Objective Review Reynolds Navier Stokes Equations (RANS) Learn about General Transport equation Start with Numerics

From the previous class Reynolds Averaged Navier Stokes equations Reynolds stresses total 9 - 6 are unknown (incompressible flow) same Total 4 equations and 4 + 6 = 10 unknowns We need to model the Reynolds stresses !

From the previous class 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

From the previous class 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

Numerics

Finite Volume Method - Conservation of f for the finite volume Divide the whole computation domain into sub-domains One dimension: n h W P dx E dx w e s Dx l e w - Finite volume is a fixed space in the flow domain with imaginary boundaries that allow the fluid to flow in and out. - Integral conservation of the quantities such as mass, momentum and energy. f

General Transport Equation -3D problem steady-state H N W P E S L Equation for node P in the algebraic format:

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: If Vx>0 and If Vx<0 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:

1D example - uniform mesh After substitution a), b) and c) into I): We started with partial differential equation: same and developed algebraic equation: We can write this equation in general format: Unknowns Equation coefficients

1D example multiple (N) volumes N unknowns 1 2 3 i N-1 N Equation for volume 1 N equations Equation for volume 2 …………………………… Equation matrix: For 1D problem 3-diagonal matrix

3D problem Equation in the general format: H N W P E S L Wright this equation for each discretization volume of your discretization domain A F 60,000 elements 60,000 cells (nodes) N=60,000 x = 60,000 elements 7-diagonal matrix This is the system for only one variable ( ) When we need to solve p, u, v, w, T, k, e, C system of equation is larger

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