So, welcome back to the lecture series of
Finite Volume and where will continue our
discussion where we left in the last lecture.
So, PISO is kind of a combination of this
to solver.
So, what happened to simple c algorithm , now
again as I said this is a modified version
of simple algorithm and it is derived by assuming
the velocity correction at point c is some
sort of an, so the velocity correction at
c is some sort of an weighted average correction
.
So, I mean taking care of that some weighted
average corrections of the neighboring elements
. So, one mathematics becomes then V c prime
which was calculated as overall the cell a
F V V H prime over F NB c a F V which will
get you summation of F over all the cell a
F V V F which will be V c prime F NB c a F
V .
Now we use our H operator that we have done,
so, this guy can be written as like F NB c
a F V V F prime by a c V equals to V c prime
summation of F over c a F V by a c V equals
to H c V prime equals to V c prime H c 1 and
then instead of neglecting the a c bar V prime
term. So, in simple what we have done that
it neglects the term is H c bar V prime
So, instead of now in this case we do not
neglect this term rather it is replaced. So,
in this particular case this term is replaced
by an equation , which can be corrected as
1 plus H c 1 V c prime equals to minus D c
V delta p prime c which will lead to V c prime
equals to minus D c V delta p prime c . So,
now what it does? So, from here one can used
to derive the pressure correction equation
and the same result can be achieved by adding
and subtracting the term of F NB c a F V v
c from the momentum equation.
So, use this one in the momentum equation
and then you once you do that the combining
everything it will lead to the a c V plus
summation of F NB c a F V v c plus F NB c
a F V v F minus V c equals to minus V c delta
p c plus b c V which in turn in a compact
form one can write V c plus H c V minus V
c equals to minus d c delta p c plus V c .
So, this you can be used for the velocity
collections as V c prime equals to minus H
c V prime minus V c prime minus D c V delta
p prime c . So, the term this term is dropped
which is equivalent to the approximation of
the previous equilibrium in a modified velocity
correction is used driving the pressure correction
. So, this is dropped and only this written
this term .
So, this is a better estimate in simple c
the relaxation of the pressure becomes unnecessary
and as compare to simple. So, in this case
we may not required to be the relapse the
pressure and the result in velocity corrections
will satisfy better the momentum equation
but consequently at higher convergence rate
is obtained. So, the convergence rate is higher
for this case so that is important.
Now the second another algorithm which sits
in between is called the prime algorithm , prime
is stands for pressure implicit pressure implicit
momentum explicit . So, the momentum equation
solve explicitly and the explicit treatment
is justified by the mall contribution to the
convergence on the other hand the pressure
corrections or pressure equation is solve
implicitly.
So, since the momentum equation is solved
explicitly one can get the new or intermediate
velocity corrections like minus H c V n minus
D c V delta p n then c plus B c V . And the
velocity correction is applied to direct the
pressure correction equation.
So, the correction fields are now the correction
fields one can obtained as V c double star
equals to V c star plus [vocalized-noise c
prime and P c star equals to p c n plus P
c prime and the corrected field this will
satisfy the equation V c double star equals
to minus H c V double star minus D c V delta
p star c plus B c V which is minus H c V star
plus V prime minus D c V delta p n plus p
prime which is c plus b c V .
Now, which will lead to the expression relating
to the pressure and velocity felid like V
c prime equals to minus H c V star minus V
n plus H c V prime minus D c V delta p c prime
. So, this particular term I mean if you substitute
in the continuity equation which will get
you back minus F nb c rho f D f delta p f
dot S f equals to minus summation F nb c m
dot star f plus summation n b c rho f H f
V star minus V n plus H bar V prime dot S
f .
Now the underline term which is here this
can be neglected or typically this is neglected
. So, the term which is neglected in the prime
this underline term can become smaller and
the, but then the term neglected in the simple
. The simple algorithm , the neglected term
was H c V prime . So, this was not that smaller
and can lead to the I mean sure convergence
rate , but in this prime algorithm this term
is quite smaller compared to that and the
neglection of that term does not have too
much of difference and it actually leads to
better convergence .
So, now using this simple c and prime the
algorithm which actually works is a PISO algorithm
. So, PISO algorithm is an interesting it
stands for again the in between of this simple
and these things and what happens in the PISO
algorithm that the which one neglected term
in the simple algorithm like H c V prime it
is partially recovered in second corrector
step .
So, the PISO is essentially based on some
sort of an predictor corrector kind of approach
and so what one can look at it that because
we have in the simple we are computed V prime
and then neglected this guy the continuity
satisfying the V star star the quantity V
star star and pressure p star
So, which were used to recalculate the coefficients
of the momentum equation and then to solve
it explicitly the new velocity field V star
star is also used to calculate the m star
star star at the element faces using Rhie
Chow interpolation and this guy a c V prime
is also partially recovered and the equation
that does is that equals to V c triple star
plus V c double prime which is minus H c V
double star this is double star minus D c
V double star delta p star c plus V c double
prime which will be minus h c double star
V V star plus V prime minus D c V double star
delta p star c plus V c double prime .
