher-order partial terms of the

velocity perturbations in all three directions, (u′, v′, w′).

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 flow field 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 flat plate at

120,000 ≤ Re ≤ 380,000 [40] assumed two different 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 first case

(continuous wave interaction), and using the notation in [40], the total disturbance can be

represented as

u = A cos(αix− ωt) +B cos(ωt) (4.6)

where A is the T-S wave amplitude, B the sound wave amplitude, αi the instability wave

number, and ω 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

ω is

C =(A2 +B2 + 2AB cos(αix)

)1/2(4.7)

and when B A (the sound wave amplitude is much larger than that of the T-S instability),

Eq. (4.7) can be expanded to

C = A cos(αix) +B +O(A2)

(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 = ωxU

(where ω is the

disturbance frequency, x is streamwise location, and U the free stream velocity) was used

to classify the Orr-Sommerfield equation into two regimes: s ≤ O(1) and s > O(1). It was

found that the Orr-Sommerfield equation was valid only for s > O(1), in which case the T-S

wave and sound wave were found to be independent. For 0 ≤ s ≤ O(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, flat 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 different 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 (ω = 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

pi (x, y, t) = f (y) ei(αix−ωt) (4.9)

where f(y) is the instability amplitude distribution across the boundary layer, αi the insta-

bility wave number, and ω the instability frequency. The pressure associated with the sound

wave can be expressed as

ps (x, y, t) = B · g (y) ei(αsx−ωt) (4.10)

30

where αs is sound wave number (in the x-direction) and B the amplitude of the sound wave,

and the function g(y) is amplitude distribution of the sound wave across the shear layer.

Normally, αs > αi so that there is no interaction; however, if B is allowed to vary in x, then

ps can be expressed then as

ps (x, y, t) = B (x) · g (y) ei(αsx−ωt) =

∫ ∞−∞

B (k) · g (y) · ei[(k+αs)x−ωt] 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+αs).

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 flow velocity profile.

31

Chapter 5

Methods

5.1 Wing Models

All tests were performed on wings with an Eppler 387 profile (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 fit 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′ = 0.1,

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) profile view of the lid of the wing. Six spanwise rows of holesare 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 2Hz ≤ f ≤ 200Hz in

the velocity range 5m/s ≤ U ≤ 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 floor of the wind

tunnel, which connects to a custom force balance (described in detail in [60] and [36]) placed

below the wind tunnel floor. 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 deflection 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 gaugesetup.

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 different moment arms, and the force balance measurement has

an expected uncertainty of 0.1 mN, confirmed through calibration. The expected friction

coefficient on a flat plat can be given by

Cf =1.328√Rex

(5.1)

and the friction drag by

Ff = Cfq∞S (5.2)

35

where q∞ is the freestream dynamic pressure. For a flat 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, α is varied from −10 to 20 in steps of 1, and for some of the

acoustic studies, α is varied in steps of 0.1 inside the hysteresis loop. At least three tests

are performed for both increasing and decreasing α 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, δt = 100−300µs. 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 1µm paraffin-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 fitted 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 flow visualization as well as quantitative measurements of velocity

fields and profiles, and spanwise vorticity, ωy.

PIV processing uses a variant of the custom CIV algorithms described in [10] and [9].

A smoothed spline interpolated cross-correlation function is directly fit 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 field is reinterpolated back onto a rectangular grid with

36

the same smoothing spline function [24]. The spline coefficients are differentiated 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% in u,w and approximately

10% in ωy.

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 60Hz−15kHz, 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-

fied by an adjustable gain Pyle Pro PCA1 2 x 15 W stereo power amplifier with a frequency

response of 20Hz−40kHz ±3dB 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 400Hz−1000Hz ± 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 amplified with a Kramer VA-16XL

balanced adjustable gain stereo audio amplifier that has a frequency range of 20Hz−40kHz.

A 4944 1/4” B&K pressure field microphone, which has a pressure-field response of ±2

dB between 16Hz−70kHz, is used to obtain acoustic measurements for the wind tunnel

resonance study. A 4954-B 1/4” B&K free field microphone, which has a free-field response

37

of ±3 dB between 9Hz−100kHz, 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 profiles 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)(φ′′(z)− k2φ(z)

)− U ′′(z)φ(z) =

1

ikRe

(φ′′′′(z)− 2k2φ′′(z) + k4φ(z)

). (5.3)

Far from the boundary, derivatives in the velocity field are small, whereby the term in Eq.

(5.3) containing U ′′(z) can be neglected as z → ∞. Then, the asymptotic form of the

eigenfunction φ(z) can be given by

φ(z) = Aekz +Be−kz + Ceγz +De−γz. (5.4)

The exponential solutions in k are homogeneous (inviscid) solutions, and the exponential

solutions in γ are the particular (viscous) solutions, where γ is given by

γ =[k2 + (ikRe) (U − c)

]1/2. (5.5)

Far above the boundary (z → +∞), only the decaying solutions exist, so A = C = 0. At

the boundary (or “wall”), the boundary conditions are

38

z = zwall = 0,

φ(zwall) = 0,

φ(zwall)′ = 0.

(5.6)

A shooting method is implemented to solve the two-point boundary valued problem.

The ordinary differential equation is integrated downwards from an initial point z = +zi far

above the boundary, as well as upwards from the boundary at z = 0. The fourth-order o.d.e.

is simplified into a set of first order o.d.e.’s, yielding four solution vectors, two at z = zi

and two at z = 0. At both limits in z, an inviscid (homogeneous) solution φi and a viscous

(particular) solution φv exist. The solution vectors are integrated towards a matching point,

z = zm, 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 and k. 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, ω = kc, is real.

The velocity profiles at different chordwise locations for the E387 wing at Re = 60,000

and α = 9, which come from PIV results, are used in the Orr-Sommerfeld solver to obtain

the initial most amplified instabilities in the boundary layer. The frequency associated with

the value of k = kr + iki that contains min(ki) at kr > 0 is the most unstable frequency for a

given profile. 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

at Re = 30k. Local two-dimensional drag coefficients are measured at 35 span locations and

six α 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 flow is negligible. Variations in the spanwise profile drag,

cd(y), are related to local flow 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 flow. In the mid portion of the wing where the flow 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 profile and

induced drag components is realized, along with estimations for inviscid and viscous span

efficiency factors, ei and ev, 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 at Re = 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 flow visualization and corresponding measurements

of flow velocity and spanwise vorticity are obtained from PIV methods. Optimum forcing

frequencies (the frequencies that yield the largest improvements in aerodynamic efficiency,

L/D) are shared between the two test Re, although the range of effective frequencies is much

greater at the higher Re. 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 flow 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 flow around

the wing is effectively modified such that large, stable vortical structures appear in the sep-

arated shear layer. Correlation between the effects 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 profile 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

different lift and drag behavior than from the known behavior of the same profile wing.

Blocking air passage through the holes - either by filling 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 effect 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 different

spanwise rows of holes at various chord locations changes the mean chordwise separation line

location. PIV measurements show that the local flow at a given span station on the wing is

affected by the presence of a nearby open or closed hole, implying the ability to locally and

passively control flow separation. The wing with open holes (‘perforated wing’) is compared

to a woodwind instrument, and practical consequences for passive flow control strategies are

discussed together with potential problems in measurements through pressure taps in such

flow regimes.

6.4 Paper IV

This paper is the main paper of this thesis and details the successful local control of flow

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 amplifiers 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 flow visualization of local flow 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 differences in the separation line and

the presence of vortical structures depending on the activation of a local speaker. The effect

internal acoustic excitation and the effect of acoustic resonance from open holes are combined

and show significantly different results than those from the individual effects. Previously

reported studies found in the literature on internal acoustic forcing are questioned as those

studies do not distinguish the effects 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 flow instability properties of experimental velocity profiles and the results thereof. A

spatial stability problem using parallel and 2-dimensional flow assumptions is solved, and

the preliminary results indicate that the flow in the boundary layer is initially bounded

and stable. The least stable frequencies, which are associated with the minimum spatial

growth rates, ki, can be determined for profiles at different 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 verification due

to potentially improper flow assumptions and an oversimplification of the problem. More

studies can be done to vary different parameters in the solver, obtain better interpolations

43

of the experimental data, and even compare experimental velocity profiles to Falkner-Skan

profiles.

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 DepartmentUniversity 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 105 are both difficult and increasingly important in ap-

plication to small-scale aircraft. Not only are the aerodynamics strongly affected by the

dynamics of the unstable laminar boundary layer, but the flow is decreasingly likely to be

two-dimensional as Re decreases. The spanwise variation of the nominally-two-dimensional

flow along a two-dimensional geometry is often held to be responsible for the large varia-

tions in measured profile drag coefficient, cd at this scale. Here, local two-dimensional drag

coefficients are measured along a finite wing using non-intrusive PIV methods. Variations

in cd(y) can be related to local flow variations on the wing itself. Integrated values can

then be compared with direct force balance data, and the dynamical significance of spanwise

variability will be re-evaluated.

46

Nomenclature

AR = aspect ratio

b = wing half span (m)

c = chord (m)

cd = sectional profile drag coefficient

cd,PIV = sectional profile drag coefficient obtained from PIV measurements

C d,0 = minimum total drag coefficient on an infinite wing

CD = total drag coefficient on a finite wing

CD,PIV+FB = total drag coefficient on a finite wing from force balance and PIV

CD,i = induced drag coefficient on a finite wing

CD,i,FB = induced drag coefficient on a finite wing calculated from force balance measurements

CD,0 = minimum total drag coefficient on a finite wing

Cf = laminar skin friction coefficient

CL = total lift coefficient on a finite wing

cs = separation line location (m)

D ’ = drag per unit span (N/m)

ei = inviscid span efficiency

ev = viscous span efficiency

f = body force (N)

l = vertical transect in the wake region of the wing

n = normal vector to surface

p = pressure (N/m2)

47

p0 = free stream pressure (N/m2)

q = dynamic pressure (N/m2)

Re = Reynolds number

S = control surface around the wing

U = mean velocity vector (m/s)

U = mean component of velocity in x (m/s)

U0 = free stream velocity (m/s)

x, y, z = coordinates are streamwise, spanwise, and normal directions

α = angle of attack (deg)

δ = boundary layer thickness (m)

σ = standard deviation

θ = momentum integral (m)

ω = vorticity vectory (rad/s)

ωy = spanwise component of vorticity (rad/s)

Ω = normalized vorticity

〈*〉 = 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 finite 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 influence from the tip vortices. One

48

of the well-known characteristics that has not been measured for the finite wing however, is

the possible variation in sectional drag coefficient 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 coefficient in ex-

periments that combine optical flow 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 sufficient (or where

it is sufficient) and then compare the variations with the previously-reported variations on

two-dimensional geometry. Ultimately the findings will extrapolate out to small UAVs with

fixed 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 coefficients among different facilities

deteriorates as Re decreases, as seen in Figure 7.1. At Re = 60k, ∆cd(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 influence of wake rakes on

the flow [6].

The spanwise drag variation, cd(y), of an E374 wing section has been measured at various

angles of attack, α, and downstream locations, x/c, for Re = 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 different facilities replottedfrom [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 cd measurements are taken at some x/c (= 2.25 for Selig) where

pressure gradients are small and the local flow 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

sufficiency 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

in cd in Figure 7.1 are for Re < 100,000, and it is not clear how to extrapolate the results at

50

Figure 7.2: cd(y) of an E374 wing section at Re = 200,000, α = −6.4 at x/c = 1.0 (trailingedge) (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 flow properties are much

less predictable.

B. Drag Calculations

The momentum equations for a viscous flow in Einstein notation are

ρ∂ui∂t

+ ρuj∂ui∂xj

= −∂P∂xi

+ ρfi + µ∂2ui∂xj∂xj

(7.1)

where ui is the total instantaneous velocity comprised of mean, Ui, and fluctuating, ui,

parts (ui = Ui + ui), pi is the total pressure also comprised of mean and fluctuating parts

(pi = Pi + pi), ρ is the density of the fluid, µ is the dynamic viscosity, fi 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 α, Eq. (7.1)

can alternatively be written by decomposing into mean and fluctuating velocity components

to yield the Reynolds Averaged Navier-Stokes (RANS) equation:

ρ∂Ui∂t

+ ρUj∂Ui∂xj

= −∂P∂xi

+ ρfi + µ∂2Ui∂xj∂xj

− ρ∂uiujxi

(7.2)

where is the Reynolds stress tensor. In classical aeronautics applications, the flow around a

fixed wing in steady motion is assumed to be steady and inviscid. Furthermore, if a region

of flow is surveyed far from the body, then the pressure there can be assumed to be equal

51

to the constant, undisturbed free stream pressure, p0, and turbulent motion of the flow 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

ρUj∂Ui∂xj

= ρfi (7.3)

Eq. (7.3) relates the body force in any direction with the mean momentum flux in that

direction. It is then convenient to express this relationship in integral form so that forces in

one region can be related to fluxes through an enclosing control volume with surface area S

and corresponding normal vector n, so

∫S

ρUiUjnjdS =

∫S

ρfidS (7.4)

The component of the force in the streamwise direction, fx, can therefore be calculated

from the change in momentum flux between upstream and downstream surfaces, and when

the flow and body geometry are uniform in one direction, such as the span, the drag force

per unit span, D′, can be evaluated from just two line integrals:

D′ = −∫l1

ρ1U21 dz +

∫l2

ρ2U22 dz (7.5)

where ρ1 and U1 are the density and velocity of the fluid upstream of the body at a vertical

transect l1, and ρ2 and U2 are the density and velocity downstream of the wing at l2, which

contains all the wake region, W. If U1 = U0 and ρ = constant, the total positive drag on the

body with span b can be written as

D = ρU20 bθ (7.6)

where the momentum integral, θ depends only on the variation of mean velocity components

over W :

52

θ =

∫W

(U2

U0

−(U2

U0

)2)

dz (7.7)

The section profile drag coefficient cd is then

cd =2θ

c(7.8)

The drag formulation of Eqs. (7.4) − (7.8) differs 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 flow is negligible

(so average cross plane velocity components go to zero). Betz’s equation for profile drag [10]

takes the survey location only in the wake of the body by introducing a fictitious velocity

component that is non-zero only in the region of the viscous wake while the current method

does not use a fictitious velocity but rather takes the survey location far downstream. The

equivalent equation for profile 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 finite wing is

commonly described as the sum of two components:

Cd = cd + CD,i (7.9)

cd is the profile drag coefficient in Eq. (7.9), which is a function of α, and can be expressed

cd = cd,0 + cd (α) (7.10)

where cd,0 is the minimum drag coefficient for a 2-d wing section. CD,i is the induced drag

coefficient,

53

CD,i =C2L

πeiAR(7.11)

where CL is the lift coefficient of a finite wing with aspect ratio AR, ei ≤ 1 is the inviscid

span efficiency factor, which accounts for departures from the ideal elliptic spanwise load

distribution. It can be convenient and reasonable to write CD as a quadratic function of

CL, and then a viscous span efficiency factor, ev, which can be obtained through the slope

approximation of the CD-CL2 curve [3], can be used to write the total drag coefficient as

CD = CD,0 +C2L

πevAR(7.12)

where CD,0 is the minimum drag value from the CL-CD polar. The slope values from the CD-

CL2 curves differ between two-dimensional airfoils and finite 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 (profile 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 influential [13].

