33rd AIAA Fluid Dynamics Conference and Exhibit, June 23–26, 2003/Orlando, Florida

Active-Adaptive Control of AcousticResonances in Supersonic Impinging Jets

(invited)

Anuradha Annaswamy, Jae Jeen Choi, Debashis SahooMassachusetts Institute of Technology,

Cambridge, Massachusetts, USA

Okechukwu Egungwu, Huadong Lou and Farrukh AlviFlorida A & M University and Florida State University,

Tallahassee, Florida, USA

Fluid-flows exhibit pronounced acoustic resonances when a jet emerges from an orificeand impinges on an edge. This paper looks at one such problem, which consists ofpressure oscillations of a supersonic jet issuing from a Mach 1.5 nozzle and impingingon a ground plane. Reliable and uniform reduction of these impingement tones over arange of operating conditions are mandated by performance considerations of lift-loss,acoustic fatigue of nearby structures and ground erosion. Prior work has clearly shownthat microjets located at the jet nozzle serve as effective actuators for the control ofimpingement tones. In this paper, we present the results of an adaptive closed-loopcontrol strategy using the microjets. A reduced-order model of the impingement tonesthat captures some of the dominant features of the flow-field is presented. A model-basedcontrol strategy based on the Proper Orthogonal Decomposition (POD) of the pressuredistribution along the azimuthal direction is presented. This control methodology isin turn shown to improve the pressure reduction of the Overall Sound Pressure Levels(OASPL) by an additional 8-10db, compared to an open-loop control, at the desiredoperating conditions.

INTRODUCTION

SEVERAL acoustic resonances have their origin inthe instability of certain fluid motions. One of

these motions is in the context of impinging high-speedjets. Experienced by STOVL aircraft while hoveringin close proximity to the ground, impingement tones,which are discrete, high-amplitude acoustic tones, areproduced due to interactions between high speed jetsemanating from the STOVL aircraft nozzle and theground, the feedback mechanisms of which are wellknown.1

The high-amplitude impingement tones are undesir-able not only due to the associated high ambient noise,but also because of the accompanied highly unsteadypressure loads on the ground plane and on nearby sur-faces. While the high noise levels can lead to structuralfatigue of the aircraft surfaces in the vicinity of thenozzles, the high dynamic loads on the impingementsurface can lead to an increased erosion of the land-ing surface as well as a dramatic lift-loss during hover.In an effort to reduce or eliminate these tones, severalpassive2–4 and active control methods5–7 have been at-tempted over the years to interrupt the feedback loopthat is the primary cause of the impingement tones. Ofthese, the technique proposed by Alvi et al.7 appears

Copyright c© 2003 by the authors. Published by the AmericanInstitute of Aeronautics and Astronautics, Inc. with permission.

most promising from the point of view of efficiency,flexibility, and robustness. This method introducesmicrojets along the periphery of the nozzle exit whichinterrupt the shear-layer at its most receptive locationthereby efficiently impacting the impingement tones.Due to their small size, these microjets can be opti-mally distributed along the circumference and can alsobe introduced on-demand.

Alvi et al.7 showed that an open-loop control strat-egy that employs the microjets is effective in suppress-ing the impingement tones. It was also observed thatthe amount of suppression is dependent to a large ex-tent on the operating conditions.7 For example, it wasobserved in experimental studies that the amount ofreduction that was achieved varied with the height ofthe lift-nozzle from the ground-plate as well as withthe flow conditions. Since in practice, the operatingconditions are expected to change drastically, a moreattractive control strategy is one that employs feed-back and has the ability to control the impingementtones over a large range of desired operating condi-tions. In this paper, we propose such a closed-loopcontrol strategy for reducing the impingement tones.

In order to design a closed-loop control strategy,we adopt a model-based approach. A model of im-pingement tones, however, is quite difficult to derivedue to the changing boundary conditions, compress-

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American Institute of Aeronautics and Astronautics Paper 2003–3565

33rd AIAA Fluid Dynamics Conference and Exhibit23-26 June 2003, Orlando, Florida

AIAA 2003-3565

Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

ibility effects, and the feedback interactions betweenacoustics and the shear-layer dynamics present in theproblem. Since our primary goal is to model the im-pingement tone-dynamics and how they respond tomicrojet-control action at the nozzle, we will derivea reduced-order model that captures these dominantdynamics and the effect of control. For this deriva-tion, while tools based on stability theory8 can beused to obtain some of the parameters such as thetonal frequencies, they are inadequate for derivingother model-details due to the complex features of theflow-field. Instead, we use the Principal OrthogonalDecomposition (POD) method and key measurementsin the flow-field to derive the model. This model inturn is used to derive an appropriate closed-loop con-trol strategy. Both the model and the model-basedcontroller are validated using experimental results ina high-speed STOVL facility in the Fluid MechanicsResearch Laboratory (FMRL) at FSU.

The POD method is a tool used to extract the mostenergetic modes from a set of realizations from the un-derlying system.9 These modes can be used as basisfunctions for Galerkin projections of the model in orderto reduce the solution space being considered to thesmallest linear subspace that is sufficient to describethe system. In fluid systems, the POD technique hasbeen applied in the analysis of coherent structures inturbulent flows and in obtaining reduced order modelsto describe the dominant characteristics of the phe-nomena. In this paper, our goal is to use the PODmethod to extract information about the mode shapesusing pressure measurements in order to determine thecontrol input strategy.

The paper begins with a reduced-order model of theimpingement tones, which is shown to predict both thefrequencies and the frequency interval quite well. Thismodel in turn is used to derive the POD-based controlstrategy and the experimental results are presented.Concluding remarks are presented at the end of thepaper.

A REDUCED-ORDER MODEL OFTHE IMPINGEMENT TONES

The reduced-order model adopted for the control ofimpingement tones is based on the vortex-sheet jetmodel of Tam.10 Within a short distance (0.01Rj)downstream from the nozzle exit, the jet can be ide-alized as a uniform stream of velocity Uj and radiusRj bounded by a vortex sheet. Small-amplitude dis-turbances are superimposed on the vortex sheet (seeFig. 1). This neglects the effect of the shock structuredue to the microjet action and due to underdevelopedjet (if any) and the boundary effect of the ground.Let p+(r, θ, z, t) and p−(r, θ, z, t) be the pressure as-sociated with the disturbances outside and inside thejet, denoted respectively by domain Ω1 and Ω2 whereΩ1 denotes jet-core which extends from z = -∞ to z

Uj

ζ (z,θ,t)

r

Rj

z

p+

p-

Kulite sensor

Microjets(p (θ))

Nozzle exit

Lift plate

r

Rj

Ω1

2

pd-

pd+pr

µ

ζ (z,θ,t)

Ω

θ

+

Fig. 1 Vortex-sheet jet model for the impingementtones control problem. Location of microjets andpressure sensors also shown.

= +∞, Ω2 denotes the domain outside the jet-coreand (r,θ,z) are the cylindrical coordinates. Also, letζ(θ, z, t) be the radial displacement of the motion of acompressible flow. It can be shown that the governingequation and r-direction momentum equation for theproblem are:

1a2∞

∂2pd+

∂t2= ∇2pd+ (r ∈ Ω2)

1a2

j

(∂

∂t+ Uj

∂

∂z

)2

pd− = ∇2pd− (r ∈ Ω1)(1)

where a∞ and aj are the speed of sound outside andinside the jet, Uj is the main jet speed and ∇2 is theLaplacian operator.

The governing wave equation is assumed to have asolution of the form:

pd+(r, z, θ, t)pd−(r, z, θ, t)

ζ(z, θ, t)

=

pd+(r)

pd−(r)ζ

ei(kz+nθ−ωt) (2)

where n = 0, ± 1, ± 2,... and k, the wavenumber andω (ω>0), the angular frequency are as yet unspecifiedparameters. Substituting the solution (2) into (1), itleads to the following eigenvalue problem about pd+

and pd−:

d2pd+

dr2+

1r

dpd+

dr− n2

r2pd+ +

[ω2

a2∞− k2

]pd+ = 0 (3)

and

d2pd−dr2

+1r

dpd−dr

− n2

r2pd−+

[(ω − Ujk)2

a2j

− k2

]pd− = 0

(4)The solutions of (3) and (4) are given by:

pd+ = C1H(1)n (η+r) (5)

pd− = C2Jn(η−r) (6)

where H(1)n is the nth order Hankel function of the first

kind, Jn is nth order Bessel function of the first kind,η+ = (ω2/a2

∞ − k2)12 , η− =

[(ω − Ujk)2/a2

j − k2] 1

2 ,and C1 and C2 are unknown constants which are to bedetermined from boundary conditions.

The conditions that make the problem well-posedare as follows.

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Condition 1 Dynamic condition at r = Rj :

p+ − p− = ∆p δ(z − ε)ei(nθ−ωt) (7)

where p+ represents the outside pressure field, thesuperposition of the acoustic wave (pd+) travellingupstream, and a wave reflected from lift plate (pr+)propagating in the opposite direction with an equalmagnitude, i.e., pr+(r, θ, z, t) = pd+(r, θ,−z, t). p−represents the inner pressure field inside the main jet,and ∆p denotes an imposed pressure jump across theshear layer.

Condition 2 Kinematic condition at r = Rj :

v+ − v− = ∆v δ(z − ε)ei(nθ−ωt) (8)

where v+ represents the outside radial velocity field,v− represents the inner radial velocity field and and∆v represents an imposed velocity jump across theshear layer.

Mathematically, the pressure field in the entire flowfield can now be expressed as follows.

p+ = pd+ + pr+

= A1H(1)n (η+r) cos(kz)ei(nθ−ωt)

p− = A2Jn(η−r) cos(kz)ei(nθ−ωt) (9)

where constants A1 and A2 are to be obtained fromboundary conditions.

The pressure and velocity jumps ∆p and ∆v are im-posed due to the presence of the ground plane and canbe determined as follows. We model the ground effectby introducing ‘virtual’ acoustic sources on the groundplane. In particular, infinite number of monopoles areassumed to be present at z = L, along a circular lineof radius r = Rj , with strength Sw varying along theazimuthal coordinate θ. The source strength is influ-enced by the jet vortical structures impinging on theground plane and is assumed as:

Sw(θ) = κp+(r = Rj , θ, z = L) (10)

where κ is a proportional constant whose value can beestimated by comparison with the experimental data(see Fig. 2). Using Eq. (10) and Fig. 2, we then cal-culate ∆p and ∆v from the sum of pressure excitationcaused by each monopole:

∆p ei(nθ1−ωt) =∫ 2π

0

∆pmonopole(θ; θ1)dθ

=∫ 2π

0

Sw(θ; n)(−ρj

∂φmonopole

∂t

)dθ

and

∆v ei(nθ1−ωt) =∫ 2π

0

∇φmonopoledθ (11)

x

yθ

pmonopole(θ)

θ1

L

Ground plane

Lift plate z pd+

Sw

(a) (b)

Fig. 2 Acoustic source modelling ground effect:(a) pressure distribution due to jet impinging onthe ground plane, and (b) overall excitation causedby infinite monopole sources. Note that the (x, y, z)coordinate system, shown here, is used only forcomputing ∆p and ∆v.

where θ1 is the azimuthal location of interest,∆pmonopole(θ; θ1) is the pressure excitation at the col-lecting point (θ1) on the lift plate due to a monopoleplaced at θ of the ground plane, φmonopole is the veloc-ity potential due to each monopole placed on groundplane, and ρj is the density of medium in the jet noz-zle. The velocity potential in a moving media can beexpressed as11

φ = 2πe

−iωt− iωM1z

1−M21

+ iω

c(1−M21)

12

(z2

1−M21

+x2+y2) 1

2

(1 − M21 )

12

(z2

1−M21

+ x2 + y2) 1

2

(12)where ω is frequency of acoustic wave in radian, c issound speed and M1 is the Mach number of movingmedia. Therefore Eqn. (11) become

∆p einθ1 =∫ 2π

0

2πiρj ωSw(θ; n)l(θ; θ1)

×

e

− iωM1L

1−M21

+ iω

c(1−M21)

0.5 l(θ;θ1)

(1 − M21 )0.5 dθ (13)

∆v einθ1 =∫ 2π

0

−2πSw(θ; n)l5/2(θ; θ1)

r − Rj cos(θ1 − θ)

×e

− iωM1L

1−M21

+ iω

c(1−M21)

0.5 l(θ;θ1)

(1 − M21 )0.5 dθ

(14)

where L is the distance between lift plate and groundplane and l is defined as

l =

√L2

1 − M21

+ 2R2j − R2

j cos(θ1 − θ).

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To determine A1 and A2, three additional conditionsdue to geometrical restrictions are imposed near thelift plate and ground plane:

Condition 3 The zero normal flow condition on thelift plate is:

∂p+

∂n= 0 (r ∈ Ω2, z = znozzle) (15)

Condition 4 Mean velocity condition of main jet (seeFig. 3) is assumed to be of the form

Uj =

U0 : 0 < z < 0.8L

− U0

0.2Lz + 5U0 : 0.8L < z < L

where U0 is the exit velocity corresponding to a givenMach number M1.

Condition 4 is inspired from experiments done byKrothapalli et. al.,12 where the mean centerline veloc-ity of impinging jet was observed to drastically reduceto zero near the ground plane.

