Computation of three-dimensional Stokes flow over complicated surfaces (3d riblets) using a boundary-independent grid and local corrections

P. Luchini
Dipartimento di Ingegneria Aerospaziale - Politecnico di Milano
Via Golgi 40 - 20133 Milano - Italy
and
A. Pozzi
Dipartimento di Progettazione Aeronautica - Università di Napoli
P.le Tecchio - 80125 Napoli - Italy

The research of new possibilities for improving the drag-reduction characteristics of riblet surfaces induced Bechert's group in Berlin, a couple of years ago, to experiment with 3d riblet geometries, in which the riblets are periodically interrupted in an alternating pattern. The rationale was that cutting the riblets might decrease the nonlinear sloshing and allow the linear viscous drag reduction mechanism to persist up to larger riblet sizes. As soon as we were informed of these experiments, we started an effort to extend the numerical calculation of viscous protrusion heights to 3d geometries. The preliminary results of this effort, which we presented at the last EDRM in Ravello, gave the qualitative indication that an improvement (although probably a modest one) could be hoped for by changing slightly the fill-to-empty ratio of the riblets used in the experiment. However, our 3d Stokes-flow program was not at the time capable to operate with the exact geometry used in the experiments, but only with a qualitatively similar one.
The subsequent trimming and perfecting of our computation led to a general and quite unconventional numerical method for studying 3d viscous flow over complicated surfaces which can be applied to a much wider range of problems. (In particular, since this is not a boundary-integral but a volume finite-differences method, extensions to the full Navier-Stokes problem are possible.)
Several decisions were involved in the set-up of our method. First, we renounced to the boundary-integral formulation that we had adopted for 2d riblets, because in three dimensions the (iterative or otherwise) inversion of a full matrix of size n2xn2 (if n is the typical number of discretization points in each direction) leads to a much larger computational work than the iterative inversion of the very sparse n3xn3 matrix resulting from a finite-difference formulation. Second, whereas numerical calculations of flow over 2d riblet geometries have in the past been conducted by others on orthogonal curvilinear boundary-fitted grids, the complications arising from the use of a boundary-fitted grid in three dimensions (where it cannot in general be made orthogonal) induced us to opt for a boundary-independent square grid. Much of our development effort was therefore devoted to suitably interpolated boundary conditions. Third, based on our previous experience with the multigrid method (P. Luchini, J. Comp. Phys. 92, 349, 1991; P. Luchini, Int. J. Num. Meth. Fluids 12, 491, 1991; P. Luchini & A. D'Alascio, Int. J. Num. Meth. Fluids 18, 489, 1994), we decided to adapt this, very fast, solution method to the Stokes problem. Last, in order to circumvent the difficulties classically associated with the discretization of pressure and velocities on co-located grid points we opted for the magnetization formulation (Luchini, AIAA J. 29, 474, 1991) in which velocity and pressure are replaced by a new vector (which bears some mathematical resemblance to the magnetization field of magnetostatics) and a new scalar (a potential).
We shall, in this meeting, present our method, taking 3d riblets as an example. We shall also give results containing the longitudinal and transverse protrusion heights for several 3d riblet profiles, including the precise geometry used in Bechert's experiments (which he kindly disclosed to us in advance).