NASA Contractor Report 182050

ICASE Report No. 90-39

ICASE

THE STABILITY OF TWO.DIMENSIONAL WAKES AND

SHEAR.LAYERS AT HIGH MACH NUMBERS

D. T. Papageorgiou

_(_ASA-CR-]0205_) T_C STA_ILI TY O_

TWO-_IMFN_InNAL wAKES AND SH_AR-LAYCRS AT

HIGH MACH NUMBERS Fin_l Report (TCAS_)

27 p CSCL 200

G3/34

NQO--Z3oTO

uncl_s

0281001

Contract No. NAS1-18605

May 1990

Institute for Computer Applications in Science and Engineering

NASA Langley Research Center

Hampton, V'u'ginia 23665-5225

Operated by the Universities Space Research Association

IXl/kSA

National Aeronautics and

Space Adminislration

Langley Research Center

Hampton, Virginia 23665-5225

THE STABILITY OF TWO-DIMENSIONAL WAKES

AND SHEAR-LAYERS AT HIGH MACH NUMBERS

D. T. Papageorgiou 1

The Levich Institute

City College of CUNY

Convent Avenue at 140th St.

New York, NY 10031

ABSTRACT

This study is concerned with the stability properties of laminar free shear-layer flows,

and in particular symmetric two-dimensional wakes, for the subsonic through the hypersonic

regime. Emphasis is given to the use of proper wake profiles that satisfy the equations of

motion at high Reynolds numbers. In particular we study the inviscid stability of a developing two-dimensional wake as it accelerates at the trailing edge of a splitter plate. The

non-parallelism of the flow is a leading order effect, and the undisturbed state is solved numerically. The neutral stability characteristics are computed numerically and the hypersonic

stability is obtained by increasing the Mach number. It is found that the neutral stability

characteristics are altered significantly as the wake develops. Multiple modes (second modes)

are found in the near-wake (they are shown to be closely related to the corresponding Blasius ones), but as the wake develops mode multiplicity is delayed to higher and higher Mach

numbers. At a distance of about one plate length from the trailing edge, there is only one

mode in a Mach number range of zero to twenty. The dominant mode emerging at all wake

stations and for high enough Mach numbers is the so-called vorticity mode, which is centered

around the generalized inflection point layer. The structure of the dominant mode is also

obtained analytically for all streamwise wake locations and it is shown how the far-wake

limit is approached. Asymptotic results for the hypersonic mixing layer given by a tanh and

a Lock distribution are also given.

1Research was supported by the National Aeronautics and Space Administration under NASA Contract

No. NAS1-18605 while the author was in residence at the Institute for Computer Applications in Science

and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665.

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1. Introduction.

Recently there has been a renewed interest in hypersonic flows, mainly in an effort towards the

achievement of hypersonic flight. At such high Mach numbers the stability or instability of the flow

has many important consequences. For example it is desirable to reduce drag in external aerodynamical

situations in order to increase performance, while it is imperative to increase instability in internal

flows such as the scram jet-engine for example, so that mixing and thus propulsion are increased.

Both the examples cited above contain certain fundamental aspects of wake flow instabilities and in

this study a laminar two dimensional high-Reynolds-number wake is analyzed for stability with

emphasis on the hypersonic limit.

The inviscid instability of a two-dimensional laminar compressible wake at high Reynolds

number, was studied analytically and computationally by Papageorgiou (1989a). The general finding is

that the flow stabilizes considerably as the Mach number is increased, with the largest spatial growth

rates being attained for streamwise stations near the trailing edge. To give a numerical example, the

maximum growth rate at a non-dimensional distance of 0.001 from the trailing edge, decreases from

about 0.25 at zero Mach number to about 0.03 at Mach 3; at Mach 9 it decreases to 0.002. Similar

drastic reductions in the instability are observed at all streamwise locations (see Papageorgiou (1989a)).

It is clear, therefore, that the hypersonic limit merits a careful study for all streamwise locations. In

particular the neutral stability curves give a wealth of information about the dominant physics, such as

multiplicity of modes, mode crossing and dominant length-scales (for the related Blasius boundarylayer flow, see for example Mack (1987) and references therein, Cowley and Hall (1988), Smith and

Brown (1989), Blackaby et al. (1990)).

