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Cascade Gas Dynamics
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
Modeling of Flow in Turbomachines….
Cascades
The flow in cascades is fully
three dimensional.
Cascade Model is a key element
in the classical tradition of
turbomachinery design.
The essence of classical
approach is the splitting of threedimensional flow into two
separate sub-problems.
The Meridional Plane
The first sub-problem is that of the flow in the meridional plane.
The flow in plane ( in r, z coordinates ) is solved.
The principle assumption is that flow is axisymmetric.
The equation of motion in radial direction is:
1 dp V2

 dr r
For a machine having radially constant stagnation enthalpy and
entropy:
dVz V d rV 
Vz

0
dr
r
dr
Above equation holds for the flow between blade rows in an adiabatic,
reversible turbomachine in which equal work is delivered by the rotor
at all radii.
This is called as radial equilibrium theory.
Blade – to – Blade or Cascade Plane
The cascade plane is in z, coordinates.
Power Consuming and Power Generating Casades
Compressor Cascades
Turbine Cascade
Aerofoil and Flow Geometry
Inlet Flow angle : a1
Inlet blade angle : a1’
Incidence angle,i :a1 - a1’
The aerofoil chord makes a certain
angle with respect to the axial
direction, c.
This is called as blade stagger
angle.
Camber angle 
Discharge flow angle a2
Discharge Blade angle a2’
Deviation angle,d :a2 – a2’
The fluid deflection angle e = a1 – a2
Equations of Motion
Steady two dimensional flow:
Conservation of Mass:
 Vz  1 V 

0
z
r 
Transform above equation into x-y coordinate system, using
x  z; y  r
u & v : velocities
u   v 

0
x
y
Conservation of momentum: Steady inviscid flow:
X-momentum
u
u
p
u  v  x
y
x
Y-momentum
v
v
p
u  v  x
y
y
Energy equation: Isentropic flow
 u 2  v2 
 u 2  v2 
 
 
u C pT  
   v C pT  
   w
x 
y 
 2 
 2 
Irrotational Flow
Vorticity w =
v u
0
x y
A potential function is defined as:
Conservation of mass:


u
&v 
x
y
 2  2
 2 0
2
x
y
Conservation of Momentum:
2 2
2 2
2
2

 
1          
   2 




2



2
2
2 
2
2

x
y
C  x  x  y  y
x y xy 


Where,
2
2


 - 1      
2
2
a  ao - 

    
x
y
 2 
    

Series Solutions
  0  1M   2 M  3 M ......................   n M
2
4
6
2n
Where, 0 is incompressible flow solution.
0  0
2
2
2
2

1  0   u0  v0   0   u0  v0 
2

 


 1  
2
2
2  x x  u
 y y  u

2
2
2
2




1 1  u1  v1
1  u1  v1 
2

 


 2  
2
2
2  x x  u
 y y  u

2
2
2
2







u

v
u

v
1




2
n -1
n -1
n -1
n -1
n -1
1

 


 n  
2
2
2  x x 
u
u
 y y 

Potential Flow Theory : Incompressible Flow
P M V Subbarao
Professor
Mechanical Engineering Department
IIT Delhi
A mathematical Tool to invent flow Machines.. ..
THE VELOCITY POTENTIAL
• It is possible to demonstrate that the condition of irrotationality implies the
existence of a velocity potential such that

V  
• On substituting the definition of potential into the continuity equation we
obtain

2
  V  .   
•The velocity potential must then satisfy the Laplace equation and it
consequently is a harmonic function of space.
•Solution of the Laplace equation, with an appropriate set of boundary
conditions, leads then to the determination of the flow field.
•Laplace equation has been widely studied in many fields, and shows
some interesting properties. Among the latter, one of the most important
is its linearity.
•Given two solutions of the Laplace equation, any linear combination of
them (and in particular their sum and difference) is again a valid
THE STREAM FUNCTION
• In the present analysis of an irrotational plane flow, the
velocity field can be obtained in terms of a stream function
instead of a potential function.
• We can in fact define a (scalar) stream function
U

y
&
V -

x
that satisfies identically the continuity equation for the Schwarz theorem
on mixed derivatives.
Such a function is called the stream function because its isolines are
streamlines.
If we now make use of the irrotationality of the flow we obtain:
U V    
w
 2  2 0
y x x
y
2
2
• So the stream function satisfies the Laplace equation, hence
being a harmonic function of space.
• Stream function and velocity potential are both harmonic
functions of space and are related by the following equations
U



