Numer. Math. 48, 561-589 (1986) Numerische Mathematik Springer-Verlag 1986 Stream Vectors in Three Dimensional Aerodynamics Fadi E1 Dabaghi and Olivier Pironneau* INRIA, Domaine de Voluceau, Rocquencourt, F-78150 Le Chesnay, France Summary. This work deals with the decomposition of a vector field u into u= V x ~k +V~b. Non homogeneous boundary conditions on ~b or ~b are in- vestigated; applications to the computation of inviscid flows are given; finally a conforming finite element implementation is studied and tested. Subject Classifications: AMS(MOS): 65N30; CR: G1.8. Table of Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 1. The Continuous Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 - Homogeneous Boundary Conditions on ~ . . . . . . . . . . . . . . . . . . . . . . . 564 - Non Homogeneous Boundary Conditions on ~, . . . . . . . . . . . . . . . . . . . . . 567 2. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 - Wings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 - Nozzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 - Entropy Corrections for the Transonic Equation . . . . . . . . . . . . . . . . . . . . 571 3. Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 - A Conforming Finite Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 - Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 - Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 O. Introduction It is a well known result of analysis (De Rham [1]) that any solenoidal vector field u (V.u=0) is the curl of a stream vector ~k. This property is used extensively in two dimensional fluid mechanics; there ~k being perpendicular to the plane of fluid motion, on simply connected domains, it is uniquely defined from u and while u has two non trivial components, r has only one. * Also: University of Paris 13