Of great relevance to the present work, are the related studies of boundary- and shear-layer stability and in particular their hypersonic limit. Balsa and Goldstein (1989) consider hypersonic mixing

layers, and in particular tanh velocity distributions, while Smith and Brown (1989) and more recently

by Blackaby et al. (1990), consider the Blasius boundary-layer for a Chapman and Sutherland viscosity

law fluid respectively. From all these studies, it emerges that the dominant mode of instability at

hypersonic Mach numbers is the vorticity mode. This mode is localized around the generalized

inflection point and decays away from it. Consequently, the majority of the boundary layer is passive

in the description of this mode and the structure of the mode together with the neutral eigenvalues are
2

solely dependent on the asymptotic form of the basic flow far away from the wall (the inflection point

is pushed to infinity as the Mach number becomes asymptotically large). Such a localized structure was

also found by Hall and Fu (1989) in their study of hypersonic Gortler vortex instabilities. The asymptotic behavior of the basic flow at large distances from the wall becomes a function of the viscosity

law used in the compressible boundary-layer equations. Smith and Brown use the Chapman viscosity

law, while Blackaby et al. use the Sutherland model. In the former case the neutral wavespeed of the

vorticity mode behaves like di-_, while in the latter it goes to an order one constant at infinite

Mach numbers. It is also worth noting that Smith and Brown found an exact solution to the reduced

vorticity mode eigenvalue problem; the vorticity equation is a canonical equation, therefore, and applies

to a range of flows. We show how it can be used to describe the hypersonic stability of a developing

wake and also present the vorticity mode analysis for mixing layers and obtain the same result as in

the analytical study of Balsa and Goldstein (1989); comparisons are also made with the numerical

study of Jackson and Grosch (1989a).

In the present problem we work with the Chapman viscosity law. The case of the Sutherland law

gives a more complicated base-flow and is still to be studied. A cartesian frame of coordinates is used

which is fixed at the trailing edge x= 0. The flow quantifies p, u, v, w ,p ,T, I-t denote nondimensional density, velocities in the x, y and z directions respectively, pressure, temperature and

viscosity coefficient; non-dimensionalizations are made with regt_ct to free-stream values denoted by a

U,L

subscript _. The Reynolds number is defined as R, =- where L is the plate length, the Mach

V.

number is M = mU" where a is the speed of sound, and the Prandtl number is Pr = __P_"cp wherer cp is

a k

the specific heat at constant pressure and k is the coefficient of thermal diffusivity. The Reynolds

number is taken to be asymptotically large throughoutand the flow is assumed to be laminar. We also

make the assumption of a model fluid with a linear viscosity law, i.e. Pr = 1 and g = C T (Chapman's

law), where C is a constant that is taken to be 1 the equation _t = T is used.

9

2. The governing equations.

If the Dorodnitsyn-Howarth transformation g ..... is employed (see Stewartson (1964)), the

undisturbed flow which is solely in the x-direction to leading order (the transverse velocity is of order
-3-

R;t/2), satisfies the following nonlinear system :

u-=vr , uuz-¥zur=urrr , (1)

_(x,r)= I + -_(v--1)M _ (l-if) (2)

Equations(I),(2)willbe used inx > 0; (2)isan exactintegralof theenergy equationforunitPrandtl

numbers under theCrocco assumption(Stewartson(1964)).The boundary conditionsin thewake are,

u-(x,**)= i , _(x,O) _-r(x,0) = 0 , _(0+, tO = _B(Y) (3)

The function W'n represents the Blasius boundary-layer velocity just upstream of the trailing edge, and

is given by,

I _BPF Vn" + _¥B ---0 Vn(0) = VB'(0) = VB'(**) = 0 ; Wn"(0) = X = 0.332...

The stability equation for the perturbation pressure eigenfunction p(Y) is

with boundary conditions

p,,_ 22" p,__[o_+132 Ix2

_-'--"c" - "7 (u--c)2 M2 ] p = 0 , (4)

p'(O)= p(**)= 0 or p(0)= p(**)= 0 (5)

We will work with the former boundary conditionfor comparison purposes with the boundary-layer

work of Mack (1987),Cowley and Hall (1988),and Smith and Brown (1989). The physicaldifference

between the differentboundary conditionsisthatthe former describeseven disturbanceswhile the

latterdescribesodd ones (forquantitativedifferencessee Papageorgiou(1989a)). Even though odd

disturbancesprovide largergrowth ratesthan the correspondingeven ones for order one Mach

numbers, thisdistinctionis irrelevantin the hypersoniclimitwhere the structureof the mode is

obtainedfrom a localanalysisfaraway from thecenter-line(seelater).Itisalsoworth notingthatthe

nullconditionatinfinityisconsistentwith thephysicsof theproblem sincethesoniclinesdo not cross

forany Mach number and so neutrallystableoutgoingor incoming waves atinfinityarenot supported

(seePapageorgiou(1989a)and Jacksonand Grosch (1989b)).
-4-

3. Numerical solutions and results.

It can be seen that for a specified _(Y), (4) together with (5) is an eigenvalue problem that in general needs to be addressed numerically. In the neutral stability case, the value of the neutral wavespeed c, say, is also known (see below) and the problem reduces to the determination of a and 13which

are now real and determine the wavelength and angle of propagation of the neutral wave. In what follows we take 13= 0 and so are choosing to study two-dimensional disturbances.

