Objective Define Reynolds Navier Stokes Equations (RANS) Start with Numerics

From the last class 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

From the last class 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

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