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CFD for industrial turbomachinery designs 69
severely.Theimplicitscheme,ontheotherhand,isgenerallyunconditionallystable,
so that a relatively large time step can be adapted. In addition to these algorithms,
thereexistsafamilyofhybridalgorithms, whichis widelyused inthe cascade-flow
simulations.The hybrid algorithms possess the positive features ofthe explicit and
implicit algorithms, providing a relatively rapid convergence process and having
a less restricted stability constraint. The essence of this algorithm is the use of
explicit and implicit finite-difference formulae at alternative computational mesh
points. However, the nature of the basic formulae for the hybridalgorithm remains
the same as for the traditional implicit and explicit schemes.
In viscous calculations, dissipating proper ties always present due to the exis-
tence of diffusive terms. Away from the shear layer regions, however, the physical
diffusion is generally not sufficient to prevent the numerical oscillation of the
schemes. Thus, the artificial dissipation terms are usually used in order to sta-
bilize and permit a higher CFL number in most of the interactive schemes for the
compressible flow computation. As is commonly recognized, the artificial dissipa-
tion will influence the accuracy of the numerical results. It has been shown that
essentially grid-converged solutions for the high-speed viscous flows over aerody-
namic shapes can be obtained if sufficiently fine meshes for the computation are
provided. However,thecomputation basedon fine meshesis very timeconsuming,
making it difficult to assess the numerical accuracy of grid-converged solutions.
The techniquesto properlytreat dissipationterms are stillan important topic ofthe
computational fluid.An attempt was made byTurkel andVatsa [15] to improve the
accuracy of the solutions on a given grid in order to reduce the required number
of grid points for obtaining a specified level of accuracy. The essential mechanism
usedintheirmethod[15]istoreducetheleveloftheartificialviscositybyreplacing
the scalarform of theartificial viscosity witha matrix form. It hasbeenshown that
the numericalaccuracy ofthe Navier–Stokessolutions isimproved throughthe use
of the matrix-valued dissipation model. In this chapter, the dissipation terms were
not treated through modifying convective fluxes as in traditional methods. Instead,
thedissipationtermswereincorporatedintothetime-derivativetermstoformanew
timediscretizationschemetoreducethe levelofartificialviscosity.Andalso inthis
way, the eigenvalue-stiffness problem associated with the time-derivative term can
bepreventedwhenthecalculations areperformed inalowMachnumberflow.This
characteristic can partly control the effect of eigenvalue stiffness, which is well
known on the convergence of both explicit and implicit schemes. Furthermore, the
present scheme is a hybrid scheme that combines the advantages of the implicit
and explicit schemes, and isasecond-order time andspatial derivative scheme that
allowsmoreaccuratecalculation.BecausethehigherCFLnumbercanbeusedwith
the present method, it is more economic than the ordinary implicit scheme.
Theeffectofeigenvaluestiffnessontheconvergenceofbothexplicitandimplicit
schemes is well known, and normally two distinct methods have been suggested
for controlling the eigenvalues to enhance convergence. There are two kinds of
methods thatcan solvethis problem.One of themethods to overcome this problem
is to premultiply the time derivative by a suitable matrix, and the other is to use
a perturbed form of the equations in which specific terms are dropped such that