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