Recommended
More Related Content
What's hot
What's hot (20)
Similar to RANS Equations Explained
Similar to RANS Equations Explained (20)
More from SajawalNawaz5
More from SajawalNawaz5 (19)
Recently uploaded
Recently uploaded (20)
RANS Equations Explained
- 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