Harris Gold In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California Thesis STABILITY OF LAMINAR

(1)

STABILITY O F LAMINAR WAKES

T h e s i s by H a r r i s Gold

I n P a r t i a l Fulfillment of the Requirements F o r the Degree of

Doctor of Philosophy

California Institute of Technology Pasadena, California

1963

(2)

ACKNOWLEDGMENTS

The author wishes to e x p r e s s his sincere thanks and apprecia- tion to Professor L e s t e r Lees for h i s guidance throughout the course of this investigation. The author would a l s o like to acknowledge many helpful discussions with Psofe s s o r Toshi Kubota and Dr. Anthony

Demetriades. He a l s o wishes to thank Kiku Matsumoto for programming a l l the IBM computer computations and Mrs. Geraldine Van Gieson for her outstanding typing of the manuscript.

The author acknowledges the receipt of a Del Mar Science Foundation Fellowship for the 1959- 60 academic year, a Boeing Fellowship for 1960-61, and a RAND Corporation Fellowship i n

Aeronautics for the y e a r s 1961-62 and 1962-63. This project was made possible through the sponsorship and with the financial support of the U. S. Army Research Office and the Advanced Research P r o j e c t s Agency.

The author would like to e x p r e s s h i s gratitude to his wife, Sandra, for her encouragement and per serverance throughout his studies.

(3)

ABSTRACT

This investigation d e a l s with the effects of compr essibility on the hydrodynamic stability of wake flows. It is found that the effect of t e m p e r a t u r e i s two-fold: ( I ) , a s the wake c o r e t e m p e r a t u r e i n c r e a s e s , the range of Mach n u m b e r s over which n e u t r a l and self- excited subsonic disturbances c a n exist a l s o i n c r e a s e s ; (2) a s long as the r e l a t i v e Mach number i s below the c r i t i c a l M a c h number the n e u t r a l inviscid wave number will d e c r e a s e with increasing c o r e temperature, implying that a hot wake will be m o r e stable than a cool one.

The a n a l y s i s of Batchelor and Gill for the inviscid stability of axi- s y m m e t r i c incompressible j e t s h a s been extended t o the m o r e g e n e r a l problem of c o m p r e s s i b l e wakes and jets. It i s shown that the r e s u l t s a r e d i r e c t l y analogous to those obtained for the two-dimensional problem. The sinuous ( n = 1 ) mode is the m o s t unstable allowable mode. This unstable mode is observed i n a hypersonic wake.

(4)

TABLE O F CONTENTS PART

Acknowledgments A b s t r a c t

Table of Contents L i s t of F i g u r e s L i s t of Symbols I. Introduction

11. Formulation of the Problem

11. 1. Outline of the Stability Problem 11.2. Two-Dimensional F r e e Shear Flows.

Small Disturbance Equations and Boundary Conditions

11. 2, a. Viscous, Incompressible P r o b l e m 11. 2 . b. Inviscid, Compressible Problem 11. 3. Axi-Symmetric F r e e Shear Flows.

Equations and Boundary Conditions

11. 3, a. Viscous, Incompressible Problem 11. 3, b. Inviscid, Compressible Problem

II.

4. Mean Flow Model

111. Stability of Two-Dimensional Incompressible Wake Flows 111. 1. Solutions f o r Small a R

111. 2. Idealized P r o f i l e s a t Long Wavelengths 111. 3. Inviscid Limit

111. 4. Large, Finite Reynolds Nurnber Stability

PAGE ii iii i v vii viii 1 3 3

(5)

IV. Stability of Two- Dimensional Compressible Wake Flows 44 IV. 1. Inviscid Disturbance Equation

a n d Outer Boundary Condition IV. 2. Solution of the Inviscid Equation IV. 3. Numerical Results

IV. 4. The Hyper sonic Wake P r o b l e m 60 V. Stability of Axi- Symmetric Compressible Wake Flows 64

V. 1. Similarity Between the Small Disturbance Equations and Boundary Conditions f o r

Axi-Symmetric and Two-Dimensional Flows 66 V. 2. Inviscid Disturbance Equation and Outer

Boundary Condition 69

V. 3. Singularities of the Inviscid Disturbance

Equation 76

V. 4. N e c e s s a r y and Sufficient Conditions f o r the

Existence of Inviscid Subsonic Disturbances 83 V. 5. Energy a n d Vorticity Relations 88 V. 6. Inviscid Stability of Axi-Symmetric Wake

Flows 97

VI. Suggestions f o r F u r t h e r Study 104

VII. Concluding R e m a r k s 106

R e f e r e n c e s 108

Appendix A

--

Boundary Conditions for the Axi-

Symmetric P r o b l e m 113

Appendix B

- -

Two-Dimensional Wake Model 115 Appendix C

--

Method of Solution of Tatsurni a n d

3Xakutani31 f o r Small a R 120

Appendix D

- -

Solution of the Inviscid Equations f o r

Amplified Disturbances 126

Appendix E

- -

Expansion about C r i t i c a l Point-

Axi-Symmetric C a s e 130

(6)

Appendix F

- -

E x t r e m u m of Density-Vorticity

Product 1 3 3

Appendix G

- -

Viscous Corrections i n the C r i t i c a l

La ye r 1 3 5

Table I

--

Neutral Inviscid Stability Character-

i s t i c s 1 4 2

Table 11

--

Amplified, Invi scid Stability

C h a r a c t e r i s t i c s 143

F i g u r e s 1 4 5

(7)

LIST O F FIGURES

NUMBER PAGE

1 Modes of Instability 145

2 Typical Neutral Stability Curve f o r Wake-Type

P r o f i l e s 146

3 Wave Number vs. Complete Wave Velocity 147

4 F r e q u e n c y vs. Reynolds No. and Spatial

Amplification Rate 148

5 Direction of Propagation of the Wave F r o n t s 149 6 Relative Wave Velocity and C r i t i c a l Point vs.

T e m p e r a t u r e Difference 150

7 Relative Wave Velocity vs. Relative Mach Number 151 8 Inviscid ,Wave Nurnber vs. Square of Relative

Mach Number 152

9 Dimensionless Frequency vs. Spatial Amplification

Rate 153

10 Hyper sonic Axi s y m m e t r i c Wake

( a ) Mm

=

13, B a s e Diameter

=

0.400",

P

=

100 mrn. H g . , Air, Cone Angle = 16O 154

(b) n = 1 Mode of Instability 155

vii

(8)

LIST O F SYMBOLS

The symbols used i n the p r e s e n t r e p o r t a r e i n g e n e r a l those commonly used i n the l i t e r a t u r e on hydrodynamic stability. In some r e g r e t a b l e instances a symbol will r e p r e s e n t m o r e than one item. To minimize the confusion, the different definitions of the s a m e symbol will have listed the s e c t i o n of the r e p o r t i n which they appear.

Dimensional Dimensionless Reference

Quantities Quantities Quantitie s

Two- Dimensional Flows :

Positional Coordinates:

longitudinal n o r m a l Velocity

Components : C A (x- c t )

longitudinal

-

\n/. =

u* - 5;

L A t w- c t )

-

n o r m a l ys ₌V' +

v * ~

^"(7) ^4-

9 ( 4 e v "

Density

-

c d ( w - c k )

P r e s s u r e pX = px + p ^XI P(Y) c ^~ ⁽

e

^~ ¹

P,"

L A t w - ct)

T e m p e r a t u r e 7% ⁼

TX +

T * ' T ~ Y )

+

^QCV)

e 7."

