A spectral /B -spline method for the Navier -Stokes equations in unbounded cylindrical domains

In this thesis, a new numerical method to solve the incompressible, unsteady Navier-Stokes equations in unbounded cylindrical domains is presented. The method comes as a novel application of Leonard's divergence-free vector expansions approach, and therefore possesses the following characteristics: (i) exact treatment of the continuity constraint; (ii) complete elimination of the pressure variable; (iii) implicit time integration of the diffusive term at no extra cost; and (iv) reduction of the number of (velocity) unknowns from three to two. Another important feature of the method, that indeed represents the originality of the present formulation, is the introduction of mapped B-spline piecewise polynomials for the discretization of the semi-infinite radial direction.

More specifically, the spatial discretization is constructed from a combination of Fourier series, for both the longitudinal (physical periodicity of temporal evolving flows) and azimuthal (geometrical periodicity) directions, and of B-splines on a mapped unitary radial domain. The particular choice of mapping function allows for an exact representation of algebraically decaying functions, up to some finite order. Besides the imposition of proper decaying conditions in the far field, complete (finite order) regularity conditions are also imposed at the center point r = 0. These mixed spectral/B-spline expansions, used to form the divergence-free vector basis functions, yield an efficient compromise between the high uncoupling associated with the orthogonality of Fourier series and the resolution positioning flexibility that is characteristic of local methods. The local character of the B-splines furthermore allows for a radial variation of the azimuthal truncation. The resulting vector basis functions are applied to in Galerkin type weighted residual formulation that transforms the complete 3-D problem into a set of small 1-D radial ODE's that are marched in time. For that latter task, the quasi-third order, mixed explicit/implicit scheme proposed by Spalart et al. (J. Comp. Phys., 96, 297, 1991) is used. The Galerkin formulation also serves for the development of an eigenvalue solver for linear stability problems. Finally, a wall-bounded version of this method, equivalent to the one presented by Loulou et al. (NASA TM-110436, 1997), is also produced in this work.

The validation of the different Navier-Stokes and eigenvalue solvers is achieved by comparing linear stability results, and nonlinear dynamics predictions with other benchmark data. The particular flow problems considered are related to the stability of a trailing line vortex, and the stability and nonlinear dynamical evolution of a special class of zero circulation vortex that leads to the formation of a triangular vortex. On one hand, comparisons made between the unbounded B-spline formulation and the wall-bounded version of the method have shown only a marginal advantage of the former method over the latter. On the other hand, comparisons made with the data obtained by purely global expansions approximation methods prove the present spectral/B-spline method to be an advantageous alternative to these global methods for the computation of unbounded flow problems having an intrinsic axial symmetry.