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Perturbation dynamics in viscous channel flows. (English) Zbl 0946.76018
The authors investigate the effect of small perturbations on plane Poiseuille and plane Couette flows in an incompressible viscous fluid. The technique consists in finding the Fourier transform of the governing disturbance equations in the streamwise and spanwise directions only, and subsequently solving the resulting partial differential equations numerically by the method of lines. Instead of assuming travelling wave normal modes as solutions, the approach offers another means which uses specified arbitrary initial input without resorting to eigenfunction expansions. The full temporal behavior, including both early-time transients and the long-time asymptotics, can be obtained for any initial small-amplitude three-dimensional disturbance. Also with the help of a proposed optimization scheme which utilizes the orthogonal Fourier series, all the known stability results for such flows can be reproduced. It has been shown that the transient growth of the perturbation energy density is very sensitive to the presence of an initial normal vorticity perturbation. Finally, direct numerical simulation of the spatial problem has been carried out and compared with the linear temporal theory. The approach used here can be applied to other classes of problems where only a finite number of normal modes exist.

76E05 Parallel shear flows in hydrodynamic stability
76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
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