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

Outline Outline 4 Conservation Laws 4 Conservation Laws 4 Constitutive Equations 4 Constitutive Equations 4 Navier 4 Navier- -Stokes Equation Stokes Equation 4 Heat Transfer Equation 4 Heat Transfer Equation 4 Dimensionless Groups 4 Dimensionless Groups G. Ahmadi G. Ahmadi ME 639-Turbulence ME 639-Turbulence Axiom 1: Principle of Conservation of Mass Axiom 2: Principle of Balance of Momentum Axiom 1: Principle of Conservation of Mass Axiom 2: Principle of Balance of Momentum ( n ) d t ∑ = ( Momentum ) Forces n d ∫ ρ = dt dv 0 Mass is invariant Mass is invariant under the motion under the motion dt v Global Global d ∂ ∫ ∫ ∫ ∫ ∫ ρ = ρ + ( n ) ρ + ρ ⋅ = v dv f dv t ds dv 0 v ds Global Global F k k k ∂ dt t v v s v s Divergence Theorem Divergence Theorem ∂ ρ Stress Tensor Stress Tensor + ρ = ∫ ∫ ( v ) 0 = t n ds t dv ( n ) = ⋅ Local Local t n t k k , l l l l k , k ∂ t s v ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 1

Axiom 2: Principle of Balance of Momentum Axiom 2: Principle of Balance of Momentum Axiom 3: Principle of Balance of Angular Axiom 3: Principle of Balance of Angular dv ∫ ρ − ρ − = ( k f t ) dv 0 Momentum Momentum k k , l l dt Global v d ∫ ∫ ρ σ + ε = ρε dv ( r v ) dv r f dv ρ = ρ + k kmj m j kmj m j k f t dt Local Local 1 4 v 4 4 4 2 4 4 4 4 3 1 v 42 4 43 4 k l , k l dt Time rate of change of angular momentum Moment of body force ∫ ∫ ∫ + ε ( n ) + ( n ) + ρ r t ds m ds ds l d v kmj m j 1 2 k 3 k ρ dt = ρ + ∇ ⋅ f t S s s 1 42 43 1 2 3 Couple Stress Moment of surface force Body couple G. Ahmadi G. Ahmadi ME 639-Turbulence ME 639-Turbulence Axiom 4: Principle of Conservation of Energy Axiom 4: Principle of Conservation of Energy Local Q ρ σ = ρ + ε + t m & l o ρ V , , k k kmj mj k , ρ l l i V , o i P o e , P , e , o ε = i i σ = = = t mj 0 m k 0 l When W k k l kmj d + = + ( K E ) W Q { { = dt t t 1 4 2 4 3 Work done Heat transferre d Stress Tensor is Symmetric Stress Tensor is Symmetric mj jm by all the forces Time rate of change kinetic and int ernal enrgies ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 2

Local Global ρ = + + ρ e t v q r & ⎛ + ⎞ d 1 k , k k , k ∫ ∫ l l ρ = ρ ⎜ ⎟ e v v dv v f dv k k k k dt ⎝ 2 ⎠ v v q = heat flux ∫ ∫ ∫ + ⋅ + + ρ ( n ) v t ds q ds rdv r = heat source k k k k s s v G. Ahmadi G. Ahmadi ME 639-Turbulence ME 639-Turbulence ∂ ρ Axiom 5: Entropy Inequality Axiom 5: Entropy Inequality + ∇ ⋅ ρ = ( ) 0 Mass v Mass ∂ t Global ρ d q n r ∫ ∫ ∫ ρη − − ≥ dv k k ds dv 0 d v t = T ρ dt = ρ + ∇ ⋅ Momentum Momentum dt T T t f t v S v 1 4 2 4 3 1 4 4 4 2 4 4 4 3 Time rate of change Heat transfer divided of entropy by temperatur e de Energy ρ = ∇ + ∇ ⋅ + ρ Energy : r t v q ρ dt q r ρ & η − − ≥ ( k ) 0 η ρ d r Local q , k ρ − ∇ ⋅ − ≥ T T ( ) 0 Entropy Entropy dt T T ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 3

