Some problems in boundary-layer transition

BY

J. A. STEKETEE

,

,

,;

..

This work was suggested by Professor A. Robinson to whom my th anks .. ·are-è-ue-.

·-Ais-e I wi:sh t-o thank -tfie Institu,t.e ef··A.e---rophy,ic& for the offer to pubH-sht-fl:i.s- reeeal'-eh---aB'-one of their-re-llort-s " in partie~li}r,

Profe--sscGr G. N. ··P-at--teF--son-andprof-es-sor 1. 1. -Gl-as-s. La-st but not least I thank Dr. H. S. Ribner for reading the manuscript and for some interesting comments and discuSlsions.

r

SUMMARY

This investigation finds its origin in the paper of

Ho W. Emmons on boundary layer transition (Ref. 26) and contains some attempts to find theoretical support for the empirical facts which are the basis of Emmons' theory. These facts are (1) the existence of regions of turbulent flow (turbulent spots) clearly distinguished from and surrounded by laminar flow (2) the way in which these regions are transported and grow.

As laminar and turbulent flow are always discussed

separately we are unable to formulate the problem. where both regimes exist side by side. in clear mathematical terms. The investigation instead implores a few models which show some aspects of the reality.

It is believed that the investig~tion throws some light on the importance of the velocity gradient. This is due to two points. In the first place we obtained the solution of some ordinary differential equations as combinations of elementary functions which are easily computed. U sing these solutions it is shown that some solutions of the perturbation equations can be made to agree with measurements of turbulent fluctuations in a shear flow.

Secondly. a few rather surprising features were found from some elementary considerations on vortices in a shear flow.

TAB LE OF CONTENTS

Page

1. REVIEW SF- THE-TR-ANS-ITI0N- PROB-LEM-- 1

1.1 Introduction 1

1.2 Gene-ra-l Cons-iderat-iens on the- Transition

Phenomenon ₁

1.3 rvJore--De-ta-ils--of the- Trans-ition; the-Stabili ty

Problem ₃

1.4- Se-mi-Empirical Theories for the Transition 6

2. THE THEOR-Y OF EMMONS 7

2.1 Outline of Emmons' Theory 7

2.2 The Formula for the Fraction of the Time

During which a Point is Cove-red with Turbulent

Flow 8

2.3 The Experimental Evidence for the Spots and

Cones 11

2.4 Outline of the Further Content of th is Report 14 'y

3. ASPECTS OF THE EQUATIONS OF TURBULENT

MOTION 15

4. A STATIONARY DISTURBANCE ON A SEMI-INFINITE

FLAT PLATE 21

4.1 Introduction 21

4.2 Oseen's Equations of Motion 22

4.3 The Solution of the Semi-Infinite Flat Plate 25

4.4 The Three-Dimensional Disturbance 31

4.5 The -Simplest Disturbance 35

4.6 Discussion and Criticism 39

5. A STATIONARY DISTURBANCE IN A FLOW WITH

SHEAR 42

.

_..

5.1 Introduction 42

5.2 The Equations of Motion of the Disturbance 43 5.3 The Solution of the Perturbation Equations 48 5.4 The Velocity Profile of the Main Flow and the

Ordinary Differential Equations 52

5.5 Deta-iled Solutions of Special Cases 54

\ (a)

5.6 The E-quations for the Non-Stationary

Dis tur bance 66

6. THE EQUATIONS OF TURBULENT MOTION FOR A

SPECIAL KIND OF MEAN FLOW 70

6.1 Introduction 70

6.2 The Equation of Turbulent Motion for the Simple

Mean Flow 70

6.3 Further Simplifications of the Equations

Obtained in (6.2) 71

6.4 Equations Analogous to (6.3.3) and (6.3.5)

Equations 73

6.5 Discussion 74

7 . SOME CONSIDERA TIONS ON VOR TICES 7 . 1 Introduction

'1.2 Two Line Vortices Near a Wall 7 .3 Vortex Wires in a Shear Flow CONCLUSIONS REFERENCES APPENDIX 1 APPENDIX 2 APPENDIX 3 FIGURES 1 to 8 TABLE 1 75 75 76 78 82 84 87 90 94

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1. REVIEW OF THE TRANSITION PROBLEM

1. 1 Introduction

It is well known. that in many theoretical investigations concerning fluid and gas flows. the influence of the viscosity can be neglected. Only in the neighbourhood of walls and of obstacles placed in the flow. where the velocity gradients are generally large one has to take the viscosity into account. These in general narrow regions • around objects placed in the flow or along walls bounding the flow. are usually called boundary layers.

In 1904 Prandtl (Ref. 1) formulated the equations of motion for the flow in these regions and since that time considerable effort has been spent to elucidate the features of these flows (Ref. 2).

Another hydrodynamical problem which has attracted much attention is the problem of transition from laminar to turbulent flow. It was Reynolds (Ref. 3) who discovered that water flowing with sufficient speed through a cylindrical tube shows a different flow

pattern at different cross sections of the tube; while near the entrance of the tube the water flows smoothly. at some distance downstream considerable agitation occurs in the flow and this continues to be the case for cross sections further downstream. The first type of motion is usually called' laminar flow. while the second pattern is called turbulent flow.

Experiments have shown that a state of affairs similar to the flow through a pipe. exists in other cases. notably in the boundary layers mentioned before; when a flat plate or 'an aerofoil is placed in an air flow. parallel with the axis of the stream. the flow in the

boundary layer is laminar near the leading edge. but some distance downstream the laminar flow breaks down and turbulent flow develops.

To distinguish this case of transition. which is of considerable importance in aeronautical engineering. it is called boundary-layer transition.

1.2 General Considerations on the Transition Phenomenon

The transition phenomena mentioned in the introduction give rise to several questions • the most important of which is to explain the mechanism causing this change of flow pattern.

This question is the more intriguing as the simple theory does not give any indication that such a transition may possibly occur.

In this respect the transition of laminar to turbulent flow is in a position similar to other physical phenomena.

consequent collapse of the structure built from it. The theory of elasticity tells in this case which deformations will be caused in the structure by certain load distributions , but it does not .appreciate or indicate the danger of rupture and collapse to which the structure is

exposed under these loads; as ~ matter of fact it does not give any

indication that a phenomenon as ruptuire may exist. To fill this gap in the theory and make it fit for strength calculations one usually adds some empirical or semi-empirical criteria-to the theory (Ref. 4).

Another example might be the problem of the condensation of a gas. lf we take the gas to be an ideal gas, the equation of state does not tell us anythingabout the possibility of condensation, and the liquefaction would come to us as a surprise, a surprise which in the past gave rise to the distinction between gases and vapours ~ By the introduction of a more general equation of state. however, it has

become possible to include the liquid-and gas-phase and the transition between the two in one equation (Rëf. 5).

The introduction of semi-empirical hypotheses in the theory of elasticity to deal with the problem of rupture, and the introduction of a more general equation of state to deal with condensation are .satisfactory up to a certain point. lf attempts are made to get more

details of the phenomena one encounters considerable difficulties. In the two examples mentioned such attempts have been usually

accom.panied by a change of the physical picture. To understand more of rupture » the phenomenon has been discussed from the point of view

of crystal lattices (Ref. 6), for example, while for the condensation clusters of molecules have been considered (Ref. 7).

In the problem of transition from laminar to turbulent flow similar developments took place. After Reynolds discovered the two types of flow he gave also a criterion which would tell wh at flow pattern to expect under certain circumstances. The criterion is simply that the Reynolds number exceeds or is less than a certain numerical value, depending on the geometrical situation . The introduction of this criterion is analogous to the hypotheses supplementing the theory of elasticity to deal with rupture. A

development analogous to the case of conq.ensation has not occurred. In attempts to obtain more details of the transition

phenomenon considerable difficulties were met.

It is, however, interesting to point out that to obtain

these details it has not been necessary to modify the physical picture as in the two other examples.

In the next paragraph an outline wiU be given of the more detailed attempts to come to an understanding of the transition

.

'

1.3 More Details of the Transition; the Stability Problem A complete and satisfactory theory of the transition

phenomenon does not exist. This is mainJy' due~ to the difficulty of the problem and the general ignorance about the characteristics of

turbulent flow. It can be doubted whether a satisfactory theory will be created as long as the formal description of the laminar flow remains different from the description of the turbulent flow; that is as long as one remains unable to solve the equations of Navier-Stokes in their general form.

Investigators in this field, which included, except people

working exclusively in mechanics, physicists as Lord Rayleigh, H. A. Lorentz, A. Sommerfeld, W. Heisenberg, have mainly dealt with the stability question, while very few people, among which

G. I. Taylor and very recently H. W. Emmons, have made attempts to go beyond this stage of the transition process.

Very early already it was conceived that the transition might be due to small disturbances , always present in experiments but

ignored in the theory, which could be amplified in certain cases and

damped out in others; in case they 8:re amplified they may grow so much that they cause the laminar flow to break down finally. In case the disturbance damps out the flow is called stabIe for this kind of disturbance, in case the disturbance amplifies, the flow is called

unstable for the disturbance.

Mathematically the idea has been worked out in two ways. The one way is the method of the exponential time factor, the second

way is the method of decreasing positive definite integrals. Most work

has been spent on the first method.

The scheme of this method is as follows. The flow is assumed to be a steady flow to which a smal! disturbance is added. The disturbance is assumed to be so small that in the equations of motion only terms which are linear in the properties of the

disturbance have to be retained.

In this way the non-linear equations of motion are reduced to linear partial differential equations for the disturbance. It is then assumed that the disturbance is periodic in one or two space

coordinates (depending on the fact whether we consider a

two-dimensional or a more general three -two-dimensional case) and depends on

the time as ect with c complex.

With these assumptions the partial differential equations reduce to ordinary differential equations with the remaining space coordinate as independent variabIe and containing a number of

parameters among which c . If further the usual boundary conditions are imposed an eigen-value problem results . From the relation between the parameters occurring in the differential equations , which

has to exist to admit eigen functions, it is then found under which circumstances disturbances will ainplify or damp out. (That is R (c)

~ 0).

The method has been applied to a large number of cases.

If the problems of the stability of a fluid heated from below, stability

in compressible fluids and magneto-hydrodynamics are left out we have still to distinguish for the incompressible fluid the stability investigations for

1) Couette flow between rotating cylinders. (Refs. 8, 9) * 2) Couette flow between plane walls. (Ref. 10) "'

3) Poiseuille flow between plane walls. (Ref. 10) .

4) Poiseuille flow in a tube of circular se-ction. (Ref. 13) 5) Poiseuille flow in a tube of arbitrary sectiorr: (Ref. 22) ' 6) Boundary layers along plane walls. (Ref. 10) '"

7) Boundary layers along curved walls. (Refs. 11, 12) Some of these investigations were conducted for 2-dimensional disturbances, in some investigations the viscosity was neglected, in others it was not In particular this last point gave rise to controversial conclusions which have been settled only in recent years.

The mos t striking investigation is without doubt the discussion by Taylor (Ref. 8) of the Couette flow between rotating cylinders. In this case it was predicted that ring vortices of a particular wavelength would become unstable under certain presetibed conditions . These vortices immediately made their appearance in the experiment when the theoretically required circumstances were reached. This vortex pattern, however, did not show signs to break down into turbulence as long as the cylinders rotated with the same speed. Only by increasing the velocity of the inner cylinder could turbulence finally be obtained,

while more recent investigations (Refs. 14, 15) indicate thfl.t the

original vortices even persist when the flow has become turbulent. While in this case the theoretioal predictions were immediately verified experimentally, this was not so for the other cases. This lack of experimental support was detrimental to the faith in the stabilÎ.ty theory.

If we restrict ourselves to the boundary layer case, which has the largest pract~cal importance, we find the following development. The stability theory for the boundary layer along a flat plate was worked out around 1930 by Tollmien (Ref. 16) and Schlichting (Ref. 17) using . essentially the methods which Heisenberg (Ref. 18) developed before.

It lasted however till 1943 before the wavy disturbances on which the

theory is based were observed by Schubauer and Skramstadt (Ref. 19)

Since then the theory has been accepted as essentially correct, although ithas similar weaknesses for the prediction of the transtition to

*The references given are by no means complete. Many references

turbulence as the theory of Taylor has for forecasting the appearanee of

turbulence between rotating cylinders.

Liepmann (Ref. 20) mentions a case where the stability

theory tells that the boundary layer becomes unstable (that is, some

2-dimensional disturbances are beginning to amplify) a.t a Rey~otds

number (based on the distance from the learling edge) of 6 x 10 ; while

the turbulegce ma~es its n rst .appearance at a Reynolds. number of 2

to 2,8 x 10 , that IS 33 to 47 times as fat f.rom the 1eadlng edge as the

point of instability. Tay10r (Ref. 21) mentions cases where the

difference is a factor 5 or more.

From these experimental facts it follows that although the

instability of the boundary layer seems a necessary requirement for the

appearance of turbulence, it is by no means a sufficient condition; the

point 'where instability begins and the point of transition, that is the point where'.tu~bu1ence first appears, are quite distinct.

The second method to deal with the stability problem is the

method of deereasing positive definite integrals (Ref. 22). In this seheme

expressions are written down for the energy or the square of the vorticity

and the beha viour in time of these expressions is studied. The results

whieh this method gives are sufficient conditions for stability, but no

information is obtained about the beginning of instability. While this

method was used extensive1y in the earlier investigations (Refs. 23, 24)

it has been little used in recent years as it is less powerful than the

method of the exponential time factor.

