Transient instability of the flow induced by an impulsively started rotating cylinder

The onset of instability induced by transient momentum diffusion in a boundary layer over an impulsively started rotating cylinder is Examined. Kirchner and Chen (J. Fluid Mech. 40 (1970) 39) have conducted experiments and reported anomalously large Taylor numbers of up to 20 × 106, far exceeding the well-known value of 1708 for Taylor vortices in steady flow. In this paper, we argue that it is inappropriate to treat the phenomenon as a steady-state wide-gap Couette flow, because the unstable boundary layer in their experiments was very thin. The instability in the fluid induced by momentum diffusion is an unsteady-state phenomenon analogous to transient thermal instability, whose mathematical equivalence for the steady-state cases have been established by Taylor (Philos. Trans. R. Soc. London A 223 (1923) 289). We find that the onset of instability can be predicted from a transient Taylor number defined as Ta = y5(∂u/∂y)2/v2Ri. The maximum transient Taylor number is found to occur at a critical depth ymax = √5vtc, where Tamax = 1.461Ui2Ri (vt)15/v2Ri. Tamax bears a theoretical critical value of 1100 from linear stability analysis. The experimental data of Kirchner and Chen (J. Fluid Mech. 40 (1970) 39) agree remarkably well with this value. The critical time can thus be predicted with good accuracy from a critical value of the maximum transient Taylor number of 1100. The theoretical critical dimension of the toroidal plume formed after the boundary layer becomes unstable is found to be λc = 5.24√vtc, which agrees well with measurements. The average critical dimensionless wavenumber of the experiments of Kirchner and Chen (J. Fluid Mech. 40 (1970) 39) is found to be 3.05, which is very close to the theoretical value of 2.9.

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