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LECTURE SERLES “BOUNDARY LAYER THEORY”. PART' I - LAIvUNAR FLOWS. By H. Schlichting. Translation of “Vortragsreihe” W.S. 1941/42, Luft-.
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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORAYDUM

LECTURE SERLES “BOUNDARY LAYER THEORY” PART’ I

- LAIvUNAR FLOWS

By H. Schlichting

Translation of “Vortragsreihe” W.S. 1941/42,Luftf ahrtforschungsanstalt He rrnam Goring, Braunschweig

UTa shing’ton

April 1943

NACA TM No. 1217

P R E F A C E I gave t h e l e c t u r e s e r i e s "Boundary-Lqer Theory" i n tho winter semester 1941/42 for t h e menibere of my I n s t i t u t e and f o r a considerable number of collaborators f r o m t h e H e m &ring I n s t i t u t e for Aviation Research. The s e r i e s embraced a t o t a l of s i x t e e n two-hour l e c t u r e s . The a i m of t h e l e c t u r e series was t o g i v e . a survey of t h e more r e c e n t r e s u l t s of t h e thaory of viscous f l u i d s aa fas a s they a r e of importance fox* a c t u a l applications. Naturally t h e theory of t h e boundary of f r i c t i o n a l l a y e r t a k s s up t h e g r e a t e s t p a r t . I n view of t h e g r e a t volume of paterial, a complete treatment w a s out of t h e question. However, I took palns t o make concepts everywhere stand out c l e a r l y . Moreover, eeveral important t y p i c a l examples were t r e a t e d i n d e t a i l . Dr. H. Hahnemann (LFA, I n s t i t u t e f o r Motor Research) went t o considerable trouble i n order t o perfect &I elaboration of t h i s l e c t u r e s e r i e s which I examimed and gupglemented i n a few placee. Miss Hildegard M%z p a r t i c i p a t e d i n t h e i l l u s t r a t i o n . To both I owe my most sincere thanks f o r t h i s collaboration. Sch l i c h t ing Aerodynamisches- I n s t i t u t der Technischen Hochschule, Braunschweig October 1942.

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. 1217 T A B L E

O F

CONTENTS

.. . . 1 C h a p t e r 1. VISCOSITY . . . . . . . . . . . . . . . . . . . . . . . . 3 Chapter I1. P0ISEITII;LE F L O W THROUGH A PIPE . . . . . . . . . . . . . 4 Chapter I11. EQUNIONS OF MOTION OF THE VISCOUS FLUID . . . . . . . . 8 a . S t a t e of S t r e s s . . . . . . . . . . . . . . . . . . . . . . . 8 b . S t a t e of Deformation . . . . . . . . . . . . . . . . . . . . . 11 c . Navier-Stokes Formulation for t h e S t r e s s Tensor . . . . . . . 15 Chapter IV. GENERAL PROPERTIES O F NAVIXR-STOEB EQUATIONS . . . . . 18 Chapter V . REYNOIDS'LAW OF SIMILARITY . . . . . . . . . . . . . . . . 25 Chapter V I . EXACT SOLUTIONS OF THE lVAVIER43TOKES EQUATIONS . . . . . 28 a . Pipe Flow. Steady and S t a r t i n g . . . . . . . . . . . . . . . . 29 b . Plane Surface. A Surface Suddenly S e t i n Motion and an O s c i l l a t i n g Surface . . . . . . . . . . . . . . . . . . . . 31 .. Plane Stagnation-Point Flow . . . . . . . . . . . . . . . . . 34 d . Convergent and Divergent Channel . . . . . . . . . . . . . . . 39 Chapter VI1. VERY SLOW MOTION (STOKES. OSEEN) . . . . . . . . . . . . 40 Chapter VI11. PRAKDTL'S BOUNDARY LAYER EQUA!I'IONS . . . . . . . . . . 44 Chapter IX. EXACT SOLVTIONS OF TEE BOUNDARY LAYER EQUATIONS FOR THE PLANEPROBLEM . . . . . . . . . . . . . . . . . . . . 51 a . The F l a t P l a t e i n Longitudinal Flow . . . . . . . . . . . . . 51 b . The Boundary Layer on t h e Cylinder (Symmtrical Case) . . . . 61 c . Wake behind t h e F l a t P l a t e i n Longitudinal Flow . . . . . . . 66 d . The Plane J e t . . . . . . . . . . . . . . . . . . . . . . . . 73 . . . . . 79 e . The Boundary Layer for t h e P o t e n t i a l Flow U = u, x? Chapter X . APPROXIMATE SOLUTION OF THE BOUNDARY LAYER BY MEANS OF THE MOMENTUM TREOREM . (KARMAN-POHLHAUSEN METHOD. PLANE PROBLEM) . . . . . . . . . . . . . . . . . . . . . . . 83 a . The F l a t P l a t e i n Longitudinal Flow . . . . . . . . . . . . . 83 b . The Momsntm Theorem for t h e Boundary Layer with Pressure Drop (Plane Problem) . . . . . . . . . . . . . . . . . . . 90 c . Calculation of t h e 3oundary Layer According t o t h e Method of Karm-Poh L haus en -Hol s.ti n . . . . . . . . . . . . . . . 93 d . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 103 ~RODUCTION

. .

9

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. . . . . . .* . . . . . . . . a. Estimation of the Admissible Pressure Gradient . . . . . . b. Various Technical Arrangemsnts for Avoiding Separation . . C. Theory of the Boundary Layer with Suction . . . . . . . . . Chapter XII. APPENDIX TO PART I . . . . . . . . . . . . . . . . .

Chapter XI.

PREVENTION OF SEPARATION

a. Examples of the Boundary Layer Calculation According to the Pohlhausen-Holstein Method

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

- i7

104

105 109 110

114

114

TECHNICAL

MEMQRAN~XTMNO.

1217

Part I - Laminar Flows" By H. Schlichting First lecture (Dec. 1, 1941)

INTRODUCTION Gentlemen: In the lecture series starting today I want to give you a survey of a field of aerodynamic6 which has for a nmiber of years been attracting an ever growing interest. The subJect is the theory of flows with friction, and, within that field, particularly the theory of friction layers, or boundary layers. As you know, a great many considerations of aerodynamics are based on the so-called ideal fluid, that ie, the frictionless incompressible fluid. By neglect of compressibility and friction the extensive mathematical theory of the ideal fluid (potential theory) has been made possible. Actual liquids and gases satisfy the condition of incompressibility rather well if the velocities are not extremely high or, more accurately, if they are small in comparison with sonic velocity. For air, for instance, the change in volume due to compressibility amounts to about 1 percent for a velocity of 60 metere per second. The hypothesis of absence of friction is not satisfied by any actual fluid; however, it is true that most technically important fluids, for instance air and water, have a very emall friction coefficient and therefore behave in many cases edmoet like the ideal frictionless fluid. Many flow phenomena, in particular most cases of lift, can be treated satisfactorily, - that is, the calculations are in good agreement with the test results, -under the assumption of frictionless fluid. However, the calculations with 'frictionless flow show a very serious deficiency; namely, the fact, b o r n as d'Alembert's paradox, that in frictionless flow each body has zero drag whereas in actual flow each body experiences a drag of greater or smaller magnitude. For a long time the theory has been unable to bridge this gap between the theory of frictionless flow and the experimental findings about actual flow. The cause of this fundamental discrepancy is the viscosity which is neglected in the theory *"Vortragsreihe "Grenzschichttheorie. Teil A: Laminare Str8mungen." Zentrale f& wissenschaftliches Berichtswesen der Luftfahrtforshung des Generalluftzeugmsisters (ZWB) Berlin-Adlershof, pp. 1-133. Given in the Winter Semester 1941/42 at the Luftfahrtforschungsanstalt Hermann Gcking, & - a m c b e i g . The original language version of this report is divided into two main parts, Teil A and Teil B, which have been translated as separate NACA Technical Memorandums, Nos. 1217 and1218, desisated part I and part 11, respectively-

'

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it of t h e i d e a l f l u i d ; however, i n s p i t e of i t s extraordinary s&hless i s decisive f o r the course of t h e flow phenomenon. As a matter of f a c t t h e problem of drag can not be t r e a t e d a t all without t a k i n g t h e v i s c o s i t y i n t o account. Although t h i s f a c t had been known f o r a long time, no proper approach t o t h e t h e o r e t i c a l treatment of t h e drag problem could be found u n t i l t h e beginning of t h e present century. The main reason w a s t h a t unsurmountable mathematical d i f f i c u l t i e s stood i n t h e way of t h e o r e t i c a l treatment of t h e flow phenomena of t h e viscous f l u i d . It i s Professor P r a n d t l ' s g r e a t m e r i t t o have shown a way t o numerical treatment of v i s c o s i t y , p a r t i c u l a r l y of t h e technically important flows under consideration and thereby t o have opened up new v i s t a s on many important perceptions about t h e drag problem and r e l a t e d questions. P r a n d t l W a s a b l e t o show t h a t i n t h e case of most o f t h e technically important flows one may treat t h e f l o w , as a whole, as f r i c t i o n l e s s and u t i l i z e t h e s i m p l i f i c a t i o n s for t h e c a l c u l a t i o n thus made possible, b u t t h a t i n t h e immediate neighborhood of t h e s o l i d w a l l s one always had t o take t h e f r i c t i o n i n t o consideration. Thus P r a n d t l BlJbfLivides, f o r t h e purpose of calculation, t h e flow surrounding a body i n t o two domains: a l a y e r subject t o f r i c t i o n i n t h e neighborhood of t h e body, and a f r i c t i o n l e s s region outside of t h i s l a y e r . The theory of t h i s s M a l l e d "Prandtl's f r i c t i o n o r boundary layer" ha8 proved t o be very f r u i t f u l i n modern flow theory; t h e present l e c t u r e w i l l c e n t e r around it. A t t h i s point I w a n t t o i n d i c a t e a few a p p l l c a t i o n s of t h e boundaryl a y e r theory. A f i r s t important a p p l i c a t i o n i s t h e c a l c u l a t i o n of t h e f r i c t i o n a l surface drag of bodies immersed i n a flow, f o r instance, t h e drag of a f l a t p l a t e i n l o n g i t u d i n a l flow, t h e f r i c t i o n a l drag of a ship, a wing p r o f i l e , and an a i r p l a n e fuselage. A s p e c i a l property of t h e boundary l a y e r i s t h e f a c t t h a t under c e r t a i n circumstances reverse flow O C C W B i n t h e immediate proximity of t h e surface. Then, i n connection with t h i s reverse flow, a separation of t h e boundary l a y e r t a k e s place, together with a more or less strong formation of v o r t i c e s i n t h e flow behind t h e body. Thus a considerable change i n pressure d i s t r i b u t i o n , compared with f r i c t i o n l e s s flow, r e s u l t s , which g i v e s r i s e t o t h e form drag of t h e body immersed i n t h e flow: The boundary-layer theory t h e r e f o r e o f f e r s an approach t o t h e c a l c u l a t i o n of t h i s form d r q . Separation occurs n o t only i n t h e flow around a body b u t a l s o i n t h e flow through 9 divergent tunnel. Thus flow phenomena i n a d i f f u s e r , as, f o r instance, i n $he bucket g r i d of a turbine, may be included i n boundary-layer thsory. Furthermore, t h e phenomena connected with t h e m a x i m - a n l i f t of a w i n g , where f l o w separation i s concerned, can be understood only with t h e a i d of' boundaryl a y e r theory. The problems of heat t r a n s f e r a l s o can be explained only by boundary-layer theory.

AE w i l l be shown i n d e t a i l l a t e r , one must d i s t i n g u i s h between t h e two s t a t e s of boundary-layer flow - laminar a r d t u - b u l e n t ; t h e i r flow l a w 6 are very d i f f e r e n t . Accordingly, t h e l e c t u r e w i l l be divided i n t o t h r e e main parts: 1. Laminar flows, 2. Turbulent flows, 3. Laminar-turbulent t r a n s i t i o n . Although t h e boundary l a y e r w i l l be our main consideration, it w i l l s t i l l be necessary as preparation t o discuss t o Borne extent t h e

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general theory of t h e viscous f l u i d . chapter.

CHAPTER I.

This w i l l be done i n t h e f i r s t

VISCOSITY

Every f l u i d o f f e r s a r e s i s t a n c e t o a form v a r i a t i o n taking place i n f i n i t e time i n t e r v a l , which is of d i f f e r e n t magnitude according t o t h e type of f l u i d . It is, f o r instance, very l a r g e f o r syrup o r o i l , but only small f o r the technically important f l u i d s (water, a i r ) . The concept of visco8ity can be best made c l e a r by means of a t e s t according t o f i g u r e 1: Let f l u i d be between two p a r a l l e l p l a t e s l y i n g a t a distance h from each other. Let t h e lower p l a t e be fixed, while the upper p l a t e i s moved with the velocity + I, uniformly and p a r a l l e l t o t h e lower one. For mming t h e upper p l a t e a tangential force P must be expended which is

according t o experiment, where F i s the area of t h e upper p l a t e and p i s a constant of proportionality. (End e f f e c t s a r e not included). The quantity p i s c a l l e d t h e v i s c o s i t y c o e f f i c i e n t o r t h e dynamic v i s c o s i t y . Since the phenomenon i n question i s a p a r a l l e l gliding, t h e transverse v e l o c i t y component i n t h e y-direction, denoted by v, equals zero. The f l u i d adheres t o t h e upper and lower surface, respectively, a l i n e a r v e l o c i t y d i s t r i b u t i o n between t h e p l a t e s i s s e t up, t h e magnitude of which depends s o l e l y on y.

Pb,

Since f o r y = 0: u = 0, f o r y = h: u = uo. If one designates t h e t a n g e n t i a l f o r c e per u n i t area, as the f r i c t i o n a l shearing s t r e s s there fellsxs :

7,

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NACA TM

NO.

1217

Ths dimensions of p a r e accordingly kg sec/m2. A flow as represented i n f i g u r e 1, where no transverse v e l o c i t y occurs m d t h e shearing stress a t a l l points of t h e flow i s t h e r e f o r e given by equation .(1.2), i s c a l l e d simple shear flow. I n t h e s p e c i a l cas2 described, t h e shearing s t r e s s i s everywhere of equal magnitude, and equal t o t h a t a t t h e surface. Besides t h e dynamic v i s c o s i t y p t h e concept,of kinematic v i s c o s i t y v i s required, which f o r t h e density

v

For 200 C,

v

i s defined as

p F g sec21&]

= P

[m*/s]

is, f o r instance, f o r water:

v

= 1.01

x 10-6 ,/'e

f o r air:

V =

14.9 x lo4 m2/e

,.

1 lo4 7

if t h e a i r pressure has t h e standard value

m2/s

po = 760 mm hg.

CHAPTER 11. P O I S E U D FLOW 'THROUGH A PIPE The elementary empirical f r i c t i o n l a w of t h e simple ah ar f l o v derived above permits t h e immediate determination of t h e flow and t h e resistance i n a smooth pipe of c i r c u l a r c r o s s section and of constant diameter, d = 2r. A t a very l a r g e distance from t h e beginning of t h e pipe one c u t s off a piece of pipe of length 2 ( f i g . 2 ) and examines t h e cylinder of diameter 2y, t h e axis of which i s i d e n t i c a l with t h e pipe axis. According t o what has been s a i d s o far, t h e v e l o c i t y probably w i l l be again a function of y. A pressure difference p1 - p2 i s required for forcing t h e f l u i d through t h e cylinder. According t o p r a ~ t i c s lsxperience, t h e s t a t i c p r e s s w e across svery c r o s s section may be regarded as constant. Tha f l o v i s asaim& t o be steady and not izpenient on t h e 35stJax:t: from- the beginning of t k - s pipe. Equilibyium

NACA TM NO. 1217

must then e x i s t between t h e pressure and t h e f r i c t i o n a l shearing s t r e s s which attempts t o r e t a r d the motion. Thus f o r t h e cylinder of r a d i u s y t h e following equation i s v a l i d : pressure f o r c e difference a c t i n g a t t h e c r o s s s e c t i o n s = f r i c t i o n a l force acting along t h e cylinder w a l l , o r

or

T =

p1

- p2 2

2

(2. l a )

Since f l o w p a r a l l e l t o t h e axis i s t o be expected, one takes f r o m t h e previous paragraph,

7

=

-#

as

( t h e minus sign i n d i c a t e s t h a t the

velocity diminishes with increasing distance from t h e axis; thus du/dy i s negative, t h e shearing s t r e s s e s under consideration, however, m e p o s i t i v e ) , and, a f t e r separation of the variables, du becomes:

and, on integration:

t h e velocity i s supposed t o be u(y) = 0 follows t h a t t h e constant of integration C has t o be C = r2/4. Thus:

From t h e f a c t t h a t for y = r

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This equation (2.3) i s P o i s s u i l l e ' s l a w f o r pipe flow. It s t a t e s t h a t t h e velocity u(y) i s d i s t r i b u t e d parabolically over t h e pipe c r o s s section. The apex of t h e parabola l i e s on the pipe axis; here t h e velocity i s g r e a t e s t , namely:

(2.4) Therewith one may write (2.3):

By P o i s e u i l l e ' s law (equation ( 2 . 3 ) ) the drag of t h e developed laminar flow (which i s proportional t o p1 - p2) i s d i r e c t l y proportional t o the f i r s t power of the velocity. T h i s statement i s c h a r a c t e r i s t i c of all kinds of laminar flow whsreas, a s w i l l be seen l a t e r , the drag i n turbulent flow i s almost proportional t o the second power of the velocity.

s

The flow volume f o r t h e present case remains t o be given. designating an s e a element, velocity paraboloid, therefore

Q

is

Q =

With

dF

u(y) d F = volume of t h e

This flow law i s often used f o r determination of t h e v i s c o s i t y , by m3asurlng the quantity flowing subjected t o a pressure gradient (usually produced by g r a v i t y i n a v e r t i c a l c a p i l l a r y t u b e ) . O f course, the s t a r t i n g l o s s e s m u s t be taken i n t o consideration which due t o t h e mixing zone (vortex formation) a t t h e pipe end a r e n o t recovered t o t h e i r full extent. A drag c o e f f i c i e n t X w i l l now be defined.' Since turbulent flows a r e more important than lamhar ones and since the drag i n turbulent flow increases about as the square of t h e velocity, X w i l l a l s o be rererreii TO u2.

For flow problems, l e t X thus be defined a s : r a t i o of the pressure drop along a t e s t section of a specified c h a r a c t e r i s t i c length t o t h e dynamic

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pressure

q = pC2/2,

with E =

across t h e c r o s s s e c t i o n ) .

r[r2 Then:

= the mean v e l o c i t y

(average taken

h . = & d

&iF P-

2

with d = c h a r a c t e r i s t i c length, thus, for t h e present cas;, t h e pipe diameter, and with h. = dimensionless quantity. For t h e present developed laminar pipe flow, according t o equation (2.5)

Thus :

or

x = -64

(2.7)

Re with t h e dimensionless quantity

s i g n i f y i n g t h e Reynolds number ' Re = a-

v

of t h e c i r c u l a r pipe. Since t h e press.ure drop which i s only l i n e a r i l y then, for laminas flow: dependent on t h e v e l o c i t y w a s r e f e r r e d t o Ti2, A-

-. I

A logarithmic p l o t of

h = f(Re)

or

= f(5) therefore r e s u l t s

U

i n a s t r a i g h t l i n e i n c l i n e d 4 5 O toward the Re-axis (compare f i g . 82 P a r t 113. After t h i s s h o r t a n a l y s i s of t h e one-dimensional case of v i s c o w f l u i d we w i l l now consider t h e t h r e d i m e n s i o n a l case.

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CHAPTER 111. EQUATIONS OF MOTION OF THE VISCOUS FLUID a. S t a t e of S t r e s s

For t h i s purpose one must know first of a l l t h e general s t a t e of stress i n a moving viscous f l u i d and must then connect t h i s s t a t e of s t r e s s w i t h t h e s t a t e of deformation. For t h e deformation of s o l i d bodies t h e r e s i s t a n c e t o t h e deformation i s put proportional t o t h e magnitude of t h e deformation (assuming t h e v a l i d i t y of Hooke's l a w ) . For f l a r i n g f l u i d s , on the other hand, t h e r e s i s t a n c e t o deformation w i l l depend on t h e deformation velocity, tha-t i s , on the v a r i a t i o n of velocity i n t h e neighborhood of t h e point under conslderation. (Solid bodies: displacement gradient = displacement per second. Fluid: velocity g r a d i e n t ) . One starts from t h e b a s i c l a w of mechanics according t o which: mass x acceleration = sum of t h e acting, o r r e s u l t a n t force. For t h e mass-per-unit volume, t h a t is, t h e density p, one may w r i t e t h e law

Dw = Dt

s u b s t a n t i a l acceleration

K - = mass f o r c e s R

-

= surface forces, composed of pressure f o r c e s norma

-F

= n e g l i g i b l e extraneous f o r c e s

t o the surface and f r i c t i o n a l f o r c e s i n t h e d i r e c t i o n of t h e surface

I n order t o formulate the surface forces, one imagines a small rectangular element of volume dV = dx dy dz c u t out of t h e flow ( f i g . 3 ) t h e l e f t f r o n t corner of which l i e s a t the point (x, y, 2). The elemsnt i s t o be very small so t h a t only t h e l i n e a r v a r i a t i o n s of a Taylor devsl3pment need t o be taken i n t o consideration; on i t s surfaces dy dz a c t t h e resultant s t r e s s s s (vectors):

-px

or

__

Ex

+

a-p, dx, ax

respectively

* Throughout the t e x t , underscored l e t t e r s a r e used i n place of corresponding G e r m s c r i p t l e t t e r s used i n the o r i g i n a l t e x t .