C. Objectives

This paper provides the first direct check on the spanwise variation of local cd 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 fields on the wing can be

checked for conditions that may cause the observed fluctuations. Direct association of the

force coefficients with the relevant flow field is still quite rare in aeronautics practice but is

important in regimes with such a rich variety of important flow behavior. The local drag

measurements come from PIV-derived velocity fields, 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 CL(CD) space

where abrupt switching between stable states can occur. Previous experimentation [15] on

the same wing and same conditions shows that the flow 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 flow in the test section. Paraffin-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 µs. Images were taken at 35 span stations spaced 1 cm apart, at

three streamwise locations (Figure 7.3), and at six angles of attack, α = 0, 2, 4, 6, 8, 10.

B. Spanwise Vorticity Measures

Persistent features in the time-averaged wake profiles 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 PIVimages were taken. The coordinate system origin is the leading edge at mid span. Thetrailing edge is x/c = 1.0. Inset: E387 profile.

magnitudes in the separation region can be related to the wake structure. Contrasting

regions of interest were studied for the ωy(x,z ) measurements at two α. For α = 0, where

the majority of the flow 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 profile. 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 significant fraction of the chord, such

as at α = 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 α = 0. The

separated region was defined by a triangle from mid-chord to the trailing edge (Figure 7.4b).

Figure 7.4: Regions of interest for spanwise vorticity fields. Top: α = 0 fore (a) and aft (b).Bottom: α = 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 ωy values were calculated

over that region to give a single r.m.s. vorticity value, 〈ωy〉i:

〈ωy〉i =

√√√√ 1

N

N∑k=1

(ωy (x, z ∈ i))2k (7.13)

where N is the total number of images. 〈ωy〉i was then normalized by the chord and mean

velocity and is denoted

〈Ω〉i =〈ωy〉i cU

(7.14)

The location of the separation point itself can be measured directly and independently

from raw particle images. At values of α 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 fluid

57

has come directly from the boundary layer where fewer tracer particles (introduced in the

exterior flow) have penetrated. cs 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 floor. 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

different moment arms. The force balance measurement has an expected uncertainty of 0.1

mN. The expected friction drag on a flat plate of the same size as the E387 wing at zero

degrees of incidence is approximately 11 mN. In force balance measurements, α was varied

from -10 to 20 in steps of 1. Three tests were performed for both increasing and decreasing

α, and results were averaged.

The force balance measures total drag (which will be labeled CD,FB) as well as lift, CL.

The profile drag values at the six different α are obtained from the PIV measurements of

momentum wake defect in the midsection of the wing (-0.4 ≤ y/b ≤ 0.4); this profile drag

component will be labeled, cd,PIV . The span efficiency (Eqs. (7.11), (7.12)) for the E387

wing is initially unknown, so CD,i,FB can be estimated by subtracting cd,PIV from CD,FB at

each α. Since CL is known at each α, a least squares fit for CD,i,FB values can be used to

solve for ei and ev from Eqs. (7.11) and (7.12). The total drag achieved by adding cd,PIV and

Ci,FB with the calculated efficiency values will be labeled CD,PIV+FB and is necessarily equal

to CD,FB. The uncertainties in CD,i,FB derive primarily from the standard deviation in CL

from force balance measurements, and the uncertainties in cd,PIV are obtained by methods

explained in the following section.

58

III. Results

A. Spanwise Drag Variation at Moderate Reynolds Numbers

Correct estimates of cd from Eq. (7.10) require that converged time-averaged profiles

exist and that contributions from dpdx

are negligible. This condition occurs at some distance

downstream, estimated to be x/c > 3 for similar conditions [16]. Mean profile integrals∫WU2(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 α. Satisfactory convergence

of θ(x) can be claimed after x/c = 2.75, and all subsequent data use streamwise averages

over x/c ∈ [3.0, 3.4].

The results of integrating Eq. (7.7) to obtain θ(y) at the six different α at the downstream

location x/c = 3.4 yield cd(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 α, which is mostly a

consequence of the non-zero mean out-of-plane momentum flux. There is also measurable

variation over the mid portion of the wing between -0.4 ≤ y/b ≤ 0.4 where the variation

with y in cd is higher than the measurement uncertainty. cd(y0) increases as α increases, but

there is no obvious variation in the absolute magnitude of ∆cd with α in the wing center.

Since the wingtip effects 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, θ, was obtained by several methods, including averaging dif-

ferent numbers of image pairs to obtain U (z ), applying different interpolation methods to

acquire the boundaries for the momentum defect regions, and using different integration

methods to calculate θ from a given mean profile The uncertainty ∆cd is the maximum dif-

ference in cd values obtained using these various methods. The greatest relative variation in

cd in the midsection of the wingspan (-0.4 ≤ y/b ≤ 0.4) is 27% at α = 0. By definition, in

this procedure, any measured variation must come from systematic and repeatable variations

in profile amplitude and width.

59

Figure 7.5: cd(y) at α = 0, 2, 4, 6, 8, 10. The symbol size is chosen to match the size of themeasurement uncertainty.

60

If there is variation in average values of θ and cd due to time-averaged variation in flow

field, then it should be possible to trace such variations upstream. cd(y) at α = 0 and

α = 8 are compared at two different downstream locations, x/c = 3.4 and 2.0 (Figure 3.6).

Figure 7.6: cd(y) at α = 0 and 8 at x/c = 3.4 (solid circles) and x/c = 2.0 (squares). Thesymbol size matches the measurement uncertainty. Arrows denote data points in subsequentsections.

At α = 0 the pattern of above-threshold spanwise cd(y) variation matches for the two

x/c locations. The difference between the two sets of data is a slight offset (the calculated

drag values are slightly higher further downstream). At α = 8 the correlation is less obvious,

but there are correlated variations, significantly 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 α = 0, instantaneous and time-averaged ωy look similar as the flow is steady and

laminar separation occurs shortly before the trailing edge (Figure 7.7 a, b). At α = 8, flow

separation is earlier and the separated shear layer has become unstable, generating coherent

structures that impinge upon the downstream portion of the suction surface. ωy and ωy 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, Rex ≈ 3300 and the boundary layer thickness δ = 5.2x√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) ωy(x, z) and b) ωy(x, z) at α = 0; c) ωy(x, z) and d) ωy(x, z) at α = 8.Arrows are fluctuating velocity vectors.

At α = 0, five y/b stations in the mid portion of the wing (-0.5 ≤ y/b ≤ 0.5) were chosen

where cd(y) varied similarly at x/c = 3.4 and x/c = 2.0 (indicated by arrows in Figure 7.6).

〈Ω〉a,b are shown at the different span stations in Figure 7.8. 〈Ω〉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. 〈Ω〉a may be expected to vary with lift

62

coefficient but may not be sensitive to changes in drag. When flow separation is mild, 〈Ω〉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. Neither 〈Ω〉a nor 〈Ω〉bvaries significantly across the span or in phase with cd(y). Apparent variations in phase with

variation in cd(y) do not rise above the measurement uncertainty. Flow separation does not

occur until close to the trailing edge at α = 0, so there is no separated region along the top

of the wing as in the case of higher α.

At α = 8 six y/b stations were also chosen where cd(y) varied similarly at x/c = 3.4 and

x/c = 2.0 (indicated by arrows in Figure 7.6). 〈Ω〉a,s and cs/c are shown at the different

span stations in Figure 7.9. 〈Ω〉s varies as cd, showing, unsurprisingly, that high local cd

is associated with high turbulence levels over the trailing half chord. There is no clear

correlation of 〈Ω〉s with cd. cs varies inversely with 〈Ω〉s so high turbulence in the separation

region is associated with earlier separation and higher local cd.

C. Estimation of Spanwise Efficiency Factors

In Figure 7.10, CD,FB is compared with values calculated from the wake measurements

that use CD,i and the least squares fit on ei (Eq. (7.11)) to match the sum of CD,i and

CD,PIV+FB. The best fit yielded ei = 0.83. This estimate uses the wake-measured variation

of cd(α) to estimate the profile drag. An alternative is to use the force balance data to

calculate the constant value of CD,0 (also sometimes known as the profile drag), and then

the implicit variation of cd with CL2 is included in the value of the viscous span efficiency,

ev, which can again be estimated by least-squares regression of CD,PIV+FB on α. The fit

is satisfactory when ev = 0.3 (Figure 7.11) [3]. This estimation by least-squares regression

essentially yields values of ev and ei that would otherwise be obtained through a linear fit

and slope calculation of the CD-CL2 curve. The estimated slope at low α is 0.14, compared

with 0.09 for a different E387 wing at the same Re in [3], where it was noted that such values

63

Figure 7.8: At α = 0, spanwise cd(y) (top, x/c = 3.15 in black, x/c = 1.75 in gray), 〈Ω〉b(middle), and 〈Ω〉a (bottom).

have limited significance when the lift-drag polars themselves have shapes very different from

model assumptions.

IV. Discussion

There is measurable variation in cd(y) on the E387 wing at Re = 30,000 over all the

tested α (0 ≤ α ≤ 10). The large differences in measured cd(y) between the mid-span and

64

Figure 7.9: At α = 8, spanwise cd(y) (top, x/c = 3.15 in black, x/c = 1.75 in gray), 〈Ω〉s(middle), and cs/c (bottom). The uncertainty in 〈Ω〉i is the maximum difference among thevalues of 〈Ω〉i obtained from averaging over different number of image pairs.

the wing tip (up to 98% at α = 10 for 0.0 ≤ y/b ≤ 0.9), are related to the non-negligible

momentum flux in out-of-plane directions, and are related directly to the lift on the finite

wing. However, spanwise cd(y) variation also occurs in the mid portion of the wing (up

to 27% variation at α = 0 for 0.0 ≤ y/b ≤ 0.2), where the local flow may otherwise be

considered to be close to two dimensional.

65

Figure 7.10: Combined force balance and PIV drag results. CD,FB (gray line + circles),cd,PIV (dashed line + diamonds), CD,PIV+FB using ei = 0.83 (solid line + squares).

The cd(y) variation trends are preserved at different x/c locations, and tracing the flow

back to on-wing conditions shows that, at α = 8,where significant flow separation occurs,

〈Ω〉s variation is directly proportional to cd(y) variation. This was not the case for 〈Ω〉a, or

at α = 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 in 〈Ω〉s corresponds to earlier

separation, suggesting that at high α, the location of the separation line cs is not uniform

and two dimensional. The location of separation affects 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 deficit, and hence the local sectional drag,

and its measurement.

66

Figure 7.11: CD,FB (gray line + circles), CD,0 (dotted line), CD,PIV+FB using ev = 0.3(dashed line + triangles).

The results of the cd(y) variation study at Re = 30,000 are quantified 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 different Re

and two different (but similar) airfoils, both show a local maximum at α ∼ 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 α. The variation of cd 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 σ decreases

as Re increases, but the increase in σ/cd with decreasing Re is consistent in both facilities.

Finally, σ for the current study at Re = 30,000 is compared with ∆cd among different

facilities at Re = 60,000 in Figure 7.14. For 0 ≤ α ≤ 10, σ at Re = 30,000 is less than ∆cd

67

Figure 7.12: σ at Re = 200,000 for the E374 from [1] (squares) and Re = 30,000 for theE387 from the current study (solid circles).

Figure 7.13: σ/cd at α = 0 (solid circles) and 5 (gray squares) at various Re from Langley[4] and Dryden. The value at α = 5 from Dryden is an average of the values at α = 4

(bottom white square) and 6 (top white square).

at Re = 60,000. Figure 7.13 shows that drag variation increases as Re decreases, so σ at Re

= 60,000 will be lower still than at Re = 30,000. The observed cd(y) variation is much less

68

than the drag variation from literature at Re = 60,000 and therefore cd(y) variation is not

the main cause of the discrepancies in measured cd among different facilities.

Figure 7.14: Drag variation from different facilities (squares) and spanwise cd(y) from currentstudy (solid circles).

The separation of drag into its different components is achievable at Re = 30,000 if ei and

ev are determined empirically. Since only the α-dependent profile drag cd 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 CD,i. At Re = 30,000, the drag measurements imply values of ei and ev 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 findings at moderate Re [3]. The existence of persistent

spanwise variation in wake defect magnitude, and hence cd, along the span, supports the

argument that a single parameter description of the departure from ideal uniform conditions

may not be a very good reflection of the detailed flow field on the wing.

69

V. Conclusion

At transitional Re flows, particularly in the sub-regime 30,000 ≤ Re ≤ 70,000, the drag

values reported by various facilities for smooth airfoils including the E387 differ significantly,

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 coefficients. 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 differences in reported results among different facilities.

70

Paper I References

[1] Guglielmo, James J., and Selig, M. S., “Spanwise Variations in Profile Drag for Airfoilsat 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 Efficiencies 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 Eppler387 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 WingMicro-Air Vehicles,” presented at the “Development and Operation of UAVs for Militaryand 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 UAVConference 1999.

[7] Selig, M. S., Guglielmo, J. J., Broeren, A. P., and Giguere, P., “Summary of Low-SpeedAirfoil 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 ARCR&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 Profile 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 Profile Drag by the Pitot-Traverse Method,” BritishARC 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 AdvancedTransit and Highways (PATH), Institute of Transportation Studies, UC Berkeley, 1994,(doi: 10.1146/annurev.fl.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 SouthernCalifornia, 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 DepartmentUniversity 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 profiles) 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 flow 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 flow around the wing is effectively modified, and large, stable vortical

structures appear in the separated shear layer. Correlation between the effects 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)

CD = total drag coefficient on a finite wing

CL = total lift coefficient on a finite wing

fe = excitation frequency (Hz)

fe* = optimum excitation frequency (Hz)

feo = uneasily excitable frequency (Hz)

fs = separated shear layer instability shedding frequency

L/D = lift-to-drag ratio

SPL = sound pressure level (dB)

St = Strouhal number

Stα = angle of attack based Strouhal number

Stθs = separation Strouhal number

Re = chord-based Reynolds number

U = free stream velocity (m/s)

Ue = edge velocity of boundary layer (m/s)

Ua = advection speed (m/s)

α = angle of attack (deg)

α0 = 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 flight regime in which many of these miniature aircraft systems

operate is where the chord-based Reynolds number, Re, lies between 104 − 105, which is

considered to be a low Re regime in aeronautics. Here, complex flow characteristics can

either favorably or adversely affect wing performance. Two main approaches can be taken

in this design space: either avoid it altogether, or manipulate and force the flow 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 flow over an airfoil susceptible to separation. When the flow

has sufficient energy to overcome the combined effects of adverse pressure gradient, viscous

dissipation, and change in momentum, the flow remains attached. Conversely, when the flow

has insufficient energy, the flow 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 affected by the presence of a laminar separation

bubble at lower Re [1]. In the regime 30,000 ≤ Re ≤ 80,000 the E387 has complex flow

characteristics where the lift-drag curves show pre-stall hysteresis and abrupt jumps between

what appear to be multiple performance envelopes due to flow 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 CL(CD) 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 CL − CD polars for the E387 wing [4].