y
x
&
V -



x
y
•Two bi-dimensional harmonic functions that satisfy the above
conditions are said to be conjugate.
•Lines along which the stream function is constant (streamlines)
and l
•Lines along which the velocity potential is constant (isopotential
lines) always intersect at right angles.
THE COMPLEX POTENTIAL
• Investigate the properties of a complex function the real and
imaginary part of which are conjugate functions.
• In particular we define the complex potential
W    i
In the complex (Argand-Gauss) plane every point is associated
with a complex number
z  x  iy  rei
In general we can then write
W   x, y   i x, y   f z 
Now, if the function f is analytic, this implies that it
is also differentiable, meaning that the limit
so that the derivative of the complex potential W in the complex z plane
gives the complex conjugate of the velocity.
Thus, knowledge of the complex potential as a complex function of z
leads to the velocity field through a simple derivative.
ELEMENTARY IRROTATIONAL PLANE
FLOWS
•
•
•
•
•
•
•
The uniform flow
The source and the sink
The vortex
The dipole
The doublet
The flow around a cylinder
The flow around a cylinder with nonzero circulation
THE UNIFORM FLOW
The first and simplest example is that of a uniform flow with velocity U
directed along the x axis.
In this case the complex potential is
and the streamlines are all parallel to the velocity direction
(which is the x axis).
THE SOURCE OR SINK
• source (or sink), the complex potential of which is
• This is a pure radial flow, in which all the streamlines converge
at the origin, where there is a singularity due to the fact that
continuity can not be satisfied.
• At the origin there is an input (source, m > 0) or output (sink, m
< 0) of fluid.
• Traversing any closed line that does not include the origin, the
mass flux (and then the discharge) is always zero.
• On the contrary, following any closed line that includes the
origin the discharge is always nonzero and equal to m.
The thick magenta line on the left is
related to the fact that the complex
potential is, in this case, a multi-valued
function of space.
At any fixed point, the potential is
known up to a constant, the so-called
cyclic constant, that in this case has the
value of i2p.
The potential is defined up to a
constant, the fact that it is a multivalued function of space does not
create any problem in the
determination of the flow field, which
is uniquely determined upon deriving
the complex potential W with respect to
z.
THE VORTEX
• In the case of a vortex, the flow field is purely tangential.
The picture is similar to that of a source
but streamlines and equipotential lines
are reversed.
The complex potential is
There is again a singularity at the origin, this time associated to
the fact that the circulation along any closed curve including the
origin is nonzero and equal to .
If the closed curve does not include the origin, the circulation will
be zero.
THE DIPOLE
• Also called as hydrodynamic dipole.
• It is created using the superposition of
a source and a sink of equal intensity
placed symmetrically with respect to
the origin.
• The complex potential of a dipole, if
the source and the sink are positioned
in (-a,0) and (a,0) respectively is :
Streamlines are circles, the center of which lie on the y-axis and
they converge obviously at the source and at the sink.
Equipotential lines are circles, the center of which lie on the x-axis.
THE DOUBLET
• A particular case of dipole is the socalled doublet, in which the quantity a
tends to zero so that the source and
sink both move towards the origin.
• The complex potential of a doublet
is obtained making the limit of the dipole potential for vanishing a
with the constraint that the intensity of the source and the sink
must correspondingly tend to infinity as a approaches zero, the
quantity
FLOW AROUND A CYLINDER
• The superposition of a doublet and a
uniform flow gives the complex
potential
Note that one of the streamlines is closed and surrounds the origin at a
constant distance equal to
Recalling the fact that, by definition, a streamline cannot be
crossed by the fluid, this complex potential represents the
irrotational flow around a cylinder of radius R approached by
a uniform flow with velocity U.
Moving away from the body, the effect of the doublet
decreases so that far from the cylinder we find, as expected,
the undisturbed uniform flow.
In the two intersections of the x-axis with the cylinder, the velocity
will be found to be zero.
These two points are thus called stagnation points.
Velocity components from w
To obtain the velocity field, calculate dw/dz.
Cartesian and polar coordinate system
Sometimes, it is more convenient to work in polar
coordnates. Let z = rei.
Grouping real and imaginary parts will give
Hence, the velocity potential and the stream function are given b
To obtain the velocity field,
Equating real and imaginary parts will give
On the surface of the cylinder, r = a, so
V2 Distribution of flow over a circular cylinder
The velocity of the fluid is zero at = 0o and = 180o. Maximum
velocity occur on the sides of the cylinder at = 90o and = o
Pressure distribution on the surface of the cylinder can
be found by using Benoulli’s equation.
Thus, if the flow is steady, and the pressure at a great
distance is pinf,
Cp distribution of flow over a circular cylinder
Terminology and Definitions
• An airfoil is defined by first drawing a
“mean” camber line.
• The straight line that joins the leading and
trailing ends of the mean camber line is
called the chord line.
• The length of the chord line is called
chord, and given the symbol ‘c’.
• To the mean camber line, a thickness
distribution is added in a direction normal
to the camber line to produce the final
airfoil shape.
• Equal amounts of thickness are added
above the camber line, and below the
camber line.
• An airfoil with no camber (i.e. a flat
straight line for camber) is a symmetric
airfoil.
• The angle that a freestream makes with
the chord line is called the angle of attack.
Conformal Transformations
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
A Creative Scientific Thinking .. ..
INTRODUCTION
• A large amount of airfoil theory has been developed by
distorting flow around a cylinder to flow around an airfoil.
• The essential feature of the distortion is that the potential
flow being distorted ends up also as potential flow.
• The most common Conformal transformation is the
Jowkowski transformation which is given by
To see how this transformation changes flow pattern in
the z (or x - y) plane,substitute z = x + iy into the
expression above to get
This means that
For a circle of radius r in Z plane x and y are related as:
Consider a cylinder in z plane
In z – plane
C=0.8
C=0.9
C=1.0
Flow Over An Airfoil
Vortex Panel Method