Indian Journal of Radio & Space Physics Vol. 34, April 2005, pp. 106-113

Absolute and convective instability study of weakly compressible mixing layers A P Roychowdhury

Aerodynamics R&D Di vision, Vikram Sarabhai Space Centre, Triu vananthapuram 695 022, India

Received I April 2004; revised 8 November 2004; accepted 24 January 2005

This is an analysis of stability study of weakl y compress ible mixing layers under non-homentropic flow conditi ons. The condition of non-zero group veloci ty required for convective instability analysis was seen to give new findings as stated below. Also, the dispersion relationships are func ti ons of the cross-stream derivatives, Mach number and chemical reaction parameters . Three modes of osci ll ations for developing mixing layer were chosen based on previous findin gs of the author and later in thi s work their possible validi ty were confirmed as follows. Jet mode (Mode 1), mi xed mode (Mode II ) and full y developed mixing layer mode (Mode Ill ) oscill at ions for developing mixing layers are considered in thi s study. Using impulse response technique it is seen that for symmetric temperature profile, Mode I is absolutely unstable, Mode II is both convecti vely and abso lutely unstable, partially each, an Mode Ill is dominantl y convectively unstable with pockets of abso lute instability. For the asymmetric temperature profile ofT= I+/.. tanh (y), Mode I is weakly convective!y unstable. Mode II is weakly absolutely unstab le and Mode Ill is convecti ve ly unstabl e for major regions of the now.

Keywords: Absolute instability, Convective instab ility, Mi xing layers, Non-isentropic formulation

J>ACS No: 44.25 +f; 47.27 Te

1 Introduction

Chemically reacting mtxtng layers are found in chemical process systems and jet engines of aircrafts and rockets. These are assoc iated with reaction and heat-transfer process, which are non-homentropic processes and reversible in nature. This is an analysis of such weakly compressible mixing layers under non-i sentropic flow conditions. In the study of compressible mixing layers using isentropic analysis 1

one found that heat ing dampened the growth of di sturbances. But our previous work in non-i sentropic analysis showed the reverse 1

• In order to look further into this analysis the present study has been made to understand how convective and absolute instabilities behave with respect to heat addition or remova l.

Impulse response behaviour of a wave packet is considered in non-homen tropic , wea kly compressible mi xing layers. The governing equations are based on linear and normal mode representation when the linear wave-packet assu mption breaks down. The resulting wave train to this impulse constitutes a finite number of primary interactions which are analysed in terms of local "absolute" or "convective" instabilities. Both these di sturbances are loca l in nature and affect only loca ll y the mean ve locity profile. Incompress ible and compressible homentropic mi xing layers have been shown to be convectively unstable2

·5

. Jets have

been shown to be absolutely unstable6. Absolute

instabilities have been linked to resonance locati ons of wave packets7

•

Linear stability studies of compress ible mtxtng layers subject to reaction and heat transfer processes have advanced two models of temperature profil es. An algebraic model is used to include the reaction effects8

. In an isentropic system, the pressure eigen­function matches the loci a 2 +yM2 =1 (where a is the x-direction wave number and y is the spec ific heat rati o)8 which fo llows from isentropic stability analys is done earlier8

, and the temperature profil e should be Case (a): T = 1 + b sechM2 (y), -1 < b < oo, b > 0 representing an exothermic process (representing a reaction of a chemical process with heat release) and b < 0 an endothermic process (representing a react ion of a chemical process without heat release, but with heat intake).

Another temperature profile based on flame-sheet approximati on9 was used to model the reacti on effects given by Case (b): T= l+ A tanh (y), - oo <A< oo, A> 0 representing exothermic process and A < 0 representing endothermic process . The temperature profiles are non-dimensionali sed with reference to a reference temperature which we bel ieve is the free stream temperature as the references8

·9 do not

explicitly mention them. Here also, y is the non-

ROYCHOWDHURY: ABSOLUTE & CONVECTIVE INSTABILITY OF MIXING LAYERS 107

dimensional coordinate. This analyisis was done to verify how the absolute and convective instabilities behave in mixing layers. For this analysis the author has considered the exponentially dying disturbances, which have the characteristics of linearly interacting modes, because an exponentially growing mode would interact in a non-linear fashion and one may lose the "linear-stability" information. Linear stability information is the incipient disturbances information, which would carry a long way in understanding the behaviour of absolute and convective instabilities. This effect would also profess itself and dominate the behaviour of non-linear transition and turbulent systems. Using these two temperature profiles representing a non-homentropic system, the impulse response behaviour of Modes I, II and III oscillations is studied.