Axi- Symmetric Flows:

Positional Coordinates:

axial r a d i a l angular

(9)

Velocity Components :

-

~ ~ ( s - r t ) tin+

axial

9: -

^w'

+

^w(*) q x i r )

e

- - -

w * : q,X

-

^0;

-

^- ^,Acy.- ^ct)^{+ c n}⁴⁾

radial

7,

=

9:

⁴

3:' qvcy~

^4-

9vCq) e -

angular

7

=

91

⁺

9;'

Angular Wave

Number:

Density ^{f *}⁼ ^{f w} ⁴

f r ' -

P r e s s u r e

px .

^p" ⁺^p''

Temperature

Time

t"

^t

Wave Length A* A

Wave Number A* = 2 q / h Y = I*/\

Di s tur banc e

-

C Propagation

c' - v e x C

Velocity

a speed of sound

b unknown constant (Section IV. 2 )

C~ drag coefficient

group velocity of disturbance

c

*/u,*

dimensionless group velocity g

local relative propagation velocity ( Eq. (5. 20) ) Section

III.

1, Appendix C )

c h a r a c t e r i s t i c body dimension exp (ia (x-ct)

+

^in#)

pq.

^(5.49)]

(10)

G n'/a2 T' n

[

^Eq. ^{(4. 18)]}

G

defined by Eq. (5. 52)

H

1

G (Appendix D)

H* stagnation enthalpy

HZ=)

Hankel functions of o r d e r 1/3, f i r s t and second kind h dimensionless static enthalpy (Appendix B)

~ ~ ( a f i r ) modified B e s s e l function of the second kind of o r d e r one

K ~ ( ~ E

r ) modified B e s s e l function of the second kind of o r d e r one k gradient of density-vorticity product (Section V) L* c h a r a c t e r i s t i c length

L(z) Lommel function (Appendix G)

M

v*/ae*

relative Mach number

Me local Mach number outside the m e a n wake M~ local Mach number of disturbance

[Eq.

^{(5. 21)]}

Moo f r e e s t r e a m Mach number

m total wave number

N~ Eq. (5. 68)

P Eq. (5. 10)

a*(>,

^t*) quantity of total flow

6*(;*)

m e a n o r steady component of flow quantity Q * ~ (;*, t*) fluctuating component of flow quantity

<*I(;*) fluctuation amplitude

(11)

q(y)

I

fluctuation amplitude for nearly parallel flow

v*L*/~

e* wake Reynolds number g a s constant

Reynolds number based on displacement thickness local external Reynolds number based on d*

local external Reynolds number based on x*

local external Reynolds number based on 0*

L*

^, position vector

r

r (Appendix F)

AT temperature e x c e s s a t centerline of wake

U transformed mean velocity component (Appendix B)

U* mean velocity component in x* direction A U velocity defect a t center of wake

V transformed mean velocity component (Appendix B)

V* c h a r a c t e r i s t i c velocity

W

1

-

U (Appendix B)

W

b a g (

3 ? Y"

) (Section V)

X transformed x coordinate (Appendix B) xi

(12)

X (1/ ?$ r T) (Appendix E)

Y transformed y cowdinate (Appendix B) Y ( 1 / ~ )( sw'

'

(Appendix E)

r

disturbance vorticity

- -

I '

mean vorticity

%

r a t i o of specific heats

A dimensionless heat t r a n s f e r coefficient (Appendix B) 6 tan' l (ar/n) (Section V)

6* net heat t r a n s f e r (Appendix B)

c

^l/(aR)1/3 (Appendix G)

Dorodnitsyn-Howarth variable Eqs. (2. 47) and (2. 56)

r

-

^rCic (Appendix G)

C

dimensionless momentum thickne s a (Appendix B) momentum thickness

viscasity

kinematic viscosity

S

Eq. (5. 10)

G

p - a

= ^{a G} (Section 111 and Appendix C) T - - J q X ' q r i Reynolds shear s t r e s s

f

^{r - r}^c

(13)

Subscripts

c c r i t i c a l point

e local condition outside mean wake I imaginary p a r t of quantity

i point a t which disturbances begin to amplify o initial values (Appendix B)

o values a t a x i s (Appendix D) R r e a l p a r t of quantity

s neutral, inyiscid values

00 f r e e s t r e a m conditions 1, 2.. , f i r s t , second solution, etc.

Super s c r i p t s

A

complex conjugate

0 , 1 ) , 2 ) zero, first, second o r d e r quantities, etc.

A bar over a quantity indicates mean value.

P r i m e s generally denote differentiation with r e s p e c t to y o r r.

The few instances where p r i m e s denote a fluctuating quantity should not cause any confusion.

xiii

(14)

I. INTRODUCTION

Experimental 2-4 studies have shown that transition i n incom- p r e s s i b l e f r e e boundary l a y e r s is preceded by a linear and non- linear wave-type instability. The linear instability can be described

5-31

by the s m a l l disturbance theory of hydrodynamic stability

.

^The

non-linear instability is a v e r y complex phenomena and is not yet understood. Recently, the problem of laminar: turbulent transition in hypersonic wakes has a l s o been of considerable interest. The purpose of t h i s p r e s e n t investigation i s to study the effects of com- p r e s sibility on the hydrodynamic stability of wake- type flows.

The stability c h a r a c t e r i s t i c s of wake flows a r e relatively

insensitive t o Reynolds number, for sufficiently high Reynolds numbers, because of the o c c u r r e n c e of a point of inflection i n the density

vorticity product 22' 32m Therefore, interesting and important r e s u l t s can be obtained by considering the "inviscid limit" of the s m a l l disturbance equations, i n which the viscosity and conductivity of the fluid can be neglected to a c e r t a i n o r d e r .

This study will be r e s t r i c t e d to subsonic disturbances, i* e*

,

disturbances whose propagation velocity i s subsonic with r e s p e c t to the f r e e s t r e a m velocity. These disturbances have amplitudes that die out exponentially far f r o m the wake axis. If such disturbances exist then the m e a n flow is unstable to s m a l l disturbances. The question of the stability of a flow to supersonic disturbances h a s not been resolved 32-35

The inviscid stability c h a r a c t e r i s t i c s of two-dimensional incompressible and c o m p r e s s i b l e wake flows a r e studied using Gaussian

(15)

distributions f o r velocity and t e m p e r a t u r e i n the Dorodnitsyn-Howarth variable. The incompressible wake was a l s o studied a t low Reynolds n u m b e r s and wave numbers. The r e s u l t s of Batchelor and Gill 36

and ~ i 1 1 ~ ~ f o r a x i - s y m m e t r i c incompressible wake-type flows have been extended t o include the effects of compressibility.

(16)

11. FORMULATION O F THE

PRQBLEM

An infinitesimally s m a l l disturbance i s imposed upon a m e a n o r a steady flow and the behavior of the amplitude of the disturbance i s examined a s t i m e p r o g r e s s e s . If, for l a r g e values of time, the disturbance is damped out, the motion is said to be stable; i f not, the motion is said to be unstable with r e s p e c t to infinitesimally s m a l l

disturbances. It is m u c h e a s i e r to prove that a motion i s unstable than stable. If the flow i s unstable to disturbances of a n y kind, even

P

-

the simplest kind, i t i s always unstable, but the flow m a y be stable with r e s p e c t to one type of disturbance and not another. In hydrodynamic stability theory, the disturbance i s a s s u m e d to have a wave- like nature. The problem is to find c e r t a i n combinations of the wave number and wave speed of the disturbance and the Reynolds number of the m e a n flow f o r which the fluid motion i s unstable, o r neutrally stable.