Which is minus H c double star V star minus
H c double star V prime minus D c V double
star delta p star c plus V c double prime
and this if you write H c double star V star
minus D c V double star delta p star c minus
H c double star H c double star minus D c
V delta p prime c plus V c double prime .
So, this equivalent to your V c triple star
, so this will become V c one triple star
plus V c double prime minus H c double star
D c V delta p prime c. So, this term represent
the portion of the H c V prime which is neglected
and it is kind of in this algorithm partially
recovered and the second velocity correction.
The second velocity correction that satisfies
that V c double prime equals to minus H c
double star V double prime minus D c V double
star delta p prime c . Now, using the Rhie
Chow interpolation between points so interpolation
between C and F these are the cell neighbors
one can obtain the new pressure corrections
equation which is n b c rho f D f delta p
f double prime dot S f equals to a minus F
nb c m dot f star plus summation of F nb c
rho f H f bar v double prime dot S f .
So, this is again a term which are again this
is neglected in the PISO also and this corrector
step may be replaced as many as time desire
so that you can lead to better convergence.
So, if you look at how it works so this actually
kind of giving you the complete PISO algorithm
and one can also look at it any finite volume
books those are mentioned is a very standard
algorithm. So, what happens you first this
is at that p plus delta t time iteration,
so this is a physical iteration . So, outer
loop as we have done in the simple this is
physical time integration .
So, what you do at the time level t plus delta
t? So, you have all m dot f n so here m dot
f n V n p n so these are all guess. So, these
steps is the guessing step so you guess everything
, then you assemble and solve for momentum
equation for all the star quantity; that means,
the first level of intermediate step where
you calculate all the star quantity then you
compute here the m dot f star using Rhie Chow
interpolation , you then after that at this
step you collect everything and solve for
pressure corrections once you are done then;
obviously, using the pressure correction the
m dot f star V star and p n is obtained.
So, these are use this correct this things
to get m dot f double star . So, at this step
you finally, get m dot f double star and V
double star and p star which is calculated
based on these values which are corrected
values using the p prime equations . Now this
portion of the algorithm particularly this
is exactly what we do in the simple algorithm
, now here we again assemble everything and
explicitly solve for the momentum and here
you solve for V triple star because already
at the second level of correction, so what
happens is that it goes in multilevel iteration
.
So, the first level you do simple kind of
algorithm and initially you started with the
guess value once you get a corrected value
you use that for the next level of algorithm
where you obtain V star V double star , then
again you calculate the m dot f triple star
using Rhie Chow interpolation then you get
pressure corrections equation here you corrections
for p double prime and then after that at
this step what you do this step you actually
get the corrections field m dot f triple star
and V triple star and p star and this correction
fields are use to calculate m dot f four star
V four star and p double star so, these are
used .
Now, if the number of corrector step exceed
it if it is yes then you move, if it is no
then you actually assign those value here
to the double star value and repeat this process.
So, this is the part of prime algorithm and
this prime algorithm is repeated , but if
that is exceeded that you do at this step
you get m dot f n equals to m dot f four star
V n equals to V four star and p n equals to
p double star . So, you assign that if it
is converged to move ahead if it is not you
go back and repeat this step. So, again you
compile simple at PISO algorithm , if it is
converge you update this thing for the next
t plus delta t iteration at once in time again
the time limit exceeded stop or you go back
and do the physical iteration .
So, it does twice kind of or two times of
the iteration in the process and now that
is why PISO is provide better stability , better
convergence and if you have a slightly non
orthogonal skewed grid and those kind of cases
actually PISO was better than simple algorithm
or simple c algorithm . Now that is why it
is special preferred and lot of the c f d
course actually is based on PISO, they provide
a better stability and better convergence
.
Now, moving ahead we have used under relaxation
factor , so now, one can look at the optimum
under relaxation factor for V and pressure
corrections. So, what one can see if you recall
the velocity corrections how we do that? We
do that minus D c into del p c and once I
am calculating the pressure field which use
some sort of and under relaxation in the velocity
corrections and we wrote in this fashion where
v c prime equals to minus H c v prime D c
delta p prime c . So, that is how we written
and now if you equate these two guys or these
two equations one can write minus D c delta
p prime equals to minus H c v prime minus
lambda which will provide lambda p equals
to this or so that gives an , so the simple
c algorithm eliminated the need to under relaxation
pressure corrections and result in an optimum
acceleration rate .
So, therefore, using an approximation introduce
with the simples will the velocity corrections
at c can be written and weighted average like
this. So, this is done in simple c and this
is the corrections for lambda for p prime.
So, the pressure correction equation this
kind of correction factor . Now when you move
ahead from the simple to simple c the kind
ofcorrections was eliminated and the velocity
corrections can be estimated like this this
already we have seen, so there is nothing
new here that we are writing .
So, if you go back to your expression for
the simple c or simple algorithm you will
find this expression and then the coefficients
can be written in this fashion invoking the
is a under relaxation factor . So, if you
simplify that you will get this and so the
role of under relaxation would limit to this.
