NASA TECHNICAL TRANSLATION . NASA TT F

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NASA TECHNICAL TRANSLATION

. NASA TT F-16566

CERTAIN METHODS AND PROBLEMS OF THE LINEAR THEORY OF HYDRODYNAMIC STABILITY f " _ J V. Ya. Shkadov

c

Translation of "Nekotoryye metody i zadachi teorii gidrodinamicheskoy ustoychivosti", Moscow State University, Institute of Mechanics, Scientific Transactions No: .25. . MoscoW, 1973, pp. 1-192

(NASA-TT-F-16566) CEP.TAIN PROBLEMS AND KETHODS OF THE LINEAR THEORY Of HYDROIYNAMIC STABILIT.Y (Agnew Tech-T.rah, Inc. r wooola.nd Hills, Calif.) 240 P HC $8.00 CSCL 20D

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NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D.C. 20546 DECEMBER 1975

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••CERTAIN

Titl. and Subtitl •

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2. Govornment Acc."sslon No.

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PROBLEMS AND METHODS OF THE LINEAR THEORY OF HYDRODYNAMIC STABILITY

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Authod\} •

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PerformIng Orllnly academician G. I. Petrov. The solution of non-linear problems involves some principal and technical difficulties. Therefore, in addition to numerical solutions, the author of this work has attempted to obtain solutions, using approximation methods. The first chapter describe.s basic methods for solving a problem for its i 1 own values in the Orr-Sommerfield equation. Chapters 2 through 4 describe the non-linear theory. Non-linear interaction of disturbances and the determination of primary wave frequencies and lengths are given considerable attention. 1 Some results are new and published here for the first time.

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PREFACE Methods and'results of the hydrodynamic. stability theory are often used in technical applications and· scientific research for explaining a number of important phenomena in flows. Scientific literature dealing with these matters is extensive. However, only a few works containing problem solution methods and a systematic description of results are available. Problems of the non-linear theory of hydrodynamic stability are particularly in need of description. This work examines certain problems which are mainly related to the non-linear theory of stability. This work does not claim to possess complete explanations in this field; it is limited by those problems studied by the author. For the most part, the problems under consideration are related to those ideas which were expressed by Academician G. I. Petrov and discussed at seminars of the Chair and Department of Aeromechanics of the Mechanical and Mathematical Faculty at ~4oscow State University. The solution of non-linear problems involves some principal and technical difficulties. As a rule, when precisely defined, they can only be numerically solved using a computer. However, it is difficult to express numerical results by means of finite correlations suitable for application. Therefore, in addition to numerical solutions, the author of this work has attempted to obtain solutions in a final from by using approximation methods. Great opportunities in this respect can be discovered through direct methods, which were first used and substantiated for stability problems by G. I. Petrov. Computer calculations are used for specifying and checking results, and also when it is not possible to obtain a simple solution in final form. The first chapter describes basic approaches to solving a problem for its own values in the Orr-Sommerfield eauation. The fundamental facts of the linear theory for the stability of interface and internal currents are considered here. Chapters 2 through 4 are devotGd to the non-linear theory. The formation of non-linear waves in thin layers of viscous liquid, non-linear wave development and the formatiun of drops in thin streams, as well as the development of disturbance in Poiseuille's and Couettets flows are investigated in these chapters.

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Non-linear interaction of disturbances and the determination of primary wave frequencies and lengths are given considerable attention. Some results are new and published here for the first time.

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The author greatly appreciates the attention and valuable advice given to this work by G. I. Petrov. A. G. Kulikovskiy and A. A. Zaytsev contributed several valuable remarks while reading the manuscript and discussing specific ~roblems. S. Ya. Gertsenshtey~, L. P. Kholpanov, L. V. F~lyand, and M. P. Ma~kova participated in the solution of several problems. The author expresses his gratitude and appreciation to all those mentioned above.

iv

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TABLE OF CONTENTS

iii

PREFACE CERTAIN PROBLEMS AND METHODS OF THE LINEAR THEORY OF HYDRODYNAMIC STABILITY • • • . • •

1

Basic Equations and a Statement of Problems Concerning the Stability of a Stationary Flow. . . . . . . . . . . . . .

1

§2.

Linear Problem Solving Methods.

11

§3.

Instability of Flows in Wakes and Streams

42

§4.

Instability in Flows with Interface • •

57

CHAPTER II

NON-LINEAR DISTURBANCES IN INNER FLOWS • . •

80

§l.

The Non-linear Development of Disturbances in Poiseuille's Plane-and-Parallel Flow..

80

The Formation of Taylor Vortices in a Flow Between Rotating Cylinders. • . • . • •

100

Non-linear Disturbances in the Couette Plane-Parallel Flow .•.••.••

III

NON-LINEAR WAVES IN LAYERS OF A VISCOUS LIQUID. • . . • • • • . . . . • ••

127

Waves in a Layer of Ideal Liquid.

127

CHAPTER I

§l.

§2. §3.

CHAPTER III

§l. §2.

Long Waves for Small Values of the Reynolds . . . . . • • •.

1,35

Long Waves in a Layer for Moderate Values of the Reynolds Number. • . . . . • . • • .

156

Waves in a Layer of Viscous Liquid on a Rotating Disk • •

181

Number.

§3.

§4.

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TABLE OF CONTENTS (Continued)

CHAPTER IV

INSTABILITY AND DISINTEGRATION OF CAPILLARY STREAMS •

••

194

§1.

Non-linear Wave Development in the Capillary Stream of a Circular Cross-section. • • • • .

195

§2.

Current and Instability in Streams of a Noncircular Section. • • . • • • . . • • • •

216

APPENDIX I APPENDIX II APPENDIX III • APPENDIX IV

























......... . .. .. ....... ........ ...









..... ...... ...... ......

235 237 238

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CHAPTER I CERTAIN PROBLEHS AND METHODS OF THE LINEAR THEORY OF HYDRODYNAMIC STABILITY

t

~ l~. ~.

v.

Ya. Shkadov

§l. Basic Equations and a Statement of Problems Concerning the Stability of a Stationary Flow. The flow of a viscous incompressible liquid is described using the Navier-Stokes system of equations and a continuity equation. (1.1) Equations (1.1) are written in the orthogonal cartesian system of coordinates x, y, z; V is the liquid velocity vector with projections u, v, w; p is pressure, and t is time. All variables in (1.1) are given in a non-dimensional form; the typical velocity Uo is taken as a velocity scale, and the typical length t is taken as a length scale. Equations (1.1) contain one dimensionless parameter which is a Reynolds number:

Re

~.

.

!

I

!

~

=

Uo l v

Each specific flow is also determined by corresponding boundary and initial conditions. If the flow is stationary, then only boundary conditions must be assigned. Let V (x,y,z), P (x,y,z) denote a certain stationary solution of system (1.1). This solution corresponds to a dynamically possible laminar flow of liquid. However, in reality, under the assigned boundary conditions, such a flow will not necessarily be observed in an experiment or in nature. Moreover, if the Reynolds number Re is sufficiently great, the stationary flow cannot be realized

*

I,

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Numbers in the margin indicate pagination in the foreign text.

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at all; experience has shown that, in this case, a more complex turbulent

flow,

characterized by the random pulsations of

all the flow characteristics, will occur. tionary flow to

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be realized, it must possess

stability proper-

ties against small disturbances which are always present in any

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In order for a sta-

flow under actual conditions.

Experience has shown that a

can be stable only where the Reynolds number values are rather small.

I

It loses this characteristic and becomes unstable when

i t approaches the critical value Re • Determining Re is one of k k the main problems in the hydrodynamic stability theory.

.-

K F

)-.

f~>

£'I

~

t

Let us consider the solution of system (1.1), which represents the sum of the stationary solution and a minor non-stationary portion •

,,=

.~

t

l

i

V(x,y,z) + ~I(x.y.z.t),

_ p = p- ( x , y , z) + p I (x. y , z , t )

~

(1.2)

I

~

Vi,



l

Values

f

stationary solution.

t

pi will be considered as disturbances of the initial Let us assign these values at the initial

f

time moment.

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~-

in the entire

Il-

otherwise it will be considered unstable.

t.

i

to ~.

I

~.

:

If with the increased time, flow

V',

p'

are decreased

field, then the initial flow is stable; Let us assume that

i t is possible. to disregard the products of the values with strokes.

non-linear members we obtain the following equations:

-at

-.

...

...

...

+(V'V)v+(v'V)V

=-

1 ' ...

grad p+--.6'"

Re

'

(1.3)

divv=O.

(strokes are omitted).

T

l'

Boundary conditions for disturbances

V,

p can be obtained

by linearization of corresponding initial boundary conditions in the problem.

2

:lit

I1

I

1

By substituting (1.2) in equations (1.1) and omitting

aV

l

j

In particular, on solid fixed surfaces we obtain

~

il I

j from adhesion conditions 1.

V = o.

1

I

Disturbance Energy Balance. -+-

Let us multiply the first equation (1.3) by V and transform the resulting equation, using the following continuity equation: ...........:.

~

-

,

}

olv12 - - _.1 oV 2 t(·oV 2 oV 2 1 ---tv(vV)V+--[(--) ... _) +(--)]= 2" 0t r?e ax ay az (a) = dlv(-pv-

1 - - 2 1 1 - 21 - Vlvl + - -- grad1vl~ 2 2 Re

Let us assume that the flow is restricted by solid walls or it can be considered periodical along those coordinates relative to which it is unrestricted. Let us integrate the resulting correlation in the entire flow field, whereby, in the case of a periodical flow, we will distribute the integration along the entire corresponding variable for one period. According to the Ostrogradskiy-Gauss formula, the integral from the right side can be reduced to the integral along the surface. In both cases under consideration,the integral will become zero: as a result, we obtain an equation for the kinematic disturbance energy.

avo--

(1.4) 2

+ (--)) dQ

oZ

,

%

"

:

If we retain non-linear terms in equations (1. 3), we will also obtain correlation (1.4), which is accurate in this sense. The entire disturbance energy is changed only as a result of the interaction of disturbances with the main flow and because

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--.i,.~.. _.

