cascade theory the theory in this lecture comes from: fluid mechanics of turbomachinery by george f....
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Cascade theory
The theory in this lecture comes from:Fluid Mechanics of Turbomachinery
by George F. WislicenusDover Publications, INC. 1965
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.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
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Decompose the velocity in the normal and the tangential direction
of the contour
2sn
22
2s
2nsn
2222
2s
2n
2
csinccoscc2cc
ccsinccoscv2sincoscc
csincccoscc
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Bernoulli’s equation
2sn2
0
2
0
csinccoscc2c2
pp
2
cpp
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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
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Forces in the x-direction
2sn2
0 csinccoscc2c2
pp
We insert the equation for the pressure, p from Bernoulli’s equation.
sincsincdsccosc
cosccoscdsccosc
cosdspdF
sn
nn
x
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Forces in the x-direction
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
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Forces in the x-direction
2nssn
22
2n
2n
32
2
s2
n
2
0x
sinccsincosccsinccsincoscds
coscc2cosccoscds
cos2
csincoscccoscccos
2
cds
cosdspdF
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Forces in the x-direction
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.
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Forces in the x-direction
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.
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Forces in the x-direction
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
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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
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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
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Forces in the y-direction
dsccdssin2
csindspdF s
2
0y
This is the force acting in the y-direction on a small element, ds of the contour.
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Forces in the y-direction
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
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Lift
dsccF sy
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Circulation
dscs
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Lift
cFy
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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.
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The law of the circulatory flow about a deflecting body
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
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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:
D
C
s2
B
A
s121 dscdsc
cs
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The law of the circulatory flow about a deflecting body
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
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The law of the circulatory flow about a deflecting body
02121
Since we do not have any body inside the hatched area:
Which gives:
21 cs
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The law of the circulatory flow about a deflecting body
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.
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The law of the circulatory flow about a deflecting body
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
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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
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A
E
s
E
A
s dscdsc
B
D
s
D
B
s dscdsc
Circulation about several deflecting bodies
From the figure we can see that:
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0321
Circulation about several deflecting bodies
E
D
s
D
B
s
B
A
s
A
E
s321 dscdscdscdsc
Circulation around 3 wing profiles in a cascade becomes:
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Cascade in an axial flow turbine
Let us look at the cylindrical section AB through the axial flow turbine.
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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
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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
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Cascade in an axial flow turbineFrom the figure we can see that:
1u
a
a
s
2u
b
b
s
cr2dsc
cr2dsc
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Cascade in an axial flow turbine
b
a
s1u
a
b
s2ui dsccr2dsccr2Z
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Cascade in an axial flow turbine
From the figure we can see that:
b
a
s1u
a
b
s2ui dsccr2dsccr2Z
a
b
s
b
a
s dscdsc
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Cascade in an axial flow turbine
1u2u cr2cr2
The circulation becomes:
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Cascade in an axial flow turbine
1u2u crcr2
Z
The change of angular momentum is related to the vane circulation by the equation:
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Cascade in an axial flow turbine
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.