Uj

L0.8 L z

U0

Fig. 3 The idealized centerline velocity

Condition 5 Equality condition at r = Rj withoutmicrojet:

∂v+

∂t= − 1

ρ∞∂p+

∂r(∂

∂t+ Uj

∂

∂z

)v− = − 1

ρj

∂p−∂r

(16)

Substituting Eqn. (9) into (7) and (8) together with(16), Conditions 1 and 2 can be written as[A1H

(1)n (η+Rj) − A2Jn(η−Rj)

]cos(kz) = ∆pδ(z − ε)

(17)and

1iωρ∞

∂p+

∂r− 1

i(ω − Ujk)ρj

∂p−∂r

= ∆vδ(z − ε). (18)

Integrating the above equation in the range of (0 <z < L), we get

A1H(1)n (η+Rj) sin(kL)

k−

∫ L

0

A2Jn(η−Rj) cos(kz)dz

= ∆p

and1

iωρ∞A1H

′(1)n (η+Rj)η+

sin(kL)

k

−∫ L

0

1

i(ω − Ujk)ρjA2J

′n(η−Rj)η−

cos(kz)dz = ∆v.

Then the pressure amplitude can be calculated fromthe following equation:[

A1

A2

]=

1F11F22 − F12F21

[F22 −F12

−F21 F11

] [∆p∆v

](19)

where

F11 = H(1)n (η+Rj)

sin(kL)k

,

F12 = −∫ L

0

Jn(η−Rj) cos(kz) dz,

F21 =

1iωρ∞

H(1)n

′(η+Rj)η+

sin(kL)

k, and

F22 =∫ L

0

−1i(w − Ujk)ρj

J′n(η−Rj)η−

cos(kz)dz.

Eqns. (9), (13), (14), and (19) provide the completesolution to the governing equations of the impingementtones problem, given by Eqn. (1).

A Control Oriented Reduced-order Model

As mentioned in the introduction, the goal is to re-duce the impingement tones using a suitable activeflow control method. In the experimental facility, ac-tive flow control was implemented using several micro-jets that are flush mounted circumferentially aroundthe main jet nozzle. The question here is to deter-mine a control strategy for modulating the microjectpressure profile in an optimal manner.

In order to determine the control strategy, the effectof the microjet has to be incorporated into the model.We note that when microjets are introduced into themain jet, Condition 5 is changed at z = znozzle, sincean additional velocity uµ is added due to the microjetaction. We therefore introduce yet another conditionbelow:

Condition 6 Equality condition at r = Rj with mi-crojet:

∂v+

∂t= − 1

ρ∞∂p+

∂r

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American Institute of Aeronautics and Astronautics Paper 2003–3565

(∂

∂t+ Uj

∂

∂z− uµ sin α

∂

∂r

)v− = − 1

ρj

∂p−∂r

(20)

where uµ is the microjet velocity and α is the microjetinclination angle with respect to the nozzle center line.The overall solution for A1 and A2 can be derived in asimilar manner as before, using Conditions 1 through6.

While determining the solutions leads to several in-sights into the underlying mechanisms, it does notdirectly provide a prescriptive methodology for obtain-ing the control solution. Instead, what is more usefulis to obtain a parametric description of the governingequation that captures the dependence of the systemvariables on the control input. Towards this goal, weexpress the equation of motion of the impingementtones, given by Eqn. (1) in the following form. Byseparation of variables, we can write for the outer areaΩ2 as

p+(r, θ, z, t) =L∑

i=1

Xi(t)Φi(r, θ, z) (21)

where Φi is a set of orthonormal functions satisfyingthe boundary conditions (Condition 1 ∼ Condition6), i.e., the first L modes of the system, and Xi are themodal amplitudes. These modes can be determinedusing the POD method. The reader is referred to Ref.9

for a general description of POD, and Ref.13 for its usein impingement tone-modeling.

We note that Φi in Eq. (21) is a function of microjetpressure pµ. Substituting Eqn. (21) in (1), and takinginner product with respect to Φi, we get:

Xj(t) = a2∞

l∑i=1

(∇2Φi, Φj)Xi(t) j = 1, · · · , L(22)

Since the modes are dependent upon pµ, we can writeequation (21) in vector form as:

X(t) = A(pµ)X(t) (23)

MODEL VALIDATIONWe now validate the model described in the pre-

vious section using the experimental results obtainedfrom the STOVL supersonic jet facility of the FluidMechanics Research Laboratory (FMRL) located atthe Florida State University.13 For the sake of com-pleteness, we briefly describe the facility below (seeFig. 4 for a schematic). The measurements were con-ducted using an axisymmetric, convergent-divergent(C-D) nozzle with a design Mach number of 1.5. Thethroat and exit diameters (d, de) of the nozzle are2.54cm and 2.75cm (see Figs. 4 and 5). The di-vergent part of the nozzle is a straight-walled conicsection with a 3o divergence angle from the throat tothe nozzle exit. A circular plate of diameter D (25.4cm

h (variable)Ground

Plate

Lift Plate

Nozzled

D

z

Fig. 4 Schematic of the impingement jet testingconfiguration at FSU.

Microjets (d = 400 µm)m

d = 27.5 mme

Kulite

Lift plate

1

2

3

45

6

(a) Geometry of the lift plate and microjets (b) Microjets feed assembly

To microjets

Secondary plenum

chambers

Primary plenum

Control valves

Fig. 5 Schematic of the impingement tone sup-pression experiment using microjets at FSU.

∼ 10d) was flush mounted with the nozzle exit, whichrepresents the ’lift plate’ of a generic aircraft planformand has a central hole, equal to the nozzle exit diam-eter, through which the jet is issued. A 1m × 1m ×25mm aluminum plate serves as the ground plane andis mounted directly under the nozzle on a hydrauliclift, and arranged so that the height h of the lift platefrom the ground plane can be varied over a desiredrange. To validate the model, this facility was run atMach 1.5, at h/D = 3.0, 4.0, and 4.5.

From Eqn. (19), it is clear that the peak in thepressure data is determined by the set (ω,k) whichsatisfies the denominator F11F22 − F12F21 = 0. Thesolution to this equation is not unique and we choosethat particular value of (ω,k) which corresponds to aphase velocity equal to the ambient speed of sound.This means that the upstream propagating acousticwave outside the jet has a phase velocity same as thatof the speed of sound in air at rest.

Using Eqns. (11), (14), and (19), the solution p+ iscompared to the actual experimental data in Fig. 6 forthe first azimuthal mode, n =1, at the lift plate. It isclear that the prediction of amplitude of the pressuresignal by the model is poor. This may be due to thefact that the velocity potential in Eqn. (12) is assumedto contain no damping effects. Therefore, no furtherinsight can be obtained by comparing the amplitude ofpredicted pressure spectrum with that observed fromexperiments in Fig. 6.

Henceforth, we focus on the ability of the model to

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80

90

100

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120

130

140

150

160

170

180

h/D = 4.0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

80

90

100

110

120

130

140

150

160

170

180

h/D = 3.0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

80

90

100

110

120

130

140

150

160

170

180

h/D = 4.5h/D=4.5

Frequency (Hz) Frequency (Hz)(a) Model Prediction (b) Experimental Data

dBdB

dB

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 110

115

120

125

130

135

140

145

150

155

160

165h/D = 3.0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 110

115

120

125

130

135

140

145

150

155

160

165h/D = 4.0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2110

115

120

125

130

135

140

145

150

155

160

165h/D = 4.5

h/D=3.0

h/D=4.0

165

160

155

150

145

140

135

130

125

120

115

1100 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

165

160

155

150

145

140

135

130

125

120

115

1100 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

165

160

155

150

145

140

135

130

125

120

115

110

410e×

410e×

410e×

h/D=3.0

h/D=4.0

h/D=4.5

410e×

410e×

410e×

Fig. 6 Frequency spectra of the unsteady pressureon the lift plate at NPR = 3.7: (a) Experimentaldata, and (b) Model prediction. Note that the am-plitude scales in (a) and (b) are different.

predict the frequency of impingement tones for dif-ferent h/D ratios. It was observed that the peakfrequency calculated from the analytical model shows“staging” phenomena similar to the experimentaldata. Fig. 7 shows a comparison of the staging phe-nomenon of impingement tones between data obtainedthrough the model and experiments. It is encouragingto observe that each tone decreases in frequency ap-proximately linearly with increase in h/D. Note thatin Fig. 7(a), the dominant frequencies observed in theexperiment closely match the edge tones1 given by thewell-known relation

N + ϕ

fN=

∫ h

0

dh

Ci+

h

Ca(N = 1, 2, 3, ...) (24)

where Ci and Ca are the convective velocities of thedownstream-travelling large structures and the speedof upstream-travelling acoustic waves, respectively, fora suitably chosen N and p.

The other relevant parameter that the model pre-dicts is the frequency interval between two tones fordifferent h/D ratios. The result is summarized inTable 1. It is again encouraging to note that the mod-elling error relative to the experimental data is lessthan 20%. Figs. 8 and 9 show the peak frequencyintervals between experimental and modelling resultsat a particular height, h/D = 6.0. Clearly, ∆f1 and

1 2 3 4 5 6 7 8 9 101

2

5

10

20Model prediction

h/D

Fre

quen

cy (

kHz)

First tone(smallest frequency)

2nd tone

3rd tone

(a) Experimental Data

1 2 3 4 5 6 7 8 9 101

2

5

10

20

Most dominant tone

2nd dominant tone

3rd dominant tone

Other tones

Freq

uenc

y (k

Hz)

h/D

(b) Model Predictionh/D

20

10

5

2

11 2 3 4 5 6 7 8 109

Freq

uenc

y (k

Hz)

Fig. 7 Variation of frequency of three impingingtones with h/D at NPR = 3.7: (a) Experimentaldata, and (b) Model Prediction.

