Feedback amplification of a plane shear layer by impingement on a downstream body

Journal of Fluids and Structures (1992) 6, 415-436

FEEDBACK AMPLIFICATION OF A PLANE SHEAR LAYER BY IMPINGEMENT ON A DOWNSTREAM BODY P. MERATI Department of Mechanical and Aeronautical Engineering, Western Michigan University, Kalamazoo, Michigan 49008, U.S.A. AND

R. J. ADRIAN Department of Theoretical and Applied Mechanics, and Mechanical and Industrial Engineering, University of Illinois, Urbana, Illinois, U.S.A. (Received 2 August 1988 and in revised form 10 June 1991) The interaction of a plane shear layer with a thin flat plate located on the centerline of a shear layer has been investigated experimentally. The shear layer velocity ratio is 0.375, and its Reynolds number is AU 0,/v = 625. It is found that disturbances from the vicinity of the disturbing plate leading edge feed back into the initial disturbance field of the shear layer to alter significantly its amplitude, increasing it at certain frequencies and suppressing it at other frequencies by negative feedback. An analytical mode1 is developed for the linear stability region of the shear layer in which the system is treated as a closed-loop, frequency-dependent amplifier with wavelength selection according to a phase condition. 1. INTRODUCTION THE DISCOVERY OF LARGE-SCALE coherent structures in turbulent mixing layers at high Reynolds numbers by Brown & Roshko (1974) has led to renewed interest in the early growth phase of shear layers, and the possibility of controlling their behavior by suitable initial excitation. Current studies of the stability of mixing layers by Crow & Champagne (1971), Michalke (1972), Liu (1981) and Morkovin (1984) describe the shear layer as a superposition of interacting instability waves that propagate and amplify as the flow develops downstream. The growth of disturbances in the shear layer occurs in two stages: (i) an initial linear stage; and (ii) a succeeding nonlinear growth, characterized by energy transfer between the fundamental frequency of oscillation and its higher harmonics. Disturbances grow exponentially in the linear stage, where the linear stability analysis by Michalke (1965) for the one-stream mixing layer and by Monkewitz & Huerre (1982) for different velocity ratios describe the growth satisfactorily. Experimental results of Freymuth (1966), Miksad (1972) and Ho & Huang (1982) indicate reasonable agreement with the theoretical investigations of Michalke (1965) and Monkewitz & Huerre (1982). The effect of the initial region of the mixing layer on downstream development has been studied by Weisbrot et al. (1982). They concluded that the rate of spread of a mixing layer was not uniquely determined by the velocity ratio of the two streams. The predominant frequency of tunnel background noise, St = f8/0, had an important role in controlling the growth of the mixing layer. The mean velocity and the momentum thickness at the separation edge of the splitter plate are denoted by u and 0, 0889-9746/92/040415+22 $03.00

0 1992Academic PressLimited

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respectively. Research on the plane turbulent free shear layer flows is reviewed by Ho & Huerre (1984). Free shear-layer flows are often modified by the presence of surrounding objects. The oscillations of the impinging free shear layers are the source of flow noise and undesirable structural loading. Some of the basic engineering applications are the impingement of shear layer flows with edge geometries such as mixing layer-edge, rectangular cavity, jet-flap, jet-plate and jet-edge. In recent years, there have been numerous attempts to simulate or model impinging shear layers. Rogfer (1978) attempted to understand the effect of incident free stream turbulence on semi-infinite flat plates, by modeling the interaction of a flat plate with an array of square-type spanwise vortices distributed uniformly throughout space. He concluded that the bisection of a vortex as it encounters the plate yields a pair of vortices which rotate in the same direction. The combined heights of these vortices are less than the height of the original vortex. Small segments of a vortex which has been cut by the plate, but not through its center, are completely absorbed by neighboring vortices. Confisk & Rockwell (1981) modeled the vortex-corner interaction using point vortices. The motivation of their study was to explain laboratory observations of the ordered interaction of vortices with the impingement surface. They concentrated on the interaction of an irregularly spaced pattern of discrete vortices with a flat plate, a bluff body, and a forward-facing step. Modeling studies by Confisk & Rockwell (1981) showed that the dimensionless amplitude of the pressure fluctuation at the impinging surface is highly sensitive to variations of the initial position of the incoming vortex, and to a large extent insensitive to variations of its strength. Most importantly, Confisk & Rockwell could determine the form of the time-averaged pressure and velocity spectra if the patterns of vortices upstream of the corner were prescribed. However, the main shortcomings of their method are associated with distributed vorticity inherent in laboratory vortices and the severing of vortices at the impingement surface. The impingement of shear-layer flows is reviewed by Rockwell (1983). Rockwell & Knisefy (1979) in their study of the impingement of a shear layer upon a cavity edge (or corner) found enhanced organization at the fundamental frequency. This enhanced organization was not focally confined to the region of the edge but extended along the entire length of the shear layer, thereby reinforcing the concept of disturbance feedback. Ziada & Rockwell (1982b) studied the central features of linear and nonlinear disturbance growth on the unstable shear layer, mechanisms of impingement of the resultant vortices on the edge, induced force on the wedge, and upstream influence in the form of induced-velocity fluctuations at separation. The details of the vortex edge-interaction are investigated by Ziada & Rockwell (1982a). The amplitude of the induced force on the edge was found to be a strong function of the transverse offset of the leading edge with respect to the incident vortex and the degree of vorticity shedding from the leading edge. The feedback mechanism of low-speed edgetones in which the action at the edge is interpreted as an acoustical source is developed by Powell (1961). A theoretical development indicating that the acoustic field is primarily due to the dipole associated with the fluctuating fluid force on the edge has been verified by Powell (1961). Previous models of impinging shear layers concentrated principally on the case in which the impinging surface was located in the nonlinear region of the shear layer. The purpose of the present study is to investigate the behavior of shear layers impinging upon sharp-edged bodies located in the region of exponential shear layer growth. This is accomplished by combining linear stability analysis of the shear layer with a feedback

