STABILITY OF AXISYMMETRIC LAMINAR WAKES

Harris Gold, Staff Scientist Research and Advanced Development Division,

Avco Corporation

I. Introduction

Early theoretical and experimental studies of wakes behind blunt and slender 5odies have been confined to low speed flows. These wakes are characterized by the existence of time-depend- ent flow fields and large-scale vortex shedding1. Recently, the wake behind a vehicle moving at hypersonic speeds has been the subject of consid- erable interest becau e it is a significant source of observabies . In this case, a steady laminar free-shear annulus, shed from the body, is established; at the apex (or "neck") of this "hollow" cone, a viscous wake is formed which persists for thousands of body diameters down- stream of the body. The wake llcore'' temperature is higher than the "edge" temperature and the flow external to botn the free-shear annulus and viscous wake is moderately supersonic for blunt bodies and hypersonic for slender bodies. The effect of compressibility is essential in deter- mining the flow field of the viscous core.

9

An important phase of the hypersonic vake problem has been the study of the onset of laminar-turbulent transition. When the body boundary layer is laminar, transition w i l l occur either in the free-shear an u us or the inner viscous wake. Experimental"' and studies show that the laminar free-shear annulus is extremely stable when the local velocity is moderately supersonic. a stability analysis shows that the proper Mach number is the one based on the velocity differ- ence (Uz - U$)+ and the speed of sound at the edge of the wake. Laminar-turbulent transition in the hypersonic wake is very similar to that occurring in the low-speed wake of a thin flat plate aligned to the flow, in which the "neck" takes the place of the trailing edge of the plate8. stown that it is possible, based upon the nature of the velocity fluctuations, to divide the trans- ition region of a low-speed, flat-plate wake into three subregions: (1) a linear region, in which small, two-dimensional sinusoidal fluctuations grow exponentially in the direction of the main flow and have amplification rates that agree very well with those calculated from laminar stability theory; (2) a non-linear region, in which the gromh rate deviates from the exponential law and the velocity fluctuations, although still two- d:mensional in nature, have distributions of amFlitude and phase that can be characterized by a double row of vortices; and (3) a three-dimen- sional region, which just precedes turbulence itself.

Downstream of the "neck",

The experiments of Sat0 and Kuriki8 have

The purpose of this present investigation is to study the effects of compressibility on the hydrodynamic stability of axisymmetric wake flows. The stability characteristics of such flows are relatively insensitive to Reynolds number, for

+ A complete list of symbols occurs at the end of this paper. Dimensional. quantities are starred; non-dimensional quantities are unstarred.

sufficient13 high Rejnolds nmbers, because of the absence of any solid boundaries in the flow field. Therefore, interesting and important result.. can be obtained b3 consideriqg the "inviscid limit" of tte small disturbapce equations, in which the viscosixy and thermal conductivity of the fluid can be neglected to a certain order. These wake flows are also dynamically unstable because of the occurrence of an extremum in the generalized density-vorticity product9,lO. This study is also restricted to subsonic disturbances, i.e., dis- turbances whose relative propagation velocity, far from the wake axis, is less than the "edge" speed of sound. These disturbances have amplitudes that die out exponentially far from the wake axis. This author has b en guided by the outstanding work of

boundary layer flows, and Batchelor and Gill'', for axisymmetric incompressible jets. L. Lees and this author12 have recently discussed the inviscid stability of compressible two-dimensional and axisymmetric wake flows and have presented detailed numerical results for the two-dimersional case. This paper will only concern itself wit? the axi- symmetric case.

Lees and Lin 5 , for two-dimensional compressible

In Section 11, the axis3;mmetrLc laminar wake stability problem is formulated and instability criteria derived. Section 111 deals with the physical interpretation of the instabilitly phenom- ena in terms of the wave motion of th.e disturbance and the transport of energy and vorticity, and Section Tv contains numerical stability calcula- tions for axisymmetric compressible wake flows. In light of these results a tentative picture of laminar-turbulent transition in hypersonic wakes is presented (Section V).

11. Formulation of the Axisymetric Stability Problem

An infinitesimally small disturbance is im- posed upon a mean (or steady flow and the behav- ior of the amplitude of the disturbance is exam- ined as time progresses. If, for large values of time, the disturbance is damped out, the motion is said to be stable; if not, the motion is said to be unstable with respect to infinitesimally small disturbances. It is much easier to prove that a motion is unstable than stable. If the flow is unscable to disturbances of any kind, even the simplest kind, it is always unstable; but the flow may be stable with respect to one type of disturbance and not another. In hydrodynamic stability theory, the disturbance is usually assumed to t a w a wave-like nature. The problem then is to find combinations of the wave number and wave speed of the disturbance and the Reynolds number of the mean flow for which the fluid motion is unstable, neutrally stable or stable.

11.1 Outline of the Stzbility Problem

The total flow consists of a time-independent or mean component an3 an infinitesimally small component -which is both space and time dependent. The total I 'lo:~ satisfies the conservation equations 01- m a s - , momentum and energy and an equation of

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, ,a te . - L "he mean f low sat isf ies t h e s teady f low equat ions o r some ap_oroximation t o them, f o r exam- > l e , t h e boundary l a y e r equat ions. The conserva- t i o n equations f o r t h e d l s x r b a n c e a r e obtained by sub t rac t ing t h e mean f low equat ions from t h e t o t a l f low eqGations. The hydrodynamic s t a - o i l i t y equaGions a r e obtained from t h i s s e t by neg lec t ing qu-adratic and h igher o rder terms of t h e d i s t u r b- ance TLantitie;.

The c o e f f i c i e n t s of t h e r e s u l t i n g l i n e a r p a r t i a l d i f f e r e n t i a l equat ions depend upon t h e mean f low q u a n t i t i e s . Time appears on ly as t h e d e r i v a t i v e ;>/at* and hence s o l u t i o n s f o r t h e d l s tu rbance ampli tudes con ta in ing an exponential time f a c t o r

+ may b e assumed. The r e s u l t i n g d i f f e r e n t i a l equat ions w i l l c o n t a i n t h e space coord ina tes as t h e o n l y independent v a r i a b l e s .

For p a r a l l e l and q u a s i- p a r a l l e l flows, i . e . , motion i n which t h e mean normal v e l o c i t y component i s zero o r very s m a l l compared t o t h e main veloc- i t y component, a l l mean q u a n t i t i e s are considered t o be independent of t h e a x i a l space coord ina te , x* ( t o o rder l /dReq , ) , and t h e angula r coordi- na te , @. Consequent y, s o l u t i o n s of t h e form

cv i(d+* + n@) q*'(x*, r*, @ ) = q*l(r*) e , (2.2)

when n i s an i n t e g e r , might b e expected. The exponent i s p u r e l y imaginary because t h e d i s t u r b- ance must be bounded f o r x* at both +(L) and -aD and. s ingle- valued with r e s p e c t t o t h e angula r coord ina te . The r e s u l t a n t set of o rd inary d i f f e r - e n t i a l equa t ions w i l l have r* as t h e on ly inde- pendent v a r i a b l e .

The d i s tu rbance ampli tude q*'(r*) and t h e The wave v e l o c i t y c* are taken t o b e complex.

main f low i s s t a b l e , n e u t r a l l y s t a b l e o r u n s t a b l e t o t h e s e waves according t o whether t h e imaginary p a r t of c* i s negat ive, zero o r p o s i t i v e , respec- t i v e l y . The r e a l p a r t of c* i s t h e phase o r propagat ion v e l o c i t y of t h e wavy d is tu rbance . The q u a n t i t y &* = 2 n / h * i s t n e wave number of t h e d i s tu rbance and i s t aken t o b e real and p o s i t i v e .

The a m p l i f i c a t i o n rate of t h e d i s tu rbance i s def ined as fol lows:

I n t h e l a b o r a t o r y , t h e experimenter has a quasi- s t a t i o n a r y problem. A s t h e d i s tu rbance propagates

+ The assumption t h a t t h e d i s tu rbance has t h e form exp (-i&*c"t*) i s known as t h e normal mode ap- proach t o hydrodynamic s t a b i l i t y . Recently, t h e i n i t i a l v a l u Casel3, Linl' and MilesT. The normal mode approach i s a p p l i c a b l e t o d i f f e r e n t i a l opera tors having d i s c r e t e eigenvalues while t h e i n i t i a l va lue method should be used with s i n g u l a r opera tors and/ o r continuous e igenva lues . However, it appears t h a t t h e modes lead ing t o i n s t a b i l i t y are assoc i- a t e d wi th t h e d i s c r e t e eigenvalues. It i s f o r t h i s reason t h a t t h e normal mode approach w i l l be used i n t h i s paper .

problem has been emphasized by

downstream of i t s o r i g i n , i t s amplitude changes i n both space and time. a m p l i f i c a t i o n r a t e , o r t h e rate a t which a d i s tu rb- ance w i l l amplify with d i s t a n c e i n t h e main flow d i r e c t i o n . The wave speed i s a func t ion of both t h e wave number of t h e d i s tu rbance and t h e Reynolds number of t h e mean flow. t h e dis turbance, i .e. , t h e v e l o c i t a t which t h e d i s tu rbance energy must propagatel: is

= c* = (d/do(*)( O(*C*)

= c* + (d*dc*/dd*) . (2.3b)

One measures t h e s p a t i a l

The group v e l o c i t y of

(dx*/dt*) g

The s p a t i a l ampl i f ica t ion rate i s then defined. by

and u*

The s p a t i a l ampl i f ica t ion r a t e i s cons tan t f o r p a r a l l e l f lows. For q u a s i - p a r a l l e l flows, it i s computed a t each streamwise s t a t i o n by using a l o c a l mean veloci ty- temperature p r o f i l e t h a t i s assumed t o be independent of X* a t t h a t s t a t i o n . The t o t a l a m p l i f i c a t i o n (or decay) of t h e d i s tu rb- ance i s found by t h e piecewise i n t e g r a t i o n of t h e l o c a l a m p l i f i c a t i o n (o r decay) r a t e s .

For wake flows, t h e t o t a l ampl i f ica t ion of a d i s tu rbance w i l l depend upon t h e decay of t h e mean v e l o c i t y . It is t h e r e f o r e necessary, f o r s t a b i l i t y cons idera t ions , t o t ransform from a coord ina te

+

system i n which t h e observer i s f ixed i n t h e body t o one i n wh:c% t h e observer Is f ixed i n a f l u i d at r e s t (Fig. 1). In t h i s l a t t e r system, t h e observer sees t h e " ve loc i ty defec t" of t h e wake wave w i l l have a propagat ion v e l o c i t y (c* - U,*) i n t h e x* d i r e c t i o n . Let t h e mean f low be dimen- s i o n a l l y represented by a c h a r a c t e r i s t i c length, L* (wake ha l f- wid th) , a c h a r a c t e r i s t i c v e l o c i t y , Ue* - Uo* ( r e l a t i v e v e l o c i t y at tl-e wake cen te r - l i n e ) , and t h e temperature, dens i ty , p ressure and v i s c o s i t y by t h e i r e x t e r n a l va lues ( s e e L i s t of Symbols). Then

and t h e

w ( r ) = (u,* - u*)/(u~* - u,*) , o(= &*L* , M = (ue* - Uo*)/ ae* , R = (Ue* - Uo*)L*h *

e

c = (c* - ue*)/(ue* - uo*) , ... (2.4) where c* (and c ) can be complex.

11.2 Small Disturbance Equations

The i n v i s c i d small d i s tu rbance equat ions can be found by tak ing t h e l i m i t ( & R h o , of t h e complete s m a l l d i s tu rbance equat ions. They are i d e n t i c a l with t h e equat ions obtained by ignoring v i s c o s i t y and thermal conduc t iv i ty . Batchelor and Gillll have found it i n s t r u c t i v e t o t rans form t h e incompresslble axisgmmetric small d i s tu rbance equat ions i n t o a two-dimensional form by a s u i t a b l e t ransformation of v e l o c i t y coord ina tes . Lees and Gold1;) g o i n t out t h a t t h i s t ransformation a l s o a p p l i e s t o compressible flow because it i s pure ly

+ The mean f low f i e l d i s s teady i n body-fixed coord ina tes .

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kinematice in nature.

Following Batchelor and Gill'', the lines of intersection of the family of surfaces

6 x + n@ = constant r = constant,

are circular helices on which the phase of the disturbance wave is constant (Fig. 2 ) . The dis- turbance amplitudes depend only on the variables r and d x + n@ and are constant on a helix of this family. The new orthogonal disturbance velocity components are (Fig. 2):

here q is the velocity component perpendicular to both a radial line and the helix of constant phase, q is the velocity component parallel to the tan Snt to the helix of constant phase, and m = 1-2 = 4 cscW is the magnitude of the total wave number. The radial component remains unchanged in this velocity transformation. The tangent to the helix of constent phase makes an angle Q = tanm1 (ar/n) with the axis of the cylinder. In this orthogonal coordinate system the small disturbance equations are:

1

Continuity:

(l/r)(rq2)' + icsca ql + i(w-c)(T/x ) = o ... (2.6) 1- Momentum:

gb(w-c)q + sin= q2w'] = -(iv/l M ~ ) c s c ~ ... (2.7) 1

2- Momentum: id2y(w-c)q2 = - W Y ~ M 2

3- Momentum:

i(w-c)q3 - c o s ~ q W' = 0. 2 Energy:

State:

(2.11)

The quantities ql and s, correspond, respectively, to the longitudinal and normal disturbance veloc- ities in two-dimensional flow. The velocity com- ponent q appears only in Eq. (2.9) and is deter- mined onJe q2 is known. "his component plays the same role as the transverse disturbance in the corresponding problem of three-dimensional disturb- ances to a two-dimensional flow. It is also inter- esting to note that the pressure perturbation does not appear in the 3- Momentum equation, i.e., the pressure gradient in the 3- direction is zero.

83y eliminating ql and from Eqs. (2.6-2.8), a second order linear differential equation in 92 is obtained

where

(2.12)

The self adjoint equation for the pressure per- turbation is

p- Tn'/(w-c)'] ' - [cA2b/q(w-c)2] = 0 . ... (2.14)

The other disturbance amplitudes are found in terms of either or T by using Eqs. (2 .6 - 2.11).

11.3 Boundary Conditions

11.3.1 The boundary conditions on the axis are

Inner Boundary Conditions (r = 0)

derived from the purely kinematic condition that all disturbance amplitudes (and disturbance vorticity components) must be finite there, re- gardless of the viscosity o r the thermal conduc- tivity of the fluid. A detailed derivation of the boundary conditions on the wake axis is given in Lees and Gold12 and w i l l not be repeated here. Instead, the boundary conditions will be summarized below, as follows:

n = O 4' q2 = q3 = v' = s ' = Q' = 0

n = l ql' = q2' = q3 = = s = Q = 0

- - n 1 ql = q 2 = q 3 = T = s = Q = O . ... (2.15)

The radial velocity disturbance amplitude is identically zero on the axis f o r n f 1 and is arbitrary for n = 1. The flow is therefore free to move normal to the axis (on the axis) only for the latter mode and has zero motion in the radial direction for all other modes. This additional "degree of freedom" seems to indicate that the n = 1 is the most unstable of all the possible allowable modes, a fact which is borne out by numerical calculations (Section IV).

