welcome 
to the second lecture on module 9 on numerical
simulation of turbulent flow in this module
we have discussed the features of turbulent
flow what are the basic elements which differentiate
the turbulent flows from laminar flows and
we also discussed briefly the numerical simulation
strategies of turbulent flow and how different
length scales and time scales which are present
in turbulent flow make our life pretty difficult
so we had a brief look at the main simulation
strategies and we had also had a look at the
time cost of simulation if you just want to
solve simulate all length and time scales
in a turbulent flow using the so called dns
or direct numerical simulation strategy
now next we are going to have a look at what
we call a rans turbulence model and in the
next lecture we would cover what we call large
area simulation so let us have a recap of
the previous lecture we discussed the basic
features of turbulent flow its dissipative
nature random fluctuations which form integral
part of turbulent flow its unsteadiness three-dimensionality
diffusive nature and its dissipative nature
we also discussed the various other features
the length scales which are present their
rough estimation and the numerical simulation
strategies namely the direct numerical simulation
which would involve the resolution of all
length and time scales requiring very fine
grid which makes dns impractical for industrial
flow simulations and we also looked at the
time scales we can use dns primarily for research
purposes to understand the basics of fluid
dynamics the turbulent flows and to refine
what we call the turbulence model used in
rans and large area simulation
we also briefly discussed the reynolds proposition
of what we call reynolds decomposition and
this way we are going to pickup form
we will start off our lecture on reynolds
averaging and rans simulation model rans stands
for reynolds averaged navier-stokes simulation
and rans simulation models are the ones which
are today the practical tool for industrial
scale simulations only recently with the advent
of very high performance computers available
with big corporations that people have been
trying to use large area simulation for final
design comparisons
for the initial design and design a traditional
cycle rans is still what we can practically
employ so we will have a look at some of the
rans models in this lecture so the outline
lecture we would first have a look at the
so called reynolds averaging procedure which
is used in reynolds decomposition
then we would derive so called reynolds averaged
navier-stokes equations in short they are
called rans equations and we will look at
few major turbulence models why do we need
these models that would become apparent when
we have a look at reynolds averaged navier-stokes
equations which have additional terms which
must be modeled and this turbulence models
they try to come up with some formulae for
those unknown additional terms in terms of
the primary quantities
now this is what we had discussed in the last
lecture the reynolds decomposition suggested
that we have flow variable phi in a turbulent
flow field it can be broken into two parts
which can be attributed to the turbulent fluctuations
which are random in nature and if a flow were
statistically stationary other part could
be called time averaged value so that is to
say that for any variable phi
we can write phi at a special location x and
time instant t it can be written as sum of
2 parts phi bar x now this particular part
the average part or mean value part it is
independent of time it only depends on the
spatial location + high prime x t so phi prime
x t that represents randomly varying fluctuating
components of the scalar phi now this phi
could be one of the velocity components it
could represent density in compressible flows
and pressure or temperature
so any flow variable which we encounter in
turbulent flows might be decomposed using
this simple strategy in terms of its mean
value and fluctuating components now this
is specific reason why we wanted in industrial
flow simulations we are interested in time
averaged quantity we are not interested in
what happens to the flow variables at every
special occasion at each instant of time doing
that would be next impossible for large scale
industrial simulation
we are interested in average quantities like
we want to know the forces drag left the pressure
drops occurring across a length of a pipe
or piping material and so on so we are primarily
interested in what we call the gross quantities
or average quantities and it does not make
much sense for us to try and resolve all the
length and time scales in our simulation that
is what we would attempt in direct numerical
simulation that would be a waste of time
so we would try to find out these time averaged
quantities through our simulations in terms
of these quantities we can calculate the values
of engineering interest that is to say the
force is acting on the surfaces that in terms
of what we call as a net drag force or lift
force or the pressure drop energy loss and
so on now we have to use this word phi or
average how do you refine these average terms
so what reynolds suggest is that if you are
dealing with statistically steady turbulent
flows that is to say the flow rate remains
constant with respect to time if you take
the average over a fairly large time period
so in that case for such flows which are formally
called statistically steady
reynolds averaged the time average and we
would define it as phi bar x=limit t tending
to infinity 1/t 0-t phi xt dt now this capital
t represents that averaging interval which
must be large compared to the typical time
scales of fluctuations otherwise we would
