Objective Reynolds Navier Stokes Equations RANS Numerical methods

Objective • Reynolds Navier Stokes Equations (RANS) • Numerical methods

Time Averaged Momentum Equation Instantaneous velocity Average velocities For y and z direction: Reynolds stresses 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: RSM: Large Eddy simulation models Reynolds stress models

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 ~[(m 2/s 3] 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 , eff Continuity 1 0 0 x-momentum V 1 + t - P/ x+Sx y-momentum V 2 + t - P/ y- g (T∞-Twall)+Sy z-momentum V 3 + t - P/ z+Sz T-equation T / l + t/ t ST k-equation k ( + t)/ k G- +GB -equation ( + t)/ [ (C 1 G-C 2 )/k] +C 3 GB( /k) Species C ( + t)/ c SC Age of air t + t Equation t = C k 2/ , G= t ( Ui/ xj + Uj/ xi) Ui/ xj S , 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 1 dimension (x): dxw P W w Dx dxe E 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 1 dimension (x): dxw P W w Dx dxe E 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 W dxe E Dx w – Central difference scheme: - Upwind-scheme: and e

Diffusion term dxw P W dxe E Dx w e

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

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

Steady state vs. Unsteady state Steady state 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