# I³MS - Kirchhart Seminar

Location: AICES Seminar Room 115, 1st floor, Schinkelstr. 2, 52062 Aachen

## Dr. Matthias Kirchhart - Vortex Methods for Incompressible Flows

AICES Graduate School, RWTH Aachen University

### Abstract

Vortex methods are numerical schemes for solving the incompressible
Navier–Stokes equations. These equations accurately describe the motion
of both gases and liquids as we encounter them in everyday life, i.e.,
at velocities far below the speed of sound and not subject to extreme
temperatures or pressures. It is hard to overestimate their importance
in engineering applications, where they can for example be used to
minimise air-resistance and thereby fuel consumption of cars. However,
current numerical schemes for these equations face severe problems when
applied to turbulent flows: stringent time-step constraints,
instabilities, or the introduction of significant amounts of artificial,
spurious viscosity make their application infeasible or render the
results unusable.

Vortex methods, on the other hand, are particle methods that are based
on the vorticity formulation of the Navier–Stokes equations. This
formulation comes with two main benefits: the pressure variable is
eliminated and the system consists of separate dynamic and kinematic
parts, which can be treated independently with semi-analytical schemes.
The dynamic part is discretised using particles, which are then
convected with the flow. This natural treatment of convection renders
the method virtually free of artificial viscosity. The kinematic part of
the equations is solved using a solver based on the Biot–Savart law,
which guarantees incompressibility in the strong, point-wise sense. In
addition, in the two-dimensional case, the resulting schemes can be
shown to also conserve circulation, linear momentum, angular momentum,
and energy. These properties make vortex methods an interesting
alternative to current, mesh-based alternatives.

In this talk we will first describe a simple vortex method in the
unbounded, two-dimensional setting to illustrate the intuition of
vortex methods. We then present basic results from their analysis,
before moving on to discuss some of the problems vortex methods are
facing in three-dimensional bounded domains. We present recent research
results on one of these problems and conclude with an outlook to further
research opportunities.

References
[1] G.-H. Cottet and P. D. Koumoutsakos. Vortex Methods. Theory and
Practice. Cambridge University Press, 2000. ISBN: 0521621860.

[2] A. J. Majda and A. L. Bertozzi. Vorticity and Incompressible Flow.
Cambridge University Press, Nov. 2001. ISBN: 0521630576.

[3] M. Kirchhart and S. Obi. ‘A Smooth Partition of Unity Finite Element
Method for Vortex Particle Regularization’. In: SIAM Journal on
Scientific Computing 39.5 (Oct. 2017), pp. A2345–A2364. ISSN: 1064–8275.
DOI: 10.1137/17M1116258.

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