FEEDBACK AMPLIFICATION

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OF SHEAR LAYER

model in which the initial disturbance seen by the shear layer at its origin is the sum of the background free stream disturbances plus a feedback signal from the impinging surface. Behavior of the non-impinging shear layer in the linear regime has been studied experimentally for the purpose of developing a full description of shear layer growth (needed for the development of the impinging layer model), and development of the impinging shear layer at downstream location of the sharp-edged body. 2. EXPERIMENTAL

FACILITY

AND PROCEDURES

Experiments were performed in a two-stream plane shear layer generated in the wind-tunnel apparatus depicted in Figure 1. The two streams were formed by dividing a single stream with a splitter plate and reducing the speed of the lower stream with fabric which created an appropriate pressure drop. The velocities of the upper and lower streams were U, = 7.31 ms-’ and Uz = 2.74 ms-‘, respectively, yielding a velocity ratio Q/U1 = 0.375. The boundary layer momentum thickness of the faster and slower streams, measured 2 mm downstream of the splitter plate separation edge where BO,= 1.1 mm and f& = 0.95 mm. Reynolds numbers based upon AU = U,- UT and & = 0”, + & were AU &,/Y = 625 and r/,&,/v = 999. The splitter plate was reduced from 3.2 mm thickness to a knife-sharp separation edge over a distance of 76 mm. The shear layer was enclosed in a 830 mm long Plexiglas test section with a 380 x 510 mm cross-section. Streamwise velocity was measured by hot-wire anemometer using a probe mounted on a 457 mm long, 4.6 m diameter horizontal probe support that was supported at its

Test

Sec,lOn

380 x 5111

+-Elasticmembrane 7I I

! ! II Paper filters

Hol(eycomb

plate

Honeycomb

L’ \

I

PWXSUre drop fabric

500

(b)

1

Figure 1. (a) Schematic of the wind tunnel; (b) the contraction cone and the test section; (c) the knife edge. All dimensions in mm.

418

P. MERATI AND R. J.ADRIAN

downstream end by a rigid, 3.2 mm thick vertical steel arm, designed to minimize flow disturbances. A 6.35 mm diameter movable brace extending across the test section well below the shear layer stiffened the vertical arm and eliminated vibrations. Most of the single-wire measurements were obtained using a sensor similar to the TSI Model 1210, but was modified to reduce aerodynamic influences of the prongs. A 4 pm wire with 1.25 mm active length was used. In circumstances where feedback disturbances from the probes were suspected to be significant (i.e., spectral measurements at x = 10 mm, y=5mm, z=O) aTS1 bound ary-layer probe was connected to the main support by a 90” bend so that the main support lay 100 mm below the measurement location. According to Hussain & Zaman (1987), free shear layer tone induced by a hot-wire probe was mostly observed when the probe support diameter was larger than five times the initial momentum thickness and the probe was oriented with an inclined angle of 20”, with its support extending across the shear layer. Their results show that the data on sensitivity of shear layers to external disturbances should be obtained with the probe placed such that no sizable part of it intersects the shear layer. In our spectral measurements, the hot-wire was located near the edge of the shear layer with its support out of the shear layer, thus minimizing the upstream effect of the probe. For comparisons of the measurements with our model, the probe location was not changed in the presence and absence of the plate. Thus, any observed upstream effect due to the presence of the plate has been real and not because of probe interference. In addition, the probe-support diameter in our measurements was 0.32 cm which was less than five times the initial momentum thickness for the present shear layer. Hot-wire signals from the TSI 1050 anemometer and TSI 1052 linearizer were low-pass filtered at 1,000 Hz, digitized in blocks of 1042 samples with 12 bit resolution at 2,500 Hz sample rate, and spectrum-analysed by conventional FFI methods. Spectra from ten independent blocks of data were ensemble-averaged to form the final spectral results. Flow visualization was performed using the smoke-wire technique with a vertical O-0762 mm diameter wire. In all experiments reported here the disturbing plate was located on the nominal centerline of the shear layer, y, = 0, and extended across the entire width of the test section. Its thickness was 2 mm, its streamwise length was 127 mm, and the shape of its leading edge was a linear taper from 0.2 to 2 mm over a distance of 20 mm. 3. NON-IMPINGING