Lees and Lin (1946), show that the neutral wavespeed is equal to the value of the undisturbed

flow at generalized inflection points. These are given by the zeros, ¥c say, of the function L(Y),

L(Y) = -_ (6)

In our problem there are two zeros symmetrically placed about the x-axis and, due to the symmetry of

the problem, they give the same value of c,. In solving (4) for Y>O, therefore, a singularity will be

encountered at Y = Yc. The integration contour is deformed around the singularity by extending _ into

the complex plane of Y. It is worth noting that for model flows such as the tanh shear-layer (Jackson

and Grosch (1989a,b)) this analytic extension can be done apriori by simply replacing Y with an

appropriate complex variable. Also in self-similar flows such as the Blasius boundary layer, the singularity in the stability equation can also be treated a priori by integration of the Blasius ordinary

differential equation along a complex contour. In either case the eigenvalues of the problem are

independent of the chosen contour, provided no branch cuts are encountered in the analytic extension.

The present problem is more complicated, however, in that the base-flow is given by the solution of

the nonlinear system (1); the numerical strategy is as follows.

First of all the undisturbed state is computed by numerical solution of (1) subject to (3). Since the

equations are parabolic in x a marching procedure is used, with an adaptive grid near the trailing edge

as dictated by the two-tiered singular structure of the flow there (a singularity appears in the pressure

gradient and the normal velocity, necessitating the triple-deck viseous-inviscid interaction region

(Stewartson (1969), Messiter (1970)). The numerical method together with results has been described

elsewhere (Papageorgiou and Smith (1989), Papageorgiou (1989b)). For a particular streamwise station the position of the generalized inflection point Yc is found from (6). Using a third order Taylor

expansion about Yc, the undisturbed velocity u-(Y) is extended into the complex plane so that the contour
-5-

of integration goes around the singularity. A rectangular deformation was used which passes below the

singularity in the complex plane. This is consistent with the zero viscosity limit solutions of the stability problem (Mack (1987)).

The eigenvalue problem was solved by a shooting method, starting at infinity (some large value

of 19 where the asymptotic forms of the solution arc used, and integrating inwards to Y = 0 using a

fourth-order Runge-Kutta method, and by employing a root-finding routine to satisfy the boundary conditions. Accuracy and convergence tests were made so that the reported results are not dependent on

either the deformed contour or the number of grid-points representing the basic flow.

The results of the calculalions are shown in Figures 1-6. The difference between the figures is

that they correspond to the neutral stability of the flow at successively increasing distances from the

trailing edge. The distances are shown on the figures in terms of the variable _ used in the base-flow

computations and which is equal to x m. Figures 1 and 2 corresponds to positions in the wake at nondimensional distances of 0.000125 and 0.001 from the trailing edge. These are near-wake stations and it

is not surprising to find that our results are almost the same as the equivalent ones for the Blasius

boundary layer presented by Cowley and Hall (1988). The only structure that is different is the zero

Math number limit. This limit is inviscidly stable for boundary-layers but not for wakes (for a discussion of this limit in boundary-layers see Gajjar (1989), and for wake-flows see Papageorgiou and Smith

(1989)). Once the flow becomes supersonic, however, the wake results shown here are a slight displacement of the boundary-layer ones. The explanation of this lies in the fact that the generalized

inflection point for compressible supersonic boundary layers lies in the main part of the boundary-layer

and so is outside the Goldstein layer (see Goldstein (1930)) which comprises a region of lateral extent

of O(xm) as x _ 0. The Goldstein layer is therefore passive and acts only to displace the inflection

point by a lateral distance of O(xl/3). At hypersonic Mach numbers agreement is even more pronounced

since the inflection point (critical layer) now lies at a logarithmically large distance from Y = 0 (see

next section also). The other feature of Figure 1 worth noting is the mode multiplicity at Math

numbers above approximately 3. This is in line with boundary-layer stability where these "second

modes" were discovered by Mack (1965), (1984). As the Mach number increases, it is clear that a

new and almost continuous mode emerges by different modes crossing. This is the wake equivalent to

the vorticity mode analyzed by Smith and Brown (1989) and is characterized by neutral wavenumbers

which grow like _. The modes below the vorticity mode decay to zero with Mach number and
they are equivalent to the acoustic modes analyzed by Cowley and Hall (1988); they decay like M -2 at

high Mach numbers.

