Biomedical Engineering Reference
In-Depth Information
convergence rate of the iterative process. We do not intend to provide the reader
with all the details of the available algorithms but to briefly indicate and describe the
modifications made to the original SIMPLE algorithm.
The SIMPLEC (SIMPLE-Consistent) algorithm follows the same iterative steps as
in the SIMPLE algorithm with the main difference being that the discretised momen-
tum equations are manipulated so that the SIMPLEC velocity correction formulae
omit terms that are less significant than those omitted in SIMPLE. Another pressure
correction procedure that is also commonly employed is the PISO (Pressure Implicit
with Splitting of Operators) algorithm. This pressure-velocity calculation procedure
was originally developed for non-iterative computation of unsteady compressible
flows. Nevertheless, it has been adapted successfully for the iterative solution of
steady state problems. PISO is simply recognized as an extension of SIMPLE with
an additional corrector step that involves an additional pressure correction equation
to enhance the convergence. The SIMPLER (SIMPLE-Revised) also falls within the
framework of two corrector steps like in PISO. Here, a discretised equation for the
pressure provides the intermediate pressure field before the discretised momentum
equations are solved. A pressure correction is later solved where the velocities are cor-
rected through the correction formulae as similarly derived in the SIMPLE algorithm.
There are other SIMPLE-like algorithms such as SIMPLEST (SIMPLE-ShorTened),
SIMPLEX, or SIMPLEM (SIMPLE-Modified) that share the same essence in their
derivations.
7.4
Numerical Solution OF ODES
The Lagrangian particle tracking equation (from Eq. 6.10 in Chap. 6) is in the form
of an Ordinary Differential Equation (ODE) and can be rewritten as
du
dt
=
f
(
u
,
t
)
(7.68)
The ODE in Eq. (7.68) defines the gradient or rate of change at any point by the
terms on the right hand side. There are many numerical methods which can be used
but in this section we present an introduction to some common methods which aims
to illustrate the main ideas of numerical solutions to ODEs.
7.4.1
Forward Euler Ethod
Euler's method is the simplest one-step numerical scheme for integration an ODE. It
solves the ODE by applying the change in the velocity term from time
t
to time
t
+
h
where
h
is a time step size
h
t.
This means that each successive approximation
u
t
+
1
can be represented through a Taylor series expansion which gives
=
d
2
u
dt
2
t
2
2
du
dt
t
u
t
+
t
=
u
t
+
+
...
truncation error
+
(7.69)
Search WWH ::
Custom Search