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Brij Bhooshan Asst. Professor B.S.A College of Engg. & Technology, Mathura (India)

Copyright by Brij Bhooshan @ 2013

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MMaatthhuurraa,, UUttttaarr PPrraaddeesshh,, ((IInnddiiaa))

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PPuurrvvii BBhhoooosshhaann

In This Chapter We Cover the Following Topics

S. No. Contents Page No.

8.1 Thickness Of Boundary Layer

Displacement Thickness

Momentum Thickness

Energy Thickness

3

4

5

6

8.2 Momentum Integral Equation 7

8.3 Solution Of The Momentum Integral Equation For Flow Over A Flat

Plate

Velocity Profile

Boundary Layer Thickness

Skin Friction Coefficient

Transverse Component Of Velocity

10

10

11

12

12

8.4 Boundary Layers With Pressure Gradient 13

References:

1. Andersion J. D. Jr., Computational Fluid Dynamics “The Basics with applications”,

1st Ed., McGraw Hill, New York, 1995.

2. Frank M. White, Fluid Mechanics, 6th Ed., McGraw Hill, New York, 2008.

3. Frank M. White, Viscous Fluid Flow, 2nd Ed., McGraw Hill, New York, 1991.

4. Fox and McDonald’s, Introduction to Fluid Mechanics, 6th Ed., John Wiley & Sons,

Inc., New York, 2004.

5. Welty James R., Wicks Charles E., Wilson Robert E. and, Rorrer Gregory L.,

Fundamentals of Momentum, Heat, and Mass Transfer, 5th Ed. John Wiley & Sons,

Inc., New York, 2008.

6. Mohanty A. K., Fluid Mechanics, 2nd Ed, Prentice Hall Publications, New Delhi,

2005.




2 Chapter 8: Boundary Layer Theory

7. Gupta Vijay, and, Gupta S. K., Fluid Mechanics & its Applications, 1st Ed., New Age

International (P) Limited, Publishers, New Delhi 2005.

8. Cengel & Cimbala, Fluid Mechanics Fundamentals and Applications, 1st Ed.,

McGraw Hill, New York, 2006.

9. Modi & Seth, Hydraulics and Fluid Mechanics, Standard Publications.

Please welcome for any correction or misprint in the entire manuscript and your

valuable suggestions kindly mail us [email protected].




3 Fluid Mechanics By Brij Bhooshan

We recall that the boundary layer model basically divides the flow of a real fluid past a

solid body to two zones: a viscous layer surrounding the solid surface, and a zero shear

stress zone beyond it. The model is applicable to high Reynolds number flows in which

the pressure distribution of the free stream is impressed on the viscous layer. A flow,

initially uniform having one component of velocity say U∞ becomes two dimensional on

encountering the solid surface. A transverse velocity component, an order of magnitude

smaller than the axial, is induced due to viscous actions.

Solution of the boundary layer equations provides the methods for estimating the

factional resistance along the wetted surface of a body.

The boundary layer is a thin layer adjacent to the solid surface in which the viscous

effects are important. Although the thickness of the boundary layer is very thin, one

cannot neglect it. Therefore, it is important to analyse the flow within the boundary

layer in details. The velocity close to the solid surface will be same as the velocity of

solid due to no-slip boundary condition. The velocity away from the surface will be

higher and therefore, there exists a velocity gradient. The velocity gradient in a

direction normal to the surface is large compared to streamwise direction.

