FLUENT Fluent Software Trainin Ncct⊙tLD TRN-99-003 Two Equation Model: Standard k-E Model Turbulent Kinetic Energy Ok aU, aUaU a ak Ox Ox: Ox:Ox ax Ox Destruction Convection Generation Diffusion Dissipation Rate ax. OxOx ax ax Convection Generation Diffusion Destruction Ok.CE, Cle, C2e are empirical constants (equations written for steady, incompressible flow w/o body forces) DIl c Fluent Inc. 2/20/01
D11 © Fluent Inc. 2/20/01 Fluent Software Training TRN-99-003 Two Equation Model: Standard k-e Model Turbulent Kinetic Energy Dissipation Rate s ,se 1e 2e k ,C ,C are empirical constants (equations written for steady, incompressible flow w/o body forces) Convection Generation Diffusion Destruction r m m s - { re þ ý ü î í ì ¶ ¶ ¶ ¶ + ¶ ¶ ÷ ÷ ø ö ç ç è æ ¶ ¶ + ¶ ¶ = ¶ ¶ 14243 144424443 14 424443i t k i i j j i i j t i i x k x x U x U x U x k U ( ) Convection Destruction Generation Diffusion 14243 1444442444443 14 424443 14243 ÷ ÷ ø ö ç ç è æ - þ ý ü î í ì ¶ ¶ ¶ ¶ + ¶ ¶ ÷ ÷ ø ö ç ç è æ ¶ ¶ + ¶ ¶ ÷ ø ö ç è æ = ¶ ¶ k C x x x U x U x U k C x U i t i i j j i i j t i i 2 1 2 ( ) e r e m m s e e r e e e
FLUENT Fluent Software Trainin Ncct⊙tLD TRN-99-003 Two Equation Model: Standard k-E Model ◆“ Baseline model”( Two-equation) Most widely used model in industry Strength and weaknesses well documented Semi-empirical k equation derived by subtracting the instantaneous mechanical energy equation from its time-averaged value E equation formed from physical reasoning Valid only for fully turbulent flows Reasonable accuracy for wide range of turbulent flows industrial flows ● heat transfer D12 c Fluent Inc. 2/20/01
D12 © Fluent Inc. 2/20/01 Fluent Software Training TRN-99-003 Two Equation Model: Standard k-e Model u “Baseline model” (Two-equation) l Most widely used model in industry l Strength and weaknesses well documented u Semi-empirical l k equation derived by subtracting the instantaneous mechanical energy equation from its time-averaged value l e equation formed from physical reasoning u Valid only for fully turbulent flows u Reasonable accuracy for wide range of turbulent flows l industrial flows l heat transfer
FLUENT Fluent Software Trainin Ncct⊙tLD TRN-99-003 Two equation Model: Realizable k-c Distinctions from standard k-e model Alternative formulation for turbulent viscosity where is now variable U a+a I(Ao, As, and U* are functions of velocity gradients a Ensures positivity of normal stresses; u;>0 Ensures Schwarz's inequality;(uu, )'su: u? New transport equation for dissipation rate, E De a Pc.SE-pc, +.,g Diffusion Generation Destruction buoyancy C Fluent Inc. 2/20
D13 © Fluent Inc. 2/20/01 Fluent Software Training TRN-99-003 Two Equation Model: Realizable k-e u Distinctions from Standard k-e model: l Alternative formulation for turbulent viscosity where is now variable n (A0 , As , and U* are functions of velocity gradients) n Ensures positivity of normal stresses; n Ensures Schwarz’s inequality; l New transport equation for dissipation rate, e: e m r m 2 k t º C e m U k A A C o s * 1 + = u 0 2 i ³ 2 j 2 i 2 i j (u u ) £ u u b j t j c G k c k c S c Dt x x D e e e e ne e r e r e s m m e r 1 3 2 1 2 + + + - ú ú û ù ê ê ë é ¶ ¶ ÷ ÷ ø ö ç ç è æ + ¶ ¶ = Diffusion Generation Destruction Buoyancy
FLUENT Fluent Software Trainin Ncct⊙tLD TRN-99-003 Two Equation Model Realizable k-c Shares the same turbulent kinetic energy equation as standard k-8 Superior performance for flows involving planar and round jets boundary layers under strong adverse pressure gradients, separation rotation recirculation strong streamline curvature D14 c Fluent Inc. 2/20/01
D14 © Fluent Inc. 2/20/01 Fluent Software Training TRN-99-003 u Shares the same turbulent kinetic energy equation as Standard k-e u Superior performance for flows involving: l planar and round jets l boundary layers under strong adverse pressure gradients, separation l rotation, recirculation l strong streamline curvature Two Equation Model: Realizable k-e
FLUENT Fluent Software Trainin Ncct⊙tLD TRN-99-003 Two Equation Model: rNG k-8 Turbulent Kinetic Energy u, S+olap where Generation Dissipation Convection Diffusion Dissipation Rate 30 u, S +olau R Additional term Convection Generation Diffusion Destruction related to mean strain turbulence quantities ak ds, Cls, C2a are derived using RNG theory (equations written for steady, incompressible flow w/o body forces) DIS c Fluent Inc. 2/20/01
D15 © Fluent Inc. 2/20/01 Fluent Software Training TRN-99-003 Two Equation Model: RNG k-e Turbulent Kinetic Energy Dissipation Rate Convection Diffusion Dissipation r m { a m - { re ÷ ÷ ø ö ç ç è æ ¶ ¶ ¶ ¶ = + ¶ ¶ 14243 1442443 i k i t i i x k x S x k U eff 2 Generation ÷ ÷ ø ö ç ç è æ ¶ ¶ + ¶ ¶ º º j i i j ij ij ij x U x U S S S S 2 1 2 , where are derived using RNG theory a ,ae 1e 2e k ,C ,C (equations written for steady, incompressible flow w/o body forces) Additional term related to mean strain & turbulence quantities Convection Generation Diffusion Destruction { R k C x x S k C x U i i t i i -÷ ÷ ø ö ç ç è æ -÷ ÷ ø ö ç ç è æ ¶ ¶ ¶ ¶ ÷ + ø ö ç è æ = ¶ ¶ 14243 1 42443 1442443 14243 2 eff 2 2 1 e r e m a m e e r e e e