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Part 2 RANS.pdf
- 1. Dr Patrick Geoghegan Book: H. Versteeg and W. Malalasekera An Introduction to Computational Fluid Dynamics: The Finite Volume Method FEA/CFD for Biomedical Engineering Week 11: CFD
- 2. RANS
- 3. • Reynolds-averaged Navier-Stokes Equations (RANS) – Mean flow is resolved, the effect of turbulence on the mean flow incorporated without resolving the turbulence Alternatives for modelling turbulence? RANS! u′ u time, t velocity, u Velocity fluctuation Mean Velocity • Solve for mean flow • Effect of turbulence on mean flow taken into account
- 4. • Derivation of Reynolds-averaged Navier-Stokes Equations (RANS) – Split the velocity into mean and fluctuating components, plug into Navier- Stokes Equations and then apply time averaging to the resultant equations Alternatives for modelling turbulence? RANS! u u t u ′ + = ) ( u′ u time, t velocity, u 𝑢𝑢𝑢 2 = 1 ∆𝑡𝑡 � 0 ∆𝑡𝑡 𝑢𝑢𝑢 2 𝑑𝑑𝑑𝑑
- 5. • Conservation of Mass (Continuity Equation) • Conservation of Momentum • Conservation of internal energy • State Equations Original Navier – Stokes equations 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 + 𝛻𝛻 � 𝜌𝜌𝐮𝐮 = 0 𝜕𝜕 𝜌𝜌𝜌 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝜌𝜌𝐮𝐮 = − 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜇𝜇𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 + 𝑆𝑆𝑀𝑀𝑀𝑀 𝜕𝜕 𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝜌𝜌𝐮𝐮 = − 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜇𝜇𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 + 𝑆𝑆𝑀𝑀𝑀𝑀 𝜕𝜕 𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝜌𝜌𝐮𝐮 = − 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜇𝜇𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 + 𝑆𝑆𝑀𝑀𝑀𝑀 𝜕𝜕 𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝜌𝜌𝐮𝐮 = −𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝐮𝐮 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 + 𝑆𝑆𝐸𝐸 𝑝𝑝 = 𝑝𝑝 𝜌𝜌, 𝑇𝑇 𝑖𝑖 = 𝑖𝑖 𝜌𝜌, 𝑇𝑇 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝐮𝐮 = 0 or We won derive this here