Table 1 Comparison of peak frequency interval(Hz) obtained from the experiment data and themodel. NPR = 3.7, and M = 1.5.

h/D ∆f1 (Hz) ∆f2 (Hz) (Model Error(Experiment) Prediction) (%)

2.0 2688.67 2525.0 6.13.0 1926.88 2250.0 16.84.0 1473.20 1666.7 13.14.5 1289.01 1500.0 16.45.0 1137.50 1350.0 18.76.0 1196.25 1125.0 6.0

∆f2 are comparable in the two figures. It can alsobe observed from Table 1 and Fig. 9 that the peakinterval in both the experimental data and the ana-lytical solution decreases as the height of the lift platefrom the ground plane increases from h/D = 2.0 toh/D = 5.0. This behavior is not observed beyondh/D = 5.0 and can be attributed to varying dynamicsgoverning the impingement tones with height. Whenthe lift plate is close enough to the ground plane, theupstream propagating acoustic waves due to impinge-ment plays a major role in shear layer excitation butits effect diminishes at larger distances. Moreover, theerror between the experimental peak interval and pre-dicted one is almost less than 20% at every h/D and

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115

120

125

130

135

140

145

150

155

160

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

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80

100

120

140

160

180

(a) Experimental result

(b) Model prediction

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

110

115

120

125

130

135

140

145

150

155

160

f1

f2

dB

dB

Hz

Hz

Fig. 8 Frequency spectra for unsteady pressureon the lift plate of NPR = 3.7, h/D = 6.0 (a) andmodel prediction of NPR = 3.7, h/D = 6.0 (b).

is around 6% at heights where the feedback loop is adominant mechanism in noise generation.

As far as we know, the present work is among thefirst to obtain analytical models of impingement tonespredicting the staging phenomenon and frequency in-terval magnitudes within tolerable limits. Althoughthe two are not sufficient indicators of the validity ofthis model, yet the model is considered sufficiently richand reliable enough to be suitably used as a mathemat-ical model for obtaining control strategy that resultsin optimal noise reduction.

ACTIVE-ADAPTIVE CONTROL:THE CLOSED-LOOP STRATEGY

AND RESULTSThe starting point for the control strategy is the

model in Eqn. (23), which in turn requires the PODmodes, Φi of the system. In order to determine Φi,as indicated by Eq. (21), the calculation of pressure

0

500

1000

1500

2000

2500

3000

0 1 2 3 4 5 6 7

Experimental Data Model Prediction

∆ f (H

z)

h/D

Fig. 9 Peak frequency interval comparison be-tween experimental data and model prediction(NPR = 3.7, M = 1.5).

at all flow points is needed. This is not feasible eitherexperimentally or computationally due to obvious con-straints. However, our main focus is to control only theimpingement tones, which stem from the large-scalestructure of the overall field. Since the key ingredientsthat contribute to the formation of these tones are theinitiation of the shear layer instability waves and theirinteraction with the acoustic waves appear to be local-ized at the jet nozzle, we derive the control strategyby focusing only on the POD of the pressure field closeto the nozzle. That is, we derive the control strategyusing the expansion:

p+(r = Rs, θ, z = znozzle, t)∆= p(θ, t) =

l∑i=1

Ti(t)φi(θ)

(25)where Rs is the radial position of the sensors on thelift plate. Note that φi’s in Eqn. (25) are, quite likely,a subset of Φi’s in Eqn. (23) which are the modes ofthe entire flow field. The state space equation corre-sponding to these reduced set of modes are given by:

Tj(t) = a2∞

l∑i=1

(∇2φi, φj

)Ti(t) j = 1, ..., l (26)

with the inner product suitably defined. In vectorform, this becomes:

T (t) = A(pµ)T (t) (27)

Once the mode shapes are determined, we simplychoose the control strategy as:

pµ(θ) = kφ1(θ) (28)

where φ1 is the most energetic mode of Eqn. (25)and k is a calibration gain. The complete closed-loopprocedure therefore consists of collecting pressure mea-surements p(θ, t), expanding them using POD modesas in Eqn. (25), determining the dominant mode φ1,and matching the control input − which is the mi-crojet pressure distribution along the nozzle − to this

