Cascade theoryThe theory in this lecture comes from:Fluid Mechanics of Turbomachinery

by George F. WislicenusDover Publications, INC. 1965

.konst2cpp2

0 =⋅ρ

+=

c = c∞+∆c

0dtdc

=

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

∆+α⋅∆+α⋅∆⋅⋅+=

⇓

∆+∆+α⋅∆+α⋅∆⋅⋅+α+α⋅=

⇓

∆+α⋅+∆+α⋅=

∞∞

∞∞

∞∞

Bernoulli’s equation

( )( )2sn2

0

2

0

csinccoscc2c2

pp

2cpp

∆+α⋅∆+α⋅∆⋅⋅+⋅ρ

+=

⇓

⋅ρ+=

∞∞

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.

( ) ( )( ) ( ) α∆+α⋅⋅⋅∆+α⋅⋅ρ−

α∆+α⋅⋅⋅∆+α⋅⋅ρ−

α⋅⋅−=

∞∞

∞∞

sincsincdsccosc

cosccoscdsccosc

cosdspdF

sn

nn

x

Flow Rate, Q Velocity in x-direction, cx

Forces in the x-direction

( )( )2sn2

0 csinccoscc2c2

pp ∆+α⋅∆+α⋅∆⋅⋅+⋅ρ

+−=− ∞∞

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

( ) ( )( ) ( ) α∆+α⋅⋅⋅∆+α⋅⋅ρ−

α∆+α⋅⋅⋅∆+α⋅⋅ρ−

α⋅⋅−=

∞∞

∞∞

sincsincdsccosc

cosccoscdsccosc

cosdspdF

sn

nn

x

Forces in the x-direction

( )( )2sn2

0 csinccoscc2c2

pp ∆+α⋅∆+α⋅∆⋅⋅+⋅ρ

+−=− ∞∞

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

( )( )( ) ( )( ) ( ) α∆+α⋅⋅⋅∆+α⋅⋅ρ−

α∆+α⋅⋅⋅∆+α⋅⋅ρ−

α⋅⋅∆+α⋅∆+α⋅∆⋅⋅+⋅ρ

+

α⋅⋅−=

∞∞

∞∞

∞∞

sincsincdsccosc

cosccoscdsccosc

cosdscsinccoscc2c2

cosdspdF

sn

nn

2sn

2

0x

Forces in the x-direction

( )( )α⋅∆⋅+α⋅α⋅∆⋅+α⋅∆⋅∆+α⋅α⋅⋅⋅ρ−

α⋅∆⋅⋅+α⋅∆+α⋅⋅⋅ρ−

α⋅

∆+α⋅α⋅∆⋅+α⋅∆⋅+α⋅⋅⋅ρ+

α⋅⋅−=

∞∞∞

∞∞

∞∞∞

2nssn

22

2n

2n

32

2

s2

n

2

0x

sinccsincosccsinccsincoscdscoscc2cosccoscds

cos2csincoscccoscccos

2cds

cosdspdF

Forces in the x-direction

( )( )α⋅∆⋅+α⋅α⋅∆⋅+α⋅∆⋅∆+α⋅α⋅⋅⋅ρ−

α⋅∆⋅⋅+α⋅∆+α⋅⋅⋅ρ−

α⋅

∆+α⋅α⋅∆⋅+α⋅∆⋅+α⋅⋅⋅ρ+

α⋅⋅−=

∞∞∞

∞∞

∞∞∞

2nssn

22

2n

2n

32

2

s2

n

2

0x

sinccsincosccsinccsincoscdscoscc2cosccoscds

cos2csincoscccoscccos

2cds

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.

Forces in the x-direction

dsccdscos2ccosdspdF

sincosccdssincos21coscdscosdspdF

n

2

0x

22n

2220x

⋅∆⋅⋅ρ−⋅α⋅⋅ρ−α⋅⋅−=

⇓

α+α⋅∆⋅⋅⋅ρ−

α−α−⋅α⋅⋅⋅ρ+α⋅⋅−=

∞∞

∞∞

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

Forces in the x-direction

dsccdscos2ccosdspdF n

2

0x ⋅∆⋅⋅ρ−⋅α⋅⋅ρ−α⋅⋅−= ∞∞

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

∫

∫∫∫

⋅∆⋅⋅ρ−=

⇓

⋅∆⋅⋅ρ−⋅α⋅⋅ρ−⋅α⋅−=

∞

∞∞

dsccF

dsccdscos2cdscospF

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 =⋅∆⋅⋅ρ−= ∫∞

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.