not get statistically averaged value of the
scalar field phi at special location x
but in case of the flow were inherently unsteady
that is to say the above fluctuations but
even the average flow rate is varying with
time suppose we have taken pipe flow and we
are steadily increasing the flow rate through
the pipe network so if the flow rate even
in average sense that is varying or changing
with time in such situations we cannot use
a time average so we have to define a different
averaging procedure
and now in this case our phi bar would not
be just a function of this spatial location
x it will also depend on time and we would
define it using what we call unsummable average
unsummable average is defined in terms of
what we call identical experiments if we had
had an opportunity to perform the experiments
in identical conditions and each time we measure
our variable pi at different locations in
time
so at a given location at a given time let
us take out n number of measurements and let
us take average of those n number of measurements
so each phi n xt that represents one measurement
we have taken n number of measurements take
their average and this n should be fairly
large number to give us what we call unsummable
average and this phi bar xt would be now the
unsteady average term and the related arrays
of equations what we call those are called
unsteady arrays
but most of the time you would be dealing
with time averaging and if you look at the
2 definitions which we had in the previous
slide which is defined in terms of integral
of the quantity phi xt over time or in terms
of this summation both of them are basically
linear operations so there are certain consequences
of this linearity
there are decomposition which we had that
was defined as sum of 2 quantities that is
a linear operator similarly the averages which
we refined those were defined in terms of
linear operators integral again is essentially
it can be represented as summation which is
a linear operator similarly an unsummable
average we had some summation so that is again
a linear operator and there is certain algebraic
rules which we can formulate for algebra of
averages and fluctuations of any two flow
variable phi and psi
the way we had defined our time averaged values
so if you take reynolds decomposition if you
want to find out average of the fluctuation
what do we get now let us briefly have a look
at whether this particular thing are we satisfied
with this phi or average would be 0
so remember we have defined our decomposition
phi xt this was defined as phi bar x+phi prime
xt for the time being for the sake of simplicity
i am going to drop the arrow operators from
x and we would presume that where we write
x that represents the 3 dimensional coordinates
of a point now how did you find our average
operator this phi bar x this was defined as
limit t tending to infinity 0-t phi xt dt
1/t
we would say that wherever we use the over
bar that represents this particular reynolds
averaging operator or reynolds averaging procedure
now let us apply this procedure to our decomposition
so let us apply this integration process or
this averaging process on both the sides of
equation 1 apply this averaging process to
both sides of equation 1 then what we get
on the left hand side we will get phi bar
and with this what we will get after averaging
if you average this phi bar xt we get simply
phi bar what would be the time average of
an average quantity remember this phi bar
does not depend on time so if you want to
take its average over the long time interval
it will remain phi bar so the first on the
right hand side what we get that remains as
phi bar x the next term which we had was this
phi prime bar t so what do we get from here
it is obvious enough that average of the fluctuations
=0
so this is first rule of our reynolds averaging
procedure now what will happen if you want
let us have a look at the next few rules on
this algebraic process this we have already
clarified that average of the average that
will remain phi bar the next thing is differentiation
and integration they commute with this averaging
process a differentiation also represents
a linear operation so that is what we say
that differentiation and averaging process
they commute
that is why we get the del phi/del s over
bar that is average of del phi/del s is same
as the derivative of del/del s of phi bar
similarly if you had an integral phi ds s
average is same as the average of phi bar
ds so in nutshell what it means is our two
linear operations for instance in this case
the differentiation and averaging they commute
similarly integration and averaging they also
commute similar
if you have sum of 2 fluctuating quantities
phi and psi the sum of that 2 that would represent
the third quantity which is fluctuating with
time how do we get the average of that quantity
that is simply given by the average of 2 quantities
so phi + phi bar=phi bar+psi bar now let us
verify these terms but when we have a product
now remember this product is no more a linear
operation so product of phi*psi its average
would be given by these 2 terms
that is phi bar + phi bar*psi bar+average
of the product of the fluctuating components
phi prime and psi prime
suppose you want to find out average of phi+psi
how do you work it out by definition this
is limit t tending to 0 1/t integral 0-t phi
xt+psi xt dt by the basic rule of algebra
we know this integral can now be broken into
2 parts integral of the first function+integral
of the second function so we get (0-t phi
xt dt+0-t psi xt dt) limit again the limit
of 2 quantities or the sum of 2 quantities=the
limit of each individual quantity
so limit t tending to 0 1/t 0-t phi xt dt+limit
t tending to 0 0-t psi xt dt and you can easily