Summarizing the stability theory one ,can Bay that the theory

describes a phenomenon which is present in the cases investigated. When,

however, ane wants to predict the point of transition, and the results of the

stability theory are used for this purpose, one may on1y expect answers

which wiU be on the safe side. The transition point and the instability

point appear to be quite distinct while the instability point is reached first.

The theory ean therefore be considered to deal only with the very initial

stages of the transition proeess.

This yàst statement can be c.larified .in the following way.

The instability point as usually understood and as mentioned

before, is the point where the Reynolds number has such a value that

for points where the Reynolds number is Iess no disturbanee of any wave

Iength amplifies. At the instability point disturbances of a particular

wavelength begin to amplify, while for larger Reyno1ds number more

and more wavelengths wiU be amplified.

It was shown by Squire (Ref. 25) that 2-dimensional

disturbances are less stabie than 3-dimensiona1 disturbanees, so that

we may expect that wUh increasing Reynoids number 3-dimensional modes

a1so make their appearance. If the spectrum of these 2-and 3-dimensiona1

modes has become sufficiently filled and the interaction of the different

modes has become sufficiently lively (non-linear terms) the resulting

transition point will he reached.

It should be noticed that the definition of transition point is rather vague as there is no clearcut distinction between a laminar and a turbulent flow.

1.4 Semi-Empirical Theories for the Transition .

Theories dealing with the transition process beyond the

instability point are very scarce. Straightforward attempts are generally exc luded due to the difficulties with the non-linear terms in the equations of motion. The few attempts which have been made try to avoid the

difficulty by introducing some suitable hypotheses.

These theories which are concerned only with the transition

in boundary layers are: .

1) Theory of Taylor (Ref. 21)

2) Hypothesis ofLiepmann (Ref. 20) 3) Theory of Emmons (Refs 26, 27)

As has been mentioned before, due to lack of exp~rimental verification, the theory of Tollmien and Schlichting was not believed for a long time. It was argued that infinitesimal disturbances are irrelevant for the transition as it was be lieved that all flows would be stable for infinitesimal disturbances. G. 1. Taylor in particular has maintained the point that the laminar flow will break down only when disturbances of sufficient size or pressure gradients of sufficient importance are acting in the flow. In the boundary layer such disturbances might be due to the turbulence in the free stream. Further it is felt intuitively that such a disturbance is more effective for causing transition when the boundary layer is thick, than when the boundary layer is thin. This lèd Taylor to the hypothesis th..at the transition point will be define.d ihrough a relation between the pressure fluctuations in the free stream and a parameter

which characterizes the thickness of the boundary layer in a particular case, for which the Reynolds number based on the displacement thickness was selected.

The formula obtained in this way for the point of transition could be made to agree with a number of experitnents.

In a wartime report H. W. Liepmann has approached the point in a different way starting from the stability theory. Among the results which the stability theory produces are the initial rates of

amplification of the infinitesimal disturbances. As there is no information available on the further be,q,aviour of the rate of amplification Liepmann assumed the rate of amplification to remain constant. Further the

hypothesis is introduced that the transition wiU occur when the shear stress produced by the disturbance has become of the same size as the shear

stress of the mean flow at the wall. From these assumptions Liepmann is able to predict approximately the Reynolds number of the transition point.

Both the theories of G. 1. Taylor and H. W. Liepmann are of semi-empirical type and may be compared to the rupture-hypothesis in the theory of elasticity.

2. THE THEORY OF EMMONS 2. 1 Outline of Emmons' Theory

The theory of Emmons is different from the other two as it tries to deal with the whole transition process and does not restrict

itself to predicting the transition point or the point of instability. The theory is therefore more ambitious than the other two, but suffers from the same lack of theoretica 1 support. First an outline of the theory will be given while more details will be discussed in the following sections .

Emmons observed that on a water tabie, when looking in a

direction perpendicular to the plane of the table, small spots of

turbulence arose in the transition region, in disorderly, random way. These spots grew while they were washed away with the flow. Some distance downstream the turbulent spots had grown so much that they covered the whole flow region and the flow therefore had become turbulent. It :was further observed that the carrying away by the flow and the expansion of the turbulent spots happened in such a way that

the envelope of the spots at different moments was a wedge shape with

a top angle of about 190. lf this envelope is plotted in a 3-dimensional reference system where the time axis is perpendicular to the plane of the table it will become a cone. For this reason the term '.'cones of Emmons" will be used several times (see Fig. 1).

Except these spots of turbulence Emmons could obtain also permanent sources of turbulence with permanent cones or wedges by

_ putting some surface disturbance on the water tabie.

Mainly from the two observations mentioned, the one about the creation of spots of turbulence, and the other about their way of propagation, Emmons developed a formula for the fraction of the time that a particulr point in the transition region will be-covered with

turbulent flow. Once this formula was obtained Emmons could compute the skin friction in a point in the transition region by assuming that during the turbulent period the skin friction may be obtained from the turbulent skin friction coefficient Cl)T ,while for the laminar period the friction may be computed from the laminar skin friction coefficient

C.DL. , so that if a point is turbulent for a fraction f of the time, the local skin friction CD is given by Cl) =:= Î C.DT

+

(1 - f) C.DL

With some additional assumptions some numerical data could be computed which could be checked with experimental results . A reasonable agreement was obtained with the experiments.

It is interesting to notice that in Emmons' theory the notion of transition point loses much of its meaning as the spots of turbulence arise in a random way in a certain r~ion, the limits of which are not clearly defined.

2.2 The Formula for the Fraction of the Time During Which a Point is Covered with Turbulent Flow.

*

The mathematical part of Emmons' theory is for a large part concerned with the derivation of the formula for the fraction of the time during which a point of the table or wall is covered by turbulent flow.

The derivation of this formula in Emmonl5! paper is felt to be rather artificial and another derivation will be given which is believed to be more straightforward and Lucid .

. Therefore we consider iirst a simple c-ase of probability theory and extend it afterwards to fit the theory of Emmons.

The problem we consider first is the following: Let there be given n simple events, and let the probability that the kth event occurs be Pk, while the probability that the ktH event does not occur is I-Pk' with Pk constant. (The sample space of each event consists of 2 points. ) The n events are independent but not mutually exclusive.

(It is noticed that the situation is c10sely related to the case of n Bernoulli trials (Ref. 28) and reduces to this if we put PI = P2

=

-- -- =p = p.) n

We ask for the probability that at least one of the first k events occurs. (There is no loss of generality in asking for a least one of the first k events, as it will always be possible to renumber the events in such a way that the ones we want to consider will be the fitst k in the new numbering system.) If the probability which we want to compute be denoted as P J. _, ok it is c1ear that we have

Continuing step by step we have

*The result of this section has appeared in the Reader's Forum of the Journalof the Aeronautical Sciences of August 1955. VoL 22, No. 8.

