Control of Free Shear Lauers

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AlAA-87-2689

Control ^of Free Shear Lauers

J . T. C . L i u and H. T. Kaptanoglu Brown University

Providence, R I

AlAA 11 th Aeroacoustics Conference

October 19-21, 1987/Palo Alto, California

For permission to copy or republish, contact the American Institute

of

Aeronautics and Astronautics 1633 Broadway, New York, NY 10019

Downloaded by BILKENT UNIVERSITY on February 1, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.1987-2689

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CONTROL OF FREE SHEAR LAYERS+

J. T. C. Liu* a n d H. T. Kaptanoglu**

Brown University Providence, Rhode Island

T h e f u n d a m e n t a l aspects of controlled multiple coherent mode prescncc in turbulent shear flows is f i r s t discussed, including the supplementary averaging procedures in addition to the Reynolds average a n d the nonlinear cnergy t r a n s f e r mechanisms coupling the coherent modes, mean flow a n d fine-grained turbulence. We then specialize to the problem of a f u n d a m e n t a l mode a n d its subharmonic i n a developing mixing layer, the prototype problem of subharmonic cascade. An integral method is presented which allows the dctcrmination of the cohercnt wave envelope or amplitude simultaneously with the mean f l o w growth rate a n d turbulence energy. T h i s is then generalized to the presence of multiple subharmonics using a binary-frequency interaction argument. Free shear layer control is discussed in terms

\J of i n i t i a l coherent mode amplitudes, dimensionless initial frequencies, phase angle between the modes a n d fine-grained turbulence levels, in particular, how these parameters could enhance or supress the shear layer spreading rate a n d the levels of fine-grained turbulence.

Presented as Paper 87-2689 a t the AIAA 1 l t h Aeroacoustics Conference, Sunnyvale, CA., October 19-21, 1987.

+The a u t h o r s dedicate the present modest work to the memory of i e s t c r Lces (1920-1986), l a t e Professor of Aeronautics, California Institute of Technology, Pasadena.

*Professor of Engineering, Division of Engineering a n d the Laboratory f o r Fluid Mechanics, Turbulence a n d Computation. Also, Visiting Fellow in 1987/88, Department of Mathematics, Imperial College, London.

**Graduate Student, Department of Mathematics, University of Wisconsin, Madison

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Zopyright @ American Institute of Aeronaulics and .istronautics. Inc.. 1987. A l l rights reserved.

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Nomenclature

A = wave amplitude or envelope

c = constants in turbulence shape assumption D = used in substantial derivative

E = energy density f = physical frequency

I = energy exchange integrals, dissipation integrals m = a n integer

N = total number of sampling n = sampling number; a n integer q = a n y overall f l o w q u a n t i t y Q = mean f l o w q u a n t i t y

r = modulated fine-grained turbulence stresses R = velocity ratio

‘ J R e = R e y n o l d s n u m b e r t ⁼ time

T ⁼ longest period

u = velocities; streamwise velocity component U = mean f l o w velocity

v = normal velocity component w = spanwise velocity component

x = spatial coordinates; streamwise coordinate y = normal coordinate

z = spanwise coordinate

@ = fluctuation dissipation integral; eigenfunction of local linear theory

@ = mean flow dissipation integral v = kinematic viscosity

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n

= rescaled normal coordinate, y/6(x)

d

B

= dimensionless frcquency parameter

6 = local half-maximum vorticity thickness

Superscripts

-

⁼ mean f l o w q u a n t i t y

-

⁼ ^{odd mode}

A = even mode

1 = fine-grained turbulence

Subscripts

f = f u n d a m e n t a l

I = coordinate a n d velocity component designation

'4

.

1: streamwise, 2: normal, 3: spanwisc

j = coordinate a n d velocity component dcsignation 1: streamwise, 2 normal, 3: spanwise

0 = i n i t i a l condition

rs = Rcynolds stress production sn = the n t h subharmonic wt = wave-turbulence exchange p = production f r o m mean flow 21 = even-odd mode exchange

-

⁼ upper f r e e stream

--

⁼ lower f r e e stream

- 3 -

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Introduction

d T h e experimental discovery of "coherent structures" in f r e e turbulent shear flows'-5 have lead to their theoretical interpretation as a hydrodynamic instability in an inflectional mean flow, superimposed upon fine-grained turbulence.' T h e practical modelling of the development of such flows, using ideas f r o m nonlinear hydrodynamic instabilities,6-8 have since been exploitedQ''2

T h e problem, of course, is not one of simply using t h e argument that inflectional mean flows are dynamically unstable and thus inviscid linear stability theory prevails. A given spectral compomcnt of the large-scale coherent structures is f o u n d to be modellcd well by (1) a ''slowly varying" wavc cnvclopc, which must be solved simultaneously with thc development of the mean flow a n d the fine-grained turbulence through nonlinear coupling; a n d (2) under the wave envelope is the "rapidly oscillating" wave, which enters the wave envelope upstream through its initiation a n d exits downstream, if a t all. T h e rapidly oscillating wave characteristics a r e modelled well by a local lincar stability theory. T h e curve f i t s of local experimental d a t a by the local linear theory confirms

* d

T h e method of modelling the nonlinear wave envelope, accounting f o r the full participation of the fine-grained turbulence in a developing mean flow, is first described by Liu a n d Merkine." Other studies of a monochromatic component of large-scale coherent structure i n a turbulent flow include those f o r the mixing a n d the round jet.18 Mode-mode interactions have also been studied f r o m a similar point of

vie^.'^-'^

I n contrast t o weakly nonlinear t h e ~ r y , ' ~ , ' ~ or simply local inviscid t h e ~ r y , ' ~ the present method allows f o r the simultaneous calculation of the spatial development of (1) amplitude or wave envelope of the large-scale coherent motion; (2) the energy level of the fine-grained turbulence and (3) the spreading rate of the mcan shear flow. As such, the coherent motion with strong amplification, affecting the mean flow development, is

d

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included

e T h e actual f i n a l numerical problem to be solved a r e first ordcr nonlincar, ordinary d i f f e r e n t i a l equations with one f o r each spectral component of thc cohcrcnt structure (this can be relaxed to include, in addition, one f o r each of the phase anglcs), onc f o r the overall fine-grained turbulence energy and one f o r the mean shcar flow thickness.

