We are talking about three-dimensional flows in actual flow compressors. In the last class,
we had a look at various aspects of physics of the three-dimensional flow in actual flow
compressors, and its various ramifications of the understanding of how actual flow compressor
works, and how it impacts on the performance of actual flow compressor. In today's class,
we will try to capture much of the three-dimensional flow in mathematical form. These mathematical
forms are very useful in the sense that eventually, they would be used for various computational
analysis purposes, which aids the design process, and in the process of this coupling between
design and analysis, the design time is substantially cut down.
We do not have to go through a very costly experimental analysis to finalize the design.
So, the design time has indeed been cut down by years with the help of these analytical
tools with the help of CFD. And in today's class will try to capture some of the mathematical
form of the physics of the flow that we have done in the last class. So, let us take a
look at some of these issues that we are about to capture in mathematical form. So, today's
class is about three-dimensional flow analysis in actual flow compressors.
Now, as we have seen many of the blade theories are indeed actually based on two-dimensional
understanding, and those two-dimensional understanding actually ignore the effect of three-dimensionality
of flow while the fluid is passing through the blade passage. Now, the picture below
we have seen in the last lecture and it gives us a fairly detailed idea about the three-dimensional
nature of the flow. There are so many things going on inside the blades, specially inside
the rotating a row of blades that it is really a not a right thing to do is ignore some of
these three-dimensionalities; so, we will today try to capture some of these things
in mathematical form. So, that much of it is actually included in the design and analysis
process. So, we will keep this picture in mind which
we have done in the last class, and try to move forward to see what all things in some
summary form in a overall form is captured in mathematical formation.
Now, we see that one of the things that is normally ignored or assume to be not present
is radial component of the flow, and we had a quick look at a simple three-dimensional
radial equilibrium, what it actually means is that the radial motion of the fluid particles
inside the blade is sort of assumed to be or pretended to be not present at all. Now,
we see what are the reasons why radial flow would almost invariably be present in actual
flow compressors. Then, the first reason is a centrifugal action of the rotational motion
which is passed on to the fluid, and this centrifugal action would invariably try to
impart certain amount of radial flow to the fluid as it is passing through the blade specially
the rotating blade. Then the convergence of the annulus flow track,
we will you know discuss a lot of things about the flow track little later in this lecture
series. But, the flow track in the modern compressors is quite often converging type
and especially in the highly loaded model aircraft, engine compressors and this kind
of annulus flow track are geometry introduces the radiality anyway, then of course, we have
the twist and the taper and various three-dimensional blade shapes, the solid body of the blade
often has highly three-dimensional shape these days they are. In fact, becoming more and
more three-dimensional and as a result of which are certain radial component of the
fluid is invariably introduced into the fluid motion as it passes through the blades.
And then of course, we have the tip clearance effects. Now, the tip clearance effects we
have studied in the last lectures, and we know that there is a substantial cross flow
through the open tip of the rotor and this introduces a three-dimensional radial flow
and definitely introduce a certain amount of radial component to the fluid flow.
Then, if we look at the fact that you know in the first picture that you we looked at
even today that a passage vortex is formed inside the blades. Now, this formation of
passage vortex creates radial flow a strong component of radial flow in the main flow
itself, which is passing through the blade passage. So, as a result that is another contribution
from the various things that are happening to the introduction of radial flow. Then,
through the blades quite often, we assume in our theories that the temperature and then
the enthalpy and then entropy gradients are you know zero.
Now, that is an assumption in a real blade quite often, we will find that there is a
certain gradient in the rear stages of a multistage axial flow compressors, we would find a temperature
gradient from root to tip or even from blade to blade, there would be enthalpy gradient,
we assume that the enthalpy is a uniform along the length of the blade from hub to tip, and
we are also assume that the work done is constant from hub to tip in the earlier simple theories
and then of course, there is a general assumption in various theories that we have propounded
that the entropy gradient in the radial direction is absent; that means, there is no gradient.
Now, these are assumptions in reality some kind of gradient would invariably appear as
the flow goes through the various stages, and those will have to be taken into account
when you are analyzing fluid flow. Now, taking them into account means bringing in certain
amount of thermodynamics and those thermodynamic issues would invariably be present in the
real flow as mentioned specially in the latest stages of a actual flow compressor, where
the temperature is higher, the flow has become more three-dimensional and as a result temperature,
enthalpy and entropy gradient start appearing in the fluid flow passing through the blade
passage. Then, we have the blade solid body thickness
which is a blockage; now, this includes that camber and the stagger you know more the camber
higher the thickness, higher the stagger, this blockage to the main flow is likely to
increase, usually the flow is basically an actual flow and the moment you have blade
there, 8 solid blade it obviously has a blockage, and this blockage will change with these essentially
with these three parameters, the thickness of the blade the camber and the stagger, now,
you see we have already noticed that the thickness the camber and stagger, all three of them
actually vary from root to the tip of a blade. Now, if they are varying from root to the
tip of the blade,. it it stands to reason that the blockage effect would also vary from
root to the tip of the blade. And this variation would also then introduce on other aspect
of radial flow or three-dimensionality into the flow. And then of course, we have the
end wall boundary layers. And then, now these boundary layers are developed, because of
the flow essentially being in a adverse pressure gradient in a compressor and these boundary
layers over the casing and the hub internal surfaces actually create a certain amount
of blockage, these blockage is a fluid mechanic blockage and this blockage often has a tendency
to deflect the main flow in wards. From the casing and from the hub and this
inward deflection of course, introduces three-dimensionality, it introduces a radial flow in addition to
the fact that it has a tendency this blockage has a tendency to reduce the main flow rate
at in the actual flow rate that you would see would be less than what was assumed initially
probably during the design. So, these are some of the effects of actual realistic flow
on the formation of radial and three-dimensional flow features inside actual flow compressor
blading.
Now, we have created a radial equilibrium theory earlier a simpler version; now, this
is based on the premise or assumption that the radial gradient of the forces which are
experienced by the fluid, and we had taken a fluid element earlier of an appropriate
shape and size and that a fluid element experiences radial gradients, and those things contribute
to the radial movement of the flow, and if they are to be checked; that thing the radial
movement of the flow or not to be allowed, it it becomes incumbent that those forces
must be balanced by the static forces exerted by the pressure gradient which also created
inside the flow. So, that at any instant of time the fluid
system or the fluid particle or the fluid element is in radial balance of forces that
is in radial equilibrium. It is this balance of forces between the dynamic and the static
forces that we would move forward take, forward today to create a more comprehensive radial
equilibrium theory, much different from much more complex and much more involved than the
simple radial equilibrium theory that we had created in the last class. So, this balance
of forces is one of the issues that axial compressor, turbo machinery designers quite
often need to do during the process of design and analysis.
Now, let us try to lay down a few simple conditions mathematical and geometrical conditions that
are going to help us in formulating the mathematical theory of three-dimensional flow. We assume
that we have motion of a particle with respect to 2 coordinate systems, and this particle
P is moving in an arbitrary path within the 2 coordinate systems. Now, the two coordinate
systems are XYZ. And then with origin o and then x y and z with origin o prime, so, we
have two origins and two coordinate systems and the small xyz coordinate system is a moving
coordinate system. In fact, with respect axial flow compressor,
it would be a rotating coordinate system which means it is fixed on the rotor body. So, it
is a it is a body fix coordinate system which rotates with the rotor whereas, we have capital
XYZ which is a fixed coordinate system, and static coordinate system and the stated analysis
could be done with respect to that coordinate system. But the rotor needs to be analyze
with respect to the rotating or body fix coordinate system.
So, we do have 2 coordinate systems to deal with in this theoretical formulation, and
we need to keep that in mind. The moving coordinate system is at a distance R from the origin
of the fix coordinate system, and the particle P is at a distance rho dashed or rho prime
from the moving coordinate system origin and is at a distance small r from the fix coordinate
system origin. So, we have 3 distances over here. One is this coordinate system is at
a distance capital R from the fix coordinate system, and then the particle p itself here
a distance a rho dashed from the moving coordinate system origin and is at a distance small r
from the origin of the fix coordinate system. So, this is how we have put together the particle
that is moving in an arbitrary path.
Now if we move forward, the two reference systems have relative motion which is represented
by r bar r dash r bar, and that is it is the motion of the position vector r with respect
to fix origin o, omega is the rotation of the particle with respect to the moving access
system xyz and V xyz is the translational motion of the particle with respect to the
moving it access system xyz. Hence, the velocity of the particle p with respect to the fixed
access system capital XYZ based on this figure would be v, xyz equal to d r bar d t of xyz
capital XYZ. So, this is the basic definition or description of the fluid particle that
we are tracking in a motion involving 2 coordinate systems.
Now, velocity of the particle p with respect to small xyz that is the moving x y, z is
V small xyz and that would be d rho dashed d t as per the diagram that we had given.
Now, the vectorially; the motion of particle P is summation of the motion of the moving
coordinate system with respect to the fix coordinate system, and the particle P with
respect to the moving system. So, vectorially r bar would be equal to capital R bar which
is the distance of the moving coordinate system to the fix coordinate system, and rho bar
which is the distance of the particle with respect to the moving coordinate system.
So, from which we can say that d r bar d t, xyz that is capital XYZ would be equal to
d r d capital R bar d t plus d rho dashed bar d t and these are of course, the velocities
of the respective coordinate system. So, we can write that down that the velocity of p
with respect to the fix system would then be capital V, X capital XYZ that would be
equal to r dot plus v, xyz that is small xyz plus omega that is the angular velocity into
rho dashed. So, that gives us the victorial summation of the velocity field that we have
in front of us for the tracking of the fluid particle through the two coordinate systems.
Now, if we define the accelerations, the acceleration of P with respect to fix coordinate system
that is capital XYZ is given as a capital XY Z equal to d V capital XYZ d t with respect
to xyz and the acceleration of p with respect to the rotating or moving coordinate system
that is small xyz and that is a small xyz equal to d V small xyz d t with respect to
the small xyz coordinate system. These are the two accelerations which we will be contending
with, because we have seen before the acceleration is is what contributes to the creation of
dynamic forces that is mass into acceleration. So, we need to write down what the acceleration
terms are going to be for this three-dimensional fluid flow.
So, let us move forward, then we can write down that the total acceleration of P with
respect to the fix coordinate system capital XYZ would be a capital XYZ and that is d V
xyz d t with respect to the capital XYZ and that would then be equal to d V small xyz
d t plus r double dot, this is the acceleration term the earlier one r dot was the velocity
term of the position vector r and then d of omega rho dashed d t; now, this is of course,
the due to the rotational component of the coordinate system is small xyz itself with
respect to capital XYZ. So, this is acceleration term that we can write down with respect to
the coordinate system that we have defined earlier.
Now, the acceleration with respect to the body fitted rotating coordinate system small
xyz. Now, this is a summation of its translation and rotating motion at the moment, we are
assuming that it has translational motion and rotating motions; and this we can write
down in terms of d V small xyz d t and that would be equal to our d V small xyz d t plus
omega into V small xyz. Now, you see the rotating component is the the second term over here.
Now, if we write that down over here, we see here d omega rho dashed d t xyz would be can
be split up in two terms that is omega into d rho dashed d t and d rho d t into omega
dashed. Now in the first term d rho dashed d t which
is a differential of the position vector rho dashed can be written down in terms of d rho
of dashed d t is equal to d rho dashed d t into small that is of the small xyz plus omega
into rho dashed. So, rotation in xyz can now be captured in the form of these equations
with respect to the position vectors that we have defined before.
Now, acceleration of the particle P with respect to the fixed origin o may be written down,
now we have seen that we need to write down the acceleration terms and these lines be
edited in terms of a xyz that will be equal to a small xyz plus r double dot plus twice
of omega into V xyz plus omega into omega into a rho dashed dot plus rho dashed rho
dot plus omega dot omega dot into rho dash dot. Now, if we assume, now this is the full
the first gives the full equation that we get of the acceleration term with respect
to the fix coordinate system. Now, if we assume that the angular velocity
omega is constant, and it does not change then the rho omega dot term would of course,
immediately go and as a result of which we get a little simpler a version of that acceleration
term and that is a capital XYZ and that would be equal to a small xyz plus twice omega V
xyz plus omega into omega rho dashed. So, this is the terminology that we finally, get
for acceleration assuming that the flow is a constant angular velocity which is very
fair, because most of the time the axial flow compressors are rotating with constant angular
velocity except when they are in transition during acceleration or deceleration of the
machine.
Now, if we look at for a compressor blade passage, the flow velocities that we are looking
at are V small, xyz and now this is the relative velocity you remember, we had two velocities
that we had defined one was relative velocity, and another is the absolute velocity. Now,
we can relate those two known velocities with respect to what we are doing right now today.
