I 

I 

NASA Contractor Report 181855 

ICASE Report No. 89-32 

ICASE 

INVISCID SPATIAL STABILITY OF A 

COMPRESSIBLE MIXING LAYER. 

PART III. EFFECT OF THERMODYNAMICS 

T. L. Jackson 

C. E. Grosch 

Contract No. NAS1-18605 

June 1989 

Institute for Computer Applications in Science and Engineering 

NASA Langley Research Center 

Hampton, Virginia 23665-5225 

Operated by the Universities Space Research Association - ', 

(#A SA -CB- 1 8 1 8 5 5) 

OP h COMPRESSIBLE MIXIlG LAYER. ,PABT 3: 

EFFECT OF THERHODYNAHZCS Final Report (ICASE) 47 p CSCL 20D Unclas 

I NV ISQ D SP AT1 hL ST ABILITY N89-27112 

G3/34 0229100 

NASA 

National Aeronautics and 

Space Administration 

Langley Research Center 

Hampton. Virginia 23665-5225 
INVISCID SPATIAL STABILITY OF A COMPRESSIBLE MIXING LAYER 

PART III. EFFECT OF THERMODYNAMICS 

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. We report the results of a comprehensive comparative study of the inviscid spatial stability 

of a parallel compressible mixing layer using various models for the mean flow. The models are (a) the 

hyperbolic tangent profile for the mean speed and the Crocco relation for the mean temperature, with the 

Chapman viscosity-temperature relation and a Prandtl number of one; (b) the Lbck profile for the mean 

speed and the Crocco relation for the mean temperature, with the Chapman viscosity-temperature relation 

and a hdtl number of one; and (c) the similarity solution for the coupled velocity and temperature equations using the Sutherland viscosity temperature relation and arbitmy but constant Prandtl number. The 

purpose of this study was to determine the sensitivity of the stability characteristics of the compressible 

mixing layer to the assumed thermodynamic properties of the fluid. It is shown that the qualitative features 

of the stability characteristics are quite similar for all models but that there are quantitative differences 

resulting from the difference in the thermodynamic models. In particular, we show that the stability characteristics are sensitive to the value of the Prandtl number. 

lhis work was suppolted by the National Aeronautics and Space Administration under NASA Contract NAS1-18605 while the authors 

welc in residence ; the Institute for Computer Applications 

23665. 

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

! 
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1. Introduction. The study of the stability of compressible shear flows is somewhat more complicated 

than that of incompressible shear flows in that the thermodynamics of the compressible fluid is of major 

importance. As a consequence of compressibility the types of disturbances which can exist are quite varied 

they can be subsonic, sonic, or supersonic modes and these can be either vorticity or acoustic modes (Mack, 

1989); there can be multiple unstable modes with the same ftequency; and finally three dimensional modes 

are of great importance because they may be more unstable than two dimensional modes; the characteristics 

of these modes are dependent upon the thermodynamics choosen. 

The basic formulation of the theory of the stability of compressible shear flows, both wall bounded 

and free. is due to Lees and Lin (1946). and Dunn and Lin (1955) first showed the importance of three 

dimensional disturbances. The major additional effort in this area appears to be directed towards understanding the stability characteristics of compressible boundary layers (see Mack; 1965, 1984, and 1987 for 

comprehensive reviews) with much less effort devoted to understanding the stability characteristics of 

compressible free shear layers. 

The earliest calculations of stability characteristics of compressible boundary layers used realistic thermodynamic relations (Brown, 1962, Lees and Reshotko, 1962, and Mack, 1965) and this has continued to be 

the practice to the present day. The situation with regard to stability calculations for free shear layers is 

quite different. Most studies of the stability-of compressible free shear flows have been based on assumed, 

somewhat arbitrary. mean velocity and temperature profiles which are not solutions to the mean flow equations but do satisfy the boundary conditions. If both the velocity and temperature profiles are arbitrary it 

might be conjectured that the solutions of the stability problem could be unrelated to those of the physical 

problem. If they are, at least, rough approximations to the actual solutions, one might conjecture that the 

solutions of the model stability problem would approximate those of physical problem. These solutions of 

the model problems might be useful in elucidating qualitative features of the true stability problem because 

they are easier to treat. Some examples of studies of this type are that of Gill (1965). who studied the temporal stability of "top hat" jets and wakes and those of Blumen (1970); Blumen, Drazin and Billings (1975). 

and Drazin and Davey (1977) who examined the temporal stability of a compressible mixing layer with the 

mean velocity profile assumed to be given by a hyperbolic tangent and a constant temperature. 

In most compressible free shear stability studies the thermodynamics used was that of the model fluid 

(Stewartson, 1964). This model was originally introduced in order to simplify the thermodynamics of the 

flow in compressible boundary layers. In addition to obeying the perfect gas law (valid for real gases at 

temperatures less than a few thousand degrees K), the model fluid has a unit Prandtl number so that the 

rates of diffusion of heat and momentum are equal, and the Chapman (1950) viscosity law with the viscosity proportional to the temperature is assumed to be valid. One class of approximate solutions involves 

modeling the mean velocity profile by a hyperbolic tangent and using the Crocco relation for the mean temperature profile. This approximation has been used by a number of authors, including Ragab and Wu 

(1988), Tam and Hu (1988), and Zhuang, Kubota and Dimotakis (1988). We have also used this model in a 

comprehensive study of the spatial stability of a compressible mixing layer (Jackson and Grosch, 1988; 

hereafter referred to as Part I) as well as in a similar study of a reacting compressible mixing layer (Jackson 

and Grosch, 1989). 

Another class of models can be defined as those which use the Lock profile (Lock, 1951), the similarity solution for the velocity profile with the viscosity proportional to the temperature, a Prandtl number of 

i 

. 
i 

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unity and various temperature profiles. Lessen, Fox and Zien (1965, 1966) in temporal stability calculations 

used the Lock profile and assumed that the flow was iso-energetic so that the temperature of the stationary 

gas was much greater than that of the moving gas even at moderately supersonic speeds. Gropengiesser 

(1969) used a generalized hyperbolic tangent profile (see his equation (2.27)) to approximate the Lock 

profile and used the Crocco relation for the temperature in spatial stability calculations. 

The final class of solutions to the mean flow equations are those where the Prandtl number is not 

necessarily one and a reasonably realistic viscosity-temperature relation, such as the Sutherland law, is used. 

The velocity and temperature profiles are exact similarity solutions of the mean flow equations. Quite 

surprisingly, there are few hear stability calculations for compressible mixing layers with these more realistic mean velocity and temperature profiles. The only published results, of which we know, are those of 

Ragab and Wu (1988) for spatial stability and those of Macaraeg, Street and Hussaini (1988) for temporal 

stability. It seems that the main interest of Ragab and Wu was to determine the dependence of the maximum growth rate of the disturbances on the velocity ratio of the mixing layer for subsonic flows. They 

concluded that the maximum growth rate depends on the velocity ratio in a complex way, with the maximum growth rate appearing at a particular nonzero velocity ratio. Macaraeg, et a1 studied the temporal stability of a compressible mixing layer for a few selected values of the Mach number (I 4). and a few values 

of the free stream temperature ratio. They were the first to point out the sensitivity of the stability characteristics of this class of flows to variations in the F’randtl number. 

Very recently there has been a revival of interest in the stability of compressible free shear layers, 

particularly at supersonic speeds. An unanswered question is the accuracy of the results of those calculations in which various models of the mean flow thermodynamics have been used. It is clear that there will 

be quantitative differences in, for example, phase speeds and growth rates of the disturbances depending on 

the model of the mean flow. The important questions are first, the extent to which qualitative predictions 

of recent, and older, studies are dependent on the models of the mean velocity and temperature used, and 

second, the magnitude of the differences in the quantitative predictions as a function of the mean flow 

model. 

In order to answer these questions we undertook a comprehensive systematic study of the stability of 

one free shear flow, the compressible mixing layer. In this study we calculated the stability characteristics 

of the compressible mixing layer using a number of representative models of the mean velocity and temperature profiles over a wide range of Mach numbers. The models are (a) the hyperbolic tangent profile for 

the mean speed and the Crocco relation for the mean temperature, with the Chapman viscosity-temperature 

relation and a Prandtl number of one; (b) the Lock profile for the mean speed and the Crocco relation for 

the mean temperature, with the Chapman viscosity-temperature relation and a Prandtl number of one; and 

(c) the similarity solution for the coupled velocity and temperature equations using the Sutherland viscosity 

temperature relation and arbitrary but constant Prandtl number. In section 2 we formulate the problem, 

including defining the thermodynamic models. Our results are presented in section 3. Finally, we summarize our conclusions in section 4. 

2. Formulation of the Problem. 

2.1. The Mean Flow The problem considered here is the inviscid spatial stability of the steady two 

dimensional flow of a compressible mixing layer which lies between two streams with different speeds and 
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temperatures. We take one of the streams to be moving at i- and the other to be stationary at -. The 

equations are nondimensionalized by the values of the density, temperature, and speed in the moving 

stream. The length scale is a characteristic length of the mean flow, and the time scale is the ratio of the 

length and speed scales. The x axis is along the direction of flow, the y axis is normal to the flow direction, and the t axis is in the cross stream direction. U and V are the velocity components in the x and y 

directions, respectively. p is the density, and T the temperature. We assume that the equations governing 

the mean flow are the compressible boundary layer equations (Stewartson, 1964). 

These equations are transformed into the incompressible form by the Howarth-Dorodnitzyn 

mation 

i Y 

Y=[pdy, v=pV+U pxdy. 

Because the pressure gradient is zero, 

pT=l. 

transfor- 

(2.la,b) 

(2.2) 

Assuming that the mean flow is given by a similarity solution. we transform to the similarity variable 

Y q=- 2 -5- 

the equations for the mean flow quantities U and T reduce to 

r 'I' 

AT' +2 fT'+(y- 1)M2 .E cf")*=o, 

[pr I [TI 

(2.3) 

where the primes indicates differentiation with respect to q. Here p is the viscosity coefficient, Pr is the 

Prandtl number, y is the ratio of specific heats of the gas, and M the Mach number of the moving stream. 

The mean flow field is found by solving equations (2.5) and (2.6) subject to the boundary conditions, 

f '(+) = 1, T (i-) = 1, (2.7a,b) 

(2.8a,b) 

with pT the ratio of the temperature in the stationary gas to that of the moving gas. If pT is less than one, 

the stationary stream is relatively cold compared to the moving stream, and if PT is greater than one it is 
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relatively hot. 

It should be noted that equations (2.5) through (2.8) constitute a fifth order boundary value problem, 

but that there are only four boundary conditions. Ting (1959) has shown that an appropriate boundary condition can be obtained by matching the freestream pressures on either side of the mixing layer if at least one 

of the streams is supersonic. However, Klemp and Acrivos (1972) have shown that this condition is incomplete if both streams are subsonic. These conditions are equivalent to a specification off (-). This value, 

a specification of the stream function, can be varied so as to ensure that f (0) takes on any particular value. 

This will not effect the physics of the flow, only the location of the origin of the coordinate system. 

The structure of the mean flow clearly depends on the variation of p, and Pr with temperature and 

pressure. In general, both p and Pr are very weakly dependent on pressure and can be taken to be independent of pressure. The Randtl number is somewhat dependent on the temperature but, for this study, will be 

assumed to be constant, with calculations being carried out over a range of Pr between 0.7 and 1.0. Finally, 

the dependence of viscosity on temperature is quite important and the choice of that dependence leads to 

several thermodynamic models discussed in the following section. 

2.2. Flow Models Given a value of Pr and p( T ) the mean flow is detennined by the solution to 

equations (2.5) and (2.6) with the boundary conditions (2.7) and (2.8) and a specification of > (0). We will 

consider three models for the mean flow. 

If it is assumed that the viscosity is proportional to the temperature, that is we use the Chapman 

viscosity law. equation (2.5) is uncoupled from (2.6). With Pr = 1, equation (2.6) can be solved in closed 

form to give the Crocco relation, which with the boundary conditions (2.7) and (2.8). is given by 

T = 1 - (l-b)(l-U) + M2U (14). (2.9) 2 

The first model of the mean flow involves using (2.9) for the mean temperature profile and appmximating the mean velocity profile by a hyperbolic tangent 

(2.10) 1 

2 U = - (1 + tmh(T)). 

We will call this approximation the Tanh model. The results for this model were presented in Part I. Some 

of these results will be reproduced here for ease of reference and comparison purposes. 

The second model of the mean flow again uses (2.9) for the mean temperature profile, and the solution to (2.5) with p, proportional to T and Pr = 1, for the mean velocity profile. We will call this the Lock 

model. 

The third model is one in which the Prandtl number is constant but not necessarily one and a reasonably realistic viscosity-temperature relation is used. For temperatures greater than about 100 KO, a Sutherland type of relation 

p= Q T”l (b + T), Q = 1 + b, b = 110.4KO / T’, (2.11) 
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with T' the reference temperature, is reasonably accurate. The mean velocity and temperature profiles are 

the solutions of (2.5) to (2.8) and p given by (2.11). We will call this the Sutherland model. 

23. The Stability Problem The stability problem can be formulated independently of the detailed 

form of the U and T profiles. The flow field is perturbed by introducing two dimensional wave disturbances 

in the velocity, pressure, temperature and density with amplitudes which are functions of q. For example, 

the pressure perturbation is 

P = ~(q)erp[i (a - ot)l. (2.12) 

with I7 the amplitude, a the complex wavenumber and o the frequency which is taken to be real because 

we are only treating the spatial stability problem. Substituting the expression (2.12) for the pressure permbation and similar expressions for the other flow quantities into the inviscid compressible equations yields 

the ordinary differential equations for the perturbation amplitudes. It is straightforward to derive a single 

equation governing n, given by 

n" - - " n' - a2T [T - M2 (U-C)~] ll = 0, (2.13) u-c 

where c is the complex phase speed 

(2.14) 

and primes indicate differentiation with respect to the similarity variable q. Since a is complex, the real 

part of a is the wave number in the x direction, while the imaginary part of a indicates whether the disturbance is amplified, neutral, or damped depending on whether q is negative, zero, or positive. The phase 

speed, cph, is given by o / 4. If a, is zero, c = cN is the phase speed of a neutral mode. 

The boundary conditions for I7 are obtained by considering the limiting form of equation (2.13) as 

q + fm. The solutions to (2.13) are of the form 

n + f=P t+&q), 

where 

Q2 + - a 2 [I - M~(~-C)~], Q2- = a2PT [PT 

We define c* to be the values of the phase speed for which Q2, vanishes. 

- 

dPT c+=1-- 1 

c- = - M' M* 

(2.15) 

- M2c2]. (2.16) 

Thus, 

(2.17) 

Note that c+ is the phase speed of a sonic disturbance in the moving stream and c- is the phase speed of a 

sonic disturbance in the stationary stream. At 

M=M. El+& (2.18) 

- 

c* are equal, and this value is denoted 2 I 
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The nature of the disturbances and the appropriate boundary conditions can now be illustrated by 

reference to Figure 1, where we plot c* versus M. In what follows we assume that a', > These 

curves divide the c, -M plane into four regions, where c, is the real part of c. If a disturbance exists with 

a M and c, in region 1, then a'+ and Q2- are both positive, and the disturbance is subsonic at both boundaries. In region 3, both a'+ and Q2- are negative and hence the disturbance is supersonic at both boundaries. In region 2. Q'+ is positive and Q2- is negative, and the disturbance is subsonic at +oo and supersonic at -, and we classify it as a fast mode. Finally, in region 4, Q', is negative and Q2- is positive so 

the disturbance is supersonic at +oo and subsonic at -, and we classify it as a slow mode. 

One can now see that the appropriate boundary condition for either damped or outgoing waves in the 

moving and stationary streams are, respectively, 

-ill- II+~-~+", if c, >c+, II +e , if c, c c+, (2.19a) 

if c, < c-, + 1 J-n2-. . if c, > c-. (2.19b) 

To solve the disturbance equation (2.13). we first transform it to a Riccati equation by setting 

G=A (2.20) aTn' 

Thus, (2.13) becomes 

2U' T' G'+aTG2-[---IC =a[T -M2(U-c)'1. U-c T 

The boundary conditions can be found from (2.19) and (2.20). 

(2.21) 

The stability problem is thus to solve equation (2.21) for a given real frequency o and Mach number 

M. The eigenvalue is the wavenumber a. Because this equation has a singularity at U = cN, we integrate 

it along the complex contour (q-,-l) to (0,-1) and (q+,-l) to (0,-1) using a variable step Runge-Kutta 

scheme. In our calculations we have taken q* to be 6 for the Tanh and Lock profiles. However, we found 

that q- needs to be larger for the Sutherland model because the decay at -00 is slower than for the other two 

models, with the rate of decay decreasing as PT is decreased (Mack, 1989). We choose an initial value of a 

and compute the boundary conditions from (2.19). We then iterate on a, using Muller's method, until the 

boundary conditions are satisfied and the jump in G at (0,-1) is less than lo4. All calculations were done 

in 64 bit precision. 

3. Results. In this section we present results for the regularity condition, phase speeds, growth rates, 

and eigenfunctions of the stability problem. In all of our calculations we have taken y= 1.4 and 

0 I M I 7. The choice of a maximum value of 7 for M is based on the results of our previous calculations 

for the Tanh profile wherein we took 0 I M I 10, (Part I). It is clear from these results that the asymptotic 

( M + QO ) behavior of the solutions to the stability problem are apparent by Mach 7. Finally, we have 

taken the reference temperature T' to be 1500KO. We have also carried out stability calculations for 

values of T' of 500KO and 1OOOK" and found at most only a 2% change in the eigenvalues. Thus, we 

conclude that the actual value of the reference temperature T' is not the important parameter, rather it is the 
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ratio of the temperatures in the stationary stream to that of the moving stream PT which is the important 

parameter. 

3.1 The Regularity Condition. The Lees and Lin (1946) regularity condition is given by 

Let qc be a root of S(q). and define E = U(q,). If E lies in region 1 of the c, - M diagram, then Lees 

and Lin (1946) have shown that E = cN is the phase speed of a me neutral mode. The corresponding neutral wave number and frequency must be determined numerically. These modes are called subsonic neutral 

modes. It has been used by number of authors, for example, Lessen, et al. (1965). Gropengiesser (1969). 

Jackson and Grosch (1988). and Macaraeg, et al. (1988). to find the phase speed of subsonic neutral modes. 

If E lies in regions 2.3, or 4 of the c, - M diagram, then E does not correspond to the phase speed of a 

true neutral mode. The phase speeds in these regions must be found numerically. 

Figures (2-5) are plots of S (q) versus q for the three thermodynamic models, particular values of PT 

and Pr, and Mach numbers of 3 and 5. All of these curves have been normalized so that their maximum is 

one. All of these plots show similar behavio~ S has a single root at Mach 3, but three roots at Mach 5. 

This is quite apparent for the Tanh, Lock, and Sutherland profiles with Pr = 1 but is not so readily apparent 

for the Sutherland profile with Pr = 0.7 because of the scale. However, there are a pair of zeros of S in the 

interval [-2,0]. 

We have shown in Part I, for the Tanh profile, that S is a cubic in tanh(r\) and therefore has either 

one or tht& real roots. There is one real mot for M cM, and three real roots for M 2M,, where M, is 

given by (3.6) of part I. This is illustrated in Figure 6 for = 1, with the dashed curves the sonic curves. 

We see that one root, that which exists at Mach zero, has a constant value of E = 0.5. The other two roots 

form a "bubble" for Mach numbers greater than M,. The bubble is symmetric because of the symmetry of 

the mean profiles. We note here from Part I that if PT > 1, the value of E for the single root is a monotonic curve greater than 112, with a bubble below it. Also, if c 1, the value of E for the single root is 

also a monotonic curve less than 1/2, with the bubble now appearing above it. Thus at PT = 1 there is a 

saddle point in the E -M plane. This value of pT, which we denote by PT, plays a critical role in the 

behavior of the solutions of the stability problem, as will be shown below. The corresponding value of the 

root which appears at Mach zero is denoted by 6. 

We have not been able to demonstrate these properties analytically for the Lock and Sutherland 

profiles, but have been able to do so numerically. Some of the numerical evidence is presented in Figures 

7-9 where we plot E versus M for the different thermodynamic models and values of PT. The dashed 

curves are the sonic curves, shown for reference. Results are shown for the Lock profile in Figure 7 for the 

saddle point value of & = 0.57753. The surface is not symmemc about the line c^ = 0.4318 because the 

Lock profile is slightly asymmetric about q = 0. Figures 8 and 9 show results for the Sutherland model with 

Pr = 1 (Figure 8) and 0.7 (Figure 9). In both figures the monotonic curve and the bubble are visible with 

asymmetries cad by the asymmetry of the Sutherland profile. In both cases the value of PT was chosen 

to be close to the saddle point value &. Since the mean profiles are now coupled through the viscosity, the 

exact saddle point value is difficult to determine numerically. From Figures 8 and 9 it is clear that the 

value of & is strongly dependent on the value of the Praxxitl number. The dependence on Pr is shown very 
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clearly in the results presented in Figures 10 and 11. Figure 10 is a plot of IsT for the Sutherland profile as 

a function of the Prandtl number. As the F’randtl number is decreased from one, the saddle point value 

decreases dramatically. Figure 11 shows the variation of the corresponding c^ with F’randtl number. This 

value also decreases as the Prandtl number decreases. 

For all three thermodynamic models the three real roots of S always lie in regions 2, 3, or 4 for two 

dimensional modes. Therefore, for a two dimensional mode, only the single root which lies in region 1 is 

the phase speed of a true neutral mode. However, the sonic speeds c* are functions of the angle of propogation of the waves (M is replaced by M cos 0 in (2.17)). and as the angle is increased, the sonic curves 

shift towards higher Mach numbers thus increasing the extent of region 1. It is therefore clear that there 

will be some angle of propagation for which all three zeros of S lie in region 1. For this angle and all 

greater angles less than 90’. all three zeros of S yield the phase speed of true neutral subsonic modes. 

3.2 Neutral modes. In this section we present the phase speeds, hquencies, and wavenumbers of the 

neutral modes for the three models with pT of 2.0, 1.0, and 0.5 and Pr = 0.7 for the Sutherland model. 

Figure 12 shows plots of the phase speeds of the neutral modes as a function of Mach number for 

PT = 2 obtained by using the Tanh, Lock, and Sutherland models. The results for all three models show 

qualitatively similar behavior and small quantitative differences. It should be noted that PT of 2.0 is considerably larger than the saddle point value of PT for all three models. As the Mach number is increased past 

M,, the Mach number at which the phase speed equals that of the sonic wave, the subsonic neutral mode in 

region 1 is transformed into a fast supersonic neutral mode in region 2. At M, a slow supersonic neutral 

mode appears in region 4. There are some quantitative differences between the results using the different 

models. For example, in region 1, the neutral mode of the Tanh model has the lowest phase speed and that 

of the Lock model the highest with the phase speed of the Sutherland model in between. However there is 

only about a 10% maximum difference in the phase speeds. The same ordering with respect to magnitude 

of the phase speed is also Due for the fast supersonic neutral mode in region 2. The phase speeds of the 

slow supersonic neutral iodes in region 4 are very nearly the same for all of the models over the range of 

Mach numbers shown. The corresponding wave numbers of the neutral modes are plotted in Figure 13 and 

the corresponding frequencies in Figure 14. The magnitudes of aN and oN are similar for all of the models 

for the slow modes appearing in region 4. The neutral wave numbers and frequencies of the fast modes for 

all models eventually increases with increasing Mach number, while those of the slow modes decrease in 

this range of Mach numbers. 

Similar results are shown in Figures 15-17 for PT = 1.0. The phase speed of the neutral modes are 

plotted in Figure 15 and one can see a marked difference between the results for the Tanh model and the 

others. This is due to the fact that PT of one is the saddle point value for the Tanh model while [j~ occurs 

at a considerably lower value for the other models. Thus the phase speeds of the neutral modes increases 

for the Lock and Sutherland models as M increases, while that of the Tanh model is constant. At M,, the 

phase speeds of the Lock and Sutherland models are transformed from subsonic to fast supersonic neutral 

models. In conmt, the subsonic neutral mode of the Tanh profile is split at M, =Me into both fast and 

slow supersonic neutral modes. Figure 16 is a plot of the corresponding wavenumbers of the neutral modes. 

For M > MI, the Tanh model yields values of uN which are equal for both the fast and slow supersonic 

neutral modes. This is due to the fact that the mean profiles of both velocity and temperature are symmetric 

about q = 0. Beyond this, all of the models yield neutral wavenumbers which have the same general 
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behavior with increasing Mach number. Finally, the variation of the hequency of the neutral modes, oN, as 

a function of Mach number is shown in Figure 17. If Ma. then oN is essentially independent of the thermodynamic model. For M>M., all three models yield similar behavior. 

Finally, Figures 18-20 show the variation of the phase speed, wavenumber, and frequency of the neutral wave with Mach number for PT = 0.5. This value of PT is less than the saddle point value for the Tanh 

(= 1.0) and the Lock (= 0.57753) models but substantially greater than the saddle point value of the Sutherland model (= 0.145). Because of this, the results from the Tanh and Lock models are similar and both 

differ from those of the Sutherland model. In region 1 of Figure 18 the neutral phase speed obtained from 

the Tanh and Lock models is less than 2 because a pT of 0.5 is less than the saddle point value for these 

models. The opposite is me for the Sutherland model. Further, in this region the CN value for the Tanh 

model is smaller than that of the Lock model because a PT of 0.5 is considerably smaller than the saddle 

point value for the Tanh model and only slightly smaller than that of the Lock model. Because the PT of 

0.5 is smaller than the saddle point values for the Tanh and Lock models the subsonic neutral modes of 

these models are transformed into slow supersonic neutral modes at M, and fast supersonic neutral modes 

appear at M.. Conversely, the subsonic neutral mode for the Sutherland model is transformed to a fast 

supersonic neutral mode at M,, while a slow supersonic neutral mode appears at M.. In regions 2 and 4 it 

can be seen that the phase speeds of the fast and slow supersonic neutral modes are quite similar for all 

models. 

The variation of the neutral wave number with Mach number is shown in Figure 19. It is apparent 

from the results shown in this figure, and those of Figures 13 and 16, that aN increases in the subsonic 

region with decreasing values of pT. The Tanh model has the largest rate of change, followed by the Lock 

model with the SutherIand model having the smallest rate of change. In contrast to the results for the other 

values of pT, the slow supersonic neutral mode of the Tanh model has a larger wavenumber than that of 

the fast supersonic neutral mode. The wavenumbers of the neutral supersonic modes are roughly equal for 

the Lock model. At larger values of the Mach number, aN increases with increasing Mach number for all 

cases except that of the slow supersonic mode of the Sutherland model. Also, Figure 20 shows the change 

in the frequency of the neutral modes with Mach number. The behavior is qualitatively similar to that 

found for other values of PT. 

Figures 21 thru 23 are plots of selected two dimensional slow supersonic neutral eigenfunctions for 

PT = 1 and Mach 5. These plots show the variation of II with ql on the contour qi = -1. All of these have 

been normalized so that the maximum of the absolute value of l7 is unity. All show a rapid phase shift 

near the center of the mixing layer. Because these are slow supersonic neutral modes they have exponential 

decay at q = -0 and oscillate with constant amplitude and linear phase at q = 00. The exponential decay is 

the slowest in the Sutherland model and most rapid in the Tanh model. The Sutherland model also yields 

the slowest rate of change in the supersonic region. Despite these quantitative differences, the eigenfunctions obtained from all the models are qualitatively similar. 