Since flow separation and reattachment both strongly affect wing performance, separation

control is of clear practical significance. 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 flaps, blowing, bleeding,

and synthetic jets. The basis for efficient 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 flow. When the initial disturbances are sufficiently

77

large, they can grow nonlinearly and result in turbulent flow. 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 fluid from the external flow is

swept into the region close to the airfoil surface [7]. Since the possible flow reattachment is

critical to the selection of bi-stable state alternatives, proposed flow 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

flow and control separation around a wing at various Re and in various flow states. External

forcing at single frequency tones has been shown to effectively 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 flow from low- to high-lift states [2], diminishing the size of the hysteresis loop

in lift-drag curves [2], reducing the tendency toward flow separation [8, 9], and changing

the basic behavior of the laminar separation bubble and turbulent boundary layer [11]. The

effects 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 α show the dependence of optimum excitation frequencies on Re, α, and the dom-

inant intrinsic instabilities in the flow. The optimum excitation frequency has been found

to increase with increasing Re or increasing α [11], while the range of effective excitation

frequencies has been found to increase with increasing Re or decreasing α [11, 13]. It has also

been suggested that the optimum excitation frequencies correspond to the most amplified

instabilities in the separated region. For pre- and immediately post-stall α, 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 α, 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 effectively reduced when the

parameter St/Re1/2 was between 0.02 and 0.03 based on the excitation frequency [11, 12].

Recalling that the boundary layer thickness over a flat plate, δ ∝ Re1/2, the St/Re1/2 scaling

is directly related to the growth of the boundary layer thickness, δ.

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 α, which act like bluff bodies [20]), then the projected height of

the airfoil is used as the length scale, and a Strouhal number that takes α into account can

be given by

Stα =fc sin(α)

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

Stθs =fθsUe

(8.3)

where θs is the momentum thickness of the separated region, and Ue is the edge velocity

of the boundary layer [18]. An optimum range of Stθs is reported to be between 0.008 and

0.016 [18].

This paper provides a study on the effects of external acoustic excitation on the forces and

flow fields 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 (CL, CD) 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 first of a series of experiments to

unambiguously establish the basic flow 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 significant 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 2Hz ≤ f ≤ 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, and z 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 60Hz −15kHz 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 amplified by an adjustable gain Pyle Pro PCA1

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2 x 15 W stereo power amplifier with a frequency response of 20Hz − 40kHz ±3dB 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 field microphone, which has a pressure-field response of ±2dB

between 16Hz ≤ f ≤ 70kHz, was used to obtain acoustic measurements for the wind tunnel

resonance study. A 4954-B B&K free field microphone, which has a free-field response of ±3

dB between 9Hz ≤ f ≤ 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 floor. 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 different moment arms. The electromechanical force balance measurement has an

expected uncertainty of 0.1 mN. The expected friction coefficient on a flat plat can be given

by

Cf =1.328√Rex

(8.4)

and the friction drag by

Ff = Cfq∞S (8.5)

where q∞ is the dynamic pressure. For a flat 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, α 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, δt = 100−300µs. The

two coaxial laser beams were converted to sheets of slowly varying thickness through a series

of convergent-cylindrical-cylindrical lenses. The flow was seeded with 1µm 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 fit 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 field was reinterpolated back onto a complete rectangular grid with the

same smoothing spline function [24]. The spline coefficients are differentiated analytically to

yield velocity gradient data. The uncertainty does not depend on velocity magnitude but is

fixed in fractions of a pixel, but when rescaled to conditions reported here, we may expect

uncertainties of 0.5−5% in u,w and about 10% in gradient-based quantities, such as the

spanwise vorticity

ωy =∂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 effects of different excitation frequencies, fe, on lift and drag forces at Re = 40k

and 60k were examined. At each Re, a value of α0 immediately preceding the hysteresis

loop was chosen. For Re = 40k, α0 = 10, and for Re = 60k, α0 = 8. For the acoustic

study at constant amplitude, fe from the waveform generator was varied while keeping both

the waveform generator peak-to-peak voltage amplitude and the power amplifier volume

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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 amplifier was kept at a constant

volume setting while the peak-to-peak voltage amplitude levels from the waveform generator

were varied at each fe 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 fe*, 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 oe . After fe* 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 flow field 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 off (allowing the flow 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 field microphone was placed inside

the empty wind tunnel test section without flow 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 amplifier volume

and waveform generator peak-to-peak voltage level were kept constant, and four excitation

frequencies were used: two values of fe* and two values of f oe . 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 α 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 flow and reattached flow conditions over the suction surface of

the wing cause the jumps between bi-stable states, as observed from PIV flow field data (not

shown here). More than an 85% difference in L/D can occur over a 0.1 change in α (at Re

= 40k, the difference 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 fe* = 525Hz and 660Hz for both Re = 40k and 60k, but

at 800Hz there is no improvement for Re = 40k. The range of fe* 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 fe = 525Hz.

The maximum ∆(L/D) at Re = 60k is only 56% at fe = 800Hz. The corresponding CL(CD)

and L/D curves at the indicated fe* values are shown in Figure 8.6.

When the flow is excited at fe*, hysteresis is largely eliminated for both Re = 40k and

60k. Excitation at fe* removes most, but not all, of the drop in L/D(α) at moderate α,

and the magnitude of the improvement varies with α. The higher Re case shows the closest

achievement of a flat, high ∆(L/D) over a broad range of α (Figure 8.6d). The original PIV

images with densely-seeded flow show dark lines that are the path lines of fluid originating

in the relatively particle-poor boundary layer.

85

Figure 8.4: CL(CD) curves (a) and L/D(α) curves (b) for the E387 at Re = 30k (circles),40k (triangles), 50k (squares), and 60k (diamonds). Error bars are the standard deviationfrom multiple tests.

86

Figure 8.5: ∆(L/D) for different fe at constant amplitude at α = 10 at Re = 40k (top) andat α = 8 at Re = 60k (bottom) with fe* values indicated.

87

Figure 8.6: CL(CD) for Re = 40k (a) and Re = 60k (b) and L/D for Re = 40k (c) and Re =60k (d) for unexcited flow (dotted gray line + circles), fe = 525Hz (black line + triangles),fe = 660Hz (gray line + squares), and fe = 800Hz (black line + diamonds).

88

Figure 8.7 compares the unforced (left column) and forced (right column) flows for α

= 8 at Re = 60k. The unforced flow separates at a well-defined location about c/4 from

the leading edge. In the forced flow 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 flow 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 different chordwise location than the well studied laminar

separation bubble that appears on the SD7003 airfoil [21] and which, by contrast, has almost

no dynamical significance.

Figure 8.7: Separation and spanwise vorticity for flow over E387 at α = 8 and Re = 60kwithout forcing (left) and with forcing at fe = 800Hz (right). Vorticity fields are superim-posed on a fluctuating velocity vector field with unity scaling.

89

C. Acoustic Excitation at Constant SPL

The effects of fe on ∆(L/D) at constant SPL at Re = 60k and 40k together with the SPL

variation with fe 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 reflected waves in the tunnel test section which has no special acoustic treatment

of the walls. In acoustic wave fields, the gradient of pressure, |5p|, is maximum where the

pressure fluctuations cross the zero line, and this consequently is where the maximum induced

particle velocity is found, 90 out of phase with the pressure fluctuations. When the acoustic

wave field 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 fluctuates between maximum and minimum

but where the pressure gradient is close to zero. The anti-resonance regions in fe are where

∆(L/D) is highest. The flow is most easily switched to its high L/D state when the acoustic

wave induced velocity field 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 α at the

three most excitable frequencies of Figure 8.8b are shown in Figure 8.9. Hysteresis is again

eliminated, although the original dips in L/D (gray curves in Figure 8.9) are not completely

eliminated. Not all α experience the same magnitude of L/D improvement, similar to the

results from acoustic excitation at constant amplitude. Of the three values of fe* (415Hz,

520Hz, and 675Hz), the lower two produce better overall lift-drag curves. Qualitative results

of the flow field and corresponding spanwise vorticity field at two values of fe* (415Hz and

520Hz) and two values of f oe (445Hz and 550Hz) are shown in Figure 8.10 and Figure 8.11,

respectively.

The particle images for the two f oe (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: Effect of fe on ∆(L/D) at Re = 60k (a) and 40k (b) and corresponding windtunnel SPL response measured at (x = 0, y = 0, and z = 0) (c).

conditions. At the two fe* (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: Effect of fe on CL and CD (a) and L/D (b) at Re = 40k for unexcited flow(dotted gray line + circles), fe = 415Hz (black line + triangles), fe = 520Hz (gray line +squares), and fe = 675Hz (black line + diamonds) at constant SPL = 75.5dB

92

Figure 8.10: Raw PIV images for flow at Re = 40k with acoustic excitation at fe* (a, c) andf oe (b, d) (SPL = 75.5dB).

Figure 8.11: Spanwise vorticity fields for flow at Re = 40k with acoustic excitation at fe*(a, c) and f oe (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

fields (Figure 8.11), obtained from the raw PIV images (Figure 8.10), reveal that exciting

the flow at f oe has no effect on the flow, which remains separated over the aft half of the

airfoil, but excitation at fe* produces a region of circulation over the front half of the wing,

corresponding to a reattached flow state.

D. SPL Dependence

The results of varying SPL on wing performance for a value of fe* (520Hz) and f oe

(550Hz) are shown in Figure 8.12. Figure 8.12 shows that varying the forcing amplitude at

fe* changes the magnitude of ∆(L/D) and the range of α over which the change is seen.

Changes in L/D can be obtained by forcing at f oe but require a much higher amplitude. For

flow at Re = 40k and α = 10, a 77% L/D improvement is achieved with an SPL = 77.8dB

at fe* = 520Hz, but the same improvement requires a much higher SPL of 91.8dB at f oe =

550Hz. A 14dB change in SPL is a 0.1mPa change in pressure.

Figure 8.12: L/D(α) for acoustic excitation at fe* = 520Hz (a) and f oe = 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 α, 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 fe* = 520Hz. The separated and reattached flow states are indistinguishable

from those achieved with varying α.

Figure 8.13: Hysteresis of L/D and SPL at Re = 40k and α = 10. fe is held constant at520Hz. Spanwise vorticity superimposed on a fluctuating velocity field is shown for the fourindicated 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 fe* cases are shown in a and b and the two f oe

95

Figure 8.14: Wind tunnel resonance in the yz -plane (normal to the chord and the meanflow).

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 significantly (by 15dB) over length scales that are comparable to the span

(b ≈ 6c) in the y, z 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 reflections will come from the

opposite wall in z−. The corresponding distribution in x, z (Figure 8.15) is more uniform,

with only minor variation in x, where there are no direct reflectors. There are also smaller

variations in z, which suggests that the large variations in y, z of Figure 8.14 come also

96

Figure 8.15: Wind tunnel resonance in the xz -plane (parallel with the chord and the freestream).

from reflections in y. The wind tunnel test section is octagonal and so this is expected. In

Figure 8.16 the x, y plane lying coplanar with the wing chord at α = 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 SPLmin occurs at y = 0 over all x in Figure 8.16 for fe*.

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 SPLmin is associated with a

maximum efficiency of flow modification. 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 thespan).

the wing center (and flow measurement point) for the frequencies f oe that are least effective

in disturbing the flow.

IV. Discussion

Previous literature results suggest that the values of fe* correspond to the most amplified

instabilities in the separated shear layer (K-H instabilities) for pre- and immediately post-

stall α. If the values of fe* are used in the calculation of St in Eq. (8.1) and then normalized

by Re1/2, the optimum range of St/Re1/2 for Re = 60k is approximately 0.015 ≤ St/Re1/2 ≤

98

0.035, and for Re = 40k the optimum range is 0.025 ≤ St/Re1/2 ≤ 0.045, as shown in Figure

8.17. The reported optimum range of St/Re1/2 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 in St/Re1/2

suggests that the correct scaling has not been identified.

Figure 8.17: ∆(L/D) as a function of St and St/Re1/2.

A shear layer frequency, fs, can be obtained given a mean advection speed, Ua, and

the spatial separation of the vortical structures in the free shear layer, xs (observable from

instantaneous spanwise vorticity fields):

fs =Uaxs

(8.6)

99

where xs is the average separation between two adjacent vortical structures in the x -direction,

and Ua, 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 xs yields fs = 445 ± 125Hz, where the uncertainty comes from using different

adjacent vortices and the uncertainty in location of the vortex centers. While this range of

fs encompasses the observed fe* = 520Hz, the large uncertainty in fs suggests that vortical

structures in the shear layer are not shed regularly when the flow is separated. Although

the vortical structures can be detected from vorticity fields, none can be clearly seen in the

raw PIV images.

In contrast, when the flow is forced at fe*, distinct structures are evident in the raw PIV

images, like Figure 8.10c. If xs is the spatial separation between the dark patches, and Ua is

calculated from the time-averaged velocity field at the x, z location of the corresponding

structures, then another shedding frequency can be calculated from Eq. (8.6). For the case

of forcing at fe* = 520Hz, the average fs is equal to 1110 ± 30Hz. This value of fs is a

second harmonic of 550Hz ± 30Hz, which equals the observed fe* = 520Hz. The uncertainty

in this shedding frequency also comes only from using different adjacent vortices and the

uncertainty in location of the vortex centers. The noticeably smaller uncertainty for fs for

reattached flow implies that the shedding is much steadier than for the separated case.

The agreement of estimated fs with the observed fe* suggests that forcing at intrinsic

most-amplified frequencies of the free shear layer could be the most effective way to control

the flow. However, if this were the case, the preferred St, or even range of St/Re1/2, would

not vary with Re, but they do. Moreover, the proposed physical mechanism based on a

resonance with fs entirely ignores the tunnel resonance dependence.