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dominant mode as in Eqn. (28). That is, the con-trol input adapts to the on-line eigenmode φ1, whichmay vary with flow-conditions. This active-adaptivecontrol strategy, which is denoted as “mode-matched”control, was used to determine the control input in theexperimental investigations using the STOVL facilityat FMRL, FSU.Remark 1: In order to guarantee that the closed-loopcontrol leads to a uniform reduction of the impinge-ment tones, we note that A(kφ1(θ)) must have stableeigen-values at desired locations. While an analyticalderivation of the conditions under which this can oc-cur is beyond the scope of this paper, as demonstratedin the next section, this is indeed what occurs in theexperimental investigations.Remark 2: The closed-loop control approach usedhere is distinctly different from the traditional feed-back control paradigm where the control input is typi-cally required to be modulated at the natural frequen-cies of the system (for example, see Rowley et al.14).The latter, in turn, mandates that the external ac-tuator have the necessary bandwidth for operating atthe natural frequencies. In the problem under consid-eration, the edge tones associated with the flow-fieldare typically a few kilohertz. Given the current valvetechnology, modulating the microjets at the systemfrequencies is a near impossibility. The approach pre-sented above overcomes this hurdle by modulating thecontrol input, pµ, at a slow time-scale, so that it be-haves like a parameter. If this control input is chosenjudiciously, then even small and slow changes in this“parameter” can lead to large changes in the processdynamics, as is shown in the next section.

Experimental Results

The mode-matched control strategy described abovewas implemented at the STOVL supersonic jet fa-cility of the Fluid Mechanics Research Laboratory,FSU (see13 for details). Four banks of microjetswere distributed around the nozzle exit, while pressurefluctuations were sensed using six KuliteTM tranduc-ers placed symmetrically around the nozzle peripheryplate, at r/d = 1.3, from the nozzle centerline whered is the nozzle throat diameter, as shown in Fig. 5(a).The jets were fabricated using 400 µm diameter stain-less tubes and are oriented at approximately 20o withrespect to the main jet axis. The supply for the micro-jets was provided from compressed nitrogen cylindersthrough a main and four secondary plenum chambers.In this manner, the supply pressures to each bank ofmicrojets could be independently controlled. The con-trol experiment was performed for a range of heights(of the nozzle above ground).

At each height, in addition to the mode-matchedcontrol, the active control strategy as given in (Shihet al.)15 was also implemented. In the latter case, thespatial distribution of microjet pressure around the

h/D = 2.0

h/D = 3.0

h/D = 4.0

h/D = 4.5

h/D = 5.0

First

mode

shape

Proposed

microjet

pressure

distribution

Transducer Position

(a)

Shear Layer Dynamics

Acoustics

Controller

(Mode-matched strategy)

Microjets

(Actuator) Kulite

(sensors)

TM

(b)

Mic

roje

t

pre

ssure

at

the l

ift

pla

te

Flo

w

pre

ssure

at

the l

ift

pla

te

Ground

Effect

Fig. 10 (a) The first mode shape and suggestedmicrojet pressure distribution for each height. h isthe height of the lift-plate from ground and D isthe diameter of the lift-plate. (b) Block diagramof the closed-loop control program of impingementtones.

nozzle exit was kept uniform, and can be viewed asan open-loop control procedure. To ensure a fair com-parison between the two control methods, the mainnozzle was forced to operate under constant condi-tion throughout the whole process. The calibrationconstant k in Eqn. (28) was chosen such that theminimum and maximum values of the POD modeover θ correspond to 70psi and 120psi, respectively,which ensured maximum effectiveness of the actua-tor. Fig. 10(a) shows the shape of the first modeand the suggested microjet bank pressure distributionfor several heights and Fig. 10(b) shows a block dia-gram of the active closed-loop control method. Fig. 11shows the results for the closed loop control strategy,which indicates better performance at the experimentthroughout all operational conditions, with a large im-provement at heights h/D = 4, 4.5 and 5. The reasonfor this increased pressure reduction can be attributedto the percentage of energy contained in the dominantmode, which is used in the control strategy. At heights4 to 5, the energy content of the first mode is above86%. In contrast, at heights 2 and 3, the energy leveldrops to about 50% and hence the corresponding im-provement in the closed-loop strategy also drops toabout half the db-value at heights 2 and 3 comparedto at heights 4 and 5.

CONCLUDING REMARKSIn this paper, an active-adaptive closed-loop control

strategy was proposed to suppress the supersonic im-pingement tones. This strategy is based on a reduced-order model of the impingement tones and a POD-

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2 3 4 5 6 7 8 9148

150

152

154

156

158

160

162

164

166

168

h/D

OA

SP

L(dB

)

without controlopen−loop controlclosed−loop control

Fig. 11 Overall sound pressure levels (OASPL)without control, with open-loop control and withclosed-loop control strategies at NPR=3.7.

analysis of the pressure distribution close to the nozzleexit along the azimuthal direction, and matches themicrojet pressure distribution to that of the dominantPOD mode of the pressure. Experimental results froma STOVL supersonic jet facility at Mach 1.5 showthat the mode-matched closed-loop strategy providesan additional 8-10 db reduction, compared to an openloop one, at the desired operating conditions.

