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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
x
Flow direction
line l
y
Point A
Point B
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.
HW problem
Point A
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 xy 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).
x
v
x
v xx
Infinitely small difference finite “small” difference
Numerical method
• Simulation domain for indoor air and pollutants flow in buildings
Solve p, u, v, w, T, C 3D spaceSplit or “Discretize”into smaller volumes
Capturing the flow properties
nozzle
Eddy ~ 1/100 in
Mesh (volume) should be smaller than eddies ! (approximately order of value)
2”
Mesh size for direct Numerical Simulations (DNS)
Also, Turbulence is 3-D phenomenon !
~2000 cells
~1000
For 2D wee need ~ 2 million cells
Mesh size
• For 3D simulation domain
3D space (room)
5 m
4 m
2.5 mMesh size 0.1m → 50,000 nodes
Mesh size 0.01m → 50,000,000 nodes
Mesh size 0.001m → 5 ∙1010 nodes
Mesh size 0.0001m → 5 ∙1013 nodes
supply
exhaustjet
jet
Indoor airflow
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
- Likewise
stands for any scalar: vx, vy, , vz, T, p, where:
)z,y,(x,vz)y,(x,V)z,y,(x,v 'xxx
,
d
1
Time averaged component
Vx
vx’
From this class We are going to make a difference between large and small letters
Averaging Navier Stokes equations
, ρ ρ ρ 'vVv xxx
, pP p
'vVv yyy
'TT T
'vVv zzz
Substitute into Navier Stokes equations
Continuity equation:
0z
'v
y
'v
x
'v
z
V
y
V
x
V
z
)'vV(
y
)'vV(
x
)'vV(
z
v
y
v
x
v zyxzyxzzyyxxzyx
0z
'v
y
'v
x
'v
z
V
y
V
x
V zyxzyx
Average whole equation:
Instantaneous velocity
Averagevelocity
fluctuationaround averagevelocity
0z
'v
y
'v
x
'v
z
V
y
V
x
V zyxzyx
Average of average = average Average of fluctuation = 0
0 0 0
0z
V
y
V
x
V zyx
Average
time
'0'
2121221121 '')')('(
0'
Time Averaging Operations
divdiv
)( graddivgraddiv
)( )( )( '2
'12121 divdivdiv
Example: of Time Averaging
)vdiv(gradμ z
v
y
v
x
vx2
x2
2x
2
2x
2
) vv(vv) vv(z
vv
y
vv
x
vv xxx
xz
xy
xx
divdivdiv
=0 continuity
x2x
2
2x
2
2x
2x
zx
yx
xx S
z
vμ
y
vμ
x
vμ
x
p)
z
vv
y
vv
x
vv
τ
vρ(
xMxxx S)vdiv(gradμ
x
p))vdiv(v
τ
vρ(
Write continuity equations in a short format:
kvjvivv zyx
Short format of continuity equation in x direction:
Averaging of Momentum Equation
xxxx S)vdiv(gradμ
x
p))vdiv(v
τ
vρ(
xxxx S)vdiv(gradμ
x
p)vdiv(v ρ
τ
vρ
τ
Vρ
τ
Vρ
τ
)v'V(ρ
τ
)v'V(ρ
τ
vρ xxxxxxx
averaging
0
z
vv
y
vv
x
vv
)k)vvjvviv(v()k)vjvi(vv()vv(
'z
'x
'y
'x
'x
'x
'z
'x
''x
'x
'x
'z
''x
'x
''x
yy divdivdiv
z
vv
y
vv
x
vv)V(V )vv()V(V )vv(
'z
'x
'y
'x
'x
'x
x''
xxx
divdivdivdiv
)V div(grad)V div(grad)vdiv(grad xxx
Time Averaged Momentum Equation
x
'z
'x
'y
'x
'x
'x
2x
2
2x
2