It is most likely that the variations in effective 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

benefit in disentangling the various contributors whose relative influence 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 affect 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 reflected

acoustic waves, and this could be arranged outside of a tunnel in free flight 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 reflections from low amplitude, local forcing are not strongly influenced by reflection.

V. Conclusion

The distinct jumps between bi-stable states and pre-stall hysteresis of the E387 wing,

particularly in the Re regime 40,000 ≤ Re ≤ 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 flow over the

wing by reattaching the formerly separated flow and forming a laminar separation bubble.

Improvement by acoustic excitation is a function of fe and SPL, and in this experiment,

optimum excitation values fe* correlate with wind tunnel anti-resonances. While fe* and

associated St* 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 similar Re, corroborating

and extending the original observations also found in the literature. Further experiments

using local on-wing forcing may help to distinguish the different effects in an open flow.

101

Paper II References

[1] McGhee, R. J., Walker, B. S., and Millard, B. F., Experimental Results for the Eppler387 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., “Effects of Acoustic Disturbances on LowRe Aerofoil Flows.” In Fixed and Flapping Wing Aerodynamics for Micro Air VehicleApplications, 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 Efficiencies 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 DragMeasurement 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,” AmericanInstitute of Aeronautics and Astronautics, Vol. 34, No. 8, 1996.

[8] Zaman, K. B. M. Q., Bar-Sever, A., “Effect of Acoustic Excitation on the Flow Over aLow-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 LayerFlows by Sound,” AIAA Paper No.1983-0726, 1983.

[10] Ahuja, K. K., Burrin, R. H., 1984, “Control of Flow Separation by Sound,” AIAA Paper84-2298, Oct. 1984.

102

[11] Zaman, K.B.M.Q., McKinzie, D.J., “Control of Laminar Separation Over Airfoils byAcoustic Excitation,” American Institute of Aeronautics and Astronautics, Vol. 29, No.29, 1991.

[12] Yarusevych, S., Kawall, J. G., Sullivan, P. E., 2002, “Influence of Acoustic Excitationon Airfoil Performance at Low Reynolds Numbers,” 23rd International Council of theAeronautical Sciences Congress, 813 September, Toronto, Ontario, Canada.

[13] Hsiao, F. B., Jih, J. J., and Shyu, R. N., “The Effect of Acoustics on Flow Passing aHigh-AOA Airfoil,” Journal of Sound and Vibration, Vol. 199, No. 2, 1997, pp. 177-188.

[14] Zaman, K.B.M.Q., “Effect 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 SouthernCalifornia, Los Angeles, CA, 2007.

[17] Huang, L.S., Maestrello, L., Bryant, T. D., “Separation Control Over an Airfoil at HighAngles of Attack by Sound Emanating From the Surface,” AIAA Paper No. 87-1261, 1987.

[18] McAuliffe, B. R., Yaras, M. I., “Transition Mechanisms in Separation Bubbles UnderLow- and Elevated-Freestream Turbulence,” Journal of Turbomachinery, Vol. 132, No. 1,2010.

[19] Zabat, M., Farascaroli, S., Browand, F., Nestlerode, M., Baez, J., “Drag Measurementson 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.fl.15.010183.001255)

[20] Roshko, A., “On the Drag and Shedding Frequency of Two-Dimensional Bluff Bodies,”NACA Report TN 3169, 1954, pp.1-29.

[21] Radespiel, R., Windte, J., and Scholz, U. “Numerical and Experimental Flow Analysisof Moving Airfoils with Laminar Separation Bubbles,” American Institute of Aeronauticsand Astronautics, Vol. 45, No. 6, 2007, pp. 13461356.

[22] Fincham, A. M. and Spedding, G. R., “Low cost, high resolution DPIV for measurementof turbulent fluid flow,” 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.

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[24] Spedding, G.R. and Rignot, E.J.M., “Performance Analysis and Application of GridInterpolation 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 DepartmentUniversity 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 fluid 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 effect 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 effect. Acoustic resonance

in the backing cavities generates tones that interact with intrinsic flow instabilities. Prac-

tical consequences for passive flow control strategies are discussed together with potential

problems in measurements through pressure taps in such flow regimes.

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I. Introduction

An emerging generation of practical Micro-Air Vehicles (MAVs) operates at flight speeds

and characteristic length scales that brings them into an especially challenging flight regime,

where abrupt changes in flight 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 coeffi-

cient (cd, with cl the corresponding sectional lift coefficient) at moderate, pre-stall, geometric

angles of attack, α.

Experimental difficulties are exacerbated because the relatively small forces (of mN or

less) are difficult 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 profile

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 × 104 (Re = Uc/ν, where U is the flight speed, c is the wing chord, and

ν is the kinematic viscosity), there were measurable variations in cd across the span of an

E387 wing, but that they were not of sufficient magnitude to account for disparities in the

literature.

The E387 airfoil is a well-studied airfoil, originally designed for sailplanes at moderate

Re (≥ 2× 105). The E387 has been referred to as the low Re calibration standard [5], since

at low Re (< 1 × 105), it experiences laminar separation without reattachment, laminar

separation with turbulent reattachment, and turbulent trailing edge separation. Although

these phenomena that occur at low Re 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 effect.

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 × 104) [6], [7]

showed that at moderate α = 4−8 the flow can be in either one of two states. In one state,

a lower envelope of cl(cd) curves marks the performance characteristics where trailing edge

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separation gradually moves forward on the wing, and cd increases rapidly for α > 4, while

cl 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 flow reattaches before the trailing edge. This high-lift state will be referred to as SII.

Consequently, at any given cd, cl has a value about 40% higher than in SI. The lower-lift

envelope of SI is occupied by all points at Re ≤ 2 × 104, and the upper envelope of SII is

characteristic of polars for Re ≥ 8× 104. In between, and at interim 4 ≤ α ≤ 10, the flow

can separate either close to the leading edge, or further aft, and the global aerodynamic force

coefficients can match with either state SI or SII. The state switch SI–SII has hysteresis so

the flow at any α depends on the time history, or previous state.

The peculiar (when contrasted with the usual, simple C-shape curves at higher Re) shape

of the lift-drag polar of the E387 at intermediate Re is not actually restricted to this single

profile shape, but is quite characteristic of a class of airfoils that have smooth rounded leading

edges with some minimum profile thickness. 74 of 94 airfoils in [1], [2] and [3] at Re < 105

have this polar shape, with non-unique values of cl over some range of cd, and 18 of 31

profiles 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 cd and increase in cl [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 configuration at intermediate Re (E61 at Re = [25,

35, 50, 60] ×103) was noted by [9], who also showed a dependence on the background

acoustic environment. It was further demonstrated that the most effective 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 α 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

effective forcing frequencies.

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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 profile,

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 flow onto the high lift state SII.

II. Materials and Methods

A. Wing Model

The aluminum wing had an Eppler 387 profile 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 fit 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 flushed with a dielectric fluid, which also regulates the discharge and

keeps the wire and metal cool. The tolerance on the wire EDM is ±0.05 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 2 ≤ f ≤ 200

Hz in the velocity range 5 ≤ U ≤ 26 m/s. Measurements on flow 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)profile view of the lid of the wing.

flow is known to be sensitive to small disturbances (e.g., [3, 30]) and placing the mount point

at one tip confines this particular disturbance to a region that at moderate α is dominated by

the induced flow 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 figure, coordinate axes x, y, z run streamwise, spanwise and vertically,

respectively, with origin at the midspan leading edge at α = 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 flow visualization and estimation of veloc-

ity components u,w in x, z. 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, δt = 100 − 300µs. A series of convergent-

cylindrical-cylindrical lenses converted the two laser beams into slowly-varying thickness

laser sheets. The flow was seeded with 1µm 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 off of the customized CIV algorithms described in [12] and

[13] in which a smoothed spline interpolated cross-correlation function was directly fit 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 field was reinterpolated back onto a complete rectangular grid with the

same smoothing spline function [14]. The spline coefficients are differentiated analytically to

generate velocity gradient data. The uncertainty is in fractions of a pixel, and when re-scaled

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to the test conditions reported here, expected uncertainties are 0.5–5% in u,w and ≈ 10%

in gradient-based quantities, such as the spanwise vorticity:

ωy =∂w

∂x− ∂u

∂z.

D. The Imperfect Test Environment

No physical experiment conforms perfectly to its nominal configuration. Here we docu-

ment some of the departures from ideal test conditions. Comparative tests indicate that none

of these are critical in influencing the basic phenomena described here, but flows at transi-

tional Re 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 reflections 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 flow 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 flow 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.

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Figure 9.3: Acoustic power spectra in an empty wind tunnel with flow at U = 6.7m/s (Re= 40k), and without flow. Note the change in axis scales.

Wing Tip Deflections

The maximum absolute deflection amplitude, z′, 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′(α) 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 deflection z′/c ≈ 0.11%. At Re = 60k, a difference in wing tip

deflection is observed when SI-SII transition occurs. In the low-lift (SI) state, z′/c ≈ 0.2%,

and after transition to the high-lift (SII) state, z′/c ≈ 0.07%. The difference is likely due

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to the increased influence 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 deflections normalized by the chord (z′/c) at Re = 40k and 60k for thewing with all holes closed (ie, solid wing).

Surface Roughness

Surface roughness of sufficient height can act as a boundary layer trip, and wings with

different roughness can act as though they operate at different Re. The nontrivial effect

in flow separation and transition at these Re has been well documented [15]. The surface

roughness near the wing tip (y/b ≈ −0.99) was measured at different points along the chord

by a Ambios Technology XP Stylus Profiler 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 fit 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, hr was 6.3 microns,

At x/c ≈ 0.9, where the wing is the roughest, the maximum measured hr ≈ 15 microns.

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The roughness may be compared with the likely boundary layer thickness, δ, which for a flat

plate at zero incidence is,

δ = 5

√xν

U, (9.1)

where x is the distance from the leading edge. At x/c = 0.3, δ = 1.3 mm, and hr/δ = 5×10−3.

At x/c = 0.9, δ = 2.1 mm, and hr/δ = 0.01. Thus, the surface roughness is small compared

with a boundary layer thickness.

We will be considering the effect 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 effect can be clearly distinguished from the small-scale distributed

manufacturing roughness. Subsequent tests described later will isolate the geometric surface

and volumetric effects.

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. At Re = 40k, the baseline wing has the characteristic jump from

SI to SII at α = 12 (Fig. 9.5a, b). This transition is marked by an increase in CL from 0.7

to values above 1, and a reduction in drag coefficient by about 10%. At Re = 60k, the jump

from SI to SII occurs at α = 9 (Fig. 9.5c, d), with similar CL 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 α. Consequently, before the SI-SII transition for the

baseline wing (0 ≤ α ≤ 12 at Re = 40k and 0 ≤ α ≤ 9 at Re = 60k), the perforated

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wing has a higher efficiency, as measured by L/D, than the baseline case. The performance

improvement is entirely passive, with no active energy input.

Figure 9.5: The effect of open holes on the aerodynamic performance of an E387 wing atRe = 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 differently. In

one case, the holes were filled 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 ratio w/h = 1/2. The resulting forces on the wing for the two cases show that a wing

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Figure 9.6: Small surface cavities have no effect 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 effect of the unsealed cavities on the flow

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 fluid flows that then affect 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 sufficient 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 given x/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

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(−0.83 ≤ y/b ≤ 0.83). The CL(CD) and L/D curves for the wing with selected rows of open

holes located at xo = 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 both Re cases, single rows of open holes at specific chord locations allow the envelope

of CL(CD) and (L/D)(α) 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 xo = 0.1, there is no abrupt jump in either L or D and the curves lie

close to SII over all α. At Re = 40k, the curves once again show sharp jumps between states

when xo ≥ 0.2, and for xo ≥ 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 xo ≥ 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. 9.5−9.8 were for the

wing with open exit ports. Figure 9.9 shows the L/D curves for the wing with different 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 α ≤ 7 in the pre-stall regime. If

the effect of the holes is to trigger separation (and then re-attachment) towards the leading

edge, then they are less effective when the large cavity volume is closed off. 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 effect of opening and closing the exit port

affects 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 flow instabilities could be amplified as the acoustic

waves impinge upon the boundary layer or separated shear layer.

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Figure 9.7: Single rows of holes at varying x/c allow the envelope of performance curves tobe populated between SI and SII at Re = 40k.

119

Figure 9.8: Similar variation in performance curves between SI and SII occur for differentrows of open holes at Re = 60k.

120

Figure 9.9: L/D for different spanwise rows of open holes with open and closed exit portsat 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 flow while leaving the sensitive SI–SII transition unchanged. In tests on a same-sized

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E387 wing under acoustic excitation at similar Re and pre-SI-SII transition α [7], the place-

ment of a microphone near the top surface of the wing prevented the flow from reattaching

when it otherwise normally would, thus keeping the flow in SI.

Since the characteristics of SI-SII transition can be monitored here, one can search for

arrangements where it is not affected 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 different distances from the wing surface (in the z-direction)

at y/b = 0 and x/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 α = 10 for the wing with open holes atx/c = 0.1 and open exit port measured at various distances, z/c, from the top surface.

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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), α = 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 at x/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 flow field. Similarly,

the 400 Hz peak, which is highest when the wing is in low-lift SI, is reduced in amplitude

when the flow is controlled to state SII, and it could mark a harmonic of a flow 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 fluctuations within the cavities and corresponding velocity fluctuations at

the orifices due to acoustic resonance. If this is so, then varying the cavity volume/geometry

should affect the resonant frequencies in predictable ways. In the original wing design, the

cavities were not fluid-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 α. However, in the

range 0 ≤ α ≤ 12, L/D values for the wing with isolated cavities are slightly lower than

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Figure 9.11: Power spectra at Re = 40k. (a) No wing; (b) wing with closed holes; (c) wingwith open holes (x/c = 0.1) and open exit port; (d) wing with open holes (x/c = 0.1) andclosed exit port. For (b)−(d), α = 10.

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Figure 9.12: Schematic of cavities interconnected by a channel. Each cavity was designed tohouse speakers with an adjacent channel for wiring.

Figure 9.13: CL(CD) (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 difference 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 different internal cavity volumes. Part of the volume directly below the holes was

125

reduced by approximately 24%. The resulting CL(CD) 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

between −4 ≤ α ≤ 12. A decrease in performance in the same α range was also observed

at Re = 60k.

Figure 9.14: CL(CD) (left) and L/D (right) for a E387 wing with 15 open holes at x/c = 0.1and two different internal cavity volumes.

