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

2. 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

3. 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:

4. Averaging of Momentum Equation

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

6. Time Averaged Continuity Equation
Instantaneous velocities
Averaged velocities
Time Averaged Energy Equation
Instantaneous temperatures and velocities
Averaged temperatures and velocities

7. Reynolds Averaged Navier Stokes equations
Reynolds stresses
total are unknown
same
Total 4 equations and = 10 unknowns
We need to model the Reynolds stresses !

8. 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

9. 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

10. 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

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

12. 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

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

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

15. 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

16. 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

17. 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

18. 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.

19. 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.

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