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Fletcher, C. A. J.; Fleet, R. W.
doi: 10.1002/fld.1650040502pmid: N/A
The Dorodnitsyn boundary later formulation is given a finite element interpretation and found to generate very accurate and economical solutions when combined with an implicit, non‐iterative marching scheme in the downstream direction. The algorithm is of order (Δ2u, Δx) whether linear or quadratic elements are used across the boundary layer. Solutions are compared with a Dorodnitsyn spectral formulation and a conventional finite difference formulation for three Falkner‐Skan pressure gradient cases and the flow over a circular cylinder. With quadratic elements the Dorodnitsyn finite element formulation is approximately five times more efficient than the conventional finite difference formulation.
Srinivas, K.; Fletcher, C. A. J.
doi: 10.1002/fld.1650040503pmid: N/A
The time‐split finite element method is extended to compute laminar and turbulent flows with and without separation. The examples considered are the flows past trailing edges of a flat plate and a backward‐facing step. Eddy viscosity models are used to represent effects of turbulence. It is found that the time‐split method produces results in agreement with previous experimental and computational results. The eddy viscosity models employed are found to give accurate predictions in all regions of flow except downstream of reattachment.
Tanguy, P.; Fortin, M.; Choplin, L.
doi: 10.1002/fld.1650040504pmid: N/A
A finite element simulation of the dip coating process based on a discretization of the continuum with discontinuous pressure elements is presented. The algorithm computes the flow field from natural boundary conditions while an extra condition provided by the existence of free surface is employed to displace the meniscus location towards the actual position. The process is iterative and uses a pseudo‐time stepping technique coupled to a cubic spline fitting of the free surface. Numerical predictions exhibit good agreement with experimental data for Newtonian fluids in the case of flat plate dip coating as well as in the case of wire dip coating.
Tanguy, P.; Fortin, M.; Choplin, L.
doi: 10.1002/fld.1650040505pmid: N/A
We apply in this paper the augmented Lagrangian method to the study of various non‐Newtonian fluid flow problems, and in particular the dip coating process. We only present in this second part the treatment specific to the non‐linearities involved in the constitutive equations, the first part having largely been concerned with the general description of the approximation used. Two rheological models illustrating different rheological behaviours are used to simulate dip coating process: the Carreau‐A model for shear‐thinning properties of the viscosity and a truncated second‐order model for a Newtonian behaviour in viscosity with elastic properties. Numerical predictions show a very good agreement with experimental data for the second‐order model. The discrepancy observed in the other case can be explained qualitatively by the elastic properties exhibited by the shear‐thinning fluids used: this elasticity is not taken into account in the Carreau‐A model.
Vorozhtsov, E. V.; Yanenko, N. N.
doi: 10.1002/fld.1650040506pmid: N/A
We consider a problem on shock wave localization in the numerical solution of one‐dimensional unsteady problems of gas dynamics in Eulerian variables obtained on the basis of finite difference shock‐capturing schemes. An optimization method for strong discontinuity localization proposed previously by Miranker and Pironneau is investigated by means of methods of classical variational calculus. This method may be difficult to implement when the entropy condition is included in the formulation of Miranker and Pironneau's optimization problem as an active constraint. In this connection we suggest an alternative optimization problem using artificial viscosity in the variational principle. It is shown theoretically that the application of such a variational principle yields a trajectory which coincides with the true discontinuity trajectory in the case of a shock wave moving at a constant speed. On the basis of this modification one more algorithm is proposed which reduces the shock localization problem to a problem of minimization of a univariate function. Numerical tests corroborate completely the theoretical conclusions. In particular, a higher shock localization accuracy is obtained on the basis of the proposed algorithms as compared to the original Miranker‐Pironneau method.
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