,., ..

r

-~!·--I 1 l

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of the viscosity effect.

The increased energy can be caused

only by the first term on the right side of (1.4), which is related to the exchange of energy between the main flow and the disturbance field.

The second term on the right side of

(1.4), which characterizes the dissipation of disturbance energy by viscosity forces, is always negative.

The total

disturbance energy can increase, if the velocity gradients in the main

flow

are sufficiently great, and the Reynolds

number is large enough so that the viscous dissipation of energy remains small. Equation (1.4) clearly shows the basic physical reasons which cause the instability in the flow. The problem with initial data for developing a small arbitrary initial disturbance in time is problem under consideration.

analogous to the physical

The flow's

instability could

have been judged according to the changed character of the -+

2

total disturbance energy In Ivl dQ. It is very difficult, however, to obtain a solution to the problem, and there are no general methods for its solution [6].

I

On the other hand, as

i i

has been shown by Petra [8], if one considers random disturbance fields and determines a sign of derivative from the energy, with

j

respect to time, on the basis of formula (1.4), then the values

1

of Re

determined in this manner cannot be, in principle, sufk ficiently accurate. Therefore, when studying stability we gen-

erally limit the problem to considering particular solutions for equations (1.3) which correspond to definite types of disturbances.

In the case of plane-and-parallel flows, the

low oscillation

method has been used most widely.

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: 2.

1

Low Oscillation Method. Let us assume that the main flow has only one non-zero

velocity component which is relative to the variable y, u

= U(y),

V = 0, w = O. In this case the coefficients of disturbance equation (1.3) will be related only to y~ therefore, a particular solution can be sought: .....

v . v(y)exp

P

=

i~,

.I \,

p(y}exp

c; = aX +

(1.5)

(: I

{3z - wt.

By substituting (1.5) in (1.3) we obtain the following equations for amplitude functions: ia(U-c}u+U'v ='-iap+-L (u" -y 2 u),

Re

ia{U-C)v

1 2 =-P'+R; (v".-y v),

la(U-c)w

- - 'f3 1 (w " .-y 2) I P +-w

(1. 6)

Re

i aut i f.w +

in which the designation

Y

2

= a 2 +f3 2 ,

Let us introduce a new variable u I U .=

V

I

'

= 0 ,

w

= a"C

is adopted.

using the correlation

aut {3w.·

I

Then the contiilUiwequation will take the form ~U. ,

+ v

= o.

(1.7)

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r

,

,1

L ____" ___1__________

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f

I

By multiplying the first equation by a

, the third one by

~

and

/8

combining, we obtain i a ( U- c ) u . + a U I

V

I

=- i 'Y 2 P+J._ (U ~/_y 2 u . ) Re

-

I

(1.8)

I'

.

Let us differentiate (1.8) by Y and substitute U1 from (1.7) and pi from the second equation (1.6); we then obtain the

f~llowing

equation (1.9) Formally, any function of U(y) can be considered as part of (1.9); however, according to the deviation, U(y) must satisfy the initial equations (1.1). This limits the application of equation (1.9). There are several important particular cases where U(y) satisfies (1.1), exactly or approximately.

In the case of internal_ flows

in a plane channel, with the walls at y

=~

1, we have Poiseuille's

flow U(y)

= 1 - y2

(1.10)

= y.

(1.11)

or Couette's flow U(y)

We obtain boundary conditions for v from adhesion conditions

v The

flow

= 0." = O.

(y

=±1 ) •

(1.12)

in the boundary layer on a flat plane is not

strictly parallel; however, the relation to longitudinal coordinate x is weak [4].

If a typical disturbance scale along x (wave length)

is on the order of a layer thickness, we can ignore this relationship.

The velocity function in the boundary layer is

where Uoo is the velocity in the external

6

flow,.

\

r

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I

, .1

I f

(n)is determined from Blazius' equation

fl/l+~

=0

ff".

(1.13)

and satisfies the conditions of f ( 0) = O. f'. ( 0)

Here U

1

'

y,

= O.

,

f' -.1. y. -00

are dimensional, physical values.



(1.14)

In place of the

asymptotic condition (1.13), the condition at the ultimate distance from the wall can be approximated: f'

= 1.