(3.2) ___

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9

(The index x s i g n i f i e s t h a t the s t r e s s tensor a c t s on a surface element normal t o t h e x-direction). Analogous terms r e s u l t f o r t h e surfaces dz dx normal t o t h e y-axis and dx dy normal t o t h e z-acis, i f x i n equation (3.2) i s replaced everywhere by y o r z, respectively. From t h i s t h e r e r e s u l t s as components of t h e r e s u l t a n t force: Force on t h e surface element normal t o t h e x-direction:

?EX dx

dy dz

dX

aP Force on t h e surface element normal t o t h e y-direction:

-9 dy

Force on t h e surface element normal t o t h e z-direction:

dz aZ

dz dx

aP

The t o t a l r e s u l t a n t surface force s t a t e of s t r e s s is therefore:

E

dx dy

per u n i t volume caused by the

(3.3)

aZ

as

ax

-2

,

p p and p a r e vectors which can be f u r t h e r decomposed i n t o -y -2 -x components. In t h i s decomposition t h e components normal t o every surface element, t h a t is, t h e normal s t r e s s e s , a r e designated by u ( i n d i c a t i n g by t h e index the d i r e c t i o n of t h i s normal s t r e s s ) ; t h e other components ( t a n g e n t i a l s t r e s s e s ) a r e denoted by T (with doubk iqdex: t h e f i r s t i n d i c a t e s t 9 which axis the surface elemmt i s perpendicular, t h e second, t h e axial d i r e c t i o n of t h e s t r e s s 7 ) . With these symbols t h e r e is:

P

+

ju

Y

+

kT

?

(3.4)

YZ I

This s t a t e of s t r e s s represents a tensor with nine vzc-tor components, which can be chnracterized by t h e s t r e s s matrix ( s t r e s s t e n s o r ) :

10

NACA TM No. 1217

It can rezdily be shown that those of the six tangential stresses which have the same indices, although in interchanged sequence, must be equal. This follows for a homogeneous state of strese from the equilibrium of the small cube dx dy dz with respect to rotation: is the force attempting to rotate the cube counterclockwise about the z-axis, (seen from above in fig. 3), with the lever arm dx, and since, correspondingly, the force -T dx dz attempts to Yx rotate the cube clockwise about the z-axis, with the lever arm dy the balance of moments requires: Since

T

Xy

T

dy dx

xg

dy dz dx

- T~

dx dz dy = 0, thus T~

=

Correspondingly, because of freedom from rotation about the x-exis and because of freedom from rotation about the y-axis 7 = T YZ ZY’ 7 = T the nine components of the stress tensor are reduced to six zx XZ’ and the stress matrix (equation (3.5)) is converted into the stress matrix symmetrical ‘withrespect to the principal diagonal:

(stress matrix)

i

(3.6)

For the frictional force one obtains according to equation ( 3 . 3 ) by insertion of the components from equation (3.4) m d by reduction to the six remaining terms according to equation (3.6):

NACA TM NO.

1217

+ y-component

= x-component

+

z4omponent

For t h e case of t h e f r i c t i o n l e s s

(ideal)

f l u i d- a l l shearing

s t r e s s e s disappear

and only t h e normal s t r e s s e s remain, which i n t h i s case a r e a l l equal. Since t h e normal stresses from within toward t h e outside a r e denoted as positive, the normal s t r e s s e s equal t h e negative f l u i d pressure:

ux

=

uY = uz

= -p

(3.9)

The s t a t i c pressure equals t h e negative arithmetic mean of the normal stresses:

- (ox

-p = 1

3

+

uy

+

“2)

b. E-Late of Deformation The s t a t e of s t r e s s t r e a t e d so f a r is, alone, n o t very useful. Therefore we w i l l now consider t h e s t a t e of deformation ( t h a t i s t h e f i e l d of velocity v a r i a t i o n s ) and then s e t up t h e r e l a t i o n s between s t a t 3 of stress .md state of deformtion.

NACA TM No. 1217

12

Let t h e v e l o c i t y xA with t h e components uA, vA, wA i n t h e directions of t h e axes e x i s t a t the point A t h e coordinates of which are XAJ YAJ 'A' I f one l i m i t s oneself t o t h e pointa x, y, z i n t h s immsdiate neighborhood o f A with the v e l o c i t y w_ = i u + jv + l p , and i f one limits oneself - as a l s o i n s e t t i n g up t h e s t a t e of deformation - t o l i n e a s t e r m only, one obtains f o r t h e deformation t h e r e l a t i v e change z per unit i n position between t h e points x, y, z and x A' 'A' A time, t h a t is, tho difference of t h e v e l o c i t i e s a t t h e p o i n t s x, y, z 2 and x AJ 'A' A'

dw = i du + j dv + _k dw = d i s t o r t i o n of t h e f l u i d region i n the neighborhood of t h e point A. Omitting the index

A

one obtains therefore:

) + & (g

dx

+

a Y dy

+

5

dz)

Thus the v e l o c i t y v a r i a t i o n (and hence, on integration, the v e l o c i t y i t s e l f ) i n t h e neighborhood of t h e point A i s known i f t h e nine p a r t i a l d e r i v a t i v e s of t h e v e l o c i t y components with r e s p e c t t o t h e space coordinates a r e known. Corresponding t o t h e s t r e s s matrix, one may form a deformation matrix:

NACA TM

NO.

13

1217

deformation matrix

The friction forces of the viscous fluid are given by a relation (which will have to be determined) between these two matrices. First, the deformation matrix is to be somewhat clarified. 1. Case of pure elongation.

One assumes u

- ua =

a(x

- xA),

with a =

ax

=

constant. Let all

other terms of the matrix disappear; the matrix will then appear as follows:

Then the velocity variation is simply du = a dx, and u = ax. A l l points of the y e i s remain at rest, the points to the right and

left of it are elongated or compressed, according to whether a > 0 or a < O (fig. 5 ) . The equation u = ax therefore represents an elongation or expansion parallel to the x-axis. Corresponding relations apply for the other terms of the principal diagonal of the matrix.

2. Case of pure translation A l l terms disappear; the matrix then reads:

14

NACA TM No. 1217

In this case, which, as a matter of fact, should have been mentioned first, u - ua = 0, du = 0; u = constant. The velocity component parallel to the x-axis is uniform (correspondingly for the other axes).

3. Case of

angular deformation.

One assumes u

- ua

= e(y

- ya),

with

as

e = aU = constant.

other terms equal zero, and the matrix reads:

All

du = e dy and u = ey; that is, all points of the x-axie retain their position; all points of the y-axis shift to the right (left), when e > 0 (e < 0 ) ; for e > 0 the y-axis is rotated clockwise by the angle E (because of the linearity). The y-axis is simultaneously elongated. The phenomenon in question is therefore a shearing (fig. 6), with tan E = e. Correspondingly there results for v

dv = f ax;

v =

- va =

f (x

- xA)

and

IX

All points of the y-axis retain their position; the points of the x-axis m e rotated by the angle 6; tan 6 = f (fig.7). Terms outside of the principal diagonal of the matrix result therefore in a deformation of the right angle with axis-elongation (shearing). The right angle between the x- and y-axes is, therefore, for e > O and f > O deformed by E

+ 6

+

=

ax

= 7 ay

xy

= Deformation about the z-axis

-

Correspondingly :

+

aZ ax

= -bw

y

YZ

ay

+

aZ

= deformation about the y-axis

= deformation about the x-axis

(The deformation angles axe herein regarded as small so that the tangent may be replaced by the argument).

c. NavierStokes Formulation for the Stress Tensor ,

One now proceeds to relate the stress matrix (equation (3.6)) with the deformation matrix (equation (3.12)). The former is symmetrical with respect to the principal diagonal, but not the latter. However, one obtains a symmetrical defomtion matrix by adding to equation (3.12) its reflection in the principal diagonal. Furthermore, one first splits off the.pressure p (contribution of the ideal fluid) from the stress matrix and sets the remaining stress matrix, according to Stokes, proportional to the deformation matrix made symmetrical:

T x y Txz

I

7x2

7y z

uz

F r o m equation (3.13) each stress component may be given immediately by coordinating the homologous parts of the matrices to each other. For instance:

ox

=

-p

+ 2

p

a = static pressure + pressure due to ax

velocity variation, or:

16

NACA TM No. 1217

uX

=

- p + 2 paus ;

u = 9

-p +

av

2~-3

as

for one-dimensional flow

az =

T

XY =

L

-p+

7sx

aw

2px

ds

Furthermore, there follows from equation (3.13):

or "0

3

x

+ u + u Y

4

=-p

(3.14)

because

av + aw = div 3 = 0 ay aZ ax -

-all+

'for the incompressi-,e flows free of sources and sir-s un-3r coneideration.* Thus for the viscous incompressible flow, as for the ideal fluid the pressure equals the arithmetic mean of the normal stresses. With these results the components of the friction force may be expressed according to equation (3.7) as follows:

* The compressibility manifests itself a8 normal stress, since it can be interpreted as a pressure disturbance, for instance due to variation in density, which attempts to spread in all directions considered infinitesimally.

-

NACA TM No. 1217

17

or, since div w - = 0,

R

9

RZ in which Au =

=--

= -

; t

pAv

ht aZ

pAw

(3.15)

a%, a2u aZ, ax2 ay2 az2 +

If one finally designates the mass forces by K = p(iX + jY + kZ), and assumes the decomposition of the substantial derivative into a local and convective part as known frornlhler’s equation, one obtains for the components of the equation of motion of the non-etationary, incompressible, and viscous fluid from equation (3.1):

av

p ( - + u z +av v-

at

av as

”) aZ

+w-

In addition, the continuity equation

= p Y - - ap

as

+

pAv

18

NACA TM NO. 1217

i s used. Written i n vector form, t h e NavierStokes d i f f e r e n t i a l equation and t h e equation of continuity read

div

(3.19)

= 0

Due t o the f r i c t i o n terms, therefore, terms of t h e second order e n t e r the d i f f e r e n t i a l equation. Boundary cond.itions a r e attached t o these equations. If a l l f r i c t i o n terms on t h e r i g h t side a r e cancelled, t h a t i s V = 0 , t h e d i f f e r e n t i a l equations become equations of t h e f i r s t order and =boundary condition i s s u f f i c i e n t , namely the boundary condition of t h e p o t e n t i a l flow: vn = 0 on t h e bounding w a l l s . This means t h a t t h e normal component vn of t h e v e l o c i t y a t the bounding surface must disappear on t h e surface i t s e l f whereas t h e f l u i d s t i l l can g l i d e p a r a l l e l t o t h e boundary ( t a n g e n t i a l v e l o c i t y vt p a r s l l e l t o t h e surface # 0).

For viscous flow where t h e d i f f e r e n t i a l equation i s of t h e second order, two boundary conditions a r e required, namely:

vn = 0 and

-

vt = 0

(condition of no s l i p )

(3.20)

t h a t i s , t h e f l u i d must i n addition adhere t o t h e surface. Second l e c t u r e (Dec. 8, 1941)

These NavierStokes d i f f e r e n t i a l equations represent together with t h e equation of continuity a system of four equations f o r the four unlrnown q u a n t i t i e s u, v, w, p. On t h e l e f t side of t h e NavierStokes d i f f e r e n t i a l equations a r e t h e i n e r t i a terms, on t h e r i g h t side the mass forces, t h e pressure forces, and the f r i c t i o n forces. Since Stokes' formulation is, of course, a t f i r s t purely a r b i t r a r y , it i s not a p r i o r i c e r t a i n whether t h e NavierStokes d i f f e r e n t i a l equations describe t h e motion of a f l u i d c o r r e c t l y . They therefore require v e r i f i c a t i o n , which i s possible only by way of experimentation.

NACA TM

NO.

19

1217

Unfortunately, due to unsurmountable mathematical difficulties, a general solution of the differential equation is not yet known, that is, a solution where inertia and friction terms in the entire flow region are of the same order of magnitude. However, known special solutions ( f o r instance, the pipe flow with predoninmt viscosity or cases with large inertia effect) agree so well with the experimental findings, that the general validity of NavierStokes differential equation hardly seems questionable, The plane problem:

By far the greatest part of the application of Navier-Gtokes differential equations concern "plane" caees, that is, the cases where no fluid flows in one direction. The velocity vector 1 is then given by

since w = 0. The equation system (equations (3.16) and transformed into the 3 equations

.G

au+u-+v")= au ax

as

p x - a ap x+

with the three unlmown factors u, v, p of the mass force K, per unit volume).

.( ax2

&A+&

(3.17)) then

is

h2)

(X and Y are the components

After various minor transformations the equation system may be written as a single equation. To this end one introduces the rotational which for the plane m s e ~ R cn1y S zmipoiierrt not vector rot equalling zero:

-12 rot2-w = mZ

(4.3)

NACA TM No. 1217

20

Furthermore, the mass force in equation (4.2) is put equal to zero. This is permissible in all cases where the fluid is homogeneous and no free surfaces are present. In order to introduce uZ into equation (4.2), the first equation of (4.2) is differentiated with respect to y, and the second with respect to x; then the first is subtracted fro= the second and one obtains:

or

With this transformation the pressure term have been eliminated. Equation (4.5) may now, with cc/p = V , be written: Dol, = Dt with

(o

= wz

VLLO

(vorticity transport equation)

being denoted as the vorticity.

T h i s equation signifies: The convective (substantial) variation of the vortex strength equals the dissipation of vorticity by friction.

Equation (4.6) forms with the equation of continuity a system of two equations with tvo unknowns, namely u and v, the derivatives of which define 0).

By introducing a flow function ~(x,Y) one may finally introduce a single equation with the unknown \Ir. The flow function represents the integral of the equation of continuity. One sets: That is, therefore, the equation of continuity is identically satisfied by 9 .

(4.7)

NACA TM N o . 1217

21

Moreover,

That i s : The Laplacian of the flow function l e exactly minus two times as l a r g e as t h e v o r t i c i t y (angular velocity). With t h i s r e s u l t e q u a t i m (4.5) becomes, a f t e r division by P:

or expressed o n l y i n

$

with equation (4.7):

(4.10) This one equation with t h e unknown $ i s t h e v o r t i c i t y t r a n s p o r t equation, but w r i t t e n i n terms of Jr. The i n e r t i a terms a r e again on t h e l e f t , t h e v i s c o s i t y terms on t h e r i g h t side. Equation (4.10) i s a d i f f e r e n t i a l equation of t h e f o u r t h order for t h e flow function. Again, i t s general s o l u t i o n i s extremely For very slow (creeping) motions d i f f i c u l t because of the non-linearity. t h e f r i c t i o n t e r n very strongly predominate. Then one may s e t : M$= 0

(4.11)

This s i m p l i f i c a t i o n i s permissible only because t h e d i f f e r e n t i a l equation remains of t h e f o u r t h order, so t h a t no boundary condition i s l o s t . However, being l i n e a r , t h i s equation i e a t l e a s t solvable. It appears a l s o i n t h e theory of e l a s t i c i t y where it i s designated as the b i p o t e n t i a l equation. There e x i s t s a solution of equation (4.11) by Stokes f o r moving dropletsrwhich w a s extended by Cunningham t o very small drop diameters (comparable t o t h e mean f r e e path of t h e molecules). Herewith we shall conclude the m r e general considerations and t u r n

te the b e - m d ~ yl a y e r p r ~ b l e r nproper, i i d t i n g ourseives t o f l u i d s of very elr?sll viscosi.ty

V.

A few preparatory considerations w i l l l e a d up t o t h e boundary l a y e r problem. One might conceive the notion of simply eliminating a l l t h e f r i c t i o n terms of the NavierStokes' d i f f e r e n t i a l eqilation i n t h e case of small v i s c o s i t y . However, t h i s would be fundamentally wrong as w i l l be proved below.

NACA TM Ro*

22

An equation which is completely analogous t o equation i n t h e theory of heat t r a n s f e r :

+

cp

(” at

u

(4.5) occurs

(5-)as2

as + v 2)= h. a 9 + a29 &x

bs

1217

(4.12)

where t h e v e l o c i t y components a r e r e t a i n e d xhereas t h e r o t a t i o n o replaces the temperature 9, t h e density p t h e s p e c i f i c h e a t cp per unit volume, and t h e v i s c o s i t y p t h e thermal conductivity A . t h e l e f t of equation (4.12) stands t h e temperature change due t o convection, on t h e r i g h t t h e change due t o heat t r a n s f e r .

On

The temperature d i s t r i b u t i o n around a heated body inrmersed i n a flow with the f r e e stream v e l o c i t y uo ( f o r instance f i g . 8) i s determined by t h e d i f f e r e n t i a l equation (4.12). One perceives i n t u i t i v e l y t h a t for small uo the temperature increase s t a r t i n g from t h e body extends.toward t h e f r o n t and a l l s i d e s far i n t o t h e ffow ( s o l i d contour) whereas f o r l a r g e uo t h i s influence i s mainly l i m i t e d t o a t h i n l a y e r and a narrow wake (dashed contour). The analogy of equations (4.12) and (4.5) i n d i c a t e s t h a t t h e friction-rotation d i s t r i b u t i o n i n question must be s i m i l a r : For omall f r e e stream v e l o c i t y t h e r o t a t i o n i s noticeable a t l a r g e distance from t h e body, whereas for l a r g e uo t h e r o t a t i o n i s l i m i t e d t o t h e immediate neighborhood of the body. Thus f o r rapid motions, that is, l a r g e Reynolds nurdbers (compare next section), one expects t h e following solution of N a v i e H t o k e s ’ d i f f e r e n t i a l equations: 1. I n t h e region outside of a t h i n boundary l e y e r p o t e n t i a l flow 2, Inside t h i s t h i n boundary l a y e r

Therefore, one must n o t set f o r small viscosity.

o)

= 0

o

#

01)

= 0,

t h a t is,

0, thus no p o t e n t i a l flow.

i n t h i s boundary layer, even

It i s t r u e t h a t t h e p o t e n t i a l flow i s a l s o a solution of NavierStokes’ d l f f e r e n t i a l equations, b u t it does not s a t i s f y t h e boundary l a y e r condition vt = 0. Proof:

The p o t e n t i a l flow may be derived from p o t e n t i a l

O(x,y,z)

as : w = grad a, with AO =

.-

a2a ax2

+ -a2Q =o

+

ay2

az2

(4.13)

However, i f A3 = 0, then a l s o grad A0 = A grad@= 0, t h a t is, Ay = 0 f o r p o t e n t i a l flow. According t o equation (3.18) t h i s f a c t signifies t h a t i n t h e Navier-Stokes d i f f e r e n t i a l equations t h e f r i c t i o n terms vanish i d e n t i c a l l y , and hence t h a t the p o t e n t i a l flow a c t u a l l y s a t i s f i e s t h e Navier-Stokes d i f f e r e n t i a l equations. However, it s a t i s f i e s only t h e one boundary condition vn = 0. Thus, f o r the l i m i t i n g case of small v i s c o s i t y , one obtains u s e f u l s o l u t i o n s f o r t h e l i m i t i n g procese v + O n o t by cancelling t h e f r i c t i o n terms i n t h e d i f f e r e n t i a l equation, since t h i s reduces i t s order ( t h e d i f f e r e n t i a l equation o f the f o u r t h order f o r the flow function would t u r n i n t o an equation of t h e second order; the NavierStokes d i f f e r e n t i a l equations would change from the second t o t h e f i r s t order), so t h a t one can s a t i s f y only correspondingly fewer boundary conditions. Thus t h e l i m i t i n g process V - 0 must n o t be performed i n t h e d i f f e r e n t i a l equation i t s e l f , b u t only i n i t s solution. This can be c l e a r l y demonstrated on an example ( r e f e r r e d t o f o r comparison by P r a n d t l ) of the solution of an ordinary d i f f e r e n t i a l equation. Consider t h e damped o s c i l l a t i o n of a mass point. The d i f f e r e n t i a l equation

m

ddX + k dx + CX = 0 at

dt2

(4.14)

a p p l i e s i n which m represents the o s c i l l a t i n g mass, k t h e damping constant and c t h e spring constant. (x = elongation, t = time). Let f o r instance the two i n i t i a l conditions be:

t = 0;

x

= 0;

dx/dt = 1

J

I n analogy t o t h e cas8 i n question one considers here t h e l i m i t i n g case of a very small mass m, since then the term of t h e highest order tends toward zero. If one would simply put m = 0, one would t r e a t nothing b u t t h e d i f f e r e n t i a l equation k -dx+ c x = o dt which by a s s d n g the solution t o be of t h e f o m x = A eAt i s transformed i n t o k A. + c = 0, whence X = -c/k. That i s , the solution reads:

(4.15)

24

NACA TM No. 1217

(4.16) However, t h e two i n i t i a l conditions x = 0 and dx/dt = 1 a t t h e time t = 0 cannot be s a t i s f i e d with t h i s solution. But i f one t r e a t s t h e complete d i f f e r e n t i a l equation (4.14) i n t h e same manner t h e r e r e s u l t s : mh.

2

+ k h + c = O

and hence :

or t h e square root might be developed i n t o a s e r i e s and ( s i n c e now t h e l i m i t i n g process m + O i s t o be performed) broken o f f a f t e r t h e second term:

x1y2 Thus

X1

-k

+

k ( 1 - 9 ) 2m

t h u s X1 =

-

m

k

corresponds t o t h e previous solution of t h e f i r s t order

d i f f e r e n t i a l equation, where, however,

small my h2 2 -k/m;

h2

had been l o s t .

For very

therewith the general solution becomes, by

combination of t h e p a r t i c u l a r solutions,

Since for = - A1,

t = 0,

x

i s a l s o supposed t o equal zero, t h e r e follows:

thus:

(4.18) This equation i s p l o t t e d schematically i n f i g u r e 9. The f i r s t term of equation (4.18), which alone cannot s a t i s f y t h e boundary conditions,

NACA TM

TJO.

1217

25

starts from t h e value A1 a t the t i m e t = 0 and decreases expon e n t i a l l y . The second term 18 important only f o r s m a l l +values and plays no r o l e f o r l a r g e t. It is very r a p i d l y v a r i a b l e and assures t h a t t h e t o t a l solution ( s o l i d l i n e ) s a t i s f i e s t h e boundary conditions. The slowly v a r i a b l e solution ( i n X1) corresponds t o t h e p o t e n t i a l flow, t h e second, r a p i d l y v a r i a b l e p a r t i c u l a r solution ( i n h. ) i n d i c a t e s , as 2 it were, t h e narrow region of t h e boundary layer; t h e d l e r m, t h e narrower t h i 8 region. Of

Herewith w e s h a l l conclude t h e general remarks and t u r n t o t h e l a w Similarity.

