Page 211 - Fluid Mechanics and Thermodynamics of Turbomachinery

192 Fluid Mechanics, Thermodynamics of Turbomachinery
geometry and flow conditions, its application to the design of axial-flow compressors
has been rather limited. The extensions of actuator disc theory to the solution of
the complex three-dimensional, compressible flows in compressors with varying
hub and tip radii and non-uniform total pressure distributions were found to have
become too unwieldy in practice. In recent years advanced computational methods
have been successfully evolved for predicting the meridional compressible flow in
turbomachines with flared annulus walls.
Reviews of numerical methods used to analyse the flow in turbomachines have
been given by Gostelow et al. (1969), Japikse (1976), Macchi (1985) and Whitfield
and Baines (1990) among many others. The literature on computer-aided methods
of solving flow problems is now extremely extensive and no attempt is made here
to summarise the progress. The real flow in a turbomachine is three-dimensional,
unsteady, viscous and is usually compressible, if not transonic or even supersonic.
According to Macchi the solution of the full equations of motion with the actual
boundary conditions of the turbomachine is still beyond the capabilities of the most
powerful modern computers. The best fully three-dimensional methods available are
still only simplifications of the real flow.

Through-flow methods
In any of the so-called through-flow methods the equations of motion to be solved
are simplified. First, the flow is taken to be steady in both the absolute and relative
frames of reference. Secondly, outside of the blade rows the flow is assumed to
be axisymmetric, which means that the effects of wakes from an upstream blade
row are understood to have “mixed out” so as to give uniform circumferential
conditions. Within the blade rows the effects of the blades themselves are modelled
by using a passage averaging technique or an equivalent process. Clearly, with these
major assumptions, solutions obtained with these through-flow methods can be only
approximations to the real flow. As a step beyond this Stow (1985) has outlined the
ways, supported by equations, of including the viscous flow effects into the flow
calculations.
Three of the most widely used techniques for solving through-flow problems are:
(1) Streamline curvature, which is based on an iterative procedure, is described in
some detail by Macchi (1985) and earlier by Smith (1966). It is the oldest and
most widely used method for solving the through-flow problem in axial-flow
turbomachines and has with the intrinsic capability of being able to handle
variously shaped boundaries with ease. The method is widely used in the gas
turbine industry.
(2) Matrix through-flow or finite difference solutions (Marsh 1968), where computa-
tions of the radial equilibrium flow field are made at a number of axial locations
within each blade row as well as at the leading and trailing edges and outside
of the blade row. An illustration of a typical computing mesh for a single blade
row is shown in Figure 6.14.
(3) Time-marching (Denton 1985), where the computation starts from some
assumed flow field and the governing equations are marched forward with time.
The method, although slow because of the large number of iterations needed to
reach a convergent solution, can be used to solve both subsonic and supersonic