Figure 3, corresponds to the wake-station _ = 0.4 (x = 0.064). Even though we are still near the

trailing edge, the basic flow is no longer sharply split into the two-tiered structure of Goldstein (1930),

but the layer where the flow adjusts to symmetry about Y = 0 grows to order one values. The quantitative features of the neutral curves become much different from those of the Blasius boundary-layer.

Multiple modes still occur but are now delayed to Mach numbers of approximately 4 and over. The

vorticity mode is still clearly seen and so are the decaying acoustic modes. The maximum of the first

mode increases and in general, for a given Mach number, the neutral wavenumber is increased as compared to the previous wake-station.

The fate of the multiple or higher modes, begins to become more evident in Figures 4 and 5

which correspond to _ = 0.6,0.8 (x= 0.216,0.512) respectively. A second mode appears at a Math

number of approximately 5.5 and 7 respectively and at the same time the first and second mode come

together. In Figure 4 the first and second modes almost cross at a Mach number of about _.5. The

presence of the vorticity and acoustic modes is still seen, even though their growth or decay respectively is slowed down somewhat and one needs to go to higher Mach numbers to see it. The vorticity

mode (which is the dominant one at high Mach numbers) is described analytically in the next section.

Figure 6 shows a collection of the neutral curves computed for the wake-stations

= 1.2,1.4,1.6,1.8 (x = 1.728,2.744,4.696,5.832). For all these eases the Mach number range 0 to

20 was scanned for second modes but none were found. The second modes are present at much higher

Mach numbers and are probably of little physical value then. It is worth noting that computations on

the compressible tanh shear layer or the Lock profile (Jackson and Grosch (1989a,b)) do not reveal any

second modes at moderately large Mach numbers at least. The reason for this is that both these mixing ....

layer flows model the far-wake development of two compressible boundary-layers, with different slip

velocities, coming together and mixing at the trailing edge. Undoubtedly, by analogy with the present

problem, such modes exist in the developing region of the mixing layer before full mixing occurs. In

fact the dominant neutral hypersonic mode that emerges for both symmetric wakes (the present problem) and the far-wake development of non-symmetric ones (tanh or Lock profiles), is the vorticity

mode analogous to the one described by Balsa and Goldstein (1989) for mixing layers and, Smith and

z

i
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Brown (1989) for boundary layers. This mode is analyzed in the next section for the wake and in an

Appendix for the tanh and Lock shear layers.

4. The vorticity mode in the wake.

In this section we will show that the structure of the vorticity mode is essentially the same as that

obtained for the Blasius boundary layer (see earlier references). The only complication is the fact that

the undisturbed velocity in the streamwise direction does not possess a uniforrniy Valid self-similar

form, and so each streamwise station should be treated separately. As the hypersonic limit is

approached, however, the critical layer gets pushed out to the edge of the wake boundary layer and so

only the large Y behavior of the flow is required for any given position in the wake. As has been noted

by previous investigators (Smith and Brown (I989), Cowley and Hall (1989), BIackaby et al. (I990)),

the physical significance of the critical layer in the hypersonic limit, is that it provides a thin layer

inside which the undisturbed temperature adjusts from an order M2 value (note that M :_ 1) just below

the layer, to its undisturbed 0(1) value just above it. The large temperatures near the plate are a consequence of heating due to the large Mach numbers involved, and for a boundary-layer on a semi-infinite

flat plate for example, this increase in temperature takes place for all streamwise stations. The situation

is different for a wake, however. When the no-slip condition at the trailing edge is removed, the

undisturbed temperature has its maximum value at the wake center-line. A glance at the temperature

equation (2), shows that as the far-wake limit is approached (the velocity _ tends to its undisturbed

value of 1), the base-flow temperature tends to its ambient unit value independent of the Mach number.

The adjustment layer will no longer be thin but will occupy the whole wake. This limit is more subtIe

than the description at order one wake-stations and we describe it first. It will also be shown how the

far-wake limit, which appears to have a different structure, produces the vorticity mode seen upstream.

This is done by taking the zero limit of the local scaled far-wake streamwise coordinate as shown

below.

For x :_ 1, then, the basic flow takes the form

A M2 e_r¢/x _'=l-_e -_/_+ "'" , T=l+7 (v--l) + ..- , (7)

where A is a positive constant (see Rosenhead (1963), Schiichting (1948)). These results are readily

obtainable from (1), (2) by linearizing _-and T about unity. As explained above, the vorticity mode is
m

obtained in regions where T is of order one. The following scalings are introduced, therefore.

x = M2 yc , Y = M2 ]" , y2 / x = _,21yc = _2 ,

where tilde variables are now of order one. The basic flow becomes

1 e_#/:i T= 1 + _-wz (7--1) e-_2/f

_= 1 M 2 _tc2

The wavespeed c also expands as

(8)

(9)

Cl

c=l---_+ ''' ,

a result that follows from the evaluation of fi in the critical layer. Substitution into the Rayleigh equation (4) gives

d2p _ 4]_e -I'2/'f dp_ _ iX2 (1 + _-l/2(T--1)e -_2/f) p = 0

dY 2 .23/2(c1 - _2-I/2e-I'2/t) dY

(10)

The wavenumber tx in (4) has been scaled according to

(y.=m

M 2 '

and _ is the order one quantity that appears in (10) above. Equation (4.4) is to be solved subject to

decay as I_ _ +**.

