2. Reynolds Method to Diagnosize Symptoms of Infected Flows.... P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Reynolds Averaged NS Equations

3. Averaging of x-momentum Equation

6. Reynolds Averaged Steady Turbulent Momentum Equations Reynolds averaged x-momentum equation for steady incompressible turbulent flow

7. The Reynolds View of Cross Correlation

8. Reynolds averaged y-momentum equation for steady incompressible turbulent flow Reynolds averaged z-momentum equation for steady incompressible turbulent flow

9. Reynolds Averaged Navier Stokes equations

10. Reynolds Stress Tensor This is usually called the Reynolds stress tensor Reynolds stresses : total 9 - 6 are unknown Total 4 equations and 4 + 6 = 10 unknowns

11. Time averaged Infected Navier Stokes Equation For all the Three Momentum Equations, turbulent stress tensor:

12. Reynolds stresses Performing the Reynolds Averaging Process, new terms has arisen, namely the Reynolds-stress tensor: This brings us at the turbulent closure problem, the fact that we have more unknowns than equations. –Three velocities + pressure + six Reynolds-stresses –Three momentum equations + the continuity equation To close the problem, we need additional equations to solve infected flow.

13. Derivations of Reynolds-stress conservation Equations Derivation of Conservation Equations for Reynolds Stresses Introduces new unknowns (22 new unknowns)

14. Simplified Reynolds Averaged Navier Stokes equations 4 equations 5 unknowns → We need one more ???

15. Modeling of Turbulent Viscosity Fluid property – often called laminar viscosity Flow property – turbulent viscosity MVM: Mean velocity models TKEM: Turbulent kinetic energy equation models