This paper presents the results of a study of the application of the Preconditioned Conjugate Gradient algorithm to some equations arising in three-dimensional turbulent flow prediction. The merits of several of the standard preconditioning are discussed, and some of the difficulties which are encountered when trying to solve such problems efficiently on a vector computer are described. The results show that the coefficient matrices which arise have important differences from those which derive from many standard test problems. Efficient solution of the problems is still possible, but the correct choice of preconditioning is important. In particular, the application of truccated power series expansions to the preconditioning phase is shown to produce substantial gains.