It is shown that there exists a family of supersonic neutral modes for a compressible mixing layer in an unbounded domain. These modes have zero wavenumber and frequency with nonzero phase speed. They are analogous to the supersonic neutral modes of the compressible vortex sheet found by Miles. The results presented give a more complete picture of the spectrum of the disturbances in this flow

Grosch, C. E.

Jackson, T. L.

English

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NASA Contractor Report 187476
ICASE Report No. 90-82
ICASE
ZERO WAVENUMBER MODES OF A COMPRESSIBLE
SUPERSONIC MIXING LAYER
T. L. Jackson
C. E. Grosch
Contract No. NAS 1-18605
December 1990
Institute for Computer Applications in Science and Engineering
NASA Langley Research Center
Hampton, Virginia 23665-5225
Operated by the Universities Space Research Association
N/ A
N_lionnI Aeronaulic._ and
Space Adminislralion
L_ngley Re_enrch Center
I_tampton, Virginia 23665-5225
0 4-
Oc"J
https://ntrs.nasa.gov/search.jsp?R=19910004356 2020-03-19T20:56:38+00:00Z

ZERO WAVENUMBER MODES OF A COMPRESSIBLE
SUPERSONIC MIXING LAYER
T. L. Jackson
Department of Mathematics and Statistics
Old Dominion University
Norfolk, Virginia 23529
C. E. Grosch
Department of Oceanography and
Department of Computer Science
Old Dominion University
Norfolk, Virginia 23529
Abstract. In this paper we show that there exists a family of supersonic neutral
modes for a compressible mixing layer in an unbounded domain. These modes have zero
wavenumber and frequency with non-zero phase speed. They are analogous to the super-
sonic neutral modes of the compressible vortex sheet found by Miles. The results
presented here give a more complete picture of the spectrum of the disturbances in this
flow.
This work was supported by the National Aeronautics and Space Administration under NASA Contract
NAS1-18605 while the authors were in residence at the Institute for Computer Applications in Science and
Engineering, NASA Langley Research Center, Hampton, VA 23665.

-1-
1. Introduction. In recent studies (Jackson and Grosch, 1989, 1990), we considered
the inviscid spatial stability problem for the compressible mixing layer in an unbounded
domain. The key to understanding the stability characteristics of this flow is the under-
standing of the different parameter regions for which various types of instability modes
can exist. In Figure 1 we show a plot of the sonic phase speeds c+ versus the Mach
number, where
1 _/_-_-T
c+=l M' c =13t/+ M
Here, M is the Mach number of the fast stream, 13u the ratio of the speed of the slow
stream to that of the fast stream, and _r the ratio of the temperature of the slow stream to
that of the fast stream. These curves divide the phase speed-Mach number plane into four
regions. If a disturbance exists with a Mach number and phase speed in region 1, the dis-
turbance is subsonic at both boundaries, and we classify it as a subsonic mode. In region
3, the disturbance is supersonic at both boundaries, and we classify it as supersonic-
supersonic mode. In region 2, the disturbance is subsonic in the fast stream and super-
sonic in the slow stream, and we classify it as a fast mode. Finally, in region 4, the dis-
turbance is supersonic in the fast stream and subsonic in the slow stream, and we classify
it as a slow mode. We found that there is only a single subsonic neutral mode for two
dimensional waves. Beyond a critical Mach number M s, the Mach number at which the
phase speed equals that of a sonic wave, the subsonic neutral modes are transformed into
supersonic neutral modes which are subsonic in one freestream and supersonic in the
other. In addition, another supersonic neutral mode appears at M, > M s, the Mach
number at which the sonic speeds of the fast and slow streams are equal. This supersonic
neutral mode has the opposite behavior than the previous. That is, if the continuation of
the subsonic neutral mode is supersonic in the fast stream and subsonic in the slow
stream, this new mode is subsonic in the fast stream and supersonic in the slow stream.
In these studies we did not find any neutral or unstable modes in region 3 which
were supersonic in both streams. In this paper we now show that there exists a family of
neutral modes in region 3 having zero wavenumber and frequency with non-zero phase
speed 1. The approach taken here follows that of Papageorgiou and Smith (1989) and
Papageorgiou (1990), who considered long waves in the wake behind a flat plate. We
show that these modes are analogous to the supersonic neutral modes of the compressible
vortex sheet found by Miles (1958). The results presented here give a more complete pic-
ture of the spectrum of the disturbances in a compressible mixing layer.
2. Formulation and Results. We consider here a compressible mixing layer in an
unbounded domain with mean velocity U(rl) and temperature T(rl), where 1'! is the simi-
larity variable in terms of the downstream distance and the Howarth-Dorodnitzyn variable.
With suitable normalization, U varies between 1 in the fast stream (rl = oo), and [_u < 1 in
This possibility was suggested to us by R. Klein.
-2-
the slow stream01=-oo). The temperatureT varies between 1 in the fast stream and
13_-> 0 in the slow stream. The results presented here are independent of the detailed form
of U and T.
The mean flow is perturbed by introducing two-dimensional wave disturbances. The
disturbance equation for the normal velocity perturbation _p is given by (Lees and Lin,
1946)
[ ](v - c)(- v'_ - a2(u - c)¢ = o, (1)G
where primes denote differentiation with respect to the similarity variable 11, and G is
given by
G =T[T-M 2(U-c)2]. (2)
Here, ot is the wavenumber of the disturbance in the downstream direction and c is the
corresponding phase speed.
Since we are interested in long wavelength perturbations (or--_ 0), the appropriate
expansions are found to be
I_ = _0 + 0t_)l + 0_2¢2 + " " " (3)
c =c o+otc 1+0_2c2+ "'" (4)
1 1
G Go
+ OtGl + ot2G2 +
where
G O = T [T - M2(U - c0)2],
(5)
(6)
G I=-2clM2G_ 2(U -co)T. (7)
Let us now define the linear operator L :
L_F) = (U - Co)W'- U'W =- (U - Co) 2 (8)
Substituting the above expansions into (1) gives the following three leading-order prob-
lems:
-3-
d (G6_l L(O0)) = 0,
dR
(9)
• d
d (G_d [L(¢l) _ c1¢0]) + .____ (G1L(¢o)) = 0,dR
(10)
d
dR
_(G_ "1 [L (¢2) - c2¢ 0 - c1¢1] ) + (G1 [L(01) - c1¢0] )
U|l
+ d (G2L(¢o))-(U-co)¢o =0"
a J!
(11)
The appropriate solution to (9) that is bounded for large 1"1is given by
¢o = Ao(U - Co), (12)
where A o is a constant. Based on this solution the following three remarks can be made:
[11 Since solutions in region 2 of Figure 1 are subsonic in the fast stream and
supersonic in the slow stream, we see that _0 is a solution if we take the
leading-order wave speed to be
c o = 1. (13)
f2| Since solutions in region 4 of
supersonic in the fast stream,
leading-order wave speed to be
Figure I are subsonic in the slow stream and
we see that ¢_0 is a solution if we take the
Co = _u. (14)
[3] Since solutions in region 3 of
slow streams, we see that ¢0
wave speed lying in the range
Figure 1 are supersonic in both the fast and
is a possible solution for some leading-order
c_ < Co < c+, (15)
where c+ are the phase speeds of the sonic neutral modes. Thus, the actual
value of c o is not determined at this order, and can only be found by using
the higher orders terms in the a expansion.
To determine the leading-order wave speed c o for region 3, we consider the expansions at
the higher orders. The solutions to (10) and (11) are given by
Ol=_ClAo+Al(U_co)I Go
(U - Co) 2 dR,
(16)
-4-
(_2 = -C2Ao + C l (U - Co) f ¢1
(U - Co) 2 drl
_al(U_co)_ G2GI
(U - Co) 2 d'q + A2(U - Co) _
Go
d
(U - Co) 2
lq
Go [ f(U-co)2ds ] d1]. (17)
+ A°(U - Co)f (V -_ Co) 2
We note here that both (_l and 42 could contain another term that is proportional to %,
which can be ignored by a suitable renormalization of A 0.
Inspection suggests that the solution diverges as 1"1----)+o,,. Thus, these expansions
will break down when Ill ] = O ((x-l). There are tWO outer regions, one in the fast stream
and one in the slow stream, which will be considered separately, For the outer region in
the fast stream, the appropriate scalings are
for the independent variable, and
1] = O_-1 Z (18)
@F =20+0t21+ ''" (19)