.

'

and generally

k

-lT

(l-p.) . 1 ' "':1

In partieular we have for k

=

n

IYV P

= 1 -

lT

(l-p/. 1, n IJ : I (2.2.1) (2.2.2)

This is the answer to our question but to conneet it with the expressions of Emmons we proeeed as follows:

We introduce a _quantity Y""rn whieh is defined.

as ~ - p - p

m - 1, m

+

1 1 _, m Then we have immediately

m

~m

=

Pm

+

1

Jr

(1 - Pi)' (2.2.3)

v:1

Further, we have from the expression for P

1 k that

m '

1 - P

1 _,m

=

_~=

lT

_I (l-p₁ .),

The L. H. S. of this expression may be written

= 1

Substituting this in expression for ~m

m-I

- L

-.r.

k"

=

I k we find

+

P_{1rfI -1} - P, m \ \ (2.2.4) =

We suppose now fhat a volume

V

is given and let this volume be

divided in n ce Us. Let the volume of the i fh eell be denoted by l:::..-v~

and let -Pi be a point in this eell and representing this eell.

Eaeh event of our problem ean then be assoeiated with a eell

in the volume

-V-

.

The probability that the i th event occurs h

becomes in this new situation the probability that the event in the i t ce 11

oecurs and we write instead of p. the expres sion g (P.) AV;.

~ 1 1

If we put Bk

=

rr

(l-Pi) (2.2.5)

I.=-I

this expression beeomes in our new notation

.. Bk"

-n- {

1 - g (Pi)

t::..'Z}

We ha ve then further

k

:·In Bk

=

Bk = Exp.

:L

~

t

1 - g (P.)

A

V-' }

[ "

~

_:-

.

"Ç_k,

_./Vn..

P

f

L

_{1 - g} 1 _{(Pi) A} " }

~

] Remembering that we have Bk - g (Pi)

~ ~

}

k:

= expo

,

L

f

,,=1

l

DO

=

L

l

l'Yl :. I and in partieular PI, n = 1 - expo

~

{ -

~I [~~)

iV:

J

'}2.2.

6)

We then suppose that the number of events or eells in

Y

inereases indefinitely in sueh a way that the 6. \,ft. '5

beeome smaller and smaller. The seeond and higher terms in the series for the Iogarithm ean then be negleeied eompared to the first term as they are of seeond and higher order small in ~

VÎJ

while the sum

~

beeomes an integral over

-V-(.,=1

finally

When we further write f (P) instead of PI we obtain ,n

(2. 2.7) This is the formula of Emmons.

To see how our formulas are related to the ones in Emmons' paper we have the following tabie: .

g (Po)

dV

~ (P Po) d-\[ f (P)

Further we have that (2.2.4) eorresponds to Emmons' integral equation. The analogue of the e.xpression (A. 4) of Appendix 1 in Emmons' paper (Ref. 26) is easily found from (2.2.3)

So far we have not explained the physieal meaning of all the terms, which is what remains to be done.

We have seen that a turbulent spot sweeps out a cone in a

3-dimensional diagram where the time -axis is perpendicular to the

plane of the water table which wiU be the xl X

2 -plane, where xl is the flow direction. Let this cone be caUed the.forward cone of P where P is the vertex.

If the generators of the cone are continued in the opposite

direction of the vertex, we eaU this cone the backward cone of P. We

further assume that the cones swept out by the different spots of turbulence

are identical and that the occurrence of a tur.bulent spot in P does not have

any influence on the possibility of turbulent spots occurring in other points

(independent events). It is th en clear that a point (xl x

2 ) on the water table will be covered with turbulent flow on time t if there is at least

one spot of turbulence in the backward cone of P(x1 x3t). If th is

probability is denoted as f (P) it is clear from our former considerations

that

f (P) = 1 - (2. 2. 7)

Where g (Po) d

V

is the probability that a turbulent souree occurs

in d

V

surrounding Po' and

Y

is the backward cone of P.

The advantage of the derivation given here is that there is

no need for the introduction of the

Y

or

"î.

(P P ) dY m "'" 0 _ to obtain the result. Emmons introduces the . ...L... (P Po) d \[

functions to eliminate "overlaps", but reaches only part of his purpose

as the conditions imposed on

':t

(P Po) are only concerned with

the xl coordinate while the x3 and t coordinates are left out altogether.

It seems therefore that the ~ (P Po) functions only e liminate the

"overlaps" as far as the xl coordinate is concerned.

-A lthough the derivation given here is more straightforward than the one in Emmons' paper, it is not a rigorous derivation as the

limit transition from the finite number of events to the continuous' infinity

of events cannot be discussed in detail as long. as g (Po) is unknown.

In case the flow outsJde the transition region is stationary one is justified, from a physical point of view, in assuming that g (Po)

will be independent of the time.

As a consequence f(P) wiU then also become independent

of the time. For physical purposes one is again justified in interpreting f(P) as the fraction of the time that the point (x1x3) on the table will be

turbulent.

2.3 The Experimental Evidence for the Spots and Cones

It is clear that numerical computation for f(P) can be made

only if we know g(P ) and the volume

V

Of both these quantitieE\,

Emmons assumes g(P 0 ) to be constant; as there is no preference for any particular choice from a physical point, due to our ignorance, the mathematically simplest choice is made. As the idea of these spots is new and due to EIDmons himself .00 Qther results exist to

give a clue as to the real form of g(P 0 ). A th-eoretical discus sion pf g (P) seems out of the question as long as we remain ignorant of the exact difference between a laminar and a turbulent flow.

The shape of the turbulent spots defines the cross section of the cone, but on the shape of the spots there is little agreement. The spot shape which is drawn in Emmons' paper (Ref. 26) may be considered equal in general appearance to the spot in Fig. 1 of Mitchner's article (Ref. 29) (see Fig. 2). The s pots of Emmons' paper and the spots in Fig. lof Mitchner are observed on water tables. 'The spot in Fig. 2 of Mitchner. however, which was observed in a wind tunnel is totally different in shape. (It may be remarked U;lat the spot shape as discussed here has only to do with the projection of the spot on the xl x3 --plane. Discussion of the extension of the spot perpendicular to the- table (the 'c'thickness") have not been made and only Fig. 2 of Mitchner gives a suggestion how it may look).

A theoretical attempt to obtain some information on th~ shape of the spot was attempted by Pearson (Ref. 30). The equations of motion, however, do not appear in this discussion, and as only the projection of the spot on the x1x3-plaoe _is considered it is not clear what the physical importance of his re;;ult is. This result is that the spots will finally approach an asymptotic form independent of the initial shape provided the propagation or expansion of the spot occurs according to a certain postulated law.