So, V small xyz is the relative velocity and V capital XYZ is the absolute velocity C.
We have seen earlier that sometimes the relative velocity is the larger component, and that
is typically at the inlet to a rotor sometimes the absolute velocities the larger component
and that could be true at the exit of the rotor. So, V capital XYZ that is absolute
velocity is of course, a summation of the other two vectors that is V small xyz plus
omega rho dashed. Now, this can be written in terms of V plus
omega r that we had written earlier r being the radius at which the fluid particle is
rotating with respect to the rotating coordinate system, and this earlier we had written down
as u. So, V plus u then becomes the victorial summation which gives us C and this is what
we had done earlier very quickly in our two-dimensional flow theories. So, we come back to what we
had done in two-dimensional flow theories, in the sense we were nothing particularly
wrong with them except that they were two-dimensional in their nature.
Now, if we differentiate this situation what we get is acceleration terms, and now we see
that a capital XYZ would be equal to d V capital XYZ d t and then d omega rho dashed d t with
respect to capital XYZ. Now, the first term then is the translational motion and the second
term is due to the rotational motion. So, the both the terms would need to be taken
into account for finding the total acceleration. And then the final acceleration then is a
capital XYZ is equal to a small xyz and then twice of omega V xyz plus omega into omega
rho dashed, now this is what we have done earlier if you remember in the last slide.
So, we get back. So, this equation with constant angular velocity is what we again get back
here using the concepts that we had used earlier in our two-dimensional flow theories.
So, we get the same acceleration term, a combination of acceleration with respect to rotating motion,
and combination of the two and the translational motion. Now, the third term is often sometimes
referred to has carioles force on carioles acceleration, and we have done that before
also, and this is what typically comes out of a full mathematical formulation that we
are attempting in this class today.
Now, if we consider that equilibrium of forces along arbitrary flow direction s. we get between
any two actual stations separated by a small distance delta S where A i is constant, A
i being that area at that particular station. So, if we take two stations of an axial flow
compressor longitudinally. The during which we assume that A i is constant. A i being
the area annulus area at that particular station. And we assume that is constant and in this
longitudinal flow field which is a small distance delta S. And in this we try to track what
is happening in the flow field in this small distance delta S that we have defined here.
Now, over this distance what we see here is the pressure force that is working, this is
of course, the static pressure force delta p in to A i which is constant now, over this
flow field, and then we have A i into rho, rho now is density please understand we have
always used rho as density and that is why we used rho dashed earlier has a position
vector. So, this rho is density and delta S which is the distance. So, A into this is
the volume and rho is the density. So, that is the mass and this is acceleration. So,
mass into acceleration is the force that is the dynamic force that has been created and
this is the axial force that has been created by delta p and that is the pressure differential
or pressure will change over this distance small distance delta S.
If we resolve this, from this what we get is one by rho delta p by delta S equal to
acceleration a xyz. So, this is now the pressure gradient along the distance delta s. Now,
we can write down that the acceleration equation from the last slide, now can be written down.
In terms of this acceleration term can now be replaced by one by rho delta p, and then
we get the D V D t plus omega square r position vector rho dashed is now replaced by r which
we are familiar with r being the radial distance of a particle from the axis of rotation of
the moving body that is the axial flow compressor access, and twice omega V which we had called
sometimes in some books is referred to as cordials force.
So, this is what the equation we get now. So, acceleration term has now been replaced
by pressure terminology or pressure term, pressure gradient along the distance s or
[de\delta] delta s.
So, with this we can now write down that as the flow in compressor blade is a diffusing
flow, we need to think that D V D t is likely to be negative; that means, the flow to the
blade passage is actually decelerating flow. So, the D V D t is likely to be a negative
in nature, and as a result of which the equation can be slightly recast as minus 1 by rho into
delta p equal to a D V D t minus omega square r plus two twice two cross omega cross v.
So, that is here in Victoria representation of the corialise force.
So, we get compressor blade; now, positioned signifying that D V D t is most likely to
be negative in nature.