T h e coefficients of nonlinear terms involving the coherent structure rcflcct integrals of known functions that a r e tabulated functions of the local shcar layer thickness or thc properly scaled local Strouhal frequcncy. Once these and other intcraction integrals a r e evaluated, the amplitude equations a r e straightforward to solve numerically. T h e present method, though not intending to replace numerical simulations, is most convenient to use in the study of f r e e shear layer control becausc this study nccessarily involves the variation of a large number of parameters, such as, initial amplitude a n d phase anglc of each of the coherent modes, the frequency content, thc initial turbulcnce level in thc shear layer, the initial mean flow thickness, and the like.

d

It is virtually impossible to control the fine-graincd turbulence once it is generntcd, either in transitional or in turbulent shear layers. However, because the large-scalc coherent structures a f f e c t the mean flow a n d the fine-grained turbulence, control of the entire shear flow development, including the fine-grained turbulence, could be madc possible through perturbing the coherent structures. T h a t this is now possible lies in thc recognition t h a t such structures a r e a manifestation of hydrodynamical instabilities. As in transitional flows, the instabilities a r e sensitive to environmental and initial conditions, within certain amplitude thresholds and Strouhal frequency ranges and a r c therefore susceptible to control.

T h e technological consequences of the possible control of turbulence dcvclopment is, of course, enormous. These being motivated by the enhancement of fine-graincd turbulence in mixing regions f o r cfficicnt combustion, control of vehicle d r a g and modifications of its observablcs.

2

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General Considerations

i/ We a r e considering, f o r purposes of shcar layer control, the interactions betwcen mcan motion, fine-grained turbulence a n d various coherent modes. T h u s other averaging procedures a r e necessary in suplementing the usual Reynolds average, in order to cxtract the coherent modes f r o m each other a n d f r o m the fine-grained turbulcnce. We begin by splitting a n y overall flow quantity, denoted by q, into a Reynolds-averaged mcan flow Q , coherent disturbances (ti+:) a n d fine-grained turbulence q ^I:

Here, denotes the odd coherent modes a n d

6

the even modes. Evcntually, we specialize thcse i n t o a single subharmonic a n d its fundamcntal, respectively. At this stage, thc modes a r e considered as (Strouhal) frequency modes a n d the flow develops in the downstream spatial direction.

For such spatial problems, as is usually f o u n d applicablc in the laboratory, the Rcynolds average is a time average taken over with the longcst period T (associated with the lowest f r e q u e n c y 0):

d

T h e supplemental average to sort out the coherent oscillations f r o m the turbulence is denoted by

< >,

conditioned upon the lowcst frequency R, known as the phase average:

n N

n=O

Here, this procedure implics f i x i n g the phase of the coherent modes with frequcncics largcr t h a n a n d equal to

13.

Such a n average was used originally i n the laboratory f o r

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the perturbation, a t a single frequency R, of a turbulent channel flowz5 a n d of a turbulent jet.26 T h e average ( 2 ) will retain all the coherent mode contributions f r o m all the frequencies mR, where m is a n integer. T h u s the phase average of linear turbulence quantities is zero,

Q

< q ' > = 0

while <Q> = Q. Subjecting the overall flow quantity to such a n average gives

< q > = Q

+ (a+;)

( 3 )

T h e overall sum of coherent modes is then obtained f r o m

< q >

-

Q =

(a+;)

⁽⁴⁾

We denote f u r t h e r a similar phase average, associated with the second lowest frequency 213, by the symbol

<< >>.

T h u s this avcrage over quantities linear in the odd modes vanish, <<4>> = 0. T h e even modes a r e then obtained upon performing the

<< >>

-

average in (4),

v/

T h e

<<

>)-average recovers all the m(2R) contributions, where m is a n integer.

modes a r e then obtained by subtracting (5) f r o m (4).

T h e odd

For flow quantities t h a t occur linearly, (5) is equivalent to performing the

<<

>>-average on the total flow quantity a n d subsequently subtracting out the mean flow,

< < q > >

-

Q =

G.

However, f o r nonlinearly occurring flow quantities, such a direct procedure is undesirable as it gives rise to the presence of the 2R-partially-modulatcd fine-grained turbulence stresses, in addition to the overall R-modulated stresses. This

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would be a n unnecessary complication. T h e procedure indicated by (4) a n d (5) will thus

4 be followed.

T h e modulated turbulent stresses, which will occur in the coherent mode momentum cquations, come f r o m products of the fine-grained turbulence velocity fluctuations, ui u j

.

T h e

<

>-average produces

I I

T h e modulated stresses a r e also split into odd a n d even modes. Upon

<<

>>-averaging of

(G),

t h e even modes a r e obtained

Similarly, the odd modes F i j a r e obtained by subtracting ( 7 ) a n d (6). I n this procedurc,

the Reynolds average of < u i u j

>

is identical to u i u j .

procedure described earlier, <<u. u '

>>

^fu ' u . would be introduced.)

1 J 1 J

\4

-

I 1 I 1

(We note t h a t in the undesirable

-

I 1 I 1

For purposes of f i x i n g ideas, we h a v e set our discussion of averaging on the basis of coherent frequencies. In the situation where there exist three-dimcnsional geometrical coherent modes, such as spanwise periodicities in a two-dimensional mean shcar f l o ~ ~ ' ~ ~ * or helical modes in a round jet," the averages already discussed must necessarily be supplemented by those related to the spatial pcriodicities.

Energy Balance

T h e 'continuity a n d momentum equations f o r the mean flow, the odd a n d even coherent modes a n d the fine-grained turbulence a r e derivable f r o m the overall continuity a n d Navier-Stokes equations according to the splitting a n d averaging procedure described

J

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in t h e previous section. I n this section, we shall state a n d interpret kinetic energy equations, obtained f r o m the momentum equations, of the various components of flow.

T h e kinetic energy equation f o r each of the flow components, suitably Reynolds averaged, are:

Mean flow

-

D

u.

2/ 2 = - - D t 1

transport exchange

2 2

a 2 u i / 2 ax2 + v

J Odd modes

-

U'U'

-

1 J

axj

v

transport exchange

2 2

+ v a2iii/2

-

v [ ]

a 2

j Even modes

transport

(9)

exchange

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(11)

Fine-grained turbulence

t

-

J J 1transport 1

- 1

-

exchange

T h e symbol D / D t on the left of (8)

-

(11) denotes the substantial derivative following the mean flow. T h e f i r s t group of tcrms on the right of (8)

-

(11) represent the transport of energy a n d is of divergence form. T h e y give zero contributions whcn integrated over a sufficiently large volume.