Finally, it may be m.entioned t~at in the hot wire measurements of Schubauer and Skramstadt (Ref. 19) bursts of turbulence are registered which appear and disappear very suddenly. In the terms of Emmons' theory these bursts represent spots of turbulence passing the hot wire. As the hot wire is fixed in the 'wind tunnel no information can be obtained in this way, however, on the sideways expansion of the spot, or on the growth of the spot in the flow direction.

Although there is little agreement on the detailed shape of the spots in the little experimental evidence available, more unity exists

on the shape of the projection of the cone on the x1x3 -plane. This is true in particular if it is assumed that a permanent source of turbulence may be considered as the limit case of successive occurrences of turbulent spots. The projections of the suc~essive cones on the x1x3 -plane will then coincide and in the limit this wiU become a wedge of turbulent flow

in the x1x3 t -space. (We shall most of the time discuss only the projection on the xl x 3 -plane but will adhere to thedesignation of "wedge".) These ·

wedges of turbulent flow have been observed on different occasions while there is other experimental evidence which is consistent with this picture.

Charters (Ref. 31) describes the transition from laminar

to turbulent flow by "transverse contamination" and shows sketches of

angular regions similar to the ones observed by Emmons. The angle of

19-

2.0 for the wedge, which Emmons uses is :taken from Charters'

paper. It was noticed that the angle of the wedge changes very slowly

with variations in the velocity of the mean flow. The angles of the wedge which Mitchner (Ref. 29) observed were 13, 20 for his experiments on

the water table and 17,20 _{for his experiments in the' wind tunnel.} Charters' experiments were made in a wind tunnel and it is noticeable

that the difference between the wedge angles in air is less than between the wedge angles in water and air. Mitchner mentions also the remarkably

smal! variation in wedge angles with variations of the mean flow.

Gregory and Walker (Ref. 32) investigated the effect on

transition of isolated surface excrescences in the_houndary layer. These

surface excrescences can be considered from our point of view as

permanent sources of turbulence. In their report are pictures of wedge shaped wakes behind the surface excrescences. For very small

excrescences the formation of this wedge-shaped wake is delayed in sueh

a way that the vertex of the wedge is not at the position of the excreseence

but some distance downstream from it. The angle of the wedges varies in their observations mainly from 150 - 18.50 • A few observations would

indicate that the shape is not a straight wedge but a horn-shape beginning

with an angle of 80 at the excrescence and increasing to an angle of 140 . (The observation on a flat plate which shows this phenomenon is made at

very low speed.) As a few observations were done on airfoils and not on

flat plates it may be that in some of their observa'tJons the curvature of

the airfoil and the pressure-gradient may have played a role.

Their observations are further remarkable for the fact that

they show a few pictures which give some suggestions for the detailed

structure of the flow in these wedge- regions in the initial stages; it

seems that the flow near the wall is built up from vortices which have

their axis in the flow direction. These vortices are non-stationary, they

either oscillate around their equilibrium position or they break up at a

later stage.

In experiments which Schlichting (Ref. 33) performed on the influence of roughnesses on resistance, it was found that the maximum velocity occurred exactly downstream from the obstacles on the plate.

As behind the obstacles a region with small velocities would be expected ,

the phenomenon was ascribed to secondary flow, that is to vortices which

have their axis in the flow direction.

Later Jacobs (Ref. 34) did experiments behind a single knob

and the observations of Schlichting were confirmed. Although these

experimenters do not give indicatioQ.s about wedge-shaped wakes, their

observations are consistent,withthe picture of the wedges and the strueture

From this experimental evidence on the wedges of turbulent flow which is consistent as a whole, it seems that a first theoretical approach might be directed towards obtaining these wedges from the equations of motion. If this can be done it may perhaps form a first step towards a theoretical justification of Emmons' theory. 2.4 Outline of the Further Content of this Report

As suggested at the end of the last section we should attempt to find theoretical support for the wedges of turbulence which are the wakes due to permanent sourees of turbulence. In the further sections of this report attempts in this direction wiU be described . It has to be pointed out at this stage that no general methods are available to deal with problems of the kind encountered here .so that the work has a

tentative and preliminary character.~.' &'B.xing ,lejigths ·the-or,ies. wh:ich have been used to describe turbulent wakes behind bodies have not been used so far. They have been omitted because the perturbation methods which were used are felt to penetrate more to the basis of the questions

involved than the mixture lengths theories do.

A summary of the foUowing sections of !he report may foUow.

First, we derive the equations of turbulent motion and show th at a mathematical approaçh along these lines is impossible. The possibility to do something with these equations has to wait tiU more experiments have been made. One can, of course, start also with the introd\lction of certain hypotheses but it is doubtful whether the

experimental evidence available is sufficient to make such a hypothesis more than a guess.

In the next paragraphs two models are discussed where the turbulence is considered as a smaU disturbance superimposed on the main flow. The main simplification is the linearization. Through this artifice it becomes possible to attack the situation with standard

mathematical methods.

The first model is introduced along the lines of Oseen's tQeory and was chosen not so much for physical reasons as for the fact tlÎat the mathematics of this method is rather wel! known.

In the second model the viscosity is negleoted but the ve locity gradient of the mean flow is contained. The mathematics is

I

more interesting than for the first case. In both cases, however, no wedge-shaped wake is found.

From the perturbation equations , used to discuss the two models, equations can be constructed whieh are formally identical with the relations for the Reynolds stresses obtained before, provided these relations are suitably simplified. The discussion of this analogy is the subject of paragraph 6.

In the last paragraph we finally discuss a few points which are all based on elementary vortex-theory. This section shows certain features of the physical problem but from a mathematical point of view it is hardly satisfactory. To unify this piece of the report seems very hard and so far there are no suitable methods available to do this 0

3, ASPECTS OF THE EQUATIONS OF TURBULENT MOTION

It is well known that the theoretical solution of a problem in fluid dynamics comes down to finding solutions of the equations of motion whtch satisfy certain initia I and boundary conditions .

If we restrict ourselves to incompressible fluids these equations are in the notation of Cartesian tensors:

where

d

I

d

P:

:

+

-(u.u.)

= -

~ U;("f 1 J

f ()

'X:.

t

X. - -Car-te-6i-an coordinate 1

Ui - Velocity component in xi direction

p

.:

.

-

Component of ·stre·s-s tensor IJ

p -

DensHy The range of the indices -i-s 1, 2. 3.

If the -stres-s tensor is -tak-en to be '" - P

Si

i' .

+

~

l

ë)

u.',

Q ~1Ct

where Kronecker-.delta

~ -Visc-ósity

the equations (3.2) are called the e-quations of Navier-Stokes.

(3.1)

(3.2)

(3.3)

Another equation of importance is the energy equation which is easily obtained from (3.2) if we multiply with ui and use the equation of continuity. The equation then reads:

( 1..+u.

_()t

_4"

.

:L

_{Cl ~.}

\/.!..u..u..)

₎

_l..2..

ti t.

