### Reynolds-averaged Navier

```Lecture Objectives:
- Define turbulence
– Solve turbulent flow example
– Define average and instantaneous velocities
- Define Reynolds Averaged Navier Stokes equations
Fluid dynamics and CFD movies
• http://www.youtube.com/watch?v=IDeGDFZSYo8
•
http://www.dlr.de/en/desktopdefault.aspx/tabid-6225/10237_read-26563/
•
http://www.youtube.com/watch?v=oOGXEfgKttM
•
http://www.youtube.com/watch?v=IFeSZZ49vAs
•
http://www.youtube.com/watch?v=o53ghmaSFY8
HW problem
The figure below shows a turbulent boundary layer due to forced convection above the flat plate.
The airflow above the plate is steady-state.
Consider the points A and B above the plate and line l parallel to the plate.
Point A
y
x
Flow direction
Point A
Point B
line l
a) For the given time step presented on the figure above plot the velocity
Vx and Vy along the line l.
b) Is the stress component txy lager at point A or point B? Why?
c) For point B plot the velocity Vy as function of time.
Method for solving of Navier Stokes
(conservation) equations
• Analytical
- Define boundary and initial conditions. Solve the partial
deferential equations.
- Solution exist for very limited number of simple cases.
• Numerical
- Split the considered domain into finite number of
volumes (nodes). Solve the conservation equation for
each volume (node).
v
v
x
x
Infinitely small difference

x
x
finite “small” difference
Numerical method
• Simulation domain for indoor air and pollutants
flow in buildings
3D space
Split or “Discretize”
into smaller volumes
Solve p, u, v, w, T, C
Capturing the flow properties
2”
nozzle
Eddy ~ 1/100 in
Mesh (volume) should be smaller than eddies !
(approximately order of value)
Mesh size for direct Numerical
Simulations (DNS)
~1000
~2000 cells
For 2D wee need ~ 2 million cells
Also, Turbulence is 3-D phenomenon !
Mesh size
• For 3D simulation domain
2.5 m
Mesh size
Mesh size
4m
5m
3D space (room)
0.01m → 50,000,000 nodes
Mesh size
Mesh size
0.1m → 50,000 nodes
0.001m → 5 ∙1010 nodes
0.0001m → 5 ∙1013 nodes
Indoor airflow
jet
exhaust
supply
jet
turbulent
The question is:
What we are interested in:
- main flow or
- turbulence?
We need to model turbulence!
Reynolds Averaged Navier Stokes
equations
First Methods on Analyzing
Turbulent Flow
- Reynolds (1895) decomposed the velocity field into a time average
motion and a turbulent fluctuation
v x (x, y, z, t )  V x (x, y, z)  v x (x, y, z, t )
'
vx’
Vx
- Likewise
f   f
,
f stands for any scalar: vx, vy, , vz, T, p, where:

t  t
1
t
t
f dt
Time averaged component
From this class
We are going to make a difference
between large and small letters
Averaging Navier Stokes equations
p  P p
,
ρ ρ
ρ
,
Substitute into Navier Stokes equations
v x  Vx  v x '
Instantaneous velocity
v y  Vy  v y '
fluctuation
around
average
velocity
v z  Vz  v z '
Average
velocity
T T T'
Continuity equation:
v x
x

v y
y

v z
z

time
 (Vx  v x ' )
x