When a real fluid flows past a solid boundary, a layer of fluid which comes in contact

with the boundary surface adheres to it on account of viscosity. Since this layer of fluid

cannot slip away from the boundary surface it attains the same velocity as that of the

boundary. In other words, at the boundary surface there is no relative motion between

the fluid and the boundary. This condition is termed as no slip condition. If the

boundary is moving, the fluid adhering to it will have the same velocity as that of the

boundary. However, if the boundary is stationary, the fluid velocity at the boundary

surface will be zero. Thus at the boundary surface the layer of fluid undergoes

retardation. This retarded layer of fluid further causes retardation for the adjacent

layers of the fluid, thereby developing a small region in the immediate vicinity of the

boundary surface in which the velocity of flowing fluid increases gradually from zero at

the boundary surface to the velocity of the main stream. This region is named as

boundary layer. In the boundary layer region since there is a larger variation of velocity

in a relatively small distance, there exists a fairly large velocity gradient [du/dy]

normal to the boundary surface. As such in this region of boundary layer even if the

fluid has small viscosity, the corresponding shear stress = (du/dy), is of appreciable

magnitude. Farther away from the boundary this retardation due to the presence of

viscosity is negligible and the velocity there will be equal to that of the main stream.

The flow may thus be considered to have two regions, one close to the boundary in the

boundary layer zone in which due to larger velocity gradient appreciable viscous forces

are produced and hence in this region the effect of viscosity is mostly confined; and

second outside the boundary layer zone in which the viscous forces are negligible and

hence the flow maybe treated as non-viscous or inviscid. The concept of boundary layer

was first introduced by L. Prandtl in 1904 and since then it has been applied to several

fluid flow problems. In the following paragraphs the characteristics of flow in the

boundary layer are discussed.

8.1 THICKNESS OF BOUNDARY LAYER

The velocity within the boundary layer increases from zero at the boundary surface to

the velocity of the main stream asymptotically. Therefore the thickness of the boundary

layer represented by is arbitrarily defined as that distance from the boundary surface





in which the velocity reaches 99% of the velocity of the main stream. In other words, the

boundary layer thickness may be considered equal to the distance y from the boundary

surface at which u = 0.99U∞. This definition however gives an approximate value of the

boundary layer thickness and hence is generally termed an nominal thickness of the

boundary layer. For greater accuracy the boundary layer thickness is defined in terms of

certain mathematical expressions which are the measures of the effect of boundary layer

on the flow. Three such definitions of the boundary layer thickness which are commonly

adopted are the displacement thickness *, the momentum thickness θ and the energy

thickness E.

Displacement Thickness

The displacement thickness, * is the distance measured perpendicular to the solid

boundary by which the boundary would have to be displaced in a frictionless flow to give

the same mass flow rate as with the boundary layer flow.

The presence of the v component will cause deviations of the stream lines. In Diagram

8.1, a uniform stream U∞ was approaching the plate without any angle of incidence. The

plate surface therefore constituted the = 0 stream line.

Diagram 8.1 Divergence of Stream Lines within the Boundary Layer.

With increasing y, both u and v components increase and the stream lines diverge away

from the plane of the plate. The consequence is that some mass moves out of the

boundary layer region. If we imagine a plane O-O parallel to the plate then the mass

flux through a section x shall be less than through an upstream section at (x x). As

the mass flux will decrease in the downstream direction, other extensive properties such

as momentum and kinetic energy will also decrease between the parallel planes.

The downstream decrease in mass flow, between the plate and a parallel plane, due to

viscous effects can be visualized as equivalent to the "blockage of the flow passage" by a

thickness (x) whereas the velocity profile is maintained uniform. The equivalence is

shown in Diagram 8.2 * is termed as the displacement thickness and its value is

evaluated using the mass flux equivalence indicated in Diagram 8.2.

Diagram 8.2 Physical Model of Displacement Thickness.

(a) Real viscous flow (b) Equivalent non-viscous flow with

equal mass flow rate

*

U∞

h

m

0

m

0

y dy

y u

U∞

U∞

= 0 U∞

x x x

x x x 0 0

v u





To address the concept of displacement thickness, let us consider the flow of a fluid

having free stream velocity U∞ over a thin, smooth, stationary flat plate as shown in

Diagram 8.2. Let us also consider an elementary strip of thickness dy at a distance y

from the plate as shown in Diagram 8.2.

The mass flow rate through the elemental strip

= Density × Velocity of flow × Area of elemental strip

= × u × dA = uwdy

where w is the width of the plate.