11. 1 . Outline of the Stability Problem

-

The total flow c o n s i s t s of a time-independent o r m e a n component,

-

Q*, and a n infinitesimally s m a l l component, Q*', which i s both space

and t i m e dependent:

Q"

(F*,

t") =

-

Q* (?*) S Q*' ($*, t*) where

( ~ * i /lad

< < 1

The total flow, Q*, s a t i s f i e s the conservation equations of m a s s ,

momentum and energy and a n equation of state. The m e a n flow,

s*,

s a t i s f i e s the steady flow equations o r some approximation to them,

(17)

for example, the boundary l a y e r equations. The conservation equations f o r the disturbance a r e obtained by substituting e x p r e s s i o n s of the form, Eq. ^{( 2 .}l), for the flow v a r i a b l e s into the total flow equatisns and

subtracting out the m e a n flow equations. The hydrodynamic stability equations a r e obtained f r o m t h i s s e t by neglecting quadratic and higher o r d e r t e r m s of the disturbance quantities.

The coefficients of the resulting linear p a r t i a l differential equations depend upon t h e m e a n flow quantities. Time a p p e a r s only a s the derivative (a/&*) and hence solutions containing a n exponential t i m e factor

hl

-

^ia*c*t*

Q*T

(y*,

^t*) = q*' (;*) e

m a y be assumed. The resulting differential equations will contain the space coordinates a s the only independent variables.

This study will be limited to parallel o r quasi-parallel flows, i. e., motion i n which the m e a n n o r m a l velocity is z e r o o r v e r y s m a l l compared to the m a i n velocity component. The following o r d e r of magnitude relations f r o m boundary layer theory apply to the m e a n flow quantities:

where

longitudinal coordinate i n two-dimensional flow o r axial coordinate i n axi- s y m m e t r i c flow

nor e i n two-dimensional flow o r radial coordinate, r*, i n axi- s y m m e t r i c flow

(18)

6*

m e a n velocity component in x* direction

*

m e a n velocity component i n y* direction

R6* Reynolds number based upon displacement thickness.

The m e a n flow quantity,

G*

is a function of the position coordinates, x* and y*. Expand Q)k about the point x* = x*

-

P ^'

The region adjacent to the point under consideration is taken t o be of the o r d e r of a few wavelengths of the disturbance i n the x* direction

Then f r o m Eqs. (2. 3b) and (2. 4b)

and f o r l a r g e values of aR6*

,

Eq. (2.4a) becomes

- -

Q* (x*, y*) = Q*(x _P

**

^Y*)

[

¹

+

⁰^(l/aRsL^)] ^(2.4d)

Therefore a l l m e a n quantities can be considered to be independent of the n o r m a l ( o r r a d i a l ) space variable to o r d e r (l/aR6+).

By considering disturbances that a r e spatially periodic, both i n the direction of flow and i n the direction perpendic'ular to the plane

e t r y of the m e a n motion, Squire 38 h a s shown for incompressible that two dimensional disturbances a r e l e s s stable than t h r e e -

(19)

dimensional disturbances. F o r c o m p r e s s i b l e flow, Dunn and Lin 39 have shown, by neglecting dissipation t e r m s and some t e r m s involving the fluctuating viscosity and t h e r m a l conductivity (which a r e valid a t m o d e r a t e Mach numbers), that a n equivalent two-dimensional disturbance i s not possible, but that the t r a n s f o r m e d three-dimensional disturbance equations a r e of the s a m e f o r m a s those for two-dimensional disturbances.

In particular, by neglecting viscosity and t h e r m a l conductivity, the three-dimensional disturbance equations a r e exactly of the s a m e f o r m as the two-dimensional ones. Therefore, important f e a t u r e s of the stability problem can be obtained by considering two-dimensional disturbances alone. F o r parallel and q u a s i - p a r a l l e l flow, the coefficients of the equations a r e independent of x* and consequently solutions of the f o r m

rV q*' (x*, y*)

=

might be expected. The exponent is purely imaginary since the dis- t u r bance m u s t be bounded f o r x* a t both 4- co and

-

^go. F o r two- dimensional flows a disturbance of the f o r m

Q*' (x*, y*, t*) = q*I(y*) e ia*(x*- c*t*)

will reduce the s e t of l i n e a r p a r t i a l differential equations to a s e t of o r d i n a r y differential equations i n y*.

T h e r e i s no d i r e c t analogue of S q u i r e ' s r e s u l t for axi- s y m m e t r i c parallel and quasi-parallel flows. However, Lessen, et a12' and P a i 2 7 indicate that for rotationally s y m m e t r i c disturbances, the incompressible disturbance equations, except for the obvious coordinate scale factors, a r e s i m i l a r to those f o r two dimensional disturbances. Batchelor and

(20)

Gill 36 show, by a suitable velocity transformation, that for d i s t u r - bances with a n angular dependence, the incompressible equations a r e exactly analogous to the two-dimensional ones, again, except for the obvious coordinate s c a l e factors. In Section V. 1 t h i s l a t t e r r e s u l t i s

,

extended to the c o m p r e s s i b l e case. The m e a n flow quantities for p a r a l l e l and quasi-parallel flows do not depend on the angular coordinate,

$b,

and depend only on the coordinate n o r m a l t o the direction of the m e a n motion, r*, t o o r d e r ( l / a ~

.

Since the amplitude of the

6*

disturbance m u s t be single-valued with r e s p e c t to the angular coordinate, a disturbance of the f o r m

Q* (r*,

0,

^x*,^t*)

=

q*l(r*) e ia*(x*- c*t)+inq)

where n i s a n integer, m a y be assumed. The resultant s e t of ordinary differential equations will have r* a s the only independent variable.

I n Eq. (2.6) o r Eq.

[

^- ( 2 . 7)] the disturbance amplitude qw(y*) [ o r q*'(r*) and the wave velocity c* a r e taken to be complex.

I

The m a i n flow i s stable, neutrally stable, or unstable to t h e s e waves according to whether the imaginary p a r t of c* is negative, zero, o r positive, respectively. The quantity a* i s the wave number of the disturbance and is taken to be r e a l and positive. The r e a l p a r t of c*

is the phase o r propagation velocity of the wavy disturbance.

-ia*c*t*

The assumption that the disturbance h a s the f o r m e

pq.

^{( 2 .}²⁴i s known a s the n o r m a l mode approach to hydrodynamic stability. If t h e r e a r e some values of a*c*, with cI*

>

⁰

,

s u c h that a non- t r ivial solution satisfying the disturbance equations and boundary conditions exists, then the flow is said to be unstable; i f not, i t is

(21)

stable. Recently, the initial value problem h a s been emphasized by

Case 45

40-44 and Lin An a r b i t r a r y s m a l l perturbation is introduced into the flow at time, t*

=

0, and i t s subsequent motion i s followed by m e a n s of a n o r m a l mode expansion, If t h e r e is a single mode with

c

* >

0 the perturbation grows exponentially with t i m e and is said to

I

be unstable. The n o r m a l mode approach should be equivalent to the initial value method although t h e r e s e e m s to be some inconsistencies between the two r e s u l t s a s the Reynolds nurnber becomes infinite 40-42

4 5

L i n h a s pointed out that t h e s e c a n be resolved by considering the limit of a n o r m a l mode i n the viscous theory. This limit i s

-

not the n o r m a l mode i n the inviscid theory, and vice v e r s a , The n o r m a l mode approach i s applicable t o differential o p e r a t o r s having d i s c r e t e eigen-

I

values while the initial value method should be used with singular

o p e r a t o r s and/or continuous eigenvalue s. However, i t appear s that the m o d e s leading to instability a r e associated with the d i s c r e t e eigenvalue s.

It is f o r t h i s r e a s o n that the n o r m a l mode approach will be used i n t h i s text.