SHEAR

LAYER

3.1. THEORETICAL BACKGROUND The behavior of the non-impinging shear layer is needed to serve as a reference case with which to compare the behavior of the impinging layer, and to provide empirical data used in the mathematical model of the impinging layer. To this end, the theory for shear layer instability and our observation of the behavior of the shear layer will be described in this section. The theoretical studies of Michalke (1965) and Monkewitz & Huerre (1972) and the experimental results of Freymuth (1966), Miksad (1972) and Ho & Huang (1982) indicate that the initial region of a free shear layer is satisfactorily described by linear stability analysis of spatially growing disturbances. Beyond this “linear” region there is a region of a nonlinear amplification, followed by growth of smaller scale threedimensional disturbances, pairing of transverse vortices, vortex breakdown and transition to turbulent flow (Ho & Huerre 1984).

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From linear stability analysis the spatially growing response of a free shear layer to an initial complex disturbance n, = cSc--rror can be represented

(1)

by the complex velocity as indicated by Michalke (1965) U(X, t) =

EF$+)&cm-w”,

(2)

where the frequency is real and the wave number I_Y = a;(w) + LX;(W) is complex. The function G(y) defines the stream-function of this flow field $I = %{ $(y)el(Oix-O’J}.

(3)

As written in equation (2) # has been normalized so that #‘(O) = 1. The eigenfunction, $, and the eigenvalues, o(o), are found by solving the eigenvalue problem for the Rayleigh equation obtained by linearizing about a basic, steady, parallel flow which is usually taken to have a hyperbolic-tangent profile. Michalke (1965) treated the one-stream mixing layer for this problem, and Monkewitz & Huerre (1982) have calculated wave speeds and growth rates for various frequencies at different velocity ratios UJ U, . Equation (2) may be conveniently recast into the form u(x, W, r) = u,(r)H*(x,

w),

(4)

where H* = @‘(y)e’“.

(5)

H* is the complex conjugate of the transfer function H(x, w). This form emphasizes the fact that the linear region of the shear layer acts as a frequency-dependent linear amplifier with complex transfer function H(x, w) and a weighing function h(q, x). The disturbance velocity U(X, t) is given by += U(X, t) =

I -a

WV, x)u,(r - r~) dl;l.

From equation (6), the temporal power spectrum velocity is shown by Bendat and Piersol (1971) to be UW

x) = IW,

(6)

S,(o, x) of the disturbance

~)I’&,(~),

(7)

where SUSis the power spectrum of the initial disturbance field. From equation (7) the spatial amplification rate of the shear layer instability at any frequency o can be evaluated from 1

I@‘(YJ 1 11nIi$‘(yz)l+21n

-~(w)=(X,-X*)

W2, Yz, w) [ &&,,Y,, w)

II ’

(8)

provided, that x2 and x1 are each located in the region where linear stability theory applies, i.e., the region of exponential growth. Equation (8) will be simplified if @‘(yJ = @‘(y2). This equality holds if the nondimensional values of y, and y2 are equal. Thus if y, = y2 = y , equation (8) becomes -a;(w)

1

= 2(x2

-x1)

ln

&(x2, [ UXll

1

Y, @) Y, w) .

(9)

The nondimensional vertical coordinate in a shear layer is usually represented by (y - y,,J/fI where 8 is the local momentum thickness; y,., is the vertical coordinate

420

P.

-

16 x lo-’

$ % I? g

12 x 10-s

f g In & g &

MERATIAND

R. J. ADRIAN

8 x lo-’

4 x 10-s

0

50

loo

Frequency (Hz) Figure 2. Spectra of the fluctuating streamwise velocity in the faster stream of the contraction (a) x = -5 cm, (b) x = -2 cm. (c) x = 0, and y = 2 cm throughout.

cone for

where the mean value of the streamwise velocity is equal to the average streamwise velocity. y,., and 8 are functions of the streamwise position. 3.2. THE INITIAL Spectra of the fluctuating streamwise velocity in the faster stream within the contraction cone are shown in Figure 2. The same spectral peaks at 5,20 and 40 Hz were observed in the slower stream, indicating that all of the disturbances originated from the same upstream source. Spectra of the tunnel vibrations measured using an accelerometer attached to the various positions of the test section showed that the frequencies that were multiples of 20 Hz coincided with the fundamental 20 Hz blade-passage frequency of the wind-tunnel blower and its harmonics. Comparison with spectra of the streamwise fluctuating velocity in the contraction region and spectra of the tunnel vibration showed that the shear layer was perturbed by the fundamental and higher harmonics of the motor blade rotation. To investigate the growth of the disturbances in the initial region of the shear layer, spectra of the fluctuating streamwise velocity were measured near the edge of the shear layer (y = O-6 cm), Figure 3. As the flow develops from x = 0 to x = 1-Ocm and further downstream, spectral peaks at 160 and 18OHz appear and grow. Among the background disturbances, the two disturbances with 160 and 180Hz frequencies are most amplified by the shear layer. The nondimensionalized root-mean-square values of the amplitude of the velocity fluctuations u’(f)/&, are shown in Figure 4 as the flow develops downstream. The value of G(f) at each spectral peak with a frequency f is obtained from u’(f) = {i:;