When the plate were not there, the fluid would have been flowing with the free stream

velocity U∞.

Then the mass flow rate through the elemental strip would have been


= × U∞ × dA = U∞wdy

Reduction of mass flow rate through the elemental strip is then

= U∞wdy uwdy = (U∞ u)wdy

Total reduction in mass flow rate due to presence of plate can be found by integrating

the elemental mass flow rate over the boundary layer thickness and is given by

Let * be the distance by which the solid boundary is displaced in the normal direction

and the velocity of flow for the displaced distance * is equal to the free stream velocity,

i.e., u = U∞. Then the mass flow rate flowing through the distance * becomes

= Density × Velocity of flow × Area

= U∞w* [8.1b]

Equating Eqs. (8.1a) and (8.1b), one can write

For constant density, the above equation yields to

Since for flow over a flat plate, free stream velocity U∞ is constant.

Momentum Thickness

The momentum thickness, θ is the distance measured perpendicular to the solid

boundary by which the boundary should be displaced to compensate for the reduction in

momentum due to the formation of the boundary layer.

Diagram 8.3 Concept of Momentum Thickness.

(b) Ideal flow for momentum equivalence

m

(a) Real flow

0

θ

u y

h

0

m

U∞

U∞





The flow passage is considered to be 'blocked' by an additional thickness θ, such that the

momentum crossing [h (* + θ)] in an ideal flow is equal to the momentum efflux over

the whole height h in a real flow (see Diagram 8.3).

Let us consider the flow of a fluid having free stream velocity U∞ over a thin, smooth,

stationary flat plate as shown in Diagram 8.2. Let us also consider an elementary strip

of thickness dy at a distance y from the plate as shown in Diagram 8.2.



= × u × dA = uwdy

The rate of momentum flowing through the elemental strip

= Mass flow rate × Velocity of flow

= uwdy × u = u2wdy

Momentum of the same fluid in absence of the plate (without boundary layer) is

= Mass flow rate × Free stream velocity

= uwdy × U∞ = uU∞wdy

Reduction of rate of momentum flowing through the elemental strip is then

= uU∞wdy u2wdy = u(U∞ u)wdy

Total reduction in rate of momentum flow is given by

Let θ be the distance by which the solid boundary is displaced in the normal direction

and the velocity of flow for the displaced distance θ is equal to the free stream velocity,

i.e., u = U∞.

Loss of rate of momentum flowing through the distance θ becomes

= Mass flow rate × Velocity of flow

= U∞wθ × U∞ = wθ [8.3b]



Since for flow over a flat plate, free stream velocity U∞ is constant, the above equation

becomes

Energy Thickness

The energy thickness, ** is defined as the distance measured perpendicular to the solid

boundary by which the boundary should be displaced to compensate for the reduction in

kinetic energy of flowing fluid due to the formation of the boundary layer.

Consider the flow of a fluid having free stream velocity U∞ over a thin, smooth,

stationary flat plate as shown in Diagram 8.2. Let us also consider an elementary strip

of thickness dy at a distance y from the plate as shown in Diagram 8.2.







= × u × dA = uwdy

The rate of kinetic energy flowing through the elemental strip

= (Mass flow rate × Velocity of flow)/2

= uwdy × u2/2 = u3wdy/2

Kinetic energy of the same fluid in absence of the plate (without boundary layer) is

= (Mass flow rate × Free stream velocity) /2

= uwdy × × 1/2= u

wdy/2

Reduction of rate of kinetic energy flowing through the elemental strip is then

= u wdy/2 u3wdy/2 = u(

u2)wdy/2

Total reduction in rate of momentum flow is given by

Let ** be the distance by which the solid boundary is displaced in the normal direction

and the velocity of flow for the displaced distance ** is equal to the free stream velocity,

i.e., u = U∞.