=

(3,.4)

The L. H. S. represents the change of kinetic-ener gy of a

fluid element. The first term on the R.H.S. represents the work done by the stresses while the second term represents the dissipation of energy if (3.3) is used for Pij'

It is usually assumed that the equations of Navier-Stokes

hold as well for turbulent as for laminar flow. Since the time of Reynolds. however. the discussion of turbulent flow proceeds mostly in a somewhat different way; as the velocity components in turbulent flow make relatively small but rapid fluctuations it is usual to write

I

U. ₁

=

u.'

_l; -t-

u..

_(; (3.5)

where

u·

"

=

mean velocity component

lL~

₌

_{fluctua tion}

~

and U! L

=

0

Similarly we write

Pij

=

PL~

.

+

p\j (3.6)

The mean velocity and stress are obtained by averaging the velocities and stresses in some appropriate way (space ave rage • time ave rage • ensemble average) . The definition of the mean velocity and the

separation in a mean velocity (and stress) and a fluctuating velocity (and stress) raises already a considerable number of mathematical

problems which will not be discussed. (See for example (Ref.

46».

Introduction of (3.5) and (3.6) in (3.1) and (3.2) gives

d ( -

I)

- Lt.' +-Lt..' =0 Ol:.. I. Cl t.. (3.7) -

d

é-

(J..

+

u...

')

ot

(;

...

o

Averaging these equations gives (3.9)

o ("

I ,

I I)

"

,

+ -

\A' \A'

+u.. ".

"

-O'X.

t

t.

f

I.

v-(

-

p

(3.10)

It is usual to write the last equation as

(3.11)

The second term between the brackets on the R. H. S. is the Reynolds stress tensor, and represents the influence of the fluctuations on the me an flow.

If we consider the set of equations (3.9) and (3.11) we have "

4 equations for the unknowns üi and P as we assume that (3.3) holds for

Pij. Due to our separation in mean flow and fluctuation and the non- r

linear terms in the equations of motion, however, the 6 components of the Reynolds stress tensor appear in our equation and each of these 6

components is unkno~n so far. It seems, therefore, necessary to write

down 6 additional equations for the Reynolds stresses. We shall see,

however, that these 6 equations contain a considerable number of new

unknowns.

To obtain the equations of the Reynolds stresses we proceed as follows: We subtract the equations (3.9) and (3.10) from (3.7) and (3.8) respectively. This gives

(),,~ C\

f-~

+

~

U·

U. I.

+

U..

u.'.

ot

o

'X:

t

l

I.

t

t

I.

(3. 12)

-}

~'

I I I I (J I .

+

U·""

u..[).. -

I I..

t.

""t -

I. (

-f

Q'i:,.

t

(3.13)

We multiply (3.13) with uik then taking the equation analogous to (3.13) for uilt we multiply it with uIL and add the two

equations. This gives af ter some manipulation with (3 ~ 12) and (3.9)

the following re sult:

UI uI

d~~

+

uI, uI

d~k

k ' - l '

J

o'lc..'

J

ox.'

r

~

+

2

(ui, u I, uI) - uI

d

(ui., u I,) - u I, () (u Ik u IJ') ::

ox. .

1 J k k ':I 1 J 1 :\

t

u~

u~

~

d

I

~~

d

I

=

P

Ic

O"'t

(Pij)

+

P

dl(

J

(Pkj) (3.14)

The only thing leÏt to be done is to average (3 .14j. This gives:

d

(I I)

I I

dlt·

\

,

_'è)

t.t~

d

l

u.llJ.~

)

+

'è)

t

!..t L IJ.I(-

+

u."

U-

_t

~

+

U' lL' +U' O?c · I.

&

àx,'

d"

Cl '):.

~

t

d

(-

-

•

.!.::

(3.15)

d

l

I I ' ) _I

f

.d",

i

(/"~

f) }

+ -

u.

i.

!..tI(-

Ut

-

-

lL~

L

(PI' -)

₊

_UI I

o")c·

_ft

_oXi

_l,q

Co

~

This is the equation for the Reynolds 'stre-s-s and we notice that there appear amongst others in this set of 6 equations another 10 unknown terms

U'~ u~

[Ad

.

If equations would be written down for

these triple products new unknowns would appear etc.

In this way the problem can never be made complete as the

number of unknowns wiU continue to exceed the number of equations .

A rigorous approach to our problem along these lines is therefore excluded.

The only way to get around this difficulty is to introduce at some stage some assumptions for the unknowns, to use experimental data or most likely a combination of these two things.

The first problem we face is then to find the necessary experimental data. Here the difficulty is that the experimental data we

need for the question of the wedges do not exist yet. In recent years

experimental work on turbulence in shear flows is continuing steadily (Refs. 35. 36, 37, 38) ~ but the,'worK which has béen -done' so far has'.

been concerned with completely developed turbulence under

geometrically simple conditions so that terms which are essential for

the discussion of tile wedges could be ignored in the experiments so far performed. Along these lines little can be done therefore at this

+

+

Before we leave these equations a few other things which give additional difficulties may be pointed out.

Contracting equation (3.15) and multiplying with 1/2 gives us the equation for the energy of the turbulent fluctuations per unit mass. The equation reads:

( èl

-+u

-

·

-

d )(,

-

_ll'U'

I ' )

₊

_U: I _LV I

ut

(o~. l... t.. IJ "~

d' _(3.16)

The first term on the L. H. S. is the rate of change of the kinetic energy of the fluctuations for a fluid element moving with the velocity of the mean flow. The second term represents the gain of energy of the fluctuations from ±he mean flow.if we put ,it ón the R .H. S.

of the equali1:y sigh. The term is usuallY'called the produciion term.

The energy gained by the' fluctuatións is 10st by the,'mean flow which is

easily checke.d on writing dOWIl the energy equation for the mean flow.

The third term on the L.H.S. represents the transport of kinetic

energy due to the fluctuations themselves. The two terms on the R .H.S. represent respectively the work performed by the fluctuating stresses and the dissipation of the fluctuations .

Of particular interest for our problem is the production

term'of the turbulence energy as in the problem we are attempting to attack the turbulence is still in the initial stages and gaining energy.

If w-e write out the term in detail it reads:

u.

1_.2..

dl..l,

+-

Ui

u.. ,

()'C:

1

+

I

u. '

()~I

-

U-,

₊

,

o::x:.

1 I ...t o?C~ ~ o'X~

u.;

U. I é)u.~

+

LL'2.

'dUa.

I I

BÜ:~

+

I 2-

-

+

_{u..~ u..;~}

'0

'X:. , .2-o?C..a.

à

'x

ol

u..'

u..'

o~

₊

Lt.'

u.. ,

oiL~

+

u...'

4.