3.3 Growth rates. The growth rates of the unstable modes are presented in this section and compared as a function of Mach number and pT for the *-models. In addition the variation of the growth 

rate with frequency for PT = 2.0 is presented at selected values of the Mach number. It should be noted 

that the phase speeds of the unstable modes lie in the vicinity of the phase speed of the neutral mode in 

region 1 and between the phase speeds of the neutral modes and the corresponding sonic curves in regions 2 
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and 4. Thus we see that at any given Mach number there is only a small band of phase speeds of the 

unstable modes. 

Figure 24 shows the maximum growth rates versus Mach number for PT = 2.0. The general variation 

is similar for all of the models. The maximum growth rate is largest at Mach zero and decreases by a factor of 5 to 10 as the Mach number increases from zero to M. and approaches a limiting value as the Mach 

number is further increased. At low Mach numbers the maximum growth rates for the Tanh model are the 

largest, followed in magnitude by those of the Sutherland and Lock models. At Mach numbers greater than 

M. the Lock model has the largest growth rates of the fast supersonic modes while those of the Tanh and 

Sutherland models are roughly equal. The second group of unstable modes, the slow supersonic modcs, 

appear just below MI. The growth rate of the most unstable of these modes first increases over a small 

range of Mach numbers and then decreases, approaching a limiting value at larger values of the Mach 

number. At this value of PT the maximum growth rates of these slow supersonic modes are about equal. In 

all cases, the maximum growth rate approaches a limiting value in this range of.Mach numbers. 

Similar results for PT = 1.0 are shown in Figure 25. In region 1 the growth rates obtained from the 

Tanh model are significantly larger than those of the other models. The maximum growth rates of these 

later two models are virtually identical in this region. Because this value of pT corresponds to the saddle 

point value for the Tanh profile, we see that the maximum growth rate first decreases as the Mach number 

approaches M., levels off and then begins to increase with increasing Mach number. Since the saddle point 

values for the other two models are lower than one, their behavior is the same as in the previous case; the 

maximum growth rate decreases as the Mach number approaches M. and then levels off for higher Mach 

numbers. Finally as in the previous case, the second group of unstable modes, those which appear at M., 

have maximum growth rates which are approximately equal and have similar behavior. 

The maximum growth rates for PT = 0.5 are plotted versus the Mach number in Figure 26. Note the 

change in scale of the maximum growth rate as PT is decreased. It is important to realize that PT = 0.5 is 

less than the saddle point value for both the Tanh and Lock models but greater than that for the Sutherland 

model. Thus for the Tanh and Lock models the maximum growth rate decreases up to M., levels off and 

then increases with increasing Mach number. However, for the Sutherland model the behavior is different 

because this value of PT is larger than its saddle point value. Therefore, the variation with Mach number is 

the same as in the previous two cases. However, if the stationary gas were to be sufficiently cooled, we 

would expect that the maximum growth rate of the Sutherland model would behave in the same manner as 

the other two models at higher Mach numbers. Finally, as in the previous two cases, the second group of 

unstable modes have maximum growth rates which are approximately equal and have similar behavior in 

this range of Mach numbers. 

Further insight into how the choice of thermodynamic model effects the growth rate of the unstable 

modes is provided by the results shown in Figures 27 and 28. In these figures we show the variation of the 

growth rate of both the fast and slow unstable supersonic modes for PT = 2.0 and M = 2.5 (Figure 27) and 

M = 5.0 (Figure 28) with the frequency of the disturbance. The results at Mach 2.5 show that the slow 

unstable supersonic modes exist in a very narrow range of frequencies compared to that of the unstable fast 

supersonic modes. The shape of the -a; versus 61 curves for the slow modes is similar for all of the 

models. The widest range of frequencies of the unstable slow supersonic modes and the maximum growth 

rates are those of the Tanh model, followed by those of the Sutherland model and then the Lock model. 
- 12- 

These results are somewhat different for the unstable fast supersonic modes, for which the Lock model has 

the largest maximum growth rates and the widest range of unstable frequencies. In terms of the maximum 

growth rates and range of unstable frequencies the Sutherland model is the next largest, followed by the 

Tanh model, Similar results for Mach 5 are shown in Figure 28. Note the change in scales between these 

two figures. All of the features shown in the previous figure appear in this figure. The only exception is for 

the slow supersonic modes in which now the Lock model has a slightly larger growth rate than the Sutherland model. 

It is apparent from the results shown in Figure 27 that the curves of growth rate versus frequency for 

the fast supersonic modes have two local maxima at this value of Pr and Mach number. As the Mach 

number increases the frequency of the maximum growth rate shifts towards higher frequencies. This is 

shown in the results of Figure 29, where we show the variation of the growth rate with frequency for PT = 

2.0 and Mach numbers of 2.3, 2.4, and 2.5 for the Sutherland model. The maximum of the growth rate 

occurs at a value of o of about 0.05 at Mach 2.3. At Mach 2.4 a second maximum, at an o of about 0.11 

has appeared which is roughly equal to that at 0.04. Finally, at Mach 2.5, the maximum growth rate is at 

an o of 0.12. Thus, as the Mach number increases there is a discontinuous jump in the frequency of the 

maximum growth rate at some Mach number. 

