Stratified flows in axially rotating pipes

We study experimentally the effects of a pipe axial rotation on a stratified displacement flow, using two miscible fluids of different density, in a long inclined pipe. Increasing the pipe rotation speed induces transverse mixing and results in a complete removal of the displaced fluid above a critical transition, described by $\text{Rb}\approx 5/2\pi \text{Fr}{\left(1-\text{Fr}\right)}^{-1}$, relating the Rossby number (Rb) to a modified Froude number (Fr). In addition, we quantify the variation of the leading and trailing displacement front velocities $({\widehat{V}}_f,{\widehat{V}}_t)$, as a function of the mean displacement velocity (${\widehat{V}}_0$), showing that for $\text{Rb}/\text{Fr}\gtrsim 5/2\pi$ we find ${\widehat{V}}_f\approx 1.3{\widehat{V}}_0$ and ${\widehat{V}}_t\approx 0.8{\widehat{V}}_0$. These findings offer insights to greatly improve displacement efficiency using rotational motion.

Figure 1

Schematic view of experimental setup.

Figure 2

(a) Evolution of ${\widehat{V}}_t\left(\widehat{t}\right)$ (blue curve) and ${\widehat{V}}_f\left(\widehat{t}\right)$ (red curve) with time, $\widehat{t}$, for a typical experiment: $[\beta ,\text{At};{\widehat{V}}_0;\widehat{\omega }]=[{83}^{\circ };{10}^{-2};30\text{mm}/\text{s};40\text{rpm}]$. Dashed lines show the nearly steady front velocities reached at longer times. (b) Comparison between our experimental front velocities ($V_f^{†}$) and those of the literature ($V_f^{\ell }$) in a static pipe ($\widehat{\omega }=0$): $[\beta ;\text{At};{\widehat{V}}_0]=[{83}^{\circ };3.5\times {10}^{-3};10–60\text{mm}/\text{s}]$ ($▾$) [10]; $[{83}^{\circ },{10}^{-2},10–60\text{mm}/\text{s}]$ ($●$) [19]; $[{60}^{\circ };{10}^{-2};0–20\text{mm}/\text{s}]$ ($■$) [19]; $[{83}^{\circ };3.5\times {10}^{-3}\text{and}{10}^{-2};0]$ ($▴$) using $V_f^{\ell }=0.7\sqrt{\text{At}\widehat{g}\widehat{D}}$ provided by [7]. Dashed line represents ${\widehat{V}}_f^{†}={\widehat{V}}_f^{\ell }$.

Figure 3

Spatiotemporal diagrams of isodensity displacements ($\text{At}=0$) in a rapidly rotating pipe ($\widehat{\omega }=90\mathrm{rpm}$). The corresponding Reynolds numbers (Re) are approximately equal to (a) 31, (b) 50, (c) 76, (d) 122, (e) 139, and (f) 198. The vertical line in each image represents distorted pixels due to the presence of the pipe support, here and elsewhere.

Figure 4

Sequence of experimental snapshots at $\beta ={83}^{\circ }, {\widehat{V}}_0=30\mathrm{mm}/\mathrm{s}$, and $\text{At}={10}^{-2}$, with increasing pipe rotation speeds: (a) $\widehat{\omega }=0$, (b) 20, (c) 30, (d) 40 rpm. Experimental times in seconds, $\widehat{t}$, are indicated beside each snapshot. Field of view in each snapshot is $19.05\times 3000{\mathrm{mm}}^2$, covering the whole pipe (note that the images are vertically stretched for visualization). Displacement is from right to left in each snapshot. Gate valve and pipe supports are marked by G and S, respectively.

Figure 5

Variation of ${\widehat{V}}_t$ ($■$) and ${\widehat{V}}_f$ ($▴$), as a function of $\widehat{\omega }$, for ${\widehat{V}}_0=30\mathrm{mm}/\mathrm{s}, \beta ={83}^{\circ }$, and $\text{At}={10}^{-2}$. Error bars are small, so they are dropped, here and elsewhere. The superimposed snapshots show the displacement for the corresponding experiments. The field of view in each snapshot is $19.05\times 3000{\mathrm{mm}}^2$, covering the whole pipe.

