Objective Reynolds Navier Stokes Equations (RANS) Numerical methods

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

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

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

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

Two equation turbulent model k~[(m/s)2] Kinetic energy Energy dissipation (proportional to work done by smallest eddies) =2/eijeij ~[(m2/s3] From dimensional analysis Deformation caused by small eddy constant We need to model Two additional equations: kinetic energy dissipation

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

Question – Discussion from previous class !

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

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

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.

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.

Convection term dxw P dxe W E Dx – Central difference scheme: - Upwind-scheme: and and

Diffusion term W dxw P dxe E Dx w e

Summary: Steady–state 1D I) X direction If Vx > 0, If Vx < 0, Convection term - Upwind-scheme: W P dxw dxe E and a) and Dx w e Diffusion term: b) When mesh is uniform: DX = dxe = dxw Assumption: Source is constant over the control volume c) Source term:

General Transport Equation unsteady-state Fully explicit method: Or different notation: Implicit method For Vx>0 For Vx<0

Steady state vs. Unsteady state We use iterative solver to get solution Unsteady state We use iterative solver to get solution and We iterate for each time step Make the difference between - Calculation for different time step - Calculation in iteration step