Now, this gives us a situation that we now need to look at representing the flow in a
manner that gives us a generalized flow path of the particle, while the position of the
particle is being tracked, it as its rotational motion captured in the form of omega, and
then we have the three coordinate system which we are familiar with the radial coordinate
system is r the axial coordinate system is a and a tangential or the peripheral coordinate
system as w which we call the whirl component. Now, this is what we have been doing earlier
in our two-dimensional flow theories where we had the radial component, we had the actual
component and we had the whirl component, and we call the three velocities for example,
V a, V r, and V w. Now, V r had been neglected earlier in all two-dimensional flow theories,
but we are going to bring that back here, and V w remains the whirl component or the
peripheral component, and we would now go back to those terminologies which we had used
earlier, and are somewhat familiar to us from the earlier theorizing that we have done.
So, as we can see there is nothing particularly different about the notation that we had used
earlier, and the notation that we have introduced in today's lecture; you can find very easily
the parallels between the two, and the equality between the two sets of notations. So, you
may have to really just sit down and look at these notations and find that these things
actually mean one and the same thing. So, with this we look at the new axis of notations
system.
Now, the assumptions made here are that the fluid is frictionless, the rotor is rigid
and rotates with constant angular velocity an assumption we made just a little while
earlier; the flow is steady relative to the rotor, we assume that the flow is not experiencing
any unsteadiness. We will formalize it in our mathematical formulation and the radial
variation of density is neglected. Now, this is something which is an assumption in many
flows that may not actually be true, but radial variation of density to begin with for design
purposes, and for design analysis look that assumption is reasonable and a valid assumption
and maybe proceeded with. Now, this leaves is still enough scope for
formation of viscosity, then entropy radiands and stagnation enthalpy gradients in the flow
field from hub to the tip of a typical compressor blade. So, these gradients and the formation
of the passage vortex, and the tip vortex, and the trailing edge vortex all these things
would indeed bring in the three-dimensionality which actually impacts the real flow, and
we are trying to see how much of these things can be captured a priory in a mathematical
formulation, so that it can aid the design and analysis process of axial flow compressor.
So, these are those assumption that we have to live within our mathematical formulation.
The other things we are going to bring in are the definition of the unit vectors which
we have written down here are in terms of D i r i being the unit vector in the radial
direction. And then D i w which is one in the whirl component or the whirl direction,
so the overall V is a V r into unit vector i r V w into unit vector i w and V a into
unit vector, i a those in part the directionality with which is the flow is proceeding in or
the flow has three components in radial whirl component and the actual component.
The other thing we are in to do is look at the fact that a capital DDT is a total operator.
Now, from your basic mathematics you know that the total operator can be split up in
two terms, one is the operator with respect to the space coordinate where s is the length
in any direction; so, the same term can be written down in terms of total operator DDS
and that plus d s d t, now the d s d t of course, is the unsteady terminology, t being
the time; now, as a result of which d s d t is an unsteady terminology if the flows
are shown to be steady and at the moment we are assuming the flow to be steady, then the
d s d t is equal to zero. Hence, that term would be neglected and as a result will be
left with only the total operator with respect to the space of the distance s, and the with
respect to time it is considered to be zero.
So, the equation for the flow inside a compressor, we would now like to recast using r theta
z coordinate system, because simply because it is a more appropriate for a rotating coordinate
system also, because of the fact that. There is a variation of parameter in the circumferential
direction and from blade to blade, and this blade to blade circumferential variation is
quite often an important variation. We know that the two surfaces of the blades are indeed
actually dissimilar, one is a suction surface and on[e] one is a pressure surface.
So, theta variation from suction to pressure surface is quite often an important variation
it sometimes it is caught a lot in highly loaded, and we shall we be know that that
variation also changes from hub to the tip of the blade. So, variation from one blade
surface to the other in the passage is an important variation, it everything varies
the pressure varies, the velocity varies, the temperature varies, the enthalpy varies
from one surface to the other in one passage. So, that can be tracked by variation of theta
that is the angular circumferential variation from one surface to the other, the other is
of course, from root to tip, and the third is in the z direction that is axial coordinate,
axial distance. So, if we use r theta z coordinate system,
we see that DV Dt can be written down, now in terms of unit vector i r into D V r D t
V w d theta d t plus unit vector i w DV w Dt plus V r d theta d t plus unit vector i
a DV a Dt. Now, these coordinate systems and the flow velocities in the relative frame
[weto\we] we always have considered as V v its components are being shown again as V
a, V r and V w as we have done before in a 2-D flow theories, where we have used use
V a and V w we did not use V r as at that time, we assume that V r does not exist. So,
we see here that the acceleration term can be split up in three components.