recognize these 2 elements they are nothing
but the respective averages phi bar+psi bar
now let us have a look at what happens if
you want to find out the average of the product
of the 2 functions so we want to find out
what will happen to phi to psi now let us
use our reynolds decomposition
so phi can be expressed in terms of phi bar+phi
prime and psi can be expressed as psi bar+psi
prime and we want to apply this over averaging
operator to both of it now let us multiply
an open product so we phi bar psi bar+phi
bar psi prime+psi bar phi prime+phi prime
psi prime so here we want to take average
of these quantities which are being separated
by additional operator so this overall summation
can be broken by the average of the separate
quantities
so we get this phi bar psi bar average+phi
bar psi prime average+psi bar phi prime average+phi
prime psi prime average now remember the way
we have defined this average of phi bar and
average of psi bar is independent of the time
so this would essentially be the product of
phi bar psi bar
similarly from the next term phi bar comes
out and we get psi prime bar+psi bar and phi
prime bar+phi bar psi prime whole over bar
we have just learnt earlier that the average
of the fluctuations that is 0 so these terms
which contain the average of fluctuating terms
only this becomes 0 so we get the average
of the product of 2 functions phi psi that
is given as the product of the individual
averages+average of the product of fluctuations
so look carefully the average of product of
the fluctuations that will not be 0 that will
depend on how these 2 quantities are correlated
now we can use these rules and apply these
rules to our navier-stokes equations to obtain
the so called reynolds averaged navier-stokes
equations so the summary of the equations
on the slide and now let us see how do we
actually obtain these equations
so first one we would like to find out we
would apply reynolds averaging procedure to
our continuity equation
so reynolds averaged navier-stokes equations
and for the time being we will discuss the
case of incompressible flows 
for which density is constant this is slight
difference procedure for obtaining this reynolds
averaging process which is referred to forward
averaging for compressible flows or variable
density flows that is something i would leave
you to explore from the literature
for the time being let us focus on the continuity
equation for our incompressible force this
will be del rho/del t term that vanishes so
we get del/del xi=0 vi is our velocity vector
so apply averaging process to it so apply
reynolds averaging to the above equation 
now remember the rule which we had learnt
earlier that differentiation and averaging
process they commute so this will be simply
del/del xi of vi bar
so thus our time averaged 
continuity equation becomes del vi bar/del
xi=0 now sometimes in our calculations we
keep densities together densities are constant
so that it does not really matter if you want
to put it inside the brackets so del vi bar/xi
so these are 2 forms which are frequently
used in numerical simulations both of them
are basically identical for incompressible
flows next let us work on our navier-stokes
equations and we will look at each term separately
and find out its average
so navier-stokes equations were given by del/del
t of rho vi+del/del xj of rho vi vj=-del p/del
xi+del/del xj of tau ij+rho vi this was our
time dependent navier-stokes equation so now
let us apply averaging process to each terms
so apply reynolds averaging 
to each term the first term contains a time
derivative so del/del t of rho vi we want
to find out the average of it remember again
that averaging and differentiation that commute
so del/del t of rho vi average
rho was basically a constant for incompressible
flows we can also write this as del/del t
of rho vi bar so this is our first step the
tricky part is the next term which is convective
term del/del xj of rho vi vj over bar so the
averaging process that will commute with our
differentiation operator so this is del/del
xj of rho vi vj over bar now remember what
we have learnt earlier recall that formula
recall that the product of phi and psi their
average is given by the product of phi bar
psi bar+phi prime*psi prime average that is
average of the product of fluctuating components
so use it in equation 3 so therefore 
equation 3 becomes del/del xj of rho vi vj
over bar=del/del xj of rho vi bar vj bar+del/del
xj of rho vi prime vj prime over bar similarly
the two terms on the right hand side they
are linear operators so we can straight away
apply the averaging process
bi is a source term or what we call body force
term which would actually not depend on the
turbulence it will be fixed quantity
so thus reynolds averaged navier-stokes equation
becomes del/del t of rho vi+del/del xj of
rho vi bar vj bar+del/del xj of rho vi prime
vj prime over bar=-del p bar/del xi+del/del
xj of tau ij bar+rho vi now what we normally
do is transfer this term which involves fluctuations
to the right hand side and club it with our
stress term so rearrange del/del t of rho
vi bar+del/del xj of rho vi bar vj bar=-del
p bar/del xi+del tau ij bar del xj-del rho
vi bar vj bar del xj+rho bi
so this is what is referred to as reynolds
averaged navier-stokes equation on the left
hand side you see this equation is basically
a transferred equation for time averaged or
reynolds averaged velocity components vi bar
so we have the first term is our temporal
term this will vanish if the flow is statistically
steady next term is convective term in terms
of the averaged components vi bar and vj bar
the first term on the right hand side that
is gradient of time reynolds averaged pressure
field the next term is delta ij bar/del xj
