Nonlinear evolution of disturbances entrained in a circular pipe

The nonlinear evolution of free-stream vortical disturbances entrained in the entrance region of a circular pipe is investigated using asymptotic and numerical methods. Attention is focused on the long-wavelength disturbances which induce streamwise elongated streaks. A pair of vortical modes with opposite azimuthal wavenumbers is used to model the free-stream disturbances and their amplitude is assumed to be intense enough for nonlinear interactions to occur. The formation and evolution of the streaks are described by the nonlinear unsteady boundary-region equations written here in cylindrical coordinates for the first time. Supplemented by appropriate initial and boundary conditions, this initial-boundary-value problem is solved numerically by a marching procedure in the streamwise direction. Numerical results show the stabilizing effect of nonlinearity on the intense algebraic growth of the streaky structures. For a high free-stream turbulence level, all the disturbances attenuate sufficiently downstream with the exception of a pulsating mode. A parametric study is carried out to evince the effect of Reynolds number, streamwise and azimuthal wavelength, and radial characteristic scale on the nonlinear evolution. Qualitative agreement between our numerical results and the limited experimental data is obtained.