Following [22]
our goal is introduce a
second-order central difference scheme for incompressible flows,
based on *velocity variables*.
The use of the velocity formulation
yields a more versatile algorithm.
The advantage of our proposed central scheme in its
velocity formulation is two-fold:
generalization to the three dimensional case is straightforward,
and the treatment of boundary conditions associated with general
geometries becomes simpler.
The result is a simple fast high-resolution method, whose
accuracy is comparable to that of an upwind
scheme. In addition, numerical experiments show the new
scheme to be immune to some of the well-known
deleterious consequences of under-resolution.

We consider a two-dimensional incompressible flow field, , so that
. The equations of
motion for a Newtonian fluid in conservation form are

where *p* is the pressure, is the kinematic viscosity, and
subscripts denote partial derivatives.
The functions and
are components of the fluxes of the conserved quantities
*u* and *v*.

The computational grid consists of rectangular cells of sizes
and
; at time level
, these cells, ,
are centered at .
Starting with the corresponding cell averages,
, we first
reconstruct a piecewise linear polynomial approximation
which recovers the point values
of the velocity field, .
For second-order accuracy, the piecewise linear reconstructed
velocities take the form,

As before, exact averaging over a staggered control volume yields

and a similar averaging applies for .

An exact computation yields

The incompressible fluxes, e.g., ,
are approximated in terms of the midpoint rule , which in turn
employs predicted midvalues which are obtained from half-step Taylor
expansion. Thus our scheme starts with a *predictor step* of the form

Note that the predictor step is nothing but a forward Euler scheme; conservation form is not essential for the spatial discretization at this stage.

This is followed by a *corrector step*

Note that the
viscous terms are handled here by the implicit
Crank-Nicholson discretization which
is favored due to its preferable stability properties.
Here, we ignore the pressure terms;
instead, the contribution of the pressure will
be integrated by enforcing zero-divergence fluxes at the last
*projection step*.

Compute the potential
solving the Poisson equation

Then, the pressure gradient at is being updated,

and finally, it is used to evaluate the divergence-free velocity field,

In Figure 1.5.15, we plot vorticity contours for
two shear layer problems studied in [5]:
the inviscid ``thick'' shear layer problem
corresponding to with ,
and a viscous ``thin'' shear layer problem (with ),
corresponding to with .
As in [5], both plots in Figures 1.5.15a and
1.5.15b are recorded at time *t*=1.2, and are subject to
an initial perturbation , with .

Further applications of the central schemes for
more complex incompressible flows (with 'variable' axisymmetric
coefficients, forcing source/viscous terms, ...), can be found in
[20],[21].

**Figure 1.5.15:** *Contour lines of the vorticity, ,
at t=1.2 with initial ,
using a grid. (a) A ``thick'' shear layer with
, and . The contour levels range from -36 to 36
(cf. Figure 3c in Ref. [5]).
(b) A ``thin'' shear layer with , and .
The contour levels range from -70 to 70
(cf. Figure 9b in Ref. [5]).
*

Mon Dec 8 17:34:34 PST 1997