bar that is in terms of our reynolds averaged
velocity components as far we would compute
del ij bar but we have got additional term
here that is bi prime vj prime over bar we
do not know what are these components these
are unknown components and in fact this represents
a second order terms which will have 9 components
so the averaging process has introduced an
unknown tensor quantity which must be modeled
and that reasons of reynolds stress models
which must be used to ensure what we call
the closure of these equations so similarly
if you had a scalar transport equation we
can again apply our averaging procedure left
hand side remains in the form very similar
to our normal equation del/del t of rho phi
bar+del/del xj of rho vj bar phi bar=del/del
xj gamma of del rho phi/del xj+qj bar
this is an additional term we normally use
this symbol tau ij superscript r and we call
this as reynolds stress as defined by –rho
vi prime vj prime so this is fluctuating components
they give rise to a stress like term this
was the reason why we call them as reynolds
stress similarly this is a turbulent flux
which is in terms of the fluctuating components
of velocity would lead to additional scalar
transport and thus the reason why it is referred
to as turbulent flux
let us have a look at the physical significance
of these reynolds stress terms
so these stresses the way we saw they involve
time correlation that is why we had this vi
prime vj prime over bar that is the time correlation
of the fluctuating components velocity the
main consequence of this velocity fluctuation
turbulence is to enhance shear stresses and
thus the transport of momentum within the
flow so the reynolds stress terms contains
components that are velocity fluctuation correlations
and you can verify this is a symmetric tensor
and has 9 components
so we will have basically 6 unknown components
for this symmetric stress tensor and those
have to be somehow modeled in terms of reynolds
averaged velocity field so this is what majority
of turbulence models attempt to ensure closure
of reynolds averaged navier-stoke equations
similarly the turbulent fluxes they arise
from the convective transport due to turbulent
fluctuations fluctuations of the velocity
field as well as fluctuations in scalar component
so when this equation is time averaged the
influenced fluctuations over averaging time
periods include via these additional flux
terms which represent enhanced heat transfer
or enhanced mass transport and via result
of the reynolds averaging applied to our conservation
equation and their sole purpose is to incorporate
the effect of turbulence in enhancement of
the transport of the respective queries
now let us have a look at some of what we
call rans turbulence model reynolds averaged
navier-stokes turbulence models and we can
classify these turbulence models broadly into
2 categories the first one is what we call
eddy viscosity models and these models they
employ what we call eddy viscosity assumption
based on boussingsq preposition
the preposition of boussingsq was that we
can incorporate these enhanced effects coming
from the fluctuating components via an additional
or enhanced viscosity and since that viscosity
is basically caused by the eddy’s or what
we call fluctuating components of the velocity
field so that is why this term eddy viscosity
is used here other model is what we call reynolds
stress for the velocity field that is for
navier-stokes equations and reynolds flux
model for our scalar transport equation
so now these imply additional transport equations
for reynolds stress tensor and turbulence
flux
now let us have a brief look at few other
categorization this turbulence models can
also be categorized based on the number of
additional transport equations which we must
solve for instance if you are solving a time
dependent problem or incompressible flow problem
we have to solve continuity equation and 3
momentum equations so we have to solve 4 partial
differential equations in laminar flow
for turbulent flow to take care of these fluctuating
components or what we call rans components
we have to use additional number of partial
differential equations so how many number
of additional transport equations which we
need to employ a categorization of rans models
can be based on that as well so the number
of pd additional pds which we must need to
enforce closure
the simplest one would be we do not use any
additional partial difference equation we
somehow get an estimate of this eddy viscosity
based on the time averaged solution for the
velocity field and this popularly referred
to as prandil’s mixing length model and
since there are no equations no additional
pds are involved this is referred to as 0
equation model then we can have one additional
pd for instance one model known as spalart-allmaras
model that implies 1 additional partial differential
equation to compute the eddy viscosity
so that is why it is referred to as one equation
model but the most popular in industry are
what we call are 2 equation models for example
k-epsilon model and k-omega model are algebraic
stress models which imply 2 additional transport
equations 2 additional partial differential
equations for instance k-epsilon model you
will have 1 equation for this kinetic energy
k and 1 equation for the dissipation term
epsilon
if you go for reynolds stress model we have
basically 7 equations so reynolds stress model
is referred to as 7 equation model so you
can just realize that if you want to solve
incompressible flow problem we had four equations
1 corresponding to continuity equation and
3 for momentum equations and to incorporate
these reynolds stresses we need to solve for
7 additional equations which will lead to