Loss of rate of momentum flowing through the distance ** becomes

= (Mass flow rate × Velocity2)/2

= (w** × U∞) = w** /2 [8.5b]



or

8.2 MOMENTUM INTEGRAL EQUATION

In Diagram 8.4, a free stream flow at U∞, approaches a surface whose leading edge

coincides with x = 0. x is measured along the surface and y perpendicular to it (x) is the

thickness of the boundary layer at a location x.

Diagram 8.4 Control Volume Analysis of the Boundary

1-2-3-4 define a control volume whose faces 1-2 and 3-4 are parallel to toe solid surface

and the other two faces are perpendicular to the surface. The height of the face 1-4 or 2-

3 is l, and l is greater than the thickness of the boundary layer.

U∞

y

x (x)

1 2

4 3

dy

p

u

x





Fluid masses enter through faces 1-4, 1-2 and 2-3 carrying with them the momentum

prevailing in the respective neighbourhood. No mass enters through 3-4, the face being

coincident with the solid wall. The face 3-4, on the other hand, experiences the wall

shear stress and is x long. A unit depth perpendicular to the plane of 1-2-3-4 is being

considered.

Conservation of Mass

The axial velocity at a location y on 1-4 face is u, and the mass entering through a strip

dy is udy. Thus mass inlet through the whole of 1-4 is

mass leaving through 2-3 is written by using Taylor's expansion as

Since at steady-state no change of properties takes place within the control volume, the

excess of outflow through 2-3 is replenished through 1-2. In other words, the mass

coming from the free stream zone into the control volume is

Conservation of Momentum

The momentum in-flow through a strip dy is u2dy, and through the face 1-4 is

Outflow through 2-3 is

Inflow through 1-2 due to the mass coming from the zone of U∞ is

Combining (i), (ii), and, (iii), we obtain the net efflux of momentum through the control

surface as

The face 1-2 being in the free stream zone, no shear stress acts on it. The pressure on

face 1-4 is p, and is independent of y by boundary layer theory.

The external forces acting on the control volume are hence

Combining Eqs. (8.7a) and (8.7b), we write the momentum balance for the control

volume as





In order to evaluate the pressure gradient we can move into the free stream zone and

use Bernoulli's equation

or

Equation (8.7c) is now written using Eq. (8.8)

or for incompressible flow

Consider the differentiation

or

Thus Eq. (8.9a) can be rewritten as

or

The limits of integration 0 to l can be split up into 0 to and to l. In the free stream

region of to l, however, u U∞ and each of the integrand is zero Hence, Eq. (8.9b) is,

effectively,

It will be seen later that the two terms in the l.h.s. of Eq. (8.10) represent variations of

significant physical parameters.

In its present from at (8.10) the integral equation for momentum can represent both

laminar and turbulent flows, since no assumption has yet been made for the shear

stress, w.

In case U∞ = C, such as it happens when a uniform flow continues past a flat plate at

zero incidence, the second term on the l.h.s. is zero, since the pressure gradient or

dU∞/dx is zero.

Then, above equation will be





Above equation is termed as Von-Karman momentum integral equation.

8.3 SOLUTION OF THE MOMENTUM INTEGRAL EQUATION FOR FLOW

OVER A FLAT PLATE

The steps involved in solving Eq. (8.11) are:

(i) choosing a velocity profile that satisfies all essential and some additional

boundary conditions,

(ii) evaluating the integrals and reducing the l.h.s. to a differential expression on

,

(iii) postulating the law of shear stress for w, depending on the flow regime. For

laminar flow w = (u/y)y = 0 by Newton's law of shear stress,

(iv) solving the differential equation for , and

(v) tracing back to w to estimates the skin friction in flow along the both surface.

We shall illustrate the method of solution by considering an incompressible, laminar

steady flow of a Newtonian fluid along a flat plate at zero incidence.

Velocity Profile

The essential conditions to be satisfied by the boundary layer velocity profile are:

(i) y = 0, u = 0, v = 0, no slip at the wall

(ii) y = , u = U∞, free stream velocity at the edge of the

boundary layer

(iii) y = , u/y = 0, no shear stress at the edge of the boundary

layer.

(iv)

Since the most important objective of the exercise is to estimate the wall shear stress as

correctly as possible, while pursuing the approximate integral solution, we force the

velocity profile to satisfy the differential boundary layer equation (exact) on the wall.

This then becomes the additional boundary condition of first priority. Thus

For a flat plate dp/dx = 0, and the condition reduces to

(v)

It is then proposed that the boundary layer velocity profile can be written in terms of a

polynomial in y with the number of terms equalling the number of boundary conditions

to be satisfied. For the four conditions listed by us, we choose

u = A + By + Cy2 + Dy3 [8.12]

the degree of the polynomial in u can be increased by postulating additional boundary

conditions. It is customary to verify the convergence of an approximate integral solution

by comparing the results of the velocity profiles, one a degree higher than the other.

From Eq. (8.12), we obtain

and





Substitution of the four boundary conditions results in:

(i) 0 = A

(ii) U∞ = B + C2 + D3

(iii) 0 = B + 2C + 3D2

(iv) 0 = 2C

Solutions of the four algebraic equations yield A = 0;

; C = 0;

, and the

velocity profile is, by substitution,

Boundary Layer Thickness

The integral momentum Eq. (8.10) for flow over a flat plate reduces to

The integral in Eq. (8.14) is evaluated, by substituting for u/U∞. from Eq. (8.13) to be

39/280, or

We now impose the laminar flow condition, for which

or

Thus the integrated momentum equation for the flat plate results in

or

which upon integration yields

The hydrodynamic boundary layer starts growing from the leading edge when the free

stream flow first encounters the solid surface. We can therefore chose = 0, at x = 0,

resulting in C = 0. We should, however, recall that the boundary layer equations are of

course not valid in the immediate vicinity of x = 0, for Rex is low.

Thus from Eq. (8.17b), we obtain

or





Note that the information of the order of magnitude analysis of Chapter 7, that (/x)

1/ is corroborated by a formal solution of the viscous layer. The constant of

proportionality C in the expression /x = C/ varies slightly with the choice of the

velocity profile. Solution of the differential form of the boundary layer equation for a flat

plate geometry results in C = 5.

Skin Friction Coefficient

The wall skin friction coefficient is defined as

Substituting for w from (8.16), we get

or

where C = 4.64 for the 3rd degree polynomial we chose for the velocity profile. We thus

have

or

The coefficient in Eq. (8.20) becomes 0.664 for the exact solution when C = 5.

The average value of skin friction coefficient over a plate of length L is Plained by

integration:

or

or

Expression (8.21a) signifies that the average value of Cf is twice the local value at the

end of the plate. Note that when a plate is wetted on both sides, the friction will be twice

the value obtainable through (8.21).

Transverse Component of Velocity

The viscous action gives rise to the v component of velocity. We can estimate its value at

a position y within the boundary layer by making use of the continuity relationship:





Since

or

or

using Eq. (10.12a), or

But

is the local transverse velocity.

v is zero at y = 0. At the edge of the boundary layer it equals:

or

and is its maximum value.

Recall that while performing the order of magnitude analysis in Chapter 6, we arrived

at the condition

We may verify this by considering Eq. (8.24).

or

Since O(*) =

, Eq. (8.25) is proved.

8.4 BOUNDARY LAYERS WITH PRESSURE GRADIENT

The flat-plate analysis of the previous section should give us a good feeling for the

behavior of both laminar and turbulent boundary layers, except for one important effect:

flow separation. Prandtl showed that separation like that is caused by excessive

momentum loss near the wall in a boundary layer trying to move downstream against





increasing pressure, dp/dx > 0, which is termed as an adverse pressure gradient. The

opposite case of decreasing pressure, dp/dx < 0, is named as a favorable gradient, where

flow separation can never occur. In a typical immersed-body flow, the favorable gradient

is on the front of the body and the adverse gradient is in the rear.

