1. Objective
Reynolds Navier Stokes Equations (RANS)
Numerical methods

2. Time Averaged Momentum Equation
Instantaneous velocity
Average velocities
Reynolds stresses
For y and z direction:
Total nine

3. 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. 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. 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

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

8. Two equation turbulent model
k~[(m/s)2]
Kinetic energy
Energy dissipation
(proportional to work done
by smallest eddies)
=2/eijeij
~[(m2/s3]
From dimensional analysis
Deformation
caused by
small eddy
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. Question – Discussion from previous class !

11. 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

12. 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

13. 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

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. 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.

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

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

18. 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:

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

20. Steady state vs. Unsteady state
We use iterative solver to get solution
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