T h e second group of terms on the right of (8)

-

( 1 1 ) a r c the energy exchange

.&

mechanisms among the various f l o w components. In (8), if

then net energy is transferred f r o m the mean flow to all the fluctuations. For the coherent modes this is the mechanism f o r their amplification within certain frequencies.

T h e r e is, of course, the possibility that individually

- au:

a n d

(12)

in which case, t h e coherent motion is in t h c "damped" regime, where energy is returned to the mean flow, as is f o u n d in the downstream of a forced mixing layer.29

-J

On the right of (9) if

then energy is transferred f r o m the odd modes to the fine-grained turbulence through the work done by the modulated stresses -7;j against the rates of s t r a i n aiii/axj.

Similarly f o r t h e same mechanism on the right of ( I O ) . We see t h a t such terms havc opposite signs in the turbulent kinetic energy equation (11). T h e energy t r a n s f e r between a single coherent mode a n d fine-grained turbulence has been intensively

T h e inter-mode energy t r a n s f e r is given by the t h i r d term among the "exchange"

group of terms on the right of (9) a n d (10). T h i s mechanism has opposite signs in (9) a n d

'a

^{(10). If}

then energy is transferred f r o m the o d d modes to the even modes. T h e interaction between coherent modes in developing shear flows has been the subject of recent

T h e viscous e f f e c t s on the right of (8)

-

(11) a r e f a m i l i a r a n d we shall omit their discussion here.

Because of the nonlinear interconnection among the various components of flow, it appears possible t h a t we a r e i n a position to e f f e c t control of the fine-grained turbulence, through the control of the coherent motion. T h e problem, however, remains unclosed. In

4'

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the following sections we shall direct our attention to carrying o u t the practical synthesis of numerous ideas f r o m hydrodynamic stability so a s to enable the execution of quantitative but simple, "rapid" calculations directed towards f r e e shear f l o w control.

Interacting Nonlinear WavcEnveIope Eqnations

To f i x ideas, we shall consider the two-dimcnsional mixing layer formed by the upper U, a n d lower U7- streams. I n order to make use of the ideas discussed earlicr, we integrate the kinetic energy equations across the shear layer. But, in order to f u r t h e r simplify matters, we f i r s t make the boundary layer approximation to the viscous d i f f u s i o n of kinetic energy, neglecting the streamwise e f f e c t a n d retaining the normal viscous diffusion. T h e latter integrates t o zero across the shear layer. I n t h c cncrgy exchange bctween the mean flow a n d the fluctuations we retain only the dominant mechanism d u e to the mean shear rae of strain, aU/ay. We f u r t h e r specialize to only two f r e q u e n c y modes: the f u n d a m e n t a l (even) a n d subharmonic (odd) modcs. T h e integrated f o r m of the kinetic energy equations a r c thcn:

.--/

fundamental mode (even, 26):

.-

1 - Â Â Â

1

^U;(u ^{+v )dy}⁼^{I P}

⁺

^{I z l}

^-

^{I w t}

^- ⁶

d x

_ a

subharmonic mode (odd, 6) ^:

I

-

^I

-

1 -

J

^U (ij2+Vz)dy =

+ -

I,,,~

-

4 d x

.a

Y

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fine-grained turbulence:

mean flow?

-

d x

1 ^A ^-

U

-

( ~ ' ~ + v ~ ~ + w I ~ ) d y = 1;

+

(Iwt+Iwt)

- @ '

d x

_ m

J

[

U i ( U 2 - U 2 ^{_ m})dy +

[

^U (Uz-U2)dy] ^m = -(fp+fp+I$)

-

^Q

_ m

I n the above, we have taken the coherent structures to be the commonly observed two-dimensional structures, with vorticity axes perpendicular to the f r e e strcams. (We refer to Liu" f o r the discussion of three-dimensional coherent structures). T h e energy exchange integrals a r e denoted by I a n d these are:

i/

(1) Energy exchange between fluctuations and the mean flow:

fundamental:

f p

⁼_J-m

I

-uv - d y

⁼ ^:

. m

- au aY

subharmonic:

-

I p =

1

^-ii? ^{- d y}

_ m

I

au

turbulence: I p =

1 ^-uIv) aY

^{- d y}

_ m

- 1 3 -

(15)

(2) Energy exchange between fundamental a n d subharmonic:

h

- a; a6 a;

121 = -I2] =

1

^[(ii2-V2) ^-^ax

⁺

i j J [ a ,

+

g ] ] d y

J-m

( 3 ) Energy exchange between coherent modes and turbulence:

fundamental: A I wt = J- m

T h e mean flow dissipation integral involves only the (aU/ay)' because of the boundary layer approximations, whereas the fluctuation dissipation integrals involve the f u l l rates of strain a n d stresses.

v

I n the above (12)

-

(21) and subsequently we deal with dimensionless quantities

-

referenced to the average velocity U, = (U,+U-,)/2, length scale 60, which is the initial, half-maximum vorticity thickness. T h e Reynolds number, which occurs as the denominator is the viscous dissipation integrals, is R e = 60Um/u. T h e remaining dimensionless parameters in the problem are: the "velocity ratio" R = (U-m-Um)/(U-m+Um), the Strouhal frequency I3 = 2nf6/iJ, where 6 = 6(x) is the local half-maximum vorticity thickness a n d f is t h e physical frequency.

-

I n order t o obtain closure of the problem, (12)

-

(15) must necessarily be accompanied by shape assumptions in what amounts , t o be a n integral method6s8-'2 f o r the nonlinear wave envelopes (or amplitudes). Because of strongly amplified coherent modes i n ' f r e e shear flows, their must be jointly solved with the mean flow problem. For turbulent shear flows, this involvcs the mean vclocity and the Reynolds-nverngcd stresses.

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Following earlier work,'-'' the cohercnt modes are assumed to take thc f o r m of thc product of a n unknown amplitude, Ai(x), with a shape distribution across t h e shear layer given by t h e local linear stability theory. T h e local shape assumption using t h c linear thcory was f o u n d to have experimental support?' a n d this is confirmed i n recent curve f i t s of experimental d a t a using t h e local linear t h ~ o r y . ' ~ ' ' ~ T h e dimensionless coherent modc velocities (and pressure) a r e assumed to take t h e f o r m

d

A A

where n = 1 refers t o t h e subharmonic (ti,?) a n d n = 2 refers t o t h c f u n d a m e n t a l (u,v).