11.3.2 Outer Boundary Conditions (r-m )

The outer boundary conditions must be deter- mined from either Eq. (2.12) or (2.14). As r--.q), w - 0 (exponentiall: ), T -+ 1 (exponentially) and Eq. (2.14) takes the limiting form

(2.1611)

In order to define p uniquely, introduce a "cut" along the negative real axis of the complex a- plane, as suggested by Lees and Ling, so that the real part of will always be positive as long as - w L argR c TT . The asymptotic form of the solution of Eq. (2.16) is

(l/F) exp 2 (3 r as R l ( ) r - r c o . (2.17) From physical considerations, f o r large values of r and must therefore behave like

must be bounded

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the condition expressed by Eq. (2.17) with the positive (+) sign as r +a , or

p' +p* = 0 as r---.o

when

(2.18a)

(2.18b)

The physical significance of these asymptotic solutions is discussed in Section 111.1. At the present, it suffices to say that neutral disturb- ances will be classified as subsonic, sonic and supersonic, according to the relation

-eR y < 1/M . 11.4 Singularities of the Disturbance Equations

11.4.1 Singularity at w = c

The disturbance radial velocity equation [Eq. has a regular singularity at the annulus,

{ 0 in the complex r-plane. (2.12)] w z e, r = r solution of %he equation in the neighborhood of this "singular annulus" is obtained by the method of Frobeniusl6. amplitudes in the neighborhood of r = rc is:

The

The behavior of the disturbance

93 J n [ (l/-f ) + k loge)C] - nQ -ns(2.20a) (2.20b)

The coefficient b is not determined by this meth- od. In going from $ 7 0 to 7 L 0 the proper path lies below the point r = rc for w ' 7 0 and hence, for proper analytical continuation, the term log, 17 I -

1

c ( 7 7 0) must be modified to (log,

i v ) ( 7 4 0) in Eq. (2.20a)9,11.

It will be shown in the following section that k must vanish for some -w = -c solution of Eq. (2.12) is to exist For neutral subsonic disturbances. However, even if k van- ishes, the disturbance amplitudes q3 (n # 0), 8 and s have hyperbolic discontinuities at r = r . The singularity in q is smoothed out by introgu - ing viscosity in a tdin annulus of order (O(R)-l73 in thicknessll,l6; the sing-ularities in 8 and s are moothed out by introducing thermal onduc- tivity in thin annuli of order (a(RPr)-'73 in thickness l - 7 ~ ~ ~ . The disturbance is inviscid in the sense that the gross features of the disturb- ance amplitudes can be found outside of the vis- cous layer without considering the effects of viscosity.

< 1/M if a

11.4.2 There is another singularity at the origin,

r = 0 when c = -1. that the only non-trivial solution for this case is the stationary deviation cA= 0, n = 0, c = -1 and \y = W-C.

Singularity at r = 0

Batchelor and Gill'' showed

11.4.3 The quantity, 3 [ Eq. (2.13)1 , can be re-

Singularity at the Sonic Point

mitten in terms of the local relative Mach number, ML [Section 111.1, Eq. ( 3 4 1 of the disturbance

At the sonic point, % = -1 and 3 -a. this is only an apparent singularity of Eq. (2.12), as Eq. (2.14) for the pressure disturbance shows. The coefficients of this latter equation are finite at this point.

However,

11.5 Stability Criteria For convenience the self-adjoint equations

and boundary conditions for the radial and pressure disturbance amplitudes are repeated here:

Radial Disturbance

( f?')' - [P + (d2/Tr)]* = 0 (2.22a)

y ( 0 ) = 0 'y' + p v = 0, r+co (2.22b)

n = o ~ ' ( 0 ) = o i ? f o 7 t ( o ) = 0

+ PIT = 0, r+=.(2.23b)

This system of equations and boundary conditions constitute a Sturm-Liouville system analogous to that treated by Lees and Ling for two-dimensional compressible boundary layers, and by Batchelor and Gillll for misymmetric jet or wake-type flows. Guided by the work of these authors, this author has extended their results to compressible axi- symmetric wake flows12,16. The main results of this latter study are presented here.

* + Multiply Eq. (2.23a) by , integrate across the wake applying the boundary conditions, and take the imaginary part of the resulting equation. The following equation is obtained

If a self-excited disturbance is to exist, then (w-e,) must change sign in the infinite interval. or 0°C. -eR c 1, For subsonic disturbances, '

0 C -eR c 1/M. Another conclusion regarding the existence of self-excited disturbances can be made by considering the radiak disturbance equation. Multiply Eq. (2.22a) by 9 ; multiply y by the complex conjugate of Eq. (2.22a), and subtract the resulting equations to obtain

(2.25) 4

where both r = 0 and r+ m [Eq. (2.22b)r for subsonic disturbances, E' must change its sign in the interval (0 c r L Q, ). Therefore, a necessary condition for the existence of self-excited sub- sonic disturbances(cI 7 0) is that the ( h a g 4 and/or (Imag p) must change their sign in the in- terval 0 < r & . For incampressible flow, Imag 5 = 0 and I m a g p = (cI/Iw-c~~)

= h a g ( 3 Y '4''). Since W vanishes at

)

( SRW') must

+ The symbol ( * ) denotes the complex conjugate of a quantity.

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vanish at some interior point of the interval if amplified disturbances are to existll. ional statements for compressible flow are diffi- cult tb obtain.

Any addit-

For a neutral disturbance, hag 3 = Imag p = 0, 1' = 0 [Eq. (2.25)] and E must be constant except possibly at the critical annulus, r = r o r , in other words, the Wronskian of the solution Wf 5 The jump W across r = re is found by integrating Eq. (2.25J across the wake and taking the limit c -c 0. The only contribution comes from the I second term in which the integrand behaves like a delta function in the above-mentioned limit. The jump in E is16

C'

is constant outside the critical annulus.''

(2.26)

for we' 7 0.++ points, a necessary condition for the existence of inviscid, neutral subsonic disturbances is that k must vanish from some -w = -c 4 1/M for # 0. It is easy to show that 9 0 even if 4 changes

Since 2 must vanish at the end

sign in the infinite interval E . When f does not change sign in the infinite

interval, it can be shown that the vanishing of k is not only necessary but sufficient for the existence of inviscid, neubral subsonic disturb- ances, and also sufficient for the existence of "adjacent" amplified disturbances. similar to those given by Lees and Ling.

The proofs are

By the oscillation theorem of Sturmlg, if [p + (o( 2/Tr)] is positive and f 7 0, then montonic; if it is negative, is oscillatory. Since \y must vanish at the end points, an addi- tional necessary cohdition for the existence of neutral subsonic disturbance is that

is

This condition restricts the number of possible modes that can exist, i.e., it imposes an upper bound on the value of n2 (provided that K vanishes for sone -w = cR 4 If p changes sign in the interval there will be a number of "possible" neutra-l modes having different wave numbers, for the same value of CR = cb.

l/M).

111. Physical Mechanism of Instability

In order to understand the instability phenomena, it is important to examine the propag- ation of the wavy disturbances and the transport of energy and vorticity between the mean flow and the disturbance.

111.1 Wave Propagation

(l/m ) exp (- pr ) [Eq. (2.17)Ipf the inviscid equations and the exponential part of the disturb- ance,

The product of the asymptotic solution,

exp [ i d (x-et) + in@ ],represents progressive

+ The constant for r-re 7 that for r-rc 4 0.

