Technical Papers and Presentations

The present author’s specialty is Aerodynamics, a field in which problems in unbounded fluid domains are common. The COMSOL Multiphysics® simulation software's support of Infinite Elements is attractive as a tool for addressing such problems. Be that as it may, neither the COMSOL Multiphysics^® core package nor the CFD Module supports the use of Infinite Elements. On the bright side, there are many practical situations in which it makes sense to suppose that the fluid velocity is a solenoidal vector field, the gravitational field is uniform in space and constant in time, the effects of viscosity are unimportant and the curl of the velocity field—i.e. the vorticity—is nonzero only in a bounded subdomain. In such situations there is an analogy between the equations that relate, on the one hand, the magnetic flux density vector, the magnetic vector potential, and the electric current density and, on the other hand, the equations that relate the fluid velocity vector, the velocity vector potential, and the vorticity, respectively. Sir Horace Lamb solved a problem in classical Hydrodynamics in which the motion is independent of time, vorticity is nonzero only in the interior of an impermeable cylinder, the fluid exterior to the cylinder is in irrotational motion (i.e. has zero vorticity), and in which the exterior motion tends asymptotically to a uniform stream. The present simulation models the exterior fluid motion by simulating its electromagnetic analog, specifically by means of the Magnetic Fields physics interface in the AC/DC Module. The magnetic potential in that subdomain satisfies Laplace’s equation. In the interior, by contrast, the relevant partial differential equation is the Helmholtz equation. One may reduce the latter equation to a pair of ordinary differential equations by the method of separation of variables. The ordinary differential equation for the corresponding radial function is homogeneous and subject to homogeneous boundary conditions at two points. A nontrivial solution is possible only for a special value of an adjustable parameter, i.e. an eigenvalue. The present simulation solves this two-boundary value problem by means of a one-dimensional simulation of the Helmholtz equation in combination with the Optimization Module to determine the eigenvalue. With modest mesh refinement the COMSOL Multiphysics^® results replicates those of Lamb’s analytic solution accurately.