Fluid Mech CV

CHAPTER 6Derive differential Continuity, Momentum and Energyequations form Integral equations for control volumes.

Simplify these equations for 2-D steady, isentropic flow with variable density

CHAPTER 8Write the 2 –D equations in terms of velocity potentialreducing the three equations of continuity, momentum andenergy to one equation with one dependent variable, the velocity potential.

CHAPTER 11Method of Characteristics exact solution to the 2-D velocity potential equation.

vectora isscalar a of scalar a is vector a of

kz) (j

y) (i

x) ( Gradient

scalar a is a) ( :re whe vola)d((a)dS

vectora is )V( : where vold)V()dSV(

integral volumea into integral surface a ms transforTheorem Divergence -Theorem sGauss'

S vol

S vol

∇∇

∂∂

+∂∂

+∂∂

=∇

∇=

∇=

∫∫ ∫∫∫

∫∫ ∫∫∫rrr

CONTINUITY EQUATION CONSERVATIVE INTEGRAL FORM

vector velocity ,V

dS control volumeopen thermodynamic systemregion in space

( ) form ive)(conservat integralin Equation Continuity vold ρt

dS V

volumecontrol theinside massin change vold ρt

) is inflow mass convention(by

volume.control theleaving mass ofnet dS V ρ

volS

vol

S

∫∫∫∫∫

∫∫∫

∫∫

∂∂

=ρ

∂∂

+

−

CONTINUITY EQUATION CONSERVATIVE INTEGRAL FORM

( )

term,outflow massnet the toTheorm sGauss' applying. is inflow mass conventionby

dS V vold ρt

outflow massnet mass) volume(control ∆zyx

here, w voldVdSV

integral volumea into integral surface a ms transforTheorm sGauss'

Svol

S vol

+

ρ−=∂∂

=

∂∂

+∂∂

+∂∂

=∇∇=

∫∫∫∫∫

∫∫ ∫∫∫

r

rr

CONTINUITY EQUATION CONSERVATIVE INTEGRAL FORM

( ) ( ) ( )

( )

0zw

yv

xu

zw

yv

xu

tρ

z andy in x, xuρ

xρu

xu ρ,ngsubstituti

0z wρ

y vρ

xu ρ

tρ

density variablefluid,any D,-3 unsteady,

(6.50) 0)V(ρtρ

vold )V(ρ vold ρt volvol

=

∂∂

ρ+∂∂

ρ+∂∂

ρ+

∂ρ∂

+∂ρ∂

+∂ρ∂

+∂∂

∂∂

+∂∂

=∂

∂

=∂

∂∂

∂+

∂∂

+∂∂

=∇+∂∂

∇−=∂∂

∫∫∫∫∫∫r

r

( )

( )( )

( ) ( ) dS dS p vold f ρ voldtV ρVdS V ρ

Force ViscousForce PressureForceBody angeMomentumCh

vold tV ρ with timeMomentum of Change

VdS V ρ volume theinside change Momentum

dS Force Viscous

dS p Force Pressure

constant forcebody theis f where

, vold f ρ ForceBody

Forces

momentumin changedtmVdF

SSvolvolS

vol

S

S

S

vol

∫∫∫∫∫∫∫∫∫∫∫∫

∫∫∫

∫∫

∫∫

∫∫

∫∫∫

τ+−=∂

∂+

++=

∂∂

τ

−

==

control volumeopen thermodynamic systemregion in space

vector velocity ,V

dS

MONENTUM EQUATION CONSERVATIVE INTEGRAL FORM

MONENTUM EQUATION CONSERVATIVE INTEGRAL FORM

( ) ( )

( )

( ) ( )