 (Vy  v y ' )
y
 ( Vz  v z ' )

z

Vx
x

x

Vy
y

Vz
z

v x '
x

v y '
y

y
v z '
z
0
Average of average = average
Vx
x

Vy
y

Vz
z

v x '
x


Vz
z
v y '
y


v x '
0
0
0
Average whole equation:
Vx
Vy
v z '
z
Average of fluctuation = 0
0
x

v y '
y

v z '
z
0
Average
Vx
x

Vy
y

Vz
z
0
Time Averaging Operations
 
f' 0
 f'  
f ' 0
f 1f 2  (  1  f '1 )(  2  f ' 2 )   1  2  f '1 f ' 2
div f  div 
div (f1f 2 )  div (  1  2 )  div (f1f 2 )
'
div ( grad f )  div grad 
'
Example: of Time Averaging
Write continuity equations in a short format:
ρ(
v x
τ

 vx

v x
 vy
x

v x
y
 vz
v x
z
)
p
x
 vx
2
μ
x
2
 vx
2
μ
y

v  vx i  vy j vz k
vx
v x
x
 vy
 vx
v x
2

x
2
 vz
y
 vx
v x
2

y
2


 div ( v x v )  v x div v  div ( v x v )
z
=0 continuity
 vx
2

z
2
 μ div(grad
vx)
Short format of continuity equation in x direction:
ρ(
v x
τ
 div(v

x

v ))  
p
x
 μ div(grad
v x )  SM x
2
 vx
2
μ
z
2
 Sx
Averaging of Momentum Equation
ρ(
v x
τ
p

 div(v
v ))  
x
 μ div(grad
x
v x )  Sx
averaging
ρ
v x
τ
 ρ div(v
p

x
v)  
x
 μ div(grad
vx )  Sx
0
ρ
v x
ρ
τ
 ( V x  v' x )
ρ
τ

 ( V x  v' x )
τ


ρ

div ( v x v )  div (V x V )  div ( v v )  div (V x V ) 
'
x

div ( v v )  div ( v (v
'
x
v x v x
'

div(grad
'
x
'
x
v x v y
'
'

'
y
v x )  div(grad

iv
'
x

'
y
ρ
τ
v x v x
'
x
Vx
τ
v x v y
'
'


j  v k) )  div ( (v v i  v v
'
x
'
z
V x )  div(grad
Vx)
'
x
'
y

'
z
v x v z
'

'
Vx
'
x
v x v z
'

z

'
y
'

j  v v k) ) 
'
x
'
z
Time Averaged Momentum Equation
Instantaneous velocity
ρ(
v x
 vx
τ
v x
v x
 vy
x
y
 vz
v x
p
)
z
 vx
2
μ
x
x
 vx
2
μ
2
y
 vx
2
μ
2
z
2
 Sx
Average velocities
ρ(
Vx
τ
 Vx
Vx
x
 Vy
Vx
y
 Vz
Vx
z
)
P
x
 Vx
2
μ
x
2
 Vx
 Vx
2
μ
y
2
2
μ
z
2
v x v x
'
ρ
x
v x v y
'
'
ρ
'
v x v z
'
ρ
y
'
z
 Sx
Reynolds stresses
For y and z direction:
ρ(
Vy
τ
ρ(
 Vx
 Vz
τ
Vy
 Vx
x
 Vy
 Vz
x
Vy
 Vy
y
 Vz
 Vz
y
Vy
 Vz
z
)
 Vz
z
P
x
)
 Vy
2
μ
P
x
x
2
 Vy
2
μ
 Vz
2
μ
x
2
y
2
 Vz
 Vy
2
μ
2
μ
y
2
z
2
 Vz
v y v x
'
ρ
2
μ
z
2
'
ρ
'
ρ
x
v z v x
'
x
v y v y
'
y
v z v y
'
'
ρ
y
v y v z
'
ρ
'
'
z
v z v z
'
ρ
Total nine
z
 Sy
'
 Sz
Time Averaged Continuity Equation
Instantaneous velocities
v x
x
v y

y

v z
z
0
Averaged velocities
Vx
x

Vy
y

Vz
z
0
Time Averaged Energy Equation
Instantaneous temperatures and velocities
ρc p (
T
τ
 Vx
T
x
 Vy
T
y
 Vz
T
z
 T
2
)k
x
2
 T
2
k
y
 T
2
k
2
z
2
Φq
Averaged temperatures and velocities
ρc p (
T
τ
 Vx
T
x
 Vy
T
y
 Vz
T
z
 T
2
)k
x
2
 T
2
k
y
2
 T
k
z
2
T v x
'
2
ρ
x
'
T v y
'
ρ
y
'
T v z
'
ρ
z
'
Φq
Reynolds Averaged Navier Stokes
equations
Vx
x
ρ(

Vx
τ
Vy
y
 Vx

Vz
z
Vx
x
Reynolds stresses
total 9 - 6 are unknown
0
 Vy
Vx
y
 Vz
Vx
z
)
P
x
 Vx
2
μ
x
2
 Vx
2
μ
y
2
 Vx
2
μ
z
2
v x v x
'
ρ
x
v x v y
'
'
ρ
'
y
v x v z
'
ρ
'
z
 Sx
same
ρ(
ρ(
Vy
τ
 Vz
τ
 Vx
 Vx
Vy
x
 Vz
x
 Vy
 Vy
Vy
y
 Vz
y
 Vz
 Vz
Total 4 equations
Vy
z
 Vz
z
)
)
and
P
x
P
x
 Vy
2
μ
x
2
 Vz
2
μ
2
μ
x
2
 Vy
y
2
 Vz
2
μ
2
μ
y
2
 Vy
z
2
 Vz
'
ρ
2
μ
z
4 + 6 = 10 unknowns
We need to model the Reynolds stresses !
2
v y v x
'
x
v z v x
'
ρ
x
v y v y
'
ρ
'
y
v z v y
'
'
ρ
y
v y v z
'
ρ
'
'
z
v z v z
'
ρ
z
 Sy
'
 Sz
Modeling of Reynolds stresses
Eddy viscosity models
v x v x
'
ρ
'
x


x
'
'
( ρv x v x )
Average velocity
Boussinesq eddy-viscosity approximation
' '
ρv i v j Is proportional to deformation   V i   V j 
μt
 x
j

 x i 
 Vy
ρ v y v y   μ t  2
 y
 2
  ρk
 3

Coefficient of proportionality
 Vx  2
ρ v x v x  μ t  2
  ρk
 x  3
 Vx Vy
ρ v x v y  ρ v y v x   μ t 

x
 y




 Vx Vz 
ρ v x v z  ρ v z v x  μ t 



z

x


 Vz Vy
ρ v z v y  ρ v y v z   μ t 

z
 y
  Vz  2
ρ v z v z  μ t  2
  ρk

z

 3




k = kinetic energy
of turbulence
'
k 
'
'
vxvx
2

'
vyvy
2
'

'
vzvz
2
Substitute into Reynolds Averaged equations
Reynolds Averaged Navier Stokes
equations
Continuity:
Vx
1)
x

Vy
y

Vz
z
0
Momentum:
2)
ρ(
3)
ρ(
4)
ρ(
Vx
τ
Vy
τ
Vz
τ
 Vx
 Vx
 Vx
Vx
x
Vy
x
Vz
x
 Vy
 Vy
 Vy
Vx
y
Vy
y
Vz
y
 Vz
 Vz
 Vz
Vx
z
Vy
z
Vz
z
)
)
)
P
x
P
x
P
x




x

x

x
[( μ  μ t )
[( μ  μ t )
[( μ  μ t )
S Tz  S z  s tz  S z 
Similar is for STy and STx

z
4 equations 5 unknowns
Vy
x
Vy
x
Vy
x
]
]
]
[( μ  μ t )
→

y

y

y
[( μ  μ t )
[( μ  μ t )
[( μ  μ t )
v x
x
Vy
y
Vy
y
Vy
y
]
]
]
 (μ  μ t )

z

z

z
v y
y
[( μ  μ t )
[( μ  μ t )
[( μ  μ t )
Vy
z
Vy
z
Vy
z
 (μ  μ t )
We need to model
μt
]  ST x
]  ST y
]  ST z
v z
z
]
Modeling of Turbulent Viscosity
μ
μt
Fluid property – often called laminar viscosity
Flow property – turbulent viscosity

 constant  t
MVM: Mean velocity models
MVM


mixing
length


TKEM: Turbulent kinetic energy

 One - Eq.





 Free






1  Layer





High
Re
wall






 2  Layer




bounded

 3  Layer


k -  






Models based on μ t 


 Low Re



TKEM  Two
 Buoyancy





Eq.

 Curvature




k -




k - l





 k - kl


k  f





......


equation models
Additional models:
LES:
RSM:
Large Eddy simulation models
Reynolds stress models
```