W)

df)L’*,

(IO)

where S,(f) is the spectrum of the streamwise fluctuating velocity. The limits of the integral in equation (10) are chosen as f - 10 and f + 10 since spectral peaks are generally observed at every 20 Hz. The value of G(f)/& at 80 Hz is not evaluated due to the absence of a clear peak at this frequency. The two most amplified disturbances with frequencies 160 and 180 Hz grow exponentially up to x = 5 cm, defining the extent of the linear region. It is followed by nonlinear distortion, involving energy transfer between the fundamental and its higher

FEEDBACK

AMPLIFICATION

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I

(a)

421

LAYER I

(cl

(b)

(4

2

2

2

2 N

0

2

;

N

1t

E

1 x 10-4

1,

L-i

OL 0

2*.k_

300 0 Frequency (Hz)

300

I

I

300

0

300

Figure 3. Spectra of the fluctuating streamwise velocity in the linear region of the shear layer for y=06cmand(a) x=0,(b) x=lcm,(c) x=2cmand(d) x=3cm.

In the nonlinear region, the two disturbances equilibrate into finite amplitude oscillations. In order to use the theoretical amplification rates obtained by Monkewitz & Huerre (1982) in the present model, the validity of the approximations made in the theory such as assuming a hyperbolic-tangent profile for the mean velocity profiles in the initial region of the shear layer and neglecting the effect of the splitter-plate wake on the development of the shear layer will be given some consideration. The mean velocity profiles near the initial region of the shear layer are not well

harmonics.

c

1 -

0 v

0 0 x 0

20 Hz 40 Hz

+- 60 Hz

1nP I 1)

A

i

I

I

1

2

4

6

8

I

0 .V(cm)

1

2

I

I

6

8

k

4

100 Hz 120 Hz 140 Hz I60 Hz 180 Hz

Figure 4. Streamwise variation of disturbance amplitude at different frequencies for the non-impinging shear layer.

I

measured at y = 0.6 cm

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P. MERATI

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Flow

*

(4

(b)

(4

y=o -

Figure 5. Instantaneous

photographs of the visualized flow of the shear layer.

FEEDBACK

AMPLIFICATION

OF SHEAR

LAYER

423

approximated by a hyperbolic tangent profile, due to the presence of boundary layers emanating from either side of the splitter plate and forming a wake. In the present study, the mean velocity profiles of Figure 6, to be discussed in Section 3.3, indicate the presence of the boundary layers up to x = 4 cm. Miksad (1972) showed that the presence of the wake did not change the most amplified frequency and only affected the amplification rate in the low frequency region. He approximated the measured mean velocity profile by a trapezoidal profile and concluded that there are two classes of instability: the Class I instability due to the lower inflexion point, and the Class II instability due to the upper inflexion point. Frequencies of the Class I instabilities are extremely low and the growth rates of disturbances are small. The dominant instabilities are from Class II, and they are similar to the instabilities calculated by Michalke (1965), assuming an hyperbolic tangent profile.

Figure 6. Streamwise mean velocity profiles at different locations in the test section of the plane shear layer tunnel.

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P.

MERATlAND

R. J. ADRIAN

3.3. DEVELOPMENT OF THE SHEAR LAYER Instantaneous photographs of the shear layer are shown in Figure 5. The separation edge

of the splitter plate located at the left side of the flow photograph of Figure 5 is marked with an arrow. Flow photographs were taken at random times. The instability waves, their roll-up into vortices and their growth are shown clearly. Vortex pairing is evident in Figure 5(c) where an upstream vortex catches a downstream one. The two small vortices upstream of the larger vortex in Figure 5(d) begin to align vertically before they merge. The new vortex formed by merging of the smaller vortices has a passage frequency of one-half of the original frequency. The lower frequency resulting from coalescence has been detected in the region of flow well upstream of merging by Ho & Nosseir (1981). By means of the merging process, the shear layer entrains some of the potential flow and thus grows as it develops downstream. The streamwise mean velocity profiles, U/U,, of the shear layer from x = 0.2 cm to x = 11 cm are shown in Figure 6. Mean velocity profiles change with downstream distance immediately after the trailing edge of the splitter plate. At x = O-2 cm, the two boundary layers merge into a wake how which persists up to x = 4,O cm. The variation of y,,,, with respect to the streamwise direction n is shown in Figure 7; y,,, is the vertical coordinate where the mean value of the streamwise velocity is equal to the average streamwise velocity. As flow develops downstream, y,,,, decreases faster in the initial 4 cm of the shear layer in comparison with the rest of the shear layer. The values Of yn.5 are usually used to obtain the nondimensional vertical coordinate in a shear layer. The streamwise turbulence intensity profiles u,/AU from x = O-2 cm to x = 11 cm are shown in Figure 8(a). The location of the maximum turbulence intensity moves toward y = 0 as the flow develops downstream. The growth of the shear layer is indicated by 0.5