0 can be different than

++ This result can also be obtained by considering the expansions around the critical annulus [Eq. (2.20a)L In fact, it is just the discontinuity in -v (and ql) across r =--re that ieacts to € ~ - - j i ~ ~ p - - in W.

waves with the direction of propagation of the wavesdependent upon the frame of reference of the observer. If a wave propagates outward and in the negative x direction with respect to an observer fixed in a fluid at rest, it will propagate inward and in the positive x direction to an observer fixed in the body, and vice versa (Fig. 1). amFlified disturbances, cI 7 will correspond to an outgoing wave with an expon- entially damped amplitude as r + m , in coordinates fixed in the fluid.+ ances), then

For 0, the disturbance

If cI = 0 (neutral disturb- fl I = 0 and

A, = l-M2cR2 = 1- (e* - Ue*)2/ ae 2 . The disturbance is classified according to whether

AR +, 0 ( o r Ue* - cR* 5 ae*). Corresponding to subsonic, sonic and supersonic disturbances, re- spectively. This classification has the fol.lowing interpretation. Consider two successive positions on the helix of constant phase, at time t and t + A t (Fig. 3). ity is given by

The local relative propagation veloc-

cp* = (eR* - U*) sin G . The local relative Mach is defined as the ratio propagation velocity of local speed of sound

number of the wave front of the local relative the disturbance to the

The wavy disturbance is supersonic with respect ML? -1, or

locally subsonic, sonic or to the mean flow if

fi $ -M (cR-w) sinG . (3.4)

For large values of r, ML --~r Mc for subsonic disturbances far f r h the axis, is -c 4 1/M. Thus only disturbances wbose relative propagation velocity is less than the ambient speed of sound, a,*, w i l l have amplitudes that vanish exponentially as r - in . This does not imply that supersonic disturbances may not exist within the wake when the disturbances are subsonic far from the wake axis. In fact, for n = 0, if

and the condition

R

an imbedded region of relative supersonic propag- ation velocity exists near the wake axis. For n f 0, e= 0 and ML 7 0 [Eq. (3.3)] at r = 0; a locally supersonic region if it exists, is bounded on both sides by locally subsonic regions.

For n 0, tkAe wave front will propagate in a direction almost normal to the x- axis, near the axis, and will gradually turn in the direction of the x- axis as r is increased. As r - a , ~ 4 T l / 2 and the wave front will propagate virtually sarallel to the x- axis (Fig. 4). two-dimensional flow) the no& to the wave front is parallel to the x- axis for all values of r. It is to be emphasized that the component of the propag- ation velocity in %he x- direction is always cR* - Ue* in a coordinate system fixed in a fluid at rest.

+ In this coordinate system and the quantities A, and b1 take the sign of CI. ++ ML a n d c fixed in a Fluid at rest.

For n = 0 (and also for

cR is less than zero

are negative in a coordinate system

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If CI = 0 and 4~ C 0 (-eR > 1/M) (neut ra l subsonic disturbances), the pressure distvrbance i s composed of both incoming and outgoing waves, in general, of uneaual amplitudes. Unless the S o m r f eld radia t ion condition (pure incoming o r outgoing waves) i s imposed f a r from the bake axis, the cha rac t e r i s t i c values of the eigenvalue problem w i l l not be d i sc re t e . This problem and t h a t of neu t r a l sonic disturbance are discussed i n Lees and Ling and will not be t r ea t ed here.

111.2 & e r a Transport Qual i ta t ive ly , the mechanism of k ine t i c

energy t r ans fe r between t h e mean f l o w and the d is- turbance can be described by the following r e l a t ion ( i n dimensionless form):

d/dt J ( p/2) (q i2 + G2 + q i2 ) r d r n " .* = j r ( d w / d r ) r d r - [viscous d iss ipa t ion]

0 ... (3.6a)+

where

t = Reynolds shear s t r e s s = - y s r 4,'

d/dt = 3/3t + U 3 / 3 x (3.6b)

and the ba r indica tes an average over one wave length i n x and one period i n #. The term on the left-hand s ide of Eq. (3.6a) represents t he rate of increase of t he k ine t i c energy of t he disturb- ance, while t he f i r s t term on the right-hand s ide represents t he conversion of energy from t h e bas ic f l o w t o the disturbance by t h e ac t ion of t he . Reynolds shear s t r e s s .

For a neu t r a l disturbance, t he time r a t e of charge of t he k ine t i c energy over one cycle must vanish and the viscous d iss ipa t ion must exactly balance the energy conversion term associated with the Reynolds stress. In order f o r a disturb- ance t o be unstable t h e mean flow must feed energy i n t o t h e disturbance. Clearly, i f t he re is t o be any i n s t a b i l i t y , t he Reynolds shear stress must have the same s ign as t h e ve loc i ty gradient of t he mean flow.

In t h e l i m i t of zero v iscos i ty , t he dissipa- t i o n terms vanish and the r a t e of change of k ine t i c energy must exact ly balance t h e energy conversion term. If t h i s Reynolds stress term i s pos i t ive , energy w i l l be t ransfer red from t he mean flow t o the disturbance and t h e flow will unstable; i f it i s negative, t he mean flow W i l l absorb energy from t h e disturbance and the flow will be s table ; i f it i s zero, t he re i s no ex- change of energy between t h e m e a n flow and t h e disturbance and the flow is sa id t o be neu t r a l ly s t ab le .

16 The Reynolds shear stress is given by

where

+ The primes ( ' ) under t h e bars indica te f luctua- t i o n quant i t ies .

G = ( lyl2 w' /Tr ) h a g [ ( l /w-c)(L f r T ) -

Batchelor and Gill' ' have shown t h a t the Reynolds shear s t r e s s is composed of two components; one perpendicular t o the he l ix of constant phase, - Y q ' , and the o ther p a r a l l e l t o the same he l ix An 3: constant, - 3 42' q3' , i , e .

I f _W i s a necessary and su f f i c i en t condition f o r neutral , subsonic disturbances, then - - cT -0, the s t r e s s component, - l i k e a d e l t a function near r = vanishes f o r n = 0; t h i s f i n i t e contribution t o the Reynolds s t r e s s term i n Eq. (3.6a) e r n f 0 and i s always a des tabi l iz ing f ac to r 1171 . amplitude d ies off exponentially as r -.,a [Eq. (2.17)] , In t h i s case the re w i l l be no energy t ranspor t i n t o o r out of the wake by the disturb- ance but t he re w i l l be a net exchange of energy between the mean flow and the disturbance within the wake because of the s ingu la r i ty of the

t i o n must alance t h i s ne t excess of energy.

111.3 Vor t ic i ty Transport

with the q ve loci ty component a t the c r i t i c a l point can &.so be in terpre ted in terms of t he t ranspor t of the mean and disturbance v o r t i c i t y across the annulus r = re, w = c.

2 and 3 d i rec t ions and the mean v o r t i c i t y compon- en t s i n the 1 and 3 d i rec t ion are:

0 which, as was shown i n Section (11.5),

behaves

For subsonic neut ra l disturbances, t he pressure

f q2' q2' stress component. Viscous dissipa-

The quanti ty k and the s ingular i ty associated

The disturbance v o r t i c i t y components i n t h e

O( = -id q3 / s inu

3 i a 2 q 2 (ql sine)' 2 s i n g C O S G q

r - r, = - cos c w1 -

(3.9) r2 = - s i n cr W' , respectively. Eliminate T- between Eqs. (2.7) and (2.8) and (rq2)I between the resul t ing equa- t i o n and the continuity equation; t he r e su l t i ng 3-disturbance v o r t i c i t y equation i s

y s ina (w-c ) [ r + ( 2 s i n u cos v q /r)] 3 3

- y s i n r ( w - c ) [ ( ~ ~ / ~ ) q l + F / U ) GI - = i rq2 ( f s i n q r r)' . (3.10) 3

The disturbance vo r t i c i ty equation i n t h e 2 direc- t i o n i s j u s t the 3- momentum equation [Fq. (2.9)1 -

s i n cr (w-C) r2 = rl 92 . (3.11)

The terms on the le&-hand s ide of Eq. (3.10) represent t he mean transport ( i n the 1- d i r ec t ion ) of a disturbance vor t ic i ty ; t h i s t ranspor t vanishes

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at w = c. The term in brackets on the right-hand side is a kind of mean angular momentum in the 3- direction that is transported radially; at w = c this quantity is identical to (Tw')'-k. It is clear that if an inviscid neutral disturbance is to exist the gradient of this generalized density- vorticity product must also vanish at w = c, provided that (rq2)w = f 0.

The interpretation of Eq. (3.11) for neutral disturbances is as follows: to an observer rid- ing with the disturbance the transport of the radial disturbance vorticity by the mean flow in the 1- direction exactly balances the radial transport of the mean vorticity in the 1- direc- tion, except at the point w = c. At this point, if h 2 I w = ity is singufar for n f 0. The fluid viscosity, however small, must be taken into account to smooth out this discontinuity.

# 0, the disturbance radial vortic-

To summarize, these considerations only emphasize the fact that even though the vanishing of the gradient of the density-vorticity product insures the existence of a solution to the neutral inviscid equations, the effect of viscosity, however small it might be, must still be taken into account to smooth out the discontinuities in the disturbance velocity and vyticity components and the Reynolds shear stress.

IV. Numerical Results

The stability characteristics of compressible axisymmetric wake flows were obtained using the method described in reference 16. The inviscid equations were solved numerically on the IBM 7094 at the Research and Advanced Development Division, Avco Corporation, by a predictor-corrector integra- tion procedure.

The mean flow model of Gold2' was used in the stability calculations, namely

w = -exp (-12)

T = 1 + AT exp (-q2) where qdy = rdr/T, 4 T is the temperature excess parameter

(4.1)

(4.2)

and r2 = q2 + 1 - T + dT.++ These analytical wake profiles were obtained by an asymptotic expansion procedure and are, strictly speaking, only valid "far" do stream of the "neck'' in the region where bT, z(( 1. Quantitative inform- ation about the stability characteristics of wake profiles near the "neck", where compressibility effects dominate the flow field, was obtained by letting AT and M2 become very large.

The necessary and sufficient condition for the existence of neutral subsonic disturbances is ~~ ~

17,18 + Thermal conduction is also required

++ The incompressible wake profile is obtained by setting AT = 0, T = 1.

.

that the gradient of the density-vorticity product vanish for some -c 4 1/M (Section 11.5). In terms of the new indepenzent variable,? ,

d/dq2 [ (sin20/T2)(dw/dq2) ] = 0 (4.3)

for some -w < 1/M, which leads to the relation rc2 = ( ~ ~ ~ / 2 - T ~ ) [ 1 + (~(r~/n)'] -I. (4.4)

The wave speed, in addition to being a function of AT, is also a function of the neutral wave number of the disturbance, dS; the neutral wave number in turn depends upon M. This is unlike the two- dimensional case in which the wave speed is a f'unction of AT alone, and does not depend upon M2. For n = 0, r = 0 and neutral subsonic disturbances are impossibfe

For the Gaussian profile and n = 1, -CR was determined as a function of As A T increases, cR increases algebraically towards zero and the phase velocity approaches the wake-edge velocity; the critical point also approaches the edge of the wake. For large values of $T, the wave speed is relatively inde endent of M and therefore is insensitive to d! (as in the two-dimensional easel-2). Since -cR decreases with increasing AT, the critical relative Mach number (M sonic diszurbances exist, also increases with A T (Fig. 5). For large values of and the critical relative Mach number iRcreases like AT. At a given value of M, there is a value of disturbancewith non-vanishing wave numbers are impossible.

A T and M2 (Fig. 5).

= - l/cR), below which unstable sub-

AT, -c 4 1/ 4 T

A T (Fig. 5) below which neutral subsonic

The neutral inviscid wave number, dS, de-

AT'= 1.5 is reached (Fig. 6); beyond creases very slightly with increasing AT until a value of this value the inviscid wave number increases with

at first, can be explained in terms of the special property of the Gaussian profile, i.e.,

2 7, e-7c / CR- 1,.

AT. The latter result, although quite surprising

2 d S - (dw/dl )/CR ... (4.5)

where 9 increases with increasing AT. When M' = 2, the slope of the temperature-inviscid wave number curve is infinite at A T = 0; as AT in-- creases, d s will also increase (not shown). For M2 > 2, the inviscid wave number is a minimum at a fixed value of M2). are of primary interest here, the cal ulations must be begun at AT = (AT),, when M' > 0. For a fixed value of hT, the inviscid wave number decreases with increas'ng M2 (not shown), confirm- ing the results of Lin .

A T = 0 and increases with increasing dT (for Since subsonic disturbances

5 By the oscillation theorem of Sturm (Section

II.5), a necessary condition for the existence of a neutral subsonic disturbance is that Max [ p + (d*/Tr)l < 0 [Eq. (2.27)1 o r

where

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2 - T ) r 2 - T% + ( o(r/n) exp ( q2 - qc2) - 1 t ... (4.6)

4r2 N =

~3 [I + (dr/n12]

for K2 = 0 For incompressib e flow, the maximum d is approx-

imtely equal to 4.7 at r2 = 1.812,1c Therefore, tbe only possible allowable modes of instability are those for n = 1, 2; all other modes are stable to s m a l l disturbances. In fact, it w i l l be s h o m that the n = 1 mode is the only unstable mode of oscillation. For compressible f l o w it is also expected that the maximum value of N2 will occur near ( d/n)2 = 0. cal point, ') = 7 c ( - 2 is somewhat lower that the maxmum Val; i t 6 , but should give an indication of the behavior of N2 with AT. limit ( d/n)2 4 0, Eq. (4.4) becomes

value of N 2 occurs for (d/n)'d 0

The value of N2 near the criti-

In the

2 r = Tc2/2 - T = 7 + 1 - Tc + A T ; (4.7)

at the critical point and in the same limit, Eq. (4.6) reduces to

2 2 N = 4T / ( 2 - Tc) . (4.E')

Eq. (4.7) is a parametric equation relating T,, Tc and AT (Fig. 7a). is shown in Fig. as a function of AT. The number of possible allowable unstable modes in- creases very rapidly as AT increases, for A T > 1. For & T 4 1, Fig. (3) indicates that the only allowable modes are n = 1, 2, as in the incompress- i le case. More precisely, the maximw. value of N obtained from Eq. (4.6), in less than nine for

The quantity N2 [4. (4.8)]

E! A T < 0.8.

The n = lmode represents a sinuous-type instzbility in which the nodal points for the radial disturbance are spaced 1.80~ apart (Fig. ea). This is the only mode for which the distirhance radial velocity is not equal to zero on the axis [Eq. (2.15)]. to anti-symetrical oscillations in two-dlmensional flow. The n = 2 mode represents a varicose-,type instability having four nodal points spaced goo apart (Fig. 8b). The radial velocity is zero on the axis for this case.

This type of instability is similar

The inviscid wave number is shown in Fjg. 6 as a function of AT (M2 = 0) fo r various unstable modes of oscillation. For n 2 2, the inviscid wave number increases from zero for increasing values of AT. Figs. 6 and 8b indicate that al- though there may De a large number of possible allowable modes of instabilit) for a given value of 4 T, there will only he a few modes of this number tl-.at are actually unstable. For example, when AT = 3 there are 5 possible modes of in- stability, n = 1 , 2, 3, 4, 5 (Fig. 8b). However, only the n = 1, 2, 3 modes will he unstable (Fig. 6). is the only unstable mode of oscillation. The inviscid ?rave numbers for the higher modes of oscillation are smaller than the n = 1 mode for a given valw of AT; therefore, it is ewected that these higher modes are more stable than the n = 1 mode. This conclusion was confirmed by direct r unerical calculations.

In the incompressible case, the n = 1 mode

The dimensionless amplification rate, 0s cI, is shown in Fig. 9 as a func+,ion of wave nwioer for various values of A T and Id2. For M2 = 0,

the maximum amplification rate increases with in- creasing wake "core" temperature until a vdue of A T = 3 is reached. Beyond this value the maximum amplification rate starts to decrease very slowly; for instance, when AT = 8 and M2 = 0 (not shown), ( d cI) = 0.0566 (at d = .60), which is still above the ncompressible value. For a given value o f A T , ( d c decreases markedly with in- creasing va,u:L% M. m e maximum amplification rate for the n = 2 mode is about three times small- e r than the maximum lification rate for the n = 1 mode (AT = 2, 9- M - O), even though the neutral wave numbers for the two modes are ahout equal (see Figs. 6 and 9).

The increased instability of Gaussian wake flows and the relative insensitivity of the maximum amplification rate to core temperature is in direct contrast with the two-dimensional case in which ( 4 ~ ~ ) ~ ~ markedly decreases with increasing core temperature; the increased stability with relative Mach number is common to both cases. These results will Fe partially explained in the following para- graph by using "too-hat" profiles. imolies that the history o f the velocity profiles and conseqLently, M, governs the stability of axisymmetric hypersonic wake f lows; the history of the temperature profile is of minor consequence in tkis calculation. Of course, the velocity and temperature profiles are related through the con- servation equations of momentum and energy and are certainly not independen: of each other. However, f o r a given velocity and temperature profile, the maximum amFlification rate seems to be relatively independent of wake core termeratme.

This result

The stability characteristics of cylindrical and planar vortex sheets should be very similar to those of a slowly-varying wake profile (for example), a Gaussian w a K e ) when the wave length of the dis- turbance is very muci. larger than the characteristic length associated with each of the aforementioned

21 wake profiles; tne details of the wake profiles are unimportant on the scale of the disturbance . When tl-e wave length is small compared to the characteristic lengths, the details of the profile are jnportant in calcLlatlng $he amplification rate of the disturbance. Tne results obtained for the Gaussian wake can be partially explained by considering the stability of "top-hat" wake profiles, i.e.,profiles in which the velocity and speed of sound are constant in two different flow regions

-1, 1' < 1 (4.9)

(0,l r 7 1 9

w, a =

where 3 is the density ratio across the wake ( f >, l), in the limit of small wave number,d -+ 0. he eigenvalue equakion is 22:

c = (-1 + i )/(1 + L) (4.10)

where

(cylindrical)

L = ( Po/ P1) ctanh co (planer anti-symmetrical)

(4.lla)

( 4. l l b )

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and

(4. llc )

for - W 4 arg p C IT . Eq. (4.10) is a transcendental equakion for c R, c and o( with 'p and M* as parameters. The quanti& cI as a func- tion of d. is shown for a cylindrical vortex she6 (Fig. 10) and for a planar vortex sheet subject? t anti-symmetrical disturbances (Fig. 11) for 8 M = 0 and different values of the density ratic, -4'. For axi-symmetric wakes, the value of CI at o( = 0 decreases very slowly with increasing values of 3 ( c 1 - P /: +s , o( -0); the slope of the cI - a( curve is zero at d = 0 and increases with increasing wave number for -f # 1. For large values of o( , CI also decreases very slowly with increasing 7 . Peak values of cI are reached when the wave number is of the order of unity; these values also decrease very slowly with in- creasing density ratio. value of rapidly for increasing relative Mach number.++ The shability characteristics of two-dimensional flows subjected to anti-symmetrical disturbances exhibit quite a different behavior. For small values of OC , c --.m and increases with increasing d unZi1 an asymptotic value (eI- f i /l + y ) is reached. with increasing M2 for a given value of f (cot shown in Fig. 11). increasing -9 , i.e. the phase velocity approaches the external wave velocity. These results are consistent, for small values of o( , with those found by using Gaussian profiles for the temper- ature and velocity.

When M2 # 0, for a given f , the quantity cI will decrease rather

The quantity CI decreases

The value of -eR decreases with

Summarizingt1)the effect of increasing the wake core temperature is to increase the range of relative Mach numbers over which neutral (and adjacent amplified) disturbances can exist and also to increase the number of possible allowable unstable modes; 2)the effect of increasing the relative Mach number is to markedly decrease the maximum amplification rate, at a given temperature, as long as M < Mer; 3) the maximum amplification rate is relatively insensitive to wake core temp- erature (for Gaussian profiles); and 4) the n = 1 mode is found to be the most unstable of all the possible allowable modes for the cases that have been considered.

V. Laminar-Turbulent Transition in Hypersonic Wakes

The upstream motion of transition in the hypersonic wake of a blunt or slender body as the free stream Reynolds number (or ambient pressure) is increased can be tentatively explained in light of the experimental an theoretical inves- tigations of Sato and Kuriki8 concerning transi-

+ Note the difference in scale of c in the two figures . I

++ Additional calculations for large values of M indicate a more significant decrease than the case shown in Fig. 10 [ f = 2, M2 = 1 ] .

tion in the low speed wake of a thin flat plate aligned with the main flow and the hypersonic cylinder and slender wedge experiments of hetriades23 and hetriades and Behrens2'.

In the linear region of the flat plate wake (see Introduction), the growth of the root-mean- square velocity fluctuations along the wake axis follows an exponential law. As the non-linear effects become important, the growth rate deviates from the exponential law, reaches a maximum, and then starts to decrease. The position of the maximum growth rate occurs fairly close to the end of the linear region. out that the "initial point of the non-linear region might be called the 'transition point' since the non-linearity is an essential feature 05 t'le turbulent motion". They also find that the stream- wise variation of the wake centerline velocity follows the classical theoretical curve of Goldstein25 for relatively small values of x*, then starts to increase rather rapidly as x* is further increased, reaches a maximum, decreases,and then increases again gradually. The point of deviation of the centerline velocity from the theoretical curve is very close to the maximum of the rms velocity fluctuation curve. This result is also consistent with t e low speed experiments of Sato for separated flows2' and jets27.

Sat0 and Kuriki8 point

The hypersonic wake experiments 23,24 indicated similar trends for the growth of the rms flilctuations along the wake axis. As the ambient pressure of the wind tunnel was increased, the energy peak moved systematically closer to the body. The stream- wise value at which the peak occurred was arbitrarily defined as the transition point. Wake thickness measurements were made with a hot-wire, using "turbulent" as an indicator. The rms intensity of the "turbulence profiles" wa highest near the transition point. McCart&, using pitot and static pressure probes, and diffusivity as a "transition indicator" corroborated the transition results of Demetriades'3.

The free flight ballistics ranges ha-re accounted for a good deal of the hypersonic wake transition data that are available today291 30,31. The transition point has been obtained b y opl;ical methods. A shadowgraph of an misymmetric 'nwer- sonic slender body wake is shown in Fig. 12. Downstream of the neck the laminar wake is practic- ally straight for about 10-15 base diameters and then starts to oscillate in what appears to be the n = lmode. The wave lengt'n of the disturbance is of the order of the wake diameter and decreases as the wave progresses downstream. The wake becomes turbulent at about 25 base diameters downstream of the neck. The ballistics range experimentalists define the transition point as that downstream position at which the boundary of the wake becomes very irregular. In this shadowgraph, this distance occurs at about 20 base diameters downstream of the trailing edge of the cone. Although tnis defini ion

+J -

is r ther arbitrary, the exgeriments c'_ted above 6 ' , 23,et indicate ' that this transition poinz is near the position of the m a x b m rms value. Thic Fhoto- graph (and many others like it) suggest ",at transi-

+ This photograph was taker! at t,he Naval Ordnance Laboratory, Wnite Oak, Md., and was obzained through the courtesy of Dr. A. J. Pallone of tke Avco Corporation, Wilmington, Mass.

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tion is preceded .by a wave-like laminar instability, which oscillates in the mode predicted by hydro- dynamic stability theory, and that transition occurs not long after the motion becomes non-linear.

The upstream motion of the non-linear region with increasing pressure should be indicative of the upstream motion of transition if the fluctu- ations grow rapidly in the initiaL portion of the non-linear region. The amplification ratio of a disturbance of a given frequency at two stream- wise stations xi* and x* is[Eq. (2.3d) 1:

. a * (5.1) I

where x* is the lacation at which this particular frequency becomes unstable and d* is a character- istic body dimension. Near a hypersonic slender body, the velocity defect and wake thickness are practically constant independent of Reynolds number29. If the spatial amplification rate (assumed to be the inviscid value) is constant, independent of downstream distance, then Eq. (5.1) can be integrated to give

(x*/d*) - (@/a*) = constant. i 24 , The transition data of Demetriades and Behrens

for slender wedges, follows the condition ex- pressed by Eq. (5.2). Far from the body, the wake half-width L* is given by

(5.3)

while the velocity difference Ue* - Uo* is given by

(Ue*-Uo*) L*/Je* = R2 ea* 'D ( -p,"/ 9e*)/4Ke ... (5.4)

For the hypersonic wake the appropriate drag coefficient is not the total drag of the body but the value of CD in the inner laminar wake. drag coefficient is approximately equal to the initial drag coefficient at the "neck" and is given by

The

(5.5)

If the spatial amplification rate is assumed constant, Eq. (5.1) becomes

where C and C are constant and D = (0.20) ( d c ~ / 'Fg). &. (5.5) indicates that there w i l l be a Reynolds number for which (x*/d*)-m . This Reynolds number can be considered to be a mLnhum critical Reynolds number below which all disturbances are stable. In the intermediate region, the history of the disturbance and, the mean flow must be known in order to perform similar amplification calculations. However, the movement of the non-linear region with body

Reynolds number must fall between the two limits expressed by Eqs. (5.2) and (5.6). similar calculations with wake transition data is found in Lees and Gold12 and Kr0nauer3~. predicted by small disturbance theory for the on- set of the non-linear region agree remarkably well with the wake transition experiments.

A comparison of

The trends

IV. Concluding Remarks

The inviscid stability of axisymmetric com- pressible flows has been examined in great detail. The axisymmetric compressible small disturbance equations have been transformed into a two-dimen- sional form by a suitable transformation of veloc- ity coordinates. !he physical mechanism of in- stability has been discussed in terms of the propag- ation of wavy disturbances and the transport of vorticity and energy between the mean flow and the disturbance. Numerical stability calculations have been performed for Gaussian and "top-hat" wake velocity and temperature profiles. maximum amplification rate is relatively insensi- tive to wake core temperature but markedly de- creases with increasing relative Mach number. The effect of increasing the wake core temperature is to increase the range of relative Mach numbers over which neutral (and adjacent amplified) dis- turbances can exist and also to increase the number of possible allowable unstable modes. The sinuous n = 1 mode appears to be the most unstable allow- able mode of oscillation. Laminar-turbulent transition in hypersonic wakes is discussed in terms of the history of the root-mean-square fluctuations along the wake axis. The motion of the non-linear region as a function of Reynolds number is derived for the two limiting cases of the near wake, where the velocity and wake-width are approximately constant, and the far wake, when the wake profiles are Gaussian in nature.

The

Acknowledgements

This paper was prepared under the sponsorship and with the financial support of the U.S. Air Force Ballistics Systems Diyision, Air Force Systems Command, Contract No. M04(694)498; this research is part of Project REST sponsored by the Air Force Ballistics Systems Division. wishes to thank Professor Lester Lees of the California Institute of Technology for his guidance throughout the course of this investigation. The author would also like to acknowledge many helpful discussions with Professor Toshi Kubota of the California Institute of Technology, Dr . Anthony Demetriades of Aeronutronics Division of the Fhilco Corporation, and Dr. Adrian J. Pallone of the Research and Advanced Division, Avco Corporation, and to thank Mr. N. F. Doherty and M r . J. A. Pavelcak of Avco FAD for the IEM 7094 computer axisymmetric wake stability calculations.

The author

Symbols

The symbols most frequently used in the present paper are defined below and are, in general, those most cmonly found in the literature on laminar-flow stability. Less common terms are explained as they appear in the text. Dimensional quantities are starred; non-dimensional quantities are unstarred.

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Positional coordinates: Axial Radial Angular

Time Mean quantit ies:

Velocity ( f lu id coordinates )

Density Temperature Pressure Speed of sound

A x i a l velocity Radial velocity Angular velocity Density Pressure Temperature

Wave length Wave numbers:

A x i a l

Disturbance amplitudes:

Angular

X

i t

Total Disturbance propagation

- veloci t ies : Axial (e*) C

Relative a x i a l (c* - ue*) C

Wave f ront

d* k L* M ML N P 91 92 93 61"' r* R

-

AT

U*

W

Y

A* 5

7 'y

- P

Q

n Subscripts:

cr e

C

L. P

Reference quantitie:

L* L* L* L*/ (Up; )

ue*L uo* 2 % Pe* ae* e

ue* - uo* ue* - uo*

Y e Pe*

Ue**- uo*

3: l/L*

1/L*

'e*

ue* - Uo*

ue* 'e* - Uo*

Characterist ic body dimension Gradient of density-vorticity product Wake half-width Relative Mach no. (Ue* - Uo*/+*) Local r e la t ive Mach no. (cp*/a*) Defined by Eq. (4.6) Defined by Eq. (2.13)

9er + sin qx

44 - cos O- qx 2 n 9 Disturbance amplitude Radius vector '

Wake Reynolds no. [ (Ue*-Uo*)L*/3 .*] Local Reynolds no. based upon displacement thickness, character- i s t i c body dimension, and stream- wise position, respectively Temperature excess parameter

Mean velocity in body fixed coordinatg s

[Eq. (4.211

u n ?(q w y ' ) Ratio of specific heats brodnitw-Howarth variable ( 1 = rdr/rp) Kinematic viscosity Defined by Eq. (2.13) tan-' r /n Reynolds shear s t r ess

r% 1- M2c2

Evaluated at c r i t i c a l point Sonic disturbance Evaluated at wake edge

i I

R S

a

0

Superscript: A

I n i t i a l value Imaginary par t of quantity Evaluated a t wake axis Real par t of quantity Neutral disturbance Evaluated in f ree stream

Complex conjugate

Primes generally denote di f ferent ia t ion with respect t o r. indicate a fluctuating quantity should not cause any confusion.