,=

(y-

(1.14' )

0),

The thickness of the boundary layer 0 is a conditional value

= 8,8

to a certain extent: if we assume that 0

~: ' then f' (n)

can be calculated out to seven valid places. Let us select the thickness of a boundary layer ~ = 0 and Uo = Uoc as scaled values. Then, for the velocity of the main flow we obtain

the velocity in the external

U(y) = f'

flow

(yofV: ). vX

(1.15)

By approaching the layer's outside boundary U" -+ 0: therefore, equation (l.9) changes to an equation with constant coefficients, and its damping solution has the following form:

v

= Aexp(-oy)+Bexp(-yy),

(j

r

> 0

If the Reynolds number is large, we can use exp (cry) =0, and the boundary conditions for v(y) will be v=O,'v'=O. v'+aV·= 0,

(y = 0); (y

=1 \ ,

(1.16)

7

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I

" ~~-

I

J

j,

~:

1! ~ ~t

must be Stric tly spea king , the secon d cond ition (1.16 ) satis fied wher e y ~ 00 city prof iles It is poss ible to cons ider simi larly othe r velo iles or appr oxiin the boun dary laye r, such as self- simi lar prof are used in the mate prof iles in the form of poly nomi als, whic h integ ral meth od of solu tion [4]. !

S are usua lly cons idere d pres et real In equa tions (1.5) a, l solu tions (1.9) valu es. By mean s of these equa tions , perio dica spon ding boun dary for x and z are intro duce d, whic h satis fy corre dary cond ition s cond ition s. Inasm uch as the equa tion and the boun of v(y) can for distu rban ces are simi lar, non- trivi al solu tions the integ er exis t only with defi nite value s (eige nvalu es) of itude (1.5) will c = c r + ic ~.• If c.~ > 0, the distu rban ce ampl will 'beco me unincre ase with time as exp c.~ t, and the flow we obta in If c < 0, the distu rban ces dq.mp, and if c i = 0 stabh~. i areas of stab ility a case of neut ral distu rban ces whic h divid es the when c i becom es and insta bilit y. The smal lest numb er for Re k , zero , is calle d the criti cal Reyn olds numb er. " , , ' ,

:~

;

r;.-

,,

t ~

I

9;

~

~

t

f r,

i

! f

t

tt ~i'

f

J

.,f

iii

f.;

[

,i .

f

!

:~

~~

~

whic h corre spon ds to two-d ime!l sd.on al distu rban ces. equa tion Then (1.9) will chang e into :the Orr-S omm erfeld Let

S

= 0,

IV 'V

'2

/1

_

...

-2a v +a~v-ia Re(( U-c)( vl-a2 v)+U "v] =

o.

S

"

y

,~1

.;,.,;,~ ,;.>"~. ,-'- . . . . . ~ ....~.,~

1

I

I

Re- a Re . I

1

(1.18 )

J

1

I;,

f._

1

¥

in (1. 9)

- -r "; U

j

1

(1.17 )

ver, it 0 the distu rban ces are three -dim ensio nal; howe ed to a twois easy to demo nstra te that this case can be reduc Indee d, let us intro duce a new Reyn olds numb er dime nsion al one.

when

I

8

r ,I

i

I

I

Now

equ~tion

(1~9)

coincides entirely with the Orr-Sommerfeld

equation, if in (1.17) we replace Re with Rei 1

;1, ,

Thus the eigenvalues of c, calculated for two-dimensional disturbances with

I

I

the parameter values Rei' Yf can also be used

for a three-dimensional case, but they will correspond to dif-

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:~

.' ,:;;

Reynolds number fdr lthree-dimensionaldisturbances (loS) will

I

be greater than for two dimensional disturbcmces.

1

ferent wave and Reynolds numbers. 'I

and a with y.

than a when'(3 .; 0, then Re > Rei

~

Since y is always greater therefore; the critical

"

, I I

'I

11

.

This is the

essence of Squire's equation [12], which makes it possible, ~;'-f

when investigating stability in a number of cases, to limit the class of disturbances under consideration. The disturbances determined under conditions of assigned

a, (3 grow or damp with respect to time. Another approach is also possible when disturbances grovving in respect to space are ~ought. Let us assign a real value w, then equation (l.S) will represent periodical time functions. In equation (1.17) a is considereq an unknown parameter. Eigenvalues a for which a non-trivial solution of a uniform boundary value problem for v (y) exists, will 9,enerally be complex i·'· a = a + i'a ,. The real portion a determines the disturbance r 1 r wave length for x, while the imaginary portion shows whether the disturbance amplitude damps or increases with an increase in x. real values

I1 j

~

I

If, given a certairi Reynolds number, we obtain low values c

l

'

j

1

ai' then both· approaches ate:equivalent, and the dis-

turbances grqwi,ng in space

of simple relationships [13]. (3 = 0 and

I1

and in 'time can be joined by means Let us assume that in (l.S)

a, ware complex values joined by the

relationship

>/.\,

F

(a,w)

= o.

(l.19 )

, 9

.1

l

r•

-J'

I

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}

,I

If (1.19) determines the analytical function w Kashi-Riman equations will be satisfied:

~~!. 00.

Ou.)"

;'cv r

= 00.,

r

I

00.,

I

OU>,1. ___

=-

00.

I

= w(a),

the

(1.20)

r

w Let us assign values a, using index 1, where a. ~ 0, ~ values a, w, using index 2 where w. = 0 and consider the ~ conditions for which a = const~ ai' Si are lesser values

r

(1.20) according to a" ~

wm « 1 .

By integrating

we obtain

UJ

rl

= W r2 ai2

w

"t

=

a.ja -w.jw ~ r ~ r

of the same order:

il

-- J 0

a i2 + J

O(.u'i'

:aa-;

0 'dUJ __ L

da

'dar

i

do.,

!

I

(1.21 )

,

From the fiirst equation of (1.21), allowing for accepted assump, . f . dW r t~ons, ~t 01 1 ows that wrl = wr2 + 0 ( wm 2 ). By expand~ng dar into Taylor's series and taking (1.20 into consideration, we

i

find

j

I

OUJ r

Cla

where a,~

r

ow

= (--!: ) 'do. r

I

:'d 2 UJ.

(---.!.)



'da2 r

(a -a ) + i.



(1.22)

1

j

is a certain intermediate value of a., 0 < a,* 0, dk=O, if k < 0, dk=:l, if k .2:.. 0, n = 0,2,4, ••• ,2 (N~2).. r

l

1

i .;

j

Boundary conditions (1.12) show:

1

1 .~

2N

(2.9')

In order to determine N + 1 expansion coefficients have a system of N + 1 equations, which includes N

(2.8), we

1 equations

,-.

"

~; "~

t

I

11

1 16 i

:'..._a'E. ·,.·I ,

r-·"I

:~

__

J

w __ . . . .-

'~'-------'T

.-~

. .-

,

L~L __.,"__ . ,. ~. _,._ (2.9) and the two equations (2.9'). The consideration of an asymmetric solution is similar. , In equations (2.8), (2.9), .;tnd (2.9') the sum must betaken according to odd index values from n = 1 to n = 2 N + 1. In the system of equations (2.9) and (2.9'), n also takes odd values n = 1.3, ••• , 2 N - 3. -

'i

: 1

Let us consider some results of numerically investigating a spectrum of eigennumbers in systenis :(2.9) and (2.9') at various approximations. Table 1.1 shows the values of the first eigennumber for a symmetrical mode when a - 1 and Re - 10,000. TABLE 1.1

-

N+1

c

i ---.

: 1

.1

I'

,

1

I

l I

f

.1

I

14 20 26 50

0,237l!375l 0,2375:2676 0,23752648 0,23752649 I I

+ + + +

0.00563644 0.00373427 0.00373967 0.00373967

i i i i

I

1 1

: The eigenvalue does not change, beginning with the 26th approximation. This value may be considered the eigenvalue of the boundary value problem (1.17), (1.12) with an accuracy to ~.4ecimal points. Asa numerical t.est will show, in order to achieve such an accuracy, the solution of the system (2.9) -12 mus~l?e carried put with an accuracy to 10 to minimize the round-off error. In calculating the following eigenvalues of the spectrum, errors will increase/so it is important to ca19u_latethe first eigenvalues with a definite margin for accuracy. Table 1.2 shows the first 7. eigenvalues, arranged in decreasing order c., calculated for symmetrical and non.J. symmetrical modes, where a = 1, Re = 10 000.

1

1

j

1

I

1 j

I

~

1 17

-I .

.,

!

, i

J~

.i

.• ~,

_".;.-~

.. ",,-,

"'e;v,-,~-~

J

~

"' ~j

~~~'''''_''''-_~'''''\;2 and )1, U ; then U" (U-c ) -1> 0, and therefore, with the sign of r max r the first integral (3.3), a positive value will remain, and i t cannot be reduced to zero.

Consequently, a critical point

Yc must exist, in which U-c = O.

The solution (3.1) in the

region y=yc has the characteristic

v - U~(U;)

-I

In(y-y) c

,

which may be eliminated only after the viscosity is taken into account in the equation.

44

The existence of a viscous instability

~

.

l

~- ----~r

0 ..

_._._

..

,.-

i

.II

) i

char acte risti c. has been linke d main ly to the exist ence of this (3.1) has a The secon d state ment is to the effe ct that if incre ase in solu tion with osci llati ons (a i = 0, c i > 0) that ction , U" = o. time , the velo city func tion has a poin t of infle of the secon d In actu ality , if a i '- 0, then with the symb ol will rema in, integ ral (3.3) , the value U" [ (U-c r ) 2 + c~~ ] -2 efore , the inwhos e symb ol coin cides with the symb ol U". 'Ther chan ges its sign tegr al may be reduc ed to 0 if U" at some poin t tive and nega and with in the regio n of integ ratio n assum es posi the prese nce of tive valu es. This asse rtion estab lishe s that is a nece ssary this poin t of infle ction in the velo city func tion To11 mien has cond ition for the exist ence of insta bilit y. As is also sufshow n [25], in a numb er of case s, this cond ition wake s In part icula r, this is true for prof iles in the ficie nt. and strea ms depi cted in figur e 1.7.

j

!

1

1

j

symm etric, If the velo city func tion of the basic flow is h damp wher e two solu tions of equa tion (3.1) may exis t, whic . For these solu y + ± oo~ they are symm etric and anti- symm etric be satis fied are, tion s to be reali zed, the cond ition s whic h must

1

;

resp ectiv ely,

-1

v' v

= =

0

0

=

0), v

+

0

(y + 00) ,

(y - 0), v

+

0

(y

(y

+

00)

(3.4) (3.4' )

i 1

I

1

of probl em It is easy to show that if v is the solu tion valu e, then v (3.1 ), (3.4) or (3.4 '), and c being the eigen Ther efore , an will also be a solu tion, and c the eigen valu e. Let us con~ O. indic ation of a non- visco us insta bilit y is c.~

I I

side r some exam ples.

'1

.~

45

rr

I

,

I

I'

I~:

--I

i

,

I

"

r4

--"~~

I

Let the velocity function given by the formula be U =

(ch y) -2

(3.5)

In order to study the problem with random a's, it is necesLet us write (3.1) in the form

sary to apply a numerical method. i'

!

of a system of two first-order equations: v'

w'



I0

- w,

2 I U" ( U--

c)

-I

(3.6)

1v.

Let us begin !the numerical integration from point y - y

o

in which

the velocity U is close to the constant and it is possible to assume the asymptotic expression v = A exp(-ay). The initial values for the system (3.6) will be: v

o

=A

exp(-ay ),

-av . 0

0

(3.7)

The integration is performed in a direction toward the point

y

=0

by the Runge-Kutta method or another numerical method.

The eigenvalue

c

is selected so that where

ditions of (3.4) or (3.4') are met.

If

c.

l.

y ~

=

0, the con-

0, then the de-

nominator in (3.6) is not reduced to zero, and integration is easily carried out.

Figure 1.5 depicts the results of calcula-

tions performed for this case [43J. The first two and the last two columns of c

r and c.l. cor-

respond to symmetric solutions, and the intervening columns correspond to non-symmetric solutions.

It is apparent that

symmetric disturbances exist where 0 < a < 1. The boundary value problem (3.4), and (3.4') for the velocity function (3.5) has two precise solutions corresponding to

neutral oscilla.tions:

46 !

~_::_"

""'~"~-~-~

-'.~-

,IIe

!iII :1

i

V

-

'I

::

1

,

-(

2,

c

, ch 2 y

rI

1,

c

ch 2 y

2 ';

thy

,

(3.8)

2

(3.8')

~{

TABLE 1.5

cr

a

cj

cr

C.I

/40

t--

-t

cJ

Ci

0,1

0.086

0,216

0,889

0,104

1.. 2

0.520

0.12:·

0,2

O~

166

0,257

0,826

0 .. 121

1,4

0,559

0.085

0.3

0.228

0,267

0.780

0..119

0.4

0.280

0.263

0 ..745

0,.108

1.6

0,597

0.053

0,5

0.823

0.252

0 .. 728

0.082

1.. 8

0,632

0.024

0.6

0.362

0 ..236

O,'DO

0,074

2,0

0.667

0,8

0.424

0,198

1,0

0.475

0,159

0 .. 66

0

0

These solutions are obtained at extreme points of the instability intervals according to

The formulas (3.8) and (3.8') may

Q.

be extended and a whole set of similar equations may be Obtained. Let the velocity function be given using the expression

i

i

I

'

,,

(3. 9)

iI

,

Then equation (3.1) has the two following precise solutions n - 12k 2:8 I)

v·= 1

I

k= 12k

11 j

a

= 2,

ii

(3.10)

1

n

I I

j

. , . ---""'--"'-r

v

2

=thY(2: k=1

C

2k

-

l

2n-l) l),

a

= 1 .,

'1

, 1

i

I

the coefficients in (3.9) and (3.10) are defined by the formulas: 1 A2 -:

2'

J

n(2n 1 1),

I

,~

'j

47

i

r

I .