( ) ( ) f ρVV ρptV ρ

,atingdifferenti

vold vold p vold f ρ voldtV ρ voldVV ρ

.inteagrals volume tointegrals surface threeeconvert th to

vola)d((a)dS and voldASdA

(6.1), Therom s Gauss' using

dS dS p vold f ρ voldtV ρVSd V ρ

volvolvolvolvol

S volvolS

SSvolvolS

+τ∇−∇−−∇=∂

∂

τ∇+∇−=∂

∂+∇

∇=∇=

τ+−=∂

∂+

∫∫∫∫∫∫∫∫∫∫∫∫∫∫∫

∫∫ ∫∫∫∫∫∫∫∫

∫∫∫∫∫∫∫∫∫∫∫∫

MOMENTUM EQUATIONSunsteady, 3D, any fluid, variable density

( ) ( )

zzzxzxz

yzyyyxy

xzxyxxx

fzyxz

wwyvv

xwu

zpw

t

fzyxz

vwyvv

xvu

ypv

t

fzyxz

uwyuv

xuu

xpu

t

f ρVV ρptV ρ

ρ+

τ

∂∂

+τ∂∂

+τ∂∂

−

∂∂

ρ+∂∂

ρ+∂∂

ρ−∂∂

−=ρ∂∂

ρ+

τ

∂∂

+τ∂∂

+τ∂∂

−

∂∂

ρ+∂∂

ρ+∂∂

ρ−∂∂

−=ρ∂∂

ρ+

τ

∂∂

+τ∂∂

+τ∂∂

−

∂∂

ρ+∂∂

ρ+∂∂

ρ−∂∂

−=ρ∂∂

+τ∇−∇−−∇=∂

∂

( )( )( ) Vµ

32

xw2µ τ

Vµ32

xv2µ τ

Vµ32

xu2µ τ

dxdu D, 1for

distance.with velocity of change the-fluid theofn deformatio ofrate theoffunction linear a is stress fluids thefor which

fluidsNewtonian oequation t momentum thetingrestric

zz

yy

xx

∇+∂∂

−=

∇+∂∂

−=

∇+∂∂

−=

µ=τ

∂∂

+∂∂

−==

∂∂

+∂∂

−==

∂∂

+∂∂

−==

zu

xwµτ τ

yw

zvµτ τ

xv

yuµτ τ

yxxy

zyyz

yxxy

ENERGY EQUATION CONSERVATIVE INTEGRAL FORTM

( )

( )

( )

Tc Uenergy, Internal

dSq addition Heat

vold 2

Veρt

volumecontrol theinsideenergy in Change

2VedS V ρ volumecontrol intoEnergy Net

VdS Work

V vol)d f (ρ Work

VdS p Work

0WVelocityForceWork

WWWW∆EW∆EQ LawFirst

v

S

2

Vol

2

S

Sviscous

Volbody

Spressure

shaft

bodypressureviscousshaft

=

+

∂∂

+

τ−

−

=×=

++++=+=

∫∫

∫∫∫

∫∫

∫∫

∫∫∫

∫∫

( ) ( ) ( )

( ) ( ) ( )

