Darmaev T.G.  

Numerical research of bifurcations of periodic modes of boundary layer flows

There is a lot of works devoted to research evolution of hydrodynamic disturbances, a great number of experiments carried out, a lot of various approaches created, but the transition mechanism from a laminar to turbulent flow is not fully investigated yet and still causes interest of a great number of researchers. Linear theory of hydrodynamic stability is well developed and is reduced to eigenvalue problem for Navier-Stokes equations linearized over small perturbations. The main goals of numerical research are the search of neutral stability and constant growth curves, calculation of eigenvalues and functions for the given pair of positive values of wavenumber and Reynolds number.
The first who engaged in attempt of the impact assessment of non-linearity was Landay L.D. He showed that non-linearity is able as well to stable growing perturbations by producing new stable flow regime as cause an increase of linearly stable perturbations. Nowadays such directions of nonlinear perturbations research are in progress as weakly nonlinear theory, parabolized equations, direct numerical simulation, basic flow contour perturbations, interaction of perturbations.
In this paper the method of invariant finite-dimensional projection of the Navier-Stokes equations by B. Yu. Skobelev is applied to plane-parallel flow of viscous incompressible liquid over flat semi-infinite plate. Herewith the initial-value problem for perturbations of the main flow is reduced to finite-dimensional system of ordinary differential equations, whose right-hand side is found from recurrence system of linear boundary value problems. Sufficient advantage of the method of invariant projection is assurance of valid description of asymptotic behavior of solutions and properly taking into account discrete and continuous spectra of the perturbations. Universal algorithm proposed in this paper allows numerically determining the amplitude surfaces of stable and unstable regimes and the points of tangent bifurcation of periodic regimes for arbitrary frequencies and Reynolds numbers. The comparison of numerical results with experiments is carried out.