So welcome back to the lecture series of Finite
Volume and what we have discussing is now
after finishing the incompressible fluid flow
problem, we are now in the middle of the compressible
fluid flow problem and once we do that, then
we will discuss some of the special topics
and there I mean essentially we will just
patch up on those special topics how to discretize
and implement in the finite volume context
.
So, let us go back where you left in the last
lecture is in the compressiblepressure based
corrections .
And where once you try to look at the pressure
correction equation , this is where we started
with the pressure density relationship, then
we used the Taylor Series expansion to get
the density where one component is the star
component. Again this case like our incompressible
case you have a correction component which
is prime and the intermediate component which
will be calculated like n star , then the
density prime was obtained like this and once
you put it back in the discretize continuity
equation, this is your discretized continuity
equation, you get this equation and where
m dot f is m dot like this and these term
here which is can be treated as intermediate
m dot f star and this is m dot f prime the
second order correction term like rho f prime
v f prime dot s f prime s f .
So, which is these term the second order correction
terms these are neglected . So, then because
these term is typically smaller. So, once
you neglect that the approximation doesnt
influence too much the iterative process or
the solution process which you .
Now, use the Rhie Chew interpolation . So,
similarly Rhie Chew interpolation is applied
and of the flux the mass flux at the cell
face can be obtained like rho star f v star
f bar dot s bar and then . So, this is equivalent
your rho f star v f star dot s f and here
m f prime is rho f star v f prime bar dot
S f minus rho f star d f v minus pressure
corrections equation and then, one contribution
come from the CP and if you do the algebra,
finally you get this expression and again
the second order term which is shown here
the under line term these are actually neglected
in the iterative process .
So, now the I mean the second order term here
it is neglected and the under lined term which
is here that actually present some difficulties
as was done in the incompressible algorithm
. So, typically this one also sort of dropped
or neglected and if you neglect this term,
the correction to the mass flux or m dot f
prime begins rho f star dot S f and this much
. So, now your corrected mass flux is looks
like that in the first term here is similar
to what arising at the incompressible case.
So, if you look at the first term this is
exactly with the kind of term that you have
seen in the incompressible case and second
term is due to, so this is due to due to density
correction. So, this is due to density correction
. Since it is a compressible case, the density
corrections do appear in the system .
Now, in the compressible system the second
term is quite important because it transforms
the pressure corrections equation to an elliptical
system than 2 and 1 also in a other format
. Now at the same time we can devise our simple
algorithm or compressible simple algorithm
based on this .
Now, one can do some sort of a normalization
of these term like m dot f star dot S f into
CP f by rho f star. If this kind of normalization
is done of weighting factor one for the for
P f prime term which will be proportional
to now becomes an weighting factor P f prime
term which will be proportional to 1 by mach
number square where m is the mach number now
and that case the m dot f prime will be r
t rho f star by this and like this expression
.
So, when you have a low mach number value,
the delta P correction term dominants and
that returning equation is actually elliptically
nature, but on the other hand when your mach
number is too high , so the P prime correction
and can no longer be neglected which will
given rise to a hyperbolic character to the
correction equation . So, these combined behavior
actually allows the prediction of the fluid
flow of all the speed. That means, there is
a strong correlation between mach number and
pressure . So, and that changes the system
behavior accordingly when mach number is low
or the lower mach number cases rather incompressible
cases your delta P prime equation behave like
an elliptical system , but when you go down
to high mach number cases, they behave like
an hyperbolic system .
So, now once you substitute everything back
in the continuity and compressible flow of
the pressure correction equation, so this
will be the transient term V c by delta t
c P P c prime summation over cell faces and
then, you density correction. So, you can
put them together nicely in a compact form
and same thing once you put the another treatment
of the underlined term yields, the variant
of simple .
So, how you actually treat this term? This
gives you a various simple algorithm which
we have already seen in the case of incompressible
flow where simple simple c prime or piso it
depends how we take care of this correction
term . Now, dropping the underline term the
pressure correction equation for the simple
algorithm can be modified like Vcc rho by
delta P and so this is a transient term . This
is conviction like term summation over all
the faces due to density diffusion like term
and then which will be a right hand side you
have a source and corrections. So, that is
a source term.