-

o~

.=. I

0')(.1

~ 0

'0):..2., ~

o

?C.a

We first notice that if the Reynolds I stress tensor is isotropie the sum of the above 9 terms disappears. (The skew terms are zero, the terms of the main diagonal are equal and as each of these is multiplied with a term which occurs in the equation of continuity of the mean flow their sum is zero too.) This makes it sufficiently plausible

that the turbulence in the stage of development wiU be non-isotropic.

Further we notice that the production of turbulence energy

in a point P is only possible if there is already turbulence in P. This

remark is significant as it excludes from the" equations derived the

possibility that at some moment turbulence in a point P"is generated

spontaneously. * This can stiU be clarified as follows:

In Emmons' theory we have to do with regions of turbulent flow and regions of laminar flow. In the light of the equations derived so far this means that there are regions in the flow where the Reynolds I

stresses and the other fluctuation terms are different from zero and

other regions where they are equal to zero. The statement that the

fluid in a point P becomes turbulent is in our description identical with saying that the Reynolds' stresses etc. and the turbulence energy become different from zero.

From a physical point of view there are a priori two in

principle different ways in which th is may take place (Compare 30).

(a) The flow is _, unstable and at a certain moment the flow breaks locally down into turbulence. (This is the process which creates the sources of turbulence in

Emmons' theory and which was described at some length at the end of 1.3 and in 1.4).

(b) The turbulence in P is due to the fact that turbulent fluid from points near P is transported to P.

If the propagation and expansion of the spots of turbulence

and the wedges of turbulence are due to the mechanism (a) the equations

derived so far cannot give us information, as there are no terms which

represent this mechanism; in our description of turbulence (a) would

be the spontaneous local generation of Reynolds i stresses etc. and a

spontaneous increase in the energy of the turbulent fluctuations ,

starting from zero, but we have just seen that a necessary requirement for the Reynolds' stresses and turbulence energy to gain energy is that they are already different from z.ero. If this mechanism therefore has to be incorporated in our problem it will have to figure in the initial and boundary conditions which wiU be necessary to complete the problem in some way.

*The statement just made wiU strictly hold only if the laminar flow is

defined as

~

flow in which aU the

~erms

u.~ u~

).&

.

(u!tu.~)

)

&-:

(IJ.LI:)'~f)

and hl. ~ are equal to zero. A flow in which I;:eriodic distuf-bances

1-1.,r CP'-}

are pre'S ent would therefore be considered as turbulent in our case J

this is usually not done. In general one thinks that it is the random character of the flow which distinguishes the turbulent flow from the

If on the other hand mechanism (b) is the important thing it

seems that we may expect some information because there are terms in the equations we have derived which represent the transport of

Reynolds I stresses and turbulence energy by the mean flow and by the

fluctuations themselves. (Triple products). This information,

however, is so mixed up with other unknown terms that it is impossible

to isolate it.

A second point which we have to face has been slightly

touched upon while discussing the first point. We saw that in our problem we have to do with regions of laminar flow, where we use

Navier-Stokes I equations and regions of turbulent flow where we use

a modified form of these equations

of

motiön in ·which the Reynolds I

stresses occur. The question arises what wiU happen at a boundary where a laminar and a turbulent region touch.

It is clear that several possibilities for discontinuities are

created at such a boundary, discontinuities which are probably

meaningless from a physical poinL9.! view, but which cannot be ignored when we stick to the description of turbulent" flow-as given before.

Before, however, attempting to apply consj.derations of this kind to the wedges of turbulence it seems necessary to do prelim inary work and apply it to geometrically simpler situations .

Such a simple case might.pe, for example, the stationary flow between parallel flat plates or through a pipè which changes at some section suddenly from laminar to turbulent flow 0 These examples

seem the more suitable as measurements of the Reynolds' stresses and

other terms have been made for the complete 1y developed turbulence in

these cases. (Refs. 37, 38).

4. A STATIONARY DISTURBANCE ON A SEMI-INFINITE FLAT PLATE 4. 1 Introduction

In th is paragraph we shall deal with a very much sim plified

picture of the wedges of turbulence. The following mode 1 wil! be

considered; we have a stationary parallel flow of an incompressible

viscous medium with velocity U in the xl - direction. In this flow we

put a semi-infinite flat plate given by xl ~O x2 = 0 - CO ~'X~ , +00. On

this flat plate a smal! disturbance wil! b l planted and the resulting flow

pattern wil! be computed.

Along the flat plate the flow wil! be retarded and a

boundary-layer will be created. This boundary-layer can be computed in the ordinary way but to keep open the possibility to superpose solutions we

linearize. .

The possibility to superpose'solutions is used to add the

The theory of this kind of approxim ation was originally developed by Oseen (Ref. 39) for incompressible flows and has recently been used by Lagerstrom and others: (Rèf ~"'4ö;:::to-'dêà'r,~ltfl "flS:éous

compressible fluids. .

In the sections which make up th is paragraph we discuss the following points:

In section 4.2 the approximate equations of motion are developed in the way of Oseen.

In section 4.3 the solution of the flat plate problem which is given in (Ref. 40) is derived. In section 4.4 the 3 -dimensional

disturbance is constructed in Us general shape. In section 4.5 the simplest possible disturbance is worked out. In section 4.6 the solution and the model are discussed and criticized.

4.2 Oseen's Equations of Motion

The general equations of motion for an incompressible viscous medium are J as we have already seen in 3 J

(4.2.1)

Ou,'

- \ ; (4.2.2)

~t

Most of the symbols in this equation are known. The only ones which are new are

Following Oseen we write

u =U+ u l 1 1 1 u₂

=

u 2 1 u 3

=

u3 1 p

=

p

l

J

kinematic viscosity Laplace operator (4.2.3) 1 1 1 1

supposing that U is const~nt and that u

that t-erms which contain -more lthan.one Qf these sy;mhols èanbe

neglected. Substitution of (4.2.3) in th~ equations of motion and

performing the indicated approximations give-s

'0

_u.',

IJ

-

_'Ot

dU.~ ~

=-

0 (4.2.4) o rx:.~ I

+

V

A.

u..~ (4.2.5)

'Ou.'.

As we are-only interested in thee stationary case the-term

~ wil!

pe

negl-eeied andas there is no further danger t-o -mix up

different terms the dashes wil! be omitted also so that the equations of

our problépl are:

() IJ:

----!:!..

=

0 _(4.2.6-)

(4.2.7)

To obtain solutions of thesé equations we write with Oseen

U)

t..a.~

u.

=

u.

+

u. (4.2.8)

1 1 1

and require that

u~)

wi11 satisfy

Ou..~

=

0 _(4.2.9) ~'X.~

u

'dJ.')

I. I

op

l

ox.,

p-

o

?Co • '-(4.2.10) while

i

_ratS)

ill sa-tisfy

t.

=

0 () ')C. ~ (-4. 2. 11)

U

d

u..~2.)

I..

