CHAPTER - II

APPROXIMATE METHODS FOR THE

SOLUTION OF THE

BOUNDARY LAYER EQUATIONS

31

1. Introduction s

If we are interested to study the practical applica­tion of the boundary layer theory, then we have to solve the differential equations of the boundary layer theory which are nonlinear parabolic differential equations. These equations are very difficult and lengthy to be solved. There­fore, it becomes essential to devise approximate methods which may be much quicker to apply, although less accurate but may yield acceptable results.

One of the earlier and most widely used approximate methods is the Karman-Pohlhausen method. It is based on the momentum integral equation of Karman.

J.L. Bansal studied the 'Asymptotic Suction temperature profiles in a laminar boundary layer over a porous flat plate.1 In this note it has been shown that in

,Uvj£Tthe case of laminar boundaryjpver a flat plate with homogeneous suction as we have the 'asymptotic suction velocity profiles' there exists also 'asymptotic suction temperature profiles' for various values of the Prandtl number (Pr) . The recovery factor in such a case is found to be independent of Pr and has a constant value 1.

G.N.Sarma and D.P.Singh [_24J investigated 'The

effect of viscosity-temperature law in unsteady boundary

32

%

layer on a flat plate'. They studied the effect of viscosity- temperature law when the wall is an arbitrary motion with steady main stream velocity, Prandtl number being unity.The dependence of coefficient of viscosity p on temperature T is assumed to be

Tw+1\i = Constant -----

T + C

where w and C are constants.This is the generalization of the work of Sarma (1965) from a linear law to the more realistic nonlinear law between viscosity and temperature.

R.P.Agarwal [lj studied 'nonlinear two point boundary value problem'. In this problem he obtained existance and uniqueness of the solutions of third order nonlinear differential equation with boundary conditions prescribed at two points.

G.C.Paini and R.N.Atolia [jL8j considered the problem to study the plane couette flow of waiters B" liquid with uniform suction at the stationary plate.

D.P.Singh ^25j investigated the 'compressible boundary layer flow in a converging canal.' In this paper he studied two-dimensional compressible boundary layer flow

33

in a converging canal when the main stream is steady and

the body is moving with velocity.

Sutapa Mukherjee and R.C.Choundary ^26~J studied

'Magnetohydrodynamic boundary layer with suction along a

porous wall in Howarth's flow'. In this paper he obtained

solution for the laminar magnetohydrodynamic boundary layer

of an incompressible fluid of small electrical conductivity

along a porous wall in Howarth's flow with uniform suction.

S.Prasad jj.9j investigated 'Boundary layer with

suction along the porous wall in Tani's flow'. In this paper

an investigation has been made into laminar incompressible

boundary layer with continuous suction along a porous wall in

Tani's flow for which the potential flow velocity is given

x2by U (x) = U (1 - —-- ) . Following M.R. Head UE0 the

o

momentum and the kinetic energy integral equations have been

used by S.Prasad with the aid of an 8th degree polynomial

velocity profile to obtain a step-by-step numerical solution.

Ram Deo Matho [22j studied the 'Boundary layer with

suction over a porous elliptic cylinder'. In this paper the

method suggested by Head has been used. The momentum and the

kinetic energy integral equation for two dimensional boundary

layers have been rederived and have been used with the aid

of the Schlichting's velocity profiles to obtain a step-by-

34

step, solution for the boundary layers with suction in the

region of adverse pressures gradient over a porous elliptic

cylinder with the ratio of the major to minor axis is four.

R. Sharma |27j explained 'A two parameter method

for calculating the two dimensional boundary layer with

suction or injection'. In this paper he applied the method

developed by Curie for flows without suction to the flows

with suction or injection. Detailed calculations of the

boundary layer parameters made by this method indicate that

the errors are within 5% of the exact values.

R. Sharma |_28J obtained the 'Exact solution of the

incompressible laminar boundary layer equations with zero

pressure gradient and variable suction'. In this paper a

numerical solution of the boundary layer equations with zero

pressure gradient and a general distribution of suction

velocity as a power series with positive power of x, has been

obtained. A set of differential equations has been very

accurately integrated numerically and principle characteristics

of boundary layer flow have been determined.

through narrow slots'. In this investigation an analytical

treatment is given by him to the laminar boundary layer on a

flat plate with a narrow suction slot. When this flow is

35

expanded in a certain perturbation series about the Blasius solution, it becomes possible to determine the first-order solution in integral form. This solution tends uniformly to the Blasius solution as the suction velocity or slot width tends to zero. An approximation method is obtained for the wall shear stress distribution in terms of the incomplete Gamma function and the comparisons are made between the results of the present analysis and Rheinboldt's numerical solutions. Further it is found that the first order solution is quite accurate if the suction parameter is small.

Kizlov, L.F. and Losinska.ja, T.I. j_ll~j investigated 'Integration on an electronic computer of the boundary layer equations of Tani's flow in the presence of suction*. In this investigation they obtained results by using power series for the stream function, numerical integration method and the system of differential equations of the boundary layer for a flow of Tani type in the presence of suction.

Husnutdinova, N.V. studied mathematical problems in the boundary layer control by means of suction.

Sutapa Mukherjee £29] investigated 'Magnetohydrodynamic boundary layer with suction along a porous wall in a sink flow'. In this paper an attempt has been made to find the solution of the MHD boundary layer equations for the two

dimensional steady laminar flow of a viscous incompressiblefluid of small electrical conductivity in the presence of anexternally applied transverse magnetic field along a porouswall placed radially in a sink flow. Suction velocityV (x) 1/x and transverse magnetic field tL(x) l/xo ohave been imposed and a similar solution has been obtained with the help of similarity substitution. The solution will be valid after a certain distance from the leading edge of the wall in the sink flow. A solution valid at a large distance from the wall and a series solution valid near the wall have been obtained and the two solutions have been joined at a suitable point.

Holt# M. and Modarress, D. ^73 studied the 'Application of the method of integral relations to laminar boundary layers in three-dimensions'. It has been traditional among fluid dynamicists to employ some numerical means (such as the finite difference techniques) to solve the two-dimensional nonlinear compressible boundary layer equations. But as an alternative to this numerical procedure, the bouhdary layer equations have been more successfully solved in an integral form, for example, with the classical Karman-Pohlhausen momentum integral method. The main principle of the method of integral equation is based on the idea of representing the streamwise velocity gradient (normal to the wall) as a simple

37

algebraic function of the streamwise velocity itself. These

authors extended the method of integral relations to the

problem of three-dimensional compressible boundary layer

flows with and without separation. By reducing the equations

of motion to a quasi-incompressible form they solved the

resulting hyperbolic partial differential equations.

In this Chapter in Section 2 we have studied the

boundary layer with suction along the porous plate by using

Tani's flow. In Section 3 we made an attempt to study the

laminar incompressible boundary layer along the porous

circular cylinder. In Section 4 of this chapter we investi­

gated the boundary layer flow Tani's flow past a wedge.

Section 5 deals with the study of boundary layer Tani's

flow along the wall of convergent Channel, Lastly in Section 6

we have given the list of major developments in the field

of application of approximate method to the bodies of

revolution and three dimensional boundary layer and made

discussions on these problems.

Before discussing Section 2 we enlist some basic

equations and the characteristic boundary layer parameters

required for our problems to be discussed.

(a) Karman momentum Integral equation :

The Prandtl boundary layer equations for a steady

38

two-dimensional incompressible flow are

^ u "“b uu---- + V---

~B x b ydU "B 2u

u — + y —- ... (i)dx B y^'B u 'B v———— + —— = 0 ... (2)x "By

With the boundary conditions are

y = 0 : u = v = 0y = 6(x) s u = u(x) ... (3)

where 6(x) is the boundary layer thickness.

Due to equations (2) and (3) equation (1) can be reduced to the form

2 d$2 ^ T.U --- + C26o + 6-.) U--- = --- ... (4)dx 2 dx ^

This equation is known as the "Kantian momentum integral

equation" for two-dimensional steady incompressible boundary layer, where

6Si ~So (1 u

U- ) dy, (displacement thickness)

6 26 u uC --- (1 - - ) dy, (momentum thickness)o U u

39

,~T, ^ uand Id = u (---- ) ,

b y °(shearing stress on the wall)

density of the fluid

(b) Ka'rma'n-Pohlhausen Method :

We can write the velocity profile as

u

Uf 01)

4z:i=o

o 41! 4. iu-- =1 / for T| ^ 1U

where T| = y/6 ... (5)

To calculate, ag, a^, a2t ag and a4 Pohlhausen

prescribed the following five boundary and compatibility

conditions*

^ 2u dU0IIg•1 V ( -—2 ) = - U----b y dx

^ u 2u* u = U, (^T)s = 0, ( ---* ) c^ y2 6 0 ...(6)

The coefficients ag to a<j. by these conditions are

given by

3g — 0, a^ ~ 2 + A _ A----- / So — ~6 2

a~, = - 2 + , and a„ = 1 - -~0 2 ^ 6 (7)

■ f p

AM A

40

62 dUwhere A- = -- --- (shape factor) ... (8)v a*

Hence the fourth degree velocity profile which satisfies all the boundary conditions in (6) is given by

u—. = f(rj) = fOi) + A G(rj) ... (9)U

where F(t]) = 2t| - 2r)3 + T]4 ,

n 3and G (rj-> = - ( 1 - T|) ,6

The shape factor A is limited to the range

- 12 C A. 4 12 ... (10)

In view of equation (9) the vailues of 6^# 62 and are given as follows j

Displacement thickness (61)

5 u61 = J (1 - 5 > dYO u

3 A= 6 ( - *>- )10 120

Momentum thickness (62) $

315 945 9072

(11)

(12)

41

Shearing Stress ( 'to ) s

to = ]i, ( ----)

, A--- ( 2 + -£-)6 6

(13)

Holstein and Bohlen ^loj solved the Karman momentum

integral equation by considering momentum thickness62 as unknown function instead of boundary layer thickness 6.For this purpose they wrote the Karman momentum integral equation in the form

U62“T

d6 2 , . 62----- + ( 2 + —-dx

) — V62 dU

dx p, u- (14)

This equation had been simplified by introducing the following parameters.

