Drag Reduction in Bubbly Taylor-Couette Turbulence

In Taylor-Couette flow the total energy dissipation rate and therefore the drag can be determined by measuring the torque on the system. We do so for Reynolds numbers between $\mathrm{R}\mathrm{e}=7\times {10}^4$ and $\mathrm{R}\mathrm{e}={10}^6$ after having injected (i) small bubbles ($R=1\mathrm{m}\mathrm{m}$) up to a volume concentration of $\alpha =5\%$ and (ii) buoyant particles (${\rho }_p/{\rho }_l=0.14$) of comparable volume concentration. In case (i) we observe a crossover from little drag reduction at smaller Re to strong drag reduction up to 20% at $\mathrm{R}\mathrm{e}={10}^6$. In case (ii) we observe at most little drag reduction throughout. Several theoretical models for bubbly drag reduction are discussed in view of our findings.

Figure 1

The drag coefficient $c_{ϵ}(\alpha ,\mathrm{R}\mathrm{e})$ for volume concentrations $\alpha$ up to 0.05. The error bars in this and the subsequent figures result from an estimated relative error of 1% in the electronically controlled $\Omega$ and in $T$, and from an estimated absolute error of 0.5% for $\alpha$. For clarity, error bars are given only for the first and the last data points of each curve. The inset shows $c_{ϵ}(\alpha =0,\mathrm{R}\mathrm{e})$ for the single phase flow case ($\alpha =0$) together with its fit Eq. (3) which nicely described the measured data. The fit parameters are $\beta =3000$ and $c_{ϵ,\infty }=3.4\times {10}^{-4}$.

Figure 2

Ratio $c_{ϵ}(\alpha ,\mathrm{R}\mathrm{e})/c_{ϵ}^{\mathrm{f}\mathrm{i}\mathrm{t}}(0,\mathrm{R}\mathrm{e})$ vs Re for $0\le \alpha \le 0.05$. The inset shows the ratios $R/\eta$ (lower) and $R/y_0$ (upper) as a function of Re.

Figure 3

Drag coefficient $c_{ϵ}(\alpha ,\mathrm{R}\mathrm{e})$ vs Re for single phase flow (solid line), bubbly flow with $\alpha =0.05$ (dashed line), and buoyant particle flow with $\alpha =0.05$ (dotted line).

Figure 4

Ratio $c_{ϵ}(\alpha ,\mathrm{R}\mathrm{e})/c_{ϵ}^{\mathrm{f}\mathrm{i}\mathrm{t}}(0,\mathrm{R}\mathrm{e})$ vs $\alpha$ for various Re.