Reynolds-averaged Navier
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Transcript 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
iv
'
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
Φq
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
'
Φq
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