∂∂

+∂∂

τ+

∂∂

+∂∂

τ+

∂∂

+∂∂

τ−

∂∂

τ+∂∂

τ+∂∂

τ−

∂∂

+∂∂

+∂∂

∂∂

−

∂∂

+∂

∂+

∂∂

−=

∂∂

+∂∂

+∂∂

+∂∂

ρ

τ+τ+τ

∂∂

+τ+τ+τ∂∂

+τ+τ+τ∂∂

−

ρ

∂∂

+ρ∂∂

+ρ∂∂

−

∂∂

+∂

∂+

∂∂

−

+ρ

∂∂

+

+ρ

∂∂

+

+ρ

∂∂

−=

+ρ

∂∂

•ρ+•τ•∇−•∇−•∇−

+∇∇−=

+ρ

∂∂

=+−τ−+

+

∂∂

+

+∇=

++++=

++++=+=

∫∫∫∫∫∫∫∫∫∫∫∫

yw

zv

xw

zu

xv

yu

zw

yv

xu

zw

yv

xu

TpT

zq

yq

xq

zTw

yTv

xTu

tTc

wvuz

wvuy

wvux

wz

vy

uxz

qyq

xq

2VTcw

y2VTcw

y2VTcu

x2VTc

t

)Vg()V(Vpq2

VTc ρ2

VTct

(2.20a) V vol)d f (ρdSV pVdS vold2

Veρt2

VedS ρQ

WWWW∆E∆EQ

WWWW∆EW∆EQ LawFirst

yzxzxyzzyyxx

p

zyxv

zzzyzxyzyyyxxzxyxx

zyx

2

v

2

v

2

v

2

v

2

v

2

v

volSS

2

vol

2

S

bodypressureviscousshaft volumecontrolin change

volumecontrol

innet

bodypressureviscousshaft

EQUATION SUMMARY - 3D, viscous, variable density

∂∂

+∂∂

τ+

∂∂

+∂∂

τ+

∂∂

+∂∂

τ−

∂∂

τ+∂∂

τ+∂∂

τ−

∂∂

+∂∂

+∂∂

∂∂

−

∂∂

+∂

∂+

∂∂

−=

∂∂

+∂∂

+∂∂

+∂∂

ρ

ρ+

τ

∂∂

+τ∂∂

+τ∂∂

−

∂∂

ρ+∂∂

ρ+∂∂

ρ−∂∂

−=ρ∂∂

ρ+

τ

∂∂

+τ∂∂

+τ∂∂

−

∂∂

ρ+∂∂

ρ+∂∂

ρ−∂∂

−=ρ∂∂

ρ+

τ

∂∂

+τ∂∂

+τ∂∂

−

∂∂

ρ+∂∂

ρ+∂∂

ρ−∂∂

−=ρ∂∂

−

=

∂∂

+∂∂

+∂∂

+

∂∂

+∂∂

+∂∂

+∂∂

yw

zv

xw

zu

xv

yu

zw

yv

xu

zw

yv

xu

TpT

zq

yq

xq

zTw

yTv

xTu

tTc

ENERGY

fzyxz

wwyvv

xwu

zpw

t

fzyxz

vwyvv

xvu

ypv

t

fzyxz

uwyuv

xuu

xpu

t

directions zy,x,MOMENTUM

0zwρ

yvρ

xuρ

zρw

yρv

xρu

tρ

CONTINUITY

yzxzxyzzyyxx

p

zyxv

zzzxzxz

yzyyyxy

xzxyxxx

EQUATION SUMMARY - 3D, viscous, variable density

∂∂

+∂∂

τ+

∂∂

+∂∂

τ+

∂∂

+∂∂

τ−

∂∂

τ+∂∂

τ+∂∂

τ−

∂∂

+∂∂

+∂∂

∂∂

−

∂∂

+∂

∂+

∂∂

−=

∂∂

+∂∂

+∂∂

+∂∂

ρ

ρ+

τ

∂∂

+τ∂∂

+τ∂∂

−

∂∂

ρ+∂∂

ρ+∂∂

ρ−∂∂

−=ρ∂∂

ρ+

τ

∂∂

+τ∂∂

+τ∂∂

−

∂∂

ρ+∂∂

ρ+∂∂

ρ−∂∂

−=ρ∂∂

ρ+

τ

∂∂

+τ∂∂

+τ∂∂

−

∂∂

ρ+∂∂

ρ+∂∂

ρ−∂∂

−=ρ∂∂

−

=

∂∂

+∂∂

+∂∂

+

∂∂

+∂∂

+∂∂

+∂∂

yw

zv

xw

zu

xv

yu

zw

yv

xu

zw

yv

xu

TpT

zq

yq

xq

zTw

yTv

xTu

tTc

ENERGY

fzyxz

wwyvv

xwu

zpw

t

fzyxz

vwyvv

xvu

ypv

t

fzyxz

uwyuv

xuu

xpu

t

directions zy,x,MOMENTUM

0zwρ

yvρ

xuρ

zρw

yρv

xρu

tρ

CONTINUITY

yzxzxyzzyyxx

p

zyxv

zzzxzxz

yzyyyxy

xzxyxxx

2D steady incompressible, inviscid

BOUNDARY LAYER Prandtl 1904

Divide a flow into two regions according to the forces that prevail

parallel and uniform al,irrotation

ssfrictionle ,isentropicFlow Potential

0,µ 0,τSTREAM FREE==

0yvv

xu

dyu

ρµ

dxdp

ρ1

yuv

yuu

dyρ1

dxdp

ρ1

yuv

yuu

equations,layer boundary bleincompresi D2equations momentum traverseignore

large very yuµ τlarge,

yu

forces interial asimprotant as forces viscousnear walllayer thin

LAYER BOUNDARY

2

2

yx

=∂∂

+∂∂

∂+−=

∂∂

+∂∂

τ∂+−=

∂∂

+∂∂

−

∂∂

=∂∂

EQUATION SUMMARY - 3D, viscous, variable density

∂∂

+∂∂

τ+

∂∂

+∂∂

τ+

∂∂

+∂∂

τ−

∂∂

τ+∂∂

τ+∂∂

τ−

∂∂

+∂∂

+∂∂

∂∂

−

∂∂

+∂

∂+

∂∂

−=

∂∂

+∂∂

+∂∂

+∂∂

ρ

ρ+

τ

∂∂

+τ∂∂

+τ∂∂

−

∂∂

ρ+∂∂

ρ+∂∂

ρ−∂∂

−=ρ∂∂

ρ+

τ

∂∂

+τ∂∂

+τ∂∂

−

∂∂

ρ+∂∂

ρ+∂∂

ρ−∂∂

−=ρ∂∂

ρ+

τ

∂∂

+τ∂∂

+τ∂∂

−

∂∂

ρ+∂∂

ρ+∂∂

ρ−∂∂

−=ρ∂∂

−

=

∂∂

+∂∂

+∂∂

+

∂∂

+∂∂

+∂∂

+∂∂

yw

zv

xw

zu

xv

yu

zw

yv

xu

zw

yv

xu

TpT

zq

yq

xq

zTw

yTv

xTu

tTc

ENERGY

fzyxz

wwyvv

xwu

zpw

t

fzyxz

vwyvv

xvu

ypv

t

fzyxz

uwyuv

xuu

xpu

t

directions zy,x,MOMENTUM

0zwρ

yvρ

xuρ

zρw

yρv

xρu

tρ

CONTINUITY

yzxzxyzzyyxx

p

zyxv

zzzxzxz

yzyyyxy

xzxyxxx

2-D, steady, inviscid (isentropic), variable density

∂∂

+∂∂

∂∂

−

∂

∂+

∂∂

−=

∂∂

+∂∂

ρ

∂∂

ρ+∂∂

ρ−=∂∂

∂∂

ρ+∂∂

ρ−=∂∂

−

=

++

+

yv

xu

TpT

yq

xq

yTv

xTuc

ENERGY

yvv

xvu

yp

yuv

xuu

xp

directions zy,x,MOMENTUM

0dydvρ

dxduρ

dydρv

dxdρu

CONTINUITY

p

yxv

0τ0w

0z) (

0t) (

==

=∂∂

=∂∂

VELOCITY POTENTIAL – reduce to one equation

( ) ( )

( ) ( )

( ) ( )ydxΦ

xdyΦ

yΦv,

xΦu ng,substituti

xv

xv

0dxdyxv

xvdl V

Theorem, Greens:CHECK

yΦv,

xΦu

function potential velocity Φ, as defineqauntity,scalar same theof

functionsare vandu

22

SC

SC

∂∂

=∂∂

∂∂

=∂∂

=

∂∂

=∂∂

=

∂∂

−∂∂

=

→

∂∂

=∂∂

=

∫∫∫

∫∫∫( ) ( )

( ) ( )

( ) ( )ydlV v,

xdlVu ,comparisonby

jdy u idx u )dlVd(

dyy

ldVdxx

ldV)ldVd(

dyy

dxx

) d( aldifferentiexact

positionon only dependent al,differentiexact an path oft independen is ldV

0µ 0, τ,isentropic

flow alirrotationfor 0ldVC

∂∂

=∂

∂=

+=

∂∂

+∂

∂=

∂∂

+∂∂

=

==

=∫

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

0ρΦρdydρ

dxdρ

0xΦρ

xΦρ

dydρ

yΦ

dxdρ

xΦ

ΦxΦ

dydv ,Φ

yΦ v

ΦxΦ

dxdu ,Φ

xΦu :substitute

potential velocity of in termsequation continuty

0dydvρ

dxduρ

dydρv

dxdρu

density varableinviscid, steady, D-2 EQUATION CONTINUITY

yyxxxx

2

2

2

2

xx2

2

x

xx2

2

x

=+Φ+Φ+Φ

=∂∂

+∂∂

+∂∂

+∂∂

=∂∂

==∂∂

=

=∂∂

==∂∂

=

=

++

+

2 variables, density will be eliminatedby the momentum equations.

