Numerical Solution of the Azimuthal-Invariant Thin-Layer Navier-Stokes Equations

NUMERICAL solutions have been obtained for a twodimensional azimuthal- (or circumferentially) invariant form of the thin-layer Navier-Stokes equations. The governing equations which have been developed are generalized over the usual two-dimensional and axisymmetric formulation by allowing nonzero velocity components in the invariant direction. The equation formulation along with the solution method is described, and results for spinning and nonspinning bodies are presented. Contents The three-dimensional flow field equations are frequently simplified for flowfields which are invariant in one coordinate direction. In the usual axisymmetric approximation, the azimuthal velocity is assumed to be zero, and the momentum equation in that direction can be eliminated. Thus, only four equations are required to be solved for four unknowns. However, for a variety of interesting flowfields, the velocity component in the invariant direction (here taken as TJ) is not zero although the governing equations are still twodimensional. Examples include viscous flow about an infinitely swept wing, the viscous flow about a spinning axisymmetric body at 0-deg angle of attack, and axisymmetric swirl flows. Each of these flows can be solved as a twodimensional problem although all three momentum equations have to be retained, and source terms replace the derivative of the flux terms in the rj-direction. Azimuthal-invariant equations are obtained from the threedimensional equations1 by making use of two restrictions: 1) all body geometries are of axisymmetric types and 2) the state variables and the contravariant velocities do not vary in the azimuthal direction. Here, TJ is used for the azimuthal coordinate, and the terms azimuthal and rj-invariant will be used interchangeably. A sketch of a typical axisymmetric body is shown in Fig. la. In order to determine the circumferential variation of typical flow and geometric parameters, we first establish correspondence between the