Large axisymmetric surface deformation and dewetting in the flow above a rotating disk in a cylindrical tank: Spin-up and permanent regimes

In this paper, we aim at characterizing the spin-up process and the permanent regime of a rotating flow with free surface. The motion is created by the quasi-impulsive rotation of a disk located at the bottom of a cylindrical tank partially filled with a Newtonian water-glycerol mixture at a fixed initial aspect ratio (height to radius) of 0.25. Two experimental setups with different radius sizes and two numerical codes with distinct formulations are used, allowing for numerous comparisons and cross validations. Time-dependent surface height profiles and azimuthal velocity profiles are accurately measured with a laser line sensor and Laser Doppler Velocimetry, respectively. These measurements are compared to numerical results showing an overall excellent agreement and whenever a discrepancy exists, explanations are given. The spin-up and permanent regime for two typical flow cases are detailed: (i) one for which the angular speed moderately deforms the free surface and (ii) one at higher angular speed for which the disk becomes partially dewetted. The shape of the free surface in the permanent regime for intermediate rotation speeds is also reported. The flow parameters are chosen in such a way that large deformations of the free surface are achieved, yet remaining under the threshold of appearance of rotating polygonal patterns. This enables us to characterize realistic base flows on which this instability may develop.

Figure 1

Scheme of the experimental setup.

Figure 2

Sketch of the experimental setup with the control and measuring systems.

Figure 3

Schematic representation of the different time constants used in the experimental protocol.

Figure 4

Snapshots of the flow field in the liquid phase from top to bottom at $t=3.92$, 23.52, and 47.04 (dimensional time $t=0.5$ s, 3 s, and 6 s) during the spin-up of the experiment for the case $G=0.25,\text{Fr}=0.88,$ and $\text{Re}=3304$: large setup at $\mathrm{\Omega }=7.84\mathrm{rad}/\mathrm{s}$ and $H=35\mathrm{mm}$. (a), (c), (e) Isocontours of $V_{\theta }$ with 21 equispaced contours between 0 and 1. (b), (d), (f) Streamlines of the meridional circulation consisting of isocontours of $\psi$ with 21 equispaced levels between 0 and ${\psi }_{\text{max}}$ (in the liquid phase), ${\psi }_{\text{max}}=0.003,0.053,0.056$, respectively. The black lines in all the graphs represent the numerical air-liquid interface.

Figure 5

Case $G=0.25,\text{Fr}=0.88,$ and $\text{Re}=3304$ (large setup at $\mathrm{\Omega }=7.84\mathrm{rad}/\mathrm{s}$ and $H=35\mathrm{mm}$). Comparison between unsteady simulation (solid lines) and experimental measurements (dots). (a) Radial profiles of the liquid surface every 3 s from the start of disk rotation, that is $t=0$, 23.52, 47.04, 70.56, 94.08 in dimensionless scale (red for the steady state). (b) Time evolution of the surface height at $r=0$, 0.25, 0.5, 0.75, and 1. For readability, the experimental signals have been resampled every 0.1 s instead of using the original temporal resolution of 0.01 s.

Figure 6

Same parameters as in Fig. 5. (a) Probe locations for the azimuthal velocity measurements. The interface shape measured at steady state is represented by the solid red line. (b), (c) Comparison between the numerical (solid) and experimental (symbols) transient evolutions of $V_{\theta }$ at two probe locations $(r,z)$. (d) Amplitude spectrum in the permanent regime for the two time signals s1 and s2 between $t=118$ and $t=432$.

Figure 7

Same parameters as in Fig. 5. Comparison between numerical and experimental steady azimuthal velocities $V_{\theta }\left(r\right)$ at four different heights $z=0.05$ (red), 0.14 (blue), 0.17 (magenta), and 0.22 (black). Colors are coherent with those used for the probes in Fig. 6. Solid lines refer to numerical solutions and symbols to measurements.

