  We introduce a novel method for the implementation of shape optimziation in
fluid dynamics applications, where we propose to use the shape derivative to
determine deformation fields with the help of the $p-$ Laplacian for $p > 2$.
This approach is closely related to the computation of steepest descent
directions of the shape functional in the $W^{1,\infty}-$ topology. Our
approach is demonstrated for shape optimization related to drag-minimal free
floating bodies. The method is validated against existing approaches with
respect to convergence of the optimization algorithm, the obtained shape, and
regarding the quality of the computational grid after large deformations. Our
numerical results strongly indicate that shape optimization related to the
$W^{1,\infty}$-topology -- though numerically more demanding -- seems to be
superior over the classical approaches invoking Hilbert space methods,
concerning the convergence, the obtained shapes and the mesh quality after
large deformations, in particular when the optimal shape features sharp
corners.
