On monotonic stability of elliptic pipe flow

Abstract

It was previously shown that the linear stability of fluid flows in pipes significantly depends on their cross-sectional aspect ratio. The linear stability analysis allows for judging the asymptotic behavior of the basic flow disturbances; however, it says nothing about their possible transient growth, which can cause the so-called subcritical laminar–turbulent transition. The lower limit of the Reynolds numbers at which the growth of the kinetic energy of disturbances is possible is the energy critical Reynolds number. In the present work, for the Poiseuille flow in a pipe of axially uniform elliptic cross-section the dependence of the energy critical Reynolds number on the pipe aspect ratio A is computed for 1≤A≤5, based on the energy stability method. The dependence is non-monotonic under scaling providing the same flow rates at the same Reynolds numbers. In particular, at A≈2.3 the critical Reynolds number reaches its maximum, but then monotonically decreases with increasing A, becoming less than in a circular pipe, and tends to the energy critical Reynolds number of the plane Poiseuille flow under an appropriate scaling as A→∞. A qualitative explanation of the obtained dependence is proposed based on the analysis of the critical disturbances corresponding to the critical Reynolds number and their kinetic energy balance. The obtained dependence suggests that the change in the pipe aspect ratio may be a promising tool for the passive control of the laminar–turbulent transition in pipe flows and can be used together with other known approaches employed for this purpose.