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

Fluid dynamics and CFD movies

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.

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

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

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

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

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 m → 5 ∙10 13 nodes

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

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 - 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

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 = Average time

Time Averaging Operations

Example: of Time Averaging =0 continuity Write continuity equations in a short format: 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 Time Averaged Energy Equation Instantaneous velocities Averaged velocities Instantaneous temperatures and velocities Averaged temperatures and velocities

Reynolds Averaged Navier Stokes equations Reynolds stresses total are unknown same Total 4 equations and = 10 unknowns We need to model the Reynolds stresses !

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

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)

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: