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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