The amplification r a t e of the disturbance is defined a s follows:

and

Q*' = Qi*' exp

i"

_ti* ^a*^{cI* dt*} ^*

In the laboratory, the experimenter h a s a quasi- stationary problem. As the disturbance propagates downstream of i t s origin, i t s amplitude

changes i n both space and time. One m e a s u r e s the spatial

(22)

9

amplification r a t e , o r the r a t e a t which a disturbance will amplify with distance i n the m e a n flow direction. The wave speed i s a

function of both the wave number of the disturbance and the Reynolds number of the m e a n flow. If the parallel and quasi-parallel assumptions a r e made, then the group velocity of the disturbance, i. e., the velocity a t which the disturbance energy m u s t propagate

46 ,

is

The spatial amplification r a t e i s

and

The spatial amplification r a t e i s constant for parallel flows. F o r quasi-parallel flows, the spatial amplification r a t e is computed a t each s t r e a m w i s e station by using a m e a n velocity profile that i s

a s s u m e d to be independent of x a t that station. The total amplification ( o r decay) of the disturbance a s i t m o v e s downstream i s found by the piecewise integration of the local amplification ( o r decay) r a t e s .

F o r wake-type flows, i t is expected that the spatial amplification r a t e s depend upon the decay of the m e a n velocity profile. Therefore, i t i s convenient to t r a n s f o r m f r o m a coordinate s y s t e m i n which the o b s e r v e r i s fixed i n the body to one i n which the o b s e r v e r is fixed in a fluid a t r e s t . In this l a t t e r system, the o b s e r v e r s e e s the tfvelocity

(23)

defect" of the wake. [ S e e Sketch 2. 1 J

Body- Centered Coordinates Coordinates Fixed i n Fluid a t Rest

Sketch 2. 1

In t h i s coordinate s y s t e m the wave h a s a propagation velocity (c*

-

^Ue*). Let the m e a n flow be dimensionally r e p r e s e n t e d by a c h a r a c t e r i s t i c length,

L*,

and a c h a r a c t e r i s t i c velocity, V*, and the t e m p e r a t u r e , density, p r e s s u r e and viscosity by their external values,

so that [see L i s t of Symbols. :

V* and

L*

will be taken to be the velocity defect a t the centerline

[w(o)

=

I ] and the half-width of the wake, respectively [section 11. 4

1 .

The disturbance equations and boundary conditions will be derived for two-dimensional [section 11. 2'1 and axi-symmetric [section 11.31 flows.

The outer boundary conditions for the compressible problem will be

(24)

discussed in detail in Section IV. 1 (two-dimensional c a s e ) and

Section V. 2 (axi-symmetric case). The quantities a r e defined in the L i s t of S p b o l s .

11.2. Two-Dimensional F r e e Shear Flows. Small Disturbance Equations and Boundary Conditions

11. 2. a. Viscous, Incompressible Problem

The incompressible, s m a l l disturbance equations a r e a limiting c a s e of the complete c o m p r e s s i b l e equations. This c a s e i s obtained by neglecting the viscous dissipation and heat conduction t e r m s i n the conservation equations and assuming that the m e a n temperature, p r e s s u r e , density, viscosity and t h e r m a l conductivity a r e constants.

In addition, i f the gradient of the t e m p e r a t u r e fluctuation vanishes at the a x i s and the outer edge of the m e a n flow, the t e m p e r a t u r e fluctuation and hence the density fluctuation can be s e t equal t o zero32. The

dimensionless s m a l l disturbance equations a r e 3 2*

Continuity

x- Momentum

y- Momentum

*

P r i m e s indicate differentiation with r e s p e c t to y for the two- dimensional c a s e and differentiation with r e s p e c t t o r for the axi-

s y m m e t r i c case.

(25)

This s y s t e m c o n s i s t s of t h r e e linear disturbance equations i n the t h r e e dependent perturbation amplitudes fa

6

^and^n,where the m e a n velocity w is determined f r o m the m e a n o r steady- state equations. The

s y s t e m i s of the fourth o r d e r i n the dependent variables.

If the m e a n velocity profile i s s y m m e t r i c a l [wl(0) = 01

,

then the disturbance amplitudes can be decomposed into even and odd p a r t s , each p a r t satisfying Eqs. (2. 12)

-

^{( 2 .}14). The even p a r t of the

longitudinal velocity disturbance, f a c o r r e s p o n d s t o anti- s y m m e t r i c a l ( o r sinuous) oscillations and the odd p a r t to s y m m e t r i c a l ( o r v a r i c o s e )

oscillations. I

The anti- s y m m e t r i c a l oscillations a r e analogous to two p a r a l l e l rows of equally spaced v o r t i c e s i n a l t e r n a t e positions (Sketch 2. 2. a)

( d r m i n vortex s t r e e t ) and the s y m m e t r i c a l oscillations to s y m m e t r i c a l l y placed vortices (Sketch 2. 2. b). ~ a r r n ~ n ~ ~ I h a s shown that the s y m m e t r i c a l

(a) Flow i n Alternate Vortex Street ( b ) Flow i n Symmetrical Vortex Street

Sketch 2. 2

vortex s t r e e t i s unstable f o r a l l values of the spacing r a t i o ( r a t i o of t r a n s v e r s e t o longitudinal dimension) a n d will tend to r e a r r a n g e itself

/

into the a l t e r n a t e vortex s t r e e t . The a l t e r n a t e position i s stable for only one spacing r a t i o and unstable f o r all others. Physically, the a n t i -

(26)

symmetrical disturbances a r e observed m o r e often than the symmetrical ones; for example, i n the wake behind a c i r c u l a r cylinder a t very low Reynolds numbers. This fact suggests that the anti- symmetrical

disturbance i s m o r e unstable than the symmetrical one and the minimum c r i t i c a l Reynolds nurnber, below which a l l disturbances a r e damped, will be lower for the f o r m e r .

Using Eqs. (2. 12) and (2. 13) the boundary conditions a t the a x i s a r e :

Anti- symmetrical oscillations (Sketch 2. 3. a)

Symmetrical oscillations (Sketch 2. 3, b)

--

r9

^"t,

c' L*

(a) Anti- symmetric oscillations (b) Symmetrical oscillations Sketch 2 . 3

(27)

The quantities f and n can be eliminated f r o m Eqs. (2. 12)

-

(2.14) and a fourth o r d e r equation in

b

can be found

The boundary conditions for l a r g e values of y a r e obtained f r o m Eq. (2. 17). As y

-

^m

,

^{w ---+}⁰exponentially and Eq. (2. 17) becomes

where

P2 =

^a2

-

^iaRc

.

The solutions of Eq. (2. 18) a r e :

Now $D m u s t be bounded for l a r g e values of y (y

>

0). If the r e a l p a r t of f3 i s positive then the solutions with the positive exponent m u s t be

rejected and the outer boundary conditions f o r Eq. (2. 17) a r e

and f r o m Eqs. (2. 12) and (2. 13):

11. 2. b. Inviscid, Compressible P r o b l e m

If the solution of the disturbance equations is a s s u m e d to be of the f o r m

(28)

and the limit

a R

^---poo i s taken, the resulting equations for the zeroth approximation, q(O! a r e called the inviscid s m a l l disturbance equations. They a r e identical with the equations obtained by ignoring viscosity and t h e r m a l conductivity. The dimensionless inviscid

3 2 . equations a r e

.

Continuity ¹

9' +

^{i f} ⁼ ( T t / T )

Q) -

i (w-c) ( s / p ) x- Momentum

p [ i (w-c) f t w'

$1 = -

( ~ B / % M 2 )

2

Momentum

Energy

State

-

This is a system of five equations in the five disturbance variables f,

$,

^a,s, and 8, where the mean flow quantities p, T, and w a r e determined f r o m the mean equations of motion. Upon eliminating four out of the five dependent variables, the system is seen to be of the second order.

Again, as i n the incompressible case, if the mean velocity and

(29)

t e m p e r a t u r e profiles a r e symmetrical, then the disturbance amplitudes can be decomposed into even and odd parts, each p a r t satisfying

Eqs. ( 2 . 2 3 )

-

( 2 . 27). The boundary conditions a t the a x i s a r e : Anti- s y m m e t r i c a l oscillations

Svmmetrical oscillations

The

outer boundary conditions will be derived i n Section IV. 1 but will be included h e r e f o r completeness. It i s

where

11.

-

3. Axi-Symmetric F r e e Shear Flows. Small Disturbance Equations and Boundary Conditions

The s a m e assumptions regarding the derivation of the axi-

s y m m e t r i c s m a l l disturbance equations apply a s f o r the two-dimensional c a s e and will not be repeated here.

*

In this equation, n equals 3. 141.

. . . .

(30)

11.3. a. Viscous, Incompressible Problem

/

3 6 . The dimensionless s m a l l disturbance equations a r e

.

Continuity

x- Momentum

r - Momentum

6-

Momentum

This i s a s y s t e m of four l i n e a r equations i n four dependent variables qr

,

q g

,

qX and a. This s y s t e m is of the sixth o r d e r i n the dependent variables since q ^@and q b ' c a n be eliminated algebraically f r o m

Eqs. (2. 321, (2. 3 4 ) and (2. 35).

F o r the a x i - s y m m e t r i c wake, the boundary conditions on the axis a r e kinematic in n a t u r e (do not depend upon viscosity) and can be derived f r o m the inviscid equations (Appendix A). All the disturbance amplitudes and the vorticity disturbance m u s t be finite on the axis.

The boundary conditions a r e :

(31)

n = O q r = o q @ = o q n a r e a r b i t r a r y

x

'

n f

^(I q x " o 3 w = O

=

l q@ =

-

^aqr ^q^si s a r b i t r a r y n > l q q J = o

,

q r = O

.

The n = 0 and n = 1 m o d e s a r e shown i n F i g u r e 1.

F a r f r o m the axis, the boundary conditions should be the s a m e a s for the two-dimensional c a s e (Section V. 2). These conditions can a l s o be derived by taking the l i m i t of Eqs. (2. 32)

-

(2. 35) a s r

---.

^co

.

The outer boundary conditions a r e :

where the r e a l p a r t of

p

is positive and

1x1. 3 . b. Inviscid, Compressible P r o b l e m

The dimensionless inviscid s m a l l disturbance equations a r e : Continuitv

x- Momentum

(32)

r - Momentum

6-

Momentum

i p (w-c) q+ =

-

^(ins/ar^{% M}2 ⁾

Energy

State

This i s a system of six linear equations i n six dependent variables qr ^s q X ⁹ q @ , s, 8

,

and s

.

As i n the two-dimensional inviscid, compressible case, the system i s of the second order.

The boundary conditions on the a x i s do not depend on the viscosity o r compressibility of the fluid and a r e Appendix A

I

:

n = 0

,

s , 8 , s a r e a r b i t r a r y

n # o (2.44)

n = 1 9$

' -

^a9, q is a r b i t r a r y r

The outer boundary condition i s the s a m e a s in the two- dimensional inviscid, compressible c a s e and will be derived i n

(33)

Section V. 2 . It is:

where

2 2

Q = ~ - M c - n < a r g

fi

^{< n}

II. 4. Mean Flow Model

The wake i n back of a blunt o r slender body can be divided into two regions of i n t e r e s t ; the "near" wake and the l f f a r " wake. The "near"

wake r e f e r s to the region n e a r the body and the ''far" wake to the region f a r downstream of the body. At low Mach n u m b e r s (Ma

< <

l), the n e a r wake i s c h a r a c t e r i z e d by the formation of v o r t i c e s and unsteady phenomena over a wide range of Reynolds numbers4' [sketch 2.4. a]

.

( a ) M m

<

C 1 (b) M

> >

1

I a

Sketch 2. 4

F o r Mm

<

i 1 the boundary l a y e r assumptions do not give a n a c c u r a t e description a t the n e a r wake because Re is low and the gradient in the

X*

(34)

s t r e a m w i s e direction (a/ax) is not s m a l l compared to the n o r m a l gradient

(a/ay).

^{The flow}is unstable a t low Reynolds nurnbers and becomes turbulent. Thus, t h e r e i s no region of laminar flow when the Reynolds number becomes large. F o r flat plates a t low Mach numbers, the v o r t i c e s and unsteadiness "disappear" and the wake becomes

laminar a t low Reynolds numbers49. The boundary layer assumptions apply i n t h i s c a s e and a n analytical solution can be found for the far wake 50 *

F o r M

>

1, o r m o r e specifically, M

> >

1

,

the near wake i s

00 00

c h a r a c t e r i z e d by two f r e e s h e a r l a y e r s ( o r a n annulus) shed f r o m the body surface, that converge into a "necktt5'. F o r blunt bodies the Mach number external t o the s h e a r layer is "frozen" a t about three, while for slender bodies i t is of the o r d e r of the f r e e s t r e a m Mach number.

2 3

Theoretical and experimental 52' 53 studies show that a laminar shear l a y e r is r e m a r k a b l y stable for super sonic external Mach numbers.

T h i s s a m e r e s u l t applies i n the neck region. Therefore, all o r p a r t of the "inner" wake will be laminar over a wide range of Reynolds

n u m b e r s 1 , 5 4

.

The boundary layer approximations a r e valid i n the inner wake region, except v e r y n e a r the neck where the m e a n profiles

change v e r y rapidly. In the far field, Oseen-type approximations can be used to l i n e a r i z e the equations and relatively simple analytical expressions can be obtained. Since the m a j o r t r e n d s of the stability problem a r e of i n t e r e s t here, these analytical expressions will be used i n the stability analysis.

5 5

Kubota' s solution for the two-dimensional compressible wake with z e r o p r e s s u r e gradient will be used (Appendix B):

(35)

where

In Eq. (2. 11) the c h a r a c t e r i s t i c length

is related to the half-width of the wake, and the characteristic velocity

i s the maximum velocity defect i n the wake. The quantity d* i s a c h a r a c t e r i s t i c body dimension. The Reynolds number of the wake

i s constant. The y coordinate in the stability equations m u s t be

"stretched" by the temperature (Eq. ( 2 . 47) ). In the derivation of

Eq. (2.46), i t i s assumed that

A T

and M a r e v e r y small ( A T , M

< <

1).

However, since the relative effects (and not the absolute effects) of temperature and Mach number a r e desired, values of AT and M g r e a t e r than unity will be used in the numerical calculations.

F o r the two-dimensional incompressible wake

(36)

A transformation analogous to Kubotals d o e s not exist f o r the axi- s y m m e t r i c c o m p r e s s i b l e wake56. The viscous s t r e s s t e r m i n the momentum equation does not t r a n s f o r m to the equivalent incompressible f o r m and t h e r e f o r e the momentum and energy equations m u s t be

integrated simultaneously. F o r the axi- s y m m e t r i c incompressible wake 50.

.

The momentum thickness

1

or d r a g coefficient

,

-

Drag

C~ - ]

i s given by

f

P e * ~ e * 2 ~ d *

CO

@a2

= CD ( ~ r d * ~ / 2 ) = 2 r

1 [

¹

-

^(U*/ue*)] ^{r* dr*}

0

and

/T

The Reynalds number for axi- s y m m e t r i c flow i s

-.

and v a r i e s a s the r e c i p r o c a l of the square root of the distance downstream of the origin.

The m e a n profiles f o r the axi- s y m m e t r i c compressible c a s e a r e a s s u m e d to be Gaussian i n the Dorodnitsyn-Howarth variable,

T ,

(37)

T = 1 + A T e

3'

where

7.

These m e a n profiles, although not s t r i c t l y valid (as mentioned p r e - viously),will be used to i l l u s t r a t e the stability c h a r a c t e r i s t i c s of

slowly varying axi- s y m m e t r i c velocity and t e m p e r a t u r e m e a n profiles.

(38)

111. STABILITY O F TWO-DIMENSIONAL INCOMPRESSIBLE WAKE FLOWS

Wakes belong to the c l a s s of two-dimensional q u a s i - p a r a l l e l flows known a s I1free boundary layers"', i. e . , flow f i e l d s i n which solid boundaries a r e not p r e s e n t . Usually flows which belong to t h i s c l a s s of q u a s i - p a r a l l e l flows have one o r m o r e points of inflection i n t h e velocity profile. The p r e s e n c e of a n inflection point indicates that the flow is dynamically unstable in the limiting c a s e of vanishing viscosity,

and that i t would become unstable a t r e l a t i v e l y low Reynolds n u m b e r s . ,

Hence the c l a s s i c a l m e t h o d s of solution f o r l a r g e Reynolds number, o r m o r e p r e c i s e l y l a r g e aR, cannot be used to d e t e r m i n e the minimum c r i t i c a l Reynolds number. The quantity, I / ( ~ R )

,

i s a m e a s u r e of the diffusion d i s t a n c e f o r vorticity during one period. New m e t h o d s of solution f o r the O r r - Sommerfeld equation have to be found f o r s m a l l values of aR. In addition, the asymptotic methods developed by

~ o l l r n i e n ~ ~ ' , ~ e i s e n b e r ~ ' ~ and U n Z 2 have to be modified f o r l a r g e values of aR.

Another problem a r i s e s i n that the q u a s i - p a r a l l e l flow assurnp- tions leading to the Or r

-

Sommerf eld equation a r e not valid throughout the e n t i r e flow field, since the t r a n s v e r s e m e a n velocity component i s of the s a m e o r d e r a s the longitudinal m e a n velocity i n c e r t a i n r e g i o n s of the field f o r the s m a l l v a l u e s of Reynolds number of i n t e r e s t i n t h i s problem. Near the t r a i l i n g edge of a flat plate, f o r instance, the t r a n s v e r s e m e a n velocity m u s t be taken into account if a "precise"

prediction of the flow stability i s d e s i r e d . However, g e n e r a l quantitative r e s u l t s a r e of i n t e r e s t h e r e , and the q u a s i - p a r a l l e l a s s u m p t i o n s will be

(39)

r e t a i n e d f o r all values of

aR.

The s u b t l e t i e s of i n c o m p r e s s i b l e wake-type flows will be d i s c u s s e d i n t h i s section. Section 111. 1 will d e a l with the solutions of the O r r - S o m m e r f e l d equation f o r s m a l l values of

aR.

A m i n i m u m c r i t i c a l Reynolds number of 39, based upon the length of a f l a t plate, i s found f o r a n t i - s y m m e t r i c a l disturbances. The stability c h a r a c t e r - i s t i c s of a smoothly varying profile a t long wave lengths c a n be found by using discontinuous velocity p r o f i l e s [section 111. 23 In Section 111. 3 the inviscid stability of a n i n c o m p r e s s i b l e G a u s s i a n f l a t plate wake i s d e t e r m i n e d by n u m e r i c a l methods. T h e s e t h e o r e t i c a l r e s u l t s a g r e e v e r y well with the e x p e r i m e n t a l r e s u l t s of Sato and Kuriki 2

.

The effect of viscosity on the eigen-value equation f o r l a r g e , but finite,

a R

flows c o m e s i n through the "inviscid solutions'' a n d d o e s not depend on the

"viscous solutionstf. However, the calculation of t h e d i s t u r b a n c e a m - plitudes m u s t include the "viscous solutions", since the "inviscid solutions" a r e singular a t w

=

c. [section 111.41

.

The O r r - S o m m e r f e l d equation c a n be d e r i v e d by eliminating the p r e s s u r e a n d longitudinal velocity perturbation a m p l i t u d e s f r o m Eqs.

( 2 . 12)

-

^(2.14) ( o r by considering the d i s t u r b a n c e vorticity equation),

The boundary conditions a t the axis a r e Eq. (2. 29) and Eq. (2. 3 0 )

I

:

(40)

= 0 anti- s y m m e t r i c oscillations

(3.3) 0 s y m m e t r i c oscillations

.

The boundary condition f o r l a r g e values of y is obtained f r o m Eq. (3. 2).

A s y

-

^ca

,

^w^-⁴⁰exponentially, and Eq. (3. 2) becomes The solutions of Eq. (3. 4) a r e

Now $?I m u s t be bounded f o r l a r g e values of y ( y

>

0 ) ; if f o r definiteness, we take

solutions with the positive exponent m u s t be rejected, and

Eq. (3. 1) together with the boundary conditions Eq. (3. 3) and Eq. (3. 7) constitute a c h a r a c t e r i s t i c - v a l u e problem. The c h a r a c t e r i s t i c - values ( o r eigen-values) a r e d e t e r m i n e d by the usual s e c u l a r determinant, leading to a r e l a t i o n of the f o r m

E ( a , c , R )

=

0 ( 3 . 8 )

w h e r e E i s s o m e g e n e r a l function of the argurnents.

(41)

If

the i m a g i n a r y p a r t of c is positive, the d i s t u r b a n c e s will

amplify with t i m e and the motion is s a i d t o be unstable. If i t i s negative, the d i s t u r b a n c e s will eventually be damped out. If c is z e r o , the d i s -

I

t u r b a n c e s a r e c o n s i d e r e d to be neutral. At e a c h Reynolds number, the s p a t i a l amplification r a t e s c a n be computed. A typical n e u t r a l stability c u r v e f o r wake type flows i s sketched i n F i g u r e 2. The h i s t o r y of a d i s t u r b a n c e a s i t p r o g r e s s e s d o w n s t r e a m is indicated i n t h i s figure.

F o r G a u s s i a n wake profiles, the Reynolds number is constant

[Eq. (2. 501) a n d the wave number is proportional t o [Eq. (2.50)]

.

The d i s t u r b a n c e will be amplified within the n e u t r a l stability c u r v e and damped outside of it.

111. 1. Solutions f o r S m a l l a R

Wake-type flows a r e v e r y unstable because of the o c c u r r e n c e of a point of inflection i n the m e a n velocity profile, i. e . , the m i n i m u m c r i t i c a l Reynolds number, below which all d i s t u r b a n c e s a r e stable, i s r e l a t i v e l y low. The effect of v i s c o s i t y is not confined to a thin l a y e r , a s in the boundary l a y e r c a s e , but i s felt throughout t h e e n t i r e flow field.

M o r e precisely, the r e l a t i v e d i s t a n c e that t h e d i s t u r b a n c e i s diffused i n one period is proportional t o the r e c i p r o c a l of s o m e power of a R, and f o r s m a l l v a l u e s of a R, t h i s distance is of the o r d e r of one, o r the full extent of the wake.

An e n e r g y balance shows that the r a t e of i n c r e a s e of the kinetic energy of t h e d i s t u r b a n c e is equal t o the conversion of e n e r g y f r o m the basic flow into the d i s t u r b a n c e by the Reynolds s h e a r s t r e s s , m i n u s the viscous dissipation 22

.