G. Spanwise Variation

The influence of spanwise uniformity of the open holes was examined. In Figs. 9.5 - 9.14,

open holes spanned from −0.83 ≤ 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)(α) 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 figure shows example time-averaged spanwise vorticity fields from five

spanwise planes under the same conditions. The span sections are at y/b = 0.0, -0.15, -0.30,

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-0.44, -0.59. The first 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 flow corresponds locally

to state SII. The last two stations at y/b = −0.44,−0.59 lie either side of an open hole, but

both show a short separation bubble that does not reattach. The effective separation point is

much closer to the leading edge and the wake is much wider. This flow state corresponds to

SI. The flow at station #4 is controlled not only by the most proximate hole state, but also

by the flow in adjoining station #5. Raw PIV images also provide qualitative information

about the flow behavior by the presence of a dark separation line, caused by the presence of

a shear layer where the fluid 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) flow, as well as at y/b = −0.59 (station #5), where

the separation line is farther from the surface, corresponding to a locally separated (SI) flow.

Figure 9.15: CL(CD) (left) and L/D (right) for a E387 wing at Re = 40k with mid portionholes open (−0.48 ≤ y/b ≤ 0.48) at x/c = 0.1.

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Figure 9.16: Spanwise vorticity fields, ωz(x, z) are averaged over 20 independent samples togive a map whose strongest features are steady, but an indication of the unsteady structureremains. This is superimposed on similarly time-averaged fluctuating velocity vector fieldsat spanwise sections y/b = 0.0,−0.15,−0.30,−0.44,−0.59 for the E387 at Re = 40k andα = 11 with open holes at x/c = 0.1 and 0.48 ≤ y/b ≤ 0.48. Every third vector is plotted.Raw PIV images at y/b = 0.0,−0.15 show dark separation lines that distinguish betweenlocally attached and locally separated flow.

128

IV. Discussion

On a wing that is characterized by abrupt jumps in lift and drag coefficient between

what we have termed SI, where the flow 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 different perturbations. In a study originally

aimed at acoustic forcing of the flow, it was found, quite by accident, that the presence of the

holes themselves in the wing suction surface was sufficient to affect the global behavior. When

holes are present, the flow 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 flow characteristics.

The effect on the wing is approximately local, so that chordwise strips can affect local

sectional cl and cd. Since each local flow state is either in SI or SII, one may think of the

wing as a device whose lift and drag coefficients can be manipulated under digital control.

Local states can be on or off, 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 flow. In general, cavity flow can be categorized into three main types, as detailed

in [16]: (a) fluid-dynamic, where oscillations come from the instability of the cavity shear

layer and are enhanced through a feedback mechanism, (b) fluid-resonant, where oscillations

are strongly coupled with resonant (standing) wave effects, and (c) fluid-elastic, where os-

cillations are linked to solid boundary motion. It is also possible to have combinations of

different types of cavity flow.

In purely fluid-dynamic cavity flow, the feedback mechanisms that enhance the oscilla-

tions are driven by the presence of the cavity downstream edge [16]. This type of cavity flow

includes the flow over a cavity covered by a perforated plate, whose uses include acoustic

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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 inflow velocity. Without the perforated plate, the shear layer was separated with an

inflection point, while with a perforated plate, the shear layer was bounded with no inflection

point. It was also suggested in [19] that unified, large-scale motion occurs through the plate

perforations and induces jet flow 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 , where f is a shedding or oscillation frequency. The reported values of fL/U

are on the order of 0.5-0.6 [20]. If the chord of the wing is used as an effective plate length,

the calculated fL/U ≈ 8 is an order of magnitude larger.

In classical cavity flows, the cavity length must be several times larger than a boundary

layer thickness at the upstream lip in order for amplification to occur of the instabilities in

the cavity shear layer. Estimates of δ 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 finite depth (equal to twice its diameter), by itself, has no effect

on the performance of the wing. Changes in flow occur only when the holes are connected to

a larger chamber and when the chamber volume is varied, suggesting the presence of acoustic

resonance effects.

B. Helmholtz Resonance

Fluid-resonant cavities include Helmholtz resonators, which are distinguished by the very

large ratio of cavity volume to cavity orifice area [16]. In general, a Helmholtz resonator is a

device in which a volume of compressible fluid is enclosed by rigid boundaries with a single

small opening and can be modeled by a second-order mass-spring system where the fluid in

the orifice has an effective mass and the compressibility of the fluid in the chamber is the

stiffness [21]. The resonator has a natural frequency, and when the instabilities in the flow

match the natural Helmholtz frequency, flow-excited resonance occurs. In such cases, small

pressure disturbances can produce large velocity fluctuations at the orifice and large pressure

fluctuations inside the resonator [21].

130

The Helmholtz resonant frequency is

fH =a

2π

√κ

V, (9.2)

where a is the speed of sound, κ is a quantity that represents the resistance of fluid passage

through the orifice, and V is the cavity volume [22]. For a circular orifice, the resistance is

κ =πr2

h+ 1.697r, (9.3)

where r is the orifice radius, and h is the orifice thickness (neck height) [22].

For multiple orifices in a single cavity, the resistance terms can be summed, yielding a

resonant frequency

fH =a

2π

√κ1 + κ2 + ...

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 mm3. Solving for fH yields fH = 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,

fi =Uaxs, (9.5)

where Ua is the streamwise advection speed, and xs is the average separation between vor-

tices, as observed in PIV data (e.g. Fig 9.17). At Re = 40k and α = 10 for the case of

15 open holes at x/c = 0.1, the calculated instability frequency is fi = 630 ± 90 Hz, where

the uncertainty comes from using different pairs of neighboring vortical structures and the

uncertainty in locating the vorticity centers. The calculated fi is equal to the calculated fH .

For the reduced volume case where the volume is approximately 10800 mm3 (Fig. 9.14), the

131

calculated Helmholtz resonance is fH = 750± 3 Hz. The calculated instability frequency is

fi = 730± 50 Hz, which again equals fH .

Figure 9.17: Instantaneous spanwise vorticity fields over the aft portion of the wing (begin-ning at x/c = 0.5) at Re = 40k and α = 10 with 15 open holes at x/c = 0.1. The timebetween successive vorticity fields is 0.1 s.

The calculation of fH in Eq. 9.2 is independent of flow speed. For a given resonator

volume and orifice radius, the same resonant frequencies should be generated for varying

flow speeds. This was confirmed by calculating fi at Re = 60k and α = 8 for the case of

15 open holes at x/c = 0.1. For the original volume, fi = 650± 90 Hz, and for the reduced

volume, fi = 710 ± 50 Hz, which are equal to the values of fi at Re = 40k for the two

different 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 fH = 925± 3 Hz. At Re = 60k and

α = 8, the calculated instability frequency is fi = 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 different 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 different than a simple resonator [22].

For the current wing, opening and closing the exit ports yielded different results, which were

especially prominent at the lower Re. The ratio of the calculated fi for open and closed exit

ports at Re = 40k is fi,open/fi,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 amplified frequency that grows

first in the still-attached boundary layer, and then subsequently after separation. The peak

in power spectra at f = 600 Hz disappears when the flow is reattached and the wing is in

SII. 600 Hz is close to the calculated fH and equal to fi, suggesting that a matching of the

Helmholtz resonance with the naturally occurring instability frequency in the boundary layer

promotes flow 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 transitional Re, the same Helmholtz resonant frequencies in perforated wings can be

expected over the entire Re range, and should yield similar results as this study at Re = 40k

and 60k. Key parameters that change the resonant frequency are orifice radius, orifice

neck height, and cavity volume. Additional parameters such as the number of orifices 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 flute. The distinct frequencies of woodwind instruments

depend on various parameters, including instrument size and geometry, selection of open and

closed finger holes, finger hole size and spacing, type of mouthpiece, and even the angle at

which the air flows 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 different spanwise and chordwise arrays, similar to the finger holes on

a woodwind instrument. The wing is much like an ocarina when the exit port is closed,

and a transverse flute 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 fH in Eq. 9.2 is independent of cavity geometry, which

explains why ocarinas can be made into so many different shapes, as long as the internal

volume is controlled. The orifice geometry of Helmholtz resonators was shown in [27] not

to be a significant 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 different 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 at Re = 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 in L/D. One of the optimum

134

excitation frequency values (660 Hz) matches the calculated fH for the perforated wing.

The calculated values of fi, also obtained from the PIV data, are the second harmonics

of the excitation frequencies, fe. Results from [7] indicate that the flow over the wing is

altered when external acoustic excitation induces maximum fluctuating velocities on the

wing (at tunnel anti-resonances). Minimum pressure and maximum velocity also occur at

the embouchure of a flute, where the air flow occurs [28]. The current study suggests that the

open holes act similarly to embouchures for Helmholtz resonators that amplify the natural

frequency of the fluid instabilities through traveling acoustic waves. That contrasts with

external acoustic excitation, where the most amplified 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 fluctuations at the orifices

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 flow 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 flow instabilities, or those induced at the orifice 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 of fH from Eqs. (9.2) and (9.3) can be made for different tube lengths (neck

heights), h, assuming constant orifice/tube radius, or constant cavity volume. Assuming an

orifice 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 essentialcomponents 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 orifice diameter (tube radius) gives the curves in Fig. 9.19b.

fH 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 orifice/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 effects from pressure taps, although

there still must be a balance to avoid attenuation of the pressure response.

Given the similar configuration of a pressure tap system to a Helmholtz resonator, the

presence of pressure taps may modify the flow over a wing, preventing SI–SII transition. The

flow 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 same Re regime [30] included sectional

pressure coefficients, obtained by 129 0.5 mm diameter pressure orifices on the wing surface.

The data show significant spanwise variation in cd at Re = 100, 000, and the lowest values of

cd were measured at the location of the pressure taps. These findings parallel those from the

136

Figure 9.19: Relationship between Helmholtz frequency, fH , and tube length (neck height),h for constant orifice/tube radius and variable cavity volume (a), and for constant cavityvolume and variable orifice/tube radius (b).

current study, which show higher CL and lower CD for configurations with open holes (Figs.

9.5, 9.7, 9.8, 9.9, 9.15). The local sectional flow state (SI or SII), local wing circulation, and

section drag have been shown to be correlated for the E387 wing in this Re regime [4]. Here,

it is clear that the presence of open holes induces spanwise variation in wing circulation and

hence section drag coefficient.

Measured lift and drag coefficients for the E387 vary widely throughout the literature,

especially at lower Re. The relative difference in measured cd for the E387 airfoil at Re = 60k

among different facilities [1] can be expressed as a fraction of the maximum cd,

∆Cd,lit =cd,max − cd,min

cd,max

. (9.6)

137

The result is given in Fig. 9.20. The data include experiments in which some of the

models had pressure orifices and some did not. The variation in CD for the E387 wing from

the current study at Re = 60k with all open holes and all closed holes is shown as

∆CD,USC =|CD,open − CD,closed|

CD,closed

, (9.7)

and is also plotted in the same figure. The values of ∆Cd,lit are higher than the values of

∆CD,USC, but they are not orders of magnitude apart, unlike a similar measure derived from

the natural spanwise variation tests in [4]. The disruptive effect of holes in pressure tap

measurements upon the global flow properties could be a significant 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 affect 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 orifices suggest

that very slightly porous wings may also operate permanently in the high lift SII state. Little

attention has yet been given to the effects of porosity in feathered wings of birds, for example,

which could be important. However, small geometric cavities by themselves will not affect

wing performance, implying that several types of surface mounted MEMS probes should not

be problematic for this particular flow 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 significantly change the

overall aerodynamic performance. This discovery led to an independent study of the effects

of open holes on the forces and local flow dynamics of the E387 wing at the transitional

Re = 40k where flow 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) flow. The large effect of the

138

Figure 9.20: ∆Cd,lit for the E387 airfoil at Re = 60k among different facilities [1], and∆CD,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 effects 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-SpeedAirfoil 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 ofLow-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., Summaryof Low-Speed Airfoil Data Vol. 3, Soartech Publication, Poole (1997)

[4] Yang, S. L., and Spedding, G. R., “Spanwise Variation in Wing Circulation and DragMeasurement 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 atLow 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 Efficiencies 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 Excitationon a Finite Wing at Low Reynolds Numbers,” American Institute of Aeronautics andAstronautics, 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., “Effects of Acoustic Disturbances on LowRe Aerofoil Flows.” In Fixed and Flapping Wing Aerodynamics for Micro Air VehicleApplications. Vol. 195, pp. 91112. Reston, Virginia: American Institute of Aeronautics& Astronautics, (2001)

[10] Zabat, M., Farascaroli, S., Browand, F., Nestlerode, M., Baez, J., “Drag Measurementson a Platoon of Vehicles,” Research Reports, California Partners for Advanced Transitand 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 SouthernCalifornia, 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 ImagingVelocimetry Algorithms,” Experiments in Fluids, 29, pp. 13-22 (2000)

[14] Spedding, G.R. and Rignot, E.J.M., “Performance analysis and application of gridinterpolation techniques for fluid flows,” 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 PastCavities,” Journal of Fluids Engineering, 100, pp. 152-165 (1978)

[17] Guess, A.W., “Calculation of Perforated Plate Liner Parameters From Specified 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 PerforatedSurface: a Self-Excited Large-Scale Instability,” American Institute of Aeronautics andAstronautics, 14, 12, pp. 4444-4447 (2002)

[19] Celik, E. and Sever, A.C. and Rockwell, D., “Self-Sustained Oscillations Past Perforatedand Slotted Plates: Effect of Plate Thickness,” American Institute of Aeronautics andAstronautics, 43, 8, pp. 1850-1853 (2005)

[20] Ekmekci, A. and Rockwell, D., “Self-Sustained Oscillations of Shear Flow Past a SlottedPlate 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 andImplications for Acoustics,” Annual Review of Fluid Mechanics, 43, pp. 529-550 (2011)

[22] Strutt, J. W. S., “On the Theory of Resonance,” Scientific 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 theAcoustical 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 theAcoustical Society of America, 44, 4, pp. 983-992 (1968)

[27] Chanaud, R.C., “Effects 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 Introducinga 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 Wakefor Flow Control,” in Low Reynolds Number Aerodynamics and Transition, (2012)

[30] McGhee, R. J., Walker, B. S., Millard, B. F., Experimental Results for the Eppler387 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,” inTechniques 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 DepartmentUniversity of Southern California

Los Angeles, California 90089-1191

American Institute of Aeronautics and Astronautics, 2013. In review.

At transitional Reynolds numbers (104 − 105), many smooth airfoils experience laminar

flow separation and possible turbulent reattachment, where the occurrence of either state is

strongly influenced 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

flow reattachment and the appearance of a separation bubble close to the leading edge.

