Outline Outline 4 Conservation Laws 4 Conservation Laws 4 - - PowerPoint PPT Presentation

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• G. Ahmadi

ME 639-Turbulence

• G. Ahmadi

ME 639-Turbulence

Outline Outline

4 4Conservation Laws Conservation Laws 4 4Constitutive Equations Constitutive Equations 4 4Navier Navier-

• Stokes Equation

Stokes Equation 4 4Heat Transfer Equation Heat Transfer Equation 4 4Dimensionless Groups Dimensionless Groups

• G. Ahmadi

ME 639-Turbulence

Axiom 1: Principle of Conservation of Mass Axiom 1: Principle of Conservation of Mass Mass is Mass is invariant

invariant under the motion

under the motion dv dt d

v

= ρ

∫

dv t

s v

= ⋅ ρ + ρ ∂ ∂

∫ ∫

ds v

) v ( t

k , k

= ρ + ∂ ρ ∂

Global Global Local Local

• G. Ahmadi

ME 639-Turbulence

Axiom 2: Principle of Balance of Momentum Axiom 2: Principle of Balance of Momentum

∫ ∫ ∫

+ ρ = ρ

s ) n ( k v k v k

ds t dv f dv v dt d

∑

= Forces ) Momentum ( dt d

F

) n (

t

n

t n t ⋅ =

) n (

Stress Tensor Stress Tensor

∫ ∫

=

v , k s k

dv t ds n t

l l l l

Global Global Divergence Theorem Divergence Theorem

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ME 639-Turbulence

Local Local

∫

= − ρ − ρ

v , k k k

dv ) t f dt dv (

l l

Axiom 2: Principle of Balance of Momentum Axiom 2: Principle of Balance of Momentum

l l , k k k

t f dt dv + ρ = ρ

t f v ⋅ ∇ + ρ = ρ dt d

• G. Ahmadi

ME 639-Turbulence

Axiom 3: Principle of Balance of Angular Axiom 3: Principle of Balance of Angular Momentum Momentum

∫ ∫ ∫ ∫ ∫

ρ + + ε + ρε = ε + σ ρ

s couple Body s k Stress Couple ) n ( k force surface

• f

Moment S ) n ( j m kmj force body

• f

Moment v j m kmj momentum angular

• f

change

• f

rate Time v j m kmj k

ds ds m ds t r dv f r dv ) v r ( dt d 3 2 1 l 3 2 1 43 42 1 4 43 4 42 1 4 4 4 4 3 4 4 4 4 2 1

Global

• G. Ahmadi

ME 639-Turbulence

Local l l

l &

, k mj kmj k k

m t + ε + ρ = σ ρ

mk

k k

= = = σ

l

l

When

t mj

kmj

= ε

Stress Tensor is Symmetric Stress Tensor is Symmetric jm mj

t t =

• G. Ahmadi

ME 639-Turbulence

Axiom 4: Principle of Conservation of Energy Axiom 4: Principle of Conservation of Energy

{

{

d transferre Heat forces the all by done Work enrgies ernal int and kinetic change

• f

rate Time

Q W ) E K ( dt d + = + 4 3 4 2 1

i i V

, ρ , , V

• ρ

W Q

• e

, P , e , P

i i

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ME 639-Turbulence

∫ ∫ ∫ ∫ ∫

ρ + + ⋅ + ρ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ρ

v s k k s ) n ( k k v k k v k k

rdv ds q ds t v dv f v dv v v 2 1 e dt d

Global

• G. Ahmadi

ME 639-Turbulence

Local

r q v t e

k , k k , k

ρ + + = ρ

l l

&

q = heat flux r = heat source

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ME 639-Turbulence

Axiom 5: Entropy Inequality Axiom 5: Entropy Inequality

dv T r ds T n q dv dt d

e temperatur by divided transfer Heat v S k k entropy

• f

change

• f

rate Time v

≥ ρ − − ρη

∫ ∫ ∫

4 4 4 3 4 4 4 2 1 4 3 4 2 1

Global

T r ) T q (

k , k

≥ ρ − − η ρ&

Local

• G. Ahmadi

ME 639-Turbulence

) ( t = ρ ⋅ ∇ + ∂ ρ ∂ v

t f v ⋅ ∇ + ρ = ρ dt d

T

t t =

r : dt de ρ + ⋅ ∇ + ∇ = ρ q v t

T r ) T ( dt d ≥ ρ − ⋅ ∇ − η ρ q

Mass Mass Momentum Momentum Energy Energy Entropy Entropy

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ME 639-Turbulence

Continuum Continuum Thermodynamics

Thermodynamics

η − = ψ T e

Helmholtz Helmholtz Free Energy Free Energy

T r ) T q ( ) T e ( T

k , k

≥ ρ − − ψ − η − ρ & & &

Entropy Entropy

T T q v t ) T ( T 1

k , k k , k

≥ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + η + ψ ρ −

l l

& &

Clausius Clausius-

• Duhem

Duhem

• G. Ahmadi

ME 639-Turbulence

Constitutive Postulates Constitutive Postulates

) T , d , , T (

k , kl

ρ ψ = ψ

• ∂

ψ ∂ + ∂ ψ ∂ + ρ ρ ∂ ψ ∂ + ∂ ψ ∂ = ψ

k , k , k k

T T d d T T

l l

& & & &

ρ ∂ ψ ∂ ρ = ρ ∂ ψ ∂ − =

− 2 1

p

kk

d ρ − = ρ &

kk

d p ρ − = ρ ρ ∂ ψ ∂ &

Assuming Pressure

• G. Ahmadi

ME 639-Turbulence

Entropy Equation Entropy Equation

T T q d d T T d ) p t ( T ) T ( T 1

k , k k k k , k , k k k

≥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ∂ ψ ∂ ρ − ∂ ψ ∂ ρ − δ + + η + ∂ ψ ∂ ρ −

