### A new formulation for turbulent eddy viscosity based on

```A new formulation for
turbulent eddy viscosity based
on anisotropy-I equation
model for inhomogeneous
flows.
By:
Ritthik Bhattacharya.
05NA3001.
Under:
Prof.Hari Warrior.
The objective:

Traditionally the effect of reynold stresses
on the mean flow has been captured by
the use of eddy viscosity. This eddy
viscosity is calculated using the stability
function approach. The work done here is
to find out eddy viscosity using a different
approach which uses the second invariant
of anisotropy. The effect of inhomogeneity
on the eddy viscosity near the boundary of
the flow is also shown.
Points to touch upon:
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Importance of turbulence.
Turbulence modeling.
Reynolds Stress model.
Formulation based on invariant of anisotropy.
The idea.
Importance of inhomogeneity.
Final model.
Results.
Importance of turbulence:
Turbulence is all around us:
 Air flowing in and out of lungs.
 The natural convection in the room in which we
sit.
 Winds and gusts.
 Drag on aeroplanes ,ships , bridges and
buildings.
 Atmospheric and oceanic flows.
 Even the liquid core of earth.
Turbulence Modeling:

Fluid motion is governed by two main
equations:
1.Equation of continuity:
div (u)=0
for incompressible flow.
u: velocity of fluid.
This is basically a statement for
conservation of mass.
2.Navier Stokes Equation:
ρ*(Du/Dt) =–grad p + ρ*ν*div2(u)
which essentially means that the rate
of change of momentum of a fluid
element is equal to pressure gradient
force plus the frictional force in absence
of body forces. Here the frictional force is
modelled assuming newtonian fluid.
NS equation in terms of vorticity:

Now the navier stokes equation for
velocity leads to a velocity distribution
with time as:
u=U+u’
u: total velocity.
U: Mean velocity.
u’: randomly fluctuation velocity .
Putting U+u’ in place of u makes the
navier stokes equation the reynolds stress
equation:
ρ*(DU/Dt) =–grad P + ρ*ν*div2(U)
+ρ div( reynold stress).
Reynold stress being of the form:
<-u’I u’j >
That is the TIME AVERAGE of product of
fluctuating velocity components.

Closure problem:


When the equation is deterministic the
velocity profile has a mean and fluctuating
part.
Now when Reynold equation gives a
formulation for the well behaved mean
velocity then it contains statistical terms
which don’t have exact governing
equation!!
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This is called the closure problem.
The whole point of turbulence modeling is
to model the behaviour of these statistical
terms using empirical equations.
Essentially it is a sophisticated exercise in
interpolating between data sets.
Types of Turbulence models:

Eddy viscocity models.
In these models the reynolds stresses are
assumed to have the same effect as the
mean shear stresses just with an
augmentation by a factor vt /v. v is the
coefficient of molecular kinematic viscosity
while vt represents a much higher “eddy
viscosity”.
-<u’I u’j >
=2vt Sij –(2k/3)*δij
Sij :Mean shear stress.
δij : Dirac Delta function,such that the sum
of tarces of reynolds stress be equal to 2k.(k:being the kinetic energy).
δij=1 for(i =j)
=0 otherwise

One equation models.
In these models apart from the Reynolds
equations another transport equation for
turbulent kinetic energy is used.

Two equation models:
In addition to the reynolds equation two
additional transport equations are used
One for turbulent kinetic energy and the
other for energy dissipation (ε) is used.


where Prt is the turbulent Prandlt
number for energy and gi is the
component of the gravitational vector in
the ith direction. For the standard and
realizable - models, the default value of
Prt is 0.85.
β is the coefficient of thermal exapnsion.
Reynolds Stress Models.

In reynolds stress models the two
transport equation used in addition to the
reynolds averaged equation are the one
for reynolds stresses and another for rate
of dissipation of kinetic energy.(ε)
The equation for reynolds stresses:
d<u’I u’j>/dt=
Pij+Rij-εij+dij
Here:
Pij is the production of reynold stress
modelled by
U j
U i
Pij   uiuk
 u j uk
xk
xk

This is equal to:
-(4/3)*k*Sij
 Rij is the pressure rate of strain term which
distributes reynolds stress values.
It is divided into two terms:
1.Slow pressure rate of strain. (Rijs)
2.Rapid pressure rate of strain.(Rijr)

Rate of slow pressure strain is important in the
process of return to isotropy, where the
turbulence decays thus reverting from its
anisotropic state to isotropy. When a system is
excited into turbulence through the effect of
mean shear or buoyancy, the turbulence
develops into an anisotropic mode. When such
an anisotropic turbulence is allowed to decay the
system returns to isotropy.

Rotta proposed the famous linear model
Rijs=-2CRεaij
Where CRis rotta’s constant, while aij is
the invariant of anisotropy defined as
aij=<u’I u’j >/2k - δij/3
The rapid pressure strain rate is modeled
as:
Rijr= -0.6*( Pij- Pii*δij/3 )
For inhomogeneous flows.
