An efficient method of solving the Navier-Stokes equations for flow control

A new method of solving the Navier{Stokes equations e ciently by reducing their number of modes is proposed in the present paper. It is based on the Karhunen{Lo eve decomposition which is a technique of obtaining empirical eigenfunctions from the experimental or numerical data of a system. Employing these empirical eigenfunctions as basis functions of a Galerkin procedure, one can a priori limit the function space considered to the smallest linear subspace that is su cient to describe the observed phenomena, and consequently reduce the Navier{Stokes equation de ned on a complicated geometry to a set of ordinary di erential equations with a minimum degree of freedom. The present algorithm is well suited for the problems of ow control or optimization, where one has to compute the ow eld repeatedly using the Navier{Stokes equation but one can also estimate the approximate solution space of the ow eld based on the range of control variables. The low-dimensional dynamic model of viscous uid ow derived by the present method is shown to produce accurate ow elds at a drastically reduced computational cost when compared with the nite di erence solution of the Navier{Stokes equation