H e r e

bn

denotes t h e eigenfunction of t h c local linear stability theory a n d is f u n c t i o n of t h e f r e q u e n c y nR (n = 1 or 2) a n d t h e rescalcd vertical or normal coordinate R = y/S(x), primes on 4 denote d i f f e r e n t i a t i o n with respect to R; 8 is t h e relative phase angle between t h e f u n d a m e n t a l a n d the subharmonic a n d C.C. denotes t h e complex conjugate.

Here, 8 is taken as constant f o r simplicity a n d An(x) a r e real.

W

T h e dimensionless mean velocity is taken to bc

U = 1

-

R t a n h R (23)

so t h a t t h e single parameter characterizing the mean flow is S(x).

of t h e turbulence is represented

T h e Reynolds stresses

where E(x) is t h c turbulent kinetic energy density such, that

S(x)E(x) =

-

2

a n d it bears t h e entire burden of the history of t h e nonequilibrium. nonlinear

d

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interactions; the constants c i j would reflect the proper ratio betwcen thc energy and thc Reynolds shear stress as well as the necessary normalization to render E(x) the turbulent energy density.

From general considerations of the Reynolds a n d modulatcd ~ t r c ~ ~thc e ~ , ~ ~ , ~ ~ ~ ~ ~ ~ ~ ~ problem of iii,Fij a n d u i , r i j a r e coupled through the action of Fij,rij on the respectivc

momentum problem even on a linear level. T h u s F.. a n d r " a r e given by their respective transport equations.16 Consistent with the shape assumptions f o r iji,ui, shape assumption f o r the modulated stresses necessarily take the f o r m

A A A

A

1J 1J

h

where rij,n would be given by a local linear theory jointly with @n(n;nB), with n = 1,2 again representing the subharmonic and fundamental, respectively. T h e transport cquations f o r rij,n a r e lengthy and we refer to Liu and Mcrkine15 f o r f u r t h e r elaborations. It suffices to say that inflectional point, dynamical instability argument^'^

lead to a n "inviscid" description of

bn

as given by the Rayleigh equation; a n d t h a t such cigenfunctions a r e then used in the linearized transport equations of rij,n to e f f e c t a n algebraic solution f o r their shape across the shear layer. This allows the evaluation of the energy transfer between the coherent structure a n d fine-grained turbulence.

W

We need to emphasize t h a t the linear eigenfunctions of @,(n;nB) nceds to be normalized locally so as to give the consistent physical meaning to the wave envelope or wave amplitude that its square is the local energy density,

A2(x)S(x) = En(x) n

where

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E l = [ G 2 ] d y

J _ m

a r e the sectional-energy contents of the subharmonic a n d fundamental, respectively. T h e streamwise evolution of such quantities have formed the basis of quantitative experimental measurements with mode interactions in developing shear layer^.^^,^'

Upon substituting the shape functions (22)

-

(26) into (12)

-

(15) a n d rearranging, wc obtain:

fundamentalr

d 6A2

-

³

..

¹

I 2

z- ^-

^I^'S2^A²

⁺

^12]A2A¹ ²

^-

^I^Wt2^EA2²

^{- -}

^{R e}^{I A2/6}^{$2 2}

subharmonic

d 6A2

- I A2

-

^121A2A

-

I EA2

- -

I I A2/S

-

1

1 1

r -

^'SI ^I ¹ ² ^{W t l} ¹ ^{R e} ^@1¹

turbulence:

dSE

I t

-

d x = I;,E

+

( A ~ I , ~ ~ + A ~ I 2 1 Wtl )E

-

1 $ , ~ 3 / 2

mean flow

2 I

-

d 6

I

-

= I A2 + I A

+

I ' E

+

^-10/6

d x ?2 2 "1 1 rs R e (33)

-17-

(19)

T h e integrals originally defined in (16)

-

(21) a r c related to the interaction integrals in (30)

-

(33) as follows:

h I p = A21 ( 6 ) 2 'S2

Integrals involving coherent structures a r e functions of 6(x) through their dependence on the local frequencyparameter B(x).

d T h e advective integrals, occurring on the lcft side of (12)

-

(14), now bccomc F1(6AT), I2(6A$) a n d It(6E), respectively; both 11 and 12 a r e "slowly varying" functions of 6 within thc frequency range of B(x) of interest a n d a r e replaced by a constant mean value,

r1

a n d r2. T h e mean flow energy defect advective integral, on the lcft side of (15), becomes - r 6 , where

i

=

2R2(3/2-Qn 2). Equation (33) thus reflects a change in sign f r o m the original (15). Integrals i n v o l v i n g t h e m e a n flow andturbuIencequantities,suchasi,Irsand theirdissipation integrals ( I $ , ,I+) a r e constants. T h e coherent mode dissipation integrals Ion a r e again functions of 6(x). T h e coherent mode integrals a r c now tabulated functions of 6 ( x ) or Wc should mention a word about the fine-grained turbulence closure. This occurs a t two levcls.

One is a t the Reynolds-averaged level. This is already partially discussed in conjunction with the shape assumption (24). T h e standard local equilibrium argument f o r large Reynolds numbers have been followed in arriving a t the E3/2 f o r m f o r the viscous dissipation integral in

(32). the (integrated f o r m of the)

-

I

Because of the shape assumption (24) a n d the use of

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turbulcnt energy equations, no explicit assumptions about t h c pressure-rates of strain

-' corrclation was needed i n t h e "amplitude" equation f o r t h c turbulent kinetic cncrgy dcnsity (E). At t h e level of wave-modulated stresses, rijn, which is needcd i n thc evaluation of wave-turbulence encrgy exchange intcgral Iwtn, additional closurc assumptions were needed i n t h e transport equations f o r r" even a t t h e local linear Icvcl.

These include ( I ) t h e usual assumptions about t h e wave-modulatcd pressure-ratcs of strain correlation terms i n t h e f o r m of the tendency of return to isotropy; ( 2 )

usual viscous dissipation rate. Only t h e "correct" functional f o r m of such rclations a r e sought, but not t h e "accuracy" o r appropriatcncss of t h e closure constants.

However, r i j n occur, together with @,, under t h e cross-sectional integral a n d a r e shown, through trials, not to be sensitive to variations of closure c o n s t a n t ~ . ~ ~ ' ~ ~ T h e f u n c t i o n a l form, AnE, in t h e wave- a n d turbulence amplitude equations a r c independent of t h e turbulcncc closure.

1Jn

t h c

2

T h e f o u r first-order nonlinear d i f f e r e n t i a l equations (30)

-

(33) a r e subject to the initial conditions A2(0) = A&, Af(0) = A1O, 6(0) = 1 a n d E(t) = EO a n d the specification of 4(0) = 410 a n d 20(0) = 1320 a n d Re. T h e specification of 010 (and t h u s 420) essentially give us t h e choicc as to what p a r t of t h e linearized (dispersion) amplification vs 13 curve we wish to s t a r t thc initiation of t h e disturbance. Since Ro = 2nf60/6, where 60 here is t h e dimensional initial half-maximum vorticity thickness, t h e changes i n 00 c a n be effected through the forcing frequency f , initial shear laycr thickness a n d thc average f r c c stream velocity.

2 2

.../

We have t h u s formulated the problem of forcing two frequency modes in a f r c c turbulent shear layer.

Multiple Subharmonics

A kinematical model of dcvcloping shcar layers, i n absence of finc-graincd turbulence has been suggested by Ho31. A f u n d a m c n t a l a n d its succccding

4

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subharmonics, each peaking a t f u r t h e r downstream locations, a r e responsible f o r thc

4 streamwise development of the shear layer. It is a n experimental fact3' t h a t ( I ) T h e f u n d a m e n t a l being of higher frequcncy, peaks earlier in the streamwise dircction than the f i r s t subharmonic. T h e i r measured cncrgy levels, or wave envelope d o not switch abruptly b u t rather, perform a fade-in a n d fade-out ovcrlap in the strcamwise direction;

a n d (2) T h e observed lower frequency modes peak f u r t h e r a n d f u r t h e r downstream.

F r o m these observations, f o r purposes of constructing a dynamical model of multiple subharmonic evolution in a developing shear layer, one can a d v a n c e the idea t h a t o n l y binary-frequency interactions need to be taken into account in the streamwisc development of wave envelopes. For the multiple subharmonic evolution model, this amounts to saying t h a t only the interaction between immediate spatially neighboring wave envelope need to be accounted. Considcr, then, the flow disturbance begins with the f u n d a m e n t a l ; i t acts as a n "even" mode to the f i r s t subharmonic which in t u r n is the

"odd" mode i n the f i r s t binary interaction. T h e f i r s t subharmonic, which has twice t h c frequency of the second subharmonic; then enter into another even a n d odd binary-modc interaction, a n d so on. In this case, the binary interaction integral, 121, once tabulatedz1 can be used f o r such successive interactions.

'd

I n constructing such a multiple-subharmonic model, we introduce the following easily reconizable notation: let the subscript f denote the f u n d a m e n t a l a n d the subscript sn (n = 1,2, ...) denote the subharmonics. T h u s their respective Reynolds stress, production integral becomes, respectively, IrSf a n d I,,,,. T h e binary interaction mechanism, which was denoted by,

2 I 2 1A1 A 2

in (30), ( 3 l ) , uses the subscript 2 to denote even a n d 1 to dcnotc the odd mode. T h e rate of s t r a i n is provided by the even mode while the stresses by the odd mode is discusscd i n the section on the Energy Balance. T h i s accounts f o r the powers of the amplitude

-,,

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occurring as A I 2 a n d A2.

4 If we make t h e "high frequency" cutoff a t the f u n d a m e n t a l (although higher harmonics c a n certainly be taken into account through binary-frequency interactions), then t h e f u n d a m e n t a l mode will have only one interaction term, t h a t with t h e first subharmonic. In this case, t h e wave-envclopc cquation f o r the f u n d a m e n t a l is similar to (30), but with t h e notation changed according to t h e discussion above,

T h e f i r s t subharmonic wave-envelope equation, similar to ( 3 1 ) , but with a n added interaction term to connect with t h e second subharmonic (As& then appear as

'.d

T h e nth subharmonic wave-envelope cquation then becomes

If t h e practical streamwise region of interest precludes consideration o subharmonic, then the n-(n+l) interaction term would be absent i n (36).

( 3 6 )

say, the ( n + l ) -

T h e modifications to t h e fine-grained turbulence a n d mean f l o w equations are straightforward. T h u s the second term on t h e right of (32) is now replaced by

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a n d terms one a n d two on the right of (33) a r e replaced by

T h e i n i t i a l conditions a r e similar to the two mode interaction problem earlier. We are now i n a position to discuss quantitatively the problem of shear layer control through forcing multiple subharmonics.

Prior to discussing the quantitative results we shall present a guide t o the nomenclature used in the Figures. In the calculations, only three subharmonics arc included f o r illustrative purposes.

Nomenclature and Parameters Used in Figures

I n the subsequent sections the detailed discussion of control of f r e e shear layers a r e presented i n terms of Figures 1 through 8. I n the f i r s t of each of the Figures, labeled (a), the dimensionless shear layer thickness is plotted as a f u n c t i o n of the dimensionless streamwise distance (both referred to ^60),showing the variations of the controlling parameter in question. I n the subsequent parts of each Figure, labeled (b), (c), a n d so on, the energy levels of the various components of the f l o w a r e shown as a f u n c t i o n of t h e dimensionless streamwise distance, a n d a r e indicated as follows:

I-_/

E/Eo, ratio of fine-grained turbulent energy to initial value

A f / A f O , ratio of f u n d a m e n t a l energy density t o its initial value ^’

2 2

--__________---

2 2

A S I / A s I ~ , ratio of first subharmonic energy

_ _ _ _ _

Y

to its initial value

As2/%20, ratio of second subharmonic energy density to its initial value

2 2

- - - - - - - -

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2 3

- - -

As3/AS30, ratio of third subharmonic energy density to its initial value

A "standard" case is computed as the basis f o r comparison w i t h results f r o m variations of controlling parameters. T h e s t a n d a r d case is f i x e d as follows: RO = 0.4985 now denotes the dimensionless frequency of the f u n d a m e n t a l mode, which is almost most amplified model a t R = 0.69

(BO

f o r all subharmonics a r e halved). T h e Reynolds number is R e = 968.

