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To: 10. Conclusions |
The fact that the vorticity redistribution method computes the individual redistribution fractions makes it easier to obtain certain desirable properties. In particular, it allows the conservation laws to be satisfied exactly. Even when resolution is very poor, such as initially for a diffusing point vortex, at least no false circulation or linear and angular momentum will be created.
A random walk procedure conserves only circulation exactly. While corrections are possible that conserve the center of vorticity [41,155], subgroups of vortices can still perform an appreciable net motion without a physical mechanism causing it. The particle scheme proposed by Fishelov [78] is not conservative unless a corrected rule is used to perform the integrations in her convolution, but the potential high-order of accuracy may make this unimportant. The particle methods do not satisfy conservation of center of vorticity exactly when the particle distribution becomes nonuniform.
Another advantage of the redistribution equations is that they tend to localize the errors in velocity that result from the numerical diffusion. To be precise, the redistribution equations ensure that the leading order decay terms of the error in velocity vanish exactly.
Another desirable property is the positivity of the redistribution fractions; it assures that regardless of numerical inaccuracy, no false reversed vorticity is created. Whether particle methods satisfy this constraint depends on factors such as the choice of smoothing function and of the time discretization. For example, Fishelov's method does not satisfy positivity and can generate reversed vorticity, although the amount should be very small if the vorticity distribution is sufficiently smooth. The integral constraints given by Degond and Mas-Gallic [72] show that third-order accurate particle schemes do not satisfy positivity.
It may seem surprising that for the Stokes equations the vorticity redistribution method can achieve any order of accuracy with positive fractions, while the particle methods cannot. The reason is that the vorticity redistribution method discretizes convection for a finite time-step, rather than an infinitesimal one. In particular, if we let the time-step tend to zero in the vorticity redistribution method, while keeping the location of the vortices fixed, the scaled spacing between the vortices would tend to infinity. In Appendix A it is shown that the redistribution equations do not have a positive third-order solution if the scaled spacing is more than a finite value. Thus high order of accuracy can only be achieved for a finite time-step.
For the Navier-Stokes equations a finite time-step is a mixed blessing; the splitting into viscous and inviscid steps should not introduce an error larger than the spatial order of accuracy. Note however that the time-step is an order smaller than the spatial resolution. In an unbounded domain, Strang splitting with reversal of the order of the steps [19] would be fourth-order accurate with respect to space.
In the next chapter, we will present the conclusions about our work and also suggest some areas for further work on our vorticity redistribution method.