2 Linear stability operator

For compressible, perfect gas a statistically steady state is assumed for the mass-conservation equation. This assumption holds for low Mach number (~ 0.2) and un-stratified flows. Hence the time averaged mass-conservation equation becomes

V. (p' U) + V. (p u') = 0 ... ( 1)

Here u' and p' are the perturbation velocity vector and density and U _and p are the mean velocity vector and density, respectively. The perturbed field is defined as

u' = (P~ IP) (o\jf'/oy) and

v' = (-P~ IP) (o\j(lox)

where, P~ is the free stream normalizing density and \jl

1 is the perturbation stream-function. Thus Eq. (1) becomes V. (p u') = 0, which means that V. (p' U) = 0. For this to happen the mean flow stream-function should be defined as U = (p~'/p')

(o\jfloy), V = (-P~'/p') (o\jJ/ox), where P~' is any reference density, say a r.m.s. value and \jf is the mean flow stream-function. The linearised Eulerian form of the stability equation are

Continuity: o/ox (pu') + oloy (pv') = 0

X momentum: p [ou'/ot + V1 oU/oy + U ou'/ox]

=- (llyM2) (op'tox)

y momentum: p [ov'/ot + u ov'lox] =- (ltyM2

) (op'toy)

Energy: p [oT'Iot + U oT'!ox + v' oT/oy] =- (y-1) p (ou'/ox +a v'!oy)

Equation of state: p'l p = p' /p +TIT

Combining the linearised Eulerian form of energy equation and the equation of state one gets

p' (ou'/ox + ov'loy) << 1

where p' is the perturbation pressure and T the mean temperature. The linearised Eulerian form of the energy equation becomes

(oT'Iot) + U (oT'!ox) + v' (oT/oy) = 0 .. . (2)

where, T' is the perturbation temperature. Equation (2) is equivalent to DT' /Dt = 0, or T' =constant along a streamline (D = d0t + U o/ox + V d0y), where V = 0 is the y-directional mean velocity. U and V are the x andy directional total velocity components.

The stability equation can be obtained from the x and y directional linearised Eulerian form of momentum equations in the form

A linear operator for the stability equation can be defined as

Impulse response of the linear operator L may be given in terms of a Green's function g(x,y,t;y0 ), which is the real part of the complex Green ' s function G(x,y,t;y0 ), where the subscript 'o' defines the location of the impulse. Here, G is the Hilbert transform of g. This satisfies the condition of exponential decay as y --7 ± oo. Then

L[GJ = { [(otot) + ucatax)JV2G-(d2 VIdl) cactax) l

= [8(x) + (il'f,x) ]o(y- Yo) o(t)

= [8(x) + (i/'f,x)] * g

with G = [8(x) + ( il i;x)] * g ... (4)

where * represents the convolution operator and 1; = 22/7. The Green's function is written in the following form5

108 INDIAN J RADIO & SPACE PHYS, APRIL 2005

where, D stands for the dispersion relationship given

by Eq. (7) and a the wave number. The factors \jJ.+

and \j( are the two solutions of Eq. (3). The method of

steepest descent is applied along with stationary phase

[8 = (a.x-wt)], d8/da = 0 to Eq. (5). The original path

F in the complex a plane, chosen so as to maintain

causality with the path of integration r in the complex

w plane, (defining a branch cut6) is deformed into two

steepest descent paths emanating from the saddle

point (a*, w\ given by clw(a*)/da·= x/t . As t ---7=, the

impulse response leads to6

(2 / )1/2 in /4 - ( ') +( ') i{a'x -w't] . . _ n e \jf y0 ,a \jf y, a e G(.x ,y, t ,yo) = 111