We a r e rescaling the energy levels so that the new level denotes E5/(8R 2 ). I n terms of this rescaling, the initial conditions a r e expressed a s EfO = 2.1 x 10- 4 , Es10 = 0.925 x

10- 5 , Es20 = Es30 = 10' 5 , EO = IO' 5

.

T h e relative phase angles a r e chosen to be 9i =

180°, so t h a t energy transfers f r o m high to lower frequencies f o r R = 0.69. T h e initial conditions a r c applied a t ~ 0 / 6 0 ^zi 7. T h e flow conditions here would correspond approximately to Urn = 363 cm/s, U-- = 1980 cm/s, f ⁼750 Hz, So = 0.124 cm in air."

T h e s t a n d a r d case is shown in Figure I(a) f o r

BO

= 0.4185 a n d in F i g u r e l(b). All conditions correspond to the "standard" case, except f o r the variations of the controlling parameter indicated.

, J

Control through Initial Strouhal Frequency Parameter,

Bo

T h e dimensionless frequency parameter is R = 2af6/U,.

-

with

BO

refcring t o the

In this case, the increase or decrease of 40 It is easy to envision a f l o w in T h e T h u s there is only a single initial half-maximum vorticity thickness 60.

implies the adjustments made through 60,f a n d

fi,

which 60 a n d

6,

are f i x e d a n d 00 is adjusted via the forcing f r e q u e n c y f.

subharmonics would have successively halvcd frequencies.

frequency to adjust, the rest follows.

As mentioned previously, the "standard" case is shown as the solid line in Figurc I(a)

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dcnotcd by i30 = 0.4485 (meaning Rfo). From the energy density evolution of this casc i n Figure l(b), we see t h a t within the x/60 region shown, there a r c two distinct pcaks in thc energy density associated with the f u n d a m e n t a l a n d subharmonic. T h e shear layer growth exhibit the two-step structure associated with these peaks. We r e f e r to Nikitopoulos a n d Liu2' a n d Liu1'f'2 f o r detailed discussion of the response of shear layer growth to the r a t e of energy extraction f r o m the mean f l o w by flow disturbances. T h e second subharmonic goes through a broad "peak" a n d thus d o not give rise to a steplikc s t r u c t u r e in 6 / 6 0 . T h e fine-grained turbulence rapidly achieves a nearly equilibrium level a n d causes the shear layer to sprcad more or less linearly. T h e t h i r d subharmonic is still growing within this region. It is experimentally a n d theoretically known t h a t low f r e q u e n c y coherent modes peak f u r t h e r downstream. Consequently, when the forcing f r e q u e n c y 130 is subject t o half its original value, all acitivites a r e shifted downstream as shown i n Figure I(c), with the f u n d a m e n t a l a n d first subharmonic peak considerably boradened. T h i s is reflected in 6 / 6 0 in Figure I ( a ) f o r this frequency, the first step a t a b o u t X / ~ O

*

75 coinciding with the peaking of the fundamental. T h e combined energy extraction f r o m the mean flow by the f i r s t a n d second subharmonics a n d the fine-grained turbulence effects a larger growth of the shear layer.

d

.d

Still decreasing the forcing frequency f u r t h e r to 130 = 0.1246 causes the f u n d a m e n t a l to peak a b o u t X/60

*

^140,which is where the first stcp in 6/60 occurs. Because of the slower extraction of energy by the coherent modes, the fine-grained turbulence is able to achieve a higher level t h a n the previous two cases, a n d give rise to a more rapid shcar layer growth rate. Thus, forcing a t low the frequencics, appropriate to the strcamwise length scale of interest, can e f f e c t a higher mean flo,w sprcading rate a n d higher levcls of fine-grained turbulence.

Effect of Finc-grained Turbulence Level

It has been theoretically shown that the level of fine-grained turbulence directly

4

- 2 4 -

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a f f e c t s t h e downstream growth rates of the cohercnt

mode^."^'^.

T h u s f o r E t 0 = 10 - 3

4 the growth of the sheat layer is depressed in Figure 2(a) because of the general much lower levels of the coherent modes shown in Figure 2(c). thc shear layer growth is enhanced, this is because of the higher levels achieved by the coherent modes. In this case, although the eventual spreading rates a r e a b o u t the same, the magnitude of 6 / 6 0 attained depends on the relative levels of amplification achieved by the coherent modes. T h i s is not surprising, as 6 is a n x-integral of the energy extraction rate.

In contrast, f o r Et0 =

T h e higher the coherent mode energy levels, the larger the "integrand".

I n order to achieve high spreading rates, the forced coherent modes must somehow be made to "pierce" this fine-grained turbulence bearer.

The Effect of Reynolds Number

As shown i n Figure 3(a), there is no d i f f e r e n c e in the mean f l o w thickness development between R e = 968 a n d R e = 968,000; though not shown, this is also the case f o r the development of the energy levels. As the Reynolds number is decreased to 96.8, as with subsequent lower Reynolds numbers of 48.4 a n d 24.7, the step-like s t r u c t u r e in 6/60 is shifted downstream. T h e overall energy level development is s h i f t e d downstream, as shown in Figure 3(b)

-

(d). There is a noticable viscous damping of the energy levels f o r low-Reynolds numbers. For the lowest Reynolds number case, the initial shear layer spread is d u e viscosity.

'4

I t is already known f r o m the work of KO, et a18 on wakes t h a t as the Reynolds number is increased the peaking of disturbance s h i f t s upstream. T h e Reynolds number e f f e c t is included here f o r completeness. It, is not expected that such effects a r c used f o r practical control purposes.

The Effect o f Relative Phasc Angles

In the "standard" case, the relative phase angles have been chosen to be

e l

⁼⁸²⁼O 3

d

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= 180°. For R = 0.69, this phase angle gives rise to energy t r a n s f e r f r o m the higher

d to the lower frequency mode in a binary interaction. T h i s is advantageous for purposes of achieving maximal shear layer spreading rate. T h i s is so because lower f r e q u e n c y modes a r e reinforced by higher frequency modes. T h e opposite is t r u e f o r 6i = 0 " a n d the consequential e f f e c t on the spreading rate a n d energy densities is shown i n Figure 4(a) a n d (e). T h e moderate case of 8i = 90" is shown ind Figure 4(a) a n d (b). I n this ease, the lower frequency mode transfers energy to the higher frequency modes, b u t eventually changes sign in this process. T h e effect, however, is modest.