So, this is your P prime equation for compressible
case . Now one can note that because that
convergence the correction fields always leads
to 0. So, the order of scheme used todiscretize
this convection like term is of no consequence
on the accuracy of the final results. So,
however there is not the case for m dot f
star. The use of higher order schemes in its
evaluation can improve the capture of socks
in the algorithm. So, to enhance the robot
robust nest and its helpful to use the append
scheme for the discretization of this convection
like termone can actually neglect the non-orthogonal
contribution for the diffusion like term and
then, the pressure correction coefficients
would become like this .
So, a c P c prime summation over all this
would give rise to this source equation, that
is my discretized equation for p prime. In
compressible case a p prime would be minus
rho F d f minus mass flow rate m dot f c f
rho f a C P prime would be Vcc rho by delta
t summation overall interior faces c p m dot
f and then correction and rho this will be
behaving like an. So, you have one along the
face; another one the tangential. So, this
is like non orthogonal corrections and this
can be neglected, but this will only once
you neglect that your convergence should be
slower, but eventually since its iterative
process you will end up getting a converge
solution. Now, once you do that your corrections
components can be estimated. Once you get
the p prime from this, once you get p prime,
then your v c star star equals to v c star
plus v c prime where v c prime equals to minus
d c v delta p prime c and p c star is p c
m and lambda.
So, here you use some sort of a under relaxation
and the factor is lambda because lambda p,
then rho c start start equals to rho c star
plus lambda p c. I mean this also for some
sort of a under relaxation for lambda rho
and then finally calculate m dot f double
star equals to m dot f star plus m dot prime
and m dot prime is estimated like this . So,
you have different under relaxation term and
you get the corrections term . Now, top of
the it one has to also discretize the energy
equation . Now, before doing that why I mean
to get the discretize energy equation we can
look at the different terms which are part
of the energy equation . For example, first
thing that we can look at the specific heat
term .
So, the other terms in energy equation, so
the specific heat term that if you look at
the volume integral of that, it will get corrections
and the previous iteration value and you can
write that term like this where if you see
that is a d c p by d t setting there which
is a material derivative or substantial derivative
of the specific heat . Now, as long as your
flow is incompressible, you can actually neglect
these terms or the contribution due to this
become essentially zero in incompressible
crisis , but since we are discussing the compressible
case we cannot neglect this term and this
leads to some sort of and transient and special
derivative of that .
Similarly, the pressure term or the substantial
derivative term which for the pressure if
you do that, this will be d p by d t star
at c v c. Again this will be written some
transient plus spatial component . So, you
need to take that also into account .
Now, you have this is your viscous dissipation
term . So, the viscous dissipation term if
you do the volume integral, this will only
get you back a volume integral with the corrections
and the other term is the if you have any
source or sink like term , then you get a
volume integral of that which will written
like that nowthis would be source or sink
term. So, this guy is the dissipation term
and this is your viscous dissipation term.
So, correct that thing . So, what first term
is the only dissipation term , this is the
viscous dissipation term where mu is setting
there and this will be the source term which
will be integrated. This we have already seen.
The only thing which is the new term arising
even then if you have a viscous dissipation
term all the it only actually gives rise to
the velocity gradient. Apart from that the
integration over a cell element is not that
difficult , but the difficulties arises from
this d p by d t term and d c p by d theta
.
Now, if you took the energy equation and this
will be your discretized form of the energy
equations. Discretized form again it looks
like similar to our momentum or anything else.
So, a c t c summation over cell a f t f, but
the superscript p stands for the temperature
coefficients a f where minus k f e f by d
c f minus mass flow rate c p f a c t is a
c dot minus here again dot . This guy is the
transient contribution.
So, this is a transient contribution minus
this and the transient contribution is rho
c c p c v c by delta t and a c not and the
subscript from t minus delta t and without
superscript ist level which can be calculated
like d c p by d t and the source term we have
some over the faces then c p temperature deferred
correction. This is a deferred corrections
where you take the high resolution and the
append corrections previous time integration
and this . So, now if you look at the coefficients
here they are bit involved and why they are
bit involved, the reason is because of the
compressibility since it is a compressible
flow. The pressure density and the other quantities
like c p they are connected and one cannot
neglect from one other . Top of that you come
across like this kind of spatial I mean the
material derivative or substantial derivative
term in your discretized equation .