[a· U (y0 )- w' J(ClD(o), a')! Clw )( d2

~· tl­da

(iln)\jf - (yo,O) \jf+( y ,O) el-iw(OJtl

+ -.------~~--~~-----------

[u (y0 )- c] (oD I oc )(c(O), O)[x - c(O)t]

00 .(6)

Thus, amongst all the wave numbers contained in

the impulse source, the flow selects along ray x/t ==

constant, one particular wave number given by

dw*/da* = x/t. The group velocity is real and equal to x/t. The Green 's function takes the form of a wave

packet in the x-t plane and this wave packet

propagates in a wedge bound by two rays. This wave packet includes the ray defining zero group velocity dw*/da* = 0. Outside this wedge all disturbances are

decaying and given by second term of Eq. (6). The wave packet which represents dying modes can be

obtained from first tertT'J of Eq . (6) by real analysi s and hence here a real wave-packet analysis has been done. The line xlt must lie also within this wave packet. For x/t = 0, it follows that any disturbance for which w(a) > 0 (upper half of the w real-plane) represents absolute in stability. The criterion w < 0 a:> a means of identifying a convective instabilitl for frozen flow should be modified . This is because convective instability should be associated with a condition that group velocity is not zero. Hence,

d8/da = x - (dw/da) t = 0, dw/da * 0. In this frame work w < 0 establishes a convective instabi lity criteria when dCJ?Ida * 0. An absolute instability criteria IS

represented by w > 0 for dw/da = 0.

3 Stream functions for this problem

For the two models of non-homentropic flows in mixing layers with different temperature profiles, the absolute and convective instability be!laviour are studied. This study is with reference to three modes of oscillations obtained from isentropic stability analysis 10

, to which developing mixing layers are subjected. They are

(i) n(y) = sech y, which represents a jet mode;

(Mode 1),

(ii) n(y) = sech y tanh y, which represents a

mixed mode of jet and mixing laye r types

(Mode II) and

(iii ) n(y) = sech2 (y), which represents a fu ll y

developed mixing layer mode (Mode III).

where, n(y) is the normal mode pressure fluctuat ion eigen function. For Case (a) temperature profile, the perturbation stream-eigenfunction obtained from the compressibility equations are of the form

(i) For jet mode; (Mode I)

\jf' = (l/yM2) { (l/a2

) sech Yo - i[2 ln(si n Yo)- In

(cosh Yo)]}

(ii) For mixed mode; (Mode II) 7 2 7

- \jf' = ( 1/yM-) { ( 1/a ) [ l-2 tanh- Yo] cosech Yo

+ i[2 (i+2) tan·' (sinh Yo) + 1/sinh Yo]} and

(iii) For mixing layer mode; (Mode III)

\jf' = ( l/yM2) { ( -2/a2

) sech2 Yo-:- (2 i/a 2 -1 ) In Yo

- i Yo2/a2]

where, Yo = (l+b) y

For Case (b) temperature profile, stream-eigen functions are of the form

(i) For jet mode; (Mode I)

\jf' = ( l/yM2) (l/a2

) sech y/(1 +A tanh y) +

(i/yM2) {[A./(1-A.)] [(1/A.) In (l+A tanh y )­

(l/2A.) In (sech2 y) -( l/2) In [(1-t-A. tanh y) /

(1-A. tanh y)]] }

(ii) For mixed mode; (Mode II)

\jf' =- (l/yM2) (lla2

) (l-2 tanh2 y)

cosech y/(1 +A tanh y)

- (i/yM2) J {[- (i+2) sech y + coth2 y sec h y ll

(l +A. tanh y) } dy and

ROYCHOWDHURY: ABSOLUTE & CONVECTIVE INSTABILITY OF MIXING LAYERS 109

(iii) For mixing layer mode; (Mode III)

\jf' = (l/yM2) [2 sech2 yl( 1 +A. tanh y)] (l/a2

)

+(l/yM2) { [2i In [tanh y/(1 +A. tanh y)]

-(2i/A.) [A. tanh y -In ( 1 +A. tanh y)]

-In [tanh y/(1 +A. tanh y)]}

Using these forms of stream-eigen functions, the dispersion relationship is obtained from Eq. (3). It takes the following form

w =a (tanh Yo+ BJAo)

awJaa may be estimated as

awJaa =tanh y + {Ao Bo +a (Ao dBolda­

BodAo/da)}/[Aof

... (7)

... (8)

The explicit forms are not given here, but they are not difficult to obtain.

For Cases (a) and (b), the coefficients A0 and Bo corresponding to the three modes of oscillations are obtained from the rel ations of stream-eigen functions. Using this dispersion relationship, the impulse response behaviour has been studied. In solving the dispersion relation , a polynomial equation for a, generally of order 10 or less is encountered. The roots

-2

OA

0.0

t -0.4

~

-o.s

0.4

l\1 I

-2 o.o 1\ r t / v .... -0.4

'"' -0.8

4 5

Y-

4 5

y-

of the polynomial equation are obtained using a real root finding algorithm.

4 Results and discussion

A convective frame with dw/da = 1.0 and an absolute frame with dw/da = 0.0 are chosen. Mode I, II and III oscillations for Cases (a) and (b) temperature profiles are studied in both absolute and convective frames. Non-dimensional coordinates have been used for representing the results. Further, as seen later on in this paper the resolution was not so clear so as to allow for a quantitative analysis of the growth rates. This is because the analysis was done for dying disturbances and the non-dimensional representation causes it. Further analysis using this method incorporating these deficiencies needs to be done. Figure 1 represents the study of Mode I oscillations for Case (a) temperature profile under absolute and convective frames. In an absolute frame for Mode I oscillations, it is observed that w > 0 for major regions of the flow , indicating dominant absolute instability of jet flows. Exothermic processes make the flow less absolutely unstable. In a convective frame, for Mode I oscillations, w > 0 for major regions of the flow , indicating that major regions of the flow are not convectively unstable. Hence, jet

0.4

-2 0.0

'-0.4

~

-0.8

0.4

-2 0.0

f -0.4

3

-O.S

\ \

\

2 4 5

Y-

~=-1 .0

\~----~ , z"7 4 5 " /

... ...._ __ ... _....-

Y-

~:=-0.0

Fi g. I -Compari son of a, and w loci for Case (a) temperature profile and Mode I osci ll ati ons in absolute and convective frames

[------ b = -0.95. -- b = 0.0,--- b = 2.0]

110 INDIAN 1 RADIO & SPACE PHYS, APRIL 2005

mode is dominantly absolutely· unstable. Exothermic processes make the flow less absolutely unstable. Hence, jet mode has dominant form of absolute instability and exothermic processes reduce this instability.

Figure 2 represents the study of Mode II oscillations for Case (a) temperature profiles under absolute and convective frames. In an absolute frame, for Mode II oscillations only some regions of the flow have w > 0, where y varies from 1.5 to 2.0. Exo­thermic processes marginally enhance the absolute instability behaviour. This cannot be seen clearly in the picture, as the scale for which they are plotted does not resolve this abovesaid effect. Under convective frame, for Mode II oscillations it is found that in some portions of the flow between y = 0.5 and 1.0 a convective instabil ity exists. It may be observed that the zones of absolute and convective instabilities do not overlap each other. Hence, Mode II oscillations have a mixture of absolute and convective instabilities. Figure 3 represents the study of Mode III oscillations in absolute and convective frames . In an absolute frame it is observed that some local pockets of absolute instabilities are observed fbr y between 1.5 and 2.5 and from 2.8 to 3.2. In a convective frame, it is found that for major portion of the flow ( + y direction), w < 0, which indicates a dominant con­vective instability behaviour. Hence, Mode III oscillations have a dominant convective instability

0.4 0.4

-2 0.0

~· 5 _,, \ 4 5

I -.' -O'l \

Y-"'o ·0.4

0 ' >< - '

>< \ .... 3 -0.8 ' ·0.8 d<.>

-1.0 -1.0 d<-=1.0 _j

0.50 4

-2 0.0 \' 5 -2, 0_0 v 2 4 5 ) - "' 0 Y-~ -0.50 - -4 >< ~ >< ~ \ 3

'6

-1 .0 '

~=DO -8

d« "

Fig. 2- Comparison of a, and w loci for Case (a) temperature profile and Mode II oscillations in absolute and convective frames . [ ------b = -0. 95 , __ b = 0.0, ___ b = 2.0)

behaviour with local pockets of absolute instabilities, representing the roll-up process of the shear layer.