The interesting limit that matches the solutions of (10) to those upstream is obtained by taking

J2_ 0. As _ becomes small, the value of I_ also becomes small, and its functional dependence is

obtained from the balance

,i-tae-_'2/t O(1) (I 1)

This balance makes all the terms in (10) of the same order, which is essential in producing the vorticity mode upstream. If we solve (11) we find that to leading order I_ goes to zero like _/-(1/2)_ tog(_).

The next order approximation leads to the thickness of the layer and a new local coordinate Y-given by

= L + _ L = "4-(I/2)log(Yc _/2) L '

Substitution into (10) yields

d2p 4e -2Y dp $2 _2 (1 + (V--1)e-2Y)p = 0 (12)

d_ c1 - e-2Y dY (-Ydog(Y¢I/2) )

=_
-9-

The boundary conditions are that the disturbance vanishes as I_ --->_. The following rescaling of

puts (12) into the canonical vorticity layer equation (Smith and Brown (1989)),

l_ = (-xl°gxln)ia av

The solutions of (12) have been given by Smith and Brown and we need only quote the results here. If

the transformation s = (_-l)e -2_ is made, together with p = saVp(s), the exact solution P = e -'t4 is found

when c I takes on its neutral value from the inflection point condition, when otv is given by

1

_V = _"

Smith and Brown show also how higher order corrections to the neutral wavespeed can be generated

and the reader is referred to that paper for a complete description of the vorticity mode.

If we now write the unscaled wavenumber a in terms of av we find

1 _ (13) (X = (XV

M 2

Now the local scale _ was defined in (8); it can be seen, therefore, that for x to be of order (in other

words to match with the solutions upslrearn), the scaling _a M-2 is relevant. Equation (13) now

gives

1

a = _ _ , (14)

which is the vorticity mode result of Smith and Brown.

To summarize, we considered the stability problem of the far-wake in the double limit M :_ x >>1.

The other limit x > M :_ 1 is much easier and is discussed below for completeness even though the

result has already been found by letting J_--->0 above.

For large x, therefore, the basic flow takes the form (7) above, where the vertical coordinate is

"q = Ylx 112. The generalized inflection points are the zeros of

T= 2 ,

where primes denote 11 derivatives and x is effectively a parameter. The inflection point occurs as

11--->_, and it is easy to show that its position is given by the solution of
- 10-

Notethatwe assume that M s :_ xla; the case when these are in balance has been discussed above. To

study the vorticity layer we write

t i M+e-r + ]

_TI=F+ 2F ' I/'-'_ x = '

as in Smith and Brown (1989), Cowley and Hall (1988). The inflection point then occurs at

rl, = log(A(y-1)). Expanding c as before and making the transformation Ae-_ = e-" together with the rescaling ct = Fay gives the canonical vorticity equation described above. The exact neutral solution of

Smith and Brown is as above and the final result is

Ms 1 I/2

_=_

'[,O,x-+ J (15)

It is exactly this mode structure that was obtained in the previous case in the limit of small ._.

At order one distances from the trailing edge, the behavior of the basic flow at large Y has the

quadratic exponential decay of (7) for example. This is enough to dictate that the vorticity mode structure has wavenumbers that scale with _.

To see the importance of the asymptotic behavior of the flow at infinity in determining the dominant hypersonic modes of instability, we include in an Appendix the hypersonic stability of two

profiles that are widely used in mixing layer studies, the hyperbolic tangent and Lock velocity distributions. It is shown that completely different results arise for the two cases.