( ) ( )( ) ( ) α⋅∆+α⋅⋅⋅∆+α⋅⋅ρ−

α⋅∆+α⋅⋅⋅∆+α⋅⋅ρ−

α⋅⋅−=

∞∞

∞∞

cosccoscdsccosc

sinccoscdsccosc

sindspdF

sn

nn

y

Forces in the y-direction

dsccdssin2csindspdF s

2

0y ⋅∆⋅⋅ρ−⋅α⋅⋅ρ−α⋅⋅−= ∞∞

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

Krefter i y-retning

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

∫

∫∫∫

∞∞

∞∞

⋅∆⋅⋅ρ−=

⇓

⋅∆⋅⋅ρ−⋅α⋅⋅ρ−⋅α⋅−=

dsccF

dsccdssin2cdssinpF

sy

s

2

0y

=0 =0

dsccdssin2csindspdF s

2

0y ⋅∆⋅⋅ρ−⋅α⋅⋅ρ−α⋅⋅−= ∞∞

Lift

∫∞

∞ ⋅∆⋅⋅ρ−= dsccF sy

Circulation

∫∞

⋅∆=Γ dscs

Lift

Γ⋅⋅ρ−= ∞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.

The law of the circulatory flow about a deflecting body

∫∞

⋅=Γ dscs1

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

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

∫∞

⋅=Γ dscs2

The law of the circulatory flow about a deflecting body

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:

∫∫ ⋅+Γ−⋅+Γ=Γ −

D

Cs2

B

As121 dscdsc

cs

The law of the circulatory flow about a deflecting body

2121 Γ−Γ=Γ −

From the figure we can see that:

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

∫∫ ⋅−=⋅D

Cs

B

As dscdsc

cs

The law of the circulatory flow about a deflecting body

02121 =Γ−Γ=Γ −

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

Which gives:

21 Γ=Γcs

The law of the circulatory flow about a deflecting body

21 Γ=Γ

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.

The law of the circulatory flow about a deflecting body

∫∞

⋅=⋅=Γ 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.

∫∫∫ ⋅+⋅=⋅=ΓA

Es

E

As

AEFs1 dscdscdsc

∫∫∫∫∫ ⋅+⋅+⋅+⋅=⋅=ΓA

Es

E

Ds

D

Bs

B

As

ABDEs2 dscdscdscdscdsc

∫∫∫ ⋅+⋅=⋅=ΓB

Ds

D

Bs

BGDs3 dscdscdsc

∫∫ ⋅−=⋅A

Es

E

As dscdsc

∫∫ ⋅−=⋅B

Ds

D

Bs dscdsc

Circulation about several deflecting bodies

From the figure we can see that:

0321 Γ=Γ+Γ+Γ

Circulation about several deflecting bodies

∫∫∫∫ ⋅+⋅+⋅+⋅=Γ+Γ+ΓE

Ds

D

Bs

B

As

A

Es321 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.

Cascade in an 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)

∫∫∫∫ ⋅+⋅+⋅+⋅=Γ⋅=Γb

as

a

as

a

bs

b

bsi dscdscdscdscZ

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

1u

a

as

2u

b

bs

cr2dsc

cr2dsc

⋅⋅Π⋅−=⋅

⋅⋅Π⋅=⋅

∫

∫

Cascade in an axial flow turbine

∫∫ ⋅+⋅⋅Π⋅−⋅+⋅⋅Π⋅=Γ⋅=Γb

as1u

a

bs2ui dsccr2dsccr2Z

Cascade in an axial flow turbine

From the figure we can see that:

∫∫ ⋅+⋅⋅Π⋅−⋅+⋅⋅Π⋅=Γ⋅=Γb

as1u

a

bs2ui dsccr2dsccr2Z

∫∫ ⋅−=⋅a

bs

b

as dscdsc

Cascade in an axial flow turbine

1u2u cr2cr2 ⋅⋅Π⋅−⋅⋅Π⋅=Γ

The circulation becomes:

Cascade in an axial flow turbine

1u2u crcr2Z

⋅−⋅=Π⋅Γ⋅

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

Cascade in an axial flow turbine

1u12u2

1u2u

cucuE

2ZcrcrE

⋅−⋅=⇓

Π⋅Γ⋅

=⋅⋅ω−⋅⋅ω=

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.