Figures 4, 5 and 6 represent Mode I, II and III oscillations studied in both absolute and convective frames, for Case (b) temperature profi I e. Mode I oscillations in an absolute frame (Fig. 4) show that the flow (y from 1.5 to 2.5) is absolutely unstable. Exothermic processes make the flow less absolutely unstable. In a convective frame, Mode I oscillations (Fig. 4) have large regions of convective instabilities. Hence, Mode I oscillations are dominantly convectively unstable with some pockets of absolute instabilities. Mode II oscillations in an absolute frame (Fig. 5) show that large regions of weak absolute instability exist (w > 0 but small in magnitude). Exothermic processes make the flow lose its absolutely unstable character. In a convective frame (Fig. 5), it is seen that the flow is convectively unstable for endothermic case, but loses its convectively unstable behaviour for the exothermic case. Mode II oscillations are thus weakly absolutely unstable. Mode III oscillations in an absolute frame (Fig. 6) show that the flow is not absolutely unstable as it is seen that w < 0 for both exothermic and

0.5

0 _4

0 _2

-2 t 0 0 ~

'(/ -o 2

-0_4

1.2

0_8

0_4

-2 0.0

l - 0_4 ~ ....

-0_8

-1.2

ii I I

I\

(\ ' ' I I I I I I

' I I ' ' ' I ' I{\ I II 1

" ' ii ,_

.. -2

'-~- -5

' ' Y-I I

' ' I ' ' I d<-> ' do(=D.O I

1.0

0.8

3 0.4

-2 0.0 4 5

-0_4 Y-

-0_8

1.2

0_8 \ 3

0.4

-2 0_0

-0_4

-0.8

-1 .2

Fig. 3- Comparison of a, and (I) loci for Case (a) temperature profile and Mode Ill oscillations in absolu <e and convective

frames. [ ---='--b = ---0.95 , __ b=O.O, ____ b = 2.0]

ROYCHOWDHURY: ABSOLUTE & CONVECTIVE INSTABILITY OF MIXING LAYERS 111

endothermic cases. In· a convective frame (Fig. 6), Mode III oscillations show large regions of convectively unstable flow and endothermic processes enhance this instability. This is in accordance with earlier work8

.

5 Conclusion

Since the temperature profiles represent symmetri"c and asymmetric (dancing flame) types, the impulse response behaviours are markedly different for t~ese . profiles. ·Furthermore, the asymmetric temperature profile describes a flame front reaction unlike the symmetric one. Hence, the symmetric profile should display characteristics of result of previous researchers who have not considered a combustion process. For non-combusting heat transfer processes, the symmetric profile shows that the jet mode (Mode I) exhibits a dominant absolute instabilities behaviour, mixed mode (Mode II) exhibits a mixed behaviour of absolute and convective instabilities, while fully developed mixing layer mode (Mode III) exhibits a dominant convective instability behaviour with pockets of absolute instabilities. These pockets of absolute instabilities describe the roll-up

Fi g. 4 - Comparison of a, and w loci for Case (b) temperature profile and Mode I oscillations in absolute and convective frames.

[ ------ ), = - 0.5 , --), = 0.0,--- A.= 0.5]

phenomenon of the m1xmg layer. The mixed mode represents an intermediate stage between jet mode and fully developed mixing layer mode. These results confirm ihe previous results on fully developed mixing layers, which are found to be convectively unstable5 and jets are absolutely unstable2

.

For both types of temperature profiles (symmetric and asymmetric) it is seen that . for weakly compressibi~Y flows, endothermic process enhances and exothermic process reduces the growth rates respectively, for only convective · instabilities. The convective instability behaviour is however like our previous study of fully compressible flows studied under non-isentropic, non-homentropic considerationt ,to.t t. This analysis could help in further understanding transition and turbulence 12

·13

.

