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Lecture Objectives:

Lecture Objectives:. Compare Navier Stokes equations and Reynolds Averaged Navier Stokes equations Define Reynolds stresses, Kinetic energy and Dissipation Solve example CFD software. Time Averaged Momentum Equation. Instantaneous velocity. Average velocities. Reynolds stresses.

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Lecture Objectives:

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  1. Lecture Objectives: • Compare • Navier Stokes equations and • Reynolds Averaged Navier Stokes equations • Define Reynolds stresses, Kinetic energy and Dissipation • Solve example • CFD software

  2. Time Averaged Momentum Equation Instantaneous velocity Average velocities Reynolds stresses For y and z direction: Total nine

  3. Time Averaged Continuity Equation Instantaneous velocities Averaged velocities Time Averaged Energy Equation Instantaneous temperatures and velocities Averaged temperatures and velocities

  4. Reynolds Averaged Navier Stokes equations Reynolds stresses total 9 - 6 are unknown same Total 4 equations and 4 + 6 = 10 unknowns We need to model the Reynolds stresses !

  5. Modeling of Reynolds stressesEddy viscosity models Average velocity Boussinesq eddy-viscosity approximation Is proportional to deformation Coefficient of proportionality k = kinetic energy of turbulence Substitute into Reynolds Averaged equations

  6. Reynolds Averaged Navier Stokes equations Continuity: 1) Momentum: 2) 3) 4) Similar is for STy and STx 4 equations 5 unknowns → We need to model

  7. Kinetic energy and dissipation of energy Kolmogorov scale Eddy breakup and decay to smaller length scales where dissipation appear

  8. Two equation turbulent model model Energy dissipation Kinetic energy From dimensional analysis constant We need to model Two additional equations: kinetic energy dissipation

  9. One equation models: Prandtl Mixing-Length Model (1926) Vx y x l Characteristic length (in practical applications: distance to the closest surface) -Two dimensional model • -Mathematically simple • -Computationally stable • -Do not work for many flow types There are many modifications of Mixing-Length Model: - Indoor zero equation model: t = 0.03874  V l Distance to the closest surface Air velocity

  10. Modeling of Turbulent Viscosity Fluid property – often called laminar viscosity Flow property – turbulent viscosity MVM: Mean velocity models TKEM: Turbulent kinetic energy equation models Additional models: LES: Large Eddy simulation models RSM: Reynolds stress models

  11. Reynolds Averaged Navier Stokes equations Continuity: 1) Momentum: 2) 3) 4) General format:

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