61
2.2 Linear Cascades
blades at the mean radius and by unrolling it into a plane. A linear cascade or blade
row is built by setting up prismatic blades with the profiles obtained. In principle,
an infinite number of blades are required in order to keep the periodicity of the
blades in the original machine. The two-dimensional flow is representative for the
real three-dimensional flow. The expressions for axial momentum are the same in
both configurations. Moment of momentum with the three-dimensional flow corresponds to tangential momentum with the cascade (tangential = circumferential).
Constant tangential momentum along the blade height with the linear cascade is
then equivalent to constant angular momentum (v r
u = constant) along the radius in
the real machine. This means then that the three-dimensional flow is a so-called free
vortex flow. The term means swirling flow with constant angular momentum. In the
pioneering time, axial turbomachines were built with free vortex flow because of
this theoretical correspondence. They show certain disadvantages discussed later
(Chap. 13: axial compressors; Chap. 15: axial and radial turbines). Present-day machines are designed with some degree of forcing of the vortex (v r
u ≠ constant)
along the radius. The deviation to the free vortex distribution is not very significant,
however. With free vortex blades in a cylindrical axial turbomachine, an exact transformation exists between the flow in the machine and the linear cascade (of course
with top and bottom walls added). This does no longer apply with some forcing on
the vortex. For instance, no cylindrical streamsurfaces exist anymore. The analogy
between the three-dimensional flow within the machine and the two-dimensional
flow within the linear cascade is only met approximately. Nevertheless, studying
the characteristics of linear cascades is very instructive as the flow within a linear
cascade is still representative for the flow in an average cylindrical section of a real
machine.
2.2.2 Cascade Geometry
The cascade is determined by the shape of the blades, their position to the axial direction and their spacing (tangential distance between the blades). The term pitch is
often used for spacing, but pitch may also mean, similarly to the term with screws,
the axial distance covered by the flow when it makes a full turn of 360º. This applies in particular to propellers and wind turbines with adjustable rotor blades. Then
often, also the term pitch angle is used. Results are also a function of the positioning
of the blades relative to the flow. The profile shape is, as with aerofoils, determined
by the camber line and the thickness distribution. The profile size is determined by
the chord. Solidity σ means the ratio of the chord to the spacing. Some characteristics are shown in Fig. 2.11. The camber angle f is the angle difference between
the tangents at the camber line on the leading and trailing edges. Profile orientation
relative to the axial direction is determined by the chord angle or stagger angle g.
The inlet velocity w 1 forms an angle of attack a with the chord, an angle β 1 with
the axial direction and an angle of incidence i with the tangent at the camber line on
the leading edge. The angle of incidence is positive when the flow turning increases
2.2 Linear Cascades
blades at the mean radius and by unrolling it into a plane. A linear cascade or blade
row is built by setting up prismatic blades with the profiles obtained. In principle,
an infinite number of blades are required in order to keep the periodicity of the
blades in the original machine. The two-dimensional flow is representative for the
real three-dimensional flow. The expressions for axial momentum are the same in
both configurations. Moment of momentum with the three-dimensional flow corresponds to tangential momentum with the cascade (tangential = circumferential).
Constant tangential momentum along the blade height with the linear cascade is
then equivalent to constant angular momentum (v r
u = constant) along the radius in
the real machine. This means then that the three-dimensional flow is a so-called free
vortex flow. The term means swirling flow with constant angular momentum. In the
pioneering time, axial turbomachines were built with free vortex flow because of
this theoretical correspondence. They show certain disadvantages discussed later
(Chap. 13: axial compressors; Chap. 15: axial and radial turbines). Present-day machines are designed with some degree of forcing of the vortex (v r
u ≠ constant)
along the radius. The deviation to the free vortex distribution is not very significant,
however. With free vortex blades in a cylindrical axial turbomachine, an exact transformation exists between the flow in the machine and the linear cascade (of course
with top and bottom walls added). This does no longer apply with some forcing on
the vortex. For instance, no cylindrical streamsurfaces exist anymore. The analogy
between the three-dimensional flow within the machine and the two-dimensional
flow within the linear cascade is only met approximately. Nevertheless, studying
the characteristics of linear cascades is very instructive as the flow within a linear
cascade is still representative for the flow in an average cylindrical section of a real
machine.
2.2.2 Cascade Geometry
The cascade is determined by the shape of the blades, their position to the axial direction and their spacing (tangential distance between the blades). The term pitch is
often used for spacing, but pitch may also mean, similarly to the term with screws,
the axial distance covered by the flow when it makes a full turn of 360º. This applies in particular to propellers and wind turbines with adjustable rotor blades. Then
often, also the term pitch angle is used. Results are also a function of the positioning
of the blades relative to the flow. The profile shape is, as with aerofoils, determined
by the camber line and the thickness distribution. The profile size is determined by
the chord. Solidity σ means the ratio of the chord to the spacing. Some characteristics are shown in Fig. 2.11. The camber angle f is the angle difference between
the tangents at the camber line on the leading and trailing edges. Profile orientation
relative to the axial direction is determined by the chord angle or stagger angle g.
The inlet velocity w 1 forms an angle of attack a with the chord, an angle β 1 with
the axial direction and an angle of incidence i with the tangent at the camber line on
the leading edge. The angle of incidence is positive when the flow turning increases