Now, having said that we can actually this
is our simple algorithm for compressible case
. So, we can see . So, initially you do the
guess value as we have done. You solve for
the momentum for v star, then do the rho star
calculation and then m dot f star where you
use Rhie Chow interpolation solve for p prime,
then you correct then again you solve for
temperature, then you update everything is
just like a simple algorithm. Only thing is
the density is now taken into consideration
and you take that into consideration if it
is converge move for the next time iteration.
If not, you go back and this place and repeat.
So, the algorithm wise it is exactly similar
that we have discussed in the incompressible
case. The difference comes here density c
p these are the term which are no more can
be neglected and top of that energy equation
becomes a part of your solver or the system
you cannot about the energy equation .
Now, having said that one can understand when
you talk about this algorithm. You need to
talk about the boundary conditions. We have
done some detailed discussion for the incompressible
case. Now look at the same in the context
of compressibility .
So, this is a typical boundary element and
here is the s b is the circus vector, this
is the c centroid and now in the boundary
cell this is my continuity equation rho c
star. So, this is my continuity rho c star
plus rho c prime minus rho c from the previous
time iteration mass flux correction and boundary
face. So, the boundary face velocity would
be corrector can Rhie Chow interpellation
using the pressure and this and the corrected
mass flow rate of the calculated mass flow
rate would be rho b with the previous iteration
and this is the corrections .
So, when actually now if you have a different
kind of boundary condition, this term of the
system of equation they get modified and they
look quite a bit of similarity accept the
term where your density these are coming in
for the mass flux calculation. Other than
that they are I mean they are looking quite
similar .
Now, the first thing is that inlet boundary
condition, now inlet boundary condition if
you have a subsonic flow , that means now
while you talking about the compressible case
or high speed case you can encounter two different
kind of boundary condition . One could be
subsonic; one could be supersonic. So, let
us first start with the inlet, but subsonic
flow if it is subsonic flow and then it could
be specified velocity if that is the case,
your p b is not known m dot b is not known,
but v b is specified . Now like incompressible
flow or compressible flow the density depends
on the pressure .
So, the mass flux remains unknown even with
the specified velocity . See that the big
difference when you talked about the incompressible
case as soon as you know the velocity you
could calculate this, but in this case without
knowing the density you cannot do that .
So, that means m dot b prime is equals to
rho b prime v b star dot s b which is not
0 . So, at the inlet boundary the coefficient
is multiplied with the p b prime and it could
be written as a b p b prime c p b by like
this . Now for the implementation of the pressure
correction p b prime is expressed in term
of some interior note and the coefficients
like this . So, a c p prime would be b c c
rho by delta t by c p f by rho f star m dot.
So, these are for coming from the interior
faces. This is some boundary face contribution
. So, this gets modified for the pressure
. Now, another could be you can specified
static pressure static pressure, that means
p b is known and also you this is known velocity
direction. What we do not know this and this
we do not know now in this case for static
pressure known the p b is known.
So, p b prime can be set to 0 and consequently
your rho b prime is also 0. Now the implementation
would be similar to incompressible case like
a dislet condition and the correction term
coefficients for pressure gets modified like
this . So, one would be the transient component,
then you get summation over all the faces
c p f rho f star m dot f plus rho f d f and
rho b d c. So, this is a boundary face contribution,
this is coming from all interior faces. Now
another kind of boundary condition that we
did not talk about in the context of incompressible
flow which is quite important here when you
define the specified p naught for the total
pressure and velocity direction. So, that
means p naught b is known, e v is known. So,
which means m dot b is not known, v b is not
known . For this case to calculate the magnitude
of the velocity, one has to use the stagnation
conditions where mach number at the face can
be defined at v b dot v b by gamma r t b and
the pressure which is unknown at the face
can be correlated with this standard isentropic
law which is stagnated condition is known.
So, the pressure can be correlated with gamma
. Now, here b refers to the boundary. Now
p b 1 can rearrange with gamma by m dot f
by rho b like this. So, which you can get
rid of mach number and velocity . So, it can
be terms of total pressure one can actually
rearrange this since e v is a unit vector
in the direction of the velocity.
So, once you differentiate this equation,
so differentiate this equation differentiate
this equation with respect to m dot star b
which will get you d p by this. So, that is
an algebra. One can carry out that and now
you substitute this guy into the equation
of the pressure corrections .