for the dependent variable. Substituting (18) and (19) into our original equation (1) yields,
to leading order, the equation
2o + Gd So =0 , G d =M2(1-Co) 2 1 (20)
The solution for outgoing waves is found to be
2o = Bo e-i _ z (21)
The next order equation is given by
+GJ =g 2o, gi _ =2ctM2(1-Co) , (22)
whose solution for outgoing waves is
g{Bo
21 B1 e-i G4-ff-_z + i= __ z e (23)
24G_
Expanding the solution as z _ 0, and matching with the inner solution (3) as rl --_ ,_,
yields the following relationships
-5-
B o = Ao(1 - Co) ,
B 1 =-ClAo,
A l = i Ao(1 - CO)2(G_) -1/2,
A2=i
1 -c o
(G$ )3a
c 1 (2 - M2(1 - Co)2)Ao .
For the outer region in the slow stream, the appropriate scalings are
*s = + + ...,
and z is defined as before.
order, the equation
^ tl¢0 + Gff _0 = 0, G0- = _T [M2(_U - c0) 2 _f]"
The solution for outgoing waves is found to be
-i G'_-z
_0 = Doe
The next order equation is given by
_i'+ G_- _1 = gi- _0,
whose solution for outgoing waves is
-/G_-_'z
_1 =Die
(24)
(25)
(26)
(27)
(28)
Substituting into our original equation (1) yields, to leading
(29)
(30)
gf = 2ClM215T (_U - Co), (31)
• g{Do -i C'_-_z
+l ze
2,g-do
(32)
Expanding the solution as z --_ 0, and matching with the inner solution (3) as 1"! --¢ -o_,
yields the following relationships
D O = AO(_U - Co) , (33)
D 1 =-ClAo, (34)
A 1 = i Ao(_v - c0)2(G_) -1/2, (35)
-6-
13u -- C o
A2= i -_)_-_ _I, c1(2[5r - M2(_U - CO)2)Ao . (36)
The two conditions (26) and (35) on A t yields the following eigenvalue equation for
CO:
_-,r [M2([iU - c0) 2 - _T ] (1 -- C0) 4 = [M 2 (1 - c0) 2 - 1 ] (_U - Co) 4" (37)
This equation is identical to (5.3a) of Miles (1958) if we re-express his result in our nota-
tion. Miles showed (in our notation), that for region 3:
[1] A single root of (37) exists for
I+V 
M > M, = , (38)1-13u
with phase speed
Pu+
Co - , (39)I+V 
which we classify as a constant speed supersonic-supersonic neutral mode.
Note that this solution is independent of Mach number, and corresponds to the
phase speed at which the two sonic speeds c± are equal.
[2] A double root first appears at
(1 + 13./Y3)3/2
M = 1 - 13u (40)
with phase speed
13u + 13,_/3
Co - (41)
1 + [3}rj
[3] There are three distinct real roots for
(1 -t- 1_/1/3)3/2
M> 1- Ih, (42)
One of the roots is given by (39), while the other two roots must be found
numerically. For the special case of _y. = 1, these roots are given by
-7-
l+_u ::1:: 1 I ] 1/2Co - 2 _ M2(1 - 13U)2 + 4 - 4N]M2(1 - _u) 2 + 1 . (43)
The root which corresponds to the (+) sign we classify as a fast supersonic-
supersonic neutral mode, while that which corresponds to the (-) sign we clas-
sify as a slow supersonic-supersonic neutral mode.
In Figure 2 we display the real roots of (37) which lie in region 3 for 13U = 0 and
various values of [_r- The roots are the neutral phase speeds. For completeness, we also
display the neutral phase speeds which lie in the other regions (Jackson and Grosch,
1989). The classification scheme is given in the figure caption.
The two conditions (27) and (36) on A2 yield the result that c 1 = 0. Thus, we have
A 2 = B 1 = D 1 = 0, and the final solution in region 3 is now given by
t_ = Ao(U - Co)
(1 - c0) 2 G O
l+iot ___ I (U _ co)2 drl +
o0 [ i(U_co)2ds } an+ .a2J"(v - 2 (44)
3. Conclusions. We have shown that the compressible mixing layer has a family of
supersonic neutral modes in region 3 of Figure 1, which have phase speeds which are
identical to those of the compressible vortex sheet (Miles, 1958). For the mixing layer the
wavenumbers of these modes are zero, while those of the vortex sheet are arbitrary. One
might expect this since the vortex sheet has no natural length scale, while the mixing layer
has a non-zero thickness. We note that these results can be extended to three dimensional
disturbances by rescaling the Mach number by cos 0, where 0 is the angle of propagation.
Finally, Artola and Majda (1987, 1989) showed that the nonlinear interaction of the three
supersonic neutral modes of the compressible vortex sheet can lead to instabilities of the
flow. We surmise that similar behavior might be exhibited by the interaction of the
supersonic-supersonic neutral modes presented in this paper. We are currently investigat-
ing this numerically.
Acknowledgements. The authors are indebted to R. Klein for suggesting that a
supersonic-supersonic neutral mode might exist in the mixing layer.
-8-
REFERENCES
Artola, M. and Majda, A.J. (1987) Nonlinear Development of Instabilities in Super-
sonic Vortex Sheets. Physica D, 28, pp. 253-281.
Artola, M. and Majda, A.J. (1989) Nonlinear Development of Instabilities in Super-
sonic Vortex Sheets II: Resonant Interaction Among Kink Modes. SIAM J. Appl.
Math., 49, pp. 1310-1349.
Jackson, T.L. & Grosch, C.E. (1989) Inviscid Spatial Stability of a Compressible
Mixing Layer. J. Fluid Mech., 208, pp. 609-637.
Jackson, T.L. & Grosch, C.E. (1990) Inviscid Spatial Stability of a Compressible
Mixing Layer. Part III. Effect of Thermodynamics. J. Fluid Mech., in press.
Lees, L. & Lin, C. C. (1946) Investigation of the stability of the laminar boundary
layer in a compressible fluid. NACA Tech. Note 1115.
Miles, J. W. (1958) On the Disturbed Motion of a Plane Vortex Sheet. J. Fluid
Mech., 4, 538-552.
Papageorgiou, D.T. (1990) Linear Instability of the Supersonic Wake Behind a Flat
Plate Aligned with a Uniform Stream. Theoret. Comput. Fluid Dynamics, 1, pp.
327-348.
Papageorgiou, D.T. and Smith, F.T. (1989) Linear Instability Of the Wake Behind a
Flat Plate Placed Parallel to a Uniform Stream. J. Fluid Mech., 208, pp. 67-89.
_g-
1.01
c, 0.5
Fast modes
9 = co
subsonic
2  up0rso.ic ,t" t
I 3
Sub su_ c_
/ supersonic _1= co
subsonic r/= - oo
L_____ L..... ] _L______t_______t I _J t I ...... _1
5 10
M
Figure 1. Plots of the sonic speeds c+ versus Mach number showing the four regions
in which different types of disturbances can exist.
-10-
0
O
t9
q
3.
\
/ 3
S-I I I I I
/'4
I
\
\
/
/
I S l I I I I
M
I
7
\
Figure 2. Plot of the neutral phase speeds as a function of Mach number for 13v = 0
and (a) Dr = 2, (b) 13w = 1, and (c) [3I, = 0.5. The neutral mode classification is: (1)
subsonic, (2) fast (supersonic), (3) slow (supersonic), (4) cN = 1, (5) CN = 0, (6) con-
stant speed supersonic-supersonic, (7) fast supersonic-supersonic, and (8) slow
supersonic-supersonic. The sonic curves are shown as dashed.


