Chapter 2 Reynolds Transport Theorem (RTT)

Chapter 2 Reynolds Transport Theorem (RTT)

2.1 The Reynolds Transport Theorem

2.2 Continuity Equation

2.3 The Linear Momentum Equation

2.4 Conservation of Energy

2.1 The Reynolds Transport Theorem (1)


ρ ( ) (inflow if negative) (11-41)net out in

CS

B B B b V n dA

42)-(11 ρ dV bBCV

CV

43)-(11 )(ρρ CSCV

dAnVbdV bdt

d

dt

dB :General sys


Special Case 1: Steady Flow

Special Case 2: One-Dimensional Flow

44)-(11 )(ρ CS

dAnVbdt

dB :flowSteady sys

:flow ldimensiona-One

45)-(11 ρ-ρρin

exiteach for out

exiteach for CV

iiiieeeesys AVbAVbdV b

dt

d

dt

dB

46)-(11 -ρinoutCV

iieesys bmbmdV b

dt

d

dt

dB

2.2 Continuity Equation (1) An Application: The Continuity Equation

47)-(11 )(ρρ0 CS

dAnVdVdt

d :equation Continuity

CV

48)-(11

in

iout

e mm :flowSteady

49)-(11 or AA 221 AVAVVV

:constant stream,Single

22111

2.3 The Linear Momentum Equation (1)

..50)-(11 )V(m

dt

d

dt

Vdm amF

51)-(11 ρdVVdt

dF

sys

52)-(11 )(ρρ)(

CSCV

sys dAnVVdVVdt

d

dt

Vmd

53)-(11 (ρρ )dAnVVdVVdt

dF:General

CSCV


Special Cases

54)-(11 )(ρ dAnVVF :flowSteady CS

55)-(11 V -Vρ ie

in

i

out

e

CV

mmdVVdt

dF

:flow ldimensiona-One

56)-(11 V -V ie

in

i

out

e mmF

:flow ldimensiona-oneSteady,


57)-(11 )V-V(mF :exit)-oneinlet,-(one

flow ldimensiona-oneSteady,

12

58)-(11 )V-V(mF :coordinate x Along x1,x2,x


2

) V dA

V

2

system

cv cs

shaft normal shear other

dEQ W e dV e

dt t

e u gz

W W W W W

Chapter 3 Flow Kinematics

3.1Conservation of Mass

3.2 Stream Function for Two-Dimensional

Incompressible Flow

3.3 Fluid Kinematics

3.4 Momentum Equation

3.1 Conservation of mass• Rectangular coordinate system

x

y

z

dx

dy

dzo u

v

w xA udydz

yA dxdz

zA wdxdy

surface control thethrough

outflux mass of rateNet 0

surface control theinside

change mass of Rate

x

y

z

dx

dy

dzo u

v

w xA udydz

x(left)A udydz

dydzdx

x

uu

dx

x

22

dxdydzx

u

xuudydz

2

1

x(right)A udydz

dydzdx

x

uu

dx

x

22

dxdydzx

u

xuudydz

2

1

y(bottom)A dxdz

dxdzdy

y

dy

y

22

dxdydzyy

dxdz

2

1

y(top)A dxdz

dxdzdy

y

dy

y

22

dxdydzyy

dxdz

2

1

x

y

z

dx

dy

dzo u

v

w

yA dxdz

z(back)A wdxdy

dxdydz

z

ww

dz

z

22

dxdydzz

w

zwwdxdz

2

1

z(front)A wdxdy

dxdydz

z

ww

dz

z

22

dxdydzz

w

zwwdxdy

2

1

dx

dy

dzo u

v

w

x

y

z

zA wdxdy

Net Rate of Mass Flux

x(left)A udydz

x(right)A udydz

y(bottom)A dxdz

y(top)A dxdz

z(back)A wdxdy

z(front)A wdxdy

dxdydzx

u

xuudydz

2

1

dxdydzx

u

xuudydz

2

1

dxdydzyy

dxdz

2

1

dxdydzyy

dxdz

2

1

dxdydzz

w

zwwdxdz

2

1

dxdydzz

w

zwwdxdy

2

1

CS AdV

dxdydzz

w

zw

yyx

u

xu

Net Rate of Mass Flux

CS AdV

dxdydzz

w

zw

yyx

u

xu

dxdydzz

w

yx

u

Rate of mass change inside the control

volume

dxdydzt

Vdt

V

dxdydzt

0

dxdydzz

w

yx

u

t

0

z

w

yx

u

Continuity Equation

t

0

z

w

yx

u

zk

yj

xi

ˆˆˆ

Vz

w

yx

u

0

Vt

3.2 Stream Function for Two-Dimensional

Incompressible Flow• A single mathematical function (x,y,t) to

represent the two velocity components, u(x,y,t) and (x,y,t).