Then what you get p b prime equals to c b
some constant like this. So, that will get
you back the mass flux prime like this . So,
these are the condition which you get for
the pressure correction .
Now, the modified cell coefficient which is
obtained for the m dot b from this equation,
you can apply that and the pressure corrections
equation get modified like this where this
will be transient term rho c p by delta t,
then you have a term which is interior face
term and you get a term on boundary and your
stagnation temperature would be estimated
like this . So, in a compressible case your
pressure density temperature they are linked
and wherever you do an condition with the
pressure, then you can have you need to take
care the density and temperature are the same
time . Now that talks about different kind
of boundary conditions are subsonic case.
Now, you could also have inlet boundary condition
which is supersonic in condition. That means,
mach number is greater than 1 . So, in that
case you can have again a situation where
specified static pressure ; that means p b
is known, v b is also known and t b is also
known. So, it is specified p t and v. So,
all three are defined then which is implied
condition like m dot p prime p b prime, they
would be 0 and the coefficient of the pressure
corrections equations gets modified like that.
So, it will be written the transient term
and this is the term which is the transient
term and there will be an integration over
all the interior faces . So, there we two
contribution which will come from this .
So, one can note here your inlet conditions
in compressible cases could be of two different
types. One could be subsonic; one could be
supersonic . So, now when you have a supersonic
we do not have too much of variation. You
can specified them as a dislet condition . Now
similarly you should have outlet boundary
condition and outlet boundary condition could
be also subsonic flow where again you can
have specified pressure which means p b is
specified , m dot b is not known, v b is not
known.
So, once specified pressure your p b corrections
would be 0 and the mass flow corrections can
be estimated like this where rho b star d
c p c prime p c prime is computed and since
the v b star is not known, it is necessary
to assume that v b star is equals to b c star
and the expression for the pressure correction
coefficient a c could be modified like v c
c p by delta t and then you have all this
contribution from the interior note and also
one has to note that energy equation a 0 gradient
or dell t by dell n kind of 0 at the energy
flux zero or neumann kind of boundary condition
needs to be specified because this is a supersonic
face. You cannot leave the energy equation
and connected. So, it has to be connected
with the system .
Now, similarly one can have specified m dot.
So, that means m dot f b is known, but p b
not known, v b not known. So, specified mass
flux mean m dot prime would be 0. So, you
can simply drop the pressure correction equation
with no modification and then, if setting
m dot b prime equals to 0, the coefficient
of the pressure correction equation or p b
prime would be like this. So, you can actually
calculate the pressure and density corrections
at the boundary and the last one which I would
like to just touch with the outlet boundary
conditions and which could be of the supersonic
type . So, in that case here the none of the
variables should be specified. So, p v and
all these are not known . Pv density temperature
and they should be extrapolated from the interior
value. So, m dot b and p b they are interpolated
or extrapolated from interior interior cells
and this is equivalent to applying a neumann
kind of boundary conditions on pressure corrections
which will lead to this. So, which is equivalent
to neumann kind of boundary condition for
p prime equation which will lead to this pressure
correction coefficient to be modified like
that .
So, a c prime would be v c c rho delta t.
This is transient term and these are all summation
over interior faces and . If you notice here
again pressure density temperature p rho t
these are all connected . So, when you compare
with the subsonic case with the supersonic
case, the one important difference which appear
is that the not only the formulation.
Because formulation only takes care of the
determine takes the energy and density into
system also there this pressure density temperature,
they are connected that is number 1. Then
you have to involve the energy equation that
is another involvement or inclusion and also
in the discretization you get some sort of
a term like d p by d t d c p by d t. These
are the term which appears and then, viscous
dissipation term these are the term or extra
term which appear in this kind of situation.
Now, the other thing which is possible is
that the when you come to compressible case,
your solution algorithm has to have to different
kind of definition of the boundary inlet condition
and outlet condition they could be of solve
your boundary condition can be of inlet or
outlet, could be of subsonic type and supersonic
type . So, one has to take care of that. Now
that essentially concludes the portion of
our fluid flow problem where we have discussed
both incompressible and compressible .
Now, what I would like to patch up on some
of the advance things like how you generate
bit, but this should not be done in a detailed
discussion, but one can always look at the
text book and then look at some of the turbulent
modeling issues and how you discretize that
. So, we will stop here today and will take
from here in the follow up lectures.
Thank you.