Now if you substitute this equation in the
velocity corrections approximation here if
you use this one here then you get v c prime
equals to a F v by lambda from where you get
lambda p equals to one minus lambda v . So,
what people have seen that , that simple algorithm
which is under relaxation factor satisfying
this equation provides to similar to that
kind of simple c kind of algorithm .
So, that is an optimum lambda calculation
for simple algorithm which can behave like
an simple c algorithm .
Now, one more important thing which one can
discuss is the Rhie Chow interpolation and
the treatment of the various terms in the
Rhie Chow interpolation . So, first one can
look at the under relaxation term , now the
under relaxation term in the momentum equation
this is how one has written the momentum equation
and the under relaxation term written like
that.
So, if you do some sort of an algebra this
is for momentum equation and velocity corrections
you get back this equation which is again
we have looked at while looking at the momentum
corrections equation and then the Rhie Chow
interpolation should be written in this fashion
and the this particular term one can express
that contribution from the elements c and
the contribution from element f. So, finally,
this actually leads to a some sort of an calculations
of the face variable which is written as v
f.
So, all this algebra you can carry out because
this starts with the momentum equation , the
important point here is that you get back
the surface value which is a average surface
value, then there is a delta p f and delta
p gradient and then 1 minus lambda. So, this
is how when we use under relaxation the Rhie
Chow interpolation actually gets modified
.
Similarly same equations can get modified
for the temporal discretisation, so when you
have written the temporal here these stands
for t minus delta t or previous time steps
and nothing stands for t or the present time
step. So, this is my momentum equation with
all thediscretize form then you write at the
face this is the modified and the face value
can be average like that.
And once you expand this finally, you get
this is an important calculation for the transient
term how the face value one can treat and
that case you get face value minus this and
you use this things . So, also one can do
the treatment of the under body force term
and body force term if you do the treatment
.
So, you can actually get, so and then just
let us look at one important thing for the
interpolations for all combined effects which
takes care of the under relaxation term; under
relaxation term , transient term and it also
takes care of the body force term.
So, then the surface vector v f is a average
D f v delta p f minus delta p f bar D f v
this will the term which will come due to
body force term and then this contributions
are there . So, this is how one can look at
the interpolation scheme in this kind of system
. Now have been said that that pretty much
actually takes care of the discussion on the
incompressible flow , now will extend the
discussion similar discussion where you can
also obtain the pressure corrections equation
or all this things for compressible cases
.
Now, there is a important difference between
compressible case where it is like that is
density is no more constant and mach number
is quite high or greater than one . So, that
is where the compressibility if it starts
acting and once that is happening you cannot
actually assume the density to be constant
rather density to be approximated using ideal
gas law.
So, that is why the , now in the compressible
flow solver there are two types of approach
one can have density based algorithm and one
can have pressure based algorithm and what
we are discussing so far all are pressure
based algorithm and when deriving the equation
for the incompressible case we did not bother
about density too much , though we written
the density in the term.
So, similar algorithm we can directly extend
for the compressible cases Now, the other
thing is that density based algorithm is not
the I mean scope of this particular lecture.
So, one can talk about that in details in
a separate lecture because it has different
kind of issues all together. So, right now
we will concentrate on pressure based algorithm
and can see how we can achieve .
Now, these are the set of equation this is
your continuity equation, this your momentum
equation, this is your energy equation . Now,
when you and one more equation which will
be required is the ideal gas law p equals
to rho RT which will get you calculate the
rho by p by RT . So, top of this all you need
this equation where the pressure and density
can be calculated and connected with the temperature
.
So, in compressible case when we are talking
about now if you see the difference between
in compressible case we did not really bother
about the energy equation, but when you come
down to compressible case so these are very
standard equation and any textbook on fluid
mechanics or compressible flow you can find
this equation. So, there is no point going
into the details of these discussion . So,
and rather we have done detailed discussion
in our initial lectures quite we derive the
our governing equations , now these are connected.
So, let us look at a element c which is our
standard element you have all this six faces
and these are fluxes or flux vector and then
if you actually take a volume integral of
the diffusion term which will convert to a
surface integral like this which we have already
seen and delta dot mu f will be mu f into
this and any flux gradient calculation can
we contribute one is the gc and gf gc and
gf are the geometric coefficients which we
have used earlier. Now how would you derive
actually pressure corrections equation .
So, first equation that will start with p
equals to rho RT and if you expand rho using
some sort of an use Taylors series to expand
rho then you write a rho equals to rho plus
del rho by del p so it is rho star plus rho
prime, so rho prime become c p like this . So,
similarly you have the corrections of the
velocity and pressure field . So, the pressure
previous pressure plus correction density
velocity mass flow rate , when we looked at
the incompressible case we did not bother
to look at the corrections of the density
.
Now, we need both corrections of all the density
and all these then if you put this in the
semi discretized equation this is how it looks
like where mass flow rate would be rho f star
plus rho f prime v f star plus v f prime and
which will boils down to two quantity. So,
this is the second correction term is usually
this is neglected and then we obtain the detail
correction term . Now will stop here today
and will take from here in the follow up lectures.
Thank you .