--.r

A

uS

lJ

O~I

I. (4.2.12) [.) ~2.)

We notice that u. is related to the pressure while Ui is (~

dependent on the vis cos ity . lFurther we see that the velocity field Ui is

irrotational as the R.H.S. of (4.2.10) is a gradient. The separation we

and irrotational part as originally used by Helmholtz. Due to the

equation (4.2.6) the irrotational field is also solenoidal in our case and

is therefore a harmonic field.

The problem now becomes to find solutions of the equations (4.2.9) - (4.2.12) in such a way that

(I) (.2.) ui = ui

+

ui

satisfies the appropriate boundary conditions .

Before we enter into the details of these solutions we may have a closer look at the construction of these vector fields.

The construction of the vector fie ld

~

provides no

difficulties. Each function

cp

which satisfies the Laplace equation

Á~=O (4.2.13)

d fo ld (I)b ° 0 °

pro uces a vector ie Ui y simply writlng

JI)=

'dep

1 (4.2.14)

LI) From equation (4.2.10) the pressure field associated with

ui can be obtained. This holds for the 2-dimensional case as weU as for the 3 -dimensional case.

More complicated is the situation for the vector field

I~~

. 1

When we have obtained a solution F (x x

2x3) of 1):1e equation

(4.2.12) we can construct from this solution a veclor field

Jf~

Following Oseen and Lamb we can write

(JJ

::!..

'à

F

1

u(~

= F -

v-

U

a

21; , (4.2.15) (~)

...r

Ö ~ U2 =

U

Ö?c...2.,.

f

u 3:: -

U

0

')C~

)

It is easily checked that this velocity field

u~

satisfies both the equation

of continuity (4.2.11) and the equation (4.2.12).

component

The peculiarity of this velocity fie~ is that the vorticity

.

S

=

Ou.~

_

Ot.t...1.

vanishes everywhere .

If we follow Helmholtz we write L~)

_'dF

_ra

F,4

l

u

₌

~ 1

_'à

_{'X:.~ O~~} (4.)

_d

_F I

ra

F~ u

₌

f

(4.2.16) 2 _{'0 ')(.~}

o

'X. , (~)

_e

_1=.2,. ~F, u

₌

-3 _{~ 'X:. , 'O'X:..a.,.}

where F F and F are three solutions of (4. 2 . 12) . 1 2 3

It is ea&ily checked that (4.2.11) and (4.2.12) are both satisfied by these u~~) ss.

1

If in the formulas (4.2.16) the F.'s are not all different some special cases can be obtained. 1

While the representation of the velocity field af ter the manner of Lamb holds true in the 2 -dimensional case, it is seen that the Helmholtz representation reduces to the case where the stream function

"t'

as used in 2-dimensional cases has to satisfy the equation (4.2.12) .

Af ter this preliminary discussion we turn to the detailed solutions .

4.3 The Solution of the Semi-infinite Flat Plate Problem

The solution of the Oseen equations for the flow along a semi-infinite flat plate was obtained by Lewis and Carrier (Ref. 41) and by a different method by Lagerstrom (Ref. 40). We follow here the method of Lagerstrom .

In the introduction we have assumed that the semi-infinite flat plate will be defined by

o o

As the plate extends from - 00 to

+

00 in the x direction the solution wi11 be independent of x as in each section x

3

3

₌

_{a the flow} pattern wil! be the same. We furthJr assume u3 = o.

The boundary conditions we impose on the two remaining ve locity components are the ordinary boundary conditions of viscous flows which require that at the wall both the normal - and the

tangential velocity components vanish, while we require further th at far from the plate the flow wil! be essentially undisturbed, that is

u -= U

1 u 2 = 0

The boundary value problem then becomes to find a solution of the equations

(i = 1, 2)

p

which satisfies the boundary conditions

u

= -

U 1 u

=

0 1 u

=

0 I 2 for x 2 ~

±

00 for x

=

0 and x -

+

00 2 2 (4.3.0 (4.3.2) (4.3.3)

To solve the problem we separate ui as indicated in section

(4.2) and require that J{will satisfy the boundary conditions for(u_l. It

is then noticed that

u~)

1= 0

for x140 x2

=

0 and the solution ui) is

a~ded(~o

make

u~

+

~

=

0

fo~

x l ; ' 0 x2

=

o.

It is easily checked that

ul)

+

ui sat-i&fy aH the-re-quired boundaFY conditions (4.3.3).

The equations for the field ui

afte-otf)

~

_o

= 0 'le-_I.

while the boUHdary conditions are-

-(.2.) u1 -= - U for Xl ~ {) (...t) ul

=

-

B- for x2 ---.

+

00 (4.3.1) (4.3.4) (4.3.5)

The-eolution of the-abev-e problem i-s obtaiaed by the-introduction of a function F (XfK'2) which is a solution of (4.3.4) and satisfies the boundary conditions (4.3.5). We then put following Lamb

--and Oseea (2) u

=

F -1 "" 0

F

U O'L₁ '" 'dl='

l2J

u 2 = -

-

_U

_ê)x..2,.

-l (4.3.6)

With these substitutions it is seen that the continuity equation (4.3.1) becomes the equation (4.3.4) for F, but F is a solution of (4.3.4) and so the equation of continuit~is sat~sfied. As Y and U are constant it

is immediately seen that u1 and

u2~)

satis.fy the equation (4.3.4). _ ' .

Further, we have F

= -

U for x1~O (2) X2: 0 and so ::. =0 for - ._ 'X. , ~ 0 x₂

=

O. It follows then that u1

= -

U for xl ~ 0

x

2

=

0 while it is easily checked too that the boundary conditions at irlfinity are satisfied.

With the substitutions (4.3.6) the problem has been reduced to finding the function F (xl x_{2 ).} To solve this problem we substitute parabolic coordinates in (4.3.4). lf Z

=

xl

+

i x₂ and ~

=

~

+

L

~ these coordinates are defined by Z

=

SJ..

which gives the relations.

xl

-=

5

4_

~.2..

2

~

l.

V

x;

.

+~~

+x

₁

It is easily checked that

o

corresponds with

o

o

corresponds with

Further, we shall only consider values of:J which are

It follows then from (4.3.7) that

S<

0 if x₂

<

0

S

>

0 if x

2

>

O.

The equation (4.3.4) becomes with these coordinates

)1.

F

'()

ï..

F"

=

k /

~ ~

_

"l1

~)

é)tl..i-

~~2..

Lf

l

~~ ..1a~

where 2k

=

~

The boundary conditions (4.3.5) become

F

-= -

U for ~ =0 and all values of

S

F '= 0 for..

"J --.

co

S

~

4- 00

.

, (4.3.7) ~ O. and (4.3.8) (4.3.9)

As the first boundary condition has to be satisfied for all values of

5

it seems appropr.iáteto look for sol~tions of (4.3.8) which are independent of

S

That is we look for a solution of