modeling of turbulent flows : prandtl mixing length theory
1. Development of Deterministic Methods to Solve
Stochastic Problem....
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
Modeling of Turbulent Flows
2. Simplified Reynolds Averaged Navier Stokes equations
0
z
W
y
V
x
U
U
x
P
V
U t
W
z
P
V
W t
V
y
P
V
V t
4 equations 5 unknowns → We need one more ???
3. Modeling of Turbulent Viscosity
μ Fluid property – often called laminar viscosity
t
μ Flow property – turbulent viscosity
......
-
k
-
k
-
k
Re
3
2
1
Re
-
k
Eq.
Two
Eq.
-
One
TKEM
constant
MVM
μ
on
based
Models
t
t
f
k
kl
l
Curvature
Buoyancy
Low
Layer
Layer
Layer
bounded
wall
Free
High
length
mixing
MVM: Mean velocity models
TKEM: Turbulent kinetic energy
equation models
4. • Eddy-viscosity models
• Compute the Reynolds-stresses from explicit expressions of the
mean strain rate and a eddy-viscosity.
• Boussinesq eddy-viscosity approximation
MVM : Eddy-viscosity models
The k term is a normal stress and is typically treated together with
the pressure term.
ij
ij
t
j
i
ij k
S
u
u
3
2
2
i
j
j
i
ij
x
U
x
U
S
2
1
5. • Prandtl was the first to present a working algebraic turbulence
model that is applied to wakes, jets and boundary layer flows.
• The model is based on mixing length hypothesis deduced from
experiments and is analogous, to some extent, to the mean free
path in kinetic gas theory.
Algebraic MVM
Molecular transport
Turbulent transport
dy
dU
lam
xy
lam
,
mfp
pec
lam l
v
2
1
where,
dy
dU
turbulent
xy
turbulent
,
6. Pradntl’s Hypothesis of Turbulent Flows
• In a laminar flow the random motion is at the molecular
level only.
• Macro instruments cannot detect this randomness.
• Macro Engineering devices feel it as molecular viscosity.
• Turbulent flow is due to random movement of fluid
parcels/bundles.
• Even Macro instruments detect this randomness.
• Macro Engineering devices feel it as enhanced
viscosity….!
7. Prandtl Mixing Length Hypothesis
U
X
Y
y
U
U
U
U
0
0
u
v
0
0
u
v
The fluid particle A with the mass dm located at the position , y+lm
and has the longitudinal velocity component U+U is fluctuating.
This particle is moving downward with the lateral velocity v and the
fluctuation momentum dIy=dmv.
It arrives at the layer which has a lower velocity U.
According to the Prandtl hypothesis, this macroscopic momentum
exchange most likely gives rise to a positive fluctuation u >0.
U
0
v
u
8. • Particles A & B experience a velocity difference which can be
approximated as:
dy
dU
l
y
U
l
U m
m
The distance between the two layers lm is called mixing length.
Since U has the same order of magnitude as u, Prandtl arrived at
dy
dU
l
u m
By virtue of the Prandtl hypothesis, the longitudinal fluctuation
component u was brought about by the impact of the lateral
component v , it seems reasonable to assume that
v
u
dy
dU
l
C
v m
1
Definition of Mixing Length
9. Prandtl Mixing Length Model
• Thus, the component of the Reynolds stress tensor becomes
2
2
2
,
dy
dU
l
C
v
u m
xy
turbulent
• This is the Prandtl mixing length hypothesis.
•Prandtl deduced that the eddy viscosity can be expressed as
• The turbulent shear stress component becomes
2
2
2
dy
dU
l
C
v
u m
dy
dU
lm
turbulent
2
10. Estimation of Mixing Length
• To find an algebraic expression for the mixing length lm, several
empirical correlations were suggested in literature.
• The mixing length lm does not have a universally valid character
and changes from case to case.
• Therefore it is not appropriate for three-dimensional flow
applications.
• However, it is successfully applied to boundary layer flow, fully
developed duct flow and particularly to free turbulent flows.
• Prandtl and many others started with analysis of the two-
dimensional boundary layer infected by disturbance.
• For wall flows, the main source of infection is wall.
• The wall roughness contains many cavities and troughs, which
infect the flow and introduce disturbances.
11. Quantification of Infection by seeing the Effect
• Develop simple experimental test rigs.
• Measure wall shear stress.
• Define wall friction velocity using the wall shear stress by the
relation
u
U
U
y
u
y
Define non-dimensional boundary layer coordinates.
wall
u
12.
U
y
y
U
C
y
U
ln
1
C
u
f
y ,
,
Approximation of velocity distribution for a fully turbulent 2D
Boundary Layer
13.
U
y
Approximation of velocity distribution for a fully turbulent 2D
Boundary Layer
y
U
C
y
U
ln
1
C
u
f
y ,
,
For a fully developed turbulent flow, the constants are
experimentally found to be =0.41 and C=5.0.