Here, we demonstrate control of the flow dynamics by localized acoustic excitation through

small speakers embedded beneath the suction surface. The flow 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′ Normalized chordwise coordinate (m)

CD Total drag coefficient on a finite wing

CL Total lift coefficient on a finite wing

cs Separation line location (m)

fe Excitation frequency (Hz)

fs 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)

U0 Free stream velocity (m/s)

x, y, zCoordinates in streamwise, spanwise, and normal directions

α Angle of attack (deg)

ωy Spanwise component of vorticity (rad/s)

Superscript

∗ Preferential value

I. Introduction

A growing number of micro aerial vehicles operate in a particular flight regime where the

chord-based Reynolds number, Re = Uoc/ν (where Uo is the flight speed, c the chord, and ν

is the kinematic viscosity), lies between 104 and 105. In this regime, laminar boundary layer

145

separation and then possible turbulent reattachment can either favorably or adversely affect

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 flow reattachment. A number

of intrinsic unstable modes in the laminar separation bubble problem have been identified [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 flow states.

Figure 10.1: CL(CD) and L/D curves show bi-stable states for the E387 wing at various Re.

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 flow around an E387 wing at Re = 40k and 60k at certain excitation frequencies, fe,

increased lift at certain angles of attack, tripped the flow from low- to high-lift state (SI

146

- SII), eliminated pre-stall hysteresis, and promoted flow reattachment. In an enclosed

chamber, minima in the rms acoustic power correspond to minimum fluctuating pressure

and maximum velocity fluctuations, and external acoustic excitation at values of fe 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, fs, suggesting that the flow 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 effective when applied at a point, x′/c = c′, close to or before the chordwise separation

point, cs, so that c′ ≤ cs.

The effect of acoustic forcing was found to deteriorate as c′ moved farther from cs, and

forcing aft of cs 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′ [7]. When excitation was forced near cs, f∗e was found to be equal to the separated shear

instability frequency, fs, 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 =fec

Uo. (10.1)

If the main frequency selection depends on lengthscales in the viscous boundary layer or in

the separated shear layer, then a modified Strouhal number, St∗ = St/Re1/2 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 specific to particular wing α [6, 7, 8],

and generally-applicable scaling laws may be elusive.

B. Objectives

This paper reports on the effects of internal acoustic excitation on the forces and flow fields

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 flow 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-efficient 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 floor. 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 Hz±3 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 amplified with Kramer VA-16XL variable gain

stereo audio amplifiers with a ±3 dB frequency range of 20 Hz− 40 kHz.

C. Wing Model

The wing model had an Eppler 387 profile (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 1800.5 mm diameter holes, and (b) profile view of the lid of the wing. Six spanwise rows ofholes are located at x/c = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6. 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 filled 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 affect 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

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Figure 10.4: CL(CD) and (L/D) curves for the E387 wing at Re = 40k (black) Re = 60k(gray) without excitation.

of fe* as the solid E387 wing, the two most effective frequencies being 420 Hz and 520 Hz

(St∗ = 0.028 and 0.035, respectively).

B. Excitation at Different Reynolds Numbers

In initial tests, 15 speakers were activated (phase-synchronized in time) over −0.83 ≤

y/b ≤ 0.83 and spaced ∆(y/b) = 0.12 apart at the chordwise excitation location c′ = 0.1. Lift

and drag forces were measured for a series of different excitation frequencies, fe, at a pre-SII

angle of attack, α0, for Re = 40k and 60k (as noted in Fig. 10.4). The changes in L/D from

internal forcing at c′ = 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 of fe exists between 200 Hz ≤ fe ≤ 1500 Hz (0.007 ≤ St∗ ≤ 0.055),

for ∆(L/D) ≈ 57%. L/D at fe = 500 Hz is about the same, regardless of Re. L/D(α) at

Re = 40k and 60k are shown in Fig. 10.6. Improvements in L and D occur within a small

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Figure 10.5: L/D for the E387 wing at Re = 40k and α = 11 (black), and Re = 60k andα = 9 (gray), resulting from excitation at different frequencies and corresponding parameterSt∗. 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 at fe = 500Hz (St∗ = 0.034), and b) Re = 60k with and without excitation at fe = 800 Hz (St∗ = 0.029).

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range of low, pre-stall α where the bi-stable states exist (5− 11 for Re = 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 influencing three-

dimensional transition was examined at Re = 60k and α0 = 9. Speakers located at c′ = 0.1

and spaced evenly were forced at fe = 800 Hz (St∗ = 0.029). The different 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 configurations (one speaker

at y/b = 0 and 15 speakers between −0.83 ≤ y/b ≤ 0.83), a family of curves exists and that

a broader spanwise coverage produces improved aerodynamic performance. Note that two

effects 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 flow. 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 fixed span fraction, and Fig. 10.9 shows

the variation in L/D as a function of number of speakers, or speaker density over −0.36 ≤

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 configuration yielding the greatest performance improvement would be a continuous line

source (where the limit of ∆(y/b) goes to zero), the optimum efficiency in terms of excitation

energy vs. propulsion energy requirement would be some intermediate value that exploits

the finite 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

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Figure 10.7: L/D for the E387 wing at Re = 60k for different spanwise distributions ofactivated speakers. Forcing is at fe = 800 Hz (St∗ = 0.029).

Figure 10.8: ∆(L/D) for the E387 wing at Re = 60k and α = 9 for varying number ofspeakers spaced evenly apart, forced at fe = 800 Hz (St∗ = 0.029).

the separated, or bubble shear layer, or whether it simply reflects the finite area influenced

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

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the flow at one wavelength and varying the spanwise wavelength of acoustic excitation sources

around that.

Figure 10.9: L/D for the E387 wing at Re = 60k with activated speakers at c′ = 0.1 between−0.36 ≤ y/b ≤ 0.36 with different spacings. fe = 850 Hz (St∗ = 0.031).

Figure 10.10: ∆(L/D) for the E387 wing at Re = 60k and α = 9 for varying number ofspeakers and fixed span coverage, forced at fe = 850 Hz (St∗ = 0.031).

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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′ = 0.1 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 effect on global L/D. For example, the effectiveness of single speaker forcing

when surrounded by passive diaphragms is much greater; at α = 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 at Re = 60k for different numbers of activated speakersbeneath a constant number of diaphragms. fe = 800 Hz (St∗ = 0.029). The black circles onthe wing schematic indicate activated speakers and white circles inactivated speakers.

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A separate amplitude test maintained a constant speaker and diaphragm configuration

and varied the amplitude directly from the sound amplifier. Five speakers centered around

midspan at c′ = 0.1 were forced at fe = 800 Hz (St∗ = 0.029). The speaker and diaphragm

configuration 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 A3 < A2 < A1.

Figure 10.12: Effect of varying amplitude for five activated speakers and 15 diaphragmscovering −0.83 ≤ y/b ≤ 0.83 at c′ = 0.1 and forced at fe = 800 Hz (St∗ = 0.029).

E. Localized Excitation

While maintaining the same diaphragm configuration, 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 five 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.

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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 five total diaphragms (one for each speaker), the

L/D improvements were significantly 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 at Re = 60k with activated speakers between −0.83 ≤y/b ≤ 0.83 at c′ = 0.1. Excitation with 15 diaphragms is at 800 Hz (St∗ = 0.029) and with5 diaphragms is at 900 Hz (St∗ = 0.033).

Raw particle images can be used to mark a dark separation line as particle-poor fluid

from the boundary layer is released into the otherwise uniform exterior distribution. These

images can be used to directly infer flow conditions on the wing. Figure 10.14 shows the

raw PIV images for the four cases labeled a-d in Fig. 10.13 at α = 9. Activated speakers

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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 filled 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 amplifies a naturally-occurring instability in the originally

separated shear layer (Fig. 10.15a), returning high-speed fluid close to the surface. The

forced vortical structures then move down the airfoil chord and result in a flow which is

re-attached, in the time-averaged sense (Fig. 10.15b).

The small differences 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 flows

are similar to previous measurements in globally-forced experiments [4]. Fig. 10.16a shows

that at α = 9, the natural state is SI, where the flow 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 α accounts for the increase in drag and rather small lift increments as α

increases. In Fig. 10.16b, the flow is acoustically forced at fe = 800 Hz, and has switched to

SII. A region of separated flow close to the leading edge is followed by reattachment so the

global flow sees a wing with high effective camber. This is what leads to the higher lift and

lower drag in SII.

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(a) (b)

(c) (d)

Figure 10.14: Raw PIV images for different span stations across the wing at Re = 60kand α = 9. Speakers are located at c′ = 0.1, and fe = 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 activatedspeakers with double the spacing of (b). Arrows indicate the downstream locations wherethe separation line vanishes.

F. Chordwise Location

The detailed instability mechanisms behind both forced and unforced flows described here

are not necessarily simple to specify and not necessarily general to all cases. Possible receptive

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Figure 10.15: Schematic of the separation line (a) when the flow is unexcited in the SI stateand (b) when the flow is excited in the SII state, where separation occurs slightly earlier andthe separation line vanishes earlier as vortical structures form.

sites to finite amplitude disturbances include, from upstream to trailing edge, Tollmien-

Schlichting waves in the attached laminar boundary layer, inflectional mean profiles in the

bubble shear layer, Kelvin-Helmholtz-type instabilities in the separated shear layer, and

wake profile 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 flow state [18, 1, 19, 2]. More detailed investigation of the natural and forced

mean flow profiles will follow later, but here initial evidence on the effect of chordwise-local

acoustic forcing is shown.

Six rows of five speakers over −0.48 ≤ y/b ≤ 0.48 spaced ∆(y/b) = 0.24 apart were

excited. The six rows of speakers were located at c′ = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6. An example

of this speaker configuration (at c′ = 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 α = 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 effective fe occurs at c′ = 0.1, where ∆(L/D) ≈ 50%. For rows

of speakers at c′ > 0.1, the range of effective fe is much narrower. The range of fe where

L/D > 12L/Dmax can be denoted as R, and is plotted for different c′ in Fig. 10.18a. R

decreases with increasing c′ until c′ = 0.5, where R is approximately the same between 0.4 ≤

c′ ≤ 0.6. In general, the magnitude of the maximum ∆(L/D) decreases with downstream

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Figure 10.16: ωy(x, z) superimposed on fluctuating velocity vector fields for the wing aty/b = 0 (a) with no excitation and (b) with excitation at fe = 800 Hz. Re = 60k andα = 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′ = 0.4 and 0.5 but then

slightly increases at c′ = 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 effective excitation. Measurements of ∆(L/D) were taken at c′ = 0.1 for two other

speaker configurations where a constant number of speakers was maintained, but the spacing

between speakers was varied. Figure 10.19 shows the effect of varying the speaker spacing,

∆(y/b), on ∆(L/D) for the wing at Re = 60k and α = 9 at select frequencies. For almost

all fe 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

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Figure 10.17: ∆(L/D) for various fe for five speakers spaced ∆(y/b) = 0.24 apart between−0.83 ≤ y/b ≤ 0.83 at different c′ locations. Re = 60k, and α = 9.

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Figure 10.18: (a) R, the range of fe where L/D > 12L/Dmax, and (b) ∆(L/D)max at different

x/c for five speakers spaced ∆(y/b) = 0.24 apart between −0.83 ≤ y/b ≤ 0.83 at Re = 60k,and α = 9.

Figure 10.19: ∆(L/D) at select frequencies for different speaker spacing, ∆(y/b), at Re =60k and α = 9. Five speakers are located at c′ = 0.1.

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Figure 10.20: L/D curves at Re = 40k for inactivated speakers with diaphragms and openholes at c′ = 0.1.

a larger improvement in wing performance, implying that the number of acoustic sources

alone does not influence the flow, but rather the distribution of these acoustic sources.

G. Combined Effect of Holes and Internal Forcing

Open holes in the suction surface of the wing drive the flow to the high-lift state through

passive resonance [5]. Covering the holes with thin diaphragms removes this effect, as shown

in Fig. 10.20. Individually, the effect of open holes and the effect 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 α region of interest (5 − 12), a local minimum in L/D occurs at at α = 7, as

seen in Fig. 10.20. At this α, a row of 15 speakers at c′ = 0.1 was forced at varying fe, and

the corresponding ∆(L/D) are shown in Fig. 10.21. In the case of pure internal acoustic

excitation at Re = 40k and α = 9 (Fig. 10.5), a single peak in L/D occurs at fe ≈ 500 Hz

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Figure 10.21: ∆(L/D) at various fe for the wing with open holes at c′ = 0.1 at Re = 40kand α = 7.

(St∗ ≈ 0.034). However, in the case of both open holes and internal acoustic excitation, the

range of effective fe is much wider (fe = 100Hz− 500 Hz or St∗ = 0.007− 0.034).

When the speakers were forced at fe = 200 Hz without diaphragms in the lid of the wing,

the combined effect of active acoustic forcing plus open holes gives the highest L/D increase,

and the resulting dome-shaped L/D curve is shown in Fig. 10.22. In the same figure, L/D

associated with diaphragms and no forcing, open holes and no forcing, and pure internal

acoustic forcing at c′ = 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

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Figure 10.22: Combined effect 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 flow separation is evidently different in

locations where there are speakers, diaphragms or just a solid surface. The presence of a local

acoustic source changes the flow 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 × 105) [6, 7,

8, 9, 10], and none used wing profiles having bi-stable state behavior like the E387 airfoil.

However, the bi-stable state is actually common for many smooth airfoils for Re < 105, and

during cruise conditions (at low, pre-stall α), these airfoils could naturally experience abrupt

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changes in aerodynamic efficiency. This is why airfoil selection is so critical at moderate Re,

and this paper shows how the potentially catastrophic, abrupt changes in both L and D can

be controlled, locally on the wing.

Ultimately the purpose of the local acoustic forcing is to selectively amplify intrinsic

flow instabilities so as to efficiently exert a strong influence on the flow through a small

control amplitude, measured in acoustic power, or electrical drain on a battery. We are also

interested in using the local flow 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 amplification mechanisms.

B. Facility Independence

The internal acoustic excitation results reported here differ from earlier reported external

acoustic excitation results on the same wing [4]. The most prominent differences in wing

performance can be seen in the L/D vs. fe graphs. At Re = 60k, the ∆(L/D)(fe) curve from

external forcing is discontinuous, showing particular preferential fe values that increase L/D,

while the equivalent curve from internal forcing is continuous, showing that all values of fe

within a given range will increase L/D. Results from the external forcing study determined

a dependence of optimum fe on wind tunnel resonances, which explains why only a selection

of fe values improve wing L/D. There is no such dependence for internal acoustic forcing,

so this efficiency enhancement and separation control technique can, in principle, be applied

to standard flying devices in open flight.

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

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Figure 10.23: ∆(L/D) of the E387 wing (a) with external acoustic forcing at α = 8 (replot-ted from Yang et al. [4]) and (b) with internal acoustic forcing from 15 speakers at c′ = 0.1at α = 9.

any wing with open cavities regardless of Re, since the resonance mechanism is independent

of flow speed.

