7.4 Other Methods
181
Fractional step methods have become rather popular. There is a wide
variety of them, due to a vast choice of approaches to time and space discretization; however, they are all based on the principles described above.
The major difference between the fractional-step method and pressurecorrection methods of the SIMPLE-type is that in the former, the pressure
(or pressure-correction) equation is solved once per time step, while in the latter, both the momentum and pressure-correction equations are solved several
times within each time step (outer iterations). This is largely because fractional step methods are used mainly in unsteady flow simulations while the
latter are used predominantly to compute steady flows. Since, in SIMPLEtype methods, mass conservation is enforced only at the end of a time step,
the pressure-correction equation need not be solved accurately on each outer
iteration (reduction of the residual by one order of magnitude usually suffices). Indeed, for steady flows, accurate satisfaction of the continuity condition is required only at convergence. In simulations of unsteady flows, the
pressure (or pressure-correction) equation must be solved to a tight tolerance
to ensure mass conservation a t each time step. Multigrid or spectral methods
are usually used to solve the Poisson equation for the pressure in unsteady
flow simulations in simple geometries while, for steady flows or complex geometries, the linear equations are usually solved using conjugate-gradient
methods.
If the time step is large, the fractional step method produces an error due
to the operator splitting, as shown in Eq. (7.60). This error can be eliminated either by reducing the time step or by using iteration of the kind used
in SIMPLE-type methods. However, if the splitting error is significant, the
temporal discretization error is also large. Therefore, reducing the time step
is the most appropriate means of improving accuracy. Note that the PISOmethod introduced in the preceding section is very similar to the fractionalstep method and has a splitting error proportional to (At)'.
7.4.2 Streamfunction-Vorticity Methods
For incompressible two-dimensional flows with constant fluid properties, the
Navier-Stokes equations can be simplified by introducing the streamfunction
$ and vorticity w as dependent variables. These two quantities are defined in
terms of Cartesian velocity components by:
and
Lines of constant 4 are streamlines (lines which are everywhere parallel to the
flow), giving this variable its name. The vorticity is associated with rotational
181
Fractional step methods have become rather popular. There is a wide
variety of them, due to a vast choice of approaches to time and space discretization; however, they are all based on the principles described above.
The major difference between the fractional-step method and pressurecorrection methods of the SIMPLE-type is that in the former, the pressure
(or pressure-correction) equation is solved once per time step, while in the latter, both the momentum and pressure-correction equations are solved several
times within each time step (outer iterations). This is largely because fractional step methods are used mainly in unsteady flow simulations while the
latter are used predominantly to compute steady flows. Since, in SIMPLEtype methods, mass conservation is enforced only at the end of a time step,
the pressure-correction equation need not be solved accurately on each outer
iteration (reduction of the residual by one order of magnitude usually suffices). Indeed, for steady flows, accurate satisfaction of the continuity condition is required only at convergence. In simulations of unsteady flows, the
pressure (or pressure-correction) equation must be solved to a tight tolerance
to ensure mass conservation a t each time step. Multigrid or spectral methods
are usually used to solve the Poisson equation for the pressure in unsteady
flow simulations in simple geometries while, for steady flows or complex geometries, the linear equations are usually solved using conjugate-gradient
methods.
If the time step is large, the fractional step method produces an error due
to the operator splitting, as shown in Eq. (7.60). This error can be eliminated either by reducing the time step or by using iteration of the kind used
in SIMPLE-type methods. However, if the splitting error is significant, the
temporal discretization error is also large. Therefore, reducing the time step
is the most appropriate means of improving accuracy. Note that the PISOmethod introduced in the preceding section is very similar to the fractionalstep method and has a splitting error proportional to (At)'.
7.4.2 Streamfunction-Vorticity Methods
For incompressible two-dimensional flows with constant fluid properties, the
Navier-Stokes equations can be simplified by introducing the streamfunction
$ and vorticity w as dependent variables. These two quantities are defined in
terms of Cartesian velocity components by:
and
Lines of constant 4 are streamlines (lines which are everywhere parallel to the
flow), giving this variable its name. The vorticity is associated with rotational