Optimal growth of small disturbances in pipe Poiseuille flow

A theoretical study is made of initial algebraic growth for small angular‐dependent disturbances in pipe Poiseuille flow. The analysis is based on the homogeneous equation for the pressure for which the eigenvalue problem is solved numerically. In the limit of small streamwise wave numbers asymptotic results for the eigenvalues are derived. On the basis of the modes of the system, which are all damped, the initial value problem is considered and in particular the largest possible growth of the disturbance energy density is determined following the ideas of Butler and Farrell [Phys. Fluids A 4, 1637 (1992)]. The results show that a large amplification of the disturbance energy is possible. The largest amplification is obtained for disturbances with a small streamwise wave number and with an azimuthal wave number of one. The energy growth is then only due to the growth of the streamwise disturbance component. However, for disturbances of shorter wavelength, the energy growth is also substantial and not only concentrated to the streamwise velocity component. The wall shear corresponding to disturbances with the largest energy growth also shows a large amplification and the dependence of wave numbers and the Reynolds number is the same as for the energy. However, the wall pressure of a long wavelength disturbance of the largest growth just decays from its initial value, but for disturbances of shorter wavelength, it is also amplified.