2x
2x
zx
yx
xx S
z
vvρ
y
vvρ
x
vvρ
z
Vμ
y
Vμ
x
Vμ
x
P)
z
VV
y
VV
x
VV
τ
Vρ(
x2x
2
2x
2
2x
2x
zx
yx
xx S
z
vμ
y
vμ
x
vμ
x
p)
z
vv
y
vv
x
vv
τ
vρ(
Instantaneous velocity
Average velocities
Reynolds stresses For y and z direction:
y
'z
'y
'y
'y
'x
'y
2
y2
2
y2
2
y2
yz
yy
yx
y Sz
vvρ
y
vvρ
x
vvρ
z
Vμ
y
Vμ
x
Vμ
x
P)
z
VV
y
VV
x
VV
τ
Vρ(
z
'z
'z
'y
'z
'x
'z
2z
2
2z
2
2z
2z
zz
yz
xz S
z
vvρ
y
vvρ
x
vvρ
z
Vμ
y
Vμ
x
Vμ
x
P)
z
VV
y
VV
x
VV
τ
Vρ(
Total nine
Time Averaged Continuity Equation
Time Averaged Energy Equation
0z
v
y
v
x
v zyx
Instantaneous velocities
Averaged velocities
0z
V
y
V
x
V zyx
qΦz
Tk
y
Tk
x
Tk)
z
TV
y
TV
x
TV
τ
T(ρc
2
2
2
2
2
2
zyxp
Instantaneous temperatures and velocities
Averaged temperatures and velocities
qΦz
vTρ
y
vTρ
x
vTρ
z
Tk
y
Tk
x
Tk)
z
TV
y
TV
x
TV
τ
T(ρc
'z
''y
''x
'
2
2
2
2
2
2
zyxp
Reynolds Averaged Navier Stokes equations
0z
V
y
V
x
V zyx
x
'z
'x
'y
'x
'x
'x
2x
2
2x
2
2x
2x
zx
yx
xx S
z
vvρ
y
vvρ
x
vvρ
z
Vμ
y
Vμ
x
Vμ
x
P)
z
VV
y
VV
x
VV
τ
Vρ(
Reynolds stresses total 9 - 6 are unknown
y
'z
'y
'y
'y
'x
'y
2
y2
2
y2
2
y2
yz
yy
yx
y Sz
vvρ
y
vvρ
x
vvρ
z
Vμ
y
Vμ
x
Vμ
x
P)
z
VV
y
VV
x
VV
τ
Vρ(
z
'z
'z
'y
'z
'x
'z
2z
2
2z
2
2z
2z
zz
yz
xz S
z
vvρ
y
vvρ
x
vvρ
z
Vμ
y
Vμ
x
Vμ
x
P)
z
VV
y
VV
x
VV
τ
Vρ(
same
Total 4 equations and 4 + 6 = 10 unknowns
We need to model the Reynolds stresses !
Modeling of Reynolds stressesEddy viscosity models
)vρv(xx
vvρ '
x'x
'x
'x
Is proportional to deformation 'j
'ivρv
Boussinesq eddy-viscosity approximation
ρk3
2
x
V2μvvρ x
txx
i
j
j
i
x
V
x
V
x
V
y
Vμvvρvvρ yx
txyyx
Average velocity
x
V
z
Vμvvρvvρ zx
txzzx
z
V
y
Vμvvρvvρ yz
tzyyz
ρk3
2V2μvvρ y
tyy
y
ρk3
2V2μvvρ z
tzz
z
k = kinetic energy of turbulence
2
vv
2
vv
2
vvk
'z
'z
'y
'y
'x
'x
Substitute into Reynolds Averaged equations
tμ Coefficient of proportionality
Reynolds Averaged Navier Stokes equations
xTy
ty
ty
tx
zx
yx
xx S]
z
V)μμ[(
z]
y
V)μμ[(
y]
x
V)μμ[(
xx
P)
z
VV
y
VV
x
VV
τ
Vρ(
yTy
ty
ty
ty
zy
yy
xy S]
z
V)μμ[(
z]
y
V)μμ[(
y]
x
V)μμ[(
xx
P)
z
VV
y
VV
x
VV
τ
Vρ(
zTy
ty
ty
tz
zz
yz
xz S]
z
V)μμ[(
z]
y
V)μμ[(
y]
x
V)μμ[(
xx
P)
z
VV
y
VV
x
VV
τ
Vρ(
]z
v)μμ(
y
v)μμ(
x
v)μμ[(
zSS z
ty
tx
tztzzTz
sSSimilar is for STy and STx
0z
V
y
V
x
V zyx
Momentum:
Continuity:
4 equations 5 unknowns → We need to model
1)
2)
3)
4)
tμ
Modeling of Turbulent Viscosity
μtμ
Fluid property – often called laminar viscosity
Flow property – turbulent viscosity
......
-k
-k
-k
Re
3
2
1Re
-k
Eq.
Two
Eq.-One
TKEM
constantMVM
μon based Models
t
t
fk
kl
l
Curvature
Buoyancy
Low
Layer
Layer
Layer
bounded
wall
Free
High
lengthmixing
MVM: Mean velocity modelsTKEM: Turbulent kinetic energy equation models
LES: Large Eddy simulation modelsRSM: Reynolds stress models
Additional models: