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Lecture Objectives: -Define turbulence –Solve turbulent flow example –Define average and instantaneous velocities -Define Reynolds Averaged Navier Stokes equations

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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

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Flow direction line l y Point A Point B 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. HW problem Point A a)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 xy lager at point A or point B? Why? c) For point B plot the velocity Vy as function of time.

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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 differencefinite “small” difference

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

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

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

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Mesh size For 3D simulation domain 3D space (room) 5 m 4 m 2.5 m Mesh size 0.1m → 50,000 nodes Mesh size 0.01m → 50,000,000 nodes Mesh size 0.001m → 5 ∙10 10 nodes Mesh size 0.0001m → 5 ∙10 13 nodes

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

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We need to model turbulence! Reynolds Averaged Navier Stokes equations

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First Methods on Analyzing Turbulent Flow - Reynolds (1895) decomposed the velocity field into a time average motion and a turbulent fluctuation - Likewise stands for any scalar: v x, v y,, v z, T, p, where: Time averaged component VxVx vx’vx’ From this class We are going to make a difference between large and small letters

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

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Time Averaging Operations

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

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Averaging of Momentum Equation averaging 0

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

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

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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 !

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

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Reynolds Averaged Navier Stokes equations Similar is for S Ty and S Tx Momentum: Continuity: 4 equations 5 unknowns → We need to model 1) 2) 3) 4)

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

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