Cascade Gas Dynamics

P M V Subbarao Professor

Mechanical Engineering DepartmentI 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 three-dimensional 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:

r

V

dr

dp 21

For a machine having radially constant stagnation enthalpy and entropy:

0

dr

rVd

r

V

dr

dVV z

z

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 :

Inlet blade angle : ’

Incidence angle,i :’

The aerofoil chord makes a certain angle with respect to the axial direction, his is called as blade stagger angle. Camber angle

Discharge flow angle 2

Discharge Blade angle ’

Deviation angle, :–’

The fluid deflection angle = 1 – 2

Equations of Motion

Steady two dimensional flow:

Conservation of Mass: 0

1

V

rz

Vz

Transform above equation into x-y coordinate system, using

velocitiesvu

ryzx

:&

;

0

y

v

x

u

Conservation of momentum: Steady inviscid flow:

x

p

y

uv

x

uu

X-momentum

y

p

y

vv

x

vu

Y-momentum

Energy equation: Isentropic flow

wvu

TCy

vvu

TCx

u pp

22

2222

Irrotational Flow

Vorticity = 0

y

u

x

v

A potential function is defined as:y

vx

u

&

Conservation of mass: 02

2

2

2

yx

Conservation of Momentum:

yxyxyyxxCyx

2

2

22

2

22

22

2

2

2

21

Where,

22

22

2

1

yxaa o

Series Solutions

nnMMMM 26

34

22

10 ......................

Where, is incompressible flow solution.

00

2

20

200

2

20

200

12

2

1

u

vu

yyu

vu

xx

2

21

211

2

21

211

22

2

1

u

vu

yyu

vu

xx

2

21

211

2

21

2112

2

1

u

vu

yyu

vu

xxnnnnn

n

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 solution. •The potential of a complex flow can then be obtained by superimposing potentials of simpler flows.

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

& x

Vy

U

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:

02

2

2

2

yxx

V

y

U

• 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

& yx

Vxy

U

•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

iW In the complex (Argand-Gauss) plane every point is associated with a complex number

ireiyxz

In general we can then write

zfyxiyxW ,,

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). Equipotential lines are obviously parallel to the y 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 multi-valued 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 so-called 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 by

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 = -90o.

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 – plane

C=0.8

C=0.9

C=1.0

Flow Over An Airfoil

Vortex Panel Method