Bird R.B., Stewart W.E., Lightfoot E.N. Transport Phenomena

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s4.4 Flow near Solid Surfaces by Boundary-Layer Theory

137

SOLUTION

We know intuitively what the velocity profile vx(y) looks like. Hence we can guess a form for

ux(y)

and substitute it directly into the von Karmiin momentum balance. One reasonable choice

is to let

v,(y)

by a function of y/6, where S(x) is the "thickness" of the boundary layer. The

function is so chosen that

and v,

v, at

This is tantamount to assuming

geometrical similarity of the velocity profiles for various values of x. When this assumed pro-

file is substituted into the von Khrman momentum balance, an ordinary differential equation

for the boundary-layer thickness 6(x) is obtained. When this equation has been solved, the 6(x)

so obtained can then be used to get the velocity profile and other quantities of interest.

For the present problem a plausible guess for the velocity distribution, with a reasonable

shape, is

for

6(x) (boundary-layer region) (4.4-14)

for

6(x)

(potential flow region) (4.4-15)

urn

This is "reasonable" because this velocity profile satisfies the no-slip condition at

0, and

dv,/dy

at the outer edge of the boundary layer. Substitution of this profile into the von

KArmBn integral balance in Eq. 4.4-13 gives

This first-order, separable differential equation can now be integrated to give for the bound-

ary-layer thickness

Therefore, the boundary-layer thickness increases as the square root of the distance from the up-

stream end of the plate. The resulting approximate solution for the velocity distribution is then

From this result we can estimate the drag force on a plate of finite size wetted on both sides.

For a plate of width Wand length

integration of the momentum flux over the two solid sur-

faces gives:

The exact solution, given in the next example, gives the same result, but with a numerical co-

efficient of 1.328. Both solutions predict the drag force within the scatter of the experimental

data. However, the exact solution gives somewhat better agreement with the measured veloc-

ity

profiles3 This additional accuracy is essential for stability calculations.

Obtain the exact solution for the problem given in the previous example.

Laminar Flow along

SOLUTION

Flat Plate (Exact

~olution)'

This problem may be solved by using the definition of the stream function in Table 4.2-1. In-

serting the expressions for the velocity components in the first row of entries, we get

This

problem was treated originally by

Blasius,

Zeits. Math. Phys.,

56,l-37

(1908).

138

Chapter 4 Velocity Distributions with More Than One Independent Variable

The boundary conditions for this equation for $(x,

are

B.C. 1:

aty=O, -=v,=O for

(4.4-21)

B.C. 2: aty=O,

-u,

forx2O (4.4-22)

B.C.

asy+m,

-vx

-urn for

0 (4.4-23)

B.C.

atx=O,

-vx

-u, for

0 (4.4-24)

Inasmuch as there is no characteristic length appearing in the above relations, the

method of combination of independent variables seems appropriate. By dimensional argu-

ments similar to those used in Example 4.1-1, we write

The factor of 2 is included to avoid having any numerical factors occur in the differential

equation in Eq. 4.4-27. The stream function that gives the velocity distribution in Eq. 4.4-25 is

This expression for the stream function is consistent with

Eq.

4.4-25 as may be seen by using

the relation

-d+/dy (given in Table 4.2-1). Substitution of Eq. 4.4-26 into Eq. 4.4-20 gives

Substitution into the boundary conditions gives

B.C.

and 2:

B.C.

and 4:

atq=0, f=O and ft=O

ass-+

fl+l

Thus the determination of the flow field is reduced to the solution of one third-order ordinary

differential equation.

This equation, along with the boundary conditions given, can be solved by numerical in-

tegration, and accurate tables of the solution are a~ailable.~,~ The problem was originally

solved by Blasius7 using analytic approximations that proved to be quite accurate.

plot of

his solution is shown in Fig. 4.4-3 along with experimental data taken subsequently. The

agreement between theory and experiment is remarkably good.

The drag force on a plate of width

and length

may be calculated from the dimen-

sionless velocity gradient at the wall, f "(0)

0.4696

. . .

as follows:

This result has also been confirmed e~perimentally.~,~

Because of the approximations made in Eq. 4.4-10, the solution is most accurate at

large local Reynolds numbers; that is, Re,

xv,/v

The excluded region of lower

Reynolds numbers is small enough to ignore in most drag calculations. More complete

54.4 Flow near Solid Surfaces by Boundary-Layer Theory

Fig.