The active-adaptive control strategy proposed herecontains three significant features that are worthpointing out. The first is that it is a static controlmethod in that the flow from the microjets is main-tained at a steady-value instead of being pulsed. Thiswas necessitated by the high values of the eigen fre-quencies of the impinging jets and the current valvetechnology. It is quite likely that appropriate pulsingat the impinging tone-frequencies may lead to a moreefficient reduction of the impingement tones. The sec-ond feature stems from the POD-based approach. Theprocess of shaping the microjet pressure distributionusing the azimuthal modes of the system via the mode-matched control strategy introduces well-organized,strong streamwise voriticity16 , whose strength is var-ied as dictated by the on-line measurements. It shouldbe pointed out that recent experimental results ob-tained from the STOVL facility indicated that theperformance improvement of the closed-loop strategyis a function of the amount of flow distortion that maybe present in the problem. When the incoming flowis straightened significantly, it was observed that theclosed-loop action continues to maintain good perfor-mance, though its improvement over open-loop controlis reduced.

The third and final feature of the closed-loop strat-egy proposed in this paper is its amenability to adap-tation. The mode-matched controller can be improvedfurther using a Recursive Proper Orthogonal Decom-position (RePOD) method17 which can be used to

update the POD mode φ1(θ), so as to accommodatevarying flow conditions on-line, such as during landingof a STOVL aircraft. Currently, work is being carriedout to implement an active-adaptive control methodreduce the tones in the STOVL facility as the groundplate is moved upstream towards the nozzle, by usingthe RePOD technique.

ACKNOWLEDGEMENTSThis work was supported by a grant from the Air-

force Office of Scientific Research, through the Un-steady Aerodynamics and Hypersonics program, withDr. Schmisseur as the Program Manager. We wouldlike to thank the staff of the Fluid Mechanics Re-search Laboratory, FSU, for their help in conductingthe tests, and Professor Krothapalli, FSU, for his in-valuable guidance throughout this research.

References1Powell, A., “On edge tones and associated phenomena,”

Acoustica, Vol. 3, 1953.2Glass, D., “Effect of acoustic feedback on the spread and

decay of supersonic jets,” AIAA Journal , Vol. 6, No. 6, 1968,pp. 1890–1897.

3Poldervaart, L., Wijnands, A., vanMoll, L., and van-Voorthuisen, E., “Modes of vibration,” J. Fluid Mechanics,Vol. 78, 1976, pp. 859–862.

4Elavarasan, R., Krothapalli, A., Venkatakrishnan, L., andLourenco, L., “Suppression of self-sustained oscillations in a su-personic impinging jet,” AIAA Journal (in press), 2002.

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8Huerre, P. and Monkewitz, P. A., “Absolute and Convec-tive Instabilities in Free Shear Layers,” J. Fluid Mech., Vol. 159,1985, pp. 151–168.

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10Tam, C. K. W. and Ahujae, K. K., “Theoretical modelof discrete tone generation by impinging jets,” J. Fluid Mech.,Vol. 214, 1990, pp. 67–87.

11Gottlieb, P., “Sound Source near a Velocity Discontinuity,”The Journal of the Acoustical Society of America, Vol. 32, No. 9,1960.

12Krothapalli, A., Rajkuperan, E., Alvi, F., and Lourenco,L., “Flow field and noise characteristics of a supersonic imping-ing jet,” J. Fluid Mech., Vol. 392, 1999.

13Lou, H., Alvi, F. S., Shih, C., Choi, J., and Annaswamy,A., “Active control of supersonic impinging jets - Flowfield prop-erties and closed-loop strategies,” AIAA 2002-2728 , 2002.

14Rowley, C. W., Williams, D. R., Colonius, T., Murray,R. M., MacMartin, D. G., and Fabris, D., “Model-based Con-trol of Cavity Oscillations, Part II: System Identification andAnalysis,” AIAA 2002-0972 , 2002.

15Shih, C., Alvi, F. S., Lou, H., Garg, G., and Krothapalli,A., “Adaptive Flow Control of Supersonic Impinging Jets,”AIAA 2001-3027 , 2001.

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16Alvi, F., Lou, H., and Shih, C., “The Role of Stream-wise Vorticity in the Control of Impinging Jets,” Proceedingsof FEDSM03 , Honolulu, Hawaii, 2003.

17Annaswamy, A., Choi, J., Sahoo, D., Alvi, F., and Lou, H.,“Active Closed-loop Control of Supersonic Impinging Jet FlowsUsing POD models,” Proceedings of the IEEE Conference onDecision and Control , Las Vegas, NV, 2002.

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