Φ and ρ

dxxvvdx

xuudx

xp

dxdv

dydu

,flow alirrotationfor since

dxyuvdx

xuudx

xp

dxby direction multipy x EQUATIONS MOMENTUM

∂∂

ρ+∂∂

ρ=∂∂

−

=

∂∂

ρ+∂∂

ρ=∂∂

−

( )

( )

( )

( )

( )

( )yyyxyx

yxyxxx

yx2

2

y

xx2

2

x

ΦΦΦΦdyyp

equation,direction y for the

ΦΦΦΦdxxp

ΦxΦ

dydv

ΦyΦv

ΦxΦ

dxdu

ΦxΦu

:substitute

+ρ=∂∂

−

+ρ=∂∂

−

=∂∂

=

=∂∂

=

=∂∂

=

=∂∂

=

( )

( )yyyxyx2

yxyxxx2

2

2

S

2

ΦΦΦΦay

ΦΦΦΦax

xp

a1

x

ap

pa

+ρ

=∂ρ∂

−

+ρ

=∂ρ∂

−

∂∂

=∂ρ∂

∂=ρ∂

ρ∂∂

=

( )

( )

D)2for (8.17, aΦΦ

2a

1a

1

ΦΦΦΦay

ΦΦΦΦax

equation,y contuinuit theinto ngsubstituti

2yx

yy2

2y

xx2

2x

yyyxyx2

yxyxxx2

−−Φ

Φ−+Φ

Φ−

+ρ

=∂ρ∂

−

+ρ

=∂ρ∂

−

(11.7) dvdy dx dyyyΦdx

xdyΦ

dyΦd

(11.6) du dy dxdyyxΦdx

xdxd

, dyΦ and

dxΦfor alsdifferentiexact

(11.5) 0cuv2

av1)

au1(

Φv,Φu ting,substitu

D)2for (8.17, ΦaΦΦ

2a

1a

1

yyxy

22

xyxx

2

2

2

yy2xy2

2

xx2

2

yx

yy2yx

yy2

2y

xx2

2x

=Φ+Φ=∂∂

∂+

∂∂

=

∂

=Φ+Φ=∂∂

∂+

∂Φ∂

=

Φ∂

∂∂

=Φ−Φ

−+Φ−

==

−−Φ

Φ−+Φ

Φ−

DN

dy dv 0 0 du dx

)av(1

auv2- )

au(1

dy dv 0 0 du dx

)av(1 0 )

au(1

dvdy Φ 0 dx Φ

du 0 dy Φ dx Φ

0av1

cuv2)

au1(

in equationslinear ussimultaneo 3

2

2

22

2

2

2

2

2

xy

yyxy

xyxx

yy2

2

xy2xx2

2

=−−

−−

=Φ

=++

=++

=Φ

−+Φ−Φ−

Φ

itic.characters thealong properties defines 0Dy).f(x,C solution, theof

sticcharacteri thedefines 0N

ate.indetermin is Φ0 are D and Nboth When solution, theof sticcharacteri aon

ate,indetermin is Φ

xy

xy

==

=

sticcharacterialongCK)M(sticcharacterialongCK)M(

FunctionMeyer -Prandtl theis dθV

dV1Md

au1

1a

vuauv-

)av(1

)au(1

dvdu

0)dxdvav(1dudy )

au(1

0numerator,N

2

2

2

2

22

2

2

2

2

2

2

2

2

2

++

−−

=υ−θ=υ+θ

−±=θ

−

−+

±

−

−=

=−+−

=

∫

( )

( )

−

−−±−=

−

−+

±−=

=−+

+

−

=−++−

=

2

2

22

sticcharacteri

2

2

2

22

2

sticcharacteri

2

2

2

2

2

2

22

2

22

2

2

au1

11Mauv

dxdy

au1

1a

vuauv

dxdy

0)av(1

dxdy

auv2

dxdy)

au(1

0)(dx)av(1dxdy

auv2 dy)

au(1

0atormindeno,D