Laminar–turbulent transition in pipe flow for Newtonian and non-Newtonian fluids

Abstract

A cylindrical pipe facility with a length of 32 m and a diameter of 40 mm has been designed. The natural transition Reynolds number, i.e. the Reynolds number at which transition occurs as a result of non-forced, natural disturbances, is approximately 60 000. In this facility we have studied the stability of cylindrical pipe flow to imposed disturbances. The disturbance consists of periodic suction and injection of fluid from a slit over the whole circumference in the pipe wall. The injection and suction are equal in magnitude and each distributed over half the circumference so that the disturbance is divergence free. The amplitude and frequency can be varied over a wide range.

First, we consider a Newtonian fluid, water in our case. From the observations we compute the critical disturbance velocity, which is the smallest disturbance at a given Reynolds number for which transition occurs. For large wavenumbers, i.e. large frequencies, the dimensionless critical disturbance velocity scales according to Re⁻¹, while for small wavenumbers, i.e. small frequencies, it scales as Re^−2/3. The latter is in agreement with weak nonlinear stability theory. For Reynolds numbers above 30 000 multiple transition points are found which means that increasing the disturbance velocity at constant dimensionless wavenumber leads to the following course of events. First, the flow changes from laminar to turbulent at the critical disturbance velocity; subsequently at a higher value of the disturbance it returns back to laminar and at still larger disturbance velocities the flow again becomes turbulent.

Secondly, we have carried out stability measurements for (non-Newtonian) dilute polymer solutions. The results show that the polymers reduce in general the natural transition Reynolds number. The cause of this reduction remains unclear, but a possible explanation may be related to a destabilizing effect of the elasticity on the developing boundary layers in the entry region of the flow. At the same time the polymers have a stabilizing effect with respect to the forced disturbances, namely the critical disturbance velocity for the polymer solutions is larger than for water. The stabilization is stronger for fresh polymer solutions and it is also larger when the polymers adopt a more extended conformation. A delay in transition has been only found for extended fresh polymers where delay means an increase of the critical Reynolds number, i.e. the number below which the flow remains laminar at any imposed disturbance.