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The velocity formulation

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, tex2html_wrap_inline5813, so that tex2html_wrap_inline5815. The equations of motion for a Newtonian fluid in conservation form are
where p is the pressure, tex2html_wrap_inline5819 is the kinematic viscosity, and subscripts denote partial derivatives. The functions tex2html_wrap_inline5821 and tex2html_wrap_inline5823 are components of the fluxes of the conserved quantities u and v.

The computational grid consists of rectangular cells of sizes tex2html_wrap_inline5377 and tex2html_wrap_inline5831; at time level tex2html_wrap_inline5833, these cells, tex2html_wrap_inline5835, are centered at tex2html_wrap_inline5837. Starting with the corresponding cell averages, tex2html_wrap_inline5839, we first reconstruct a piecewise linear polynomial approximation which recovers the point values of the velocity field, tex2html_wrap_inline5841. 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 tex2html_wrap_inline5843.

An exact computation yields
The incompressible fluxes, e.g., tex2html_wrap_inline5845, 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 tex2html_wrap_inline5847 solving the Poisson equation

Then, the pressure gradient at tex2html_wrap_inline5707 is being updated,

and finally, it is used to evaluate the divergence-free velocity field, tex2html_wrap_inline5851

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 tex2html_wrap_inline5853 with tex2html_wrap_inline5855, and a viscous ``thin'' shear layer problem (with tex2html_wrap_inline5857), corresponding to tex2html_wrap_inline5853 with tex2html_wrap_inline5861. 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 tex2html_wrap_inline5865, with tex2html_wrap_inline5867.
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, tex2html_wrap_inline5869, at t=1.2 with initial tex2html_wrap_inline5873, using a tex2html_wrap_inline5875 grid. (a) A ``thick'' shear layer with tex2html_wrap_inline5855, and tex2html_wrap_inline5879. The contour levels range from -36 to 36 (cf. Figure 3c in Ref. [5]). (b) A ``thin'' shear layer with tex2html_wrap_inline5861, and tex2html_wrap_inline5857. The contour levels range from -70 to 70 (cf. Figure 9b in Ref. [5]).

next up previous contents
Next: References Up: Incompressible Euler equations Previous: The vorticity formulation

Eitan Tadmor
Mon Dec 8 17:34:34 PST 1997