CHAPTER V. REYNOLDS' LAW of SI-ITY So far no general methods f o r the solution of t h e NavierStokes d i f f e r e n t i a l equations a r e known. Solutions t h a t a r e v a l i d f o r a l l values of t h e v i s c o s i t y a r e so far known only f o r a very few s p e c i a l cases ( f o r instance, P o i a e ~ l l l 6 e~ pipe flow). Meanwhile t h e problem of flow i n a viscous f l u i d has been tackled by s t a r t i n g from t h e limits, t h a t is, one has t r e a t e d on t h e one hand flows of very great v i s c o s i t y , on t h e other hand flows of very e m a l l v i s c o s i t y , since one obtains i n t h i s manner c e r t a i n mathematical simplifications. However, s t a r t i n g from these l i m i t i n g cases one cannot possibly i n t e r p o l a t e for flows of average v i scositg. The t h e o r e t i c a l treatment of t h e l i m i t i n g cases of very g r e a t and very emall viscosity i s mathematically s t i l l very d i f f i c u l t . Thus research on viscous f l u i d s w a s undertaken l a r g e l y from t h e experimental side. The Navier4tokes d i f f e r e n t i a l equations o f f e r very useful indications, which permlt a considerable reduction of t h e volume of experimental investigation. The r u l e s i n question a r e t h e so-called l a w s of Simih.Tity. The problem is: Under what conditions a r e t h e forms of flow8 of any l i q u i d s o r gases around geometrically s i m i l a r l y shaped bodies themselves geometrically similM Such flow8 a r e c a l l e d mechanically similar. Consider f o r instance t h e flows of two d i f f e r e n t f l u i d s of d i f f e r e n t v e l o c i t i e s around two spheres of d i f f e r e n t s i z e ( f i g . 10). Under what conditions a r e t h e flows geometrically similar t o each other? Obviously t h i s i s t h e case when a t p o i n t s of eimllar p o s i % i m $2 the two flow p a t t e r n s t h e f o r c e s a c t i n g on volume elements a t these p o i n t s have t h e same r a t i o . Depending on what kinds of f o r c e s a r e i n e f f e c t , various l a w s of s i m i l a r i t y w i l l r e s u l t from t h i s requirement. Most important f o r t h i s investigation i s t h e case except t h e i n e r t i a and f r i c t i o n f o r c e s are negligible. f r e e surfaces a r e t o l e present, s o t h a t t h e e f f e c t of compensated by t h e hydrostatic pressure. In t h i s case

where ell f o r c e s Furthermore, no g r a v i t y 58 t h e flow mound

NACA TM No. I217

26

t h e two spheres i s geometrically similar when t h e i n e r t i a and f r i c t i o n f o r c e s have the same r a t i o a t every point. The expressions f o r the i n e r t i a and v i s c o s i t y f o r c e s a c t i n g on t h e volume elemmt w i l l nowbe derived: t h e r e i s as f r i c t i o n f o r c e per u n i t volume p u

h.

ax

d7 = -

ds

P a2u 2 J

whereas the i n e r t i a f o r c e per u n i t volume i s

as

The r a t i o

aU

i n e r t i a force

--

P U -

ax

a2U

f r i c t i o n force

must, therefore, be the same a t all p o i n t s of t h e flow. One now i n q u i r e s as t o the v a r i a t i o n of these f o r c e s with v a r i a t i o n i n the q u a n t i t i e s c h a r a c t e r i s t i c of the phenomenon: f r e e stream v e l o c i t y V, diameter d, density p, and v i s c o s i t y P. For v a r i a t i o n of V and d t h e individual q u a n t i t i e s i n equation (5.1) a t s i m i l a r l y located p o i n t s vary as follows:

Therewith equation (5.1) becomes: i n e r t i a force f r i c t i o n force

P

P

(5.2)

The law of mechanical s i m i l a r i t y i s therefore: The flows around geometrically similar bodies s i m i l a r l y located and a l i n e d with r e s p e c t geometrically s i m i l a r stream t o t h e flow have, f o r equal p V d/p, l i n e s as well. If the flows i n question are, for instance, two flows of t h e same f l u i d of equal temperature and density ( p and p equal) around two spheres,one of which has a diameter twice t h a t of t h e other, t h e flows a r e geometrically similar provided that t h e f r e e stream v e l o c i t y f o r t h e l a r g e r sphere has half t h e [email protected] of t h a t f o r the smaller sphere.

NACA TM NO. 1217

27

The quantity p V d/p i8,as a quotient of two forces, a dimensionless number. This fact is immediately recognized by substituting for the quantities their dimensions:

PVd

-

P

m2

kg sec2 m

m4sec

kg sec

= 1

This law of similarity was discovered by Osborme Reynolds in his studies of fluid flows in a pipe. The dimensionless quantity is called after him: PV d/cc =

V d/v = Re

=

Reynolds' number

The introduction of this dimensionless quantity helped greatly in advancing the development of modern hydrodynamics. Connection between Similarity and Dimensional Considerations

As is known, all physical laws can be expressed in a form free of the units of measure. Thus the similarity consideration may be replaced by a dimensional analysis. The following quantities appearing in the NavierStokes differential equations are essential for the stream line pattern: V, d, p, p. The question is whether there is a combination

va

P7

which is a Reynolds number and therefore has the dimension 1. amounts to determining a, j3, 7, 6 in such a manner that ~ d p p 7 ~ o 0 =L0KT o = l

This

(5.3)

with K, L, T representing the symbols for force, length,and time, respectively. Without limiting the.generality a may be set equal to unity (a = 1) since any power of a dimensionless quantity is still a pure number. With a = 1 there results fro= eqidation (5.3)

28

NACA TM

By equating t h e exponents of obtains the t h r e e equations:

L, T, K

K: L: T:

47-26=0 27+ 6 = 1

The solution gives:

p=1;

1217

on t h e l e f t and r i g h t s i d e s one

7 + 6 = 0

1 + p

NO.

I

(5.5)

(5.6)

7 = 1 ; 6=-1

Accordingly t h e only p o s s i b l e dimensionless combination of i s t h e quotient

V, d,

p, p

This dimensional a n a l y s i s l a c k s t h e p i c t o r i a l q u a l i t y of t h e s i m i l a r i t y consideration; however, it o f f e r s t h e advantage of a p p l i c a b i l i t y even when knowledge of t h e exact equation of motion i s s t i l l miassing, i f t h e r e i s only known what physical q u a n t i t i e s determine t h e phenomenon. C

m VI.

EXACT SOLVTIONS OF TEE NAm-STORES EQUmIONS

In general, t h e problem of f i n d i n g exact s o l u t i o n s of t h e N a v i e p Stokes d i f f e r e n t i a l equations encounters insurmountable d i f f i c u l t i e s , p a r t i c u l a r l y because of t h e non-linemity of t h e s e equations which p r o h i b i t s a p p l i c a t i o n of t h e p r i n c i p l e of ailperposition. Nevertheless one can give exact s o l u t i o n s for a few s p e c i a l cases, mostly, whsn t h e second power t e r m vanish automatically. A few of t h e s e exact s o l u t i o n s w i l l be t r e a t e d here. One i n v e s t i g a t e s f i r s t l a y e r flows i n g e n e r a l , . t h a t i s , flows where only one v e l o c i t y conponent e x i s t s which, moreover, I s n o t dependent on t h e analagous p o s i t i o n coordinate, whereas t h e two other velocity components vanish i d e n t i c a l l y ; thus f o r instance:

NACA TM NO. 1217

29

The NavierStokes d i f f e r e n t i a l equations (3.16) a r e thereby transformed into :

at au

) :+: (

p - = p x - 4 +h,

O = p Y - 32

ay

o = p z - - ap az

while t h e continuity equation is i d e n t i c a l l y s a t i e f i e d .

a. Pipe Flow, Steady and S t a r t i n g 1. Steady pipe flow.

For t h e case of t h e pipe l y i n g horizontally t h e mass f o r c e s a r e everywhere constant, and t h e equation aystem (equation ( 6 . 2 ) ) y i e l d s :

with t h e solution p = ax + b y t cz

Thus t h e pressure i s a l i n e a r function of t h e position.

In t h ~case ~f the pipe standing v e r t i c a l l y ( f i g . 11) t h e mass f o r c e i s constant i n the y- and z-direction and increases i n the x-direction corresponding t o t h e hydrostatic pressure of t h e f i e l d of g r a v i t y , Moreover, i f one puts Y = 0 and Z = 0:

30

NACA TM No. 1217

Thus there remains from equation (6.2) -a2P =Oor-

aJ? = constant

ax

ax2

The system (equation ( 6 . 2 ) ) Then becomes, under t h e f u r t h e r assumption of steady flow:

a% + a2u = 1 &i = constant lJ ax

ay2 aZ2

One t r i e s a solution of t h e form

u = u

max

(a2+ by2

+

cz2)

(6.5)

urnax representing

with condition

u

= 0

t h e v e l o c i t y a t t h e pipe c e n t e r , With t h e a t the pipe w a l l y2 + z2 = r2 t h e r e r e s u l t s :

a = 1; b = c = -I/= 2 and therefore \

,with

This solution i s i d e n t i c a l with equations (2.3) and (2.4) If one b e a r s i n mind that i n those equations y represented t h e r a d i a l distance from t h e pipe center. Thus P o i s e u i l l e ' s pipe flow w a s found as an exact solution of t h e NavierStokes d i f f e r e n t i a l equations. 2. S t a r t i n g f l o w__ through .. a pipe. By the expression, "Starting flow through a pipe," the following problem i s meant: Let t h e f l u i d i n a c i r c u l a r pipe of i n f i n i t e length be a t r e s t u n t i l the time t = 0. A t the t i m 2 t = 0 l e t a pressure

NACA TM NO. 1217

31

difference invariable with time,

P1

across 9. piece of pipe of length following equation applies:

2,

t>0:

Here

at #

0,

ap

- =

ax

-Pp

suddenly be established

so t h a t f o r the e n t i r e pipe t h e

~1- p 2 = constant 2

(6.7)

and t h e i n e r t i a and f r i c t i o n terms balance each other.

The solution of t h i s problem, which w i l l not be f u r t h e r discussed here, was given by Szymanski (reference 1 2 ) . The phenomenon i s n o t dependent on t h e l o n g i t u d i n a l coordinate x. The v e l o c i t y p r o f i l e s a t various times can be seen from f i g . 12. It i s c h a r a c t e r i s t i c that, f i r s t , a t t h e pipe center, t h e v e l o c i t y remains l o c a l l y constant and t h e f r i c t i o n i s notlcesble only i n a t h i n l a y e r near the surface. Only l a t e r does t h e f r i c t i o n e f f e c t reach t h e pipe center. P o i s e u i l l e ' s parabolic p r o f i l e of t h e steady pipe flow i s a t t a i n e d asymptotically f o r t + 0 3 , . One must c l e a r l y d i s t i n g u i s h between t h e non-steady s t a r t i n g flow through a pipe discussed here and t h e steady pipe i n l e t flow. This l a t t e r i s t h e flow a t t h e i n l e t of a pipe. The rectangular v e l o c i t y p r o f i l e present i n the entrance c r o s s section is, with increasing distance x from the i n l e t , gradually transformed under t h e e f f e c t Of f r i c t i o n i n t o P o i s e u i l l e ' s parabolic profile.

Since here

. dU # ax

0,

t h i s i s not a l a y e r flow. This i n l e t flow w a s , f o r t h e plane problem, exactly calculated from the d i f f e r e n t i a l equations by H. Schlichting (reference 14) and f o r t h e rotationally-symmetrical problem, according t o an approximate method, by L. S c h i l l e r (reference 32). Third l e c t u r e (Dec. 15, 1941) 'b. Plane Surface; a Surface Suddenly Set i n Motion

and an O s c i l l a t i n g Surface 1. O s c i l l a t i n g Surface.

Let a plane surface of i n f i n i t e extent perform i n i t s plane r e c t i l i n e a r o s c i l l a t i o n s i n t h e x - d i r e c t i o n ( f i g . 13). The y d i r e c t i e ~i s assumed normal t o t h e surface. Let t h e o s c i l l a t i o n take place with t h e v e l o c i t y uo = A c o s n t, with A denoting t h e amplitude, n t h e frequency of t h e oscilla-tion. The f l u i d near the surface i s c a r r i e d along by t h e f r i c t i o n . Sinse the surface i s o f i n f i n i t e extent, t h e s t a t e of flow 2 s independent sf' x and z . The flow i n question i s t h e r e f o r e a non-steady p l a n e l a y e r flow f o r which

i

NACA TM N o . 1217

32 u = u(y,t);

=w

v

0;

aP ax

E

0

If, f i n a l l y , one puts t h e ma88 f o r c e s equal t o zero, t h e NavierStokes d i f f e r e n t i d equations (equation (3.16) ) a r e reduced t o

2

&=&A

at

as2

with the boundary condition y = 0, u = uo cos n t. This equation has t h e same s t r u c t u r e as t h e d i f f e r e n t i a l equation f o r thermal expansion i n a rod ( l i n e a r equation of heat conduction). It becomes i d e n t i c a l with it i f v i s replaced by t h e thermal conductivity a and u by t h e temperature (compare equation (4.12)).

I

For t h e solution of equation (6.8) one uses

u = Ae*

where k

cos ( n t

- ky)

i s a constant t o be determined.

Then

h= - k

as

Ae-ky cos ( n t

- = - 2k2Ae-ky a211

- ky) +

sin (nt

Insertion i n equation (6.8) gives

k Ae-ky s i n ( n t

- ky)

n~ = 2k2AV,

and thence

k=vA

(6.10)

2 v

Th? velocity d i s t r i b u t i o n

u(y, t)

- Q)

is therefore an o s c i l l a t i o n with

amplitxle decreasing toward t h e outside

Ae-

m;

the l a y e r a t the

33

NACA TM NO. 1217

distance

y

from t h e surface has a phase lag of

t h e motion of the surface. follows from Irx = 2x as

The wave length

X

E

y

with respect t o

i n the y-direction

(6.11) Thus t h e c o + s c i l l a t i n g l a y e r i s thinner, t h e g r e a t e r t h e frequency n and t h e smaller t h e v i s c o s i t y V . The r e s u l t l, .Y i s t o be noted. The v e l o c i t y p r o f i l e s f o r various times are given i n f i g u r e 13. 2. The Surface Suddenly Set in,Motion. Equation (6.8) y i e l d s another exact solution of t h e NavierStokes d i f f e r e n t i a l equations, namely t h e f l o w i n t h e neighborhood of a plane surface which suddenly starts moving i n i t s own plane with t h e constant v e l o c i t y uo. I n t h i s case t h e boundary conditions are

< t -0,

f o r all y:

u = 0 (6.12)

t>0,

y=o:

u=uo

An appropriate v a r i a b l e f o r t h e solution i s t h e dimensionless quantity

A s t h e solution of equation equation (6.12) one obtains

(6.8) with t h e boundary conditions

(6.14)

The c o i - r e e t ~ e s sof t h e eolution i s readily confirmed by s u b s t i t u t i o n . Ths variatior; of t h e f l o w with time i s indicated i n f i g u r e ilc. ,03

The probability i n t e g r a l appearing i n equation (6.14)

gl,

-e7*

has, for 7 = T~ = 1.9, the value 0.01. Therefore, f o r 7 = 76' The thickness of the l a y e r c a r r i e d along by t h e f r i c t l o n u = 0.01 uO. i s , tharefore,

a7

NACA

34 6 = 2 qg

pt

TM No. 1217

(6.15)

Thus here again t h e f r i c t i o n l a y e r thickness i s

6-

p

A t t h i a point w e conclude t h e discussion of l a y e r flows and t u r n t o a few other exact s o l u t i o n s of N a v i e r S t o k e s d i f f e r e n t i a l equations.

C.

Flane S t a g n a t i o n 4 o i n t Flow

The plane flow i n t h e neighborhood of a stagnation p o i n t on a smooth w a l l i s considered. With t h e coordinate system according t o f i g u r e 15 t h e corresponding p o t e n t i a l flow has t h e p o t e n t i a l CE, = a 2 (x2

- y2)

(6.16)

and t h e stream function

Q = a x y The v e l o c i t y component8 are:

u

= ax;

v

= 4 y

(a = c o n s t a n t )

(6.17)

This i s a p o t e n t i a l flow which, coming from t h e d i r e c t i o n of t h e y-is, encounters t h e s o l i d w a l l y = 0, divides, and flows o f f p a r a l l e l t o t h e x-axis. Whereas t h e p o t e n t i a l flow g l i d e s along t h e w a l l , t h e viscous flow must adhere t o it, If one designates t h e v e l o c i t y components of t h e viscous flow by u(x, y ) and v(x, y ) , t h e boundary conditions f o r them are:

u=o;

v = o

, u=u;

v = v

y=o; y =

w*

(6.18)

For t h e stream function of t h s viscous f l u i d one uses t h e equation:

NACA TM

NO.

35

1217

(6.19) Thence one obtains:

(6.20)

The boundary conditione (Equation (6.18) ) require

(6.21)

A t l a r g e distances from t h e surface, t h a t is, i n t h e p o t e n t i a l flow, t h e

pressure i s calculated from Bernoulli's equation p = po

- g w2

= Po

-

a2 (x2

+

y2)

(6.22)

with W = signifying t h e magnitude of t h e v e l o c i t y of t h e p o t e n t i a l flow, given by equation (6.17). Let po be t h e t o t a l pressure of t h e p o t e n t i a l flow.

For t h e viscous flow one formulates t h e analogous equation

With these equations one t u r n s t o t h e NavierStokes equations which read, f o r vanishing mass forces:

(6.24)

NACA TM No. 1217

36

The continuity equation has already been integrated by introduction of the stream function $. By substitution of equations (6.20) and (6.23) into equation (6.24)one obtains

These are two differential equations for the two unknown functions f(y) and F(y) which determine the velocity distribution and the pressure distribution, respectively. The component of the velocity u is of particular parallel to the surface, that is,the function f'(y), interest. Since F ( 9 ) does not appear in the first equation, one solves first the first equation and then, after substitution of this solution, the second. Thus the differential equation to be solved reads at first ft

2 - f f" = a2

+

v f'"

(6.26)

with the boundary conditions according to equation (6.21). This non-linear differential equation cannot be solved in closed form. If one introduces instead of y ths variable

(6.26~~)

E=ay and in addition the similarity transformation

Then the inhomogeneous term in equation (6.26)becomes equal to 1, and the solutions therewith become independent of the specific data for the flow. Thereby equation (6.26) become6

If one now equates

a2A2 = a2 and vAa3 = a2,

that is,

NACA TM No.

37

I217

t h e d i f f e r e n t i a l equation for cp(k)

with

f =t

y

reads:

(6.28) with t h e boundary conditions

The solution found by s e r i e s development can be found i n the t h e s i s of Hiemenz (reference lo), compare t a b l e l*. The v e l o c i t y component p a r a l l e l t o the surface is

I

It i s indicated i n f i g u r e 16. The curve cp'( t) increases linearly a t = 0 and approaches one asymptotically. For about 5 = 2.6, cp' Z 0.99; thus within about one percent of the f i n a l value. If one again designates the 'corresponding distance from t h e surface y = 6 as the boundary l a y e r thiclmess ( f r i c t i o n l a y e r thicbness), then

Thus i n t h i s flow, aa i n t h e former one8,

It i s a l e o remarkable t h a t the dimensionless v e l o c i t y d i s t r i b u t i o n according t o equation (6.29) and the boundary l a y e r t h h k n e s s according t o equation (6.30.) a r e independent of x, thus do not vary along the w a l l . T h e t a b l e s appear i n appendix, chapter X I I .

38

NACA TM N o . 1217

For later applications the characteristics important for the friction layer, displacement thickness 6* and momentum thickness 6 , are introduced here; they are defined by

u219

=

/

00

u(u

y=o

- u) ay

The di~placementthickness gives the deflection of the stream lines of the potential flow from the surface by the friction layer; the momentum thickness is a measure of the momentum loss in the friction layer. By insertion of equation (6.29) in (6.31) and (6.32) and calculation of the definite integral one finds (1

- 9')

dS = 0 . 6 4 8 2 E

=0

and hence

*:=

2.218

(6.35)

I9

The quantity 8* is indicated in figure 16. For comparison with a later approximate solution one also notes the numerical value of the g*2 dimensionless quantity dx'

One finds from equations (6.17) and (6.33)

dU = -v d x

0.4202

(6.36)

The exact solution of the NavierStokes differential equations found here gives, therefore,.for large Reynolds numbers a friction layer thickness decreasing with

and a transverse pressure gradient

39

NACA TM NO. 1217

decreasing with p a F a . t o be discussed later.

Both confirm the boundary l a y e r assumptions*

d. Convergent and Divergent Channel A f u r t h e r c l a s e of exact solutione of t h e AavierStokee d i f f e r e n t i a l equations e x i s t s f o r t h e convergent and divergent channel with plane w a l l s ( f i g . 17), as given by G. Hamel (reference 11). Without entering i n t o t h e d e t a i l s of t h e r a t h e r complicated calcul a t i o n s t h e character of t h e solutions will be b r i e f l y sketched: The v e l o c i t y d i s t r i b u t i o n s f o r convergent channels, p l o t t e d a g a i n s t distance along t h e surface f o r various included angles a and f o r various Rmuuibers appear as indicated i n figure 18. A t t h e tunnel center t h e v e l o c i t y i s almost canstant, and at t h e surfaces it suddenly declines t o zero.

In t h e caee of divergent tunnels one obtains g r e a t l y d i f f e r i n g forms far t h e v e l o c i t y p r o f i l e s , depending on t h e included angle and t h e Re-nuuiber. Here a l l v e l o c i t y p r o f i l e s have two i n f l e c t i o n points. For small R e - n d e r s and small included angles t h e v e l o c i t y is p o s i t i v e over t h e e n t i r e croes section ( s o l i d curve i n f i g . 19); f o r l a r g e r angles and l a r g e r Re-numbere, on the other hand, t h e v e l o c i t y p r o f i l e s have reverse flow a t t h e surface (dashed curve i n f i g . 19). The reverse flow i s t h e i n i t i a l phase of a vortex formation and therefore of t h e separation of t h e flow from t h e surface, Generally, t h e separation does not occur symmetrically on both surfaces; t h e flow separates from one s i d e and adheres t o t h e other surface ( f i g . 20). These examples a l s o confirm t h e theory that exact splutions have t h e same character as approximate solutions of boundary l a y e r theory; i n p a r t i c u l a r , they confirm t h a t f o r t h e convergent channel a very t h i n l a y e r with cons i d e r a b l e f r i c t i o n e f f e c t i s present near t h e surface (here also t h e c a l c u l a t i o n ahoy8 t h a t t h e l a y e r thickness -fi)and t h a t f o r t h e divergent channel reverse flow and separation occur. We here conclude t h e chapter on the exact SOlutionE of t h e NavierStokes d i f f e r e n t i a l equations and t u r n t o t h e approximate solutions. By exact s o l u t i o n s have been M d e r s t o o d t h o s e where i n t h e Iiavier-Stokee d i f f e r e n t i a l equations all terms a r e tahen i n t o consideration t h a t , i n t h e various cases, a r e n o t i d e n t i c a l l y zero. By approximate s o l u t i o n s of t h e NavierStokes d i f f e r e n t i a l equations w i l l be understood, i n contrast, s o l u t i o n s where terms of small magnitude a r e neglected i n t h e d i f f e r e n t i a l equations themselves. However, by no means a r e a l l t h e f r i c t i o n terms t o be neglected simultaneously, since t h i s would represent t h e case of p o t e n t i a l flow. q h e rotationally-symmetrical stagnation-point flow has been calculated by Homann (reference 17). Instead of equation (6.28) one obtains t h e d i f f e r e n t i a l equation (pt" + 2qq" - (pt2 + 1 = 0.

40

NACA TM No. 1217

CEUPITX? V I I .

VERY

SLOW MCrrION

(STOKES, OSEnV)

The exact solutions of t h e NavierStokes d i f f e r e n t i a l equations discussed i n the previous chapter a r e of a very s p e c i a l kind. Most of them d e a l t with flows along a plane surface, where t h e stream l i n e s a r e r e c t i l i n e a r . Most flows e x i s t i n g i n p r a c t i c e , as f o r instance flows around a r b i t r a r y bodies, cannot be c a l c u l a t e d exactly from t h e NavierStokes d i f f e r e n t i a l equations, but must be t r e a t e d by approximate methods. Two kinde of such approximations are possible: 1. For predominant viecoeity, completely neglecting t h e i n e r t i a terms euggeete i t s e l f (very small Re-number; R e < 1). 2. For very small v i s c o s i t y and t h e r e f o r e predominant i n e r t i a one takes the v i s c o s i t y i n t o consideration o n l y i n a very t h i n l a y e r i n t h e neighborhood of t h e s o l i d w a l l ; f o r t h e rest, t h e flow i s regarded a s f r i c t i o n l e s s . Here the Re-number i s very l a r g e (Prandtl'a boundary l a y e r theory).

The f i r s t l i m i t i n g case with very small Re-number w i l l be discussed i n t h i s chapter. A emall Re-number i n d i c a t e s small v e l o c i t i e s , emall body dimensions, and l a r g e v i s c o s i t y . Since t h e i n e r t i a terms depend on the square of t h e v e l o c i t y whereas the f r i c t i o n t e r m a r e l i n e a r , a l l i n e r t i a terms i n t h e WavierStokes d i f f e r e n t i a l equations are, f o r very small Rmunibers, n e a i g i b l e . It i s t o be expected t h a t an approximation w i l l thereby be obtained f o r very slow (creeping) motion, as for instance t h e f a l l i n g of a minute fog p a r t i c l e * ) or t h e s l o w motion of a body i n a very v i s c i d o i l . Neglecting all i n e r t i a terms one obtains from t h e NavierStokes d i f f e r e n t i a l equations (3.16) t h e following:

aU + -av + -a~= o as aZ ax

(7.2)

~

*For a sphere f a l l i n g i n a i r ( v = 14 x lo4 m2/sec) for inetance: Re = V d/V = 1, f o r d = 1 m; V = 1.40 cm/sec.

NACA TM

NO.

41

1217

The same boundary conditions apply to this system of equations as apply to the complete NavierStokes differential equations, namely vanishing of the normal component vn =. 0 and the tangential component v = 0 t at the bounding surfacee. The neglect of all inertia term8 in NavierStokes differential equations does not represent as serious an inaccuracy as the neglect of all friction tern when transforming the Navier-Stokes differential equations into Euler's differential equations of the frictionless flow. That is, by neglecting the inertia tern, the order of the differential equations is not lowered so that in the sfmplified differential equations the same boundary conditions as in Ithe NavierStokes complete differential equations can still be satisfied. Furthermore one obtains from the equations (7.1), taking into account the continuity, by differentiating the first with respect to x, the second with respect to y, the third with respect to z, the following equation for the pressure p

that is, for creeping motions the pressure function p(x, y, z ) potentia1 function.

Js a

The details of the calculation w i l l not be discussed more thwoughly, particularly since the creeping motion is technically not very iaportant. However, at least Stokes' famous solution for the sphere will be discussed briefly (fig. 21). The drag of a sphere for creeping motion consists of the contributions of the pressure drag (form drag) and the surface friction drag. The latter is obtained by integration of the wall shearing stress over the entire sphere surface. Stokes performed the integration of the equation system (7.1) and (7.2) f o r a sphere in a uniform flow of velocity Uo. There results, according to Stokes, for the entire drag of the sphere of radius R:

W = W

D

+ W r

R

=6fipu0

(7.4)

The drag is, therefore, [email protected] t o the first p w e r of the valcrcity. If one introduces for the sphere a drag coefficisnt cw vhich, in the customary manner, is referred to the frontal area and the dynamic pressure of the free stream velocity

42

NACA TM No. 1217

W = cW II R2

9 Uo2

(7.5)

t h e r e r e s u l t s f o r t h e drag c o e f f i c i e n t according t o Stokes' formula:

One can s t a t e immediately t h a t t h e stream l i n e p a t t e r n of t h i s creeping motlon must be t h e sams ahead of and behind t h e sphere s i n c e f o r r e v e r s a l of t h e i n i t i a l flow (sign r e v e r s a l of t h e v e l o c i t y components) t h e equation system (equation (7.1)) goes over i n t o i t s e l f . The strean-line p a t t e r n f o r t h e viscous sphere flow, as it p r e s e n t s i t s e l f t o an observer who i s a t r e s t r e l a t i v e t o t h e flow a t i n f i n i t y , i s shown i n f i g u r e 22. The f l u i d p a r t i c l e s a r e pushed a s i d e by t h e sphere i n f r o n t and come together again behind it. A s shown by a comparison of Stokes' drag formula equation (7.6) with t e s t r e s u l t s (reference 33), t h i s formula i s v a l i d only f o r t h e region R e < 1.

Correction by Oseen I n Oseen's l a t e r improvement of Stokes? s o l u t i o n f o r t h e sphere . t h e i n e r t i a terms in t h e d i f f e r e n t i a l equations a r e p a r t l y taken i n t o consideration. Oseen formulates t h e v e l o c i t y components u, v, w:

u =

uo +

u';

v = v';

(7.7)

w = w?

where u t , v', w t may be considered as disturbance v e l o c i t i e s which i n general a r e small compared with t h e f r e e stream v e l o c i t y Uo. This assumption i s n o t a c t u a l l y c o r r e c t f o r t h s immediate proximity of t h e sphere su-rface. With the formulation (equation (7.7)) t h e i n e r t i a terms i n equation (3.16) a r e divided i n t o two groups, f o r instance: aut

TJo

uo ax 9

avt

,.

..

and

u'-,

aU'

ax

ut-,

1

ax

...

The second group of second order, as compared with t h e f i r s t group, i s n e g l e c t d . Therewith m e then o b t a i n s From the N a v i e r S t o k e s e q i a t i o n s of m 3 t i o n (3.16) the following equations of motion, which a r e taken a s a basis by Osecn.

43

NACA TM NO. 1217

In addition, one uses the continuity equation: aut + -av'

ax

+ -&ft =o

(7.9)

aZ

and t h e same boundary conditions aB i n t h e N a v i e M t o k e s d i f f e r e n t i a l equations. One c a l l s t h e contributions of t h e convective terme i n these the equations that were taken i n t o consideration, f o r instance Uo

u, ax

semi-quadratic terme. These d i f f e r e n t i a l equations of Oseen and Stokes' d i f f e r e n t i a l equations a r e both linees. The stream l i n e pattern, as it r e s u l t s f o r t h i s sphere flow according t o Oseen, i s given i n f i g u r e 23. Here q a i n t h e observer i s a t r e s t r e l a t i v e t o t h e f l u i d a t l a r g e distance from t h e sphere. Thus the sphere i s dragged p a s t t h e obeerver with the v e l o c i t y UO. The stream l i n e pattern ahead of and behind t h e sphere ere now not t h e same, as w a s t h e case i n Stokes' solution. Ahead of t h e sphere e x i s t s almost t h e same displacement flow as i n Stokes' pattern; behind t h e sphere, however, the s t r e a m l i n e s a r e c l o s e r together, that i s t h e v e l o c i t y i s g r e a t e r here than i n Stokes' case. A wake i s present behind t h e sphere similar t o t h a t from t e s t r e s u l t s f o r large Reynolds number s

.

For t h e sphere drag calculated by Stokes t h e r e r e s u l t s with t h e drag c o e f f i c i e n t cw introduced i n equation (7.5) t h e formula:

cw=Re 24

UoD . 1+ 1 R e ; R e = -

(

16

)

V

The t e a t r e s u l t s (reference 33) show that Oseen's formula i s f a i r l y accurate up t o about Re = 5. With these b r i e f remarks w e conclude t h e l i m i t i n g case of emall Reynolds numbers and turn t o t h e case which i s of foremost i n t e r e s t i n p r a c t i c e : t h e case of very l a r g e Reynolds number.

44

NACA TM N o . I217

CHAPTER V I I I .

PRA"L'S BOUNDARY LAYER EQUNCIONS

The other extreme case of very s m a l l v i s c o s i t y o r of very l a r g e Reynolds number w i l l now be t r e a t e d . I n t h i s case the i n e r t i a e f f e c t s a r e predominant within the main body of t h e f l u i d whereas t h e v i s c o s i t y e f f e c t s there a r e almost negligible. A s i g n i f i c a n t advance i n t h e treatment of motion of f l u i d s f o r l a r g e Reynolds n d e r s , t h a t i s , i n general, of f l u i d s of very s m a l l viscosity, was a t t a i n e d by L. Prandtl i n 1904 (reference 7). Prandtl demonstrated i n what way v i s c o s i t y is e s s e n t i a l f o r l a r g e Reynolds numbers and how one can simplify t h e NavierStokes d i f f e r e n t i a l equations i n order t o obtain a t l e a s t approximate solutions.

L e t u s consider the motion of f l u i d of very small v i s c o s i t y , f o r instance of a i r o r water surrounding a c y l i n d r i c a l streamline body ( f i g . 24). Up t o very near the surface t h e v e l o c i t i e s a r e of t h e order of magnitude of t h e f r e e stream v e l o c i t y Uo. The stream l i n e p a t t e r n as well a s the velocity d i s t r i b u t i o n agree t o a l a r g e extent with those of t h e f r i c t i o n l e s s f l u i d ( p o t e n t i a l flow). More thorough i n v e s t i g a t i o n s show, however, t h a t the f l u i d by no means g l i d e s along t h e surface (as i n p o t e n t i a l flow) but adheres t o it. The t r a n s i t i o n from zero v e l o c i t y a t the surface t o the f u l l y developed v e l o c i t y as it e x i s t s a t some distance f r o m t h e body, i s effected i n a very t h i n l a y e r . Thus one must distinguish between two regions which, it i s t r u e , cannot be rigorously separated : 1. A t h i n layer i n t h e immediate proximity of t h e body where t h e velocity gradient normal t o t h e surface i s very l a r g e Q1

(boundary l a y e r ) . Here t h e v i s c o s i t y p, though very small, plays an e s s e n t i a l r o l e inasmuch as t h e f r i c t i o n a l shearing stress

T

= p

aw an

can assume considerable values.

2. In t h e remaining region outside of t h i s l a y e r velocity g r a d i e n t s o f such magnitude do not occur, so t h a t t h e r e t h e e f f e c t of v i s c o s i t y becomes i n s i g n i f i c a n t . Here f r i c t i o n l e s s p o t e n t i a l flow prevails.

I n general'one may say t h a t t h e boundary l a y e r i s thinner, t h e smaller the v i s c o s i t y or, more generally, the l a r g e r t h e Re-number. It was shown before on the b a s i s of exact solutions of t h e NavierStokes d i f f e r e n t i a l equations t h a t t h e boundary l a y e r thickness i s

s q / T The approximations t o the NavierStokes d i f f e r e n t i a l equations t o be made below are more v a l i d t h e thinner t h e boundary layer. Thus t h e solutions of t h e boundary layer equations have an asymptotic character f o r i n f i n i t e l y increasing Reynolds numbers.

NACA TM

NO.

1217

45

Let us now mak t h e simplifications of ,e NavierStokes d i f f e r e n t i a l equations for t h e boundary layer. To t h i s en t h e order of magnitude of t h e separate terms of the Navie-tokes d d f f e r e n t i a l equations must be estimated. One considers t h e flow around a c y l i n d r i c a l body according t o f i g u r e 24. One imagines t h e NavierStokes d i f f e r e n t i a l equations w r i t t e n nondimensionally, by r e f e r r i n g a l l v e l o c i t i e s t o t h e f r e e stream v e l o c i t y Uo and t h e lengths t o a body length 2 . The pressure w i l l be made dimensionless with p Uo2, the time with 2 / U o . F’urthermore R e = ‘02 represents the Reynolds number. Accordingly, t h e NavierStokes

V

d i f f e r e n t i a l equations become - omitting t h e mass f o r c e s according t o equation (4.2), by writing t h e same l e t t e r s f o r t h e dimensionless q u a n t i t i e s a s for t h e dimensional ones -

aU+a2u

a U + u -au + v -aU =-ap+L

at

ax

aY

a x R B2

1

ax2 as2 1/62

1

1

The estimation gives: Longitidinal velocity u i s of t h e order of magnitude 1. Dimensionless boundary layer thiclmess 8/2,

=

the length of the plate.

2

u o i t f"(0)

According to

=

Therewith the local surface shearing stress is

-

The friction drag according to equation (9.16) is therewith 2

W

= cylb U o f i

Jx=o F

= 2a

b U , d x

and therefore the drag of the plate wetted on both sides is 2W=kab

= 1.328

T i

b Uo

pp 2

If one introduces in the customary manner a dimensionless drag coefficient by the equation

cw= F

2w

(F

=

2 b 2 = wetted area)

Uo2

one ob-tainsfor the drag coefficient the formula

cw - 1.128

6

(Re =

T)

(9.19)

. 59

NACA TM No. 1217

Displacement Thickness of t h e Boundary Layer By t h e development of t h e boundary l a y e r on t h e p l a t e , which i n c r e a s e s downstream with E, t h e p o t e n t i a l flow i s d e f l e c t e d outward from t h e surface by a n amount E*, which i s c a l l e d t h e displacement t h i c k n e s s of t h e boundary l a y e r . It can b e e a s i l y c a l c u l a t e d from t h e v e l o c i t y d i s t r i b u t i o n i n t h e boundary layer, as follows: L e t y1 denote a p o i n t outside t h e boundary layer; then according t o t h e d e f i n i t i o n f o r 8*

or

According t o equation ( 9 . 5 )

P yl

Since t h e point

'1 =

T~ l i e s outside of t h e boundary l a y e r , one can put

f o r f(7) t h e f i r s t approxinstion of the asymptotic s o l u t i o n according t o equation (9.13), t h u s

T k ~ one s r i n d s f'?r the displs:emnt

thickness of t h s bcwdary layer

60

NACA TM No. 1217

The distance f r o m t h e surface y = 6* i s a l s o ghown i n f i g u r e 30. Thus t h e s t r e a m l i n e s of the p o t e n t i a l flow are, because of t h e f r i c t i o n e f f e c t , deflected outward by t h i s amount. The a c t u a l boundary l a y e r thickness 6 cannot be given accurately since the f r i c t i o n e f f e c t i n t h e boundary l a y e r thickness decreases asymptotically toward t h e outside. The component of v e l o c i t y p a r a l l e l t o t h e surface u i s asymptotically converted i n t o t h e v e l o c i t y Uo of t h e p o t e n t i a l flow ( t h e function f ' ( 7 ) asymptotically approaches t h e value 1). If one wants t o define t h e boundary l a y e r thickness as t h e point where the v e l o c i t y u = 0.99 Uo ( f u l l value), one obtains for it according t o t a b l e 2, 7 = 5.0. Therewith one has f o r the thus defined boundary l a y e r thickness

6 = 'j.O\iz

The thus defined boundary l a y e r thickness equals about t h r e e times t h e displacement thickness of t h s boundary l a y e r . Let here a l s o be introduced t h e value f o r t h e momentum thickness 19, needed l a t e r . This l a t t e r i s a measure f o r t h e momentum loss due t o f r i c t i o n i n t h e boundary l a y e r and is, as indicated before i n equation (6.32), defined by t h e equation

Jy=o The calculation r e s u l t s , because of equation (9.5), i n

P" f'(i

- f*)dv

E

= 0.664

J= ,O

Finally t h e form parameter becDm2s therewith

0

61

NACA TM NO* 1217

6* =

2.605

I9

Experimental investigations of t h e laminar boundary l a y e r on t h e f l a t p l a t e were performed by B. G. van der Hegge Zynen (reference 19) and M. Hansen (reference 20). I n a l l e s s e n t i a l p o i n t s t h e t h e o r e t i c a l r e s u l t s were well confirmed. The measurements showed t h a t t h e laminar boundary = 3.5 t o 5 x 105, l a y e r e x i s t s t o about t h e Reynolds number (Uo x/v) crit if x denotes t h e length of run of the boundary layer. For l a r g e r Reynolds numbers t r a n s i t i o n t o t h e turbulent s t a t e of flow takes place. F i f t h l e c t u r e on January 5 , 1942. b. The Boundary Layer on the Cylinder (symmetrical case) The i n t e g r a t i o n method of Blasius given i n t h e previous section w a s used by Hiemenz ( t h e s i s Cijttingen 1911) f o r c a l c u l a t i n g t h e boundary l a y e r on t h e c i r c u l a r cylinder. The same method was l a t e r f u r t h e r extended by Howarth (reference 15) t o t h e general case of a c y l i n d r i c a l body of a r b i t r a r y c r o s s section. This method w i l l be b r i e f l y presented f o r t h e symmetrical case. One considers ( f i g . 32) a c y l i n d r i c a l body with symmetrical cross section i n a flow approaching i n t h e d i r e c t i o n of t h e symmetry axis with t h e v e l o c i t y Uo. Let x be t h e a r c length along t h e contour, measured f r o m t h e f r o n t stagnation point, y t h e v e r t i c a l distance from the surface. Let the p o t e n t i a l flow U(x) be given by i t s power s e r i e s development i n X. At t h e stagnation point (x = 0 ) , U ( x ) = 0, and f o r t h e symmetrical case o n l y the odd t e r n s of t h e power s e r i e s are d i f f e r e n t f’rom zero. Therefore:

u(x) = u x + u x 3 t u x 5 + 1 3 5 The c o e f f i c i e n t s

ul,

u3 ,

. .

. . . depend s o l e l y on t h e shape of the

body and a r e therefore q u a n t i t i e s known f’rom t h e p o t e n t i a l flow, The s t a t i o n a r y boundary l a y e r equations according t o equation (8.3) a r e a l s o v a l i d f o r t h i s case with a curved surface and t h e r e f o r e read:

NACA TM No. 1217

62 From equation (9.22) one obtains f o r t h e pressure term:

u

a'= dx u1 [ulx

+ 4 ~ ~ +x (6 3 u5 + $)x5

+

..

1

(9.24)

The continuity equation i s again i n t e g r a t e d by t h e stream function:

It i s now necessary t o f i n d a s u i t a b l e formulation f o r t h e v e l o c i t y d i s t r i b u t i o n u(x, y ) , v(x, y) and therewith f o r t h e stream function $(x, y ) . I n analogy t o equation (9.22) a power series i n x suggests i t s e l f f o r u(x, y ) as w e l l , with c o e f f i c i e n t s , however, which a r e dependent on y. It i s important t o f i n d a form where t h e coeffic i e n t s ( o r functions) dependent on y have a u n i v e r s a l character, that is, need n o t be c a l c u l a t e d anew f o r each shape of body, b u t may be calcul a t e d once f o r a l l . Howarth (reference 15) succeeded i n f i n d i n g such a formulation. For t h e distance from t h e surface one introduces t h e dimensionless variable :

The expression (9.24) f o r t h e pressure term suggests t h a t t h e following equation be s e l e c t e d f o r $

This yields:

( * = d i f f e r e n t i a t i o n with r e s p e c t t o -

r-

7):

2

u = u x f ' + 4 u x 3f t + 6 x 5 u g ' + - hu3' 1 1 3 3 5 5 1 5

+.

..

(9.28)

*One o b t a i n s t h i s equation from t h a t of B l a s i u s according t o equation (9.3) by s u b s t i t u t i n g f o r Uo t h e f i r s t term of t h e s e r i e s equation (9.22).

