# Chapter 6 2009_2

• View
219

0

Embed Size (px)

### Text of Chapter 6 2009_2

• 7/27/2019 Chapter 6 2009_2

1/41

1

Chapter 6

Flow Analysis Using Differential Methods(Differential Analysis of Fluid Flow)

• 7/27/2019 Chapter 6 2009_2

2/41

2

In the previous chapter-- Focused on the use of finite control volume for the

solution of a variety of fluid mechanics problems. The approach is very practical and useful since it

doesnt generally require a detailed knowledge of thepressure and velocity variations within the control

volume. Typically, only conditions on the surface of the

control volume entered the problem.

There are many situations that arise in which thedetails of the flow are important and the finitecontrol volume approach will not yield thedesired information

• 7/27/2019 Chapter 6 2009_2

3/41

3

For example --

We may need to know how the velocity varies over the cross

section of a pipe, or how the pressure and shear stress vary

along the surface of an airplane wing.

we need to develop relationship that apply at a point,

or at least in a very small region ( infinitesimal volume)within a given flow field.

involve infinitesimal control volume (instead of finite

control volume)

differential analysis (the governing equations are

differential equation)

• 7/27/2019 Chapter 6 2009_2

4/41

4

In this chapter

(1) We will provide an introduction to the differential

equation that describe (in detail) the motion of fluids.

(2) These equation are rather complicated, partial differential

equations, that cannot be solved exactly except in a few

cases.

(3) Although differential analysis has the potential for

supplying very detailed information about flow fields, the

information is not easily extracted.

(4) Nevertheless, this approach provides a fundamental basis

for the study of fluid mechanics.

(5) We do not want to be too discouraging at this point,

since there are some exact solutions for laminar flow that

can be obtained, and these have proved to very useful.

• 7/27/2019 Chapter 6 2009_2

5/41

5

(6) By making some simplifying assumptions, many otheranalytical solutions can be obtained.

for example , small 0 neglected

inviscid flow.

(7) For certain types of flows, the flow field can be conceptuallydivided into two regions

(a) A very thin region near the boundaries of the system inwhich viscous effects are important.

(b) A region away from the boundaries in which the flow isessentially inviscid.

(8) By making certain assumptions about the behavior of the fluidin the thin layer near the boundaries, and

using the assumption of inviscid flow outside this layer, a large

class of problems can be solved using differential analysis .

the boundary problem is discussed in chapter 9.

Computational fluid dynamics (CFD) to solve differential eq.

• 7/27/2019 Chapter 6 2009_2

6/41

6

)(.)((.)(.)(.)(.)(.)(.)

.

.

problem.particularafort,z,y,on x,dependlyspecificalcomponents

velocitythesehowdeterminetoisanalysisaldifferentiofgoalstheofOne

vtz

wy

vx

utt

Da

a

z

uw

y

uv

x

uu

t

ua

dt

vd

z

vw

y

vv

x

vu

t

va

z

y

x

• 7/27/2019 Chapter 6 2009_2

7/41

7

elementtheofndeformatioangularx

v

y

u

elementtheofndeformatiolinear

z

w

y

v

x

uNote

v

ratedilationvolumetricvz

w

y

v

x

u

dt

d

,

,,:

fluid,ibleincompressanfor0

)(1

• 7/27/2019 Chapter 6 2009_2

8/41

8

flowalIrrotation0

zeroisaxis-zthearoundRotation

)(2

1assuch(6.12)Eq.From

2Define

vor

y

u

x

v

x

v

y

uww

y

u

x

vw

Vcurlvwvorticity

OBOA

z

Vcurlv

wvuzyx

kji

k

y

u

x

vj

x

w

z

ui

z

v

y

w

kwjwiw

vectorrotationtheW

zyx

2

1)(

2

1

2

1

})()(){(

2

1

• 7/27/2019 Chapter 6 2009_2

9/41

9

6.2.1 Differential Form of Continuity Equation

inout

cv

vAvA

dAnv

zyx

t

d

t

d

t

zyxd

)()(

elementtheofsurfacesthethroughflowmassofrateThe)(

)(

0

• 7/27/2019 Chapter 6 2009_2

10/41

10

s

yx

xx

xxx

x

uuu

y

x

uuu

)(,)(assuch,terms

orderhighneglecting--expansionseriesTaylor

2

)(|

2

)(|

direction-xtheinflowThe

2

2

2

zyxz

w

zyxyv

zyxx

uzy

x

x

uuzy

x

x

uu

)(direction-zinrateNet

)24.6()(direction-yinrateNet

similarly

)23.6()(

]2

[]2

[

direction-in xoutflowmassofrateNet

• 7/27/2019 Chapter 6 2009_2

11/41

11

.formaldifferentiinequationcontinuityThe

)27.6(0

outflowmassofrateNet:

0][

0)(Since

z

w

y

v

x

u

t

zyxz

wzyx

y

vzyx

x

uNote

zyxz

wzyx

y

vzyx

x

uzyx

t

dAnvd

t cscv

• 7/27/2019 Chapter 6 2009_2

12/41

12

)31.6(0

)30.6(0

0

flowibleincompressFor--

)29.6(0

0)(

)28.6(0

formIn vector

mechanicsfluidofequationslfundamentatheofOne--

z

w

y

v

x

uor

vt

const

z

w

y

v

x

uor

v

vt

• 7/27/2019 Chapter 6 2009_2

13/41

13

equation.continuityhesatisfy ttorequired,w:Determine

?

flowibleincompressanFor

26.Example

222

w

zyzxyv

zyxu

),(2

3nIntegratio

3)(2

0)()(

0

continuityofequationthefrom:Solution

2

222

yxczxzw

zxzxxz

w

zwzyzxy

yzyx

x

z

w

y

v

x

u

• 7/27/2019 Chapter 6 2009_2

14/41

14

6.2.2 Cylindrical Polar Coordinates

• 7/27/2019 Chapter 6 2009_2

15/41

15

01)(1

0)()(1

)(1

scoordinatelcylindricain

equationycontinualltheofformaldifferentitheisThis

)33.6(0

)()(1)(1

z

vv

rr

rv

r

vz

vr

vrrr

z

vv

rr

vr

rt

zr

zr

zr

• 7/27/2019 Chapter 6 2009_2

16/41

16

6.2.3 The Stream Function

)36.6(0

)2(0)(0

0

equationContinuity

y

v

x

u

flowDz

wcte

twhere

z

w

y

v

x

u

t

0)()(

eq.continuitythesatisfiesitthatso;where

xyyxy

v

x

u

xv

yu

yx

satisfiedbewillmassofonconservati

unknowoneunknowstwofunction

streamusing

v

u

• 7/27/2019 Chapter 6 2009_2

17/41

17

6.3.Example

)42.6(1

01)(1

0)()(1)(1

flow.D-2place,,ibleIncompress

forequationycontinuallthe,scoordinatelcylindricaIn

rv

rv

v

rr

rv

rz

vv

rr

vr

rt

r

rzr

• 7/27/2019 Chapter 6 2009_2

18/41

18

Examp le 6.3 Stream Func t ion

The velocity component in a steady, incompressible, twodimensional flow field are

Determine the corresponding stream function and show on asketch several streamlines. Indicate the direction of glow alongthe streamlines.

4xv2yu

• 7/27/2019 Chapter 6 2009_2

19/41

19

Example 6.3 Solution

(y)fx2(x)fy 2212

Cyx222

From the definition of the stream function

x4x

vy2y

u

For simplicity, we set C=0

22yx2

=0

01

2/

xy22

• 7/27/2019 Chapter 6 2009_2

20/41

20

6.3 Conservation of Linear Momentum

amF

amFordt

vdmFEq

VCsmallFor

FdAnvvdvtdt

vmD

dmvdvP

Pdt

Ddv

dt

D

dt

vmD

sys

cvcv

csvc

cv

sys

syssys

sys

sys

systemaforlaw2ndNewtonsThe

)44.6(

..

)44.6()()(

momentumlinearfor thet theoremtransporReynoldstheFrom

where

)(

momentumlinearFor the

..

• 7/27/2019 Chapter 6 2009_2

21/41

21

Figure 6.9 (p. 287)Components of force acting on an arbitrary differential area.

• 7/27/2019 Chapter 6 2009_2

22/41

22

Figure 6.10 (p. 287)Double subscript notation for stresses.

• 7/27/2019 Chapter 6 2009_2

23/41

23

Figure 6.11 (p. 288)Surface forces in the x direction acting on a fluid element.

• 7/27/2019 Chapter 6 2009_2

24/41

24

6.3.2 Equation of Motion

Velocitiesstresses-----Unknowns

.restatormotio

Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Science
Education
Documents
Documents