5. Conclusions. _ =;:_ +:_ : : ::::: =:= :: +_: .... - : : _: : ....

We have carried out a computational and analytical study of the inviscid instabilities present in

laminar wake flows. We have computed the variation of neutral wavenumbers with Mach number for

......... sever_ wakesstations: It: _S been found that near-w_e:supersonic modes: are essenfialiya smaff perturbation of those found for the Blasius boundary-layer, the reason being that inviscid compressible

neutral waves are associated with the generalized inflection points of the basic flow, and these points

are almost coincident for the near-wake and the Blasius profile. As the wake develops, the Mach

number range which supports a unique mode increases. Higher modes arc delayed to ever increasing
-11-

v

j_.-

7i

z

Mach numbers and for fixed but large Mach numbers the rate of growth of the vorticity mode recedes

and so does the rate of decay of the acoustic modes. This plays a role in the comparison of analytical

results of the vorticity modes and the computed ones. The agreement is excellent at near-wake stations

but as the wake develops very high Mach numbers are required to bring exact and asymptotic results

into agreement. The asymptotic result is valid at infinite Mach numbers anyway and the neutral

wavenumber ct grows fairly slowly with Mach number due to the logarithmic dependence described in

the previous section. Our results indicate that even though the vorticity mode clearly emerges as the

major hypersonic instability mechanism, its asymptotic description for wake stations far from the trailing edge comes into play at very large Mach numbers that are very dependent on the station under

consideration. This may not be the case for other flows, and indeed as we show in the Appendix, the

vorticity mode analysis fully captures the developed mixing layer dynamics.

Our results also indicate that multiplicity of modes is delayed to impractically large Mach

numbers for positions far enough from the trailing edge. At the wake station which is about one platelength away from the trailing edge, there is only one mode present for the range 0 _<M _<20. This is

certainly also true for distances further downstream. We note here that it is difficult to continue the

computations to Mach numbers beyond approximately 20, because the neutral wavespeed tends to the

free stream velocity making the solutions inaccurate.

Finally, it is interesting to consider the effects of the thermodynamics used in the model, and in

particular the relaxation of the Chapman linear viscosity law. A more accurate relationship is believed

f,+ l

to be the Sutherland law g = [T+Cj_ 2 where C is a constant approximately equal to ll0°K/T_. This

model was adopted by Blackaby et al. (1990) in their interesting study of hypersonic flow past a flat

plate. They find that the basic flow decays algebraically at infinity and as a consequence the structure

of the vorticity mode is significantly altered in that the neutral wavespeed approaches a constant in the

hypersonic limit, rather than exhibit the Mach number dependent growth obtained from the Chapman

law. It should be noted, however, that the maximum growth rate is still small and of order M'2. We

expect similar results to hold for a developing wake with the Sutherland viscosity law. Some preliminary work indicates that the far-wake has the vorticity mode structure of the Chapman law Blasius

flow, while at near-wake stations the results of Blackaby et al. are expected. The interesting regime of

order one streamwise stations must be tackled numerically, it seems, since there is no simple
representation of the basic flow.

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i

IK

1
7

iAppendix.

In this section we briefly present the analysis that leads to the hypersonic stability limit of two

model mixing layers. For the tanh mixing layer, we will arrive at the result given by Balsa and Goldstein (1989), but in a more straight-forward manner. This result is also consistent with the computations of Jackson and Grosch (1989a).

The tanh mixing layer.

We take the basic flow used by Jackson and Grosch (1989) and Balsa and Goldstein (1989), the

stability equation being (4).

1

u-(rl)= _ (1 + tanhrl) , (A1)

T(_) = 1 - (1-13T)(1-u--)+ -_(V--1)M2u-(1-u-) (A2)

Using (A1) and (A2) together with (6) we find that the inflection point position satisfies the equation

1 1 ea_h_cosh .

"_(T- 1)M2 - coshZq = _(1-_r)_ (coshrl-smhrl) (A3)

It is clear from (A3) that as M becomes infinite the value of rl must increase like logM. More precisely,

solving (A3) for large 1] we find

I. T1= 1ogM+ _',og[ 2 J

To study the behavior of the critical layer, therefore, we introduce the 0(1) variable 11 given by

= logM+ _I

The wavespeed is expanded also

c!

C = 1 - _-_" + ...,

(A4)

leigh equation (4), together with the transformation •-2g= 2e-_ gives

d2p 2e-' dp_ _I ct2(I+ (l-_')e-') 2p = 0 (A5)

dz2 cl dz 4

-- e. -z

2

where the neutral value is given by el = 2/(T--1). Substitution into (AI), (A2) and then into the Ray-
-- 14-

The Smith and Brown transformation described in section 3 applies, and so does the exact neutral solution they found. The result is

1 (A6)

2

in complete agreement with Balsa and Goldstein (1989).

The Lock profile.