Now the equation may be resolved in its three components using the r, theta z coordinate
system that we have brought in now; now the what it shows is that the three components
are 1 by rho del p del r and that would be equal to V d V r d s minus V w plus omega
r square by r that is equation S. Now, that is in the radial direction; so, the left hand
term is the pressure gradient, and the right hand term is the dynamic acceleration terminology
of the terms; the second equation is in the circumferential direction theta coordinate
and that is 1 by rho del p del theta, 1 by rho 1 by r del p del theta. So, that is the
pressure gradient in the theta direction and that would be equal to V into DV a DS plus
V w into V r by r plus twice omega V r and that is the equation two that is in the circumferential
direction. In the actual direction it is 1 by rho del
p del z equal to V DV a DS and that is in the actual direction. So, again the left hand
term is the pressure gradient and the right hand term is the acceleration term, which
gives rise to the dynamic or the kinematic forces. So, these are the three components
of the equation that we had written down in the earlier slide, and three components are
in the radial circumferential and in actual directions.
We can now denote these components in a very simple vector diagram, in which all the velocity
components are shown here, we see the velocity velocity vector V r has now appeared, we are
tracking fluid flow particle and this now has three components V r, V a which is actual
velocity. We are looking at a particular plane on which we have the fluid particle, and this
fluid particle is experiencing a rotating motion, omega it is at a distance r from the
centre of rotation and it is at a [ang\angular] at a angular distance theta from the reference
axis. So, at this position it has three velocity components V a that is axial velocity, V r
that is the radial velocity, and V w that is the peripheral velocity.
A combination of all the three is of course, the overall relative velocity with respect
to this coordinate system which write in the beginning in today's lecture, we have seen
itself may be moving with respect to a fixed coordinate system. But let us look at this
moving coordinate system analysis right now. Now, we see that a combination of V r which
is the radial component and V a the axial component also gives a velocity which can
be now described as V m, and these V m is velocity which is a resultant of V r and V
a and is on a particular plane of this plane is a radial, and axial plane a combination
of radial and axial plane. In this plane, which is perpendicular to the plane we are
looking at which is the radial and peripheral plane and, V m is on that radial axial plane
and this V m is referred to as meredional velocity.
We will come to this definition of meredional velocity. So, the velocity triangle for this
flow gives us that the C w would be equal to V w plus omega r, what we see now is that
the velocity triangles which we are all familiar with actually gave as C w that is equal to
the axial velocity equal to V w plus omega r omega being the angular velocity, and in
the earlier coordinate system that we have just seen we get a V w which is the peripheral
component a relative peripheral component with respect to the rotating coordinate system.
So, we come back to our C w which is vectorially has to be equal to V w plus omega, r in this
coordinate system, you remember this whole thing is rotating or the particle is rotating
with respect to this with angular velocity omega. So, this entire coordinate system is
a rotating with angular velocity omega. Now, the equation a and b from the two slides earlier,
we had three components from the slide 21, let us go back very quickly what we see is
we have two components of the full equation in two directions the radial and the peripheral.
If we bring forth, then we have the radial component can be recast as one by minus one
by rho d p d r equal to V into d V r D s minus C w square r by r and this rewrite as a equation
d, and the second equation is now one by rho r into the del p del theta equal to V by r
into D r C w D s. So, what we have done is we have replaced the V w is now with C w,
because that is the axial velocity component that the flow is actually having a with respect
to the fixed coordinate system. So, the equations have been slightly recast.
If we now look at the flow, we can now use the kinematic relations as V D capital operator
D D S equal to V a capital operator D V a of any parameter where V a and a are the axial
components of V and S. S being the distance and a is the axial distance and V a is the
axial velocity and V o is the overall velocity. We can now define a meredional velocity to
begin with let us define a meredional direction, and this meredional direction is D m into
unit vector i m equal to D r into unit vector i r plus D a into unit vector i a and this
is what I was talking about just a little while earlier that this meredional flow is
in a plane that is composed of radial and axial plane.
So, in this plane we can have a meredional direction defined, and this meredional direction
of a flow or a particular flow path a flow particle taking a particular path would have
would create an angle phi with respect to the axial direction. So, this phi is the angle
with which the meredional flow is proceeding in the meredional direction, which is not
the same as axial direction. Now, the equation d from the last slide can be rewritten as
minus 1 by rho del p del r equal to V m we have brought in the meredional velocity V
m and that is d V r D m and d m is the meredional direction minus C w square by r.
Now, C w of course, we are familiar with is the whirl component of the absolute velocity
are being the distance of the fluid particle with respect to the axial coordinate system.
As a result of which we get a for motion again on the left hand side, we have the pressure
gradient on the right hand side, we have the acceleration terms due to the dynamics of
the flow.
This gives us the fact that in the meredional direction, if we complete the definition of
meredional direction 10 phi is V r by V z or V a and V is V of meredional direction,
V m into sign phi and phi is defined here and as a result of which we can write down
that the force balance equation only in the radial direction can be recast or rewritten
as one by rho del p del del p del rho is equal to C w square by r minus V m square d sin
phi d m equal to minus V m sin phi d V m D m. Now, this is the full equation in the radial
direction. Now, by our definition V D D s the total operator is equal to V m D D m,
this is mathematically approved and as a result of which we can replace with respect to the
meredional velocity and using the meredional direction in this equation.
So, if we look at the flow as we [desc\describing] what we are describing is that we have other
meredional plane over here now; and this is the direction of the fluid particle of fluid
path given as phi, V r is the radial flow component, V a is the axial flow component
and V m here is the meredional flow component in a fluid path. And this is what we are talking
about an V w of course, remains the peripheral component or the whirl component that we have
talked about which means this total V would be the full complement of all the three components
put together omega of course, is the rotational angular velocity that the entire flow is experiencing
with the rotating blade.
So, given this model, given this model we can now write down that D sin phi D m would
then be equal to cos phi D phi D m, and D phi D m will be equal to minus one by r m.
Now, minus one by r m is indeed the radius of curvature of the meridional flow, and this
radius of curvature of the meridional flow is due to the fact that the meridional path
may not be linear, it may not be a curved path and if it is a curved path the radius
of curvature of that path. In the meridional direction would create a certain centrifugal
action, and hence the meridional radius of curvature of the meridional flow is being
brought into the picture. Now, this negative sign is a somewhat arbitrary,
but the axial flow compressor the flow track inside generally moves towards a lesser value
of phi as it moves actually forward that is the r m becomes higher and higher which means
the flow tends to become a later on flatter and flatter. So, it moves from a high r m
to low r m as it moves forward through the blade passage. Hence, the full radial equilibrium
that we are looking at now is 1 by rho del p del r equal to C w square by r plus cos
phi into V m square by r m minus V r D V m D m. And this is the full radial equilibrium
equation which in the say is a circumferential average from blade to blade flow properties
inside a termination blade row. Now, this radial equilibrium equation captures
the all the components of the dynamics of the flow, in terms of all the acceleration
terms that come about due to three-dimensionality of the flow. Now, this we can see is has a
more terms than what we had seen in the simple radial equilibrium equation, we had done couple
of classes earlier which tells us that more of the dynamics or kinematics of the flow
have been captured in our mathematical formulation.
And this tells us that if we now simplify backwards; that means, if we consider that
the flow is actually more or less actual then the last term is actually eliminated in the
very early designs, it was a done thing that the flow path was considered linear, and the
second term is also then neglected or it vanishes and which gives us back the simple radial
equilibrium equation which we had formulated earlier in terms of one by rho del p del r
equal to c w square by r. Now, this was the simple radial equilibrium
equation; what we need to look at now is a full radial equilibrium equation and we see
now that a the simple radial equilibrium equation was somewhat inadequate, and it becomes necessary
to utilize the full equilibrium in design and analysis of modern axial flow compressors.
Now, if we go back and summarize some of the things that we have done, we can say that
wherever the flow is not experiencing the centrifugal force the radial equilibrium cannot
be applied, and with with this understanding it was at one time assume that radial equilibrium
is not applicable between the rotor and the stator where no blades exist. However, experience
experiments have shown that in between the blade rows in the axial gap between the rotor
and the stator there could be a radial shift of the meridional path, and hence for accurate
design analysis, radial equilibrium full radial equilibrium equation should be used in that
flow path also between the rotor and the stator. For the computational purpose is further steps
need to be taken from this mathematical form, the radial equilibrium equation is transformed
into a form that contains partial derivatives of all parameters with respect to radius and
a peripheral coordinate theta, because all the parameters like pressure, and density
and enthalpy are likely to vary from root to the tip of the blade and also likely to
vary from blade to blade. So, this blade to blade variation and root to tip variation
need to be captured, I will in the next slide how the physical model or mathematical model
attempts to capture it in its computational form.