Apparently, the energy t r a n s f e r mechanism as f u n c t i o n of the relative phase angle has t o be mapped o u t f o r each R. T h e R = 031 casez1 is qualitatively d i f f e r e n t . T h e allowance of the nonlinear streamwise evolution of the relative phase angles is presently being considered by S . S . Lee a n d D. E. Nikitopoulos.

W The Effect of Forcing the Fndamcntal

With all other conditions fixed, i t is known theoretically f o r the single mode problem' that if the initial disturanee amplitude were increased, the development of the coherent mode a n d the enhanced spreading of the mean f l o w would be moved upstream. T h i s e f f e c t illustrated through 6/60 in Figure S(a), comparing the initial

amplitudes EfO

-

^IO-1 ^{a n d} I n Figure S(b) f o r

EfO

-

the relative amplification is larger f o r The f u n d a m e n t a l energy density a n d is relatively larger than the EfO

-

10-l case in Figure S(c); compare also with F i g u r e l(b). T h e exceedingly larger EfO ease effccts a choking of its own energy supply through the weakening of the mean flow. Not only the relative amplification is weakened, the streamwise life time is shortened in the high initial amplitude forcing. Though not shown here, more scvcre forcing causes the immediate decay downstream, t h u s rendering the forcing amplitude to be the maximum attachablc

with the s t a n d a r d case of

4

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amplitude.I8 T h i s also a f f e c t s the weakening of the subharmonics as shown in Figure .4 5(c). Again, the subsequent shear layer growth ratc is the samc, b u t because of thc weakening of t h e coherent modes, t h c large initial f u n d a m e n t a l amplitude case causcs the shear layer t o spread not as widely as the weaker initial f u n d a m e n t a l amplitude cases.

Effect of Forcing the First Subharmonic

T h e qualitative effects of forcing the f i r s t subharmonic is similar to t h a t of t h c fundamental: amplification is moved upstream a n d relative amplification is limitcd with increasing f o r c i n g amplitude. For the present case, all relative phase angles a r c a t 1800, where energy transfers f r o m high to lower f r e q u e n c y components. We see t h a t the f u n d a m e n t a l is weakened by a sronger subharmonic; compare Figure 5(b) to Figure l(b). F u r t h e r increases of E S l 0 causcs the f u n d a m e n t a l to decay. Because of the sclf-limitation upon ESl with increasing ES10, the second subharmonic is not very much affected. T h e mean flow spread is a f f e c t e d in two streamwise regions: one is in the vicinity of the peaking of the f i r s t subharmonic the other is f u r t h e r downstream where 6/60 spreads a t a slightly higher level t h a n the s t a n d a r d case because of the highcr levels of E,,.

W

Effect of Forcing the Second and Third Subharmonics

T h e forcing of second a n d third subharmonics a r c shown in Figures 7 a n d ^{8 .} As with the case of forcing the f i r s t subharmonic because of the imposed relative phasc angles, a strong lower frequency component takes energy a w a y f r o m the higher frequency components. T h u s Figure 7(c) a n d 7(d) shows that the higher freqency f i r s t harmonic a n d the f u n d a m e n t a l a r e both weakened by the seond subharmonic. T h i s is also the casc with higher frequency components i n the forcing of the t h i r d subharmonic. T h c subsequent resurgence of a higher frequency component is due to the resurgcncy of

4

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production f r o m the mean flow relative t o other subsiding energy sinks such as the binary f r e q u e n c y transfer mechanism to lower frcqucncics f o r 6i = 180°.

.J

While the second subharmonic forcing still shows a local e f f e c t a r o u n d its peak, the subsequent spreading rate is somewhat dramatically enhanced. We note, however, the fine-grained turbulence level always sccm to settle to a n equilibrium level within the region of interest. T h e spreading rates associated with the t h i r d subharmonic forcing is most dramatic. F o r a forcing lcvcl of Es30

-

we can see that there is a doubling of the shear layer thickness a r o u n d ^{X / ~ O}

-

200 compared to the Es30

-

loT5 s t a n d a r d case. T h e doubling in thickness is achieved much earlier, a t a b o u t X/SO In the latter case, the forcing velocity r a t i o would be about 30%, a severe case approaching that of Favre-Marinet a n d Binder'sz6 forcing of a r o u n d jet

100, when Es30

-

^IO-'.

Concluding Remarks

W We have shown how the simple method, d u e originally to Lester Lees, could be of such great utility in studying the control of f r e e shear layers. Clearly, we have only scratched the s u r f a c e in that there is much work to bc donc on including the three dimensional coherent disturbances in studies of f r e e shear layer control.

Acknowlcdgcmcnts

T h i s work is supported partially by DARPA/ACMP through its University Research Initiative; NSF G r a n t MSM83-20307; NASA/Lewis Research Center G r a n t MSM83-20307;

NASA/Lewis Research Center G r a n t NAG3-673 a n d , the UKSERC through its visiting fellowship program a t the Department of Mathematics, Imperial College. We t h a n k J. T.

S t u a r t f o r the many beneficial conversations on shear flows.

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References

-/ 'Liepmann, H. W., "Aspects of the Turbulence Problem. P a r t 11", 2. Angew. Math.

Phys., Vol. 3, 1952, pp. 407-426.

'Liepmann, H. W., "Free Turbulent Flows", Mecanique d e la Turbulence, CNRS, Paris, 1962, PP. 211-227.

S . C., a n d Champagne, F. H., "Orderly Structure in Jet Turbulence", Journal of

Fluid Mechanics, Vol. 48, 1971, pp. 547-591.

*Brown, G., a n d Roshko, A,, "The E f f e c t of Density Difference on the T u r b u l e n t Mixing Layer", Journal of Fluid Mechanics, Vol. 64, 1974, pp. 775-816.

'Davies, P. 0. A. L., a n d Yule, A. J., "Coherent Structures i n Turbulence", Jotrrnal o f Fluid Mechanics, Vol. 69, 1975, pp. 513-537; a n d "Corrigendum", Vol. 74, 1976, p. 797.

'Stuart, J. T., "On the Nonlinear Mechanics of Hydrodynamic Stability", Journal of Fluid Mechanics, Vol. 4, 1958, pp. 1-21.

'Stuart, J. T., "Nonlinear Stability Theory", Ann. Review of Fluid Mechanics, Vol. 3, '4 1971, pp. 347-370.

