CHAP_9_SEC1.PDF

7/27/2019 CHAP_9_SEC1.PDF

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57:020 Mechanics of Fluids and Transport Processes Chapter 9

Professor Fred Stern Typed by Stephanie Schrader Fall 1999 1

Cform(form drag)

Chapter 9: Surface Resistance

9.1 Introduction: drag and lift on immersed bodies

( )

+

=

S Sw

2D dAitdAinpp

AV2

11C

( )

= dAjnpp

AV2

1

1C

2L = lift coefficient

For inviscid fluid, CD= 0 since both the form and skin

friction components are zero, which is known as

DAlembert paradox. However, CL0 and can often bepredicted accurately with ideal-flow theory.

= drag coefficient

Cf(skin friction drag)


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external

flow

In general,

low Re < 1: Cf> Cform Stokes flow (lab 1)

med & high Re: 1. Cf>> Cform

t/c > Cf

t/c 1 i.e., bluff body

1. = subject of this chapter

2. = subject of chapter 11 along with CL

Topics of Chapter 9

9.2 Surface Resistance with Uniform Laminar Flow

1. parallel plates (internal flow)

2. flow down an inclined plane (open channel flow)

3. parallel plates with pressure gradient (internal flow)

In this class:

1. parallel plates

2. extend as per 3. including inclined flow

3. flow down an inclined plane

9.3 Boundary Layer Flow

9.4 Laminar

9.5 Turbulent

9.6 Transition control (brief comments)


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9.2 Surface Resistance with Uniform Laminar Flow

We now discuss a couple of exact solutions to the Navier-

Stokes equations. Although all known exact solutions(about 80) are for highly simplified geometries and flow

conditions, they are very valuable as an aid to our

understanding of the character of the NS equations and

their solutions. Actually the examples to be discussed are

for internal flow (Chapter 10) and open channel flow

(Chapter 15), but they serve to underscore and display

viscous flow. Finally, the derivations to follow utilize

differential analysis. See the text for derivations using CVanalysis.

1. Couette Flow

boundary conditions

First, consider flow due to the relative motion of two

parallel plates

Continuity 0

x

u=

Momentum2

2

dy

ud0 =

u = u(y)

v = o

0y

p

x

p=

=


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or by CV continuity and momentum equations:yuyu 21 =

u1= u2

( ) == = 0uuQAdVuF 12x

xdydy

dxyx

dx

dppyp

++

+= = 0

0dy

d=

i.e. 0dy

du

dy

d=

0dy

ud2

2

=

from momentum equation

Cdydu =

DyC

u +

=

u(0) = 0 D = 0

u(t) = U C =t

U

yt

Uu=

=

==t

U

dy

duconstant


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2. Generalization for inclined flow with a constant pressure

gradient

Continutity 0x

u

=

Momentum ( )2

2

dy

udzp

x0 ++

=

i.e.,

dx

dh

dy

ud

2

2

= h = p/+z = constant

plates horizontal 0dx

dz=

plates verticaldx

dz=-1

which can be integrated twice to yield

Aydx

dh

dy

du +=

BAy2

y

dx

dhu

2

++=

u = u(y)v = o

0y

p=


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now apply boundary conditions to determine A and B

u(y = 0) = 0 B = 0u(y = t) = U

2

t

dx

dh

t

UAAt

2

t

dx

dhU

2

=+=

+

=2

t

dx

dh

t

U1

2

y

dx

dh)y(u

2

= ( ) ytUyty

dxdh

2

2 +

This equation can be put in non-dimensional form:

t

y

t

y

t

y1

dx

dh

U2

t

U

u 2+

=

define: P = non-dimensional pressure gradient

=dx

dh

U2

t 2

zp

h +

=

Y = y/t

+

=

dx

dz

dx

dp1

U2

z2

Y)Y1(YPU

u+=

parabolic velocity profile


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t

y

t

Py

t

Py

U

u2

2

+=

[ ]

t

dyU

t

qu

udyq

t

0

t

0

==

=

+=

t

0

2

2dy

t

yy

t

Py

t

P

U

ut

=2

t

3

Pt

2

Pt+


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2

U

dx

dh

12

tu

2

1

6

P

U

u 2+

=+=

For laminar flow 1000tu

0, i.e., 0

dx

dh< the pressure decreases in the

direction of flow (favorable pressure gradient) and the

velocity is positive over the entire width

=

+

= sin

dx

dpz

p

dx

d

dx

dh

a) 0dxdp <

b) < sindx

dp

2. If P < 0, i.e., 0dx

dh> the pressure increases in the

direction of flow (adverse pressure gradient) and the

velocity over a portion of the width can become

negative (backflow) near the stationary wall. In this

case the dragging action of the faster layers exerted on

the fluid particles near the stationary wall is insufficientto over come the influence of the adverse pressure

gradient


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0sindx

dp>

> sindxdp or

dxdpsin


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Flow down an inclined plane

uniform flow velocity and depth do not

change in x-direction

Continuity 0dx

du=

x-momentum ( )2

2

dy

udzp

x0 ++

=

y-momentum ( ) +

= zpy0 hydrostatic pressure variation

0dx

dp=

= sindy

ud2

2

cysindy

du+

=


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DCy2

ysinu

2

++

=

dsinccdsin0dy

du

dy

+=+

==

=

u(0) = 0 D = 0

dysin

2

ysinu

2

+

=

= ( )yd2ysin2

u(y) = ( )yd2y2

sing

d

0

32

d

0 3

ydysin

2udyq

==

=

sind3

1 3

=

== sin3

gdsind

3

1

d

qV

22

avg

discharge per

unit width


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in terms of the slope So= tan sin

=

3

SgdV o

2

Exp. show Recrit500, i.e., for Re > 500 the flow willbecome turbulent

=

cosy

p

=

dVRecrit 500

Cycosp +=

( ) Cdcospdp o +==

i.e., ( ) opydcosp +=

* p(d) > po

* if = 0 p = (d y) + poentire weight of fluid imposed

if = /2 p = pono pressure change through the fluid


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9.3 Qualitative Description of the Boundary Layer

Recall our previous description of the flow-field regions for

high Re flow about slender bodies


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w= shear stress

wrate of strain (velocity gradient)

=0y

y

u

=

large near the surface where

fluid undergoes large changes to

satisfy the no-slip condition

Boundary layer theory is a simplified form of the complete

NS equations and provides was well as a means ofestimating Cform. Formally, boundary-layer theory

represents the asymptotic form of the Navier-Stokesequations for high Re flow about slender bodies. As

mentioned before, the NS equations are 2ndorder nonlinear

PDE and their solutions represent a formidable challenge.

Thus, simplified forms have proven to be very useful.


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Near the turn of the century (1904), Prandtl put forth

boundary-layer theory, which resolved DAlemberts

paradox. As mentioned previously, boundary-layer theory

represents the asymptotic form of the NS equations for highRe flow about slender bodies. The latter requirement is

necessary since the theory is restricted to unseparated flow.

In fact, the boundary-layer equations are singular at

separation, and thus, provide no information at or beyond

separation. However, the requirements of the theory are

met in many practical situations and the theory has many

times over proven to be invaluable to modern engineering.

The assumptions of the theory are as follows:

Variable order of magnitude

u U O(1)

v


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The theory assumes that viscous effects are confined to a

thin layer close to the surface within which there is a

dominant flow direction (x) such that u U and v


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Some important aspects of the boundary-layer equations:

1) the y-momentum equation reduces to

0yp =

i.e., p = pe= constant across the boundary layer

from the Bernoulli equation:

=+ 2ee U2

1p constant

i.e.,x

UU

x

p ee

e

=

Thus, the boundary-layer equations are solved subject to

a specified inviscid pressure distribution

2) continuity equation is unaffected

3) Although NS equations are fully elliptic, the

boundary-layer equations are parabolic and can be

solved using marching techniques

4) Boundary conditions

u = v = 0 y = 0

u = Ue y =

+ appropriate initial conditions @ xi

edge value, i.e.,

inviscid flow value!


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There are quite a few analytic solutions to the boundary-

layer equations. Also numerical techniques are available

for arbitrary geometries, including both two- and three-

dimensional flows. Here, as an example, we consider thesimple, but extremely important case of the boundary layer

development over a flat plate.

9.4 Quantitative Relations for the Laminar Boundary Layer

Laminar boundary-layer over a flat plate: Blasius solution

(1908) student of Prandtl

0y

v

x

u=

+

2

2

y

u

y

uv

x

uu

=

+

u = v = 0 @ y = 0 u = U @ y =

We now introduce a dimensionless transverse coordinate

and a stream function, i.e.,

=

y

x

Uy

( )= fxU

Note:x

p

= 0

for a flat plate


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( )=

=

= fUyy

u = U/uf

( )ffxU

21

xv =

=

substitution into the boundary-layer equations yields

0f2ff =+ Blasius Equation

0ff == @ = 0 1f = @ = 1

The Blasius equation is a 3rdorder ODE which can be

solved by standard methods (Runge-Kutta). Also, series

solutions are possible. Interestingly, although simple in

appearance no analytic solution has yet been found.

Finally, it should be recognized that the Blasius solution is

a similarity solution, i.e., the non-dimensional velocity

profile fvs. is independent of x. That is, by suitablyscaling all the velocity profiles have neatly collapsed onto a

single curve.

Now, lets consider the characteristics of the Blasius

solution:

U

u

vs. y

V

U

U

v

vs. y


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Re

x5= value of y where u/U = .99

=

xUxRe


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=U/x2

)0(fU

w

i.e.,xRe

664.0

U

2c

x2

wf

==

=

see below

==L

0fff )L(c2dxc

L

1C

=LRe

328.1

LU

Other:

=

=

0 x

*

Re

x7208.1dy

U

u1 displacement thickness

measure of displacement of inviscid flow to due

boundary layer

=

=

0 xRe

x664.0dy

U

u

U

u1 momentum thickness

measure of loss of momentum due to boundary layer

H = shape parameter =*

=2.5916

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CHAP_9_SEC1.PDF