Application of the Discrete Fourier Transform to Study the Dynamics of Wave Packets

A method for solving equations that describe the dynamics of wave packets of the Tollmien–Schlichting waves in the boundary layer is proposed. The method of splitting the initial problem into the linear and nonlinear parts at each time step is used. The linear part is resolved by using an equation for spectral components of the wave packet with a subsequent Fourier transform from the space of wavenumbers to the physical space. A system of ordinary differential equations is solved in the physical space. The Fourier transform is performed by means of the library procedure of the fast Fourier transform. As examples, the problems solved were the linear dynamics of the wave packet concentrated in the vicinity of the instability region (i.e., a set of wave vectors in the space of wavenumbers for which the imaginary part of the eigenfrequency of the Tollmien–Schlichting waves is positive) and the nonlinear dynamics of the wave packet overlapping the instability region.