We can explain flow separation with a geometric argument about the second derivative

of velocity u at the wall. From the momentum equation at the wall, where u = v = 0, we

obtain

or

for either laminar or turbulent flow. Thus in an adverse gradient the second derivative

of velocity is positive at the wall; yet it must be negative at the outer layer (y = ) to

merge smoothly with the mainstream flow U(x). It follows that the second derivative

must pass through zero somewhere in between, at a point of inflection, and any

boundary layer profile in an adverse gradient must exhibit a characteristic S shape.

Diagram 8.5 illustrates the general case. In a favorable gradient (Fig. 7.7a) the profile is

very rounded, there is no point of inflection, there can be no separation, and laminar

profiles of this type are very resistant to a transition to turbulence.

In a zero pressure gradient (Diagram 8.5b), e.g., flat-plate flow, the point of inflection is

at the wall itself. There can be no separation, and the flow will undergo transition at Rex

no greater than about 3 × 106, as discussed earlier.

Diagram 8.5 Effect of pressure gradient on boundary-layer profiles; PI = 0 point of inflection.

In an adverse gradient (Diagram 8.5c to e), a point of inflection (PI) occurs in the

boundary layer, its distance from the wall increasing with the strength of the adverse

gradient. For a weak gradient (Diagram 8.5c) the flow does not actually separate, but it

is vulnerable to transition to turbulence at Rex as low as 105. At a moderate gradient, a

critical condition (Diagram 8.5d) is reached where the wall shear is exactly zero (u/y =

0). This is defined as the separation point (w = 0), because any stronger gradient will

(e)

Excessive adverse

gradient:

Backflow

at the wall:

Separated

flow region

(d)

Critical

adverse

gradient:

Zero slope

at the wall:

Separation

(c)

Weak adverse

gradient:

No separation,

PI in the flow

(b)

Zero

gradient:

No separation,

PI at wall

(a)

Favorable

gradient:

No separation,

PI inside wall

U

u

PI

U

u

PI

U

u

u

PI

U

u

PI

U

Backflow





actually cause backflow at the wall (Diagram 8.5e): the boundary layer thickens greatly,

and the main flow breaks away, or separates, from the wall.

The flow profiles of Diagram 8.5 usually occur in sequence as the boundary layer

progresses along the wall of a body. For example, a favorable gradient occurs on the

front of the body, zero pressure gradient occurs just upstream of the shoulder, and an

adverse gradient occurs successively as we move around the rear of the body.

Diagram 8.6 Boundary-layer growth and separation in a nozzle-diffuser configuration.

A second practical example is the flow in a duct consisting of a nozzle, throat, and

diffuser, as in Diagram 8.6. The nozzle flow is a favorable gradient and never separates,

nor does the throat flow where the pressure gradient is approximately zero. But the

expanding-area diffuser produces low velocity and increasing pressure, an adverse

gradient. If the diffuser angle is too large, the adverse gradient is excessive, and the

boundary layer will separate at one or both walls, with backflow, increased losses, and

poor pressure recovery. In the diffuser literature this condition is called diffuser stall, a

term used also in airfoil aerodynamics to denote airfoil boundary-layer separation. Thus

the boundary-layer behavior explains why a large-angle diffuser has heavy flow losses

and poor performance.

Presently boundary-layer theory can compute only up to the separation point, after

which it is invalid. New techniques are now developed for analyzing the strong

interaction effects caused by separated flows.

Now,

Flow is separated.

Flow is one the range of separation.

Throat:

Constant

pressure

and area

Velocity

constant

Zero

gradient

Nozzle:

Decreasing

pressure

and area

Increasing

velocity

Favorable

gradient

Diffuser:

Increasing pressure

and area

Decreasing velocity

Adverse gradient

(boundary layer thickens)

U(x)

U(x)

(x)

(x)

Separation

point w = 0 Profile point

of inflection

Boundary

layer

Nearly

inviscid

core flow

Separation

Backflow

Dividing

stream line





The flow will not separated or flow will remain attach with the

surface.

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