chapter 2 reynolds transport theorem (rtt) 2.1 the reynolds transport theorem 2.2 continuity...

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)

2.1 The Reynolds Transport Theorem (2)

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

2.1 The Reynolds Transport Theorem (3)

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

2.3 The Linear Momentum Equation (2)

2.3 The Linear Momentum Equation (3)

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,

2.3 The Linear Momentum Equation (4)

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

flow ldimensiona-oneSteady,

12

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

2.4 Conservation of Energy

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

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

3.5 Conservation of Energy

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