Figure 6

Variation of ${\widehat{V}}_f$ and ${\widehat{V}}_t$ versus ${\widehat{V}}_0$, above and below the thick solid line, respectively. Flow parameters are (a) $\beta ={83}^{\circ }$ and $\mathrm{At}={10}^{-2}$, (b) $\beta ={60}^{\circ }$ and $\mathrm{At}={10}^{-2}$, (c) $\beta ={83}^{\circ }$ and $\mathrm{At}=3.5\times {10}^{-2}$, and (d) $\beta ={60}^{\circ }$ and $\mathrm{At}=3.5\times {10}^{-2}$ with different pipe rotation speeds: $\widehat{\omega }=0$ ($▴$), 10 ($+$), 20 ($▸$), 30 ($◂$), 40 ($★$), 50 ($⧫$), 60 ($▿$), 70 ($\mathbf{\times }$), 80 ($✶$), and 90 ($•$) rpm. The slopes of the dotted, solid, and dashed lines are 0.8, 1, and 1.3, respectively. As the maximum errors are small, the horizontal and vertical error bars are removed from the datapoints, here and elsewhere.

Figure 7

Variation of ${\widehat{V}}_f$ and ${\widehat{V}}_t$ versus $\widehat{\omega }$. Flow parameters are $\beta ={83}^{\circ }$ and $\text{At}={10}^{-2}$, with different imposed flow rates: ${\widehat{V}}_0=0$ ($▴$,$▹$), ${\widehat{V}}_0=50$ ($•$,$◯$) mm/s. The leading and trailing front velocities are illustrated by the filled and hollow symbols, respectively.

Figure 8

Exchange flow regime classification: nearly stagnant ($▾$) and nonstagnant ($●$) flows, separated by the oblique line. From left to right, images show spatiotemporal diagrams for $\text{At}={10}^{-2}$ and $\beta ={83}^{\circ }$ at $\widehat{\omega }=0$, 20, and 40 rpm, with vertical axis of $0\le \widehat{t}\le 60$ s and horizontal axis of $0\le \widehat{x}\le 1000\mathrm{mm}$. Dashed line slopes represent ${\widehat{V}}_f$ at long times.

Figure 9

Variation of ${\widehat{V}}_f$ (filled symbols) and ${\widehat{V}}_t$ (hollow symbols) with $\widehat{\omega }$ at a very low (although nonzero) mean imposed velocity with ${\widehat{V}}_0=10\mathrm{mm}/\mathrm{s}$: $\beta ={83}^{\circ }$ and $\text{At}={10}^{-2}$ ($▸$, $▹$); $\beta ={60}^{\circ }$ and $\text{At}={10}^{-2}$ ($●$, $\mathbf{◯}$); $\beta ={83}^{\circ }$ and $\text{At}=3.5\times {10}^{-3}$ ($■$, $\square$); $\beta ={60}^{\circ }$ and $\text{At}=3.5\times {10}^{-3}$ ($★$, $☆$).

Figure 10

Spatiotemporal diagrams for ${\widehat{V}}_0=10, \text{At}={10}^{-2}$, and $\beta ={83}^{\circ }$, at $\widehat{\omega }=$ (a) 0, (b) 10 rpm, (c) 20 rpm, (d) 30 rpm, (e) 40 rpm, and (f) 50 rpm. These diagrams correspond to a part of the case marked by ($▸$, $▹$) in Fig. 9.

Figure 11

Displacement flow regime classification, with ${\widehat{V}}_t>0$ ($●$) and ${\widehat{V}}_t<0$ ($■$), roughly separated by the oblique straight line (empirical) passing through $\text{Fr}=1$ and $\text{Fr}/\text{Rb}=2\pi /5$. Motionless trailing fronts (${\widehat{V}}_t\approx 0$) and nearly stagnant flows (${\widehat{V}}_f={\widehat{V}}_t\approx 0$) are marked by $▸$ and $◂$, respectively. The empirical line corresponds to flows with ${\widehat{V}}_t=0$, marking the transition between the regimes of removable and nonremovable displaced fluids.