Understanding Reynolds-Averaged Navier-Stokes Equations: Modeling and Numerical Methods

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

2. From the previous classReynolds 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 !

3. From the previous classModeling of Reynolds stressesEddy 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

4. From the previous classReynolds Averaged Navier Stokes equations Continuity: 1) Momentum: 2) 3) 4) Similar is for STy and STx 4 equations 5 unknowns → We need to model

5. 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

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

7. 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

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

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

10. General CFD Equation Values of , ,eff and S

11. Numerics

12. - Conservation of ffor the finite volume Divide the whole computation domain into sub-domains Finite Volume Method One dimension: n h P E W dx dx w e s Dx w e l - 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

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

14. 1-D example of discretization of general transport equation dxw dxe P Steady state 1dimension (x): E W Dx e w 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.

15. Convection term dxw dxe P E W Dx e w – Central difference scheme: - Upwind-scheme: If Vx>0 and and If Vx<0

16. Diffusion term dxw dxe P E W Dx e w

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

18. 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

19. 1D example multiple (N) volumes N unknowns i N 1 2 N-1 3 Equation for volume 1 N equations Equation for volume 2 …………………………… Equation matrix: For 1D problem 3-diagonal matrix

20. 3D problem Equation in the general format: H N W E P 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

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