Finally, the author would like to point out that he has recently found success in solving the self adjoint stability equation with reference to acoustic noi se prediction for highly under-expanded jets 14

• These jets were shown to reproduce the very quick self­similarity behaviour in a very short di stance from the ex it of the nozzle (within 2.5 m). This was possible, only because, he has used the " radiative boundary condition" inherent in basic solution strategy 14 used to solve the stability equations. This method could also be extended to non-self adjoint equations.

I : t ~~ I

- 2

"f -2

-4

41

.... ~ 2

,, , , 'I

-2 00

~{ -2

:l ,. - 4 :I

d<.> dtt :1 .0 4

3

· .~ -4 5 -2 00 · .. .v.-:-2 ··· i.· 5

Y- Y -

4 5

Y-

-2

-4

I ''~\ - \ 3 o. 4~ I '-

' ' \ 2 '

-2 0.0 ~5

0.4 I 0.8

Fi g. 5- Compari son of a, and w loci for Case (b) temperatu re profile and Mode II oscillations in absolute and convecti ve

frames . [ ------ ), = -0.5, __ A. = 0.0, ___ A.= 0.5]

112 INDIAN J RADIO & SPACE PHYS, APRIL 2005

dG.> 1.2 1.2 d~=l.O

\ . L.. --·- \ ~ 0.8 3 O.B

0 .4 0.4

-2 0.0 4 5 -2 0.0 5

Y--0.4 - 0.4

L..

~

-2

0.4 O.B dw

I dl( =0.0

0.2 (},4 3 .

I•

- --~

,, I

0.0 2 ' - -2 0.0 I t2 ' . 4 5 4 5 ' I I ' y~ ' ~Y-

-0.'2 I 0.4 ;., I /1 ,

/ I \' / .

\IV /. I I /' ,, ,... , I It /•_./

-0.4 ( : v 0.8 (;!· •,/

Fig. 6- Comparison of a, and w loci for Case (a) temperature profile and Mode III oscillations in absolute and convective

frames. [ ------ A= -0.5 , __ A= 0.0, ___ A= 0.5]

Nomenclature \jf' -perturbation stream function

a -x direction wave number (complex) of wave (.L) -frequency of the wave vector

vector .

-frequency of wave vector at saddle point (.L)

• -x direction wave number at saddle point a b -parameter defining exothermic and () -Dirac delta function endothermic processes in temperature profile

y -specific heat ratio of the gas flow for Case (A) model

A. -parameter defining exothermic or endothermic D -dispersion relationship obtained using the two

processes (Case B temperature profile) models

TC -normal mode eigen-function of perturbation g -Greens function

pressure p G -Hilbert transform of g

p -mean density of the mixing layer flow - imaginary part of complex number which is

p· -perturbed density component of mixing layer equal to the square root of ( -1)

flow M -Mach number of the mixing layer related to

P«> -free stream normalizing density Case (A) and Case (B) models

\jf -mean flow stream function 0 -subscript 'o' defines location of impulse

ROYCHOWDHURY: ABSOLUTE & CONVECTIVE INSTABILITY OF MIXING LAYERS 11 3

p -mean pressure

p -perturbed pressure component

-time component

T -temperature profile re lated to Case (A) and Case (B) mode ls

U -velocity vector (total component)

U -x directi onal mean ve loc ity component of velocity vector

tt -x direc ti onal perturbed ve locity component

v' -y directional pe rturbed velocity component

x -x directi on of flow mode l

y -y direc ti on of fl ow model

+ -superscript+ representing one of the two solutions of \If solution of the di spersion re lationship

-superscript- representing one of the other of two solutions of \If solution of the dispersion re lationship

Acknowledgements

The author would like to thank Drs Pradeep

Kumar, R Balu , A N Subash, A K Verma and V

Adimurthy of V S S C for he lping him with this work.

The author is very thankful to the reviewers for their

meticulous analysis of this paper and their comments

have helped him to modify and improve this paper.

Finally the author would like to thank the Director

and Chairman ISRO of VSSC for the ir

encouragement and support for this work.

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