Objective Introduce Reynolds Navier Stokes Equations (RANS) Discuss HW 1

Energy Cascade Concept in Turbulence Kinetic energy is continually being transferred from the mean flow to the turbulent motion by large-scale eddies The process of vortex stretching leads to a successive reduction in eddy size and to a steepening of velocity gradients between adjacent eddies. Eventually the eddies become so small that viscous dissipation leads to the conversion of kinetic energy into heat.

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

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

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

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

Mesh size For 3D simulation domain 2.5 m Mesh size 0.1m → 50,000 nodes 3D space (room)

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

Time Averaging Operations

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

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

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 stresses Eddy 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

Kinetic energy and dissipation of energy Kolmogorov scale Eddy breakup and decay to smaller length scales where dissipation appear

Prandtl Mixing-Length Model (1926) One equation models: Prandtl Mixing-Length Model (1926) Vx y x l Characteristic length (in practical applications: distance to the closest surface) -Two dimensional model -Mathematically simple -Computationally stable -Do not work for many flow types There are many modifications of Mixing-Length Model: - Indoor zero equation model: t = 0.03874  V l Distance to the closest surface Air velocity

Two equation turbulent model Kinetic energy Energy dissipation From dimensional analysis constant We need to model Two additional equations: kinetic energy dissipation

Reynolds Averaged Navier Stokes equations Continuity: 1) Momentum: 2) 3) 4) General format:

General CFD Equation Values of , ,eff and S Equation  ,eff S Continuity 1 x-momentum V1 + t -P/x+Sx y-momentum V2 -P/y-g(T∞-Twall)+Sy z-momentum V3 -P/z+Sz T-equation T /l + t/t ST k-equation k (+ t)/k G- +GB -equation  (+ t)/ [ (C1G-C2)/k] +C3GB(/k) Species C (+ t)/c SC Age of air t   t =Ck2/ , G= t (Ui/xj +Uj/xi) Ui/xj , GB=-g(/CP)( t/T,t) T/ xi C1=1.44, C2=1.92, C3=1.44, C=0.09 , t=0.9, k =1.0,  =1.3, C=1.0

1-D example of discretization of general transport equation Steady state 1dimension (x): W dxw P dxe E Dx w e Point W and E represent the cell center of the west and east neighbors of cell P and w, e the neighboring surfaces. Integrating with Gaussian theorem on this control volume gives: To obtain the equations for the value at point P, assumptions are used to convert the surface values to the center values.

HW Example 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.

3-D