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

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

3. 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

4. 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

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. 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 =  V l
Distance to the closest surface
Air velocity

8. Two equation turbulent model
Kinetic energy
Energy dissipation
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 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

11. Numerics

12. 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

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

14. 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.

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

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

17. Summary: Steady–state 1DI)
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:

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 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

20. 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

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