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εij is the dissipation term for reynolds
stress.
For high reynolds number flow it is found
to be:
εij =(2/3)ε δij
Where ε is the energy dissipation rate.
dij is the transport term which is non zero
for inhomogeneous flows only. It consists
of the divergence of a term.
dij=
∂/ ∂(xk) [{.22 k2/ε} ∂/ ∂(xk) (<u’iu’j>)]


The transport equation for ε is nearly the
same as that used in the k- ε model with
an anisotropic coefficient used instead of a
constant in the transport term.
The invariant of Anisotropy.

We have already defined,
aij=<u’iu’j>/2k - δij /3
and the eddy viscosity hypothesis as:
<-u’iu’j>=2vtSij – (2k/3) δij
Therefore using the two above:
aij = -vt*Sij/k

We define the invariant of anisotropy
Π=aij aji
Using the formulation of aij ,
Π =(vt)2 *(Sij)2/k2
so that,
 t   II 
1/ 2
k
S

s=√(Sij)2
This is the important step. This means that the
eddy viscosity can be calculated at any point
given that we find the value of the invariant of
anisotropy at that point and we find the mean
velocity components there.This method needs
less assumptions than the stability function
approach estimating,
vt=Ck2/ε

C being the stability function containing all
the nonlinearity of the problem.

For isotropic 2 D turbulence
Π=1/6 i.e vt=(1/√6)*k/S
For isotropic 3D turbulence
Π=0
i.e vt=0
For 1 D turbulence
Π=2/3 i.e vt= √(2/3)*k/S
The idea was to:

Use the formulations for the transport of
reynolds stress in RSM and that of k from
k-ε model and use them to find a
transport equation for
Π which can give the Π value for all times
and hence the value of vt for all times.

This was performed by Craft et al. for
was for inhomogeneous flows.
The importance of inhomogeneous
flows:
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The viscous sub layer of flow for a flow near a
boundary is inhomogeneous that is the spatial
gradients of the turbulent quantities can NOT be
neglected in the viscous sublayer.
The boundary of a jet or wake where
entrainment of ambient flow occurs into the
turbulent wake field is inhomogeneous.
We start from:
1.Dk/dt=d + P –ε
from k- ε model,
d: transport of turbulent kinetic
energy.
P: production of kinetic energy.
P=-aij *k*Sij
ε:dissipation of turbulent KE
2.d<u’I u’j>/dt= Pij+Rij-εij+dij
Using these two we get governing
equation for aij.
Multiplying the expression for daij./dt by
aij. we get expression for dΠ/dt.

Therefore the equation for Π becomes:
aij
DII
II
 2  dk  Pk     2  dij  Pij  ij   ij 
Dt
k
k

In the homogenous case the expression for
dΠ/dt is found out to be:


d II 2

II1  C R   4S (II ) 1 / 2 II  2
2
15
dt
k


In the inhomogeneous case two extra
terms need to be modeled d and dij.
Since the effect of inhomogeneity is more
on the reynold stress rather than the
turbulent kinetic energy. So d is taken to
be 0 and dij is modeled.
The term added to dΠ/dt is
2aij*dij/k

dij =(.22k2/ε)[∂2/ ∂xk ∂xk (<u’iu’j>)]
Since we are concerned with shear flow in i
direction so spatial derivatives wrt xi is
negligible. So the only spatial derivatives
taken into consideration are those wrt xj.
Therefore
dij=(.22k2/ε)[∂2(aij)/ ∂xj ∂xj] *2k
There for the term to be added,
(.44k2/ ε)[aij ∂2(aij)/ ∂xj ∂xj]
= (.44k2/ ε)
[[∂{aij ∂(aij)/ ∂xj}/ ∂xj]-(∂(aij)/ ∂xj)2]
= (.44k2/ ε)[.5*∂2(Π)/ ∂xj2 –(∂(√ Π)/ ∂xj)2]
The derivatives are calculated using finite
difference schemes

Results:

Though the results for the eddy viscosity
was not decisively better than the
previous stability function approach it was
arrived using much lesser assumptions.
Also the inhomogeneous term changed
the value of eddy viscosity near the
boundary of fluid flow in the POM.
16
IIeq
MY
Observed
14
SST
12
10
8
6
4
400
450
500
550
600
650
days
700
750
800
850

The nature of change of eddy viscosity
with depth and time varies nearly the
same way as in the stability function
model
MY
IIeq
0.10
eddy viscosity
0.08
0.06
0.04
0.02
0.00
450
500
550
600
650
days
700
750
800
0
depth (m)
-50
IIeq - day 30
IIeq - day 32.5
MY - day 30
MY - day 32.5
-100
-150
-200
-250
0.00
0.02
0.04
eddy viscosity
0.06
Thank You!!
```