• A continuous function (x,y,t) is defined such that

xyu

and

The continuity equation is satisfied exactly

0

xyyxyx

u

Equation of Streamline

• Lines drawn in the flow field at a given instant that are tangent to the flow direction at every point in the flow field.

dyjdxijuirdV ˆˆˆˆ0

dxudyk ˆ

0 dxudy Along a streamline

0

ddyy

dxx

dxx

dyy

Volume flow rate between streamlines

u

v V

21, yxB

11, yxA

22 , yxC1

23

x

y

Flow across AB

21

21

yy

yy dy

yudyQ

Along AB, x = constant, and dyy

d

1221

21

yy ddy

yQ

Volume flow rate between streamlines

u

v V

21, yxB

11, yxA

22 , yxC1

23

x

y

Flow across BC,

21

21

xx

xx dx

xdxQ

Along BC, y = constant, and dxx

d

1221

12

xx ddx

xQ

Stream Function for Flow in a Corner

Consider a two-dimensional flow field

0

w

Ay

Axu

xyu

and

yAxu

xfAxyxfdy

y

dx

dfAy

x

0

dx

df

cAxy

C:\course\fluid mechanics\Demo\Project_stagnation_inviscid.exe

C:\course\fluid mechanics\Demo\Project_stagnation_inviscid.exe

Motion of a Fluid Element

Translation

x

y

z

Rotation

Angular deformationLinear deformation

3.3 Flow Kinematics

Fluid Translation

x

y

z

Fluid particle pathAt t At t+dt

r

rdr

tzyxVVtp ,,,

dtt

Vdz

z

Vdy

y

Vdx

x

VVd pppp

t

V

dt

dz

z

V

dt

dy

y

V

dt

dx

x

V

dt

Vda pppp

p

t

V

z

Vw

y

V

x

Vu

dt

Vda p

p

t

VVV

Dt

VDa p

p

Scalar component of fluid acceleration

t

u

z

uw

y

u

x

uu

Dt

Duaxp

tzw

yxu

Dt

Dayp

t

w

z

ww

y

w

x

wu

Dt

Dwazp

Fluid acceleration in cylindrical coordinates

t

V

z

VV

r

VV

r

V

r

VV

Dt

DVa rr

zrr

rr

rp

2

t

V

z

VV

r

VVV

r

V

r

VV

Dt

DVa z

rrp

t

V

z

VV

V

r

V

r

VV

Dt

DVa zz

zzz

rz

zp

Fluid Rotation

x

y

aa'

b

b'

o

x

y

t

x

t ttoa

00

limlim

txx

ttxx

xx

xt

xtxx

toa

0

lim

t

y

t ttob

00

limlim

tyy

ututy

y

uu

aa'

b

b'

o

x

y

xx

u

yy

uu

y

u

t

ytyyu

tob

0

lim

aa'

b

b'

o

x

y

xx

u

yy

uu

xoa

y

uob

y

u

xoboaz

2

1

2

1

Similarily, considering the rotation of pairs of perpendicular line segments in yz and xz planes, one can obtain

zy

wx

2

1

x

w

z

uy 2

1

Fluid particle angular velocity

zyx kji ˆˆˆ

y

u

xk

x

w

z

uj

zy

wi

ˆˆˆ2

1

wuzyx

kji

V

ˆˆˆ VV

curl

2

1

2

1

V

2 Vorticity: A measure of fluid element rotation

rzrzr

V

rr

rV

rk

r

V

z

Ve

z

VV

reV

11ˆˆ1

ˆ

Vorticity in cylindrical coordinates

Fluid Circulation, c sdV

x

y x

x

u

yy

uu

c

y

xo

b

a

yxyy

uuyx

xxu

yxy

u

x

yxz 2

A zA zC dAVdAsdV

2

Circulation around a close contour

=Total vorticity enclosed

Around the close contour oacb,

Fluid Angular Deformation

x

y

aa'

b

b'

o

x

y

xx

u

yy

uu

dt

d

dt

d

dt

d

dy

du

x

Fluid Linear Deformation

x

y

yy

uu

a a'

bb'

o

x

y

xx

u

t

y

dt

dy

t

yy

0

limdilation of Rate

tyy

tyy

tyy

2

1

ydt

d yy

tyy

txx

utx

x

uutx

x

uu

2

1

t

xx

txx

00 limstrain of Rate

x

uxx

0

z

wzz

0

tx

u

x

tz

w

z

tzz

wtz

z

wwtz

z

ww

2

1

t

zz

tzz

00 limstrain of Rate

yy

uu

a a'

bb'

o

x

y

xx

u

t

VVV

t

0

limrate dilation Volume

zyxzyxVV

Ozyyxzx

tz

wt

yt

x

u

z

w

yx

ut

VVV

t

limrate dilation Volume0

V

zyx

zyx

zyyxzx

V

VV

Rate of shearing strain(Angular deformation)

y

u

xyxxy

zy

wzyyz

x

w

z

uxzzx

x

uVxx

2

3

2

yVyy

2

3

2

z

wVzz

2

3

2

Rate of Strain

Rate of normal strain

3.4 Momentum Equation

z

Vw

y

V

x

VuVd

Dt

VDdmFdFdFd sB

t

V

z

Vw

y

V

x

Vu

Vd

Fd

t

VVV

x

y

z

xxxy

xz

zy

zxzz

direction plane jiij

yy

yzyx

Forces acting on a fluid particle

x

y

z

x-direction

2

dz

zzx

zx

2

dy

yyx

yx

2

dz

zzx

zx

2

dx

xxx

xx

2

dx

xxx

xx

2

dy

yyx

yx

SxdF dydzdx

xdydz

dx

xxx

xxxx

xx

22

+ dxdzdy

ydxdz

dy

yyx

yxyx

yx

22

dxdydz

zdxdy

dz

zzx

zxzx

zx

22

+

Forces acting on a fluid particle

x-direction SxdF dydzdx

xdydz

dx

xxx

xxxx

xx

22

+ dxdzdy

ydxdz

dy

yyx

yxyx

yx

22

dxdydz

zdxdy

dz

zzx

zxzx

zx

22

+

SxdF dxdydzzyxzxyxxx

SxBxx dFdFdF dxdydzzyx

g zxyxxxx

Components of Forces acting on a fluid element

x-direction

Vd

dF

Vd

dF

Vd

dF SxBxx

zyx

g zxyxxxx

Vd

dF

Vd

dF

Vd

dF SyByy

zyx

g zyyyxyy

Vd

dF

Vd

dF

Vd

dF SzBzz

zyx

g zzyzxzz

y-direction

z-direction

Differential Momentum Equation

zyxg zxyxxx

x

zyxg zyyyxy

y

zyxg zzyzxz

z

z

uw

y

u

x

uu

t

u

z

wyx

ut

z

ww

y

w

x

wu

t

w

l element/Vo fluid on the acting Forces

naccleratio Fluid

Momentum Equation:Vector form

Dt

VDg

zxAyxAxxAAAAxSx kji

zk

yj

xi

Vd

dF

ˆˆˆˆˆˆ

A

A

A

zzyzxz

zyyyxy

zxyxxx

k

j

i

ˆ00

0ˆ0

00ˆ

zyxzxyxxx

is treated as a momentum flux

Stress and Strain Relation for a Newtonian Fluid

y

u

xxyyxxy

zy

wyzzyyz

x

w

z

uzxxzzx

x

uVpp xxxx

2

3

2

yVpp yyyy

2

3

2

z

wVpp zzzz

2

3

2

Newtonian fluid viscous stress rate of shearing strain

Surface Forces

zyxVd

dF zxyxxxSx

y

u

xyxyx

x

w

z

uzxzx

x

uVpp xxxx 2

3

2

zxyxxxSx

zyp

xVd

dF

x

w

z

u

zy

u

xyV

x

up

x

3

22

Momentum Equation:Navier-Stokes Equations

x

w

z

u

zy

u

xyV

x

u

xx

pg

Dt

Dux

3

22

y

w

zzV

yyxy

u

xy

pg

Dt

Dy

3

22

Vz

w

zy

w

zyz

u

x

w

xz

pg

Dt

Dwz

3

22

Navier-Stokes Equations

For flow with =constant and =constant

0 V

2

2

2

2

2

2

z

u

y

u

x

u

x

pg

Dt

Dux

2

2

2

2

2

2

zyxy

pg

Dt

Dy

2

2

2

2

2

2

z

w

y

w

x

w

z

pg

Dt

Dwz


iij

j

Dh Dp udiv k T n

Dt Dt x

Summary of Basic Equations

t

D

Dtg + p

Dh

Dt

Dp

Dtk T

u

x

ij

iji

j

div V

V'

div '

0

Chapter 2 Reynolds Transport Theorem (RTT)

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Transcript of Chapter 2 Reynolds Transport Theorem (RTT)