Nat_ai A_",aut_5 a_
SP_: e L_Or_q_StrabOC
I. Report No. 2. Government Accession No,
NASA CR- 187476
ICASE Report No. 90-82
4. Title and Subtitle
ZERO WAVENUMBER MODES OF A COMPRESSIBLE SUPERSONIC
MIXING LAYER
7. Author(s)
T. L. Jackson
C. E. Grosch
Report Documentation Page
9. Performing Organization Name and Address
Institute for Computer Applications in Science
and Engineering
Mail Stop 132C, NASA Langley Research Center
Hampton, VA 23665-5225
3. Recipient's Catalog No.
5. Report Date
December 1990
6. Performing Organization Code
8. Performing Organization Report No.
90-82
12. Sponsoring Agency Name and Address
National Aeronautics and Space Administration
Langley Research Center
Hampton, VA 23665-5225
10. Work Unit No.
505-90-21-01
11. Contract or Grant No.
NASI-18605
13. Type of Report and Period Covered
Contractor Report
14. Sponsoring Agency Code
15. Supplementa_ Notes
Langley Technical Monitor:
Richard W. Barnwell
To be submitted to Physics
of Fluids A
Final Report
16. Abstract
In this paper we show that there exists a family of supersonic neutral modes
for a compressible mixing layer in an unbounded domain. These modes have zero wave-
number and frequency with non-zero phase speed. They are analogous to the super-
sonic neutral modes of the compressible vortex sheet found by Miles. The results
presented here give a more complete picture of the spectrum of the disturbances in
this flow.
17. Key Words(SuggestedbyAuthor(s))
mixing layer, stability
19. SecuriW Cla_if. (of this report)
Unclassified
18. Distribution Statement
34 - Fluid Mechanics and Heat Transfer
59 - Mathematics and Computer Sciences
(General)
Unclassified - Unlimited
_. Securi_ Cla_if. (of this page) 21. No, of pa_s
Unclassified 12
T
22. Price
A0 3
NASA FORM 1626 OCT 86 NASA-L_Iey, 1990



Zero wavenumber modes of a compressible supersonic mixing layer

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