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Spectral methods for the Navier-Stokes equations with one infinite and two periodic directions. (English) Zbl 0726.76074
Summary: Two numerical methods were designed to solve the time-dependent, three- dimensional, incompressible Navier-Stokes equations in boundary layers (method A, semi-infinite domain) and mixing layers or wakes (method B, fully-infinite domain). Their originality lies in the use of rapidly- decaying spectral basis functions to approximate the vertical dependence of the solutions, combined with one (method A) or two (method B) slowly- decaying “extra functions” for each wave-vector that exactly represent the irrotational componant of the solution at large distances. Both methods eliminate the pressure term as part of the formulation, thus avoiding fractional-step time integration. They yield rapid convergence and are free of spurious modes in the Orr-Sommerfeld spectra. They are also efficient, although the operation count is of order $$N^ 2$$ (N is the number of modes in the infinite direction). These methods have been used for extensive direct numerical simulation of transition and turbulence. A new time-integration scheme, with low storage requirements and good stability properties, is also described.

##### MSC:
 76M25 Other numerical methods (fluid mechanics) (MSC2010) 76D10 Boundary-layer theory, separation and reattachment, higher-order effects 76D25 Wakes and jets
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