a tremendous computational burn
so thus far these reynolds stress models they
are used rather infrequently in industrial
cfd simulations
some more things which we must remember that
all these models they involve empirical numerical
constants they are called models they have
not been arrived from vigorous first principles
of continue mechanics majority of them are
based on an extensive experimental data and
empirical numerical constants have been obtained
based on the validation of an assumed model
with experimental data
these constants which we get from the experimental
data of course those will depend on the circumstances
of the problems for which the experiments
were performed so that is why these constants
are not universal in the sense that suggested
values may not yield correct results for all
turbulent flows and hence care must be exercised
in the choice of the model constants we might
have one set of model constants
let us say in one type of flow and for a different
type of flow we have to go for either a different
model or we have to fine tune or we might
have to change these model constants the fine
tuning is typically achieved using 2 means
that is we can perform extensive control experiments
that is one another way is to use direct numerical
simulation on a very small computational domain
performed very fine grid direct numerical
simulation
that can be used to fine tune the model constants
in the rans turbulence models so both of these
approaches these are very active areas of
current research in turbulent flows that is
perform direct numerical simulation on certain
set of geometrics certain flow situations
and perform a rans simulation for the same
problem for similar type of problem which
involves similar physics and then try to fine
tune the model constants used in rans model
so that the rans model gives the average quantities
which are fairly close to what we would obtain
from direct numerical simulations so it is
still a very active open area of research
now we will have a look at boussingsq preposition
which we just referred to now this boussingsq
preposition is based on 1 simple observation
which boussingsq had that looked if we had
a simple laminar flow viscous stresses they
are proportional to what they are related
to velocity gradients
so they are proportional to velocity gradients
or what we call as strain rate tensor and
normally for new term includes we say that
deviatoric components they are directly proportional
to what viscosity times the strain rate tensor
so can we do the same thing so with tremendous
experimental evidence and theoretical evidence
to suggest that the rate of mixing due to
turbulence or the turbulent ads themselves
they depend on the velocity gradients in the
flow
so based on those experimental and theoretical
observations boussingsq proposed that even
reynolds stresses they should be proportional
to the mean velocity gradients especially
the deviatoric reynolds stress it is proportional
to the mean rate of strain so in terms of
the symbols which of interest tau ijr that
is what is our reynolds stress tensor which
is =-rho vi prime vj prime over
this can be represented in terms of mu t*velocity
gradient or strain rate tensors or mu t times
del vi bar/del xj+del vj bar/del xi-2/3 rho
k delta ij so the first term on the right
hand side you can clearly say this is mean
velocity gradient the constant multiply which
we have taken in fact this is not a constant
this would depend on position of the flow
fields it will vary from point to point in
flow field and this particular symbol
since we have used the same symbol which we
used for viscosity so this is referred to
as mu t is called dynamic turbulent or eddy
viscosity the term eddy is often used to correlate
formally that look these stresses arise because
of the different eddies which are there in
the turbulent flows so that is why this mu
t is called dynamic turbulent or eddy viscosity
and the k which we have used in this relationship
that is what is referred to as turbulent kinetic
energy
that is to say we obtained mu prime square
over bar+v prime square over bar+w prime square
over bar/2 this gives us turbulent kinetic
energy per unit mass so this is the boussingsq
preposition which is the basis of eddy viscosity
based turbulence models
similarly the same concept can also be applied
to the diffusion of a scalar quantity that
is to say that we can say the turbulent fluxes
will also be proportional to the gradients
of the scalar quantity so we can say this
qjr which we define as –rho vj prime phi
prime over bar=gamma subscript t del phi r/del
xj so this del phi bar/del xj this is the
gradient of the mean value of the scalar field
phi and this gamma subscript t this is called
turbulent or eddy diffusively
remember this gamma t or mu t which we have
introduced earlier these are not material
properties these depend on the flow and they
would vary from point to point so we have
to obtain estimates of these quantities at
each point in our numerical simulation and
that is precisely what turbulence models attempt
to do so we will take a few of the popular
turbulence models in the next lecture for
the time being we will have just a brief look
at the names
eddy viscosity based rans models we will have
a brief look at mixing length model we will
briefly look at spalart-allmaras model then
standard k-epsilon model for the improvement
model like k-epsilon rng model or realizable
k-epsilon model or k-omega model i would refer
you to appropriate reference or other textbook
so today we stop here
in the next lecture we would have a bit more
detailed look at few of these turbulence models
and reynolds stress model and thereafter we
will start off with large eddy simulation
