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Lecture Objectives:

Lecture Objectives:. Define turbulence Solve turbulent flow example Define average and instantaneous velocities Define Reynolds Averaged Navier Stokes equations. Fluid dynamics and CFD movies. http://www.youtube.com/watch?v=IDeGDFZSYo8

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Lecture Objectives:

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  1. Lecture Objectives: • Define turbulence • Solve turbulent flow example • Define average and instantaneous velocities • Define Reynolds Averaged Navier Stokes equations

  2. Fluid dynamics and CFD movies • http://www.youtube.com/watch?v=IDeGDFZSYo8 • http://www.dlr.de/en/desktopdefault.aspx/tabid-6225/10237_read-26563/ • http://www.youtube.com/watch?v=oOGXEfgKttM • http://www.youtube.com/watch?v=IFeSZZ49vAs • http://www.youtube.com/watch?v=o53ghmaSFY8

  3. HW problem The figure below shows a turbulent boundary layer due to forced convection above the flat plate. The airflow above the plate is steady-state. Consider the points A and B above the plate and line l parallel to the plate. Point A y Flow direction Point A Point B line l • For the given time step presented on the figure above plot the velocity • Vx and Vy along the line l. b) Is the stress component txy lager at point A or point B? Why? c) For point B plot the velocity Vy as function of time.

  4. Method for solving of Navier Stokes (conservation) equations • Analytical • Define boundary and initial conditions. Solve the partial deferential equations. • Solution exist for very limited number of simple cases. • Numerical - Split the considered domain into finite number of volumes (nodes). Solve the conservation equation for each volume (node). Infinitely small difference finite “small” difference

  5. Numerical method • Simulation domain for indoor air and pollutants flow in buildings 3D space Solve p, u, v, w, T, C Split or “Discretize” into smaller volumes

  6. Capturing the flow properties 2” nozzle Eddy ~ 1/100 in Mesh (volume) should be smaller than eddies ! (approximately order of value)

  7. Mesh size for direct Numerical Simulations (DNS) ~1000 ~2000 cells For 2D wee need ~ 2 million cells Also, Turbulence is 3-D phenomenon !

  8. Mesh size • For 3D simulation domain 2.5 m Mesh size 0.1m → 50,000 nodes Mesh size 0.01m → 50,000,000 nodes Mesh size 0.001m → 5 ∙1010 nodes 4 m Mesh size 0.0001m → 5 ∙1013 nodes 5 m 3D space (room)

  9. Indoor airflow exhaust jet supply jet • The question is: • What we are interested in: • main flow or • turbulence? turbulent

  10. We need to model turbulence! Reynolds Averaged Navier Stokes equations

  11. 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: From this class We are going to make a difference between large and small letters Time averaged component

  12. Averaging Navier Stokes equations Substitute into Navier Stokes equations Instantaneous velocity fluctuation around average velocity Average velocity Continuity equation: time 0 0 0 Average whole equation: Average Average of fluctuation = 0 Average of average = average

  13. Time Averaging Operations

  14. Example: of Time Averaging Write continuity equations in a short format: =0 continuity Short format of continuity equation in x direction:

  15. Averaging of Momentum Equation averaging 0

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

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

  18. 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 !

  19. 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

  20. 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

  21. 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

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