Differential Forms of the Equations of devenpor/aoe3114/19 - Differential...Deriving Differential...

download Differential Forms of the Equations of devenpor/aoe3114/19 - Differential...Deriving Differential Forms 2 1 ∫†ndS =∫∇†dV r INTEGRAL THEOREMS DIFFERENTIAL FORM

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Transcript of Differential Forms of the Equations of devenpor/aoe3114/19 - Differential...Deriving Differential...

  • DifferentialFormsoftheEquationsofMotion

  • DerivingDifferential FormsDifferentialForms

    21

    2

    = dVdSn r

    INTEGRALTHEOREMSDIFFERENTIALFORM

    (POINTWISEDESCRIPTION)

    INTEGRALFORM(FORCONTROLVOLUMES)

    S V

    =S V

    dVBdSnBrrr ..

    SUBST DERIVATIVE0. =+

    CV CS

    dSnUdVt

    rr

    rrr

    BUtB

    DTDB

    +

    = .r

    SUBST.DERIVATIVE

    =+

    CSCV CS

    dSnpdSnUUdVUt

    rr).(

    =+++ dSUnpdSnUUedVUe

    rrrr )()()( 2121 =+++ CSCSCVdSUnpdSnUUedVUe

    t.).()()( 22

  • Conservation of MassConservationofMass

    0. =+ dSnUdVt

    rr = dVBdSnBrrr ..

    INTEGRALFORM DIVERGENCETHEOREM

    CV CSt S V

    SUBST.DERIVATIVE

    BUtB

    DTDB

    +

    = .r

  • Momentum and EnergyMomentumandEnergy

    =+ dSnpdSnUUdVU rr

    rrr).(

    (INVISCID)

    pUD =

    rMOMENTUM

    CSCV CSp

    t)(

    Dt OR

    (INVISCID ADIABATIC)ENERGY

    =+++

    CSCSCV

    dSUnpdSnUUedVUet

    rrrr .).()()( 2212

    21

    (INVISCID,ADIABATIC)ENERGY

    ).()( 221 Up

    DtUeD

    r

    =+

    BUtB

    DTDB

    +

    = .r

  • Physical MeaningPhysicalMeaning

    UDr

    = MASS

    UD r

    UDt

    .=

    MOMENTUM

    p

    DtUD

    =

    ENERGY

    ).()( 221 Up

    DtUeD

    r

    =+

    ENERGY

  • ResultsDerivedFromtheFundamentalEquations

  • The Stagnation Enthalpy EquationTheStagnationEnthalpyEquation

    21 Uhh +).()(2

    21 UpUeD

    r+ UD

    r

    ENERGY MASS

    20 Uhh +

    2

    Dt= U

    Dt.=

    BUtB

    DTDB

    +

    = .r

  • The Entropy (Croccos) Equation

    pUD rMOMENTUM

    TheEntropy(Crocco s)Equation

    dpdhTds =

    2ND LAW

    Dt=

    dhTds =

    rrr

    UUUtUUU

    tU

    DtUD rr

    rrr

    rr

    +

    =+

    = )(. 221

  • Croccos Shockinnozzle

    Equation

    rr

    Uh

    UUsT

    ++

    =r

    0 t0

    Bourgoing &Benay (2005),ONERA

  • ImportantSimplifications

    tp

    DtDh

    =10

    STAG.ENTHALPY

    tDt

    tUhUUsT

    ++=r

    rr0

    CROCCOs

    t0

    AirfoilatM=10

    convergecfd.com/applications/externalflow/

  • SimplerEquationsforIsentropicSteadyFlow

    0 Ur

    0= U

    0=t

  • MomentumMomentum

    pUD r

    Dt=

    0)( 221 =+ Up

    rrr

    UUUtUUU

    tU

    DtUD rr

    rrr

    rr

    +

    =+

    = )(. 221

  • Mass+MomentumMass+Momentum

    UDr

    = MASS

    UDt

    .=

    MOMENTUMMOMENTUMBU

    tB

    DTDB

    +

    = .r

    0)( 221 =+ Up

  • In Two DimensionsInTwoDimensions

    IRROT. uv

    0= Ur

    0=

    yu

    xv

    MASS+MOMENTUM

    2Ur

    r0)(.

    2. 22 = Ua

    UU

    02)1()1( 222

    2

    2

    =

    +

    vuvvvuu 0)1()1( 222

    + xayaxa