The change in the overall maximum growth rate as a function of PT at fixed Mach number is rather 

complex. This is due to the fact that for Mach numbers greater than M. there are now two unstable modes, 

one in region 2 and another in region 4. An increase in PT can result in a change in type of the most 

unstable mode. Some results bearing on this are shown in Figure 30. Here we have plotted the maximum 

growth rate as a function of for selected values of the Mach number for the three models. For example, 

at Mach 2 the most unstable mode for the Tanh model is a slow supersonic mode for values of PT up to 

1.25 while it is a subsonic mode for PT 2 1.50. For the Lock profile, again at Mach 2, the most unstable 

mode is a slow supersonic one up to PT = 0.75, a fast supersonic mode at PT = 1.0, and subsonic modes at 

values of PT greater than or equal to 1.25. Finally, the Sutherland model at the same Mach number has, as 

its most unstable mode, a slow supersonic one at PT = 0.5, a fast supersonic at 0.75, and subsonic modes for 

PT greater than or equal to 1.0. This change in mode type is related to the increase in M. with increasing 

PT independently of the thermodynamic model. Thus as PT is increased the extent of region 1 of thc 

c, - M diagram increases so that unstable modes at fixed M can undergo a change from supersonic to subsonic. 

The general trends of variation of the maximum growth rate with PT are similar for the Tanh and 

Lock models. At M = 1.5 the maximum growth rate is a monotonically decreasing function of PT over the 

range for which we have obtained results. At the other values of the Mach number there is first a decrease 

in the maximum growth rate with increasing PT and then a slight increase with a further increase in PT. 

The Sutherland model yields somewhat different results. Only at Mach 3 is the variation of the maximum 

growth rate similar to that obtained from the other two models; a slight decrease followed by a small 

increase as pT is increased from 0.5 to 2.0. At Mach 2 the Sutherland model gives a monotonically increasing maximum growth rate with increasing PT and at Mach 1.5 there is first a small increase followed by a 

slow decrease as PT is increased. 

i 

4. Conclusions. The characteristic features of the solutions to the stability problem for the compressible mixing layer are qualitatively similar for all af the thermodynamic models used in this study. However, 
- 13 - 

there are quantitative differences between the results obtained from the different models. These range from 

about 10% in the phase speeds to, in the most extreme case, about 50% difference in the maximum growth 

rates. 

Despite these quantitative differences there is an underlying similarity in the qualitative behavior of 

the solutions to the stability problem. For all three thermodynamic models the regularity condition yields a 

cubic in Mach number at fixed PT and for a critical value, BT, there is a saddle point. The behavior of the 

solutions to the stability problem depends on whether PT is larger or smaller than &. If PT is larger than 

fi~ the subsonic modes are transformed into fast supersonic modes at the Mach number at which their phase 

speed equals that of the sonic wave in the stationary stream. On the other hand, if PT is smaller than fi~ 

the subsonic modes are transformed into slow supersonic modes when their phase speed equals the sonic 

speed of the moving stream. If PT = fiT, then the phase speeds of the subsonic neutral modes are constant. 

This mode splits into a pair of fast and slow supersonic modes at M* . For any value of PT there is a single 

band of unstable subsonic modes in regionl, but there are two bands of unstable supersonic modes, one in 

region 2 and one in region 4. For all of the thermodynamic models, the second band of unstable supersonic 

modes appears when the Mach number equals Me. These second modes are slow supersonic modes if 

PT > 131. and fast supersonic modes if PT e fir. Both the fast and slow unstable supersonic modes have a 

rather small variation in the phase speed about the mean value. 

We have found that, independent of the thermodynamic model, the maximum growth rates of the 

unstable modes decrease by a factor of 5 to 10 as the Mach number is increased from zero to Me. As the 

Mach number is increased beyond Me the maximum growth rates of the fast supersonic modes approach a 

limiting value. On the other hand, the maximum growth rates of the slow supersonic modes modes first levels off for A4 > Me and then begin to increase with a further increase in the Mach number. 

In view of the results presented here we conclude that all of the thermodynamic models yield qualitatively similar results. In view of this, all previous work based on simplified thermodynamic models yields 

qualitatively correct results. Because of the analytical simplicity of the Tanh model and because it appears 

to be a reasonable approximation to the mean velocity and temperature profiles we suggest that it is appropiate for use in nonparallel and nonlinear models of the stability of the compressible mixing layer. 
- 14 - 

REFERENCES 

Blumen. W. 1970 Shear Layer Instability of an Inviscid Compressible Fluid. J. Fluid Mech., 40, 

769-78 1. 

Blumen, W., Drazin, P. G. & Billings, D. F. 1975 Shear Layer Instability of an Inviscid Compressible Fluid. Part 2. J. Fluid Mech., 71, 305-316. 

Brown, W. B. 1%2 Exact numerical solutions of the complete linearized equations for the stability of 

compressible boundary layers. Norair Rep. No. NOR-62-15, Northrop Aircraft Inc., Hawthorne, CA. 

Chapman, D. R. 1950 Laminar mixing of a compressible fluid. NACA Rep. 958. 

Drazin, P. G. & Davey, A. 1977 Shear Layer Instability of an Inviscid Compressible Fluid. Part 3. J. 

Fluid Mech., 82,255-260. 

Dunn, D. W. & Lin, C. C. 1955 On the stability of the laminar boundary in a compressible fluid. J. 

Aero. Sci. 22,455477. 

Gill, A. E. 1965 Instabilities of “Top-Hat” Jets and Wakes in Compressible Fluids. Phys. Fluids, 8, 

1428-1430. 

Gropengiesser, H. 1969 On the Stability of Free Shear Layers in Compressible Flows. (in German), 

Deutsche Luft. und Raumfahrt, FB 69-25, 123 pp. Also, NASA Tech. Translation NASA TT F12,786. 

Jackson, T. L. & Grosch, C. E. 1988 Inviscid Spatial Stability of a Compressible Mixing Layer. J. 

Fluid Mech., in press. 

Jackson, T. L. & Grosch, C. E. 1989 Inviscid Spatial Stability of a Compressible Mixing Layer. Part 

II. The Flame Sheet Model. ICASE Rep. No. 89-18. Also, J. Fluid Mech., submitted. 

Klemp, J. B. & Acrivos, A. 1972 A note on the laminar mixing of two uniform parallel semi-infinite 

streams. J. Fluid Mech., 55, 25-30. 

Lees, L. & Lin, C. C. 1946 Investigation of the stability of the laminar boundary layer in a compressible fluid. NACA Tech. Note 11 15. 

Lees, L. & Reshotko, E. 1962 Stability of the compressible laminar boundary layer. J. Fluid Mech., 

12, 555-590. 

Lessen, M., Fox, J. A. & Zien, H. M. 1965 On the Inviscid Stability of the Laminar Mixing of Two 

parallel Streams of a Compressible Fluid. J. Fluid Mech., 23, 355-367. 

Lessen, M., Fox, J. A. & Zien, H. M. 1966 Stability of the Laminar Mixing of two Parallel Streams 

with Respect to Supersonic Disturbances. J. Fluid M-ech., 25,737-742. 

Lock, R. C. 1951 The velocity dismbution in the laminar boundary layer between parallel streams. 

Quart J. Mech. and Appl. Math., 4.42-63. 
- 15 - 

Macaraeg, M. G., Streett, C. L.. & Hussaini, M. Y. 1988 A spectral collocation solution to the 

compressible stability eigenvalue problem. NASA Technical Paper 2858. 

Mack, L. M. 1965 Computation of the Stability of the Laminar Compressible Boundary Layer. In 

Methods in Computational Physics. B. Alder, S. Fernbach, and M. Rotenberg, eds. Vol. 4, 247-299, 

Academic Press. 

Mack, L. M. 1984 Boundary layer linear stability theory. In Special Course 

on Stability and Transition of Laminar Flow. AGARD Report R-709, 3-1 to 3-81. 

Mack, L. M. 1987 Review of linear compressible stability theory. In Stability of 

Time Dependent and Spatially Varying Flows. D. L. Dwoyer and M. Y. Hussaini, eds. SpnngerVerlag, 164-187. 

Mack, L. M. 1989 On the inviscid acoustic-mode instability of supersonic shear flows. Fourth Symposium on Numerical and Physical Aspects of Aerodynamic Flows, California State University, Long 

Beach, California. 

Ragab, S. A. & Wu. J. L. 1988 Instabilities in the Free Shear Layer Formed by Two Supersonic 

Streams. AIAA Paper 88-0038. 

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

University Press, Great Britain. 

Tam, C. K. W. & Hu, F. Q. 1988 Instabilities of Supersonic Mixing Layers Inside a Rectangular 

Channel. AIAA Paper 88-3675. 

Ting, L. 1959 On the mixing of two parallel streams. J. Math. Phys., 28, 153-165. 

Zhuang, M., Kubota, T. & Dimotakis, P. E. 1988 On the Instability of Inviscid, Compressible Free 

Shear Layers. AIAA hper 88-3538. 
-16- 

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Figure 30. Plot of maximum growth rates versus PT for Mach numbers (1) 1.5, (2) 2.0, (3) 

3.0 for Subsonic (o), Fast Supersonic (O), and Slow Supersonic (0) Modes; (a) Tanh, (b) 

Lock, (c) Su therland. 
Report Documentation Page 

I 2. Government Accesston No. I 3. Recipient's Catalog NO. 1. Repon No. 

7. Authods) 

NASA CR- 181855 I I 

8. Performing Organization Repon No. 

ICASE Report No. 89-32 I 

5. Report Date 4. Title and Subtitle 

17. Key Words (Suggested by AuthorW) 

MIXING LAYER. PART 111. EFFECT OF THERMODYNAMICS 1 

INVISCID SPATIAL STABILITY OF A COMPRESSIBLE 

18. Dmtribution Statement 

19. swum CW. 20. Security Clad. (of this page). 21. No. of pages (of this report) 

gnclassified Un cl a s si f i ed 46 

T. L. Jackson 

C. E. Grosch 

9. Performing Organization Name and Address 

Institute for Computer Applications in Science 

Mail Stop 132C, NASA Langley Research Center 

Hampton, VA 23665-5225 

National Aeronautics and Space Administration 

Langley Research Center 

Hampton, VA 23665-5225 

and Engineering 

2. Sponsoring Agency Name and Address 

22. Price 

A0 3 

89-32 

10. Work Unit No. 

505-90-21-01 

11. Contract or Grant No. 

NAS1-18605 

13. Type of Report and Period Covered 

Contractor Report 

I 

5. Suppbmentarv NotLangley Technical Monitor: Journal of Fluid Mechanics 

Richard W. Barnwell 

Final Report 

16. Abstract 

We report the results of a comprehensive comparative study of the inviscid 

spatial stability of a parallel compressible mixing layer using various models for 

the mean flow. The models are (a) the hyperbolic tangent profile for the mean 

speed and the Crocco relation for the mean temperature, with the Chapman 

viscosity-temeprature relation and a Prandtl number of one; (b) the Lock profile 

for the mean speed and the Crocco relation for the mean temperature, with the 

Chapman viscosity-temperature relation and a Prandtl number of one; and (c) the 

similarity solution for the coupled velocity and temperature equations using the 

Sutherland viscosity temperature relation and arbi trary but constant Prandtl 

number. The purpose of this study was to determine the sensitivity of the 

stability characteristics of the compressible mixing layer to the assumed 

theromdynamic properties of the fluid. It is shown that the quantative 

differences resulting from the difference in the thermodynamic models. In 

particular, we show that the stability characteristics are sensitive to the value 

of the Prandtl number. 

mixing layer, themodyami cs', 

spatial stability 

34 - Fluid Xechanics 

NASA FORM l(126 OCT 86 

NASA-Langley, 1989