Page 212 - Fluid Mechanics and Thermodynamics of Turbomachinery

Three-dimensional Flows in Axial Turbomachines 193
L.E.
T.E.
20

Radius, r (cm) 15

10
T.E.
L.E.
0 5 10 15
Axial distance, Z (cm)
FIG. 6.14. Typical computational mesh for a single blade row (adapted from
Macchi 1985).

flow. With the present design trend towards highly loaded blade rows, which
can include patches of supersonic flow, this design method has considerable
merit.
All three methods solve the same equations of fluid motion, energy and state
for an axisymmetric flow through a turbomachine with varying hub and tip radii
and therefore lead to the same solution. In the first method the equation for the
2
2 1/2
meridional velocity c m D .c C c / in a plane (at x D x a ) contain terms involving
r x
both the slope and curvature of the meridional streamlines which are estimated by
using a polynominal curve-fitting procedure through points of equal stream function
dx/ and .x a C dx/. The major source of difficulty
on neighbouring planes at .x a
is in accurately estimating the curvature of the streamlines. In the second method a
grid of calculating points is formed on which the stream function is expressed as a
quasi-linear equation. A set of corresponding finite difference equations are formed
which are then solved at all mesh points of the grid. A more detailed description of
these methods is rather beyond the scope and intention of the present text.

Secondary flows

No account of three-dimensional motion in axial turbomachines would be
complete without giving, at least, a brief description of secondary flow. When a
fluid particle possessing rotation is turned (e.g. by a cascade) its axis of rotation is
deflected in a manner analogous to the motion of a gyroscope, i.e. in a direction
perpendicular to the direction of turning. The result of turning the rotation (or
vorticity) vector is the formation of secondary flows. The phenomenon must occur
to some degree in all turbomachines but is particularly in evidence in axial-flow
compressors because of the thick boundary layers on the annulus walls. This case
has been discussed in some detail by Horlock (1958), Preston (1953), Carter (1948)
and many other writers.