14. Measures for Mixing Length
• Outside the viscous sublayer marked as the logarithmic layer, the
mixing length is approximated by a simple linear function.
ky
lm
•Accounting for viscous damping, the mixing length for the viscous
sublayer is modeled by introducing a damping function D.
•As a result, the mixing length in viscous sublayer:
kDy
lm
A
y
D exp
1
The damping function D proposed by van Driest
with the constant A+ = 26 for a boundary layer at zero-pressure
gradient.
15. • Based on experimental evaluation of a large number of
velocity profiles, Kays and Moffat developed an empirical
correlation for that accounts for different pressure
gradients.
0
.
1
26
abP
A
With
0
.
9
25
.
4
0
a
b
P
for
0
.
9
29
.
2
0
a
b
P
for
5
.
1
2
1
w
dx
dP
P
Effect of Mean Pressure Gradient on Mixing Length
19. Conclusions on Algebraic Models
• Few other algebraic models are:
• Cebeci-Smith Model
• Baldwin-Lomax Algebraic Model
• Mahendra R. Doshl And William N. Gill (2004)
• Gives good results for simple flows, flat plate, jets and simple
shear layers
• Typically the algebraic models are fast and robust
• Needs to be calibrated for each flow type, they are not very
general
• They are not well suited for computing flow separation
• Typically they need information about boundary layer
properties, and are difficult to incorporate in modern flow
solvers.
23. Creation of Large Eddies an I.C. Engines
• There are two types of structural turbulence that are
recognizable in an engine; tumbling and swirl.
• Both are created during the intake stroke.
• Tumble is defined as the in-cylinder flow that is rotating
around an axis perpendicular with the cylinder axis.
Swirl is defined as the charge that rotates concentrically about
the axis of the cylinder.
24. Instantaneous Energy Cascade in Turbulent Boundary Layer.
A state of universal equilibrium is reached when the rate of
energy received from larger eddies is nearly equal to the rate of
energy of when the smallest eddies dissipate into heat.
3
5
3
2
k
C
K
E K
25. One-Equation Model by Prandtl
• A one-equation model is an enhanced version of the algebraic models.
• This model utilizes one turbulent transport equation originally
developed by Prandtl.
• Based on purely dimensional arguments, Prandtl proposed a
relationship between the dissipation and the kinetic energy that reads
t
D
l
k
C 2
3
•where the turbulence length scale lt is set proportional to the mixing
length, lm, the boundary layer thickness or a wake or a jet width.
•The velocity scale is set proportional to the turbulent kinetic energy as
suggested independently.
•Thus, the expression for the turbulent viscosity becomes:
5
.
0
k
l
C m
turbulent
with the constant C to be determined from the experiment.
2
2
2
2
1
2
1
w
v
u
u
u
k i
i
26. Transport equation for turbulent kinetic Energy
u
U
μ
x
p
P
v
V
u
U
τ
u
U
ρ
x-momentum equation for incompressible steady turbulent flow:
Reynolds averaged x-momentum equation for incompressible
steady turbulent flow:
subtract the second equation from the second equation to get
u
μ
x
p
V
u
v
U
τ
u
ρ
U
x
P
z
w
u
y
v
u
x
u
u
V
U
Multiply above equation with u and take Reynolds averaging
27.
u
μ
x
p
u
V
u
v
U
τ
u
u
ρ '
Similarly:
v
μ
y
p
v
V
v
v
V
τ
v
v
ρ
w
μ
z
p
w
V
w
v
W
τ
w
w
ρ
2
2
2
2
1
2
1
w
v
u
u
u
k i
i
Define turbulent kinetic energy as:
Reynolds Equations for Normal Reynolds Stresses
28. Turbulent Kinetic Energy Conservation Equation
The Cartesian index notation is:
Boundary conditions:
0
:
at Wall
0
y
k
i
j
i
i
i
i
j
i
i
i
x
x
v
v
p
k
v
τ
τ
V
v
v
x
k
V
τ
k
2
'
29. One and Two Equation Turbulence Models
• The derivation is again based on the Boussinesq approximation
• The mixing velocity is determined by the turbulent turbulent
kinetic energy
• The length scale is determined from another transport equation
ij
t
ij S
2
5
.
0
2
1
k
lm
turbulent
2
2
2
2
1
2
1
w
v
u
u
u
k i
i
2
3
k
lmix
k
i
k
i
x
u
x
u
'
'
with
31. Dissipation of turbulent kinetic energy
• The equation is derived by the following operation on the
Navier-Stokes equation
The resulting equation have the following form
32. The k-ε model
• Eddy viscosity
• Transport equation for turbulent kinetic energy
• Transport equation for dissipation of turbulent kinetic
energy
• Constants for the model
33. Dealing with Infected flows
• The RANS equations are derived by an averaging or
filtering process from the Navier-Stokes equations.
• The ’averaging’ process results in more unknown that
equations, the turbulent closure problem
• Additional equations are derived by performing operation
on the Navier-Stokes equations
• Non of the model are complete, all model needs some
kind of modeling.
• Special care may be need when integrating the model all
the way to the wall, low-Reynolds number models and
wall damping terms.