0

1

2

3

4

5

6

7

8

x (4 Figure

7. Streamwise

variation

of the vertical coordinate where is equal to the average streamwise

the mean value of the streamwise velocity.

velocity

FEEDBACK

AMPLIFICATION

OF SHEAR

25 ’

I

42s

LAYER

I

@) 20 -

0

x

-8 0.2

-$ Figure

8. (a)

Streamwise turbulence at different locations

(%)

intensity profiles, (b) peak of the streamwise turbulence in the test section of the plane shear layer tunnel.

intensities

the existence of the thicker turbulence-intensity profiles for the flow developing downstream. The maximum value of the streamwise turbulence intensities (u,/AU),,, at each x location is shown in Figure 8(b). The increase in the peak of the turbulence intensities in the linear region (0
426

P. MERATI AND R. J. ADRIAN

x=0 v = 0.6

I -

x=2 y = 0.6

Frequency (Hz) Figure 9. Spectra of the fluctuating streamwise velocity near the edge of the shear layer in the faster stream; x and y values are in centimetres.

I

I

y = 0.6 cm

4 x 10-4 f Q z

I

---

x=0

-

x=2cm

I “r “g 3 x 10-4 ;

1 I

42 B 2 x 10-g -I B & 2

I 1 x 10-d -1 I

Fl

F2

0

MO Frequency (Hz)

Figure 10. Comparison of the spectra of the fluctuating streamwise velocity at x = 0 and x = 2 cm.

FEEDBACK

AMPLIFICATION

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LAYER

between the spectra obtained at x = 0, y = O-6 cm with those further downstream clearly indicates the growth of the disturbances in the theoretical range of frequencies. The disturbances around 160 Hz are the most amplified disturbances of the present shear layer. Second harmonics around 320 Hz and third harmonics around 480 Hz are evident in the spectra obtained at x = 3,4,.5,6 and 7 cm; they have small amplitudes. Comparison of the spectrum at x = 0, y = 0.6 cm with the rest of the spectra of Figure 9 suggests that the broad-band peaks are due to the presence of the vortices. The downstream development of these vortices can be studied by defining a mean vortex passage frequency as

(11)

To find F, and F2, spectra at any downstream location x were superposed on the spectrum at x = 0, as in Figure 10; F, and F2 are the lower and upper frequencies corresponding to the first broadband peak in each of the superposed spectra. The mean of the vortex passage frequency is shown in Figure 11. The average frequency of the disturbances and the rolled vortices remains constant in the linear region (0
(12) _

200 i-mm--

I

-

-/ 0.75

0

5

10

15

25

20

x (cm) Figure 11. Mean of the vortex-passage

frequency,

f,.

and the momentum

thickness

of the shear layer. 0.

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P. MERATI

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where U is the mean velocity at the point y in the velocity profile. In practice, the integration is extended over the thickness of the shear layer. The momentum thickness at x = 0 is obtained for the faster stream and is defined by (13) The momentum thickness of the shear layer as the flow develops downstream is shown in Figure 11. The more rapid growth of the shear layer thickness downstream of x = 5.0 cm compared to its growth in the linear region (0 5 x 5 5.0 cm) indicates the presence of the merging process in the nonlinear region. 4. IMPINGING

SHEAR

LAYER

This section describes the interaction of the plane shear layer with a thin, flat plate located downstream, and the effect that this interaction has upon the initial growth of the shear layer. In general, the impingement of a shear layer on a body necessarily creates fluctuating flow disturbances in reaction to the deceleration of the unsteady approach flow at the surface of the body. These disturbances are manifested to varying degrees throughout the flow domain, but in particular, they appear in the region of initial formation of the shear layer, where they become a part of the total initial perturbation field. Since the shear layer behaves as a frequency-dependent amplifier, certain perturbations are amplified more than others, resulting in even larger amplitudes for the feedback signals from the downstream disturbing body. If the disturbance signals arrive in-phase with the initial disturbances, they are amplified still further, but if they are out-of-phase, they will be attenuated. Since disturbances in the shear layer propagate as traveling waves, the phase relationship is determined by the wavelength, and hence, the frequency of the disturbance. Overall, the impinging shear layer acts like a feedback oscillator. In the present investigation we will restrict our attention to cases where the leading edge of the disturbing plate is in the linear growth region of the impinging shear layer. It will be assumed that the flow is adequately described in this region by linear stability analysis. Since the flow has not undergone significant nonlinear evolution, the effects of large scale turbulence in the shear layer are insignificant. The sharp leading edge of the disturbing body promotes large disturbances. Ziada & Rockwell (1982b), for example, have shown clearly the formation of large unsteady separation bubbles on each surface of a wedge as the shear layer alternates by sweeping fluid up and down across the leading edge. As shown in the previous section, the most amplified frequencies in the present shear layer are of order 170 Hz. The acoustic wavelength corresponding to this frequency is approximately 2 m, while the distances between the separation edge and the leading edge of the plate are less than 0.1 m in the present experiments, lying well within the acoustic near-field. Consequently, the time delays associated with acoustic propagation from the disturbing plate to the start of the shear layer can be ignored. Within the range of validity of this approximation, the upstream disturbances can be approximated as elliptic reactions of the potential flow field to the downstream boundary condition. The response of an impinging shear layer to a sharp edge is a complex problem, and the prediction of the resulting upstream disturbance is even more complicated. We shall treat this problem semi-empirically by defining an empirical “coupling coefficient” that relates the effective initial disturbances to the disturbance field impinging upon the leading edge of the disturbing plate.

FEEDBACK

4.1.

FEEDBACK

MODEL OF THE

AMPLIFICATION LINEAR

OF SHEAR

429

LAYER

INSTABILITIES

A schematic of the feedback mechanism in the impinging 12. In this figure, u0 is the background disturbance in disturbance at x = 0, equation (1). The total disturbance disturbance plus ur, the feedback disturbance caused disturbing plate:

shear layer the tunnel is the sum by vortices

is shown in Figure and U, is the total of the background impinging on the

LLY = U,,+ U,

(14)

In the present model, the feedback signal is specified by postulating the form

a relationship

r+(r) = K(XL)U(XL, f)?

of (13

where K(xJ is a “coupling coefficient” which can in general be complex. The implications of this assumption are that the feedback is determined by the local how interaction at the leading edge of the flat plate, and that the feedback disturbances are linearly proportional to the interaction. Experimental findings by Ziada & Rockwell (1982a, b) showed that the force on the edge is proportional to the approaching velocity and that the disturbance amplitude at separation is proportional to the force on the edge. The use of a complex coupling coefficient permits phase differences between u(xL, t) and ur. The assumption of linearity is justifiable for small disturbances. If the frequency of the feedback signal is low, the disturbances caused by unsteady vortices impinging upon the flat plate could, in principle, be calculated from potential flow theory, provided that flow separation at the leading edge could be ignored. In this theory the induced feedback velocity field would negate the onset velocity at the plate so as to satisfy the condition of no penetration. If the frequency is high, the acoustic propagation of the disturbance pressure field would need to be considered. In the first case, the assumption that the feedback velocity field upstream of the plate is proportional to the incoming disturbance velocity is clearly justified because the potential flow solution would depend linearly upon the disturbance flow at the boundary of the plate. In the second case, the feedback signal is proportional to the fluctuating pressure field at the leading edge of the plate. Expanding Bernoulli’s equation about the mean velocity and pressure shows that the fluctuating pressure field is proportional to the fluctuating velocity field to the leading order, again justifying the form of equation (15).

Uf =

ml.,

YLh&$)

u2

Figure 12. Sketch of the splitter plate, shear layer and disturbing plate

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The coupling constant K(xL) can be calculated in principle, but in the present model it is treated as an empirical constant which is obtained from comparisons of the experimental results with the model. From equation (4) the velocity disturbance at the plate leading edge is

4% m,t) = U,(f)H*(XL,0). Substituting into equations (14) and (15) and rearranging

(16) yields

uo

us =1 - K(XL)H*(XL, w) ’

(17)

from which we find

Su,(~)

s”Aw) =II- K(_Q)H*(.Q,

o)l’ ’

08)

where SUsand S,, are the temporal power spectra of u, and uo. Finally, by using equations (7) and (18), one can find the power spectrum of the impinging shear layer at any point X: (19) It is informative to compare the power spectrum of the shear layer impinging upon the plate to the power spectrum without impingement. Theoretically this ratio is given by

&(w xl

F=[SJW, X)]Kd, = I1 =

K(x,)~‘(XL,

o>l’

1 1 + Kze-2nGL- 2KcepaGLcos( a7 + xL + y + K) ’

(204 (20b)

where K = IK(xJ

IG’(rdI

and y is the phase angle of @‘(yL) with respect to $‘(yo.J and K is the phase angle for K(xL). The value of K depends upon the propagation of the disturbance from the downstream plate to the separation edge, which is ignored for the reasons argued above. It also depends upon the effective location of the disturbance. If we assume that the disturbance is concentrated at (x,, yL), then there is no contribution to K from this effect. The function F as defined in equation (20) is proportional to the square of the ratio between the initial disturbance amplitudes with and without the plate. Sarohia (1977) showed that severe phase distortion occurs in flows over shallow cavities. He showed that at a fixed streamwise location, the phase of the shear-layer velocity fluctuations relative to their phase at the shear-layer separation edge varies a great deal across the cavity flow. As one moves toward the cavity from outside (constant streamwise location), the phase of the disturbance decreases until one reaches the region across which a sharp drop in phase occurs. As one moves further inside the cavity, the phase of the disturbance increases. The present model indicates that the phase of the velocity fluctuations relative to their phase at the shear layer separation edge, at any point in the impingement region, depends on the location of that point x and the impingement xL. This dependency for an impinging shear layer is shown to have the following form

~6, w t) = [UO+ K(xL)E,e-‘Wf~‘(yl.)ei~~]~‘(y)eiOUT.

(21)

FEEDBACK AMPLIFICATION OF SHEAR LAYER

431

Thus, phases of functions G’(y), $‘(yL) and values of CYX~ and LXX determine the overall phase difference between the fluctuations in the impingement region and the separation edge. It is this overall phase difference for the first mode of oscillations which is measured by Sarohia (1977). 4.2.

EXPERIMENTAL

RESULTS

Comparisons of the model with the experimental results must be performed with the plate located in the linear region of the impinging plate in order to agree with the model assumed for the shear-layer disturbances. It is also appropriate to place the plate far away from the splitter-plate trailing edge to avoid its wake. Thus, the plate was positioned in the x-z plane, at the impingement length xL = 3 cm. Presence of the plate generally does not affect the shape of the velocity profile in the impingement region. Sarohia (1977) showed that the mean velocity profiles in the impingement region of a shear layer are similar to the non-impinging shear-layer profiles. However, the spread rate, de/&, of the impinging shear layer is different for different impingement lengths. The spread rate increases with the increase in the impingement length. A maximum spread rate of O-022 was observed by Sarohia (1977). Spatial growth of the disturbances at y = O-6 cm with plate located at xL = 6 cm, yL = 0 is presented in Figure 13. Presence of the plate enhances the oscillations at 160 Hz and does not generally affect the amplitude of the oscillations at other frequencies. In order to compare the model with the experimental results, the function F Mined in equation (20) was measured. Spectra of the streamwise velocity at x = 1.0 cm, y = 0.5 cm were measured with and without the plate. This location of the probe was selected to minimize its disturbing effect on the initial region of the shear layer. The

100 HZ

i 10-j I 0

0

20Hz

V

40 Hz

W

60 Hz

120 Hz 140 Hz 160 HZ 180 Hz

I

I

I

I

I

I

I

I

I

2

3

6

8

II

2

-1

6

8

I (cm) Figure

13. Streamwise variation edge of the impinging

of disturbance amplitude at different frequencies measured at the upper shear layer (y = 0.6 cm); plate located at x,. = 6 cm, y,_ = 0.

432

P. MERATJ AND R. J. ADRIAN 8

Exp.

t ~

Model

Frequency (Hz) Figure 14. Ratio of the spectra of fluctuating streamwise velocity obtained at x = 1-Ocm, y = 0.5 cm with and without the presence of the plate for the model and measurements; xL = 3cm, K, rO.195. +, Experiments; -, model.

0466

0.6

-iE 0.4 s .” e

0444

T

I

0.2

0.022

0

0 -0.1

I

0

I

I

I

100

200

300

, 7,

0.15

0.3

1

I

i

0.45

2nfaQ Figure 15. Theoretical

amplification rates for the present shear layer; R = 0.67 (Monkewitz 1982).

& Huerre

FEEDBACK

AMPLIFICATION

OF SHEAR

433

LAYER

spectra of the streamwise velocity at x = 1.0 cm, y = 0.5 cm with the plate leading edge located at xL = 3 cm and y, = 0 were divided by the spectra at x = 1-Ocm, y = O-5 cm without the plate. The ratio of the measured spectra at x L = 3 cm and the prediction of the model, equation (20), are compared in Figure 14. For the purpose of this comparison, the theoretical amplification rates, Figure 15, and wave-numbers, Figure 16, evaluated by Monkewitz & Huerre (1982) (R = 0.67 based on the minimum velocity of the wake at x = 2 cm) were used in the model and we have set y = 2~ and K = 0. The value of y = 6~r is obtained from Figure 13 of the paper by Ziada & Rockwell (1982b) since the velocity ratio of their work (0.35) is very close to that of the present investigation (O-375). Since most of the linear growth occurs between x = 0 and 4 cm, the momentum thickness 19used in Figures 15 and 16 is measured at the middle of the linear growth. 8 at x = 2 cm is 0.11 cm. The ratios of the measured spectral values were smoothed over frequency increments of 20 Hz. The average value of the spectra1 ratio in the 20 Hz band was obtained and the frequency of the mean value was assigned to be equal to the mid-frequency of the corresponding band. The value of the coupling constant, K,., was inferred by curve-fitting the model to the measurements. Equation (20b) (K = 0)can be rearranged in the following form K,2e-2Wl _ 2KCe-“J’ cos(aJ,~+y)=;-

1.

(22)

A least-square technique is used to infer Kc from curve-fitting the left side of the above equation to the measured values obtained for the right side of the equation. Measurements shown in Figure 14 indicate that the feedback signal from the plate increased the shear-layer amplitude by peak values of 1.40 to 2.24. The feedback amplification depends upon xL and frequency, as predicted by the theoretical model. Both the mode1 and measurements indicate that at xL = 3 cm the oscillations within the frequency band (150-190 Hz) are amplified most. Equation (20) predicts that the downstream body amplifies those disturbances whose wavelengths, I. = 21r/a;, are such that the cosine term in the denominator is unity.

fW) 0

I

I

I

0.15

0.30

0.45

i

2nffG Figure 16. Theoretical

wave number

versus frequency for the present shear layer; R = 0.67 (Monkewitz Huerre 1982).

&

434

P. MERATI

AND

R. J. ADRIAN

implying in-phase feedback. The amplification of the disturbances for the impinging shear layer depends on the amplification of the same disturbances for the nonimpinging shear layer. Those disturbances whose wavelengths are such that their frequencies are closest to the most amplified frequency of the shear layer are amplified most. The theory implies that the oscillations that are most amplified are those whose feedback signals from the disturbing plates are in phase with the wave at the splitter plate separation edge. As the plate moves downstream, the harmonics of the fundamental amplified frequency of the closed-loop shear layer begin to amplify. At each impingement length, xl>, the frequencies of the in-phase disturbances follow f=?,

n = 1.2,.

.

where 0, is the phase velocity of the disturbance convection. Magnitudes of U,. and u, the mean velocity of the shear layer, are very close. At xL = 3 cm, n = 1 results in a frequency of 164 Hz. The harmonic of this most amplified frequency is not amplified, since its gain, as given by Figure 15, is insufficient. At x1_= 6 cm, n = 1 results in a frequency of 82 Hz considerably below the value of the most amplified frequency. The second harmonic, n = 2, is amplified more due to the large gain at f = 164 Hz. The phase criterion described in equation (23) is in agreement with the measurements obtained by Gharib (1983). The experimental results and the model suggest that the oscillations that are amplified are those whose feedback signals from the disturbance plate are nearly in phase with the wave at the splitter plate separation edge. The same results were obtained experimentally by Ziada & Rockwell (1982). For the phase criteria to be satisfied, the wavelength of the most amplified frequency of the shear layer should be 3 cm at xL = 3 cm. As the plate moves downstream, the increase in the wavelength continues, until a certain impingement length is reached where the oscillations cannot sustain themselves, because their gain, as given by Figure 15, is insufficient. Then, by addition of an extra wavelength between the splitter plate and the plate leading edge, the phase condition is again satisfied. At xl<=6-O cm, the wavelength of the most amplified frequency is again 3 cm. Sarohia (1977) and Gharib (1983) performed experiments in laminar axisymmetric flows over shallow cavities. They did not observe any self-sustained oscillations for impingement lengths less than approximately 80 times the initial momentum thickness. The maximum impingement length of the present experiments is 55 times of the initial momentum thickness, 0,,,. The disagreement between the two experimental results might be due to the differences in the shear layers, one being an axisymmetric shear layer and the other is a plane shear layer. Ziada & Rockwell (1982b) have observed self-sustained oscillation for impingement lengths less than 30 times the initial momentum thickness. Hussain & Zaman (1987) have also observed self-sustained oscillation for impingement lengths larger than 20 times the initial momentum thickness for a plane shear layer. 5. SUMMARY

AND CONCLUSIONS

We have examined the behavior of a plane two-stream shear layer in which a thin flat plate located on the centreline of the shear layer in the region of exponential growth creates periodic disturbances that feedback upstream to the initial instability, thereby modifying the amplitude of the shear layer. It is shown that feedback can increase the shear-layer amplitude by as much as a factor of 2.24 at certain frequencies. Feedback,

FEEDBACK

AMPLIFICATION

OF SHEAR

435

LAYER

is effectively negative, can also suppress the shear-layer amplitude at certain frequencies. The condition for maximum amplification is essentially a resonance condition in which the disturbance frequency is such that the distance between the separation splitter-plate edge and the leading edge of the downstream plate contains an integer number of wavelengths, and the frequency lies within the amplified region of the undisturbed shear layer. The feedback model that has been developed provides a more complete description of the dependence of the shear layer amplitude upon disturbance-plate position and frequency. As the disturbance-plate location, xl,, increases, the amplification rate decreases. The next strong resonance occurs when the frequency of the next higher order resonant mode coincides with the most amplified frequency of the undisturbed shear layer. The feedback model uses linear stability analysis to describe the undisturbed (open loop) shear layer. It assumes that disturbances from the downstream plate originate from the leading edge of the plate, rather than being distributed, and it closes the feedback loop by injecting the downstream disturbances into the shear layer at x = 0. While more complex models involving distributed disturbances and distributed injection can be envisioned and probably should be pursued, it appears that the present model describes much of the physics of the feedback effect. The present results pertain only to disturbing bodies located in the initial, linear region of the shear layer. Extension of the feedback model into the nonlinear region and beyond into the turbulent region depends upon development of adequate descriptions of the natural, undisturbed shear layer in these regions. While the present work has been concerned with a specific downstream-body shape, it is not unreasonable to expect similar results and considerations to apply to a range of disturbance-producing interactions, including perhaps the impingement of a separatedflow shear layer during reattachment. ACKNOWLEDGEMENT Portions of this work were supported ATM 82-03521.

by National Science Foundation

Grant.

NSF

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R. J. ADRIAN

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MIKSAD, R. W. 1972 Experiments

on the nonlinear

stages of free shear layer transition.

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MORKOVIN, M. 1984 Guide to experiments on instabilities and laminar turbulent transition in shear layers. New York: AIAA. MONKEWITZ, P. A. & HUERRE, P. 1982 The influence of the velocity ratio on the spatial instability of mixing layers. Physics of Fluids 25, 1137-1143. POWELL, A. 1961 On the edgetone. The Journal of the Acoustical Society of America 33, 395-410.

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