- 6. • Conservation of Mass • Reynolds Equations Reynolds-averaged Navier-Stokes Equations (RANS) 𝜕𝜕 𝜌𝜌� 𝑢𝑢 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌� 𝑢𝑢𝐮𝐮 = − 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜇𝜇𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔� 𝑢𝑢 + − 𝜕𝜕𝜌𝜌𝑢𝑢′𝑢𝑢𝑢 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝑢𝑢′𝑣𝑣𝑣 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝑢𝑢′𝑤𝑤𝑤 𝜕𝜕𝜕𝜕 + 𝑆𝑆𝑀𝑀𝑀𝑀 𝜕𝜕 𝜌𝜌 ̅ 𝑣𝑣 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌 ̅ 𝑣𝑣𝐮𝐮 = − 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜇𝜇𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 ̅ 𝑣𝑣 + − 𝜕𝜕𝜌𝜌𝑢𝑢′𝑣𝑣𝑣 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝑣𝑣′𝑣𝑣𝑣 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕 + 𝑆𝑆𝑀𝑀𝑀𝑀 𝜕𝜕 𝜌𝜌� 𝑤𝑤 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌� 𝑤𝑤𝐮𝐮 = − 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜇𝜇𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 + − 𝜕𝜕𝜌𝜌𝑢𝑢′𝑤𝑤𝑤 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝑣𝑣′𝑤𝑤𝑤 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝑤𝑤′𝑤𝑤𝑤 𝜕𝜕𝜕𝜕 + 𝑆𝑆𝑀𝑀𝑀𝑀 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝐮𝐮 = 0 Viscous Stresses Convective transfer of momentum due to turbulence Reynolds Stresses = Normal Stresses + Shear Stresses In turbulent flow, Reynolds stresses dominate viscous stresses except very close to walls where turbulent fluctuations go to zero
- 7. • Original Scalar Transport Equation for an arbitrary scalar Φ (energy, mass fraction, etc) • RANS Scalar Transport Equation for an arbitrary scalar Φ (energy, mass fraction, etc) Reynolds-averaged Navier-Stokes Equations (RANS) 𝜕𝜕 𝜌𝜌𝛷𝛷 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝜌𝜌𝐮𝐮 = 𝑑𝑑𝑑𝑑𝑑𝑑 𝛤𝛤 𝛷𝛷𝜇𝜇𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔Φ + − 𝜕𝜕𝜌𝜌𝑢𝑢′Φ 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝑣𝑣′Φ 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝑤𝑤′Φ 𝜕𝜕𝜕𝜕 + 𝑆𝑆Φ Convective transfer of Φ due to turbulent eddies 𝜕𝜕 𝜌𝜌𝛷𝛷 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝜌𝜌𝐮𝐮 = 𝑑𝑑𝑑𝑑𝑑𝑑 𝛤𝛤 𝛷𝛷𝜇𝜇𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔Φ + 𝑆𝑆Φ Rate of accum. of Φ in fluid element Net rate of flow of Φ into fluid element (Convection) Rate of change of Φ due to diffusion Rate of change of Φ due to sources + + =
- 8. Model effect of turbulent fluctuations Reynolds Stresses on mean flow We are solving for U, V, W, and P • For reference the expanded form of the RANS equations Reynolds-averaged Navier-Stokes Equations (RANS) 𝜌𝜌 𝜕𝜕� 𝑢𝑢 𝜕𝜕𝜕𝜕 + 𝜌𝜌 � 𝑢𝑢 𝜕𝜕� 𝑢𝑢 𝜕𝜕𝜕𝜕 + ̅ 𝑣𝑣 𝜕𝜕� 𝑢𝑢 𝜕𝜕𝜕𝜕 + � 𝑤𝑤 𝜕𝜕� 𝑢𝑢 𝜕𝜕𝜕𝜕 = − 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 + 𝜇𝜇 𝜕𝜕2 � 𝑢𝑢 𝜕𝜕𝑥𝑥2 + 𝜕𝜕2 � 𝑢𝑢 𝜕𝜕𝑦𝑦2 + 𝜕𝜕2 � 𝑢𝑢 𝜕𝜕𝑧𝑧2 + − 𝜕𝜕𝜌𝜌𝑢𝑢′𝑢𝑢𝑢 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝑢𝑢′𝑣𝑣𝑣 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝑢𝑢′𝑤𝑤𝑤 𝜕𝜕𝜕𝜕 + 𝑆𝑆𝑀𝑀𝑀𝑀 𝜌𝜌 𝜕𝜕 ̅ 𝑣𝑣 𝜕𝜕𝜕𝜕 + 𝜌𝜌 � 𝑢𝑢 𝜕𝜕 ̅ 𝑣𝑣 𝜕𝜕𝜕𝜕 + ̅ 𝑣𝑣 𝜕𝜕 ̅ 𝑣𝑣 𝜕𝜕𝜕𝜕 + � 𝑤𝑤 𝜕𝜕 ̅ 𝑣𝑣 𝜕𝜕𝜕𝜕 = − 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 + 𝜇𝜇 𝜕𝜕2 ̅ 𝑣𝑣 𝜕𝜕𝑥𝑥2 + 𝜕𝜕2 ̅ 𝑣𝑣 𝜕𝜕𝑦𝑦2 + 𝜕𝜕2 ̅ 𝑣𝑣 𝜕𝜕𝑧𝑧2 + − 𝜕𝜕𝜌𝜌𝑢𝑢′𝑣𝑣𝑣 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝑣𝑣′𝑣𝑣𝑣 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕 + 𝑆𝑆𝑀𝑀𝑀𝑀 𝜌𝜌 𝜕𝜕� 𝑤𝑤 𝜕𝜕𝜕𝜕 + 𝜌𝜌 � 𝑢𝑢 𝜕𝜕� 𝑤𝑤 𝜕𝜕𝜕𝜕 + ̅ 𝑣𝑣 𝜕𝜕� 𝑤𝑤 𝜕𝜕𝜕𝜕 + � 𝑤𝑤 𝜕𝜕� 𝑤𝑤 𝜕𝜕𝜕𝜕 = − 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 + 𝜇𝜇 𝜕𝜕2 � 𝑤𝑤 𝜕𝜕𝑥𝑥2 + 𝜕𝜕2 � 𝑤𝑤 𝜕𝜕𝑦𝑦2 + 𝜕𝜕2 � 𝑤𝑤 𝜕𝜕𝑧𝑧2 + − 𝜕𝜕𝜌𝜌𝑢𝑢′𝑤𝑤𝑤 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝑣𝑣′𝑤𝑤𝑤 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝑤𝑤′𝑤𝑤𝑤 𝜕𝜕𝜕𝜕 + 𝑆𝑆𝑀𝑀𝑀𝑀 τxx τxy τxz τyy τyz τzz
- 9. • Solve equations for mean properties of flow (u, v, w, F) • By applying time averaging, we lose details of the turbulent fluctuations • But for engineering purposes, we usually only interested in mean properties of the flow anyway • With the Navier-Stokes Equations, we had 5 equations (4 transport equations and one state-equation) and 5 unknowns (u, v, w, ρ, p) • With the Reynolds-averaged Navier-Stokes Equations are now 6 extra unknowns (the Reynolds stresses) • Turbulence modelling is about developing equations to model the Reynolds stresses Reynolds-averaged Navier-Stokes Equations (RANS)
