Transient growth of disturbances in laminar pipe flow

Abstract: Transient growth of disturbances is studied theoretically in pipe Poiseuille flow and experimentally both in pipe and plane Poiseuille flow. The theoretical results, based on the initial value problem, show that a large transient amplification occurs for small, angular dependent disturbances in pipe Poiseuille flow although all modes are damped. The largest growth is obtained for disturbances with zero streamwise wave number and with an azimuthal periodicity of one. Only the streamwise disturbance component is then amplified and the energy density growth is proportional to the Reynolds number squared. For disturbances of finite axial wave number, also the radial and azimuthal disturbance components are amplified from their initial values but the growth is still dominated by the streamwise disturbance. When the azimuthal wave number increases the disturbance peak occurs at shorter times and the amplification decreases. However, for small times, larger azimuthal wave numbers dominate the growth. Asymptotic results for the most amplified disturbances show that two classes of modes can be distinguished and it is necessary to include modes from both classes to get amplification. The asymptotically derived propagation speeds of the two classes are in the ranges <2/3 times the centerline velocity (Ucl) and <0.818Ucl, respectively. If the nonlinear interaction of axially independent disturbances is considered, the results indicate that the amplification of the most linearly amplified disturbance is reduced. Laser Doppler measurements of the streamwise disturbance velocity due to radially induced initial disturbances of azimuthal periodicities (n) of one and five have also been conducted. The results show that the disturbance velocity in fact exhibits a transient growth. The disturbance with n=1 gives a slowly varying peak-value while the peak owing to an initial disturbance with n=5 shows a more rapid amplification to a maximum before it decays. The propagation speeds of the disturbance peaks are in the range theoretically expected for transiently amplified disturbances. By hot-wire measurements of the streamwise disturbance velocity due to two radially induced jet-like disturbances, the spatial structure of an evolving disturbance is investigated in detail. Especially the positive disturbance velocity exhibits an amplification followed by a decay. The disturbance also spreads and becomes streak-like and elongated in the streamwise direction. The spread implies that an integrated disturbance quantity grows to a peak and eventually decays. The propagation speed and the radial location of the disturbances agree with the theoretical findings for algebraically growing disturbances. In plane Poiseuille flow the symmetry properties of an evolving disturbance are investigated by means of hot-wire anemometry. The initial streamwise disturbance is designed to be symmetric with respect to the normal direction. Despite this, an antisymmetric structure develops in the region where the main amplification occurs as indicated by the theory.