The energy transfer mechanism of a perturbed solid-body rotation flow in a rotating pipe

Three-dimensional direct numerical simulations of a solid-body rotation superposed on a uniform axial flow entering a rotating constant-area pipe of finite length are presented. Steady in time profiles of the radial, axial, and circumferential velocities are imposed at the pipe inlet. Con-vective boundary conditions are imposed at the pipe outlet. The Wang and Rusak (Phys. Fluids 8:1007–1016, 1996. doi:10.1063/1.86882) axisymmetric instability mechanism is retrieved at certain operational conditions in terms of incoming flow swirl levels and the Reynolds number. How-ever, at other operational conditions there exists a dominant, three-dimensional spiral type of instability mode that is con-sistent with the linear stability theory of Wang et al. (J. Fluid Mech. 797:284–321, 2016). The growth of this mode leads to a spiral type of flow roll-up that subsequently nonlinearly sat-urates on a large amplitude rotating spiral wave. The energy transfer mechanism between the bulk of the flow and the perturbations is studied by the Reynolds-Orr equation. The production or loss of the perturbation kinetic energy is com-bined of three components:the viscous loss, the convective loss at the pipe outlet, and the gain of energy at the out-let through the work done by the pressure perturbation. Theenergy transfer in the nonlinear stage is shown to be a nat-ural extension of the linear stage with a nonlinear saturated process.