482
13 Axial Compressors
flow is averaged in the circumferential direction. This again gives a two-dimensional flow denominated flow on the S 2 -surface or the hub-to-shroud surface. An
S 2 -surface is typically a meridional plane, but it may also be a streamsurface. The
equations describing the flow on the S 2 -surface contain terms expressing the average effect of the blade forces. The S 1 -S 2 decomposition technique requires iteration
between calculations on the S 2 -surface (through-flow) and calculations on a series
of S 1 -surfaces (blade-to-blade flows). No analytical solution is possible for either
of these two-dimensional flow types. So the technique has to be a numerical one. A
Cartesian-type grid for three-dimensional calculation is represented in Fig. 13.12.
The projection of the quasi-radial grid lines and the quasi-axial grid lines on a meridional plane is shown. This hub-to-shroud surface could be an S 2 -surface and the
surfaces of revolution obtained with the quasi-axial grid lines could be S 1 -surfaces.
The S 1 -S 2 decomposition technique was proposed by C.H. Wu in 1952 as an exact decomposition of a three-dimensional flow into two-dimensional flows. But,
S 1 -surfaces are then not surfaces of revolution, but are twisted. The S 2 -surface is
twisted too. The exact decomposition leads to a time-consuming iterative calculation with, nowadays, no advantage compared to an entirely three-dimensional flow
calculation. Nowadays, the quasi-three-dimensional flow representation is used,
typically followed by a fully three-dimensional simulation.
13.3.2 Radial Equilibrium
In the space in between rotor blade rows and stator blade rows, there is no blade
force in the circumferentially averaged S 2 -flow. Therefore, the circumferential S 1 -
streamsurfaces may be determined approximately on purely geometrical grounds,
for instance by equal distances in between them. The momentum equation in the direction perpendicular to the streamsurfaces is then reduced to equilibrium between
the centrifugal force associated to the tangential velocity component and a pressure
gradient in the normal direction. A further simplification comes with the assumption of cylindrical streamsurfaces. Both forces are in the radial direction then and
the equilibrium is called radial equilibrium. From the radial equilibrium follows an
equation allowing the analysis of the variation in the radial direction of the velocity
components. This allows acquiring a qualitative image of the real three-dimensional
flow.
The equation of radial equilibrium is
Total enthalpy is
2
1
.
u
v
dp
dr
r
=
r
2
2
1
1
0
2
2
.
= +
+
a
u
h h
v
v
13 Axial Compressors
flow is averaged in the circumferential direction. This again gives a two-dimensional flow denominated flow on the S 2 -surface or the hub-to-shroud surface. An
S 2 -surface is typically a meridional plane, but it may also be a streamsurface. The
equations describing the flow on the S 2 -surface contain terms expressing the average effect of the blade forces. The S 1 -S 2 decomposition technique requires iteration
between calculations on the S 2 -surface (through-flow) and calculations on a series
of S 1 -surfaces (blade-to-blade flows). No analytical solution is possible for either
of these two-dimensional flow types. So the technique has to be a numerical one. A
Cartesian-type grid for three-dimensional calculation is represented in Fig. 13.12.
The projection of the quasi-radial grid lines and the quasi-axial grid lines on a meridional plane is shown. This hub-to-shroud surface could be an S 2 -surface and the
surfaces of revolution obtained with the quasi-axial grid lines could be S 1 -surfaces.
The S 1 -S 2 decomposition technique was proposed by C.H. Wu in 1952 as an exact decomposition of a three-dimensional flow into two-dimensional flows. But,
S 1 -surfaces are then not surfaces of revolution, but are twisted. The S 2 -surface is
twisted too. The exact decomposition leads to a time-consuming iterative calculation with, nowadays, no advantage compared to an entirely three-dimensional flow
calculation. Nowadays, the quasi-three-dimensional flow representation is used,
typically followed by a fully three-dimensional simulation.
13.3.2 Radial Equilibrium
In the space in between rotor blade rows and stator blade rows, there is no blade
force in the circumferentially averaged S 2 -flow. Therefore, the circumferential S 1 -
streamsurfaces may be determined approximately on purely geometrical grounds,
for instance by equal distances in between them. The momentum equation in the direction perpendicular to the streamsurfaces is then reduced to equilibrium between
the centrifugal force associated to the tangential velocity component and a pressure
gradient in the normal direction. A further simplification comes with the assumption of cylindrical streamsurfaces. Both forces are in the radial direction then and
the equilibrium is called radial equilibrium. From the radial equilibrium follows an
equation allowing the analysis of the variation in the radial direction of the velocity
components. This allows acquiring a qualitative image of the real three-dimensional
flow.
The equation of radial equilibrium is
Total enthalpy is
2
1
.
u
v
dp
dr
r
=
r
2
2
1
1
0
2
2
.
= +
+
a
u
h h
v
v