NACA TM NO. 1217

63 2

ax

= u1f 1 '

+

1

12u

xf"

X2f'

3

3

+

+ 4u x 3f " + 3

1 1

+

jOx4 E 5 g ;

6x5

5

u g " + - u32 h"]

5 5

3

+

h.1

ul

u1

5

... +

..

(9.29)

}

(9.30)

After i n s e r t i o n of t h e expressions (9.24) and (9.28) t o (9.32) i n t o t h e first equation (9.23) one o b t a i n s by comparison of t h e c o e f f i c i e n t s a system of ordinary d i f f e r e n t i a l equations f o r t h e unknown functions fl, , which appears as followa: h5, f3, g5,

....

terms with

g i v e s t h e d i f f e r e n t i a l equation

u x 1

f

4u u x3

4f'f' 1 3

- 3f"f - f

6f'h'

- 5f"h - f

1 3

2

-f

1

f" = 1 + f'" 11 1

1 3

f" = 1 + f'**

1 3

3

6ulu5x 5 6u 2x5

3

1 5

1 5

h" = 1 + h'"

1 5

2

5

- 8(f' 2 - f 3

f")

3 3

Formulation of t h e flow function according t o equation (9.27) has t h u s accomTlished t h e elimination o f t h e c o e f f i c i e n t s depending on body

64

NACA TM No. 1217

shape (ul, u3, fl, f3'

. . .)

from t h e d i f f e r e n t i a l e q u a t i o n s - f d r t h e functions

. . which t h u s now have a u n i v e r s a l character.

The boundary conditions f o r t h e f u n c t i o n s

fl, f

. . . follow

3'

-

from

by comparison of equation (9.28) with equation (9.22); they a r e 7 = 0 :

=f'=O; 1

f

f

3

= f ' = O ;

3

g5

=

h5=h5=0; 7

=n:

f'

1

= 1;

f1

3

1 = 4'

g; =

g;

=

O

...

(9.34)

h'=O;...

The d i f f e r e n t i a l equations (9.33) are a l l of t h e t h i r d order, and equation (9.34) gives t b e e boundary conditions f o r each. The different i a l equation f o r f (7) i s non-linear and i s i d e n t i c a l with t h e d i f f e r e n t i a l equation (6.2 ) obtained i n chapter V I f o r t h e etagnation p o i n t - cp; 7 z E , as follows by comparison of equation ( 6 . 2 6 ~ ~ ) flow: fl= with equation (6.26). A l l t h e remaining d i f f e r e n t i a l equations i n equation (9.33) are l i n e a r , with t h e c o e f f i c i e n t s determined by t h e functions of t h e preceding approximations.

8

The s o l u t i o n s of t h e d i f f e r e n t i a l equations (9.33) a r e b e s t obtsined by numerical i n t e g r a t i o n . The f u n c t i o n s f l and f 3 were already calculated by Hiemenz (reference 10). Howarth (reference 15) improved t h e t a b l e s f o r f 3 and r e c e n t l y N i l s Fr'dssling (reference 16) c a l c u l a t e d g and h as well. The which i s i d e n t i c a l with cp' according t o

5

5

fi

equation (6.28) w a s already given i n f i g u r e 16. seen f r o m f i g u r e 33, t h e functions g' and h'

5

numerical values a r e compiled i n t a b l e 3.

5

The function f from f i g u r e 34.

j

c a i be The

Concernlng t h s a p p l i c a b i l i t y of t h i s c a l c u l a t i o n method it must be mentioned t h a t f o r slender body shapes t h e s e r i e s f o r U(x) and u(x,y) converge poorly. The reason is, t h a t f o r these body shape3 U(x) h a s a vary steep ascent i n t h e neighborhood of t h e stagnation point ( f i g . 3 5 ) , while showing a r a t h e r f l a t curve f u r t h e r on. Such a function cannot be developed r e a d i l y i n t o a Taylor s e r i e s . For such body shapes many more of t h e functions of t h e d i f f e r e n t i a l - e q u a t i o n system (9.33) would be required

65

NACA TM No. 1217

than have been calculated so far. For blunt bodies, as for instance the circular cylinder, the convergence is considerably better so that one proceeds rather far with this calculation although not always up to*the point of separation. Howarth (reference 15) also performed the corresponding calculation for the unsymmetrical case where the even coefficients also appear in the power series f o r U(x). This is the case f o r a symmetrical body at an angle of attack and quite generally for any unsymmetrical body. Fr6ssling (reference 16) made the application to the rotationally symmetrical case.

Circular Cylinder The boundary layer on the circular cylinder will be treated as an example for the application of this method. Whereas Hiemenz (reference 10) took a pressure distribution measured by him as the basis for this case, we shall here calculate with the potential-theoretical pressure distribution. The velocity distribution of the potential flow reads, with the symbols according to figure 36,

The power series development gives:

(9.36) In comparison with equation (9.22): uO

u1 = 2 -; R

u3

=

- 52 3uo ; u5

=

2 uo' 5 Rsj

...

Therewith follows from equation (9.26):

It fOllOWt3 that for the velocity distribution from equation (9.28)

(9.364

66

NACA TM NO*

One f u r t h e r c a l c u l a t e s t h e p o s i t i o n of t h e

point

is, according t o equation (8.5) determined by

I217

xA which

= 0. Therewith

y=o r e s u l t s from equation (9.36): X

fY(0)

A - - f"(0) 31

R

With t h s numerical values

3

(27 + . . .

t 0 ) = 1.23264; fl(

f"(0) = 0.7246

3

(9.384

one f i n d s :

*

X

A R

= 0

=

1.60;

(PA =

92O

(9.39)

Hiemenz (reference 10) based h i s c a l c u l a t i o n on h i s experimental pressure d i s t r i b u t i o n ; he c a l c u l a t e d t h e separation p o i n t t o be a t cp = 82O, whereas h i s measurements gave cp = 81'. This result i s A A considerably d i f f e r e n t from t h a t obtained with t h e p o t e n t i a l - t h e o r e t i c a l pressure d f s t r i b u t i o n . The reason i s t h a t f o r a body as b l u n t as t h e c i r c u l a r cylinder t h e experimental and t h e p o t e n t i a l - t h e o r e t i c a l pressure d i s t r i b u t i o n i n t h e neighborhood of t h e separation p o i n t d i f f e r g r e a t l y . The method described here of c a l c u l a t i n g t h e boundary l a y e r by a power-series development s t a r t i n g from t h e stagnation p o i n t has found b u t l i t t l e acceptance because of t h e extensive c a l c u l a t i o n required. It i s , however, indispensable f o r fundamental considerations, s i n c e t h e r e e x i s t no other exact s o l u t i o n s of t h e d i f f e r e n t i a l equations of t h e boundary l a y e r f o r t h e flow about a body. Thus approximation methods came i n t o use f o r t h e p r a c t i c a l performance of boundary l a y e r calculations; they w i l l be discussed i n t h e following chapter. It i s t r u e t h a t t h e i r accuracy i s sometimes considerably lower than t h a t of t h e previously discussed exact s o l u t i o n s .

c.

Wake behind t h e F l a t P l a t e i n Longitudinal Flow

The a p p l i c a t i o n of t h e boundary l a y e r equations i s n o t absolutely li.mited t o t h e presence of s o l i d w a l l s ; They may a l s o be applied i f t h e r e i s present within t h e flow a l a y e r i n which t h e f r i c t i o n e f f e c t i s W h i s r e s u l t v a r i e s somewhat i f i n t h e series development equation (9.38a), one t a k e s f u r t h e r terms i n t o consideration. However, f o r t h i s purpose one would have t o c a l c u l a t e a t l e a s t up t o t h e term x7. I

~

NACA

TM

NO.

67

1217

predominant. This i s the case f o r instance when within t h e flow two l a y e r s of d i f f e r e n t v e l o c i t i e s adjoin, as f o r instance i n t h e wake region behind a body or a t t h e outflow from an opening. I n t h i s and t h e following chapter we s h a l l t r e a t two examples of such flows which we s h a l l l a t e r encounter again i n t h e discussion of turbulent flows. 4

The wake flow behind t h e f l a t p l a t e i n longitudinal flow i s chosen as t h e first example ( f i g . 37). A t the t r a i l i n g edge of t h e p l a t e t h e two boundary l a y e r p r o f i l e s grow together and form a "wake p r o f i l e " t h e width of which increases with distance while t h e v e l o c i t y decrement a t i t s center decreases. The s i z e of the "wake" i s d i r e c t l y connected with t h e drag of t h e body. Otherwise, however, t h e form of t h e v e l o c i t y d i s t r i b u t i o n i n t h e wake a t a l a r g e distance from the body i s not dependent on t h e shape of t h e body, whereas t h e velocity d i s t r i b u t i o n very close behind t h e body n a t u r a l l y depends on the boundary l a y e r of t h e body and on any e x i s t i n g separation. From t h e v e l o c i t y d i s t r i b u t i o n measured i n t h e wake one may calc u l a t e t h e drag by means of the momentum theorem i n t h e following manner. The momentum theorem s t a t e s : The variation of t h e momentum with time ( = momentum flow through a control a r e a f i x e d i n space) equals t h e sum of t h e r e s u l t a n t forces. By r e s u l t a n t f o r c e s one has t o understand: 1. Pressure f o r c e s on t h e c o n t r o l area, 2. Extraneous forces, which a r e t r a n s f e r r e d from s o l i d bodies t o t h e flowing f l u i d , f o r instance t h e shearing s t r e s s a t the surface which gives t h e f r i c t i o n drag. For t h e present case t h e control a r e a AAIBBl i s placed as indicated i n f i g u r e 38. Let t h e boundary AIBl which i s p a r a l l e l t o t h e p l a t e be so f a r d i s t a n t from it t h a t it l i e s everywhere i n t h e undisturbed v e l o c i t y Uo. Furthermore, t h e r e a r c r o s s section BB1 i s t o l i e so far behind t h e p l a t e t h a t t h e s t a t i c pressure t h e r e has t h e aame undisturbed value as i n f r o n t of t h e p l a t e . Then t h e pressure i s constant on t h e e n t i r e control area, so t h a t t h e p r e s s m e f o r c e s make no contribution. I n caldulating the momentum flow through t h i s control area one has t o consider t h a t , due t o continuity, f l u i d must flow out through t h e boundary AIB1, namely t h e difference between t h e l a r g e r quantity flowing through t h e cross section AA1 and the smaller quantity flowing through the c r o w The c r o s s section AB does not make a contribution t o t h e section BB1. x-momentum, since f o r reasons of symmetry t h e transverse velocity on it equals zero. The momentum balance i s given i n t h e t a b l e below, with entering momentums counted as positive, outgoing ones as negative.

NACA TM NO* I217 Cross s e c t i o n

Mass [m3/sec]

A B

0

Momentum flow i n x - d i r e c t i o n = volumexdensityxvelocity 0

b

A %

B1

-b

r

h

udy

-

P

p

.

AlBl

E=

Mass

XMomentum Flow

=w

= o

Control a r e a

The t o t a l momentum loss of t h e flow f o r t h e present caee ,equals t h e drag W of,one s i d e of t h e p l a t e . Thus one obtains

JO

Jo

The i n t e g r a t i o n t h e r e i n may be extended from y = 0 t o m, i n s t e a d of from y = 0 t o h, s i n c e f o r y > h t h e integrand i n equation (9.40) vanishes. For t h e drag of t h e p l a t e wetted on both s i d e s one o b t a i n s t h e r e f ore :

NACA TM NO. 1217

I

+O0

I n equations (9.40) and (9.41) the i n t e g r a l s a r e t o be extended over t h e wake as indicated a t a distance 'behind t h e p l a t e where t h e s t a t i c pressure has i t s undisturbed value. However, one may n a t u r a l l y apply equations (9.40) and (9.41) a l s o t o t h e boundary l a y e r on t h e p l a t e a t a c e r t a i n point x; then they give t h e drag of the p a r t of t h e p l a t e f r o m t h e leading edge t o t h i s point, The d e f i n i t e i n t e g r a l 6 i n equations (9.40) and (9.41) represent physically the momentum loss due t o t h e f r i c t i o n e f f e c t . As mentioned before, it i s customary t o introduce f o r t h i s i n t e g r a l a l s o t h e momentum lose thickness 4 by the following equation (compare equation (6.32)).

uo2 4

=I (u, u

u) dy

(9.42)

Therewith the formula f o r t h e drag may a l s o be written, by comparison with equation (9.40): 2

W=bpU0 4

(9.43)

The velocity d i s t r i b u t i o n i n t h e w a k e , p a r t i c u l a r l y a t l a r g e distance x behind t h e p l a t e i n longitudinal flow ( f i g . 37) i s t o be calculated next. This c a l c u l a t i o n must be performed i n two steps: 1. By a development "from t h e front", t h a t is, by a c a l c u l a t i o n which follows t h e f u r t h e r development of t h e Blasiua4oundary l a y e r p r o f i l e present a t the trailing edge of the plate. 2. By a development "from t h e r e a r " t h a t is, by a n asymptotic calculation f o r t h e wake, under t h e assumption t h a t t h e difference velocity

is smaii. The first calculation w a s performed by S. Goldstein (reference 21) and i s very troublesome; the second'was indicated by Tollmien (reference 3) and y i e l d s r a t h e r simple l a w s , i n p a r t i c u l a r a l s o an exact solution of t h e d i f f e r e n t i a l bo-mdarg l a y e r equation. Since such exact eolutions a r e

70

NACA TM No. 1217

comparatively rare and since, moreover, t h e asymptotic l a w f o r t h e wake a p p l i e s n o t only f o r t h e f l a t p l a t e i n longitudinal flow b u t f o r any a r b i t r a r g body shape, t h i s asymptotic s o l u t i o n w i l l be t r e a t e d here' somewhat more thoroughly. The wake v e l o c i t y ul(x, y ) introduced i n equation (9.44) i s assumed t o b e so small i n comparison t h the f r e e stream velocity Uo that t h e second-power terms (ul/LJo? are negligible r e l a t i v e t o 1. Moreover, t h e pressure term dp/dx in t h e boundary l a y e r equation i s t o be s e t equal t o zero f o r t h e f i r s t asymptotic approximation. Therewith t h e d i f f e r e n t i a l boundary l a y e r equation (8.3) becomes, f o r t h e present case:

au,

uo-=v-

a2u1

ax

(9.45)

as2

With the boundary conditions:

y = m:

u1 = 0 (u =

uo)

For t h e solution of t h e d i f f e r e n t i a l equation (9.45) one introduces as before i n the p l a t e flow according t o B l a s i u s ? equation (9.3) t h e new variable

Further, one uses f o r

u t h e equation: 1

(9.47)

The di8tanCe

x f r o m t h e t r a i l i n g edge of t h e p l a t e i s thus made

dimensionless by aividing by the p l a t e length.

The power

- -2

for

x

ie given by t h e f a c t t h a t t h e momentum i n t e g r a l which, according t o equation (9.4l),gives t h e p l a t e drag must be independent of x. With t h e second-power t e r n neglected the drag of t h e p l a t e Wetted on both s i d e s is, according t o equation (9.41):

NACA TM

NO.

1217

71

One f i n d s f u r t h e r :

1

-0

and therewith:

2W=bpU,

(9.50) -a

The c a l c u l a t i o n of t h e s i n g l e terms i n equation (9.45) gives:

-&l =

(9.51)

vx

aY

-L 21 X /

By i n s e r t i o n i n t o equation (9.45) me cbtains a f t a r d i v i ~ i o nby 1 2 c uo ( X m - x-' t h e follobing d i f f e r e n t i a l equation f o r t h e v e l o c i t y d i s t r i b u t i o n g(7) :

'

(9.52)

72

NACA TM No. 1217

with the boundary conditions: q = 0: g'

=o;

q

=a:

g=o

A single integration gives:

where, because of the boundary condition at q = 0, the integration constant K must be zero. Repeated integration gives the solution:

-I

g = e

(9.53)

where a multiplicative integration constant may be put equal to one without limiting the generality since the velocity distribution ul still contains, ,according to equation (9.47), the multiplicative free constant C. This constant C is determined from the consideration that the plate drag obtained from the momentum loss (equation (9.50)) must be the same as the frictional drag of the plate. First,

d

J

--OI

-"

and therewith fromequation (9.50):

v

'O

On the other hand, the friction drag of the plate wetted on both sides was according to the solution of Blasius (equation (9.18))

2w = 1.328 p

Therefrom follows 2C

\In-=

1.328

and

2E

uo

NACA TM

73

No. 1217

c = - 0.664

\I.

(9.54)

Thus t h e f i n a l s o l u t i o n f o r t h e wake v e l o c i t y f o r t h e f l a t p l a t e i n l o n g i t u d i n a l flow becomes:

(9.55)

'

The v e l o c i t y d i s t r i b u t i o n of t h i s asymptotic l a w i s represented i n f i g u r e 39. It i s noteworthy t h a t t h e function f o r t h e v e l o c i t y d i s t r i b u t i o n i s i d e n t i c a l with Gauss' e r r o r function. I n keeping with t h e hypothesis t h e l a w according t o equation (9.55) is v a l i d only f o r l a r g e d i s t a n c e s behind t h e p l a t e , according t o t h e c a l c u l a t i o n s by Tollmien (reference 3 ) f o r x r 32. The development of t h e wake from t h e f r o n t , performed by Goldstein i s v a l i d only f o r comparatively small x/2. However, f o r intermediate x/t both s o l u t i o n s can be joined t o 8ome extent, so t h a t one then o b t a i n s t h e v e l o c i t y d i s t r i b u t i o n i n t h e e n t i r e wake. Such a f i g u r e i s given by Tollmien (reference 3 ) . S i x t h Lecture on January 12, 1942. d.

The Plane J e t

A. f u r t h e r example of a flow without bounding w a l l t o which t h e boundary l a y e r theory i s applicable i s t h e outflow of a j e t from a hole. The problem t o be t r e a t e d i s t h e plane s t a t i o n a r y one where t h e j e t goes out from a long narrow s l o t and mixes with t h e surrounding f l u i d a t r e s t . This i s one of t h e r a r e cases where t h e d i f f e r e n t i a l boundary l a y e r equations may be i n t e g r a t e d exactly. The c a l c u l a t i o n s were performed by H. Schlichting (reference 22) and W. Bickley (reference 3 0 ) and w i l l be discussed a l i t t l e more thoroughly.

Due t o t h e f r i c t i o n e f f e c t t h e j e t e n t r a i n s a p a r t of t h e f l u i d a t r e s t and weeps it along. There r e s u l t s a stream-line p a t t e r n l i k e t h e one drawn i n f i g u r e 40. Let t h e x-direction coincide with t h e j e t axis m d +,hs migir; l i e in t h e s i o t . It then immediately becomes c l e a r t h a t t h e width of t h e Jet increases with tine d i s t a n c e x and t h e mid-velocity decreases v i t h t h e distance x. For the c a l c u l a t i o n t h e s l o t i s assumed t o be i n f i n i t e l y narrow. I n order t o make t h e volume of flow, together w i t h i t s momentum, f i n i t e , t h e v e l o c i t y i n t h e s l o t i s then i n f i n i t e l y l a r g e . Again, as ir, t h e previous example concerning t h e wake flow, t h e pressure term dp/dx may be neglected since t h e presslare v a r i e s only very l i t t l e i n t h e x d i r e c t i o n . The smallness of t h e pressure term can

e

NACA TM No. 1217

74

a l s o be shown subsequently. from t h e f i n i s h e d solution. Thus t h e differe n t i a l boupdary l a y e r equations f o r t h e present case read, according t o equation (8.3):

au

u - + v - =mzv -

ax

a2U

as

as2

y=o:

v=o;

5 = 0

y = w :

u = o

with the boundary conditions:

au

Since the pressure i s constant, t h e e n t i r e momentum f l o w i n the Idirection f o r control a r e a fixed i n space (compare f i g u r e 40) must be independent of the distance from t h e hole x. If one chooses t h e l a t e r a l boundaries of the control a r e a a t so l a r g e a distance f r o m t h e a x i s that t h e r e u = 0, then

J = p

I+*

u*Q = Constant

(9 57)

It i s t o be noted f o r t h e i n t e g r a t i o n of t h e equation of motion equation (9.56) that for t h i s problem, as before f o r t h e p l a t e i n longitudinal flow, no special length-dimension exists. Thus a f f i n i t y of t h e velocity p r o f i l e s u(x, y ) i s suggested, t h a t i s : with b signifying a suitable width of t h e j e t , t h e v e l o c i t y p r o f i l e s a r e only functions of y/b. Accordingly one uses the following expression f o r t h e stream function JI:

*

= XPf

(5) X

= XPf

(E)

(9.58)

NACA TM NO. 1217

75

-

The two a t first m o w n t h e conditions t h a t

- exponents

p

and

q

a r e d e t e d n e d from

1. t h e momentum flow f o r t h e x d i r e c t i o n i s independent of according t o equation (9.57), and that 2. t h e a c c e l e r a t i o n terms, f o r instance

u

ax' al

x

and t h e i n e r t i a

term i n equation (9.56) are of t h e same order of magnitude and hence m e t be of t h e aame degree in I. T h i s y i e l d s t h e two equations

It follows t h a t 1

Therewith t h e f i n a l equations, a f t e r addition of s u i t a b l e constant f a c t o r s , read as follows:

Therefrom one obtains, with

t h e following expressions f o r t h e velocity components'and t h e i r d e r i v a t i v e s :

NACA TM No. I217

76

1 aU = -x -

ax

(9.63)

413 1 - (27f"-f')

9

By substitution into the differential equation (9.56) there results, after cancelling the factor I

-5 13 x , 27

the following differential equation

for the flow function f(7):

I

f'*

+

ff"

+

f'" = 0

(9.64)

with the boundary conditions:

As for Blasius' plate flow here also an ordinary differential equation (9.64) was obtained from the two partial differential equations (9.56) by means of the similarity transformation equation (9.61). Here also, as in most boundary-layer problems, the differential equation is non-linear and of the third order.