The next circumferential average of those parameters is then integrated over the theta
from the pressure side of one blade to the suction side of the other blade. And then
the flow is analyzed and various axial stations with energy equation, continuity equation
and the radial equilibrium equation. So, you need all three of them to analyze the flow.
Now, this is what I was talking about that, you have a model in which you have two blades
in between the two blades, you need to create these surfaces, these are known as S 1 surfaces
which if you very S 1 from root to the tip of the blade, you have the variation of various
parameters from root to the tip of the blade, base two surfaces from blade to blade that
is from pressure surface to suction surface of the other blade. So, if you take the variation
of parameters on is two surface, and then vary them circumferentially in the theta direction
from one surface to the other, you have the variation of all the parameters in the circumferential
direction. After you integrate on the S two surface and
on the S one surface you get integrated parameters of all the parameters. So, this model has
been used by the computational people to transform radial equilibrium equation into a computational
tool for analyzing the flow through action flow compressors.
It is necessary that the flow properties obtained in this manner and various actual stations
be consistent with one another as the flow properties are evaluated from up to the tip
at each station, that mean that the radial acceleration of the fluid particle is to be
accounted for through radial equilibrium equation, which is the additional equation, you need
to use in addition to the energy equation and the continuity equation or continuity
condition for analyzing flow through action flow compressors. Now, this can be achieved
by assuming shapes for the meridional streamlines. So, you need to assume the shapes of the meridional
streamlines, consistent with the continuity condition, which expresses the radial acceleration
in terms of the streamlined slow and the streamlined curvature. So, you got to find the meridional
path of the meridional streamlined along which you can apply the radial equilibrium equation
that meridional path can be found by using the continuity condition along the meridional
path. This of course, implies that unique may need
to hydrate the method of solution, because paraphrasing to begin with you may not know
the meridional path, you may not know that streamlines that create the meridional path.
This method in which surfaces are used to build up the flow inside to missionary blade
has been widely used and this wide you widely used formulation is being used for last twenty
twenty five years for analyzing flow through action flow compressor from blade to blade
surface and those on the meridional plane, and they need to be solve separately to account
for all aspects of the three-dimensionality of the flow through axial flow compressors.
The 3-D flow contributions has provided huge assistance to the Indian designers of the
compressor design is more specifically, it is cut down the design time and it has reduced
dependence on costly experimental analysis, experimental analysis making the blade testing
it in the real is a very costly business and time consuming business; competition is cut
it down substantially. The 3-D methods have helped understand various flow phenomena such
as the secondary flow development, choking in the stages, as an when the occur aspects
end of wall flows, not to speak of problems like stalling, and those can be then sorted
out in the design itself, before the blade is actually fabricated.
However the designers use these solutions quite often in conjunction, even today in
conjunction with empirical relations specially the losses that occur are very difficult to
capture in CFD, and hence quite often empirical relations do again come back into picture,
and experimental data to account for some of the losses, and thereby predict the performance
of the design action flow compressor blade, which means that there is still a scope for
improvement of these methods. So that the empiricism, the empirical formulations are
reduced further to the minimum, some of it may have to be done in the last stage of the
design before the blades are actually fabricated and tested; however the dependence on the
empirical formula can be reduced, if this is the method is taking forward. So that more
and more of the three-dimensionality that we have seen right in the first slide is captured
in the formulation. So, today's class we have done lots of attempt
to capture a lots of three-dimensionality in the mathematical form and finally, we have
got a full radial equilibrium equation, that is substantially different and for more complex
in Bob Anderson radial equilibrium equation that we had done earlier. This allows us to
go forward and do an analysis of a modern action flow compressor and eight in the process
of the design of this modern axial flow compressor. So, that brings us to the end of today's lecture
of creating a mathematical formulation of flow through a three-dimensional flow through
axial flow compressor. In the next class, we will look at and tent to solve some of
the problems of simple 3-D flow theory is that we have done earlier, and we will try
to use those theories to solve some of the problems, use the free vortex theory and other
vortex theory to solve some of the typical problems, and show how the numbers actually
come up in solution of axial flow compressors.