- 10. • Turbulent kinetic energy is a measure of the energy in the fluctuations of the flow, not the mean flow • Procedure for deriving the transport equation for turbulent kinetic energy 1) Multiply the momentum equations of the Navier-Stokes equations by the appropriate fluctuating velocity component (x-component momentum equation multiplied by u’ etc) 2) Add the three resultant equations to form one equation 3) Repeat 1) and 2) for the Reynolds-averaged Navier-Stokes equations 4) Subtract the two resultant equations Turbulent Kinetic Energy Equation, k 𝑘𝑘 = 1 2 𝑢𝑢𝑢 2 + 𝑣𝑣𝑣 2 + 𝑤𝑤𝑤 2
- 11. Turbulent Kinetic Energy Equation, k 𝑘𝑘 = 3 2 𝑢𝑢𝑢𝑟𝑟𝑟𝑟𝑟𝑟 2 Units = 𝑚𝑚2 𝑠𝑠2 𝑘𝑘 = 1 2 𝑢𝑢𝑢 2 + 𝑣𝑣𝑣 2 + 𝑤𝑤𝑤 2 Or 𝐼𝐼 = 𝑢𝑢𝑢𝑟𝑟𝑟𝑟𝑟𝑟 𝑢𝑢 × 100% Remember It is often assumed the fluctuations are the same in all directions so often 𝑘𝑘 = 1 2 𝑢𝑢𝑢𝑟𝑟𝑟𝑟𝑟𝑟 2 + 𝑣𝑣𝑣𝑟𝑟𝑟𝑟𝑟𝑟 2 + 𝑤𝑤𝑤𝑟𝑟𝑟𝑟𝑟𝑟 2 This can all be used to calculate the turbulence kinetic energy
- 12. Turbulent Kinetic Energy Equation, k 𝜕𝜕 𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝜌𝜌𝐮𝐮 = 𝑑𝑑𝑑𝑑𝑑𝑑 −𝑝𝑝′𝒖𝒖′ + 2𝜇𝜇𝒖𝒖′𝑠𝑠𝑖𝑖𝑖𝑖′ − 𝜌𝜌 1 2 𝑢𝑢𝑖𝑖 ′ . 𝑢𝑢𝑖𝑖 ′𝑢𝑢𝑗𝑗′ − 2𝜇𝜇𝑠𝑠𝑖𝑖𝑖𝑖 ′𝑠𝑠𝑖𝑖𝑖𝑖′ + 𝜌𝜌𝑢𝑢𝑖𝑖′𝑢𝑢𝑖𝑖′. 𝑆𝑆𝑖𝑖𝑖𝑖 Rate of Change of turbulent kinetic energy (k) Rate of Change of k by convection Transport of k by pressure Transport of k by viscous stresses Transport of k by Reynolds Stresses Rate of dissipation of k Rate of production of k
- 13. • Rate of dissipation of k or viscous dissipation • Work done by smallest eddies against viscous stresses • Always negative, destroys turbulent kinetic energy • Divide by density to get dissipation rate (ε) (m2/s3) Turbulent Kinetic Energy Equation, k Rate of dissipation of k 𝜀𝜀 = 2𝜈𝜈𝑠𝑠𝑖𝑖𝑖𝑖 ′𝑠𝑠𝑖𝑖𝑖𝑖′ 𝜕𝜕 𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝜌𝜌𝐮𝐮 = 𝑑𝑑𝑑𝑑𝑑𝑑 −𝑝𝑝′𝒖𝒖′ + 2𝜇𝜇𝒖𝒖′𝑠𝑠𝑖𝑖𝑖𝑖′ − 𝜌𝜌 1 2 𝑢𝑢𝑖𝑖 ′ . 𝑢𝑢𝑖𝑖 ′𝑢𝑢𝑗𝑗′ − 2𝜇𝜇𝑠𝑠𝑖𝑖𝑖𝑖 ′𝑠𝑠𝑖𝑖𝑖𝑖′ + 𝜌𝜌𝑢𝑢𝑖𝑖′𝑢𝑢𝑖𝑖′. 𝑆𝑆𝑖𝑖𝑖𝑖
- 14. • Rate of production of k • Represents conversion of kinetic energy from the mean flow into turbulent kinetic energy (that is production of turbulent kinetic energy) • Always positive Turbulent Kinetic Energy Equation, k Rate of production of k 𝜕𝜕 𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝜌𝜌𝐮𝐮 = 𝑑𝑑𝑑𝑑𝑑𝑑 −𝑝𝑝′𝒖𝒖′ + 2𝜇𝜇𝒖𝒖′𝑠𝑠𝑖𝑖𝑖𝑖′ − 𝜌𝜌 1 2 𝑢𝑢𝑖𝑖 ′ . 𝑢𝑢𝑖𝑖 ′𝑢𝑢𝑗𝑗′ − 2𝜇𝜇𝑠𝑠𝑖𝑖𝑖𝑖 ′𝑠𝑠𝑖𝑖𝑖𝑖′ + 𝜌𝜌𝑢𝑢𝑖𝑖′𝑢𝑢𝑖𝑖′. 𝑆𝑆𝑖𝑖𝑖𝑖