The integration of this differential equation (9.64) is attained in a surprisingly simple manner. First, one obtains by a aingle integration I

ff' + f" = Constant = -

o

(9.65)

NACA TM NO*

77

1217

The constant of i n t e g r a t i o n t h e r e i n i s zero becauss of the boundary condition f " ( o ) = 0. The secmd order d i f f e r e n t i a l equation now obtained could be i n t e g r a t e d once more if a factor 2 were present i n t h e first term. This f a c t o r can be obtained by performlng t h e following f u r t h e r s i m i l a r i t y t r a n s f ormati on : One puts:

S=aq

(9.66)

f = 2a F(E)

(9.67)

a i s a f r e e constant which i s determined l a t e r . With t h e equations (9.66) and (9.67) one obtains from equation (9.65), t h e prime now signifying d i f f e r e n t i a t i o n with respect t o

(,

F" + 2FFt = 0

(9.68)

with t h e boundary conditione:

This d i f f e r e n t i a l equation can now be integrated again and y i e l d s

F' + F 2 = K The constant of i n t e g r a t i o n K i s obtained from t h e boundary conditions, equation (9.68a), as K = 1, i f one s t i p u l a t e s F1(o) = 1, which i s possible without l i m i t i n & t h e generality because a i s s t i l l present as a f r e e constant i n F. One now has for F t h e first order non-linear d i f f e r e n t i a l equation

F' + F 2 = 1 which i s a R i c c a t i d i f f e r e n t i a l equation.

dF 1-F

(9.69)

The i n t e g r a t i o n y i e l d s

-

= -I iog 1 + F = a r c tanh F 2 2 1-F

and therewith for t h e inverse function

NACA TM No. 1217 -25

F=tanhE = 1 - e -25 l + e

Furthermore,

dF = 1 - tan2 dE.

equation (9.63),

5.

(9.70)

If one i n s e r t s t h e solution found i n t o

one obtains f o r t h e v e l o c i t y d i s t r i b u t i o n

The velocity d i s t r i b u t i o n over t h e width of t h e j e t calculated f r o m t h i s equation i s represented i n f i g u r e 41. The f r e e constant a remains t o be determined. This can be done f r o m t h s condition (equation (9.57)) according t o which the momentum flow i n t h e x-direction i s independent of x. From equation (9.71) and (9.57) one obtains

Let the momentum J f o r the J e t be a prescribed constant which i s , f o r instance, d i r e c t l y r e l a t e & t o t h e exces8 pressure under which t h e j e t flows from t h e s l o t . Then a becomes, according t o equation (9.72),

(9.73)

and therewith t h s velocity d i s t r i b u t i o n

(9.74)

NACA TM NO.

79

1217

The value of t h e transverse component of t h e v e l o c i t y a t t h e edge of t h e j e t (y = Q)) a l s o i s noteworthy. From equation (9.74) one f i n d s f o r t h i s l a t e r a l inflow v e l o c i t y v,

=

1/3

- 0.550

(9.75) px

2/3

One can f u r t h e r c a l c u l a t e t h e flow volume f o r a l a y e r of unit height

Q

=

u dy.

b e finds

The f l d w volume increases downstream, since f l u i d a t r e s t i s c a r r i e d along from t h e side. The s o l u t i o n indicated here n a t u r a l l y has a singular point a t x = 0, since an i n f i n i t e l y narrow s l o t with i n f i n i t e l y l a r g e e x i t v e l o c i t y w a s assumed. Actually f o r a narrow s l o t of f i n i t e width one has immediately behind t h e s l o t opening a v e l o c i t y d i s t r i b u t i o n that i s rectangular across the j e t c r o s s section b u t which a t some distance i s transformed i n t o t h e bell-shaped d i s t r i b u t i o n found here with width - 1/3 213 and mid-velocity x b-x ~

..

F i n a l l y it should be mentioned t h a t the corresponding r o t a t i o n a l l y symmstrical problem where t h e J e t goes out from a very small c i r c u l a r hole also can be solved i n closed form. (compare H. Schlichting (reference 22)). I n t h i s case t h e width of t h e j e t i s proportional t o x and t h e midvelocity proportional t o x-'.

e.

The Boundmy Layer f o r t h e P o t e n t i a l Flow

U = u,xm.

Another c l a s s of exact s o l u t i o n s of t h e boundary l a y e r equations w i l l be discussed b r i e f l y which includes t h e p l a t e i n l o n g i t u d i n a l flow and t h e stagnation point flow a s special cases. Falkner and Skan (reference 37) have shown t h a t , J u s t a s f o r Blasius' p l a t e flow, t h e boundary l a y e r d i f f e r e n t i a l equations for t h e p o t e n t i a l flow

u(x)

=

u x" 1

(9.77)

I

80

NACA

TM No. 1217

can be reduced by a s i m i l a r i t y transformation t o an ordinary d i f f e r e n t i a l equation (ul = constant, m > 0 accelerated, m < 0 r e t a r d e d flow). For m > 0, x = 0 i s t h e stagnation point of t h e flow. For m = 0 one obtains U = u1 = Uo, t h e r e f o r e t h e p l a t e flow; m = 1 givee U = ulx, therefore the stagnation point flow according t o equation (6.17). The d i f f e r e n t i a l equations of t h e boundary l a y e r read

aU

v u -bu + v - = u - +dU

ax

as

a2U

as2

dx

The pressure term becomes

U

dx

U

=

2m-1

~x U

~

~

As a new independent variable one introduces

,

and the continuity equation i e i n t e g r a t e d by introduction of a stream function for which one uses t h e equation:

2

(9.79)

m + l

, One h m

m-1

and one obtains

NACA

TM no- 1217

81 (9.80)

After i n s e r t i o n i n t o t h e first equation of motion and d i v i s i o n by

m ul%*'

one obtains, when

-=2m

B

m + 1

t h e following d i f f e r e n t i a l equation f o r cp'n

=

- (p(p" +

cp(e):

$ (cp' 2 -,1)

(9.83)

Boundary conditions:

The d i f f e r e n t i a l equation (9.83) was solved f o r d i f f e r e n t values by Hartree (reference 38). The r e s u l t is given i n f i g u r e 41a. The corresponding values of f3 and m are given i n t h e following table.

NACA TM NO* 1217

82 m

B

-0.0654

-0.14

0

0

0.111

0.2

0.333

0.5

1

1

4

1.6

> 0 ) one obtains velocity Drofiles without inflection points, for retarded flow ( m < 0 , B C-Oj velocity pr'ofiles with inflection points. Separation OCCWB for

For accelerated flow ( m > 0 , B

j3 =

- 0.199,

that i g m =

- 0.091

-0.091 Separation takes place for the potential flow U(x) = u x , thus for very weak retardation. Compare chapter XI a, where1almost the same result is obtained f r o m an approximation calculation. Special cases: 1. Stagnation point flow:

It is obtained for

m = l ;

Then 6 =

d~

u1

y; $ =

d y -x

cp( E ) .

B = 1

These are the same expressions as

for the stignation point flow, equation (6.268) and (6.27a), a l s o (6.19) and (6.2614. The differential equation (9.83) also is transformed into the equation of the stagnation point flow (equation (6.28))

.

2. Plate in longitudinal flow: This case is obtained for

83

NACA TM No. 1217

Then

1 5 =-

6 var iab 1e according JI

=

G-\Ivv,x

PY vx

=

6

vx

y = 3- with

t o equation (9.3).

Cp(6 , ) ;

thus

cp =

f -

E

6

&themore,

If

becomes

compared with t h e expreseion

f o r t h e p l a t e flow equation (9.4).

for $

7 s i g n i f y i n g Blasius'

Because of

dCp = df ,

dE a?-= 2f'"(7), t h s d i f f e r e n t i a l equation (9.83) ftl(q), and cp'"(E) i s f o r t h i s case transformed i n t o 2 f * " ( 7 ) + f f " ( 7 ) = 0. This i s i d e n t i c a l w i t h equation (9.8).

cp"(E)

=

fi

CHAPTER X. A P F ' R O X m SOLUI'ION OF THE BOUNDARY LAYER EQUA!TION BY MEANS OF THE MOMENTUM THEOREM (KARMAN4'O"AUSEN

a.

,

METHOD, PLANE PROBLEM)

The F l a t P l a t e i n Longitudinal Flow

The examp'les of exact s o l u t i o n s of t h e boundary l a y e r equation discussed i n t h e previous chapter give s u f f i c i e n t proof of t h e rather considerable mathematical d i f f i c u l t i e s i n solving t h e d i f f e r e n t i a l equation. Yet t h e examples t r e a t e d were s e l e c t e d as simple as pos8ible. In some other caaes t h e mathematical calculations are s t i l l more d i f f i c u l t than i n those. examples. P a r t i c u l a r l y the problem of flow about a body of a r b i t r a r i l y prescribed shape cannot, i n general, be solved by exact s o l u t i o n of t h e d i f f e r e n t i a l equations of t h e boundasy l a y e r . J u s t t h i s problem, however, i s of s p e c i a l p r a c t i c a l importance, for instance f o r t h 3 c a l c u l a t i o n of t h e boundary l a y e r s on a i r p l a n e w i n g s . There exists therefore a tendency t o apply a t l e a s t approximate methods, even i f t h e i r accuracy i s sometimes not q u i t e s a t i s f a c t o r y , f o r cases where t h e exact solution cannot be obtained with a reasonable expenditure of c a l c u l a t i o n time. Such simpler approximate s o l u t i o n s can be a t t a i n e d i f one does not attempt t o s a t i s f y t h e d i f f e r e n t i a l boundary l a y e r equation f o r every p a r t i c l e o f f l u i d . I n s t e a d one s e l e c t s a p l a u s i b l e expression f o r t h e v e l o c i t y d i s t r i b u t i o n i n t h e boundary l a y e r and satisfies merely t h e momentum equation which : s obtained by i n t e g r a t i o n from t h e equation of motion. A method on t h i s b a s i s f o r t h e plane prob&em of flow about an a r b i t r a r y body w a s worked o u t by von Karman and Pohihausen (references 23 and 2 4 ) . We s h a l l t r y o u t t h i s method i n t h i s chapter a t f i r s t on t h e simpler cme of t h e f l a t p l a t e i n longit u d i n a l flow, where no presslwe v a r i a t i o n s e x i s t . For t h i s s p e c i a l caae t h e momentum theorem y i e l d s t h e statement t h a t t h e momentum l o s s of t h e flow through a c o n t r o l a r e a f i x e d i n space according t o f i g u r e 42 equals t h e friction d r q W(x) of t h e p l a t e from t h e leading edge ( x = 0 ) t o

84

NACA TM No. 1217

t h e point x. Application of t h e momentum theorem f o r t h i s case w a s discussed i n d e t a i l i n chapter IX; f o r t h e drag of t h e p l a t e wetted on one side according t o equation (9.40) it had r e s u l t e d i n t h e formula:

W W = b

(10.1) Jy=o

t h e other han the f r i c t on uag can a l s o be expressecL a s t h e of t h e surface shearing s t r e s s , namely: ,A

ntegra

X

W(X)

= b

h-.

To

(x) dx

(10.2)

In forming the i n t e g r a l (equation (10.1)) i t i s t o be noted that t h e does not make integrand outside of t h e boundary l a y e r , where u = U ,, By comparison of equation8 (10.1) and (10.2) it follows a contribution. that :

Jy=o

If one introduces moreover t h e momentum loss thickness, as defined i n equation (9.42), equation (10.3) can a l s o be w r i t t e n i n t h e form:

(10.4)

This i s t h e momentum theorem of t h e boundary l a y e r f o r t h e s p e c i a l case of t h e f l a t p l a t e i n longitudinal flow. Physically it s t a t e s t h a t t h e surface shearing s t r e s s equals t h e momentum loss i n t h e f r i c t i o n layer, since i n t h e present case t h e pressure gradient makes no contribution. The next chapter w i l l acquaint us with t h e extension of equation (10.4) t o include the more general case of a boundary l a y e r with pressure difference

.

Equations (10.4) and (10.3), respectively, w i l l now be used f o r approximate c a l c u l a t i o n of the f r i c t i o n l a y e r on t h e f l a t p l a t e i n longitudinal flow. A comparison of t h e r e s u l t s of t h i s approximate

85

NACA TM No. 1217

calculation with the exact solution according to chapter IXa will give information about the usefulness of the approximation method. For the approximate calculation one selects a suitable expression for the velocity distribution in the boundary layer in the form: u =

uo

f

(g)

=

uo

f(’1)

with ‘1 = -; 6 = 6 ( x ) 6

(10.6)

6

represents the boundary layer thickness, undetermined at first. For the flat plate it may, moreover, be assumed again that the velocity profiles at various distances from the leading edge of the plate are affine to each other. Thie assumption is contained in equation (10.5) if f(7) there stands for a function which no longer contains any free paramete’rs. Furthermore, f(7) must, for large values 7, assume the constant value 1.

The velocity distribution being given by equation (lo.?), the momentum integral in equation (10.3) may be evaluated. It yielde:

The definite integral in equation (10.7) can be calculated immediately if a definite formulation is given for f(7). Thus one put6

(10.8)

a =

Hence

or : 9 = a 6

(io. i o )

86

NACA TM No. 1217

Furthemre, the displacement thickness of the boundary layer 6* according to equation ( 9 . 2 0 ) :

becomes,

P'

lo

6" = 6

(1-f)dn=a6

(10.loa)

1

On the other hand, the shearing stress

at the surface is:

T~

(10.11)

if one introduces the further simplification $ =

f'(0)

(10.12)

By introduction of these expressions into the momentum equation (10.4) there results: $ -vuO =

6

d6 uo2 u dx

or (10.14)

The integration with the initial value 6 = 0 for x = 0 yields, as first result of the calculation:

Hence the shearing stress becomes, according to equation (10.11): (10.16)

NACA TM l

NO.

1217

and furthermore

and hence, f i n a l l y , t h e t o t a l drag of the p l a t e wetted on both s i d e s according t o equation (10.2): W = 2b 2

.

I

I



%$\ippUo 3 2

A comparison of t h e r e s u l t s f o r boundary l a y e r thickness, surface shearing s t r e s s , and t o t a l drag, which were thus found, with t h e corresponding .) shows t h a t formulas f o r t h e exact s o l u t i o n according t o equation (9. t h e approximate c a l c u l a t i o n according t o t h e momentum theorem reproduces t h e c h a r a c t e r i s t i c s of t h e formulas w i t h p e r f e c t correctness i n a l l cases, t h a t i s , t h e dependence of t h e boundary l a y e r q u a n t i t i e s on t h e l e n g t h of run x, t h e f r e e stream v e l o c i t y U,, and t h e v i s c o s i t y coeff i c i e n t V . The numbers a, 8, unknown a t f i r s t , can be determined 6nly a f t e r making s p e c i a l assumptions for the v e l o c i t y d i s t r i b u t i o n , t h a t i s , e x p l i c i t l y prescribing t h e function f ( 7 ) i n equation (10.5).

Numerical examples

1

The usefulness of t h e method o f approximation w i l l be i n v e s t i g a t e d by a few numerical examples, The acuuracy of t h e r e s d t s w i l l depend t o a g r e a t e x t e n t on a s u i t a b l e choice of the expresbion f o r t h e v e l o c i t y d i s t r i b u t i o n according t o equation (10.5). A t any r a t e t h e function f ( q ) must equal zero f o r 7 = 0 and have the constant value 1 f o r l a r g e 9. A s f i r s t example we s e l e c t t h e very rough assumption t h a t t h e v e l o c i t y d i s t r i b u t i o n i n t h e boundary l a y e r i 6 represented by a l i n e a r function according t o f i g u r e 43a. T h e :

f(7) = q

O S + l :

(10.18) q

2 1: f(7)

Hence t h e r e s u l t s f o r t h e nimbrs and (10.12) a r e :

G,

= 1

j3 according t o equation (10.8)

88

NACA TM No. 1217

The formulas (lO.lg), (10.16) and (10.17) can now be evaluated immediately. One obtains t h e r e s u l t s :

(10.20)

To =

E

1

= 0.289 p U o e

(10.21)

A velocity d i s t r i b u t i o n i n t h e form of a cubic parabola according t o f i g u r e 43b i s s e l e c t e d as second numerical example i n t h e following manner:

This s a t i s f i e s the conditions:

7 = 0:

f = 0 ;

7 = 1 : f = 1; f ' = 0;

t h a t i s , t h e boundary l a y e r p r o f i l e j o i n s t h e v e l o c i t y of t h e p o t e n t i a l flow with a continuous tangent. The c a l c u l a t i o n of t h e numerical f a c t o r s according t o equations (10.8) and (10.12) gives: a = = 39 ;

p = -3

2

(10.24)

and hence f o r t h e c h a r a c t e r i s t i c param2ters of t h e boundary l a y e r : 6 =

4.64vg

(10.25~~)

(10.25b)

89

NACA TM NO. 1217

2W = 1.29 b\lllpU,3'

(10.25~)

The exact value for the drag is, according to equation (9.18),

V G .

2W = 1.328 b The simple assumption of a linear velocity distribution therefore gives a drag too small by thirteen percent, the cubic velocity distribution a drag too small by three percent.

The calculation of the displacement 'thicknesses of the boundary layers according to equation (10.10a) results, for the 1inea.r velocity distribution, in 6* =

-I2 6,

and for the cubic velocity distribution

in 6" = 3 6. This gives, because of equations (10.15) and (lO.25a):

-

3

6* =

1.732g

(linear velocity distribution)

6" =

1.740E

(cubic velocity distribution)

The agreement with the exact value 6* =

1.728

\la

I

1

(10.26)

according to

equation (9.21) is surprisingly good; this is, however, more or less accidental. The essential characteristics of the boundary layer according to the approximate calculation described above are once more compiled with the exact solution in the table below. Characteristics of the Boundary Layer on the Flat Plate; Compari son of Approximate Calculation and Exact Solution Kind of calculation 6 E . 9 g

6" 9

$E

cw($-) u 2 1/2

Linear approximation 1.732 0.578 3.00 (fig. 43a) _ . 1.740 0.645 2.70 Cubic approximation (fig. 43b')

0.289 0.323

1.29

1.729 0.664 2.61

0.332

1.328

1 155 ._

__-

Exact solution (Blasius)

J

90

NACA TM NO. 1217

A s one can see f r o m t h i s t a b l e , the agreement, p a r t i c u l a r l y of t h e cubic approximation and the exact solution, i s r a t h e r good. On the whole, t h e r e s u l t s of t h i s calculation with t h e a i d of t h e momentum theorem may be regarded a s very s a t i s f a c t o r y , especially i n view of t h e simplicity, as compared t o the exact calculation.

Seventh Lecture (January 19, 1942) b.

The Momentum Theorem for t h e Boundary Layer with Pressure Drop (Plane Problem)

Last t i m e t h e boundary l a y e r on the f l a t p l a t e i n l o n g i t u d i n a l flow w a 8 caldulated by means of t h e momentum theorem. Today t h e general case of t h e boundary l a y e r with a pressure difference' i n t h e flow d i r e c t i o n w i l l be treated. One considers the flow along a curved surface, and measures t h e coordinate x a s a r c length along t h e surface; l e t y be the perpendiculm distance from t h e surface, U(x) t h e given p o t e n t i a l flow ( f i g . 44). The fundamental equation may be obtained by a momentum consideration as i n chapter IXc; now, however, the pressure d i f f e r e n c e has t o be'talren i n t o consideration. The same r e s u l t i s obtained by integration o f the equation of motion of t h e boundary l a y e r with r e s p e c t t o y from y = 0 ( s u r f a c e ) t o y = h, the l a y e r y = h l y i n g everywhere outside of t h e f r i c t i o n l a y e r ( f i g . 44). The d i f f e r e n t i a l equations of t h e boundary l a y e r f o r t h e steady case read, according t o equation (8.3),

with the boundary conditions: y = 0: u = v = 0; integration from y = 0 t o h gives:

y

=w

: u = U.

The

(10.28) J= ,0

JO

I n t h e first term the d i f f e r e n t i a t i o n with respect t o x and L e integrat i o n w i t h respect t o y may be interchanged, since the upper l i m i t h i s independent of x. On the l e f t side t h e second term i s transformed by integration by p a r t s :

NACA

vh

91

TM N ~ .1217

representing t h e transverse velocity outside of the boundary l a y e r .

By continuity,

av = - 2 8s

and

% = - IbuQ ax and hence :

I n s e r t i o n i n equation (10.28) gives, because of:

t

the relation:

Thia i s the s M a l l e d K a r m a n integral-condition, (reference 23).

/

/

first given by v.'Karmftn

For the Dresswe term one now introduces the p o t e n t i a l velocity V(x); furthermore, equation (10.31) i s t o be transformed s o t h a t t h e displacement thickness 5* and t h e momentum thickness 9 appear i n i t a s defined by equation (6.31) and (6.32), namely:

NACA TM No. 1217

92

( 10 321

u2q

=[

h

u

(u - u )

dy

According to Bernoulli ' 8 equation:

which c m also be written:

By substitution of equation (10.34) into equation (10.31) there results:

JO

JO

anclafter differentiation of the second term:

93

NACA TM N ~ .1217 1

The displacement thiclmess and t h e momentum thickness can now be 'introduced d i r e c t l y and one obtains:

t

or

I-

=

u2

g + (23 + 6*) u

du

,I

This i s t h e form of t h e momentum equation f o r t h e boundary l a y e r with pressure drop t h a t w i l l b e used as a b a s i s f o r f u r t h e r considerations. Since i n it 'I' i s q u i t e generally t h e s u r f a c e shearing stre88, equation (10.3%) must apply i n t h e same way t o t u r b u l e n t flow, too. We s h a l l come back t o t h a t l a t e r . For t h e s p e c i a l case of vanishing pressure 1

,

drop

3

0, equation (10.36) i s transformed i n t o equation (10.4) found

dx b e f o r e f o r t h e f l a t p l a t e i n longitudinal flow.

The f u r t h e r c a l c u l a t i o n of t h e boundary l a y e r on t h e basi13 of equation (10.36) i e performed f o r t h e laminar case according t o t h e method of Pohlhausen (reference 24) and f o r the t u r b u l e n t case according t o t h e method of Gruschwitz (reference 34) (chapter XVIII). c.