Viscous dissipation is a l w a y s a stabilizing

(42)

f a c t o r (always l e a d s t o a d e c r e a s e i n energy) and f o r s m a l l Reynolds n u m b e r s i s v e r y l a r g e . D i s t u r b a n c e s will be damped out v e r y rapidly i n t h i s region.

Tatsurni and Kakutani 31

,

anticipating a s m a l l value of t h e m i n i m u m c r i t i c a l Reynolds number, have expanded the solution i n p o w e r s of a R a s follows:

L

YI=o

w h e r e

Substituting Eq. (3. 9) into the O r r - S o m m e r f e l d equation, Eq. (3. 2), and matching p o w e r s of i a R, the following equations relating the $(n)'s a r e obtained:

w h e r e

The solutions of Eq. (3. 10a) a r e :

The solutions of Eq. (3. l o b ) c a n be found by the method of variation of

(43)

C.

I *

The g e n e r a l solution of Eq. (3. 2) is

where C

,

C2

,

C3

,

and C a r e a r b i t r a r y constants. 4

F r o m the outer boundary conditions,Eq. (3. 7),C2 = C 4 = 0 , and f o r a non-trivial solution,

6

^and

@

m u s t satisfy the following

1 3 '

c h a r a c t e r i s t i c equations [ E ~ . (3. 3)

1

^:

Anti- s y m m e t r i c Oscillations

Symmetric Oscillations

Eq. (3. 9 ) i s substituted into Eqs. (3. 13) and (3. 14) and a corn- plex eigenvalue equation is obtained a s a power s e r i e s i n (i a R).

**

*

The solution Eq. (3. 9) converges uniformly f o r the Gaussian

L 1

wake profile, w =

-

e - Y

,

w h e n a

<

1 a n d R I P C 1 Reference 31, page 270, where

I <

constant

1 u f ( ~ ) I <

^constant

**

T a t s u m i and 3Sakutani5' considered only anti- s y m m e t r i c a l

- -

oscillations. Their method is e a s i l y extended t o the c a s e of s y m m e t r i c a l oscillations (Appendix C).

(44)

T e r m s of the f o u r t h o r d e r and higher w e r e neglected, because i t w a s found that t h e s e t e r m s did not affect the f i r s t t e r m of the asymptotic behavior of the lower b r a n c h of the n e u t r a l stability curve. Appendix C,

[

Eq. (C. 18) a n d Eq. (C. 19)]

.

The eigenvalue equation w a s f u r t h e r

J

simplified by neglecting t e r m s of o r d e r

c3

and higher w h e r e

and

Since the a n t i - s y m m e t r i c oscillations a r e m o r e unstable than the s y m m e t r i c ones, the m i n i m u m c r i t i c a l Reynolds number w a s

r 3

d e t e r m i n e d only f o r t h e f o r m e r c a s e . Eq. (C. 18) Appendix C w a s solved graphically f o r the n e u t r a l c u r v e

I

_Ther e s u l t s a r e

J

indicated i n the following table f o r a Gaussian profile:

Table 3. 1 - TI --.--

+O. 40 SO. 45 4-0.50

The m i n i m u m c r i t i c a l Reynolds number is R = 4. 7 a t 'a = 0. 17. At t h i s point, a R = 0. 8 which is exactly the s a m e r e s u l t found by T a t s u m i and KakutaniS1 f o r a jet.

R

. 0 7 7 - 0 9 5

.

¹²⁵

a .-

.

¹²⁰

.

¹⁶

. 1 9

R

4. 8 4. 7 4. 7

c R

-.

⁰²²

-.

⁰³⁴

-.

⁰⁴⁶

(45)

F o r a n i n c o m p r e s s i b l e f l a t plate (length d ) G a u s s i a n wake

T h e r e f o r e the m i n i m u m c r i t i c a l Reynolds number based upon the length of the plate is Red+ = 39. T h i s value i s considerably below the experi- m e n t a l value of about 600 m e a s u r e d by Hollingdale 5

,

and about 700 m e a s u r e d by ~ a n e d a ~ ~ , f o r which oscillations w e r e observed. At low Reynolds n u m b e r s the amplification r a t e i s a strong function of Reynolds number, i. e . , v i s c o u s dissipation t e n d s t o d a m p out the d i s t u r -

bances. T h e r e f o r e , oscillations will begin to occur f a r d o w n s t r e a m of the plate and the Reynolds number a t which they a r e f i r s t o b s e r v e d will be considerably higher than the m i n i m u m c r i t i c a l Reynolds number d e t e r m i n e d by stability.

The a s y m p t o t i c behavior of the lower branch w a s found by taking the l i m i t of Eq. (C. 18) a s

$ ---

0

,

and Eq. (C. 19) a s

rI

^---+

-

^rn

.

Any other l i m i t did not produce a n y meaningful r e s u l t s . The f i r s t approximation consisted of solving the eigenvalue equations using only t e r m s up to o r d e r a R; the second approximation, using only t e r m s up to o r d e r ( a R) 2 ; etc. The r e s u l t s a r e tabulated below:

Anti- s y m m e t r i c Disturbance

-

Approximation

/

1 s t 2nd

I

^3rd ¹

a R 2 cR/a ²

3. 00 1. 48

-

1. 51

-1.50

, -1.64 -1.57 36.40

22.96

1. 29 1. 2 2

(46)

Svmmetric Disturbances

Table 3, 2

F o r the anti- s y m m e t r i c disturbances, the fourth approximation in-

volves t e r m s of o r d e r ( a ~ ) 4

,

which do not modify the t h i r d o r d e r approximation. F o r the s y m m e t r i c disturbances, the r e a l part of the eigen-

2 2

value equation h a s t e r m s of o r d e r a R to the approximations m a d e c R

- 1 . 0 5 - 1 . 2 6 Approximation

1 s t

- 2nd

(even though the imaginary p a r t h a s t e r m s of o r d e r

(aR)

5

1,

and i t i s

--

a R 0. 67 0. 47

not obvious that the 3 r d approximation will not effect the 2nd approximation.

In the l i m i t a

-

⁰

, R -

^co

,

for anti- s y m m e t r i c disturbances, the product

aR

2 approaches a constant along two branches. The lower

Sketch 3. 1

of the two branches, in a r e f e r e n c e system fixed i n a fluid a t r e s t ,

(47)

corresponds to a wave travelling in the s a m e sense a s the centerline velocity while the upper branch corresponds to a wave travelling i n the sense opposite to this velocity. The flow i s unstable if

2 2

-

^{1. 57 a}

<

^c

<

1. 22 a and stable outside t h i s region.

R

Along t h e s e two branches a ~ - F a s a ^-40

,

R

-

^m

,

thus confirming the validity of the expansion [ E ~ . ( 3 . 9 j

.

^{At the}

c r i t i c a l point a R = 0. 8; t h i s value i s higher so that m o r e t e r m s i n the eigenvalue equation should probably be retained i n o r d e r to find a m o r e p r e c i s e value of the minimum c r i t i c a l Reynolds number. However, the purpose of the computation w a s to find a n approximate value of the min- i m u m c r i t i c a l Reynolds number, and i t was felt that any additional calculations w e r e not commensurate with the a i m s of the investigation.

Moreover, a t these low Reynolds numbers, the boundary l a y e r equations themselves a r e not accurate.

In reviewing the paper of Tatsumi and ~ a k u t a n i ~ l, t h i s author found that t e r m s of o r d e r ( a R) 3

,

which w e r e neglected i n Eq. (6.4) of their paper, should have been retained. In the limiting c a s e a ---+ 0, R ---s. co

,

they a r e of the s a m e o r d e r a s the t e r m s that w e r e retained.

In addition, the second branch was not recognized by these authors.

Drazin 14 reported these additional r e s u l t s i n 1961.