The modifications 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 effects, as observed here (e.g. Fig. 10.22). Although most

previous studies were for airfoils and wings at large, post-stall α, when Helmholtz resonance

would be unimportant, at lower α, 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 flight Reynolds number, 104 ≤ Re ≤ 105, where,

at pre-stall α, the flow over the suction surface of an airfoil or finite wing can suddenly

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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 flow separation, and combinations of active and passive acoustic forcing allow

access to an envelope of L/D(α) 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 effective in modifying the flow but the mechanism is not yet clear. Future

work on calculating unstable modes of both attached and separated flow profiles 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 flying devices.

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Paper IV References

[1] Marxen, O. and Rist, U., “Mean flow 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 effect of small amplitude convective disturbanceson 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 separatinglaminar boundary layer,” Journal of Fluid Mechanics, 711, pp. 1-26 (2012)

[4] Yang, S. L., and Spedding, G. R., “Separation Control by External Acoustic Excitationon a Finite Wing at Low Reynolds Numbers,” American Institute of Aeronautics andAstronautics, 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 Effect of Acoustics on Flow Passing aHigh-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 AcousticExcitation,” American Institute of Aeronautics and Astronautics, 28, 8, pp. 1440-1486(1989).

[8] Chang, R. C., Hsiao, F. B., Shyu, R. N., “Effect 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 HighAngles 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 Effect of Acoustic Forcing onTrailing Edge Separation and Near Wake Development of an Airfoil,” American Instituteof Aeronautics and Astronautics, 1st National Fluid Dynamics Congress, (1988).

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[11] Zabat, M., Farascaroli, S., Browand, F., Nestlerode, M., Baez, J., “Drag Measurementson a Platoon of Vehicles,” Research Reports, California Partners for Advanced Transitand 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 SouthernCalifornia, Los Angeles, CA (2007)

[13] Spedding, G. R. and McArthur, J., “Span Efficiencies 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 Societyof 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] McAuliffe, B. R. and Yaras, M. I., “Transition Mechanisms in Separation Bubbles UnderLow 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 Inflectional Instability of aLaminar 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).

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Paper V

173

Chapter 11

Stability Analysis of Experimental Velocity

Profiles Using an Orr-Sommerfeld Solver

Yang, S. L.

Aerospace and Mechanical Engineering DepartmentUniversity of Southern California

Los Angeles, California 90089-1191

Empirically-acquired velocity profiles of separated flow and sound-induced reattached

flow 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 profiles 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 flow 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 affects boundary layer fluid flow.

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I. Introduction

In aeronautics, when the chord-based Reynolds number Re = UC/ν (where U is the

freestream velocity, C is the wing chord, and ν is the kinematic viscosity) lies in the particular

regime 104 ≤ Re ≤ 105, fluid flow over a surface undergoes complex and complicated flow

phenomena. In this Re regime, fluid flow in a boundary layer is extremely sensitive to

small environmental or geometry variations, causing fluid flow to be prone to separation

with possible turbulent reattachment. The sensitivity to small disturbances and changes is

a result of the onset and amplification of instabilities in the boundary layer.

A developing boundary layer contains several stages through which initially laminar flow

can transition to a fully turbulent flow. 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 amplified, T-S waves can grow

into larger, three-dimensional instabilities. When the T-S instabilities reach large enough

amplitudes, the flow 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 flow [2].

The linear transition to turbulence involves receptivity, linear stability, and nonlinear

breakdown. In the receptivity stage, where the local Re 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 flow [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

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amplified if the flow is unstable, or attenuated. Sometimes the disturbance might decay over

a considerable distance before being amplified [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 × 104 from experimentally-obtained

velocity profiles. 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 affecting the intrinsic flow over the wing was unclear. Attempts

were made to experimentally measure the most amplified frequencies in the boundary layer

over the wing surface, but the presence of a measurement probe in the shear layer affected the

nominal performance of the wing, inhibiting flow reattachment and the expected transition

from low- to high-lift states. Previous experiments in the literature also used measurement

probes to obtain the most amplified instabilities, but there was little validation reported on

the effects 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 flow 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 flow, and

the amplified frequencies are extracted and compared to the optimum forcing frequencies

observed from experimentation.

176

II. Empirical Velocity Profiles

Velocity profiles U(z) for the E387 wing at Re = 60k and α = 9 are obtained by non-

intrusive PIV methods described in [3, 4, 5]. In this coordinate system, x is streamwise, y is

spanwise, and z is normal to the wing surface, and the origin is defined to be at the leading

edge and midspan of the wing (Fig. 11.1).

Figure 11.1: Coordinate system is defined as: x is streamwise, y is spanwise, and z is normalto the wing.

Flow field images, which are taken with a 210-mm lens, have approximately 200 pixels/cm

image resolution. The flow field data are processed on a grid that yields 80 points per velocity

profile from a customized algorithm described in [13] and [14]. The free stream velocity can

be subtracted out of the total velocity profiles to leave only the fluctuating velocity, u. Figure

11.2 depicts the total and fluctuating velocity profiles, U(z) and u(z), respectively, at various

chordwise stations for the wing at Re = 60k and wing incidence α = 9. Figure 11.2a shows

velocity profiles over the wing in its nominal state (without acoustic excitation). Points of

inflection occur at x/C = x′ ≈ 0.4. For x′ > 0.4, the flow is separated, and reverse flow

occurs where ∂U∂x< 0. Figure 11.2b shows velocity profiles over the wing with internal forcing

from five speakers located at x′ = 0.1, all forced in phase at fe = 900 Hz. Under internal

acoustic forcing, inflection points occur much closer to the leading edge (x′ ≈ 0.1) where a

laminar separation bubble begins to forms over the front half of the airfoil, and the flow is

mostly attached over the back half.

For each velocity profile, the second derivative U ′′(z) is numerically solved for by a finite

difference scheme. The value of U ′′(z = 0) is not clear from the current PIV data due

177

Figure 11.2: Total and fluctuating velocity profiles, U and u, respectively, for the wing (a)without forcing and (b) with internal forcing from 5 speakers at fe = 900 Hz at x′ = 0.1.Re = 60k and α = 9. Velocity values are arbitrarily scaled.

to inadequate image resolution near the boundary. Therefore, a 1-d linear interpolation

must be done on U ′′(z), and the value of U ′′(z = 0) is initially extrapolated to zero. It

should be noted that the extrapolation to U ′′ = 0 at z = 0 is a simple assumption; an

accurate extrapolation requires more information about the flow at the boundary, which is

later discussed. Figure 11.3 shows an example profile, U(z), and corresponding interpolated

second derivative, U ′′(z), taken at midspan and x′ = 0.1 for the unexcited wing at Re =

60,000 and α = 9. U(z) and U ′′(z) are normalized by the free stream velocity, U∞, and the

vertical distance z is normalized by the displacement thickness δ∗ given by

δ∗ =

∫ ∞0

(1− U

U∞

)dz. (11.1)

178

Figure 11.3: Empirical velocity profile U(z) (left) and second derivative U ′′(z) with interpo-lation to U ′′(z = 0) = 0 (right) at x′ = 0.1 for the unexcited wing at Re = 60,000, α = 9.The original profile U(y) is obtained from PIV data taken at midspan.

III. Orr-Sommerfeld Solver

The linear instability properties of the empirical velocity profiles are determined by nu-

merical solutions of the Orr-Sommerfeld equation [12],

(U(z)− c)(φ′′(z)− k2φ(z)

)− U ′′(z)φ(z) =

1

ikRe

(φ′′′′(z)− 2k2φ′′(z) + k4φ(z)

). (11.2)

In Eq. (11.2), φ(z) is the amplitude function, k is the wave number, and c is the phase

speed. k and c are related by the frequency, ω:

c =ω

k. (11.3)

The Reynolds number in Eq. (11.2) is based on δ∗ from Eq. (11.4), such that

Re =Uδ∗

ν. (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 → +∞) 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 first order differential equations that are integrated

towards a matching point above the wing surface. Here, a matching condition is satisfied

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 ω is real. The spatial instability growth rate is given by ki, the

imaginary part of the wave number. ki > 0 signifies that the flow is initially contained

and stable, while ki < 0 signifies that the flow is initially unbounded and unstable, so the

condition associated with the minimum value of ki (or maximum −ki) 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 field are small, whereby the term in

Eq. (11.2) containing U ′′(z) can be neglected as z →∞. Then, the asymptotic form of the

eigenfunction φ(z) can be found:

φ(z) = Aekz +Be−kz + Ceγz +De−γz. (11.5)

The exponential solutions in k are homogeneous (inviscid) solutions, and the exponential

solutions in γ are the particular (viscous) solutions, where γ is given by

γ =[k2 + (ikRe) (U − c)

]1/2. (11.6)

180

Far above the boundary (z → +∞), only the decaying solutions may exist, so that A = C =

0. At the boundary (or “wall”),

z = zwall = 0,

φ(zwall) = 0,

φ(zwall)′ = 0.

(11.7)

C. A Shooting Method

A shooting method is implemented to solve the two-point boundary valued problem.

Integration of the ordinary differential equation starts from an initial point z = +zi 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 simplified into a set of

first order o.d.e.’s

d~φ

dz= F

(~φ; Re

), (11.8)

where

~φ = [φ φ′ φ′′ φ′′′] . (11.9)

The solution vectors at z = zi are

~φi =

e−kz

−ke−kz

k2e−kz

−k3e−kz

and ~φv =

e−γz

−γe−γz

γ2e−γz

−γ3e−γz

, (11.10)

and the solutions at z = 0 are

181

~φi =

0

0

1

1

and ~φv =

0

0

1

i

, (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 = zm, 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 of c and k. The Wronskian is obtained from the four eigenvectors

in Eqs. (11.10) and (11.11),

W =

∣∣∣∣∣∣∣∣∣∣∣∣

φ+i φ+′

i φ+′′

i φ+′′′

i

φ+v φ+′

v φ+′′v φ+′′′

v

φ−i φ−′

i φ−′′

i φ−′′′

i

φ−v φ−′

v φ−′′

v φ−′′′

v

∣∣∣∣∣∣∣∣∣∣∣∣, (11.12)

where the “+” corresponds to solutions marched downward from z = zi (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 example |W | < 1×10−7, 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

Wn+1 = Wn +∂Wn

∂c|n (cn+1 − cn) + ... (11.13)

where n corresponds to the current estimate of c, and n+1 corresponds to the next calculation

based on the updated estimate of c.

In order to converge on the eigenvalue, Wn+1 = 0, so an updated estimate of c is found

from the Taylor series expansion as

182

cn+1 = cn −Wn

∂W∂c|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 first provided, which is converted into ω by the relation ω = 2πf . An initial

guess is made for k = kr + iki, which is taken from pre-defined kr, ki pairs. These pairs

are formed by the relations

kr = r cos θ,

ki = r sin θ,(11.15)

where r is a radius and θ is an angle. 10 arbitrary values of r between 0.001−1.0 and 10

arbitrary values of θ between 0 − 360 yield a total of 100 kr, ki 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 each k, an iteration process is performed on c until the convergence criterion

on W is met. For each ω, 100 values of k are computed. The value of interest is the kr, ki

pair containing min(ki) for kr > 0. The general iteration process is as follows:

• Input f (in Hz) and compute ω = 2πf

• Make first guess for k = k1 using (r1, θ1) from Eq. (11.15)

– Calculate c1 = ω/k1; compute W1

– Make second guess for c = c2; compute W2 and ∂W∂c|2

– Estimate a new guess for c = c3

– Iterate on c until the convergence criterion for W is met, at which point a final

value c = cf is found

– Calculate kf,1 = ω/cf , the final k value for the first guess

183

• Make a new guess for k = k2 using (r1, θ2)

• Iterate on c; calculate kf,2

• Repeat process for all (r, θ) pairs to obtain kf,1, kf,2, ..., kf,100

• Find min(ki) for kr > 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 φi has exponential solutions away from the boundary

with growth rates ±k, which are of order O(1), while the viscous eigenfunction φv has

exponential solutions away from the boundary of order O(Re1/2), such that

Re(γ) Re(k). (11.16)

The linear independence of the two eigenfunctions is lost as the slower growing solu-

tion, φi, becomes contaminated through truncation error by the faster growing solution,

φv. 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 φi and φv undergo the pseudo-orthogonalization process.

The norms of the eigenvectors are given by

||Ni|| = φ′′i −φ′′′iφ′′′v

φ′′v (11.17)

and

184

||Nv|| = φ′′′v −φ′′vφ′′iφ′′′i . (11.18)

Each of the four components of the eigenvector undergo pseudo-orthogonalization such

that

~Φi =~φi − φ′′′i

φ′′′v~φv

||Ni||, (11.19)

and

~Φv =~φv − φ′′v

φ′′v~φi

||Nv||, (11.20)

whereby the new eigenvectors have the form

~Φi = (Φi Φ′i 1 0) , (11.21)

and

~Φv = (Φv Φ′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

~Φ then become the new values of ~φ for the next integration step.

IV. Results

Profiles for the unexcited wing located at x′ = 0.1, 0.2, 0.3, 0.4 (the red-colored profiles

in Fig. 11.4) were used in the Orr-Sommerfeld solver. At each value of f , a final value of k

meeting the condition min(ki) for kr > 0 was obtained. Figure 11.5 shows the spatial growth

rate, ki, as a function of frequency, f , for profiles at the different x locations. The first

observation to note is that ki > 0 in all cases, indicating that the flow is initially bounded

and stable. The least stable frequencies can be determined by finding the minima in the

185

ki vs. f curves shown in Fig. 11.5. At x′ = 0.1 and 0.2, the ki vs. f curves are quite

flat. Fewer points were generated for the x′ = 0.1 profile due to the elimination of obviously

incorrect, singular results. More points were generated for the x′ = 0.2 profile, for which a

local minimum in ki occurs around f = 500Hz, and even lower values of ki at f > 1000 Hz.

At x′ = 0.3, min(ki) occurs between 200−300 Hz, and at x′ = 0.4, min(ki) occurs around

200 Hz.

Figure 11.4: Velocity profiles U(z) for the wing without forcing at Re = 60k and α = 9.Red-colored profiles are at x′ = 0.1, 0.2, 0.3, 0.4.

Figure 11.5: Velocity profiles U(z) for the wing without forcing at Re = 60k and α = 9.Red-colored profiles are at x′ = 0.1, 0.2, 0.3, 0.4.

186

V. Discussion

A. Initial Comparisons of Experimental and Numerical Results

In the internal acoustic forcing experiments reported in [5], the largest range of effective

forcing frequencies (200 Hz ≤ f ∗ ≤ 1500 Hz) occurred at x′ = 0.1, as shown in Fig. 11.6,

replotted from Yang et al. At x′ = 0.2 and 0.3, a single value of effective forcing frequency

at f ∗ = 300 Hz exists, followed by a range of effective frequencies (800 Hz ≤ f ∗ ≤ 1200 Hz).