4.4-3.

Predicted and observed velocity profiles for tangential laminar flow along a

flat plate. The solid line represents the solution of Eqs. 4.4-20 to 24, obtained by Blasius

[see

Schlichting, Boundary-Layer Theory, McGraw-Hill, New York, 7th edition (1979),

1371.

analyses8 indicate that Eq. 4.4-30 is accurate to within 3% for

Lv,/v

lo4

and within 0.3%

for

Lv,/v

lo6.

The growth of the boundary layer with increasing

eventually leads to an unstable

situation, and turbulent flow sets in. The transition is found

begin somewhere in the

range of local Reynolds number of Re,

xv,/v

10'

loh,

depending on the

uniformity of the approaching streams8 Upstream of the transition region the flow re-

mains laminar, and downstream it is turbulent.

We now want to treat the boundary-layer problem analogous to Example 4.3-3, namely the

flow near a corner (see Fig. 4.3-4). If

the problem may also be interpreted as the flow

near

Corner

along a wedge

included angle

Pn-,

with

2/(2

P). For this system the external flow

is known from Eqs. 4.3-42 and 43, where we found that

This was the expression that was found to be valid right at the wall (i.e., at

0). Here, it

is assumed that the boundary layer is so thin that using the wall expression from ideal

flow is adequate for the outer limit of the boundary-layer solution, at least for small values

Kuo,

Math.

Phys.,

32,83-101

(1953);

Imai,].

Aero.

Sci.,

24,155-156 (1957).

140

Chapter 4 Velocity Distributions with More Than One Independent Variable

SOLUTION

Fig.

4.4-4. Velocity profile for

wedge flow with included

angle prr. Negative values of

correspond to the flow

around an "external corner"

[see Fig. 4.3-4(ii)I with slip at

the wall upstream of the

corner.

We now have to solve Eq. 4.4-11, using

Eq.

4.4-31 for v,(x). When we introduce the stream

function from the first row of Table 4.2-1, we obtain the following differential equation for

which corresponds to

Eq.

4.4-20 with the term v,(dv,/dx) added. It was discovered9 that this

equation can be reduced to a single ordinary differential equation by introducing a dimen-

sionless stream function f(q) by

*(x,

V'ZT~~X"'~

(4.4-33)

in which the independent variable is

Then

Eq.

4.4-32 becomes the

Falkner-Skan

equation9

This equation has been solved numerically with the appropriate boundary conditions, and

the results are shown in Fig. 4.4-4.

It can be seen that for positive values of p, which corresponds to the systems shown in

Fig. 4.3-4(a) and Fig. 4.3-5, the fluid is accelerating and the velocity profiles are stable. For

negative values of p, down to

-0.199,

the flows are decelerating but stable, and no sepa-

ration occurs. However, if

-0.199, the velocity gradient at the wall becomes zero, and

separation of the flow occurs. Therefore, for the interior corner flows and for wedge flows,

there is no separation, but for the exterior corner flows, separation may occur.

QUESTIONS FOR DISCUSSION

For what types of problems is the method of combination of variables useful? The

method of separation

variables?

Can the flow near a cylindrical rod of infinite length suddenly set

motion in the

axial direction be described by the method in Example 4.1-l?

Falkner and S.

Skan,

Phil. Mag.,

12,865-896 (1931);

R. Hartree,

Proc. Camb. Phil. Soc.,

33,

Part 11,223-239 (1937);

Rouse (ed.),

Advanced Mechanics of Fluids,

Wiley, New York (19591, Chapter

VII,

Sec.

Schlichting and

Gersten,

Boundary-Layer Theory,

Springer-Verlag, Berlin (2000), pp.

169-173 (isothermal), 220-221 (nonisothermal);

Stewart and

Prober,

Int.

Heat Mass Transfer,

1149-1163 (1962); 6,221-229,872 (1963), include wedge

flow

with heat and

mass

transfer.

Problems

141

What happens in Example 4.1-2 if one tries to solve Eq.

4.1-21

by the method of sepa-

ration of variables without first recognizing that the solution can be written as the

sum of a steady-state solution and a transient solution?

What happens if the separation constant after

Eq.

4.1-27 is taken to be

instead of

-c2?

