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Lecture Objectives:

Lecture Objectives:. - Review Compare - Navier Stokes equations and - Reynolds Averaged Navier Stokes equations Model Reynolds stresses, Kinetic energy and Dissipation Numerics. NS Equations (instantaneous velocities). y. z. x. Direct Solution → DNS .

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Lecture Objectives:

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  1. Lecture Objectives: - Review • Compare - Navier Stokes equations and - Reynolds Averaged Navier Stokes equations • Model Reynolds stresses, • Kinetic energy and Dissipation • Numerics

  2. NS Equations(instantaneous velocities) y z x Direct Solution → DNS

  3. Capturing the flow properties nozzle Eddy ~ 1/100 in For DNS mesh (volume) should be smaller than eddies ! (approximately order of value)

  4. First Methods on Analyzing Turbulent Flow Reynolds (1895) decomposed the velocity field into a time average motion and a turbulent fluctuation vx’ Vx • Likewise f stands for any scalar: vx, vy, , vz, T, p, where: Time averaged component

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

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

  7. Time Averaged Concentration Equation Instantaneous Concentration and velocities Averaged Concentration and velocities

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

  9. 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

  10. 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

  11. 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

  12. 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

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

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

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

  16. General CFD Equation Values of , ,eff and S

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