Handbook of Turbomachinery Second Edition Revised - Ventech!

Navier–Stokes equations are time-marching schemes. Both explicit and
implicit time-marching solutions are used; however, each has advantages
and disadvantages. Explicit time-marching methods are simpler, in that they
allow the solution to be directly calculated at each cell, using known
conditions at the old time level. However, they impose a stability criterion
on the size of the time step used in the marching process. That stability
criterion is a function of the grid cell size and may require the solution to
proceed very slowly in time, if small cells are present in the computational
space. In contrast, implicit solution schemes are not limited by such a
stability criterion and may use larger time steps; however, they are
formulated as a set of simultaneous equations, which must be solved as a
system. To avoid the substantial computation time required for matrix
inversion, other techniques, such as approximate factorization, are often
employed to simplify the computation for implicit methods.
To achieve better performance, other hybrid algorithms have also been
developed, which employ characteristics of both implicit and explicit
schemes. Jameson and Baker [66] introduced implicit residual smoothing in
an explicit Runge–Kutta technique to relax the stability criterion imposed
by the explicit technique. Also, Dawes [67] devised a two-step explicit and
one-step implicit variation on the implicit Beam and Warming algorithm.
Another approach to improve performance involves the use of local
time stepping. If only a steady-state solution is desired, as opposed to a true
unsteady analysis, then explicit time marching may be performed with
unequal time intervals at various cells within the computational grid. This
allows stability constraints to be locally satisfied, while still permitting the
solution to converge as rapidly as possible, on highly stretched grids.
To analyze flow phenomena that are unsteady, such as vortex
shedding, wake passing, and blade-row interaction, requires that the timemarching
solution be performed with uniform time steps, such that the
entire flow is advanced in time by the same amount with every time step. It is
also necessary that the solution be allowed to proceed long enough in time,
from an initialized state, to establish periodic flow behavior. This can
impose very long calculation times on the solution, particularly when
stability criteria governed by dense packed grids require that small time
steps be taken.
As Lakshminarayana [41] points out in his review of CFD techniques,
both explicit and implicit schemes offer similar levels of maturity and have
been used successfully for turbomachinery flow analyses. Neither approach
provides overriding advantages in terms of accuracy; however, explicit
methods tend to require less computation time and are better suited for
application on vector- and parallel-processor computers.
Copyright © 2003 Marcel Dekker, Inc.