Try

solving the problem in Example 4.1-3 using trigonometric quantities in lieu of

complex quantities.

How is the vorticity equation obtained and how may it be used?

How is the stream function defined, and why is it useful?

what sense are the potential flow solutions and the boundary-layer flow solutions

complementary?

List all approximate forms of the equations of change encountered thus far, and indi-

cate their range of applicability.

PROBLEMS

4A.1

Time for attainment of steady state in tube flow.

(a) A heavy oil, with a kinematic viscosity of 3.45

m2/s, is at rest in a long vertical tube

with a radius of 0.7 cm. The fluid

suddenly allowed to flow from the bottom of the tube by

virtue of gravity. After what time will the velocity at the tube center be within 10% of its final

value?

(b) What is the result if water at 68OF is used?

Note:

The result shown in Fig. 4D.2 should be used.

Answers:

(a)

6.4

lop2

s; (b) 0.22 s

4A.2

Velocity near a moving sphere. A sphere of radius

is falling in creeping flow with a termi-

nal velocity

through a quiescent fluid of viscosity

At what horizontal distance from the

sphere does the velocity of the fluid fall to

of the terminal velocity of the sphere?

Answer:

About 37 diameters

4A.3

Construction of streamlines for the potential flow around a cylinder. Plot the streamlines for

the flow around a cylinder using the information in Example 4.3-1 by the following procedure:

(a) Select a value of

(that is, select a streamline).

(b) Plot

(straight lines parallel to the X-axis) and

K(X~

Y2)

(circles with ra-

dius 1 /2K, tangent to the X-axis at the origin).

(c)

Plot the intersections of the lines and circles that have the same value of K.

(d) Join these points to get the streamline for

Then select other values of

and repeat the process until the pattern of streamlines is clear.

4A.4

Comparison of exact and approximate profiles for flow along a flat plate. Compare the val-

ues of v,/v, obtained from

Eq.

4.4-18 with those from Fig. 4.4-3, at the following values of

yG:

(a) 1.5, (b) 3.0, (c) 4.0. Express the results as the ratio of the approximate to the exact

values.

Answers:

(a) 0.96;

(b)

0.99; (c) 1.01

4A.5

Numerical demonstration of the von

Klirmin

momentum balance.

(a) Evaluate the integrals in

Eq.

4.4-13 numerically for the Blasius velocity profile given in

Fig. 4.4-3.

(b)

Use the results of (a) to determine the magnitude of the wall shear stress

T,,(,=~

(c)

Calculate the total drag force,

F,,

for a plate of width

and length

wetted on both

sides. Cornparme your result with that obtained in

Eq.

4.4-30.

Answers:

(a)

pv,(v,

v,)dy

0.664-

IOm

p(o,

vJdy

1.73e

142

Chapter 4

Velocity Distributions with More Than One Independent Variable

Use of boundary-layer formulas. Air at

atm and 20°C flows tangentially on both sides of

thin, smooth flat plate of width

10 ft, and of length

ft in the direction of the flow.

The velocity outside the boundary layer is constant at 20 ft/s.

(a)

Compute the local Reynolds number Re,

xvm/v at the trailing edge.

(b) Assuming laminar flow, compute the approximate boundary-layer thickness,

inches, at

the trailing edge. Use the results of Example 4.4-1.

(c)

Assuming laminar flow, compute the total drag of the plate in lbf. Use the results of Exam-

ples 4.4-1 and 2.

Entrance flow in conduits.

(a) Estimate the entrance length for laminar flow

a circular tube. Assume that the boundary-

layer thickness 6 is given adequately by Eq. 4.4-17, with

of the flat-plate problem corre-

sponding to v,,, in the tube-flow problem. Assume further that the entrance length

can be

taken to be the value of

at which

Compare your result with the expression for

cited

52.3-namely,

0.0350 Re.

(b)

Rewrite the transition Reynolds number xvm/v

3.5

lo5

(for the flat plate) by inserting

from Eq. 4.4-17 in place of

as the characteristic length. Compare the quantity 6vm/v thus

obtained with the corresponding minimum transition Reynolds number for the flow through

long smooth tubes.

Compare the result with that given in Problem 4C.l(d).

Flow of a fluid with a suddenly applied constant wall stress. In the system studied in Ex-

ample 4.1-1, let the fluid be at rest before

0. At time

0 a constant force is applied to the

fluid at the wall in the positive

direction, so that the shear stress

r,,

takes on a new constant

value

0 for

(a) Differentiate Eq. 4.1-1 with respect to

and multiply by

-p

to obtain a partial differential

equation for

ryw(

t).

(b) Write the boundary and initial conditions for this equation.

(c)

Solve using the method in Example 4.1-1 to obtain

(d) Use the result in

(c)

to obtain the velocity profile. The following relation7 will be helpful

Flow near a wall suddenly set in motion (approximate solution) (Fig. 48.2). Apply a proce-

dure like that of Example 4.4-1 to get an approximate solution for Example 4.1.1.

(a) Integrate Eq. 4.4-1 over

to get

Make use of the boundary conditions and the Leibniz rule for differentiating an integral

(Eq. C.3-2) to rewrite Eq. 4B.2-1 in the form

Interpret this result physically.

A useful summary of error functions and their properties can be found in

Carslaw and

Jaeger,

Conduction of

Heat

in Solids,

Oxford University Press, 2nd edition

(1959),

Appendix

11.

Problems

143

Fig.

4B.2.

Comparison

of true and approximate

velocity profiles near a

wall suddenly set in mo-

tion with velocity

vo.

(a)

True

solution

(b)

Boundary-layer approximation

(b) We know roughly what the velocity profiles look like. We can make the following reason-

able postulate for the profiles:

Here 6(t) is a time-dependent boundary-layer thickness. Insert this approximate expression

into Eq. 4B.2-2 to obtain

(c)

Integrate Eq. 4B.2-5 with a suitable initial value of 6(t), and insert the result into

Eq.

4B.2-3

to get the approximate velocity profiles.

(d)

Compare the values of

v,/v,

obtained from

(c)

with those from

Eq.

4.1-15

y/l/4vt

0.2,0.5, and 1.0. Express the results as the ratio of the approximate value to the exact value.

Answer

(d)

1.015,1.026,0.738

4B.3 Creeping flow around a spherical bubble. When a liquid flows around a gas bubble, circula-

tion takes place within the bubble. This circulation lowers the interfacial shear stress, and, to a

first approximation, we may assume that it is entirely eliminated. Repeat the development of

Ex.

4.2-1 for such a gas bubble, assuming it is spherical.

(a) Show that B.C.

of Ex. 4.2-1 is replaced by

B.C. 2:

and that the problem set-up is otherwise the same.

(b) Obtain the following velocity components:

(c)

Next obtain the pressure distribution by using the equation of motion:

pgh

cos

('Y)

144

Chapter 4 Velocity Distributions with More Than One Independent Variable

(dl Evaluate the total force of the fluid on the sphere to obtain

This result may

obtained by the method of 52.6 or by integrating the z-component of -[n

IT]

over the sphere surface (n being the outwardly directed unit normal on the surface of the

sphere).

48.4

Use of the vorticity equation.

(a) Work Problem 2B.3 using the y-component of the vorticity equation

(Eq.

3D.2-1) and the

following boundary conditions: at x

0 and at

v,,,. Show that this

leads to

Then obtain the pressure distribution from the z-component of the equation of motion.

(b) Work Problem 3B.6(b) using the vorticity equation, with the following boundary con-

ditions: at r

R, v,

0 and at

KR,

vO. In addition an integral condition is needed

to state that there is no net flow in the

direction. Find the pressure distribution in the

system.

4B.5

Steady potential flow around a stationary ~phere.~ In Example 4.2-1 we worked through the

creeping flow around a sphere. We now wish to consider the flow of an incompressible, invis-

cid fluid in irrotational flow around a sphere. For such a problem, we know that the velocity

potential must satisfy Laplace's equation (see text after Eq. 4.3-11).

(a) State the boundary conditions for the problem.

(b) Give reasons why the velocity potential

can be postulated to be of the form +(r, 6)

f(r) cos 19.

(c)

Substitute the trial expression for the velocity potential in

(b)

into Laplace's equation for

the velocity potential.

