Noise amplification in open Taylor-Couette flow

We present an extensive experimental and theoretical study of the flow in an open Taylor-Couette apparatus with radius ratio ${\mathit{r}}_1$/${\mathit{r}}_2$=0.738 and imposed axial through-flow. Emphasis is given to the amplification of noise observed when the base flow is convectively unstable. Parameter boundaries for absolute and convective instability with respect to axisymmetric disturbances are determined experimentally and theoretically for axial Reynolds numbers R≲4, with excellent agreement between experiment and theory. Above onset a sustained pattern of traveling Taylor vortices is observed downstream of the inlet. In the case of absolute instability, the pattern is periodic within experimental resolution as evidenced by a narrow frequency spectrum in the time series of the Taylor-vortex velocity at a fixed point. In contrast, the patterns in the convectively unstable case arise via spatial amplification of microscopic noise. There results a broadened frequency spectrum caused by the pattern phase executing a pseudorandom walk. Virtually all aspects of the observed behavior are captured quantitatively by numerically integrating a complex Ginzburg-Landau (CGL) equation with an additive, spatially distributed, stochastic term. Precise measurements of the spatial amplitude profiles were made using fluids of various viscosities. The noise power required to fit the data has a viscosity dependence consistent with thermal noise, i.e., it has a ‘‘white’’ spectrum over at least a decade in frequency. Within our experimental uncertainty, the noise power is independent of the axial Reynolds number over the range 1.5≲R≲4. Simulations of the stochastic CGL equation indicate that the noise corresponds to rms velocity fluctuations, which are smaller than typical fully developed secondary flows in our experiment by a factor of about ${10}^5$. However, numerical evaluation of a recent theoretical result for the thermal noise power in the Taylor-Couette geometry with no through-flow turns out to be about 270 times smaller than the experimental result.