Handbook of Turbomachinery Second Edition Revised - Ventech!

Navier–Stokes equations will be presented. Details of the various solution
techniques may be found in texts by Hirsch [65] and Anderson et al. [66]. In
addition, an excellent survey is provided by Lakshminarayana [41] in his
review of CFD techniques for turbomachinery applications.
In order to numerically solve the set of governing equations for a
particular turbomachinery flow field, the partial differential or integral
equations of the continuous solution domain must first be discretized into a
set of algebraic equations, which are applied at discrete points or cells in the
computational grid. These algebraic equations are then solved, using a
numerical solution technique that can be implemented as a computational
algorithm.
Depending on the formulation of the governing equations, they may
be discretized into a finite-difference, finite-volume, or finite-element
representation. The finite-difference method approximates the derivatives
in the differential form of the governing equations by difference expressions
formulated at discrete points in space. Because the finite-difference method
relies on the construction of differences between adjacent points, it requires
the use of a structured computational grid. In contrast, finite-volume
methods discretize the integral form of the governing equations. The
integral equations are applied to a small control volume around every point
in the computational grid. This approach ensures that mass, momentum,
and energy remain conserved in the discretized formulation. Because the
finite-volume method utilizes a control volume concept, arbitrary
computational grids may be accommodated. The finite-element method
also employs an integral formulation of the governing equations, as
obtained from the differential form, through the use of the method of
weighted residuals. As with the finite-volume method, finite-element
approximations may utilize computational grid cells of arbitrary shape
and size. Hirsch [65] presents a detailed discussion of each of the
discretization techniques.
Finite-volume schemes are experiencing widespread usage today for
turbomachinery CFD applications, replacing the previously dominant
finite-difference discretization. Finite-element schemes, although popular
for mechanical applications, have received less attention in computational
fluid dynamics. The increasing usage of finite-volume techniques recognizes
their improved ability to conserve mass, momentum, and energy, relative to
the finite-difference methods. The flexibility afforded by the use of arbitrary
computational grids is also a distinct advantage for the finite-volume
methods.
Once the governing equations have been discretized, the resulting set
of algebraic equations must be solved numerically. The most appropriate
solution techniques for the discretized time-dependent Reynolds-averaged
Copyright © 2003 Marcel Dekker, Inc.