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Differential Forms of the Equations of Motion
Deriving Differential FormsDifferential Forms
21
2
∫ ∫∇= dVdSn φφrINTEGRAL THEOREMS
DIFFERENTIAL FORM(POINTWISE DESCRIPTION)
∂
INTEGRAL FORM(FOR CONTROL VOLUMES)
S V
∫ ∫∇=S V
dVBdSnBrrr ..
SUBST DERIVATIVE0. =+
∂∂∫ ∫
CV CS
dSnUdVt
rrρρ
∫∫ ∫∂ rrr
BUtB
DTDB
∇+∂∂
= .r
SUBST. DERIVATIVE
∫∫ ∫ −=+∂∂
CSCV CS
dSnpdSnUUdVUt
rr).(ρρ
∫∫∫ −=+++∂ dSUnpdSnUUedVUe
rrrr )()()( 2121 ρρ ∫∫∫ =+++∂ CSCSCV
dSUnpdSnUUedVUet
.).()()( 22 ρρ
Conservation of MassConservation of Mass
0. =+∂∂∫ ∫ dSnUdV
trrρρ ∫ ∫∇= dVBdSnB
rrr ..INTEGRAL FORM DIVERGENCE THEOREM
∂ ∫ ∫CV CSt ∫ ∫
S V
SUBST. DERIVATIVE
BUtB
DTDB
∇+∂∂
= .r
Momentum and EnergyMomentum and Energy
∫∫ ∫ −=+∂ dSnpdSnUUdVU rrrrr
).(ρρ
(INVISCID)
pUD ∇−=
rMOMENTUM
∫∫ ∫∂ CSCV CS
pt
)(ρρρDt OR
(INVISCID ADIABATIC)ENERGY
∫∫∫ −=+++∂∂
CSCSCV
dSUnpdSnUUedVUet
rrrr .).()()( 2212
21 ρρ
(INVISCID, ADIABATIC)ENERGY
ρ).()( 2
21 Up
DtUeD
r∇
−=+
BUtB
DTDB
∇+∂∂
= .r
ρ
Physical MeaningPhysical Meaning
UD r∇= ρρ
MASS
UD ∇r
UDt
.∇−= ρ
MOMENTUM
ρp
DtUD ∇
−=
ENERGY
ρ).()( 2
21 Up
DtUeD
r∇
−=+
ENERGY
Results Derived From the Fundamental Equations
The Stagnation Enthalpy EquationThe Stagnation Enthalpy Equation
21 Uhh +≡).()( 221 UpUeD
r∇+ UD r
∇ρρENERGY MASS
20 Uhh +≡ρ
2
Dt−= U
Dt.∇−= ρ
BUtB
DTDB
∇+∂∂
= .r
The Entropy (Crocco’s) Equation
pUD ∇rMOMENTUM
The Entropy (Crocco s) Equation
dpdhTds −=
2ND LAW
ρDt−=
ρdhTds −=
rrr
UUUtUUU
tU
DtUD rr
rrr
rr
×∇×−∇+∂∂
=∇+∂∂
= )(. 221
Crocco’s Shock in nozzle
Equation
rr
Uh
UUsT
∂∂
+∇+
×∇×−=∇r
0 t∂0
Bourgoing & Benay (2005), ONERA
Important Simplifications
tp
DtDh
∂∂
−=ρ10
STAG. ENTHALPY
tDt ∂ρ
tUhUUsT∂∂
+∇+×∇×−=∇r
rr0
CROCCO’s
t∂0
Airfoil at M=10
convergecfd.com/applications/externalflow/
Simpler Equations for Isentropic Steady Flow
0×∇ Ur
0=×∇ U
0=∂∂t
MomentumMomentum
pUD ∇r
ρDt−=
0)( 221 =∇+∇ Up ρ
rrr
UUUtUUU
tU
DtUD rr
rrr
rr
×∇×−∇+∂∂
=∇+∂∂
= )(. 221
Mass+MomentumMass+Momentum
UD r∇= ρρ
MASS
UDt
.∇−= ρ
MOMENTUMMOMENTUMBU
tB
DTDB
∇+∂∂
= .r
0)( 221 =∇+∇ Up ρ
In Two DimensionsIn Two Dimensions
IRROT.∂∂ uv
0=×∇ Ur
0=∂∂
−∂∂
yu
xv
MASS+MOMENTUM
2Ur
r0)(.
2. 2
2 =∇−∇ UaUU
02)1()1( 22
2
2
2
=∂
−∂
−+∂
−vuvvvuu 0)1()1( 222 ∂∂
+∂ xayaxa