Stability analysis of velocity profiles in water-hammer flows

This paper performs linear stability analysis of base flow velocity profile s for laminar and turbulent water-hammer flows. These base flow velocity pr ofiles are determined analytically, where the transient is generated by an instantaneous reduction in flow rate at the downstream end of a simple pipe system. The presence of inflection points in the base flow velocity profil e and the large velocity gradient near the pipe wall are the sources of flo w instability. The main parameters that govern the stability behavior of tr ansient flows are the Reynolds number and dimensionless timescale. The stab ility of the base flow velocity profiles with respect to axisymmetric and a symmetric modes is studied and its results are plotted in the Reynolds numb er/timescale parameter space. It is found that the asymmetric mode with azi muthal wave number 1 is the least stable. In addition, the results indicate that the decrease of the velocity gradient at the inflection point with ti me is a stabilizing mechanism whereas the migration of the inflection point from the pipe wall with time is a destabilizing mechanism. Moreover, it is shown that a higher reduction in flow rate, which results in a larger velo city gradient at the inflection point, promotes flow instability. Furthermo re, it is found that the stability results of the laminar and the turbulent velocity profiles are consistent with published experimental data and succ essfully explain controversial conclusions in the literature. The consisten cy between stability analysis and experiments provide further confirmation that (I) water-hammer flows can become unstable; (2) the instability is asy mmetric; (3) instabilities develop in a short (water-hammer) timescale; and (4) the Reynolds number and the wave timescale are important in the charac terization of the stability of water-hammer flows. Physically, flow instabi lities change the structure and strength of the turbulence in a pipe, resul t in strong flow asymmetry, and induce significant fluctuations in wall she ar stress. These effects of flow instability are not represented in existin g water-hammer models.