(d) Integrate the equation obtained in (c) and obtain the function f(r) containing two con-

stants

integration; determine these constants from the boundary conditions and find

-ZJ,R[(;)

(?)i]

cos

(el

Next show that

(;)i]

cos

-v.[l

(:r]

sin 6

(f)

Find the pressure distribution, and then show that at the sphere surface

$pvi(l

sin2 8) (4B.5-4)

4B.6

Potential flow near a stagnation point (Fig. 4B.6).

(a) Show that the complex potential

-v,?~

describes the flow near a plane stagnation point.

(b)

Find the velocity components v,(x,

and v,(x,

y).

(c)

Explain the physical significance of v,.

Landau and

Lifshitz,

Fluid

Mechanics,

Pergamon, Boston, 2nd edition

(1987),

pp. 21-26,

contains a good collection of potential-flow problems.

Problems

145

Fig.

4B.6.

Two-dimensional potential flow

near a stagnation point.

'stagnation point

4B.7

Vortex flow.

(a) Show that the complex potential

(ir/27r) In z describes the flow in a vortex. Verify

that the tangential velocity is given by vo

r/2777 and that v,

This type of flow is some-

times called a free vortex. Is this flow irrotational?

(b) Compare the functional dependence of v, on

in (a) with that which arose in Example

3.6-4. The latter kind of flow is sometimes called a

forced

vortex. Actual vortices, such as those

that occur in a stirred tank, have a behavior intermediate between these two idealizations.

48.8

The

flow field about a line source. Consider the symmetric radial flow of an incompressible,

in-

viscid fluid outward from an infinitely long uniform source, coincident with the z-axis of a

cylin-

drical coordinate system. Fluid is being generated at a volumetric rate

per unit length of source.

(a) Show that the Laplace equation for the velocity potential for this system is

(b) From this equation find the velocity potential, velocity, and pressure as functions of position:

where

is the value of the modified pressure far away from the source.

(c)

Discuss the applicability of the results in (b) to the flow field about a well drilled into a

large body of porous rock.

(d)

Sketch the flow net of streamlines and equipotential lines.

4B.9

Checking solutions to unsteady flow problems.

(a) Verify the solutions to the problems in Examples 4.1-1,2, and

by showing that they sat-

isfy the partial differential equations, initial conditions, and boundary conditions. To show

that Eq. 4.1-15 satisfies the differential equation, one has to know how to differentiate an inte-

gral using the Leibniz formula given in gC.3.

(b)

In Example 4.1-3 the initial condition is not satisfied by

Eq.

4.1-57.

Why?

4C.1

Laminar entrance flow in

slit? (Fig. 4C.1). Estimate the velocity distribution in the entrance

region of the slit shown in the figure. The fluid enters at

with v,

and v,

(v,), where

(v,) is the average velocity inside the slit. Assume that the velocity distribution in the entrance

region

(boundary layer region,

(potential flow region,

(4C.

-2)

in which

and

are functions of

yet to be determined.

numerical solution to this problem using the Navier-Stokes equation has been given by

Wang and

Longwell,

AKhE

Journal,

10,323-329

(1964).

146

Chapter

Velocity Distributions with More Than One Independent Variable

Fig.

4C.1. Entrance flow into a slit.

y=B

y=o

(a)

Use the above two equations to get the mass flow rate

through an arbitrary cross sec-

tion in the region

L,.

Then evaluate

from the inlet conditions and obtain

(b)

Next use Eqs. 4.4-13,4C.1-1, and 4C.1-2 with

replaced by

(why?) to obtain a differen-

tial equation for the quantity

S/B:

(c)

Integrate this equation with a suitable initial condition to obtain the following relation be-

tween the boundary-layer thickness and the distance down the duct:

(dl

Compute the entrance length

from Eq. 4C.1-5, where

is that value of x for which

S(x)

(e) Using potential flow theory, evaluate

in the entrance region, where

is the value

of the modified pressure at

Answers:

(d)

0.104(v,)B2/v;

(e)

O-2

Torsional oscillatory viscometer (Fig. 4C.2). In the torsional oscillatory viscometer, the fluid

is placed between a "cup" and "bob as shown in the figure. The cup is made to undergo

small sinusoidal oscillations in the tangential direction. This motion causes the bob, sus-

pended by a torsion wire, to oscillate with the same frequency, but

with

a different amplitude

Torsion wire

"Bob

"Cup"

%arced

oscillation of

outer CYlinder

Fig.

4C.2. Sketch of a torsional oscillatory viscometer.