H (/\ )tl

6 2

1(A) to ^2)XU

. (15)

. (16)

.. (17)

5199A

42

In view of equations (15) to (17)# equation (14) reduces to the form

& 61 L(A)--- ( —£- ) = ----- ... (18)dx U

where L( \ ) = 2 + (H + 2 ) j ... (19)

Equation (18) is known as reduced form of Ka’rman momentum integral equation.

(C) Application of Karman-Pohlhausen method s

(i) Boundary layer over a flat plate :dU

In this case U(x) = constant. Therefore, --- = 0dx

Hence, A = 0 and equation (18) reduces to

d L(0) 0.4693---( —) =---- = ----- ... (20)dx \) U U

The solution of equation (20) with the initial value 0 at x = 0 gives the momentum thickness

0.685 v U (21)

and therefore, from equations (16) and (21), the displacement thickness 5^ is given by

43

61 = 1-75 \Mj- ... (22)

and from equations (17) and (21) the shearing stress

on the plate is given by

= 0.343 ]j,UU

IX

. (23)

(ii) Two-dimensional stagnation point flow s

In this case of two-dimensional stagnation point

flow the potential flow velocity is given by

U (x) = ax ... (24)

Therefore/ U'(x) = a and U*' = 0, we have

l)0.077

a

Thus, the momentum thickness is given by

62 = 0.278VV

... (25)

(26)

The displacement thickness 61 from equations (16) and (26)

is given by

V61 = 0.64l\/---

a(27)

and the shearing stress on the wall from equations (17)

44

and (26) is given by

■‘■u >V it(d) Energy-Integral equation t

.. (28)

By using equations (1)/ (2) and (3) we obtained the energy integral equation for two dimensional steady incompressible boundary layers in the form

where

do 2d--- (U 63) = --dx o

. (29)

u u2--- (1 - —* ) dy, (energy thickness)...u ir

... (30)^ u

( ----)dy, (dissipation ... (31)^ y integral)

2. Boundary layer with suction along the porous plateby using Tani's flow s

Tani ^34^ obtained a family of solution of the

boundary layer equations. These solutions are related to the potential flow given by U(x) = UQ - ax11 , (n = 2,3,4,... )

where UQ is free stream velocity, a is reference length.

which constitutes a generalized form of the flow along a

45

flat plate if a ^ 0 there is potential flow with rise

pressure.

Tani £34] used the series expansion method for the

stream function and obtained that separation occurred at

x/a = 0.271 in the case of potential flow given by2X.

U(x) = UQ (1 - —^ ). M.R. Head obtained the momentum

integral equation, and the kinetic integral equation. Prasad ^JL9*] studied the boundary layer with suction along porous

plate. When the velocity profile has been expressed as a

polynomial of eighth degree as follows

--- = 5”L t]1 when O <^r\ 1U i=0

= 1 if n ^1y

where T] = ----- , 6 (x) is boundary layer thickness.6 (x)

Equations :

Head j[8j| calculated the momentum integral equation

for the two-dimensional laminar incompressible boundary

layer along porous plate in dimensionless form :

dt*

dx

2

0 ^1 - (H + 2) A + 6' (1)

46

wherex = coordinate along the wall

x =

u =

Xa U(x)u_

t a is reference length.

U(x) is potential flow velocity and is free stree velocity.

A = e* dUdx

e =m

S --- u^ uouU ), (momentum thickness)

i\) =-- / kinematic viscosity.9.

et* = ( — )

a2 uo a

0<S" = — X

6

A = If-i)

V„ is suction velocity

H =6*0

6*oo

( 1 - - )4y, (displacement thickness) o

47

6^ = — , o / 0, suctionV

Vg normal velocity at the platee ^ u

1 = 1 ST >Y=0

Head j~8j established the kinetic energy integral equation for the two dimensional laminar boundary layer for an incompressible flow along porous plate in the following dimensionless form

to

ro

++

re ix mr~—\

' to ’

O

Cllli

<+ i

^* iI

S !

X CD

PCM

Id

i >i

fO 1̂5

V—-•

CM

I

CD I d

8^,0

£ 3* 3

tow0)GXu•Hr:■pxCnM0)G0)>iP

a id

coi ror-i

dipii

[mCD

II<r>x

CO£Or—I »—i O pCO(T3co■H■P•HPGOO-P-Hr~t'H■p(0a£Oop<DG*H(00)

58 i—I

!> + H °\3 II

0 >i

U, 7d in

1 1 >1

d I >i

CM I

l fOl/O

fO I/O 'J

(DIPII

M

0)S-iCD

CM IP Q}|IIe

48

Family of Velocity Profile %

The eighth degree polynomial velocity profile is

given by

u/U = f (tj )= aQ + a-jTi + --- + a8 t|8 ... (4)

ywhere r) =----- , 6 (x) is boundary layer thickness.

6 (x)

With the boundary conditions to be satisfied for a porous

plate are

T]=0su=0, y=V„i

0, where n = 1, 2, 3, 4, (5)

T] = 1 i U = U

V1 u

Therefore due to (5), the eighth degree polynomial profile

given by (4) has been obtained in the following form

uu = fOl)

= 1 - (1 - I!) 1 + ( 2.774 - K)T) +45

6 + A 0+ (3.87------- K) Tp + 1.61 T]'

9 ]- (6)

where, K is a parameter which is connected with the plate

\ Vs 6porosity A = ---,— and the pressure gradient of external

49

flow U — according compatibility condition (3) reduces dxto the form

2 15 +X 0^X (2.226 + ---- K) + —-- 2K - A = 0. ... (7)62 45 62

Tani's flow is useful to investigate the boundarylayer with suction along a porous plate fox which the

x2potential flow velocity is given by U(x) = UQ ( 1 - )

U (x) x2x.e. ———— = 1 — ——

UG

U = 1 - X^

Differentiating with x and multiplying by t* on bothsides

dUt*--- = - 2 x t*

dxThe momentum integral equation (1), the kinetic energy integral equation (2) and the plate compatibility condition (3) becomes

dt*--- = f (x, t*, He )dx

2

1 xI + 2x t* (H+2) + ——— X/ •" • (8)

L 6 J

50

«€dx

g (x, t*, H£}

t*U-x2) 2D + Hr Jl + 2x t* (H-l) +

and 62+ 15 )\ + 90)K + 10.17 + lOxt* 0

(9)

(10)

In this article we have studied the boundary layer with suction along a porous plate in Tani's flow for which potential flow velocity is given by

3U(x) = U (1 - --- )

a3

This type of flow occurs when a straight channel with parallel plate succeeded by a diverging channel with adverse pressure gradient for this flow

U(x)UO

3x

i.e. U = 1 - x2

Differentiating with respect to x and multiplying t* on both sides we get

t*dUdX

= - 3 x2 t*

um. pniprsrifito * J|^ssiiVAji university. kulmapmi

51

d*g(x, t*, Hg)

1

t* (1-x3)

6J

2D-H g £l + 3x2 t* (H-l) +

'] *!'] ... (12)

and the compatibility condition (3) will be reduced to

02 r 9 n 135 x—](A + 15 X+ 90)K + ICO.17 A I +------------- = 0ft2 t a2

... (13)

3. Laminar incompressible boundary layer along a porous

circular cylinder.

(a) Introduction ; Von Karman and Millikan £39j! first

studied the problem of separation of the boundary layers.

They gave an.approximate method for treating the boundary

layer in the region of retarded flow.

Pohlhausen f2cQ took the fourth degree polynomial

t-*

ba)

2a)L©ArHrH•rHco•H-Ptddcr0)

.H©UO'

0)AG*H3 <D 3 ftt3 (1

)

Xft *(—

5 H + to X

I to <+ * X + to +O

I®

Q, t O

uM

l I (+ I *

II tti XI

to

IlooI IX

$ba)u3b0)ua)ArHrH3:

fNc0"fip©&©H©P0>©AG•rl>101

M © G ©O•HA©G•HX©A

52

velocity profile to solve the momentum integral equation and the first wall compatibility condition.

Schlichting |j3cfj assumed one parameter family of velocity profiles. He applied his method to the entry flow on a flat plate with constant suction and to the flow past at circular cylinder.

Watson Hro"] studied the asymptotic theory of boundary layer flow of an incompressible fluid with suction by the method of series expansion.

Terril ^3fTj investigated the laminar boundary layer flow near the stagnation point with and without suction. He considered the potential about circular cylinder.

Schlichting and Ulrich calculated the boundary layers by approximate methods over elliptical cylinder whose major axis are parallel to the direction of the stream. The ratio of the major to minor axis of the cylinders ranged over a/b = 1, 2, 4, 8. In the case of a/b = 4, they found that the point of separation was given by 0 - 138° from the stagnation point.

Ram Deo Mahto studied the momentum and the kineticenergy integral equation for two-dimensional boundary layers have been rederived and have been used with the aid of the Schlichting"s velocity profiles to obtain a step-by-step,

53

solution for the boundary layers with suction in the region of adverse pressure gradient over a porous elliptic cylinder with the ratio of the major to minor axis is four.

Equations s

The momentum integral equation, the kinetic energy integral equation and the compatibility condition for the two dimensional laminar incompressible boundary layers along porous wall with suction as given by Head [91 in dimensionless

forms are respectively

dt* 0 r| I - (H + 2)A + <r~f ... (l)dx U L- J

dHdx u

-L r2D-H€ (t* u c 1

I - (H-l)yv + ^| +... (2)

and m = - /V + I <5^ ... (3)

where, x = coordinate along the wall.- x . .x = - , where a is a representative length, a

\) = ji/ ^ , Kinematic viscosity

U0 = Free stream velocity

U(x) = Potential flow velocity

u = u/uoa?

_/~(1 ~ r: ) dy, o u (displacement thickness)6* *

54

= Ya_§ , </> 0, suctionS)

Vg = Normal velocity at the wall,

Family of velocity profiles %

Schlichting, H. studied the laminar boundarylayer with suction for bodies of arbitrary shape by approximate

009= dy, (momentum thickness)o u u

X II Cn *

H II a i a>

y*—v

a/i oJ

i

xj l C l if o

o

(Vi 0/

i

•<J i

to

to l c

ca l

® i r

cd i —

v I

01 E>

I

04

a> i m*

§ 13CNCD

1 P* II

toto<Daxo•r|b+>

COS-i0)ca)a ‘Cloi

CD

|KJ

0t

CO<y a

<

m it

°f cjie: o H I

C I c

Q/l I

xj lto« II

0c CJ |

CD

OP

I C

DIII X

55

method. He gave one parameter family of velocity this problem as

--- = F-, (tj) + K F9(tj) ... (4)U

where, (t) ) = 1 - e-”^

TD]F2(t1) = Fx - Sin , 0 ^ T| <4 3o

F2(T]) = Fx - 1, T] > 3

yr] = ---- , non-dimensional y-coordinate6 (x)

6(x) is local boundary layer thickness

The compatibility condition for this problem takes the form,2 W _ r n ^

= 0 ... (5)©“ g

(K+l) —+ --- 6^ il + K (16^ 6 ^1 + K (1 - )j-A =

Thwaites 36 gave an approximate representation of the numerator in the momentum integral equation (1) as

2 [j - (2 + H)A + <r] — 0.45 — 6+ 1.28 6^+

+ 0.76 <f2 ... (6)

Integrating the momentum integral equation (1) we get the integration formula

t* = r— f U5 (0.45 + 1.28 <5% 0.76 <5"2) dx ...(7) U6 o

Thwaites integration formula (7) would give sufficiently

56

accurate value of t* in the favourable pressure gradient.

(b) Application of approximate method to the porouscircular cylinder s

In this article/ we have investigated the laminar incompressible boundary layer along a porous circular cylinder in a uniform stream UQ parallel to the x-axis.

The velocity distribution along the contour of circular cylinder is given by

U(S) = 2UQ Sin 0

where S is the length measure from the forward stagnation point, 0 is the angle measured from the stagnation point.

Taking x = a Cos ©, y = a Sin'©/ where a is radius of the circular cylinder/ equations (1), (2) and compatibilitycondition (5) for this problem will be reduced to the following forms respectively

dt*de

de

f( e, t*, H£ )

lSin 6

(I + cO + 2(H + 2)t* Cos2 6 ... (9)

g(©/ t*/ h ^ )

1(2D + d") + H £ 2 (H-l) . t*Cos2

2t*Sin 0 ...(10)

57

eand (K+l)----+

-[■l+K (1 - Tr/6) , - 2Cos^ 0 = 0 ...(11)6^ 6 - ^

'rom equation (7) the value of t* is given by

t* = - Sin 6 ... (12)Equation (12) gives values of the momentum thickness

parameter t* in the region of favourable pressure gradient

from 0 = 0 to 6 = 7i/2.

4. Boundary layer for Tani's flow past a Wedge :

A boundary layer flow of incompressible fluid past

a wedge# with the wedge angle up where o o <c 2 as shown

in figure

Pig.8 - Boundary layer flow past a wedge for incompressible fluid.

58

Let o be taken at the stagnation point. Let x-axis

be along the wall and the y-axis be perpendicular to the

wall. Thus we have the boundary layer equation

b u b u dU b 2uu ---- + v ----= U----- + y -~2b x by dx ay

b u

b x 0

(1)

(2)

With the boundary conditions

y = 0; u = 0 = v

y —OQ ? u = U (x) ... (3)

where }) is the kinematic viscosity and U(x) is the Tani1 s

flow potential velocity. Now we introduce the stream function

(x, y), which satisfies the continuity equation (2)

Thus we takeb if-

b x, V = _ ....

by using equation (4), equation (1) takes the form as

. (4)

b b ^by b x b y

b sf- b y. b x by2

dUU----+

dx

b 3(5)

The corresponding boundary conditions are

y = o? Vf bb y

0

59

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It may be noted that a = fM(0) is still unknown we have find

the values of a by substituting the values of f and f' from

equation (12) we get a linear differential equation in f"(q)

Therefore fM(q) * e“F^ #(q) ... (13)

where F (r]) = f (T)) dq o

and 0 (q) = a - J' (1 - f'2) dq3 o

Equation (13) satisfies the boundary condition at q = 0.

Then unknown parameter a is found from the equation

rH

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2 y a2

5. Boundary layer Hani-VS. flow along the wall of aconvergent channel.

o ipmrrrn /ry /ntrsyf

<T<>>rrrn > rt ? i?

Fig.9 - Boundary layer flow along the wall of convergent channel.

The potential flow velocity is given by

('*V,

U(x) = UQ (1--- - ) ‘ : • * ... (1)az !where, x is measured along the wall of channel and UQ is stream velocity, a is reference length. Now we can introduce

63

Y /~Uo

and stream function

- (2)

4" - ( >1/2 f(n)a^

U0 I/? ^and u = Uf • (ti), V * - ( 1) —- )' ---f1 Ol)_2 va A

... (3)

... (4)

and the function f(T|) satisfies the differential equation.2f'M + i 0 ... (5)

the boundary conditions are

T\ = 0 s f = 0, f* = 011 = CD : fJ = 0 ... (6)

The solution can be obtained following Schlichting, H.

6. Application of the approximate method to bodies ofrevolution and three-dimensional boundary layers.

In this article we have enlisted some major develop­ments in the field of applications of approximate methods to the bodies of revolution and three-dimensional boundary layer flow.

C.B.Millikan jjL4~j was the first person who introduced

an approximate method for the calculation of boundary layers on bodies of revolution in axial flow. Pohlhausen also studied

64

an approximate method of calculation of boundary layers on

bodies of revolution. This method is based on a polynomial

of fourth degree. S. Tomotika E?7' 3®1 had extended this

method to the body of revolution. F.W.Scholkemeyer ^33~| worked

on the bodies of revolution. H. Holstein and T.Bohlen studied

for two-dimensional momentum equation by using F.W.

Scholkemeyer1s work. Thus the momentum equation for the

axially symmetrical case is obtained by using following two

equations

b u ^ u b ui.e. -------+ u----------+ v----------

as

^ (ur) ^ (vr)

—— + ---------- = 0ox by

b 2’1 O p O "Ux - --- ----- + )J--------- IT

^ t ^ x by ^ b x by

2 & © d U q drU ------- + (2 0 + 6*)U-------- + u--------

d x dx r dx ^

We obtain the following differential equation for the•y

quantity z = 0 / ))

(1)

(2)

(3)

1 dz- U-------- +2 dx

r ~i i <3r : u2 + f, (k) k + --------- - k =* f_ (k) ... (4)

r dx : U’ 2

where '0 dUK = ----- ----- / K is second! shape factor

\) dx

f ]_ (k) = -----6*

e

3

10A120

37

315A945

At9072

, J\ is first shape

factor

65

where * denotes differentiation w.r.t. x from the shape

1 dr uof the body can evaluate the- term - ^

r dx u

The point of . separation occurred at = - 12 i.e.

at k = - 0.1567. But at the stagnation point the values of

the shape factors /\ and k are different.

If the body of revolution has a blunt nose then at

the upstream stagnation point

1 dr Ulim ( - ----- - ) = 1 ;x —* 0 r dx U'

r 1 dr UThe values of < F (k) - 2k - — —

L r dx U1is ; reduced to F(k) - 2k.

But for two-dimensional case F(k) - -2k = 0 will be a

condition to find out the initial value of k.

invo

VO

IO

l t—1

fO|

VO

, ;

>£* I

Cn |

nnr- IrH ro looi do it to +

cn !> iKd

iiHi

to

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X> *3

to * Hi

X*C

M4-1C

MII

sPm

tr p V) d 0) fl> 3 O b1 c+ 0 H- 3 ID O* H- 3 -!

?- (D Hi 0

VO(D3P

to b bin I

Qj I & X j

H

a i a

- i t—cO

N I

XI o

Di

N Cl

66

Hence, equation (4) at the stagnation point becomes

ovoT'U,

0.05708U*o

(dzdx

>o = 0 . (5)

Due to symmetry the initial slope is zero for the body of revolution. Hence at the stagnation point, U" = 0

In the case of axially symmetrical bodies N.Rott and L.F. Crabtree j~23^j used the method of direct integration

to the Thawaites equation

U 0“jr

0.470"u5'

x/ U5 dx

where 0 is momentum thickness and obtained it in the form

U 0 0.470r2 U5

xf r2 U5 dx ... (6)

They also extended this method of direct integration to the yawed cylinder.

S. Tomtika {*38} using potential and measured pressure

distribution calculated the boundary layer on a sphere for a range of Reynolds numbers. A Fage studied this problem at critical Reynolds number on a sphere and confermed.This work has been extended by W.Moeller C15>

67

#

In the case of axially symmetrical flow# the flow becomes much more complicated when there exists a tangential component in addition to the longitudinal component. This case will become similar to the three-dimensional case .e.g. the case yawed cylinder belongs to this category of problems. In such cases the momentum equation in the meridional plane (x-direction) should be supplemented with the momentum equation in the transverse plane (y - direction) by choosing the third Z coordinate in the direction perpendicular to the surface. For the case of a body rotating in an axial stream these two equations will take the following forms :

d ex dU 1 dr 9 9uz---- + U---(2 0 + 6*) + - (IT 0 + v^ e ) =dx dx x x r dx xoy

IVu° ... (7)

vn d.--- ---(U r3' 6 )r3 dx o\

... (8)

With the following boundary layer parameters

u1 o = p. ( ) ;^ z 2=0

vLyo = „ ( --- ) z=0 '

COS' (1 - )dz;z=0 ^ 0xCOf ft < 1 - “u )dz?z=0 U u

68

0© ooS172= C

U

u

V— dz ,

where v(x,z) is the tangential velocity distribution.