For x′ > 0.4, the effective range of frequencies is narrower (900 Hz ≤ f ∗ ≤ 1100 Hz). Of

particular interest are the empirical and numerical results at x′ = 0.3. The single effective

frequency at 300 Hz that was left unexplained in the internal forcing experiments is the

frequency where the minimum ki occurs in the present numerical analysis. If this frequency

is in fact the initially least stable frequency, then in the experiments the flow was excited

into a high-lift state when the acoustic frequency matched the initial, naturally occurring,

least stable frequency at x′ = 0.3. It should also be noted that x′ = 0.3 is the location of

the natural separation location for the wing at the given Re and α.

Correlation of experimental and numerical results are less obvious at pre-separation x′

locations, although aft of the natural separation point (x′ = 0.4) it seems that the values of f ∗

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, fs, or

sub harmonics of fs [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 flow 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 fe from internal acoustic forcing at different x/c locationsat Re = 60k, and α = 9, from Yang et al. [5].

188

B. Oversimplified Problem

The present comparisons are merely speculations as there are several possible sources of

inaccuracy − in particular, improper flow assumptions and oversimplification of the problem.

First, a parallel and 2-dimensional flow was assumed for the experimental velocity profiles

when the actual flow is over a curved surface that has a growing boundary layer with increas-

ing streamwise location. The wing model is also finite in span, so three-dimensional flow

behavior, especially with external disturbances, is likely to occur. For each velocity profile,

the second derivative U ′′(z) was extrapolated to zero. However, U ′′(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 flow without external forces, the momentum equation gives

U∂U

∂x+W

∂U

∂z= −1

ρ

∂P

∂x+ ν

(∂2U

∂x2+∂2U

∂z2

). (11.23)

At the boundary, z = 0,

U = 0,

W = 0,

∂2U∂x2

= 0.

(11.24)

Then, the two terms on the left-hand side and the ∂2U∂x2

term on the right-hand side of Eq.

(11.23) go to zero, which leaves the relation

∂2U

∂z2= − 1

νρ

∂P

∂x. (11.25)

If the pressure at different x locations on the wing surface is known, then Eq. (11.25) can

be used to solve for U ′′(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 same Re in other facilities (i.e., [15, 16, 17, 18]), which

could be implemented into the interpolation procedure.

C. Refined Test Parameters

The current analysis only used 100 different (r, θ) pairs to solve for k at each f . Sev-

eral computational runs at x′ = 0.1 gave obviously incorrect, singular values, which were

discarded. The incorrect values most likely occurred because even after 100 (r, θ) pairs, the

proper values k were still not found. Increasing the number of (r, θ) pairs, although increas-

ing computational time, would expand the locus of kr, ki values, most likely improving the

accuracy of the final k values.

D. Falkner-Skan Profile Matching

Another approach for solving the initial stability properties is using Falkner-Skan profiles

instead of the experimental velocity profiles in the numerical solver. For example, a family

of velocity profiles with perturbation terms to independently vary the strength of backflow

velocity and depth of reserved-flow region, as used in [11], is given by

U(η) = f ′ (η; β)− aηe−(η−η0)/η0 , (11.26)

where f(η; β) is determined by the Falkner-Skan equation

f ′′′ + ff ′′ + β[1− (f ′)2] = 0, (11.27)

with boundary conditions

f(0) = 0,

f ′(0) = 0,

f ′(∞) = 1.

(11.28)

Equations (11.26)−(11.28) produce a family of profiles for various a, η0, and β values. The

experimental velocity profiles can be compared to the family of Falkner-Skan profiles, and the

190

closest matching Falkner-Skan profile to the experimental profile can then be used in the Orr-

Sommerfeld solver. An obvious problem with this method is that there are likely no closely

matching profiles simply due to the nature of empirical results, but if experimental profiles

do closely match Falkner-Skan profiles, issues such as having inadequate flow information at

the boundary would be eliminated.

VI. Conclusion

Stability analysis is performed on empirical velocity profiles using a numerical Orr-

Sommerfeld solver. The results of the spatial stability problem describe only the initial

behavior of the boundary layer flow, rather than the fully developed flow behavior. Prelimi-

nary results from the numerical analysis indicate that the flow is initially bounded and stable

in the boundary layer. Comparisons of the numerical results to the experimental results from

which the velocity profiles are taken suggest that internal acoustic excitation at frequencies

matching the initially least stable frequencies in the boundary layer trigger the flow from a

separated, low-lift state into a reattached, high-lift state. These preliminary findings are still

questionable due to improper flow assumptions and an oversimplified problem. Further tests

can be done to obtain more accurate second derivative interpolations of the velocity profiles,

vary the test parameters in the solver, or compare experimental profiles with Falkner-Skan

profiles. 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 verification.

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 effort 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,” AmericanInstitute 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 Excitationon a Finite Wing at Low Reynolds Numbers,” American Institute of Aeronautics andAstronautics, 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 LowReynolds 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 ImagingVelocimetry Algorithms,” Experiments in Fluids, 29, pp. 13-22 (2000)

[15] McGhee, R. J. and Walker, B. S. and Millard, B. F., “Experimental Results for theEppler 387 Airfoil at Low Reynolds Numbers in the Langley Low-Turbulence PressureTunnel,” 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-SpeedAirfoil 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 ofLow-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 atHigh 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 finite 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 flow separation

and improve the aerodynamic performance of an Eppler 387 wing. Acoustic resonance was

further discovered to passively control flow separation. Numerical analysis of the instabilities

of the experimental flow profiles suggest that the mechanism by which sound interacts with

boundary layer flow 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 effects of

internal acoustic forcing on the stability characteristics of low Reynolds numbers wings and

small-scale flying devices.

194

References

[1] Ahuja, K., and Burrin, R. H. Control of flow separation by sound. In AIAA 9thAeroacoustics Conference (1984).

[2] Ahuja, K., Whipkey, R., and Jones, G. Control of turbulent boundary layer flowsby sound. In AIAA 8th Aeroacoustics Conference (1983).

[3] Boltz, F. W., Kenyon, G. C., and Allen, C. Q. The boundary layer transitioncharacteristics of two bodies of revolution, a flat plate, and an unswept wing in a lowturbulence wind tunnel. Nasa, Ames Research Center, 1960.

[4] Chang, R. C., H. F. B. S. R. N. Forcing level effects of internal acoustic excitationon the improvement of airfoil performance. American Institute of Aeronautics andAstronautics 298, 58 (1992), 823–829.

[5] Collis, S. S., Joslin, R. D., Seifert, A., and Theofilis, V. Issues in active flowcontrol: 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 inflectional instability of alaminar separation bubble. Journal of Fluid Mechanics 629 (2009), 263–298.

[8] Dovgal, A., and Kozlov, V. On nonlinearity of transitional boundary-layer flows.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 fluid flow. Experiments in Fluids 23 (1997), 449–462.

[11] Goldstein, M. Aeroacoustics. NASA SP. Scientific and Technical Information Office,National Aeronautics and Space Administration, 1974.

195

[12] Goldstein, M. E., and Hultgren, L. S. The evolution of Tollmien-Schlichtingwave 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-wavedisturbances. In Annual Review of Fluid Mechanics, vol. 21. 1989, pp. 477–488.

[14] Grundy, T. M., Keefe, G., and Lowson, M. Effects of acoustic disturbances onlow re aerofoil flows. In Fixed and Flapping Wing Aerodynamics for Micro Air VehicleApplications, T. J. Mueller, Ed. 2001, pp. 91–112.

[15] Hammond, D., and Redekopp, L. Global dynamics and aerodynamic flow vectoringof 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 jetactuators. American Institute of Aeronautics and Astronautics 50, 4 (2012), 788–796.

[19] Hsiao, F., Liu, C., and Shyu, R. Control of wall-separated flow by internal acousticexcitation. 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 performanceimprovement by internal acoustic excitation. American Institute of Aeronautics andAstronautics 32, 3 (1994), 655–657.

[21] Hsiao, F. B., Jih, J. J., and Shyu, R. N. The effect of acoustics on flow passing ahigh-aoa airfoil. Journal of Sound and Vibration 199, 2 (1977), 177–188.

[22] Huang, L., Bryant, T. D., and Maestrello, L. The effect of acoustic forcing ontrailing edge separation and near wake development of an airfoil. In 1st National FluidDynamics Congress (1988).

[23] Huang, L., Maestrello, L., and Bryant, T. D. Separation control over anairfoil at high angles of attack by sound emanating from the surface. In AIAA 19thFluid Dynamics, Plasma Cynamics and Lasers Conference (1987).

[24] Huerre, P., and Monkewitz, P. A. Local and global instabilities in spatiallydeveloping flows. Annual Review of Fluid Mechanics 22 (1990), 473–537.

[25] Kachanov, Y. Physical mechanisms of laminar-boundary-layer transition. AnnualReview 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 ofArizona, 1993.

[28] Kerschen, E. J., Choudhari, M., and Heinrich, R. A. Generation of boundarylayer 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 ofbreakdown. 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 Fluids30 (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 ofTurbomachinery 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 measurementson a low-re airfoil using particle image velocimetry. In Proceedings of the ASME TurboExpo 2005 (2005), American Society of Mechanical Engineering.

[38] McAuliffe, B. R., and Yaras, M. I. Transition mechanisms in separation bubblesunder low and elevated free stream turbulence. In Proceedings of the ASME Turbo Expo2007 (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 ofthe Royal Society of London (1980), vol. A 372, pp. 517–534.

[41] Nishioka, M., Asai, M., and Yoshida, S. Control of flow separation by acousticexcitation. 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 bluff bodies.Naca, 1954.

[45] Saric, W., White, E. B., and Reed, H. L. Boundary-layer receptivity to freestreamdisturbances 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 oscillationsand transition on a flat 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. Summaryof Low-Speed Airfoil Data, vol. 2. Soartech Publication, 1996.

[50] Shapiro, P. The influence 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 blasiuslayer. Physics of Fluids 21, 11 (1987), 2112–2113.

[54] Watmuff, J. Evolution of a wave packet into vortex loops in a laminar separationbubble. 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 andtransition at a change of surface curvature. Journal of Fluid Mechanics 439 (2001),305–333.

[57] Yarusevych, S., Kawall, J. G., and Sullivan, P. E. Influence of acoustic exci-tation on airfoil performance at low Reynolds numbers. In 23rd International Councilof the Aeronautical Sciences Congress (2002).

[58] Yarusevych, W., Kawall, J. G., and Sullivan, P. E. Airfoil performance at lowreynolds numbers in the presence of periodic disturbances. Journal of Fluids Engineering128 (2006), 587–595.

[59] Yarusevych, W., Sullivan, P. E., and Kawall, J. G. On vortex sheddingfrom an airfoil in low-Reynolds-number flows. 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 CaliforniaBerkeley, 1994.

[61] Zaman, K. Effect of acoustic excitation on stalled flows over an airfoil. AmericanInstitute of Aeronautics and Astronautics 30, 6 (1992), 1492–1499.

[62] Zaman, K., and Bar-Sever, A. Effect of acoustic excitation on the flow over alow-re airfoil. Journal of Fluid Mechanics 182 (1987), 127–148.

[63] Zaman, K., and McKinzie, D. Control of laminar separation over airfoils by acousticexcitation. American Institute of Aeronautics and Astronautics 29, 7 (1991), 1075–1083.

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Appendix A

Derivation of Sound and Fluid Flow Equation

For flow in the x-direction only, the mass conservation equation is

∂ρ

∂t+

∂

∂x(ρu) = 0 (A.1)

and the Navier-Stokes equation for incompressible flow without external forces is

∂u

∂t+ u

∂u

∂x= −1

ρ

∂p

∂x+ ν

∂2u

∂x2. (A.2)

Assume an initially unperturbed state at rest (u0 = 0) and small perturbations in density,

pressure, and velocity:

ρ = ρ0 + ρ′ (ρ′ ρ0),

p = p0 + p′ (p′ p0),

u = u′.

(A.3)

Substituting the relations from Eq. (A.3) into Eqs. (A.1) and (A.2), neglecting all high-order

ρ′ and u′ terms, and using the relationship

c2 =p′

ρ′(A.4)

yields

200

∂ρ′

∂t+ ρ0

∂u′

∂x= 0 (A.5)

and

ρ0∂u′

∂t+ c2∂ρ

′

∂x= ρ0ν

∂2u′

∂x2. (A.6)

Taking the divergence of Eq. (A.5), and applying ∂ρ0∂x

= 0 and ∂ρ0∂t

= 0, gives

∂2ρ′

∂x∂t+ ρ0

∂2u′

∂x2= 0. (A.7)

Taking the time derivative of Eq. (A.6), and applying ∂ρ0∂x

= 0 and ∂ρ0∂t

= 0, gives

ρ0∂2u′

∂t2+ c2 ∂

2ρ′

∂x∂t= νρ0

∂

∂t

(∂2u′

∂x2

). (A.8)

Multiplying Eq. (A.7) by c2 and subtracting it from Eq. (A.8) gives

∂2u′

∂t2− c2∂

2u′

∂x2= ν

∂

∂t

(∂2u′

∂x2

). (A.9)

The pressure perturbations, p′, can also be related to velocity perturbations, u′. The

time derivative of Eq. (A.5) is

∂2ρ′

∂t2+ ρ0

∂

∂t

(∂u′

∂x

)= 0, (A.10)

and taking the divergence of Eq. (A.6) is

ρ0∂

∂t

(∂u′

∂x

)+ c2∂

2ρ′

∂x2= νρ0

∂3u′

∂x3. (A.11)

Subtracting Eq. (A.11) from Eq. (A.10) and relating pressure and density by Eq. (A.4)

gives the relation

201

1

c2

∂2p′

∂t2− ∂2p′

∂x2= −νρ0

∂3u′

∂x3. (A.12)

For a pure sound wave, the wave equation is

∂2u′

∂t2− c2∂

2u′

∂x2= 0. (A.13)

Take u′ to be of the form

u′ = ϕexp [ikz − iωt] . (A.14)

It follows that

∂2u′

∂t2=(−ω2

)u′, (A.15)

∂2u′

∂x2=(−k2

)u′, (A.16)

and

∂

∂t

(∂2u′

∂x2

)=(ik2ω

)u′. (A.17)

Use the fact that

ω = ωr + iωi, (A.18)

where ωr and ωi are the real and imaginary parts of ω, respectively. Substituting Eqs.

(A.15)−(A.17) into Eq. (A.9) and separating real and imaginary components yields

ωi = −1

2νk2 (A.19)

202

and

ω2r − ω2

i = c2k2 + νωik2. (A.20)

Substituting Eq. (A.19) into Eq. (A.20) then gives

ω2r

k2= c2 − 1

2ν2k2. (A.21)

203