The f e w instances *ere primes

References

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AIAA J. 2, 417-428 (1964). 2 Lees, L., llHypersonic wakes and t r a i l s , "

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Larson, H.K., "Heat t ransfer in separated flows," J. Aerospace Sci. 26, 731-738 (1959); a lso Larson, H.K. and Gat ing, S.J., Jr., "Transition Reynolds numbers of separated flows of supersonic speeds," NASA TN D-439 (1960).

Lin, C.C., "On the s t a b i l i t y of the laminar mixing region between two para l l e l streams in a gas," NACA TN 2887 (1953).

Landau, L., "Stabi l i ty of tangential discon- t i n u i t i e s i n compressible fluid, I' Compt. Rend. Acad. Bulgare Sei. E, 139-141 (1944).

Miles, J.W., "On the disturbed motion of a plane vortex sheet," J. Fluid Mech. k, 538-552 (1958 1.

Sato, H. and Kuriki, K., "The mechanism of t r ans i t ion i n the wake of a th in f l a t p la te placed p a r a l l e l t o a uniform flow," J. Fluid Mech. 11 321-352 (1961). J

Lees, L. and Lin, C.C., "Investigation of the s t a b i l i t y of a laminar boundary layer in a cam- pressible f lu id ," NACA TN 1115 (1946).

lo Lin, C.C., The Theory of Hydrodynamic S tab i l i ty (Cambridge University Press, Cambridge, &gland, 1961), Chap. 4.

Batchelor, G.K. and G i l l , A.E., "Analysis of the s t a b i l i t y of axisymmetric je ts ," J. Fluid Mech. 2, 529-551 (1962).

l2 Lees, L. and Gold, H., "Stabi l i ty of laminar boundary layers and wakes a t hypersonic speeds. Part 1. Stab i l i ty of laminar wakes: Paper presented a t International Symposium on Fundamental Phenomena in Hypersonic Flow, Buffalo, New York (June 25-26, 1964).

l3 Case, K.M., "HHrodynamic s t a b i l i t y and the inviscid l i m i t , " J. Fluid Mech. lo, 420-429 (1961).

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l4 Lin, C.C., "Some mathematical problems in the theory of the stability of parallel flows," J. Fluid Mech. l0, 430-438 (1961).

l5 Whitham, G.B., "Group velocity and energy propagation for three-dimensional waves, " C m . hre Appl. Math. &, 675-691 (1961).

Gold, H., "Stability of laminar wakes," R.D. Thesis, Calif. Inst. Tech., Pasadena, Calif. (1963).

Reshotko, E., "Stability of the Compressible boundary layer," Guggenheim Aeronaut. Lab., Calif. Inst. Tech., Memo. 52 (1960).

l8 kes , L. and Reshotko, E., "Stability of the compressible laminar boundary layer," J. Fluid Mech. l2, 555-590 (1962).

Ince, E.L., Ordinary Differential Equations 19 (Dover Publications, Inc., New York, New York, 1956), Chap. 10.

23 Gold, H., "Laminar wake with arbitrary initial profiles," A I M J. 2,948-949 (1964).

21 Drazin, P.G. and Howard, L.N., "The instabil-

ity to long waves of unbounded parallel inviscid flows," J. Fluid Mech. &, 257-283 (1962).

22 Gill, A.E., "Instabilities of 'top-hat' jets, and wakes in compressible fluids," Dept. Math., Mass. Inst. Tech. (1964).

23 Demetriades, A., "Some hot-wire anemometer measurements in a hypersonic wake," Proceedings of the 1961 Heat Transfer and Fluid Mechanics Institute (Stanford University Press, 1961), 1-9.

WAVE FRONTS * PROPAGATION

p VELOCITY C

24 Demetriades, A. and Behrens, W., "Hot-wire measurements in the hypersonic wake of slender bodies," Grad. Aeronaut. hb., Calif. Inst. Tech., Internal Memo. 14 (1963); also Demetriades, A., AIAA J. 2, 245-250 (1964).

Goldstein, S., Modern Developments in Fluid 25

Dynamics (Oxford University Press, London 1938), Vol. 2, 571-574.

26 Sato, H., "Experimental investigation of the transition of laminar separated layer," J. Physical SOC. Japan II, 702-709, 1956.

27 Sato, H., " The stability and transition of a two-dimensional jet," J. Fluid Mech. 7, 53-81 (1960).

28 McCarthy, J.F., Jr., "Hypersonic wakes,"

Guggenheim Aeronaut. Lab., Calif. Inst. Tech., Hypersonic Res. Project, Memo. 67 (1962); also McCarthy, J.F., Jr. and Kubota, T., "A study of wakes behind a circular cylinder at M = 5.7," A I M J. 2, 629-636 (1964).

29 Pallone, A., Erdos, J. and Eckerman, J., "Hypersonic laminar wakes and transition studies," A I M J. 2, 855-863 (1964); also Avco RAD Tech. Memo. m-m-63-33 (1963). 30 Slattery, R.E. and Clay, W.G., "Laminar-

turbulent transition and subsequent motion behind hypervelocity spheres,'' ARS J. s, 1427-1429 (1962).

31 Levensteins, Z. J., "Hypersonic wake character- istics behind spheres and cones," A I M J. - 1, 2848-

32 Kronauer, R.E., I' Growth of regular disturb-

2850 (1963).

ances in axisymmetric laminar and turbulent wakes," Avco RAD Tech. Memo.-TM-64-3 (1964).

( a 1 BODY CENTERED COORDINATES

( b 1 COORDINATES FIXED IN A FLUID AT REST

64 -8111

F I G U R E 1 C O O R D I Y A T E S"STEh\S A L D DIRECTICI '" OF P R O P A G A T I O I OF A U A V E FROAT

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ax + n = CONSTANT / r

IO

f TANGENT TO HELIX 3 F 1 \CONSTANT PHASE F

I I f, I

I I I

/

I ! )

L r = CONSTANT

64-8109

FIGURE 2 ORTHOGONAL COORDINATE SYSTEM FOR AXISYMMETRIC FLOW

+ +X

I r = r

r = r2>rl

m+ I I

64-8108 FIGURE 4 SCHEMATIC REPRESENTATION OF THE CHANGE OF DIRECTION OF

PROPAGATION OF A WAVE FRONT WITH INCREASING R A D I A L DISTANCE

6

AT 4

2

X

TAN.=? '\ /

PERPENDICULAR TO HELIX OF CONSTANT PHASE

64-8110

FIGURE 3 LOCAL RELATIVE PROPAGATION VELOCITY OF A WAVE FRONT

i

I SUBSONIC DISTURBANCES I "

0 4 8 12 16 M

64-8104 FIGURE 5 REGIONS OF S U B S C N I C , S O N I C AND SUPERSONIC DISTURBANCES.

RELATIVE WAVE VELOCITY VERSUS RELATIVE M A C H NUMGER FOR A X I S Y M M E T R I C WAKE FLOW

2

0

6

A T

4

I I \ I I C

0 0.4 0.8 1.2 16 a) AT V S T, AND 9,2

3.0

b) A T VS N 2 =S

64-8103 64-8338 FIGURE 6 TEMPERATURE DIFFERENCE VERSUS NE1,TRAL ';.A'VE NU'.,8ER 'CR

UNSTABLE RIODES OF OSCILLATIO'! FIGURE 7 AXISYlr lRIETRIC COb3,PRES51EIf WAKE 'LOWS

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+

I +-

64-8339

( b i r . 2

" 0 0.4 0.8

a

64-8102

0.551 I I I I I I "

P = l , M2 =o A 0.50

045 b P.2, M 2 = I

0 40

Y I I I 2 4 6

0 35 0

a

2

0 5 r ~. c pF I I "

A

c

0 4 1 I / / 0 - E 1

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