Thermodynamics Constitutive Postulates Constitutive Postulates Continuum Thermodynamics Continuum ψ = ψ ρ ( T , , d , T ) ψ = − η Assuming e T k l , k Helmholtz Free Energy Free Energy Helmholtz ∂ ψ ∂ ψ ∂ ψ ∂ ψ • & ψ = & + ρ + + T d T & & ρ ρ q r ∂ ∂ ρ ∂ k ∂ , k l T d T & − η − ψ − − ≥ ( e & T & ) ( k ) 0 k , k l Entropy Entropy , k T T T ∂ ψ ∂ ψ = − = ρ 2 p Pressure − ∂ ρ 1 ∂ ρ ⎡ q T ⎤ 1 Clausius- - Clausius & − ρ ψ + η + + k , k ≥ ( & T ) t v 0 ⎢ ⎥ ∂ ψ p k , k l l T T ρ = − Duhem ρ = − ρ Duhem ⎣ ⎦ & d & d ∂ ρ ρ kk kk G. Ahmadi G. Ahmadi ME 639-Turbulence ME 639-Turbulence Entropy Equation Linear Constitutive Equations Entropy Equation Linear Constitutive Equations ⎡ ⎤ ∂ ψ ∂ ψ ∂ ψ = − δ + 1 • q T t p L d & − ρ + η & + + δ − ρ − ρ + k , k ≥ ( ) T ( t p ) d T d 0 ⎢ ⎥ k k k , k k ∂ l l l ∂ ∂ l T ⎢ T T d T ⎥ ⎣ ⎦ k k k ij ij l l l , k k l ∂ ψ ∂ ψ ∂ ψ = = 0 η = − q = L T ∂ ∂ T d ∂ T , k k l k kj , j q T + δ + k , k ≥ ( t p ) d 0 ≥ L d d 0 ≥ L T T 0 k k k l l l k ij ij k T l l kj , k , j ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 4

Isotropic Materials Thermodynamical Constraints Constraints Isotropic Materials Thermodynamical = + + + µ − L λδ δ µ( δ δ δ δ ) ( δ δ δ δ ) λ + µ ≥ µ ≥ κ ≥ 3 2 0 0 0 k ij k ij ki j kj i 1 ki j kj i l l l l l l = κδ L Stokes Assumption Stokes Assumption k l k l 1 2 Newtonian Fluids Newtonian Fluids = − p t λ = − µ kk 3 = − + λ δ + µ 3 t ( p d ) 2 d k ii k k l l l 1 = − δ + µ D t p 2 d D = − δ d d d = κ ij ij kk ij q T k k k l l l 3 Fourier Law Fourier Law k , k G. Ahmadi G. Ahmadi ME 639-Turbulence ME 639-Turbulence dv ρ = − + µ + λ + µ + ρ k p v ( ) v f , k k , jj j , jk k dt ∇ v ⋅ = 0 Incompressible Fluids Incompressible Fluids d v ρ = −∇ + µ ∇ + ρ 2 p v f dt Navier Navier Stokes Stokes ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 5

de Enthalpy Enthalpy = κ q T ρ = ∇ + ∇ ⋅ + ρ : r ρ t v q p d d p dp k , k = − = ρ − p pv ( ) dt = + k , k ρ ρ h e dt dt dt ρ de ρ = ∇ ⋅ κ ∇ + + ρ ( T ) t v r Heat Heat ij j , i dt dh dp Transfer Transfer ρ = + ∇ ⋅ κ ∇ + Φ + ρ ( T ) r dt dt = − + Φ t v pv ij j , i k , k Heat Capacities Heat Capacities = = dh c dT de c dT Φ = λ + µ v v 2 d v Dissipation Dissipation P v k , k i , i ij j , i G. Ahmadi G. Ahmadi ME 639-Turbulence ME 639-Turbulence dT dP Thermal ρ = ρ − β − Thermal ( 1 ( T T )) ρ = + κ ∇ + Φ + ρ 2 c T r 0 0 Expansion Expansion P dt dt [ ] ρ = − ρ − β − g 1 ( T T ) Body Force f k Body Force Incompressible Flow Incompressible Flow 0 0 dT Boussinesq Equation Boussinesq Equation ρ = κ ∇ + Φ + ρ 2 c T r d v v dt ˆ ρ = −∇ + µ ∇ − ρ β − 2 P g ( T T ) v k 0 0 0 dt Φ = µ + ( v v ) v ˆ Dissipation Dissipation = + ρ i , j j , i j , i P p gz 0 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 6

∂ ρ * Mass Mass Dimensionless Variables Dimensionless Variables + ∇ * ⋅ ρ * * = ( ) 0 v ∂ * t ρ x tU v = U ρ * = Momentum Momentum i = * t * = ∞ * x i v ρ L L * d 1 Gr 0 v ∞ 2 ρ = −∇ + ∇ − * * * * * * * P T v f * 2 dt Re Re ˆ − T T − P P f Energy Energy = f = * * T 0 = * ∞ * * P dT dP 1 Ec 2 ∆ ρ * = + ∇ * * + Φ * g T Ec T ρ 2 U 0 * * ∞ dt dt Re Pr Re 0 G. Ahmadi G. Ahmadi ME 639-Turbulence ME 639-Turbulence Reynolds Number Reynolds Number Prandtl Number Number Prandtl Concluding Remarks Concluding Remarks ρ U ∞ L µ c = Re 0 = Pr P � Conservation Laws � Conservation Laws µ κ � Constitutive Equations � Constitutive Equations Eckert Number Eckert Number Grashof Number Number Grashof � Navier � Navier- -Stokes Equation Stokes Equation 2 βρ ∆ U 2 3 g L T = = ∞ Gr 0 0 Ec � Heat Transfer Equation � Heat Transfer Equation µ 2 ∆ c T � Dimensionless Groups � P 0 Dimensionless Groups ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 7