Cascade theory The theory in this lecture comes from: Fluid Mechanics of Turbomachinery by George F. Wislicenus Dover Publications, INC. 1965.

Cascade theory

The theory in this lecture comes from:Fluid Mechanics of Turbomachinery

by George F. WislicenusDover Publications, INC. 1965

.konst2

cpp

2

0

c = c∞+c

0dt

dc

c∞

FY

FX

ds

X

Y

Contour

The contour is large compared to the dimensions of the vane

c is the change of velocity due to the vane

Decompose the velocity in the normal and the tangential direction

of the contour

2sn

22

2s

2nsn

2222

2s

2n

2

csinccoscc2cc

ccsinccoscv2sincoscc

csincccoscc

Bernoulli’s equation

2sn2

0

2

0

csinccoscc2c2

pp

2

cpp

Forces in the x-direction

The forces in the x-direction acting on the element ds can be calculated as a force coming from pressure and impulse.

sincsincdsccosc

cosccoscdsccosc

cosdspdF

sn

nn

x

Flow Rate, Q Velocity in x-direction, cx


2sn2

0 csinccoscc2c2

pp

We insert the equation for the pressure, p from Bernoulli’s equation.

sincsincdsccosc

cosccoscdsccosc

cosdspdF

sn

nn

x


2sn2

0 csinccoscc2c2

pp

We insert the equation for the pressure, p from Bernoulli’s equation.

sincsincdsccosc

cosccoscdsccosc

cosdscsinccoscc2c2

cosdspdF

sn

nn

2sn

2

0x


2nssn

22

2n

2n

32

2

s2

n

2

0x

sinccsincosccsinccsincoscds

coscc2cosccoscds

cos2

csincoscccoscccos

2

cds

cosdspdF


2nssn

22

2n

2n

32

2

s2

n

2

0x

sinccsincosccsinccsincoscds

coscc2cosccoscds

cos2

csincoscccoscccos

2

cds

cosdspdF

The change of velocity, c is very small because the large distance from the airfoil to the contour. We neglect the terms that has the second order of c.


dsccdscos2

ccosdspdF

sincosccdssincos2

1coscdscosdspdF

n

2

0x

22n

2220x

This is the force acting in the x-direction on a small element, ds of the contour.


dsccdscos2

ccosdspdF n

2

0x

By integrating around the contour, we will find the total force acting in the x-direction.

dsccF

dsccdscos2

cdscospF

nx

n

2

0x

=0 =0

d’Alembert paradox

The term cn·ds is the flow rate through the contour. If the flow is incompressible, the integral of the term cn·ds around the contour will be zero.

A body in a two-dimensional and non-viscous flow with constant energy will not exert a force in the direction parallel undisturbed flow, c∞

0dsccF nx

Forces in the y-direction

The forces in the y-direction acting on the element ds can be calculated as a force coming from pressure and impulse.

cosccoscdsccosc

sinccoscdsccosc

sindspdF

sn

nn

y


dsccdssin2

csindspdF s

2

0y

This is the force acting in the y-direction on a small element, ds of the contour.


By integrating around the contour, we will find the total force acting in the y-direction.

dsccF

dsccdssin2

cdssinpF

sy

s

2

0y

=0 =0

dsccdssin2

csindspdF s

2

0y

Lift

dsccF sy

Circulation

dscs

Lift

cFy

The law of the circulatory flow about a deflecting body

In the absence of any deflecting body inside the hatched area of the contour the force in y-direction must necessarily be zero. This leads to the theorem that:

For a flow of constant energy, the circulation around any closed contour not enclosing any force-transmitting body must be zero.


dscs1

Let the circulation around the outer contour in the figure be:

cs Let the circulation around the inner contour in the figure be:

dscs2


Let the circulation around the inner and outer contour be connected along the line A-B.

The circulation around the hatched area can now be written as:

D

C

s2

B

A

s121 dscdsc

cs


2121

From the figure we can see that:

The circulation around the hatched area can now be written as:

D

C

s

B

A

s dscdsc

cs


02121

Since we do not have any body inside the hatched area:

Which gives:

21 cs


21

cs

This leads to the theorem:

For a given flow condition (with constant energy), the circulation around the deflecting body is independent of the size and shape of the contour along which the circulation is measured.


dsccs ssm

cs

The mean velocity for the circulation around a contour having the length s is:

For a constant value of the circulation, the mean velocity, csm has to decrease if the length s increases.

The circulation is in inverse ratio to the distance of the contour

Circulation about several deflecting bodies

We have 3 wing profiles in a two-dimensional cascade and makes a contour around the whole cascade. This contour is marked ABGDEF.

A

E

s

E

A

s

AEF

s1 dscdscdsc

A

E

s

E

D

s

D

B

s

B

A

s

ABDE

s2 dscdscdscdscdsc

B

D

s

D

B

s

BGD

s3 dscdscdsc

A

E

s

E

A

s dscdsc

B

D

s

D

B

s dscdsc



0321


E

D

s

D

B

s

B

A

s

A

E

s321 dscdscdscdsc

Circulation around 3 wing profiles in a cascade becomes:

Cascade in an axial flow turbine

Let us look at the cylindrical section AB through the axial flow turbine.


By unfolding the cylindrical section AB from the last slide, we can look at the blades in a cascade

Cascade in an axial flow turbineCirculation around the blades is: (where Z is the number of blades)

b

a

s

a

a

s

a

b

s

b

b

si dscdscdscdscZ

Cascade in an axial flow turbineFrom the figure we can see that:

1u

a

a

s

2u

b

b

s

cr2dsc

cr2dsc


b

a

s1u

a

b

s2ui dsccr2dsccr2Z



b

a

s1u

a

b

s2ui dsccr2dsccr2Z

a

b

s

b

a

s dscdsc


1u2u cr2cr2

The circulation becomes:


1u2u crcr2

Z

The change of angular momentum is related to the vane circulation by the equation:


1122

12 2

uu

uu

cucuE

ZcrcrE

By multiplying the change of angular momentum from the upstream to the downstream side of a turbine runner is the torque acting on the turbine shaft with the angular velocity of the runner we will recognize Euler’s turbine equation.

Cascade theory The theory in this lecture comes from: Fluid Mechanics of Turbomachinery by George F. Wislicenus Dover Publications, INC. 1965.

Documents

contour slide

xdirection slide

c slide

circulation slide

c x slide

cscs slide

y direction

lift slide