'"·~·---'r---l

~

ir

I =

(1/41


i

by the relationship (3.15).

The solutions in all rectilinear

sections must be combined so that conditions at the breaking

Itt.

'. ;$ j

,-

points of the function U(y),

(3.14) and (3.15) and given

i'"

i,~" I

i

I

l

-"---~~~~.-

~

i

i ,

boundary conditions are satisfied.

r ~

t

As an example, let us suppose that there is a layer

., r:

O~y~l with a width of b=l,

in which the velocit.y i~creases

linearly from a quantity -U

to +U.

i'

velocity is constant and equal to -U o where y < 0, and to U o , where y > 1. Since the disturbances must damp where y +:!:. 00,

>

t,

t.

11:

I

. !

o

0

Outs,id,: the layer, the

then in solving (3.12), we shall obtain:

v

=

,

( -00 0, let us perform

may be n l . The value nl sufficiently large in module, so that the velocity distribution Figlirel.lO of the basic flow will turn out asymptotic. shows the forms of eigenfunctions in several concrete variants where r = 1/3: the curves 1, 2 and 3 correspond to the values

a - 1.0; 0.7: 0.45.

68

I

%

hw.L, . . .

The basic reciprocity of a disturbance in a mean flow originates near

~

-1.5, where

acute maximum.

w.1

has an

The results of calcu-

lating the eigenvalues are presented in figure 1.11. a

and

F,

With all values of

there exist unstable

wave disturbance, for which c.1 f O. Thus, in accounting for a mean flow, caused by a viscosity effect, interface proves to be unstable.

Fig. 1.10

There

is a mechanism for transmitting energy from a gaseous current to a mean flow and from a mean flow to c

~------~------,-------~

waves.

Curves of

c., relative to 1

equal values a

and

presented in figure 1.12.

Fare The

largest value, c.1 = 0.243, is . attained where F = 0, a = 0.25,

(J

0.25

Fig. 1.11

and the maximum velocity of increase, a c.1 = 0.131, is possessed .

1 1

by the wave disturbance where a ::::: 1.

i

I

The displacement velocity of such a wave constitutes 0.514 Uoo • 2.0 rj&·---....:---r-:------'

For example, the flow of gas above

I /58

the liquid surface; and of wind over water are usually non-homogenous.

1

j

Instability

mechanisms, related to these phenomena, also have been theoretically considered in a series of works [36].

[

!

n =

Fig. 1.12 .,

I

j

69

r

I

I,

.i

3.

The Liquid Layer on an Oblique Surface. In the case of a plane-parallel stationary flow of a viscous

liquid in a laxer on an oblique surface (figure 1.13), t,he equations in (4.2) have a solution which satisfies the condition of adhesion to the wall and the condition of the absence of tangential stress on the surface:

(4.34)

1

- 2

P ;:: 19U o cos

Fig. 1.13

I

u(l-y).

The solution in (4.34) is written in a dimensionless form. a typical length,

~,

here, the width of the layer has been selected;

therefore the surface equation will be y=l.

uo ' may be selected by various means. 'U o

~

I

j l

As

,

A typical velocity,

!1 I

Let us suppose

:-0

gl ?v-1s1nu, then the solution will assume the form

3

)

(4.34')

The liquid expenditure in such a flow in dimensional form 'is equal to q

From this we find the

1

= Uol

f Udy o

express~ons

= Uol.

for the layer width and the

o

- 1

= qv

- 1



The conversion y = n + Enh

transfers the layer to the region of

constant width ·0'::' n ::,1.

Therefore, in equation (4.15), it is

and (4.17), where n 70

= nh, and = I, it is

/59

1

!

l

(4.35)

Re ;:: U 1 v

"l 1

·1

i

Reynolds number through the flow rate

possible to accept f

I

in the boundary conditions (4.16) possible to assume f = h.

Let

I

I

~ I ! I,

'~

r! >

I

t,

I

,

l-

.. _..

-~---~~.-.-

>'

-~---.....,.~~

~

"Ll

I!

1

I

-.~.

i

,_"""""'"'*\

--

n

!

i

j

k __

I

J

us suppose that the surface of the layer is free--i.e., the effect of the environment on the flow is neglegible.

Then the

equation and the boundary conditions for the problem of flow instability in the layer with a velocity profile (4.34') are reduced to a dimensionless form: ylY -

2a2yll.+allv-iaRe[(U-c) (V"-a2v)-U"v)

vlI+[a 2-U".(U-c)

- 1

o .,

:::;

I

]y:::; 0,

I

(4.36) !!

I

yW-[3a 2 +iaRe{U-c))v'+ia{3 ctg u + a~ReW)

- 1

1

1



Two other conditions are given for a hard surface:

v : :; 0,

V':::;

0,

(T) = 0) •.

(4.37)

The boundary condition in (4.36) include!:.:' Weber's dimensionless number W, which is

e~pressed

through Re and the physical character-

J

istics of the liquid

j

W:::; pU 2 1u-

1

o

,-

:::;

y-

1

Re' 13 3-

1

1 . '1

I j

(4.38)

Through y, a combina'tion of parameters is designated, including ii

the force of gravity

I

0:

p,

V

1 (4.39)

1

Ii'

The slope angle u may be sufficiently arbitrary.

I I ,

r

important, special cases:

Let us note two

1

the layer on the surface, slightly

j

deviating from a horizontal position (u small), and the layer on the surface, situated almost vertically (U-TI/2 small).