F o r the s y m m e t r i c oscillations, a R and c approach a constant R

a s a

-

⁰

, R -

^w

.

The value of lcRl

,

however, i s l a r g e r than the maximum value of

[wl .

This behavior of the lower branch w a s f i r s t

suggested by c u r l e l ' and verified numerically by Clenshaw and Elliot 7 for the Bickley jet, w = s e c h y. L

The r e s u l t s obtained by the method of Tatsumi and Kakutani 31

(48)

suggest another type of expansion f o r s m a l l values of a R. Along the lower branch, R v a r i e s f r o m i t s minimum value to infinity while a i s always v e r y s m a l l (a

< .

2). Another type of expansion, then, would be to hold R fixed and expand c and

0

a s a power s e r i e s i n a. Howard 18 used this method f o r the Bickley jet and obtained r e s u l t s that a r e

identical to those of Tatsumi and Kakutani

I

if the additional t e r m s a r e added to Eq. (6.4)

,

Reference 31

1

This author h a s used t h i s method for the Gaussian flat plate wake and a l s o obtained the s a m e minimum c r i t i c a l Reynolds number, and the s a m e asymptotic behavior of the two lower branches (anti- s y m m e t r i c oscillations) a s by the method of Tatsumi and Kakutani.

III. 2. Idealized P r o f i l e s a t Long Wave Lengths

In b o t h t h e m e t h o d s of T a t s u m i a n d ~ a k u t a n i ~ l a n d H o w a r d 18

,

the m e a n velocity profile a p p e a r s only i n integrals, indicating that the p r e c i s e shape of the velocity profile i s unimportant f o r s m a l l wave- numbers o r l a r g e wave-lengths of the disturbances, Drazin and Howard 15 and ~ r a z i n ' ~ have used discontinuous velocity profiles to find the stability c h a r a c t e r i s t i c s of flows a t low wave-numbers. Some of t h e i r i d e a s will be presented h e r e a s applied to the wake-type flow stability problem.

F r o m Eq. (2. l l ) ,

R = ( v * L * / ~ * ) a

=

a* L*

,

where the s t a r r e d quantities r e p r e s e n t dimensional quantities, and

(49)

w(y) ⁴0 a s y

-

k

-

⁰⁰

.

The eigenvalue equation, Eq. (3. 8), leads t o a relation between c, a'

,

and a R o r (R/a), i. e.,

c = c (a, R/Q ) (3. 19)

F o r a fixed a*

,

a

-

⁰

,

( ~ / a ) = W/(a*3*) = constant a s L* ---o 0

.

Therefore, c

-

c (0, R/a )

=

f

(~*/a*a*)

and

A s L*

-

0 f o r a fixed dimensionless velocity profile, w(y),

>

then w*(y*) ^---A0 since y = ( y * / ~ * ) --p. oo (y*

<

0 )

.

T h e r e f o r e the two limits, a*

--+

0

,

L* and (R/a)

=

(v*/a*a*) fixed, a n d L*

-

⁰

,

^{a* and}

( v * / ~

^*)fixed, give the s a m e r e s u l t a --A 0

,

R/U fixed. In other words, f o r w(y) fixed, each l i m i t gives the s a m e limiting f o r m of the eigenvalue relation, Eq. ( 3 . 20), and the

stability c h a r a c t e r i s t i c s of the flow a r e the s a m e f o r both the limiting profile w*(y*) a s a*

-

⁰

.

In other words, by using the limiting profiles w*(yJF) a s

L* -

⁰

,

which m a y be discontinuous, the stability c h a r a c t e r - i s t i c s of smoothly varying profiles actual w * ( ~ * ) ] a s a*

-

0 c a n be determined.

[

~ r a z i n l * d e r i v e s jump conditions f o r

[

a, oR bounded

1

a t the points where w and/or (dw/dy) a r e discontinuous and applies them t o the c a s e of a broken line jet.

He finds that t h r e e n e u t r a l " b r a n ~ h e s ' ~ exist for a n t i - s y m m e t r i c disturbances

(50)

a R ⁴co

,

a fixed ( a ) a ,,p-tanh " ( 7

-

⁴ ⁵ ⁾

2 2

a R -constant a

-

⁰ ^{(b) a R}

-

1 . 3 4 c R - - 4 - 1 . 54 a 2

(3. 21)*

R-co

( c ) a R 2

-

^{32.4 cR}⁴^{1.21 a}²

and f o r syrnmetric disturbances

2 2

a R fixed, a ---r 0

,

c --r

-

^{[ l}

+

²ⁿ ⁽ⁿ/ a ~ ) ]

The branches (b) a n d ( c ) of Eq. (3. 21) correspond t o those found in Section 111. 1 and have approximately the s a m e limiting values. The wave speeds a g r e e t o within 2 O/o, while the product a R a g r e e s t o 2 within 10 O/o on the lower branch (b) and to within 50 O/o on the upper branch (c). Thus the u s e of discontinuous velocity profiles a t low wave n u m b e r s is justified a t l e a s t f o r qualitative purposes.

The f i r s t branch ( a ) i s not meaningful in that the assumption a R bounded i s not met. T h i s branch was not found i n Section 111. 1.

However, t h i s calculation indicates that the flow is unstable between the branches (b) and ( c ) and above branch (a). In Section III. 3, another

limit i s found, namely the inviscid limit, on which a R

-

^oo

,

^a^---e

dS .

Below t h i s branch, the flow is unstable, which l e a d s t o the hypothesis that t h e r e might be a n "island of stability" within the unstable region 14 This situation i s i l l u s t r a t e d i n the following sketch. It was suggested

9 In t h i s text, c i s defined a s the relative wave velocity and i s R

the negative of the values found i n Reference 14.

(51)

Sketch 3. 2

14 P

by Stuart that branch (a) corresponds to the limit

aR -

^constant,

F o r symmetric disturbances, a R + constant, c

R -

^constant

a s R --L

,

o

-

^014. This behavior is exactly that found in Section 111. 1 f o r the smoothly varying profile. F o r n

=

^1, cR

- -

^1.^{3 3 and}

Therefore, for low wave numbers o r long wave lengths, the stability c h a r a c t e r i s t i c s of smoothly varying velocity profiles can be found by using suitable discontinuous velocity profiles.

111. 3. Invi scid Limit

A point of inflection in the mean velocity profile indicates that the flow i s unstable i n the limiting c a s e of vanishing &scosity, and that the main f e a t u r e s of the instability mechanism can be obtained by neglecting the viscous forces2'. The effect of viscosity i s a stabilizing

(52)

one and can be taken into account once the inviscid instability mechanism i s understood.

The inviscid l i m i t is formally obtained by expanding

56

ⁱⁿ^a

power s e r i e s i n (l/aR),

substituting this expansion into Eq. (3. I), and taking the limit a R ⁴ ⁰ ⁰

.

The equation for the zeroeth-order approximation,

,

i s known a s the inviscid Orr-Sommerfeld equation and is

F o r neutral disturbances ( c

-

0) the wave speed,

I

-

^C~

'

i s equal to the

m e a n velocity, w

,

a t the point where w" vanishes. The vanishing of w"

is a n e c e s s a r y and sufficient condition for the existence of a n e u t r a l distur banceZ2

,

and the disturbance amplitudes a r e finite everywhere i n the flow region if t h i s condition holds. The inviscid neutral wave speed and wave number a r e commonly denoted by c and a s

.

2

F o r a Gaussian flat plate wake, w =

-

^e

-' ,

the inviscid neutral

1

wave speed i s c s =

-

^{e m Z}⁼^-0.606. ~ c ~ o e n ' ~ approximated the

velocity profile in t h r e e different regions and found that a