The tanh velocity distribution discussed above is only a model for the mixing layer and does not

satisfy the equations of motion. It is meant to model the solution derived by Lock (1951) in his study

of the boundary layer between parallel streams. The distribution is found as a similarity solution of the

boundary-layer equations with a similarity variable 11= Y/2x II2.

u-'Ol)=f(rl) , f" + 2ff'=O , u-(**)= l , u-(--**)=O , (A7)

T= 1 - (1-l]r)(1-u-") + l(y-1)M2u-(1-u--) , (A8)

where 13Tis the ambient temperature of the lower stream. Using (A7), (A8) we find that the generalized inflection point position satisfies the equation

(A9)

The behavior of _ as 11_ .0 is therefore the same as that for the Blasius,boundary-layer (e.g. Rosen-

\

head (1963)). _e asymptotic form is given by

_'-I- z.z(_-_- _-3+ o(_-s))exp(-__2) , _= 11- _ ,

where _,1 and _ are positive constants we need not specify here. For large T! therefore, the dominant

balance in (A9) comes from the third derivative terms and gives the critical layer scaling

,=

D

_k

it

with

e -r2 1
- 15-

To leading order, therefore, F = dO_-_, in line with the result of Smith and Brown as expected. The

neutral wavenumber is found by deriving the vorticity equation as above, and utilizing the exact soluo:= 2_ (AIO)

tion of Smith and Brown. The final result is

It is interesting to compare the theoretical results (A6) and (A10) with the computations of Jackson and Grosch (1989a). For example, their Figure 13, gives the neutral curves of cc versus Mach

number for the tanh and Lock profile as well as those for a Sutherland viscosity law, for the subsonic

fast modes which have been analyzed here. It is clear from the figure that the neutral curve for the

tanh profile is tending to a constant. The value attained at the highest computed Math number of 7 is

about 0.35-0.4, and the curve is flat and increasing slightly. The same is true for the Sutherland

viscosity law. The situation for the Lock profile is quite different, however. The neutral value of ct at

M = 7 is about 0.7 and the slope of the curve is not flat and is increasing slowly with Mach number.

The leading order asymptotic result (A10) evaluated at M = 7 gives a value of approximately 0.986.

The agreement becomes better when the Math number is increased; further more, the next order

correction to (A10) which is of order log(logM') (see Smith and Brown) is negative and enhances

logM'

agreement between the exact and asymptotic results.
- 16-

Acknowledgements.

This work was supported by the National Aeronautics and Space Administration under NASA

Contract NA$1-18605 while the author was in residence at the Institute for Computer Applications in

Science and Engineering, NASA Langley Research Center, Hampton, VA 23665.

The author also wants to thank Professor Philip Hall for many valuable discussions, and in particular for his comments and suggestions regarding the analysis of Section 4. Professor Frank Smith is

also thanked for many useful discussions.

=

v
-17-

References.

Balsa, T.F. and Goldstein, M.E. 1989 J.Fluid Mech., submitted.

Blackaby, N., Cowley, S.J. and Hall, P. 1990 J.Fluid Mech., submitted.

Cowley, S.J. and Hall, P. 1988 ICASE Report No. 88-72 and J.Fluid Mech., in the press.

Gajjar, J.S.B. 1989 In Proceedings of the ICASE/NASA Langley Workshop on Instability and Transition, May 15-June 9, 1989, ed. M.Y.Hussaini. Springer Verlag.

Goldstein, S. 1930 Proc.Camb.Phil.Soc., 26, pp. 1-30.

Hall, P. and Fu, Y.B. 1989 Theoret. Comput. Fluid Dynamics, Vol.1 No.3, pp.125-134.

Jackson, T.L. and Grosch, C.E. 1989a ICASE Report No. 89-32 and J.Fluid Mech., submitted.

Jackson, T.L. and Grosch, C.E. 1989b J.Fluid Mech., 208, pp. 208-683.

Lees, L. and Lin, C.C. 1946 NACA Tech. Note No. 1115.

Lock, R.C. 1951 Quart.J.Mech. and Appl.Math., 4, pp.42-63.

Mack, L.M. 1965 In Methods in Computational Physics, eds B.Alder, S.Fembach and M.Rotenberg,

Vol.4, 247-299.

Mack, L.M. 1984 In Special Course on Stability and Transition of Laminar Flow, AGARD Report R709, 3-1 to 3-81.

Mack, L.M. 1987 In Stability of Time Dependent and Spatially Varying Flows, eds. D.L.Dwoyer and

M.Y.Hussaini. Spdnger-Verlag.

Messiter, A.F. 1970 SIAM J.ApplAtaths, 18, pp.241-257.

Papageorgiou, D.T. 1989a Theoret. Comput. Fluid Dynamics, in the press, and ICASE Report No. 89-

66.

Papageorgiou D.T. 1989 In Proceedings of the ICASE/NASA Langley Workshop on Instability and

Transition, May 15-June 9, 1989, ed. M.Y.Hussaini. Springer Verlag.
-18-

Papageorgiou, D.T. and Smith, F.T. 1989 J.Fluid Mech., 208, pp.67-89.

Rosenhead, L. 1963 Laminar Boundary Layers, Dover Publications.