• l

l l l l

& &

T ∂ ψ ∂ − = η

d T

k k ,

= ∂ ψ ∂ = ∂ ψ ∂

l

T T q d ) p t (

k , k k k k

≥ + δ +

l l l

• G. Ahmadi

ME 639-Turbulence

Linear Constitutive Equations Linear Constitutive Equations

ij ij k k k

d L p t

l l l

+ δ − =

j , kj k

T L q =

d d L

k ij ij k

≥

l l

T T L

j , k , kj

≥

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ME 639-Turbulence

Isotropic Materials Isotropic Materials

) δ δ δ (δ ) δ δ δ µ(δ δ λδ L

i kj j ki 1 i kj j ki ij k ij k l l l l l l

− µ + + + =

l l k k

L κδ =

l l l k k ii k

d 2 ) d p ( t µ + δ λ + − =

k , k

T q κ =

Newtonian Fluids Newtonian Fluids Fourier Law Fourier Law

• G. Ahmadi

ME 639-Turbulence

Thermodynamical Thermodynamical Constraints Constraints

Stokes Assumption Stokes Assumption

2 3 ≥ µ + λ

≥ µ

≥ κ

µ − = λ 3 2

kk

t 3 1 p − =

D k k k

d 2 p t

l l l

µ + δ − =

ij kk ij D ij

d 3 1 d d δ − =

• G. Ahmadi

ME 639-Turbulence

Incompressible Fluids Incompressible Fluids

k jk , j jj , k k , k

f v ) ( v p dt dv ρ + µ + λ + µ + − = ρ

= ⋅ ∇ v

f v v ρ + ∇ µ + −∇ = ρ

2

p dt d

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ME 639-Turbulence

Navier Navier Stokes Stokes

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ME 639-Turbulence

r v t ) T ( dt de

i , j ij

ρ + + ∇ κ ⋅ ∇ = ρ

r : dt de ρ + ⋅ ∇ + ∇ = ρ q v t

k , k

T q κ =

Φ + − =

k , k i , j ij

pv v t

i , j ij i , i k , k

v d 2 v v µ + λ = Φ

Heat Heat Transfer Transfer Dissipation Dissipation

• G. Ahmadi

ME 639-Turbulence

dt dp ) p ( dt d dt d p pv

k , k

− ρ ρ = ρ ρ − =

ρ + = p e h

r ) T ( dt dp dt dh ρ + Φ + ∇ κ ⋅ ∇ + = ρ

dT c dh

P

= dT c de

v

=

Heat Capacities Heat Capacities Enthalpy Enthalpy

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ME 639-Turbulence

r T dt dP dt dT c

2 P

ρ + Φ + ∇ κ + = ρ

r T dt dT c

2 v

ρ + Φ + ∇ κ = ρ

i , j i , j j , i

v ) v v ( + µ = Φ

Incompressible Flow Incompressible Flow Dissipation Dissipation

• G. Ahmadi

ME 639-Turbulence

)) T T ( 1 ( − β − ρ = ρ

[ ]

) T T ( 1 g − β − ρ − = ρ k f

Thermal Thermal Expansion Expansion Body Force Body Force

k v v ) T T ( g P ˆ dt d

2

− β ρ − ∇ µ + −∇ = ρ

gz p P ˆ ρ + =

Boussinesq Boussinesq Equation Equation

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ME 639-Turbulence

Dimensionless Variables Dimensionless Variables

L x x

i * i =

∞

= U

*

v v

L tU t*

∞

=

*

ρ ρ = ρ

2 *

U P P ˆ P

∞ ∞

ρ − =

*

T T T T ∆ − =

g

*

f f =

• G. Ahmadi

ME 639-Turbulence

Mass Mass

) ( t

* * * * *

= ρ ⋅ ∇ + ∂ ρ ∂ v

* * 2 * 2 * * * * * *

T Re Gr Re 1 P dt d f v v − ∇ + −∇ = ρ

* * 2 * * * * * *

Re Ec T Pr Re 1 dt dP Ec dt dT Φ + ∇ + = ρ

Momentum Momentum Energy Energy

• G. Ahmadi

ME 639-Turbulence

Reynolds Number Reynolds Number

µ ρ =

∞L

U Re

κ µ =

P

c Pr

P 2

T c U Ec ∆ =

∞

2 3 2

T L g Gr µ ∆ βρ =

Prandtl Prandtl Number Number Grashof Grashof Number Number Eckert Number Eckert Number

• G. Ahmadi

ME 639-Turbulence

Concluding Remarks Concluding Remarks

• Conservation Laws

Conservation Laws

• Constitutive Equations

Constitutive Equations

• Navier

Navier-

• Stokes Equation

Stokes Equation

• Heat Transfer Equation

Heat Transfer Equation

• Dimensionless Groups

Dimensionless Groups