- 15. • Terms in the equation mathematically represent phenomena observed to occur in nature energy cascade (1)Largest eddies extract energy from the mean flow by a mechanism of vortex stretching (2)This energy is passed down to smaller eddies existing within the larger eddies, also by a mechanism of vortex stretching, and so onwards to smaller and smaller eddies (3)At the smallest eddy-scales, energy is finally dissipated as heat by friction Turbulent Kinetic Energy Equation, k
- 16. • Lets revisit this Reynolds-averaged Navier-Stokes Equations (RANS) 𝜌𝜌 𝜕𝜕� 𝑢𝑢 𝜕𝜕𝜕𝜕 + 𝜌𝜌 � 𝑢𝑢 𝜕𝜕� 𝑢𝑢 𝜕𝜕𝜕𝜕 + ̅ 𝑣𝑣 𝜕𝜕� 𝑢𝑢 𝜕𝜕𝜕𝜕 + � 𝑤𝑤 𝜕𝜕� 𝑢𝑢 𝜕𝜕𝜕𝜕 = − 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 + 𝜇𝜇 𝜕𝜕2 � 𝑢𝑢 𝜕𝜕𝑥𝑥2 + 𝜕𝜕2 � 𝑢𝑢 𝜕𝜕𝑦𝑦2 + 𝜕𝜕2 � 𝑢𝑢 𝜕𝜕𝑧𝑧2 + − 𝜕𝜕𝜌𝜌𝑢𝑢′𝑢𝑢𝑢 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝑢𝑢′𝑣𝑣𝑣 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝑢𝑢′𝑤𝑤𝑤 𝜕𝜕𝜕𝜕 + 𝑆𝑆𝑀𝑀𝑀𝑀 𝜌𝜌 𝜕𝜕 ̅ 𝑣𝑣 𝜕𝜕𝜕𝜕 + 𝜌𝜌 � 𝑢𝑢 𝜕𝜕 ̅ 𝑣𝑣 𝜕𝜕𝜕𝜕 + ̅ 𝑣𝑣 𝜕𝜕 ̅ 𝑣𝑣 𝜕𝜕𝜕𝜕 + � 𝑤𝑤 𝜕𝜕 ̅ 𝑣𝑣 𝜕𝜕𝜕𝜕 = − 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 + 𝜇𝜇 𝜕𝜕2 ̅ 𝑣𝑣 𝜕𝜕𝑥𝑥2 + 𝜕𝜕2 ̅ 𝑣𝑣 𝜕𝜕𝑦𝑦2 + 𝜕𝜕2 ̅ 𝑣𝑣 𝜕𝜕𝑧𝑧2 + − 𝜕𝜕𝜌𝜌𝑢𝑢′𝑣𝑣𝑣 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝑣𝑣′𝑣𝑣𝑣 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕 + 𝑆𝑆𝑀𝑀𝑀𝑀 𝜌𝜌 𝜕𝜕� 𝑤𝑤 𝜕𝜕𝜕𝜕 + 𝜌𝜌 � 𝑢𝑢 𝜕𝜕� 𝑤𝑤 𝜕𝜕𝜕𝜕 + ̅ 𝑣𝑣 𝜕𝜕� 𝑤𝑤 𝜕𝜕𝜕𝜕 + � 𝑤𝑤 𝜕𝜕� 𝑤𝑤 𝜕𝜕𝜕𝜕 = − 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 + 𝜇𝜇 𝜕𝜕2 � 𝑤𝑤 𝜕𝜕𝑥𝑥2 + 𝜕𝜕2 � 𝑤𝑤 𝜕𝜕𝑦𝑦2 + 𝜕𝜕2 � 𝑤𝑤 𝜕𝜕𝑧𝑧2 + − 𝜕𝜕𝜌𝜌𝑢𝑢′𝑤𝑤𝑤 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝑣𝑣′𝑤𝑤𝑤 𝜕𝜕𝜕𝜕 − 𝜕𝜕𝜌𝜌𝑤𝑤′𝑤𝑤𝑤 𝜕𝜕𝜕𝜕 + 𝑆𝑆𝑀𝑀𝑀𝑀 τxx τxy τxz τyy τyz τzz Reynolds Stresses
- 17. • Basis for most turbulence models is Boussinesq’s eddy viscosity concept – By analogy to the viscous stresses in original NS equations, the turbulent stresses are proportional to mean-velocity gradients Eddy viscosity (𝜇𝜇𝑡𝑡) is the proportionality “constant” Turbulence Modelling 𝑘𝑘 = 1 2 𝑢𝑢𝑢 2 + 𝑣𝑣𝑣 2 + 𝑤𝑤𝑤 2 𝜏𝜏𝑥𝑥𝑥𝑥 = −𝑝𝑝′𝑢𝑢′𝑢𝑢′ = 2𝜇𝜇𝑡𝑡 𝜕𝜕� 𝑢𝑢 𝜕𝜕𝜕𝜕 − 2 3 𝜌𝜌𝜌𝜌 + 𝜇𝜇𝑡𝑡𝑑𝑑𝑑𝑑𝑑𝑑𝐮𝐮 𝜏𝜏𝑦𝑦𝑦𝑦 = −𝑝𝑝′𝑣𝑣′𝑣𝑣′ = 2𝜇𝜇𝑡𝑡 𝜕𝜕 ̅ 𝑣𝑣 𝜕𝜕𝜕𝜕 − 2 3 𝜌𝜌𝜌𝜌 + 𝜇𝜇𝑡𝑡𝑑𝑑𝑑𝑑𝑑𝑑𝐮𝐮 𝜏𝜏𝑧𝑧𝑧𝑧 = −𝑝𝑝′𝑤𝑤′𝑤𝑤′ = 2𝜇𝜇𝑡𝑡 𝜕𝜕� 𝑤𝑤 𝜕𝜕𝜕𝜕 − 2 3 𝜌𝜌𝜌𝜌 + 𝜇𝜇𝑡𝑡𝑑𝑑𝑑𝑑𝑑𝑑𝐮𝐮 ? 𝜏𝜏𝑥𝑥𝑥𝑥 = 𝜏𝜏𝑦𝑦𝑦𝑦 = −𝑝𝑝′𝑢𝑢′𝑣𝑣′ = 𝜇𝜇𝑡𝑡 𝜕𝜕� 𝑢𝑢 𝜕𝜕𝜕𝜕 + 𝜕𝜕 ̅ 𝑣𝑣 𝜕𝜕𝜕𝜕 𝜏𝜏𝑥𝑥𝑥𝑥 = 𝜏𝜏𝑧𝑧𝑧𝑧 = −𝑝𝑝′𝑢𝑢′𝑤𝑤′ = 𝜇𝜇𝑡𝑡 𝜕𝜕� 𝑢𝑢 𝜕𝜕𝜕𝜕 + 𝜕𝜕� 𝑤𝑤 𝜕𝜕𝜕𝜕 𝜏𝜏𝑦𝑦𝑦𝑦 = 𝜏𝜏𝑧𝑧𝑧𝑧 = −𝑝𝑝′𝑣𝑣′𝑤𝑤′ = 𝜇𝜇𝑡𝑡 𝜕𝜕 ̅ 𝑣𝑣 𝜕𝜕𝜕𝜕 + 𝜕𝜕� 𝑤𝑤 𝜕𝜕𝜕𝜕 Added to ensure consistency in definitions