Smith, F.T. and Brown, S.N. 1989 J.Fluid Mech., submitted.

Stewartson, K. 1964 Theory of Laminar Boundary Layers in Compressible Fluids. Oxford University

Press.

Stewartson, K. 1969 Mathematika, 16, pp.106-121.

z

L
-19-

O

L

q)

Z

0.5

0.4

0.5

_:0.05

I I I I I I I I

1 2 5 4 5 6 7 8

k!cch number

9

FIGURE 1: Neutral stability ¢m-ves at _ = 0.05 (_ffi0.000125).
-20-

_5

E}

k.-

(D

Z

0.5

04

0.5

0.2

0.1 -

0.0

0

I I I I I I I I

1 2 5 4 5 6 7 8 9

Moch number

i

=

FIGURE 2: Neutral stability curves at _ = 0.1 (x-0.001).
-21-

0.6

m

ID

i.

z]

(D

Z

0.5

0.4

0.,3

0.2

0.1

0.0

0

I

2 4 6 8 10

Moch number

FIGURE 3: Neutral stability curves at g = 0.4 (x-0.064).
-22-

_5

O

k_

0)

Z

0.6

0.5

0.4

0.5

0.2

0. I

0.0

0

I I I I

2 4 6 8

Moch number

\

10

FIGURE 4: Neutral stability curves at _ = 0.6 (x-,0.216).
-23-

o

CD

Z

1.0

0.8

0.6

0.4

0.2

0.0

0

J

J

l I I I

\

2 4 6 8

Moch number

10

FIGURE 5: Neutral stability curves at _ = 0.8 (xffi0.512).
-24-

O

©

0.5

0.4

0.5

0.2

0.1

0.0

_'=1.2,1.4,1.6,1.8

(i)

- (ii) --------

(iii)/

(iv)

I I I I I

0 5 10 15 20

M

FIGURE 6: Neutral stability curves.

(iv) _ = 1.8 (x-5.832).

(i) _ = 1.2 (x-1.728), (ii) _ = 1.4 (x-2.744), (iii) _ = 1.6 (x-4.696),
Report Documentation Page Nal,3_al ,C_'Or_l_IC $ and

SPaCe _]r_SXral_

1. Report No.

NASA CR- 182050

ICASE Report No. 90- 39

2. Government Accession No.

4. Title and Subtitle

THE STABILITY OF TWO-DIMENSIONAL WAKES AND SHEARLAYERS AT HIGH MACH NUMBERS

7. Author(s)

D. T. Papageorgiou

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

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Langley Research Center

Hampton, VA 23665-5225

3. Recipient's Catalog No.

5. Report Date

May 1990

6. Performing Organization Code

8, Performing Organization Report No.

90-39

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. Supplementaw Notes

Langley Technical Monitor:

Richard W. Barnwell Submitted to Physics of Fluids A

Final Report -.\ - _

16 Abstract

_ndy is concerned with the stability properties'of laminar free shear-layer flows, and In

particular symmetric_wo-dJmens_ional wakes, for the subsonic through the hypersonle regime. Emphasls

Is given to the use of proper wake profiles that satisfy the equations of motion at high Reynolds

numbers. In paFt:_culariw_--stu_y the inviscid stability of a developing t_-_mensio_ wake as it

accelerates at the trailing edge of a splitter plate. The non-parallelism of tbe flow is a leading

order effect, and the undisturbed state is solved numerically. The neutral stability characteristics

are computed numerically and the hypersonic stability is obtained by increasing the Mach number. It

is found that the neutral stability characteristics are altered significantly as the wake develops.

Multiple modes (second modes) are found in the near-wake (they are shown to be closely related to

the corresponding Blasius ones), hut as the wake develops |mode multiplicity is delayed to higher and

higher Mach numbers. At a distance of about one plate length from the trailing edge, there is only

one mode in a Mach number range of zero to twenty. The dominant mode emerging at all wake stations

and for high enough Mach numbers is the so-called vortlcity mode, which is centered around the generalized inflection point layer. The structure of the dominant mode is also obtained analytically

For all streamwlse wake locations and it is shown how the far-wake limit is approached. Asymptotic

results for the hypersonic mixing layer given by a tanh and a Lock distribution are also given.

17, Key Words(SuggestedbyAuthor(s))

wake, mixing layer, high Mach number

limit

19. Security Classif. (of this report)

Unclassified

18. Distribution Statement

34 - Fluid Mechanics and Heat Transfer

Unclassified - Unlimited

_. SecuriW Cla_f. (of this pa_) 21. No. of pa_s

Unclassified 26

22. Price

A03

NASA FORM 1626 OCT 86

NASA-L_Iey, 1990