At the wall the local tangential velocity vQ is given by vQ = r.w.

The momentum integral equations for the yawed cylinder can be obtained from the above by putting r = constant and introducing

0xyf u vJ - ( 1----- ) dzz=0 U

where is the component of the oncoming velocity in thedirection of the cylinder generator.

W. Dienemann jjT] worked out an approximate method

for yawed cylinders which was based on the equations (7) and (8). In the case of axial flow, 0. Parr |^2lJ performed

an experiment for the boundary layer on a rotating body of revolution.

The extension of these approximate methods to the three dimensional boundary layer flow is very difficult. Recently some research workers due to the mathematical complexities involved in the problem.

69

G.Nath fc6l made detailed study of several problem

dealing with the unsteady three-dimensional flows# swirling flows through nozzles and diffusers# spin-up and spin-down phenomena in rotating fluids# and stagnation point flows with massive blowing. Warasi# Z.U.A. ^41~J studied 'Investiga­

tion of three-dimensional boundary layer separation criteria by a minimum principle'. He was concerned with the three- dimensional laminar boundary layer equations and their properties at position where both components of the skin friction vanish (critical points). Bloor# M.I.C. and Ingham, D.B. W investigated 'on the use of a pohlhausen

method in three-dimensional boundary layers'. They obtained approximate solution of a class of rotationally symmetric three-dimensional laminar boundary layer flows using the pohlhausen method. Kaplan, V.S. ^JL2^j studied 'Invariant

solutions for the equations of a three-dimensional boundary layer on developable surfaces'. The exact solutions of the three-dimensional boundary layer equation are possible when the independent variables in the equations can be reduced to two or one. B.M.Nayer j~17~j studied 'Mixed boundary value

Iproblem in three-dimensions'. He obtained integral equations by using symmetrizing kernels of classical potential theory. These problems can be tackled by using; Tani' s flow.

=000=

70REFERENCES

1. Agarwal, R.P.‘Non-linear two point boundary layer value problem1. Indian J.Pure and pppl. Math. 4(9 & 10) 1973 ; 757-69.

2. Bansal, J.L.'Asymptotic suction temperature profiles in a laminar boundary layer over a porous flat plate'.Indian J. Pure appl. Math. 3(5) : 721-28.

3. Bloor, M.I.C. and Ingham, D.B.'On the case of a pohlhausen method in three- dimensional boundary layers'.Z. Angew. Math. Phys. 28 (1977), no.2,. 289-99.

4. Dienemann W., Berechnung des Warmeuberganges an laminar umstromten korpem mit konstanter and ortsveranderlieher Wandtemperatur,ZAMM 33, 89-109 (1956).

5 Fage, A., Experiments on a phere at critical Renolds Numbers ARCR & M 1766 (1936).

6 Husnutdinova, N.V.'Mathematical questions of the control of the boundary layer means of suction'.Sibirsk. Mat. Z. 13 (1972), 485-489.

7 Holt, M. and Modarress, D.'Application of the method of integral relations to laminar boundary layers in three-dimensions'. Proc.Roy.Soc. London Ser.A. 353 (1977) no. 1674, 319-347.

m-*

71

8 Head, M.R.'Approximate methods of calculating the two-dimensional laminar boundary layer with suction1. Boundary layer and flow control, edited by G.V.Lachmann, Pergamon Press London, 2, p. 801 (1961) .

9 Head, M.R.'A approximate method of calculating the laminar boundary layer in two-dimensional incompressible flow'.ARC.R and M No. 3123 (1957).

10 Holstein, H. and Bohlen, T. s Ein einfaches verfahren zur Berech. nung laminar Reibungsschichten die dem Naherungsansetz von K. Pohlhausen genugen. Lilienthal Berechts 10, 5-16 (1940).

11 Kozlov, L.F. and Losinskaja, T.I.'Integration on an electronic computer of the boundary layer equations of Tani's flow in the presence of suction1.Vycisl Prikl. Mat. (Kiev) Vyp. 23 (1974), 127-135.

12 Kaplan, v.S.'Invariant solutions for the equations of a three- dimensional laminar boundary layer on developable surfaces', Izv. Akad Nauk SSSR Ser. Mech. Zidk. Gaza 1975, no.4,33-40.

13 Mills, R.D.'On boundary layer suction through narrow Slots'.Quart.Journ.Mech.Appl.Math.Vol.XXIV,Pt.4, 1971spp.461-71.

72

#

14 Millikan C.B., The boundary layer and skin friction

for a figure of - revolution. Trans. ASME 54, 29 (1932).

15 MoHer, W., Experimentalle Untersuchungen zur

Hydrodynamik der Kugel. Phys. Z. 39, 57 (1938) .

16 Nath, G.

'Boundary layer flow'.

Indian Institute of Science Research and Development

in Physics and Math. Sci. Sept. 1984 s pp.13-16.

17 Nayar, B.M.

'Mixed boundary value problem in three-dimensions'.

Indian J. Pure appl. Math. 6(3)s303-309,March, 1975.

18 Paini G.C. and Atolia, R.N.

'Study of the plane couette flow of waiters 0*

liquid with uniform suction at the stationary plate'.

Indian J. Pure appl. Math. 5(11); Nov 1974;1977-92.

19 Prasad, S.

'Boundary layer with suction along the porous wall in

Tani's flow.'

Ran.Uni. Math. Jour. Vol. 13(1982) s 28-35.

20 Pohlhausen, K.

'Zur naherungsweisen integration der Differential

glei chung der Laminar Reinhungsschicht'.

ZAMM 1, 235 (1921) .

21 Parr, o., Die stromung urn einen axial angestromten

rotierenden Drehkorper Jahrb Schiffbautechn.

Ges. 53 (1959). S. 260-271.

^ BAEASABEB KHARIKAR UBHAF •uiv&ji wisiiveasiTY.

73

22 Ram Deb Mahto#'Boundary layer with suction over a porous ellipticcylinder.' Rail. Uni. Math.J.Vcl.l4 (1983) : 39-^8 .

23 Rott, N. and Crabtree, L.F., Simplified laminar boundary calculations for bodies of revolution and for yawed wings. Jour. Aero. Sci. 19, 553 (1952).

24 Sarma, G.N. and Singh, D.P.'The effects of viscosity-temperature law in unsteady boundary layer on a flat plate*.Indian J. Pure appl. Math. 4(2), 1973; 161-170.

25 Singh, D.P.'Compressible boundary layer flow in a converging canal'. Indian J. pure appl. Math. 5(1) Jan.1974s101-10.

26 Sutapa Mukherjee and R.C. Choudhary,'Magnetohydrodynamic boundary layer with suction along a porous wall in Howarth's flow*.Bull. Cal. Math. Soc., 75(5), 1983: 321-329.

27 Sharma, R.'A two parameter method for calculating the two- dimensional boundary layer with suction or injection'. Bull. Cal. Math. Soc., 74, 157-67 (1982).

28 Sharma, R.'Exact Solution of the incompressible laminar boundary layer equation with zero pressure gradient and variable suction'.Bull.Cal. Math. Soc., 74, 169-78 (1982)

74

29 Sutapa Mukherjse*'M&gftfetOhydrodynamic boundary layer with suction along a porous Wall in a Sink flow'#Ranchi Uni. Math. JoUr. Vol. 13 (1982)s 52-59.

30 Schlichting, H.'An approximate method for the calculation of laminar boundary layer with suction for bodies of arbitrary shape1« NACA* Tech. MN. 1216 (1949).

31 Schlichting* H. and Ulrich, A.Zur Berechung des Umschlages laminar Turbulent Jahrbuch d. dt. Laftfahrtforschang, 1,8-35 (1942).

32 Schlichting, H.'Boundary layer theory',McGrow-Hill Series in Mechanical Engineering.. (1962) pp% 144-46.

33 Scholkemeyer, F.W.-, Die lamina re R e ibungsach i-cht an rotationssymmetrischen korpem. Thesis Braunschweig 1943.

34 Tanh, 1.'On the solution of the laminar boundary layer equation'.J.Phys.Soc. Japan 4„ (149-54),, 1949.

35 Terri 1,, R.M-.'Laminar boundary layer flow near stagnation with and without suction'.Trans* Roy.Soc.Lond.Sec.A.-253,,p.55-100 ((i960).

75

36 Thwaites, B.•The development of laminar boundary layers under conditions of continuous suction,Part II s Approximate methods of suction'.Rep.aero.Res.Lond.No. 12699 (1949).

37 Tomotika, S., Laminar boundary bn the surface of a sphere in a uniform stream. ARCR & M 1678 (1935).

38 Tomotika, S. and Imal, I., On the transition from laminar to turbulent flow in the boundary layer of a sphere. Rep. Aero. Res. Inst.Tokyo/ Imp.Univ.No.167 (1938) and S. Tomotika/ Pro. Phys. Math.Soc. of Japan 20 (1938).

39 Von Karman, Th and Millikan, C.B.'On the theory of laminar boundary layer involving separation'.NACA, TR. No. 504 (1934).

40 Watson, E.J.•The asymptotic theory of boundary layer flow with suction'. British ARC, R and MNo. 2619 (1952).

41 Warsi, Z.U.A.'Investigation of the three-dimensional boundary layer separation criteria by a minimum principle'.Z.Angew. Math. Phys. 28 (1977), no.2, 289-99.

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