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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: • 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 • 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
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.
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
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
Capturing the flow properties 2” nozzle Eddy ~ 1/100 in Mesh (volume) should be smaller than eddies ! (approximately order of value)
Mesh size for direct Numerical Simulations (DNS) ~1000 ~2000 cells For 2D wee need ~ 2 million cells Also, Turbulence is 3-D phenomenon !
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)
Indoor airflow exhaust jet supply jet • The question is: • What we are interested in: • main flow or • turbulence? turbulent
We need to model turbulence! Reynolds Averaged Navier Stokes equations
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
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
Example: of Time Averaging Write continuity equations in a short format: =0 continuity Short format of continuity equation in x direction:
Averaging of Momentum Equation averaging 0
Time Averaged Momentum Equation Instantaneous velocity Average velocities Reynolds stresses For y and z direction: Total nine
Time Averaged Continuity Equation Instantaneous velocities Averaged velocities Time Averaged Energy Equation Instantaneous temperatures and velocities Averaged temperatures and velocities
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 !
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
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
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