Typical

j

examples are flows in an open channel and the runoff of fine,

1

1

71

r

~

L

I

.,

I! .,

f

I

0'

?

I

I .I

ity. liqu id and filmy subs tance s as a resu lt of grav bring s The form ulati on of the probl em unde r cons idera tion plan e-pa ralle l to mind the probl em of stab ility in Pois euill e's boun dary laye r flow . The diffe renc e lies in the fact that the rful, mech anism has distu rban ces. This crea tes a new, more powe ce wave s. Whil e of stab ility , relat ed to the fdrm ation of surfa hting wave s occu rs the insta bilit y induc ed by the Tollm ien-S chlic , the insta bilit y only where the Re valu es are suff icien tly large e Re-O ; there fore induc ed by the surfa ce wave s alrea dy begin s wher the flow in ther e is a large inter val of Reyn olds numb ers when the layer may be lami nary , yet agita ted. 1, emer ge With smal l Re valu es, long wave s for whic h a « surfa ce tensi on. firs t, and the shor t wave s are extin guish ed by solu tions as For long wave s it is poss ible to obta in simp le inter preta tion the final resu lt. This not only faci litat es the resea rch meth ods of resu lts, but it also aids in the selec tion of If that a «1. for a non- linea r probl em. Thus , let us suppo se ld show defi nite we use this ineq ualit y, it follo ws that we shou ce tensi on. care in evalu ating a term whic h reduc es the surfa In acco rdanc e with (4.38 ) we obta in a 2 W-

1

= a2yR e-5/3 31/3

fore with mean For many liqui ds the magn itude y is large ; there a smal l a. valu es of aRe, this expr essio n is fina l, even with 0 exam ple, Re-2 7, For wate r with t=15 C, we have y=28 50, and for . 2 -1 and by reje ctand a=O. l, then a W =0.16 5. By vlrtu e of this, 2 retai n the term ing the magn itude on the orde r of a , we shal l whic h char acte rizes the surfa ce tensi on. 2 orde r of a If in (4.36 ) and (4.37 ), the magn itude s on the s for a sing le are rejec ted, then the equa tion obta ined allow ition in (4.36 ), integ ratio n whic h, with regar d to the secon d cond redu ces to the follo wing boun dary valu e probl em: 72

.

,.,",.•

I I

I .;

r



.;

I!

Tj

,

i

,I

f 1

i ( aRe) -1 v II I

-

U I V 1 (U

i{3 ctg

C)Vl

v

0,

VI

'V

Re

-1

2-1 +a W ) h.

0, (4.40)

(n

0)

iU I h. ,

v"

i{U';'c)h.,

v

Since on the basis of (4.14), v

(n

1) .

is on the order of

a

then from

/61

equation (4.2), designed for the y axis, it follows that, with mean values for wRe, the stress gradient along y is proportional to a 2 . Therefore, the physical meaning of the simplification conducted is that the stress is considered as a constant across the layer and is equal to its value on the surface of the layer. Let us search for the approximate solution (4.40) in the form (4.41) In order for the boundary conditions to be satisfied, it should i

be taken into consideration that a

=

1/4 [6 (U-c) - U"] ,

b

i

l !

=

!"

1/4[-2(U-c)+U"].

Here and later, U and U" are obtained, where

-1.

By substituting

(4.41) in (4.40), and integrating the equation from after segregating the real and imaginary portions, we . 3(oRe)

1

('J, ,r

C

r

)

3 / 5 c.+2(U c r le. I . I

n-o

to n=l, shall obtain

I

1

I

()

(4.42)

i

j J

·1

If we presuppose the c. is small, then with an accuracy to terms

2 ~ on the order of c., from (4.42) we shall find that

1

~

I

73 '.'J

I,.,

r

..

!!

~ -'~.',,-~--.- ~-.~ --~-

,

.----.-_"---.,j-

-~-, ~

I

" '"*,,,.

-

,I I

i

f

"

"If ;

l

j

I

I

I

i

[

~

l

1

(4.43 ) -1

C

j

= uRe( 3-u 2 W

-3Re

-1

ctg

2

u)

[3+1 0S/25 (aRe) J

, and from the firs t equa tion in For neut ral distu rban ces c.=O ~ ment with (4.34 ), (4.43 ), we find that cr=O . Since it is in agree equa l to 3/2, the velo city of the basic flow at the surfa ce is distr ibute d with then cons eque ntly, the neut ral distu rban ces are than the liqu id t.he phase velo city, whic h is two time s grea ter distu rban ces velo city at the laye r surfa ce. For incre asing 3. From the with smal l aRe, the phase velo city is less than tion for the expr essio n in (4.43 ),. for c.~ we obta in an equa neut ral curv e in the plane a, Rp. (4.44 ) are depi cted in Some neut ral stab ility curv es with diffe rent E 2) the laye r fi,gu re 1.14 . On the vert ical, hard surfa ce (u=~/ l '~f a 2 < 3 y -lRe 5/3 3 -1/3 . Th e be, I . ~s unsta Qlr'---r--~--~~~ neut ral curve orig inate s from the begin f1i.. ning of the coor dina tes, and there fore with any liqui d flow rate , the laye r is ·ISOC I. f)·go· 2. O.JO·

3. 9, 15' Q -

.....~-+--...,...-~

25

unsta ble to infin itesi mall y smal l disturba nces with a suff icien tly large wave leng th. Shor t wave s fade out due to surfa ce tensi on. With a redu ction in

dimi nishe s and the surfa ce tensi on cons eque ntly, the inter val of unsta b1e wave s exten ds towa rd the shor ter wave s.

Fig. 1.14

(u