'KO,

D. R. S . , Kubota, T., and Lees, L., "Finite Disturbance E f f e c t in the Stability of Laminar Incompressible Wake Behind a Flat Plate", Journal o f Fluid Mechanics, Vol.

40, 1970, PP. 315-341.

'Liu, J. T. C., "Nonlinear Development of a n Instability Wave in a Turbulent Wake", Phys. Fluids, Vol. 14, 1971, pp. 2251-2257.

"Liu, J. T. C., De-!eloping Large-Scale Wavelike Eddies a n d the Near J e t Noise Field", Journal o f Fluid Mechnnics, Vol. 62, 1974, pp. 437-

.

"Liu, J. T. C., "Interactions between Large-Scale Coherent Structures a n d Fine-Grained Turbulence in Free Shear Flows", Transition and Turbulence, Academic Press, N.Y.,

1981, pp. 167-214.

"Liu, J. T. C., "Contributions to the Undcrstanding of Large-Scale Coherent Structures in Developing Free Turbulent Shear Flows", Advances in Applied Mechanics, Vol. 2 6 ,

d

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Academic Press, N.Y., 1987, (in press).

4' "Gaster, M., Kit, E., a n d Wygnanski, I., "Large-Scale Structures in a Forced Turbulent Mixing Layer", Journal of Fluid Mechanics, Vol. 150, 1985, pp. 23-39.

14Wygnanski, I., a n d Petersen, R. A,, "Coherent Motion in Excited Frec Shear Flows", AIAA Paper 85-0539, 1985.

"Liu, J. T. C. , a n d Merkine, L., "On the Interactions between Large-Scale Structure a n d Fine-Grained Turbulence in a Free Shear Flow. Part I. T h e Development of Temporal Interactions in the Mean", Proc. of the Royal Society o f London, Series A, Vol. 352, 1976, pp. 213-247.

"Alper, A,, a n d Liu, J. T. C., "On the Interactions between Large-Scale Structure and Fine-Grained Turbulence in a Free Shear Flow. Part 11. T h e Development of Spatial Interactions in the Mean", Proc. of the R o j d Society of Londori, Series A, Vol.

359, 1978, pp. 497-523.

"Gatski, T. B. and Liu, J. T. C., "On the Interactions betwcen Large-Scale Structure a n d Fine-Grained Turbulence in a Free Shear Flow. P a r t 11. A Numerical Solution", Phil. Transactions of the Royal Society of London, Series A, Vol. 293, 1980,

d

pp. 473-509.

'*Mankbadi, R. R. a n d Liu, J. T. C., "A Study of the Interactions between Large-Scale Coherent Structures a n d Fine-Grained Turbulence in a Round Jet", Phil. Transactions of the Royal Society of London, Series A, Vol. 298, 1981, pp. 541-602.

"Kaptanoglu, H. T., "Cohrent Mode Interactions in a Turbulent Shear Layer", Brown University, Division of Engineering, Providence, RI, Sc.M. Thesis, May 1984.

"Mankbadi, R. R., "On the Interaction betweep Fundamental a n d Subharmonic Instability Waves in a Turbulent Round Jet", Journal of Fluid Mechanics, vol. 160, 1985,' pp. 385-419.

21Nikitopoulos, D. E. and Liu, J. T. C., "Nonlinear Bindary-Mode Interactions in n Developing Mixing Layer", Journal of Fluid Mechanics, Vol. 179, 1987, pp. 345-370.

d

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"Lee, S. S., "Multiple-mode Interactions in a Developing Round Jet", Brown University,

d Division of Engineering, Providence, RI, Ph.D. Thesis, Octobcr 1987.

'%tuart, J. T., "On the Nonlinear Mechanics of Wave Disturbances in Stable and Unstable Parallel Flows. P a r t 1. T h e Basic Behavior in Plane Poiseuillc Flow", Journal of Fluid Mechanics, Vol. 9, 1960, pp. 353-370.

24Crighton, D. G., a n d Gaster, M., "Stability of Slowly Diverging Jet Flow", Journal of Fluid Mechanics, Vol. 77, 1976, pp. 397-413.

25Reynolds, W. C., Hussain, A. K. M. F., "The Mechanics of a n Organized Wave in T u r b u l e n t Shear Flow. 3. Theoretical Models a n d Comparisons with Experiments", Journal of Fluid Mechanics, Vol. 54, 1972, pp. 263-288.

26Favre-Marinet, M., a n d Binder, G., "Structure des jets pulsants", Journal d e Mecaniq, Vol. 18, 1919, pp. 357-394.

"Jimenez, J., "A Spanwise Structurc in the Plane Shear Layer", Journal of Fluid Mechanics, Vol. 132, 1983, pp. 319-336.

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Thesis, 1985.

"Fiedler, H. T., Dziomba, B., Mensing, P. a n d Rosgen, T., "Initiation Evolution and Global Consequences of Coherent Structures in Turbulent Shear Flows", Lecture Notes in Physics, Vol. 136, 1981, pp. 219-251.

30Michalke, A., "Jnstabilitat eines Kompressiblen R u n d e n Freishahls under Berucksichtigung des Einflusses d e r Strahlgrenzschichtidcke", Z. Flugwissenchaften, Vol. 19, 1971, pp. 319-328 (English translation as NASA TM-75190, 1977).

31Ho, C. M., a n d Huang, L., "Subharmonics and Vortex Merging in Mixing Layers", Journal of Fluid Mechanics, Vol. 119, 1982, pp. 443-473.

32Ho, C. M., "Local a n d Global Dynamics of Free Shear Layers", Numerical and Physical Aspects of Aerodynamic Flows, Springer-Verlag, N.Y., 1981, pp. 521-533.

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Figure 1. Effect of forcing frequency

2 (a) Shear layer thickness (b)

-

(d) Energy densities:

(b) 80 = 0.4985 (c) 80 = 02492 (d) 80 ⁼0.1246

Figure 2. Effect of fine-grained turbulence level (a) Shear layer thickness

(b)

-

(e) Energy densities:

(b) EtO =

(c) = 10-7

Figure 3. Effect of Reynolds number (a) Shear layer thickness

(b) -(d) Energy densities:

(b) Re = 96.8 (c) Re = 48.4 (d) Re = 24.2

- 3 2 -

Control of Free Shear Lauers

AlAA-87-2689