Calculation of t h e Boundary Layer According t o t h e Method of Karman4ohlhauserGHolstein

For f u r t h e r c a l c u l a t i o n it i s of importance t o f i n d a s u i t a b l e expression f o r t h e v e l o c i t y d i s t r i b u t i o n i n t h e boundary l a y e r u(x, y). According t o our understanding of t h e exact s o l u t i o n s of t h e d i f f e r e n t i a l equations of t h e boundary l a y e r t h i s expression must a t least s a t i s f y t h e conditions t h a t f o r y = 0: u = 0, and f o r y = - : u = U. Furthermore t h e d e r i v a t i v e au/ay m u s t vanish f o r l a r g e y. Moreover, v e l o c i t y p r o f i l e s with and without i n f l e c t i o n points must be possible, as they occur i n t h e pressure decrease and pressure i n c r e a s e region, r e s p e c t i v e l y . / au = 0 must be p p s s i b l e i n m d e r t o have E F i n a l l y , a p r o f i l e with

',

Wy=o

separation point r e s u l t from t h e approximate c a l c u l a t i o n . One chooses f o r t h e v e l o c i t y d i e t r i b u t i o n sn expresszor, of t h e form u (x, y ) = Uf(y/Gp), and s e t s , according t o Pohlhausen,

94

NACA TM No. 1217

for

v a l i d for

f(y/fjp)


0:

U

dx

= F(K) =

F(-0,1369)

=

1.523

(11.6)

The numerical value must be substituted for F ( K ) , if the form parameters are to remain constant at the values given by equation (11.1). From equations (11.6) and (11.4) follows therewith, for the constancy of the form parameter X = -10, the conditional. equation

0.1369

E= U'

*

1.523

or

For a > 11 the boundary layer can still bear the pressure increase; for a < 11 separation occurs; for a = 11 the boundary layer always remins with X = -10, on the verge of separation. Qualitatively, the following can be immediately said about the distribution of the potentialflow velocity U(x) which gives no separation. Because of equation (11.7) a necessary condition for avoiding separation in retarded flow is:

that is, a negative velocity gradient Ut must exist, the magnitude of which decreases in the flow direction. If, therefore; the curve U(x) in figure 63 is curved downward behind the maximum (U"< 0), separation occurs in every case; if it is curved upward (u"> O ) , separation sometimes does not OCCW. The limiting case U" = 0 for U' < 0 always leads to separation. The sufficient condition f o r avoiding separation is w"/rr'* >11.

NACA TM

107

1217

NO.

One now proceeds t o c a l c u l a t e what p o t e n t i a l flow and what boundary l a y e r thiclrness v a r i a t i o n correspond t o (I = +u. From equation (11.7) f 0l.lows :

.and after integration:

l o g U' = 11 l o g U -Ut= -

C;

a6 i n t e g r a t i o n constant.

-1u

-10

10

For

x = 0,

U(x)

s h a l l be

or

c:

.._I

with

- log Ci

Repeated i n t e g r a t i o n gives: = C'

x

1 U(x) = Uo,

+ cg

(11.8)

thus

Furthermore, one puts

ci

uo

10

=

c1

(11.10)

and obtains from equation (11.8) for the p o t e n t i a l f l o w

U(11.11)

(1 + 10 C1x)

0.1

Thereby i s found the desired velocity d i s t r i b u t i o n t h a t j u s t avoids separation. The constant C 1 can be determined from t h e boundary l a y e r thickness 6, at the i n i t i a l point x = 0:

NACA TM NO* 1217

108

I*

According t o equation (11.11)

U' =

-

uo (1 + 10 clx)

I

and thence 6

I

From 6 = 6,

for x = 0

=iE-

(1 + 10

1.1

clx) 0.55

follows

(11.12)

and thus, as the final solution f o r t h e p o t e n t i a l flow and t h e boundary l a y e r thickness variation,

u

=

uo 1 +

(

100

-

-0.1

*

u;>>

6 = 6, 1

(

+

100

t'lc

0.55

(11.14)

uopo2)

The permissible r e t a r d a t i o n ( v e l o c i t y decrease) i s t h e r e f o r e comparable 10 and i s thus very small. The v e l o c i t y i s t h u s very c l o s e t o to the c.onstant velocity of the f l a t p l a t e i n longitudinal flow. For t h e present case t h e growth of t h e boundary l a y e r thiclmess 8 must t h e r e f o r e be somewhat l a r g e r than for t h e f l a t p l a t e , where 6

6

x112

.

Here

thus t h e increase i s only s l i g h t l y l a r g e r .

The flow i n a divergent channel with plane w a l l s ( t w M i m n s i o n a l problem) will be t r e a t e d as another example. In f i g u r e 64 l e t x be t h e r a d i a l distance' from t h e o r i g i n 0. The w a l l s start a t x = a, where t h e entrance velocity of t h e p o t e n t i a l flow equals Uo. The p o t e n t i a l flow i s *Compare Chapter 7x e whsre it w a s found, as exact solution of t h e d i f f e r e n t i a l equation of the boundary layer, t h a t i n retarded flow -0.091 separation occurs when U(x) = u1

NACA TM

NO.

1217

109

u(x) U'

=

=

uo 2a

a - uo 2 X

a V' = 2uo X3

Thus Ut < 0 asd v"> 0 f o r a l l x so t h a t t h e necessary condition equation (11.7a) f o r avoidirq separation i s s a t i s f i e d . However, calculat i o n of t h e dimensionless number u according t o equation (11.5) gives a = 2

(11.16)

The s u f f i c i e n t condition f o r avoiding separation, a > 11 according t o equation (11.7) i s therefore violated. For t h e divergent channel with plane w a l l s separation therefore occurs f o r any included angle. This example shows especially c l e a r l y the l o w a b i l i t y of t h e laminar flow t o overcome a pressure increaee without separation. According t o a calculat i o n of Pohlhausen (reference 24) t h e separation point l i e s a t (x/a) = 1.212 A and thJs i s independent of t h e included angle a. b.

Various Technical Arrangements for Avoiding Separation

It i s a favorable circumstance f o r technical applications t h a t f o r higher Reynolds numbers t h e boundary layer does not remain laminar b u t becomes turbulent. The turbulence consists of an i r r e g u l a r mixing motion. By t h i s mlxing motion momentum i s continuously transported i n t o t h e l a y e r s near t h e w a l l , and t h e p a r t i c l e s retarded a t the w a l l a r e c a r r i e d out i n t o t h e f r e e stream and thus r e d c c e l e r a t e d . Because of t h i s mechanism the turbulent flow i s a b l e t o withstmd, without separation, considerably higher pressure increases than t h e laminar flow; thus t h e pressure increases e x i s t i n g i n technical flows a r e made possible. A f e w technicai p o s e i b i i i t i e s f o r avoiding separation w i i l be d i sCU6S8d. 1. Blowing. For a wing p r o f i l e the separation of the boundary l a y e r f o r l a r g e angles of a t t a c k (fig. 6 5 ) can be prevented by blowing air i n the flow d i r e c t i o n from a s l o t directed toward t h e rear. The v e l o c i t y for the l a y e r near the surface i s thus increased by the energy

NACA TM No. 1217

110

supplied and t h e true t h a t i n the large J e t energy I n order t o make kept Ermall. But vortices.

danger of separation i s t h e r e f o r e eliminated. It i s p r a c t i c a l execution n o t much i s gained, because of t h e required f o r any considerable improvement of t h e flow. t h e energy output small, t h e width of t h e j e t mtlst be then t h e j e t , soon after i t s e x i t , breaks up i n t o

2. Another p o s s i b i l i t y of avoiding separation i s t h e arrangement of a s l o t t e d wing according t o figure 66. The e f f e c t depends on t h e boundary l a y e r formed on t h e s l o t AB being c a r r i e d away i n t o t h e free stream, before it separates, by t h e flow through t h e s l o t . A new boundary l a y e r develops a t C which is, however, a t first very t h i n and reaches D without separation.

The same p r i n c i p l e i s used f o r t h e Townend r i n g and NACA cowling ( f i g . 67).

3. Suction. A f u r t h e r p o s s i b i l i t y f o r t h s prevention of separation i s suction. For t h e wing, f o r instance, t h e r e t a r d e d bounikry-layer material is. sucked off i n t o t h e i n t e r i o r of t h e w i n g through one o r several s l o t s ( f i g . 68). The point of suction l i e s s l i g h t l y ahead of or behind t h e separation point so t h a t no r e v e r s a l of t h e flow can occur. A new boundary l a y e r which a t first i s very t h i n develops behind t h e suction point and permits t h e pressure t o increase f u r t h e r . Ln t h i s manner one can overcome considerably l a r g e r pressure increases and a t t a i n higher values of maximum l i f t f o r t h e wing. Many d i f f e r a n t suction arrimgemsnts f o r increasing maximum l i f t have been i n v e s t i g a t e d by 0. Schrenk (reference 28). Values f o r ca max of 3 t o 4 were obtained.

c.

Theory of t h e Boundary Layer with Suction

Suction i s a very e f f e c t i v e means f o r influencing t h 3 f r i c t i o n l a y e r on a body i m e r s a d in a flow and p a r t i c u l a r l y f o r avoiding separation. This w a s pointed out f o r t h e first t i m e i n 1904 by I,. P r a n d t l i n h i s Fundamental work on t h e boundary l a y e r . A n o t h s p o s s i b i l i t y of a p p l i c a t i o n of suction, recognized only recently, i s t o keep t h s f r i c t i o n l a y e r laminar. Here t h e boundary l a y e r i s , by suction, kept so t h i n t h a t t r a n s i t i o n t o t h e t u r b u l e n t s t a t e of flow i e avoided. The surface f r i c t i o n drag i s thereby reduced. Experimental i n v e s t i g a t i o n s of t h i s e f f e c t were c a r r i e d out by Ackeret (reference 39). The laminar f r i c t i o n l a y e r with suction can a l s o be subjected t o a numarical treatmsnt which w i l l be b r i e f l y discussed. The following assumptions a r e made. f o r t h s calculation:

ll1

NACA TM NO. 1217

1. The suction is introduced into the calculation through the assumption that the normal velocity at the wall vo(x) is different from zero. The wall is therefore assumed to be permeable. A continuous distrfbution of the suction velocity vo(x) serves the purpose of numerical treatment best. 2. The suction quantities a r e so small that o n l y the parts in the immediate neighborhood of the wall a r e sucked from the

boundary layer. This leads to a very Small ratio of suction velocity vo(x) to f r e e stream velocity u0: vo/u0 = 0.001 to 0.01.

3. The n-lip

condition at the wall u = 0 is retained with suction, likewise the expression for the wall shearing stress

The equations of motion for the boundary layer with suction therefore read

uau+ - ?au -=u~+v ax aY dx

-1

a2U 'as2

with the boundary conditions y = 0

u = 0

v = vo(x)

(11.18)

y = m u = u

vo

.O blowing.

As in chapter X b the momentum theorem is again applied to the boundary layer with suction. The momentum equation for the boundary layer with suction is obtained in exactly the sane manner as in chapter X b (compare fig. 44) provided one takes into consideration, in addition, that the n o m 1 velocity at the wall is different from zero. In chapter X b the momentum equation was derived by integration of the equation of

NACA TM No. 1217

112

motion for the xdirection over y between the limits y = 0 and y = a' , (compare equation (10.28)). One imagines exactly the sane calculation performed for the boundary layer with suction: then the expression for the normal velocity at the distance from the wall y = h is different, compared with the calculation in chapter X b. The normal velocity now becomes

The remaining calculation is exactly the same as in chapter X b and finally yields as the momentum equation for the boundary layer with suction.

lp 1

I

TD=$d9+

(24

+

6+) u -dU -vou dx

dx

(11.20)

The newly added term -v U (compared with equation (10.36)) gives the loss of momentum due to ?he suction at the wall. We shall now treat the special case of the flat plate with suction in longitudinal flow (fig. 6 9 ) (reference 29). The free-atream velocity is Uo. Equation (11.20) then becomes (11.21)

if one takes the law for the laminar wall shearing stress into coneideration. Furthermore, the assumption is made that the suction velocity (or blowing velocity) -vo along the plate is constant. In this c a m one can obtain from the momentum equation (11.21), by the following ,simplecalculation, an estimate of the variation of the momentum thickness along the plate. One puts

(%>, B 9 =

uO

(11.22)

f3 2 0 signifying a dimensionless form parameter of the velocity profile. It may be assumed, to a first approximation, that j3 varies only little

113

NACA TM NO. 1217

with the length of run x; accordingly, Theniequation (11.21) may be written

l3

will be considered conqtant.

with the initial condition 9 = 0 for x = 0. For suction (vo < 0) one obtains dd/ctx = o for

B

9- =

V (suction) -vO

(11.24)

(that is, therefore, the momentum thickness reaches, after a certain approach leqth, a constant asymptotic value given by equation (11.24)). Simultaneously, displacement thickness, velocity distribution, and a l l other boundary layer coefficients also become asymptotically independent of x. (vo > 0) the value db/dx i s , according to equation (11.23),larger than zero along the entire plate; that is, 9 ( x ) increases with the length of run x without limit so that for large values of x, one can neglect in equation (11.23) the first term on the right side 8 s compared with the second. (Ine obtains therefore, as asymptotic law, For blowing

9 , =

%

x (blowing)

On the whole, one obtains the remarkable result that for the flat plate in longitudinal flow with constant auction or blowing velocity, the boundary layer thiclmese for suction becomes constant after a certain approach length, whereas for blowing, it increases proportionally to the length of run x. In between lies the case of the impermeable wall where the boundary layer thickness increases with F. For the case of the' l&n& boundary layer with the asymptotically constant boundary layer thichees it is also possible to give immediately an exact solution of the differential equations of the boundary layer in a surprisingly simple form. In this case &/ax E 0, hence also, according to equation (ll.l7), &/by I 0 and therefore

V(X,Y)

= vo = constant

(11.26)

114

NACA TM NO. 1217

Hence there follows from equation (11.17)

and from it the solution which satisfies the boundary conditions equation (11.18)

(11.28) F r o m this equation results the displacement thickness of the asymptotic boundary layer 6*m =

V -v 0

the momentum thickness

and the form parameter

6*0

-=

2. By comparing equation (11.29) with 9 , equation (11.24) one finds the factor l3 = 1. The velocity distribution of the asymptotic boundary layer profile according to equation (11.28) is plotted in figure 70 together with the Blasius solution for the impermeable wall.

Herewith the considerations of boundary layer with suction will be concluded. CHAPTER X I I . APPEN-DIX TO PART I a. Examples of the Boundary Layer Calculation

According to the Pohlhausen-Holstein Method For the integration of the differential equation (10.58) it is best t o ase the isocline method. It is expedient t o calculate with Aimensionless quantities. The arc length 6 is made dimensionless by

115

NACA TM NO. 1217

dividing by a characteristic length of the body immersed in the flow, for 'instance, for the wing, by the wing chord t. The variable Z = *82/V is made dimensionless by multiplying bg'

3.

Thus one puts:

t

(12.1)

Hence the differential equation reads: (12.2)

The calculated example concerns a symmetrical wing profile (J 015) in symmetrical approach flow (ca = 0). The prescribed potential-flow velocity and its first derivative with respect t o the arc length is given in tpble 6. The initial values f o r the integration are calculated, according to equation (10.60), to be, for the present case: Z*,

= 0.00

149

since at the stagnation point d2U/ds2 = 0. The auxiliary function F ( K ) required for the integration is given in figure 47-8 and table 5. The calculation according to the isocline method is shown in figure 48. Here the curve IC = -0.1567 which gives the separation point c a n be calculated according to the relation:

The intersection of'the integral curve with this curve gives the separation point. As a result of the integration one obtains at first the variation of the nomentum thicheee. By means of the ~ c t i o n6*/9 = f(K) and

5=

U P

f

(K)

given in table

5 one can

a l s o calculate the displacemant

116

NACA TM No. 1217

thiclmess and t h e shearing s t r e s s . The r e s u l t of t h e c a l c u l a t i o n i s compYled i n t a b l e 6 and given i n f i g u r e 49. Moreover, the v e l o c i t y d i s t r i b u t i o n i n tll;e boundary l a y e r can be seen from f i g u r e 56.

Translated by Mary L. Mahler National Advisory Committee f o r Aeronautice

\

NACA TM NO.

1217 BIBLIOGRAPHY* P a r t A. -Laminar Flows

1. Textbooks and Encyclopedias: 1. Prandtl-Tietjens: Bydro- und Aeromechanik, 1929 and 1931.

Bd. I and 11. Berlin,

2. Prandtl, L.: The Mechanics of Viscous Fluids. Aerodgnamic Theory, Bd. 111, 1935.

3. Tollmien, W.: Laminwe Grenzschichten. Bd. IV, T e i l I, Leipzig, 1931.

I n F. W. Durand,

Handbuch der Fxperimentalphysik,

4. Goldstein, S.: Modern Developments i n Fluid Dynamics. Oxford, 1938.

Vols. I and 11.

5. Maler, 'W. : Einf;;hrung i n d i e Theorie der z&en F l u s e i g k e i t e n Liepzig, 1932.

6. Lamb, H.

: Lehrbuch der Hydrodynamlk.

Teubner, Leipzig, 1907.

2. A r t i c l e s :

7. Prandtl, L.:

Ueber Fliissigkeitsbewegung b e i sehr k l e i n e r Reibung. Verhandlung 111. Intern. Math. Kongrees, Heidelberg,1904, represented i n "Vier Abhdnlg. z u r mdrodynamik", G t t i n g e n , 1927

8. Blaeius, H. : Grenzschichten i n Flusaigkeiten m i t k l e i n e r Reibung. Zs. Math. u. Phys. Bd. 56, p. 1 (1908). 9. Boltze, E. : Grenzschichten an Rotationskzrpern i n Flzssigkeiten m i t k l e i n e r Reibung. D i s s . Gb'ttingen 1908. Die Grenzschicht a n einem i n den gleichfGmigen 10. Hiemenz, K.: ( D i 8s. F1;'ssigkeit sstrom eingetauchten geraden f i e i szylinder G&tingen, 1911). Dingl. Polytechn. Journ. Bd. 326, p. 321, 1911.

.

11. Hamel, G. : Spiralf&mlge Bewegungen &.her Fliiasigkeiten. Jahresber. Mathem. Vereinigung 1916, p. 34.

i2. Szy~f~LnBki : Quelques solutions exactes des e'quations de l'hydrodynamique du r'luide visqueux dans l e cae d'un tube cylindrique. Journ. de math. puree e t appliqu6es. Reihe 9, Bd. 11, 1932, p. 67. See also: I n t e r n . C o w . of Mechanic6 Stockholm, 1930.

* On the whole, only the more recent publications, since about'1932, m e given here. The older l i t e r a t u r e can be found f o r instance, i n Prandtl's contribution t o W a n d : Aerodynamic Theory, vol. 111.

u8

NACA TM No. 1217

13. Schlichting, H.:

Berechnung ebener periodischer Grenzschichtstromungen. Physik. Zeitschrift, 1932, p. 327.

14. Schlichting,

H.:

Laminare Kanaleinlaufstr6mung. Zs. f. angew. Math. Mech. Bd. 14, p. 368, 1934.

U.

15. Howarth, L.: On the Calculation of Steady Flow in the Boundary Layer Near the Surface of a Cylinder in a Stream. ARC Rep. 1632, 1935.

16. Fr’dssling, Nils: Verdunstung, W&meiibergang

und Geschwindigkeiteverteilung b e l zweidlmensionaler und rotationssymmetriszher laminarer Grenzschichtstr’dmung. Lunds. Vniv. Arsskr. N. F. Avd. 2, Bd. 35, Nr. 4, 1940.

17. Homann, F.:

’Der Einfluss grosser aigkeit bei der Strcmung um den Zylinder und um die Kugel. Zs. f. angew. Math. u. Mech. Bd. 16, P O

153 (1936).

18. Howarth, L.: On the Solution of the Laminar Boundary Layer Equations. 1938, p. 547. Proc. Roy. Soc. Lond. A No. 919, ~01.164,

19. v.

Thesis Delft 1924, Burgers Proc. I. Intern. Congress appl. Mech. Delft 1924.

d. Hegge-Zynen:

20. Hansen, M.: Die Geschwindigkeitsverteilung in der Grenzschicht an einer eingetauchten Platte. Zs. f. angew. Math. u. Mech. Bd. 8, p.

185, 1928.

21. Goldstein, S.: Concerning some Solutions of the Boundary Layer Equations in Bydrodynamics. Proc. Phil. Soc. Vol. 26, I, 1936. 22. Schlichting, H.: Laminare Strahlausbreitung. Mech. Bd. 13, p. 260 (1933).

Zs.

23. v. K&m&, Th. : Laminare und turbulente Reibung. u. Mech. Bd. 1, p . 235, 1921.

f. angew. Math. u.

Zs. f. angew. Math.

24. Pohlhausen, K.:

Z u r n’herungsweisenIntegration der Differentialgleichungen der ldnaren Reibungsschicht. Ze. f. angew. Math. u. Mech. Bd. 1, p. 252, 1921.

Ein vereinfachtes Verfahren zur Berechnung lamiwer Reibungsschichten, die dem Niiherungsansatz von K. Pohlhausen genGen. (Not yet published.)

25. Holstein-Bohlen:

26. Tomotika, S.: The laminar boundary Layer on the Surface of a Sphere in a uniform stream. ARC Rep. 1678, 1936. 27. PretECh,

J.:

Die lamlnarre Reibungsschicht an elliptischen Zylindem Uinf3trzmmg. Luftfaihrtforschmg Bd. 18, p. 397, 1941.

una Rotationadlipsoiden bei symmetrischer

119

NACA TM NO. 1217 28. Schrenk, 0. : Versuche m i t Absaugefl$eln. p. 16.

Luftfahrtforschung 1935,

Die Grenzschicht m i t AbsaLuftfahrtforschung Bd. 19, p. 179, 1942.

29. Schlichting, H.:

The Plane J e t . P h i l . Ma@;., s e r e 30. Bickleg, W.: Apr. 1937.

31. Prandtl, L.:

und Ausblasen.

7, v o l 23,

Anschauliche und nutzliche Mathematik.

NO.

156, P. 727,

Vorleamg W. S .

1931/32* 32. S c h i l l e r , L.: Forschungsarbeiten auf dem Gebiet des Ingenieurwesens. Heft 248, 1922.

33. R e s u l t s of t h e Aerodynamics Test I n s t i t u t e a t Gottingen. V e r e u c h s a n ~ t a l tzu Gijttingen. 2. Lieferung, p. 29.

Aerodynamischen

34. Gruschwitz:

Turbulente Reibungsschicht i n ebener Str(c;mung b e i Druckabfall und Druckanstieg. 1ng.-Arch. Bd. 2 (1931).

35. Schlichting, H., and Ulrich, A , : laminar

- turbulent.

Zur berechnung des Umschlagee (not y e t published).

36. BUSS-,

Die laminare Reibungsschicht an Joukowsky--Profilen. K.: (not yet published).

37. Falkner, V. M.,

and Skan, S. W.: Same Approximate Solutions of t h e Boundary Layer Equations. R. & M. No. 1314, B r i t i s h A.R.C.,

1930

9

38. Hartree, D. R.:

On an Equation Occurring i n Falkner and Skan's Approximate treatment of the Equations of t h e Boundary Layer. Carnbridge P h i l . SOC.,Bd. 33 (1937).

39. Ackeret, J., Pfenninger, W.,

Verhinderung des Turbulentwerdens Kas, M.: e i n e r Reibungsachicht durch Absaugung. N a t u r w . 1941, p. 622.

NACA TM No. 1217

120 TABU I.

- TEE

FUNCTION Cp

FLOW (ACCORDING TO

OF THE PLARE STAGNATIOX POINT

HIEMENZ (RB3DE3NC.X 1 0 ) ) ; TO FIGURE 16

-

-! --

'p

'p'

0 0.1 0.2

0

0

0.0060

0.3

0.0510

0.1183 0.2266 0.3252 0.4144 0.4946 0.5662 0.6298 0.6859 0 7350 0 7778 0.8149 0.8467 0 8739 0.8968 0.9161 0.9324 0 9457 0 9569 0 9659 0.9732 0.9792 0.9841 0.9876 0 9905 0.9928 0.9946 0.9960 0.9971 0 9979 0 9985 0 9988 0 9992 0.9994 0 9996 0 9997 0 9998 0 9999 0 9999 0 9939 1.oOoo

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

1.3

1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2

2.3

2.4 2.5 2.6 2.7 2.8 2.9 3.0

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3 -9 4.0 4.1 4.2 4.3

0.0233

0.0881

0.1336 0.1867 0,2466 0.3124 0 3835 0.4592 0 5389 0.6220 0.7081 0.7966 0.8873 0 9798 1.0738 1.1688 1.2650 1.3619 1.4596 1 5577 1.6563 1 7552 1.8543 19537 2.9533 2.1529 2.2528 2.3525 2.4523 2.5522 2.6521 2,7521 2.8520 2.9520 3 0519 3.1518 3.2518 3 3518 3.4518 3 5518 3.6518

-

'p"

1.23264 1.1328 1.0345 0.9386 0.8463 o 7583 0.6751 0 5973 0.5251 0.4586 0.3980 0.3431 0.2937 0.2498 0 2109

0.1769 0.1473 0.1218 0 0999 0.0814 0.0658 0.0528 0.0420 0.0332

0.0260 0.0202

0.0156 0.0119 0.0091

0.0068 0.0051 0.0036 0.0027 0.0023 0.0019 0.0014 0.0010

0.0008 0.0004 0.0003 0.0002

1,0000

0.0001

1.0000

0.0001 0

1.0000

NACA TM No. 1217 TABIZ

II. - "E FUNCI'ION f OF THE BOUNDARY LAYER

ON

TBE FUU! I?LA!TE

IN LONGITUDINAL FLOW (ACCORDING TO BLASlTTS (REFERENCE 8)) ; TO FIGURE

30 7

0.4 0.6 0.8 1.0 1.2

1.4 1.6 1.8 2.0 2.2

2.4 2.6 2.0 3.0 3.2 394

3 *6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2

5.4 5.6 5.8 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8

YE; rr,

0

0

0.00664 0.02656 0.05974

0.06641 0 13277 0.19891C 0.26471 0.32979 0,39378 0.45627 0.51676 0 57477 0.62977 0.68132 0 72899 0.77246

0.10611 0 16557

0.23795 0.32298 0.42032 0 52952 0.65003 0.78120 0.92230 1.07252 1 23099 1.39682 1.56911 1.74696 1.92954 2.11605 2.30576 2.49806 2.69238 2.88826 3.08534 3 28329 3.$8189 3.68094 3.88031 4* 07990 4.27964 4.47948 4.67938 4.87931 5 07928 5.27926 5 47925 5.67924 5.87924 6,07923 6.27923 6.47923 6.67923 6.87923 7,07923

=

f'(7)

f'

f

7

0 0.2

=

-

0.81152

0.84605 0.87609 0 90177 0.92333 0.94112 0 95552 0.96696 0.97587 0.98269 0.98779 0 99155 0.99425 0.99616 0.99748 0 * 99838 0 99898 0.99937 0 99961 0 99977 0 * 99987 0.99992 0 99996 0* 99998 0 99999

-

-

1 00000 1.00000 1.00000 1.00000 1.00000 1.00000 e

f"

0.33206 0.33199 0.33147 0.33008 0.32739 0.32301

0.31659 0 30787 0 29917 0.28293 0.26675 0.24835 0.22809 0.20646 0,18401 0.16136 0.13913 0.11788 O.og809 0.08013 0.06424 0.05052

0.03897 0.02948 0.02187 0.01591 0.01134 0 00793 0.00543 0.00365 0.00240 0.00155 0.00098

0.00061 0.00037 0.00022 0.00013

O.OOoO7 O.OOOO4 0 00002 i

0.00001 0.00001 0 0 0

122

NACA TM No. 1217

f:

f

'1

3

0 0.1 0.2

0 0 * 0035 0.0132

0

0.3

0.0282 0.0476 0.0705 0.0962 0.1240 0.1534 0.1838 0.2149

0.1734

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

1.3

1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2

2.3

0

0

0.0114

0.4 0.6 0.8

0.0405

0.1072 0.1778 0.2184 0.2367 0 * 2399 0.2342 0.2239

0.2ll2

0.1530 0.0292

0.2462

0.3140

0.2776 0.3088 0.3397

0.3132

0.0028 -0.0173

0.3107 0.3070

-0.0320 -0.0420

0.3702

0.3025

0.4002 0 * 4297 0.4587 0.4871 0.5151 0.5426 0.5698 0.5966

0.2947 0.2923 0.2871 0.2822 0 2775 0.2733 0.2695 0.2662 0.2632 0.2607 0.2586 0.2568 0.2554 0.2542

-0.0482 4.0513 4.0518 4.0506 -0.0480 -0.0444 -0.0402 -0.0358 4.0314 -0.0271

0.6230

3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4

0 0.2

0.3125

0.6492 0.6752 0.7010 0.7266 0.7520 0.7774 0.8027 0.8279 0.8531 0.8782 0.9033 0.9284 0* 9534 0.9785 1 0035 1.0285 1.0535 1.0785

3.2

0.2444 0.2688 0.2869 0.2997

0.7244 0.6249 0.5286 0.4375 0.3539 0.2780 0.1037 0.0626

2.4

3.0 3.1

0.2129

'1

3

0.3080

2.5

2.6 2.7 2.8 2.9

0.0675 0.1251

f"

-

1.1035

1.1285

--

0.2533 0.2525

0.2519 0.2515 0.2511

1.0 1.2

1.4 1.6 1.8 2.0 2.2

2.4 2.6 2.8 3.0 3.2

3.4 3.6 3.8 4.0 4.2 4.4

g5 - -~

0.0806

0.1264 0.1742 0.2218 0.2676 0.3112

0.3526 0.3918 0.4293 0.4655 0.5007 0.5352 0.5692 0.6030 0.6365 0.6700 0.7034 0.7368

__ g;

0.2123 0.2012

h

h'

h"

5

0 0.2

0

0

0.0017

-0.0131

0.4 0.6 0.8

0.0045

0.0141 0.0117

1.0 1.2

-0.0012 -0.OogO

1.4 1.6 1.8

-0.0185 -0.0286 -0.0384 -0.0472 -0.0546 -0.0604 -0.0649 -0.0681 -0.0703 4.0717 -0.0726 4.0732 4.0735 -0 0737 -0.0738 -0.0738

4.0024

0.2506

-0.0019

0.2504

-0.0014

2.0 2.2

0.2503

-0.0011

0.2502 0.2502 0.2501 0.2501 0.2500 0.2500

4.0008 -0.ooo6 -0.0004

2.4 2.6

-0.0003 -0.0002 -0.0001

--

2.8 3.0 3.2

3.4 3.6 3.8 4.0 4.2 4.4

4.0522

0.8035

0.no1

4.0194 4.0160

-0.0032

4.0431 -0.0567 -0.0580 -0.0432 4.0335 4.0245 -0.0171 4.0ll4 -0.0072 4.0043 4.0026

5

0.2508

-0.0106

0.1916 0.1839 0.1781 0.1740 0.1712 0.1694 0.1682 0.1676 0.1672 0.1669 0.1668 0.1667 0.1667

4.0230

-0.0106 -0.0085 -0.0067 4.0052 -0.0041

g;

0.6348 0.4402 0.2717 0.1408 0.0483

0.0057 0.0039

-0.0010

-0.0176 4.0330 4.0441 4.0498 -0.0503

4.0468 4.0406 -0.0331

4.0257

-0.0015 4.0010 4.oO04 4.OOO1

-0.OO01

5 0.1192 0.0249 -0.0436 -0.0783 4.0833 -0.0680

-0.0423 4.0149 +0.0088 0.0256 0.0351

0.0380 0.0361

-0.0189

0.0312

4.0133

0.0249 0.0187

4.0089 4.0058 4.0036 -0.0021 -0.0012

4.ooo6 -0.0003 -0.0001

~-

0.0132

0.0089 0.0057 0.0036 0.0022 0.0012

0.0007

NACA TM No. 1217

T B L E IV.

- TEE

FUNCTIONS

F(y/$)

and

G(y/Gp)

FOR THE VELOCITY

DISTRIBUTION IN THE BOUNDARY LAYER ACCORDING TO POHLHAITSEN

(REFERENCE 24) AND HOWARTH (REFERENCE 15)

0 0.1 0.2

0 0.1981

0.3 0.4

0.5541

0.5

0.8125

0.6 0.7 0.8 0.9 1.0

0.3856 0.6976 0 8976

0.9541

0.9856 0.9981 1

0

0.01215 0.01725 0.01715 0.0144 0.0104 0.0064 0.00315 0.00105 0.00015 0

124

NACA TM No. 1217

TABLE V.

- AUXILIARY

FUNCTIONS FOR THE BOUNDARY

CALCULATION

ACCORDING TO HOLSCEXR (REFERENCE 25) f

K

( K )

=

E

f ( 2

=

K )

5's

2.305

u u 0.345 0.351 0.354 0.356 0.354 0.351 0.346 0.340 0.338 0.337 0.335 0.333

D

2.308

0.332

Q .0021

2.309 2.312 2.314 2.316 2.318

1

0.0767 0.0760 0.0752 0.0744 0 * 0737 0 * 070.0721 0.0713 0.0706 0.0697 0.0689 0 0599 0.0497 0.0385 0.0264 0.0135

0

0

0.4698

1

0.0941 0.6948

-0.0657 -0.0814 -0.0913 -0.0946

0.0941

-0.0911

9 8 7.8 7.6 7.4 7.2

0.0920 0.0882 0.0831 0.0820 0.0807 0.0794 0.0780

4.0806 -0.0608

7 * 052

0.0770

7 6.9 6.8 6.7 6.6 6.5 6.4 6.3 6.2 6.1 6 5 4 3

15

0.0885

14 13

0.0920

12 1 1 10

_____

-

-0.0332

-0.0271 -0.0203 -0.0132

4.0051

0.0061 0.0102

0.0144 0.0186 0.0230

0.0274 0.0319 0.0365 6.0412 0 .Ob59 0.0978 0 * 1579 0.2255

79

2.279 2.262 2.253 2.250 2.253 2.260 2 * 273

2.289 2.293 2.297

2.301

0.331 0.330

0.330 0.329 0.328 0.327 0.326 0.325 0.324

2.321 2.323 2.326

2.328 2.331 2.333

0.322 0 . 3 ~

2.361 2.392 2.427 2.466 2.508

0.310 0.297 0.283 0.268 0.252

-10 -11

2.604 2.658 2.716 2.779 2.847 2.921 2.999 3.084 3.177 3.276 3 * 383

0.97

-0.1255 -0.1369 4.1474

0.5633 0.6616 0.7640 0.8698 0.9780 1 0853 1.1981 1.3078 1.4173 1.5231 1.6251

-12

-0.1567

1* 7237

3 * 500

-13

-0.1648 -0.1715 -0.1767

1.8159 1.9020 1.9821

3.627 3.765 3.920

2

-0.0140 4.0284 -0.0429 -0 * 0575 -0.0720 -0.0862 -0.0999

-1 -2

-3 IC -5

-6 -7 43 -9

-14 -15

.

-0.1130

0.3000

0.3820

0.19

0.179 0.160 0.140 0.119 0.100

0.079 0.059 0 039 0.019 0

I

0

-0.019 -0.037 -0.054

NACA TM

NO.

1217

I

I

I

I

I

nl

I

I

126

NACA TM No. 1217

Y

U

I

c t //////////;///////////’

-

Figure 1.

Simple shear flow.

Figure 2. - Hagen-Poiseuille’s pipe flow.

+-c

Figure 3.-

a p 6d

ax

The general stress tensor.

X

NACA TM

NO.

1217

Figure 4.- The shearing stress (to fig. 3).

c F

Figure 5.-

The deformation of a pure elongation.

Y ’

1

L

X

Figure 6.- Pure angular deformation (e > 0).

NACA T M No. 1217

128

M. x

Figure 7.-

Figure 8.-

Pure angular deformation (f > 0).

Analogy between heat boundary layer and flow boundary layer.

x

*t Figure 9.-

-

Types of solutions of the Navier-Stokes differential equations.

NACA TM NO. 1217

Y

#

-

Figure 10.- Reynolds’ law of similarity.

Figure 11.- Laminar pipe flow.

NACA TM No. 1217

1.0

.a .6 .4

.2

0

U

-.2

-.4 -.6

-.8 -/.

0 X

Figure 12.-

Velocity profiles of the starting pipe flow

( T

=$).

NACA TM No. 1217

-

WU,

Figure 13. - Velocity distribution on an oscillating surface.

NACA TM NO. 1217

i ///////,

-u,

U

Figure 14.- Velocity distribution on a surface s e t suddenly in motion.

Figure 15. - The plane stagnation point flow.

NACA TM NO. 1217

Figure 16.- The velocity profile of the plane stagnation point flow-

Figure 17.- The convergent and divergent channel.

Figure 18.

-

Velocity distribution in the convergent channel.

NACA TM No.

.

Reverse flow Figure 19.-

Velocity distribution in the divergent channel.

Figure 20. - Separation in the divergent channel.

Figure 21.-

Viscous flow around a sphere.

1217

NACA TM NO.

1217

Figure 22. - Streamline pattern of the viscous flow around a sphere (according to Stokes).

Figure 23.-

Streamline pattern of the viscous flow around a sphere (according to Oseen).

Figure 24. - Concerning Prmdtl's bottn,dary-layer equation. (Boundary layer thickness 6 magnified.)

NACA TM No. 1217

Y

t

n

X

0

Y

t Figure 25.-

Separation of the boundary layer. (A = point of separation.)

NACA TM

I

NO.

1217

Y

Figure 26.-

Y

Y

Velocity distribution in the boundary layer for pressure decrease

(g < 0).

Y

Figure 27.-

Velocity distribution in the boundary layer for pressure increase

(g

> 0).

I

X

Figure 28. - Concerning the cakdatiorr of the friction drag.

NACA

Figure 29.-

0

Figure 30.-

TM No. 1217

The boundary layer on the flat date in longitudinal flow.

7

2

4

5

6

Velocity distribution u(x,y) in the boundary layer on the flat plate (according to Blasius).

NACA TM NO. 1217

Figure 31.-

Figure 32.-

The transverse velocity v(x,y) in the boundary layer on the flat plate.

The boundary layer on a cylindrical body of arbitrary cross section (symmetrical case).

I

140

Figure 33.-

NACA TM No. 1217

The function fg’ of the velocity distribution in the boundary layer.

NACA TM NO. 1217

141

Circular cylinder

Arc length

x

Figure 35. - Velocity distribution of the potential flow for a wing profile.

Figure 36.-

Concerning the calculation of the friction layer on the circular cylinder.

Y

k

Figure 37.-

Wake flow behind the flat plate in longitudinal flow.

142

NACA

TM No. I217

R

Figure 38. - Concerning application of the momentum theorem for the flat plate in longitudinal flow.

LO

Figure 39.- Asymptotic velocity distribution in the wake behind the flat plate in longitudinal flow.

NACA TM

NO.

1217

I

Figure 40.-

Streamline pattern and velocity distribution of the plane jet.

Figure 41.-

The velocity profile of the plane jet.

144

NACA TM No. 1217

1 A

i E:

0

.I+

4

cd

JJ

rn

t \--

\

NACA TM no. 1217

145

Figure 42.-

Application of the momentum theorem for the flat plate in longitudinal flow.

Figure 43.-

Velocity distribution in the boundary layer on the flat plate in longitudinal flow.

(a) Linear approximation. (b) Cubic approximation f o r the velocity profile.

Figure 44.- Application of the momentum theorem to the boundary layer with pressure gradient.

NACA TM No. 1217

146

0

0.2

0.4

-

0.6

48 Y/6P

Figure 45. - The universal functions F(y/tjp) and G(y/s p) for the velocity distribution in the boundary layer according to Pohlhausen.

I

I

I

1

0.4

Figure 46. - The one -parameter family of velocity profiles according to Pohlhausen.

Figure 47(a).- Auxiliary functions of the boundary layer calculation according to Holstein (cf. table 5); A and F(K) against K .

I

148

NACA TM No. 1217

Figure 47(b). - Auxiliary functions f o r the boundary layer calculation according to Holstein (cf. table 5); f l ( IC)and f Z ( K ) against IC.

B.At1.A TM No.

-ha./

1217

1

0 gob--

q03-

z02-I I

I I

I

I

I I

I I

I

I I I

Figure 48. - Integration of the differential equation of the boundary l a y e r according to Pohlhausen and Holstein (profile J 015; ca = 0).

NACA TM No. 1217

150

2.0

t5

I

.----_ /

/

/-

I

--tI

-z

05

-05

- 10

Joukowsky profile d/t = 0.15

Figure 49.- Result of the boundary-layer calculation for the example according to figure 48 (profile J 015; ca = 0).

NACA TM NO. 1217

7

2

u=u, x

b) Plate profile

u=U,

-

3

Y/6*

a) Stagnation point profile

1

-

3

2

Y/6*

Figure 50. - Comparison of the approximate calculation according to Pohlhausen with the exact solution.

NACA TM No. 1217

Figure 51. - Potential-theoretical velocity distribution on the elliptic cylinders with axis ratio al/bl = 1, 2, 4, 8 for flow parallel to the major axis (A = laminar separation point), t' = half the circumference.

NACA TM

NO.

1217

6* c

amr

f

Separation point according t10 P4

-5/t'

Figure 52.-

Result of the boundary-layer calculation for the elliptic cylinders of axis ratio a l / b l = 1, 2,4, 8.

NACA TM No. 1217

a 4 0 w

;I

@ -1% 01 .r(

4 .r(

0

a

c

0 -r(

4

0

a,

0

+J M

G

.r(

a k

d

0 0 0

H

d

II

a

G

a

a,

+-'

3

3

8

.-i

...

4-

C)

cd

C)

k

a, 3, cd 4 G

0

c,

.r(

V

.d

k

b-l

NACA TM NO. 1217

Figure 5 4 . - Potential-theoretical velocity distribution for the Joukowsky profile J 015 for c, = 0; 0.25; 0.50; 1.0.

156

NACA

Suction side

TM No. 1217

Pressure side

6" t

0

0

Of

0.2

03

Q4

05 0-

Figure 55.- Result of the boundary-layer calculation for the Joukowsky profile J 015 (t' = half the profile perimeter).

NACA TM NO. 1217

h

m * 0-

v) .r(

4

*

--ro -wc _

I

’H

0

NACA

c

0

4

CI

h

5130

TM NO. 1217

NACA TM

NO.

1217

159 Q

m d

P

m m

a Y

r( U

a

$ a a,' a 3 r-l

a

0

d

6

NACA

TM No. 1217

I

NACA TM NO. 1217

I

162

NACA TM No. 1217

NACA TM ~ o 1217 .

,s I

I

Figure 61.

-

Danger of separation

Pressure

Pressure distribution and separation on a wing.

/ -

Figure 62.-

Boundary layer with laminar separation avoided.

uo times no separation

6

-

-

X

Figure 63.- Potential flow with separation: U’< 0; U” e 0; sometimes without separation: U’ < 0; U” > 0.

164

NACA TM No. 1217

Figure 64. - Divergent channel.

Figure 65.- Prevention of separation on wing by blowing.

Figure 66.- Prevention of separation by a slotted wing.

Figure 67.- NACA cowling f o r prevention of separation.

NACA m NO. 1217

Figuse 68.-

165

Prevention of separation on wing by suction.

Y

h=COn5t.

Figure 69.-

Flat plate in longitudinal flow with suction.

7

t

-

With suction

I

44

Figure 70.-

Asymptotic velocity profile on flat plate in longitudinal

flow with suction (I) ti* = v/-v,

(11) 6

*=

.

1.73 "0