Figure 8

For the same parameters as in Fig. 5, the steady state is computed using Sunfluidh (multicolor contours) and Rose (magenta contours). (a) Azimuthal velocity $V_{\theta }$; 21 equally spaced levels between 0 and 1. The overshoot region is highlighted (yellow area) in which the azimuthal velocity exceeds the solid body rotation by more than 1 % ($V_{\theta }>1.01r$). The steady interfaces obtained using Rose and Sunfluidh (solid black line) are both plotted but are indistinguishable. (b) Streamfunction $\psi$; positive contour values: 21 equispaced levels between 0 and ${\psi }_{\text{max}}=0.0057$ (the maximum is reached at the same location marked by the “$+$” symbol for both codes); negative contour values: one dashed contour at $\frac{1}{2}{\psi }_{\text{min}}$ for Sunfluidh (dashed blue) and for Rose (dashed magenta) with ${\psi }_{\text{min}}=-1.96\times {10}^{-4}$ (the minimum in the liquid phase is reached at almost the same location marked by the magenta “$\times$” symbols, negative contours in air phase are not represented). For Rose, the domain $[0,1]\times [0,0.25]$ was used, mapped by a curvilinear grid with $401\times 101$ points. For Sunfluidh, air parameters read as follows: ${\rho }_{\mathrm{air}}=1.205$ kg/${\mathrm{m}}^3$, ${\mu }_{\mathrm{air}}=1.82\times {10}^{-5}\mathrm{Pa}\mathrm{s}$ and the surface tension is $\sigma =65.5\times {10}^{-3}$ N/m; at the upper boundary $z=0.5$, free slip boundary conditions are imposed.

Figure 9

Comparison of surface evolution at several probes between the case “L” (red lines) and the case “S” (blue lines).

Figure 10

Comparison of radial height distributions for large (red curves) and small (blue curves) setup for the parameter values of Table 2. (a) Temporally averaged surface height for $400<t<600$ in experiments (solid lines) and numerical results (dashed lines). (b) Rescaled function $\tilde{h}\left(r\right)$ for experiments (solid lines), together with the parabolae ${\tilde{h}}_{\text{para}}$ [Eq. (6)], where ${\tilde{h}}_0=-0.155$ and ${\tilde{h}}_0=-0.1456$ are taken from their experimental values in the large and small setups, respectively. (c) Difference ${\tilde{h}}_{\text{para}}\left(r\right)-\tilde{h}\left(r\right)$.

Figure 11

(a) Time-averaged surface height distributions for cases 1–6 in Table 3. Measurements (solid lines) and numerical results (dashed lines). From top to bottom at $r=0$: $\text{Fr}=0.44,0.97,1.23,1.51,1.72,2.58$. (b) Surface height at $r=0,0.25,0.5,0.75,1.0$ as a function of Froude number. $\circ$, measurements; $+$, Sunfluidh results in the permanent regime;—, Rose results (up to dewetting). (c) Rescaled surface height distributions $\tilde{h}\left(r\right)$ as defined in Eqn (5) for the experimental cases of (a). All experiments are performed on the small setup with G8W2 as working fluid at $T\approx 28{}^{\circ }\mathrm{C}$ with a layer of initial depth $17.5\mathrm{mm}$.

Figure 12

Case $G=0.25,\text{Fr}=2.58,$ and $\text{Re}=2847$ (small setup at $\mathrm{\Omega }=19\mathrm{rad}/\mathrm{s}$ and $H=17.5\mathrm{mm}$). Comparison between unsteady simulation (solid lines) and experimental measurements (dots). (a) Radial profiles of the liquid surface every second from the start of disk rotation, that is $t=0$, 19, 38, 57, 76 in dimensionless scale (red for the steady state). (b) Time evolution of the surface height at $r=0$, 0.25, 0.5, 0.75, and 1. For readability, the experimental signals have been resampled every 0.1 s temporally and 1.5 mm spatially.

Figure 13

Snapshots of the computed flow field in the liquid phase from top to bottom at $t=38$, 57, and 72 (dimensional time 2 s, 3 s, and 3.8 s) corresponding to the spin-up of the experiment for the case $G=0.25,\text{Fr}=2.58,$ and $\text{Re}=2847$: small setup at $\mathrm{\Omega }=19.0\mathrm{rad}/\mathrm{s}$ and $H=17.5\mathrm{mm}$. (a), (c), (e) Isocontourss of $V_{\theta }$ with 21 equispaced contours between 0 and 1. (b), (d), (f) Streamlines of the meridional circulation consisting of isocontours of $\psi$ with 21 equispaced levels between 0 and ${\psi }_{\text{max}}$ (in the liquid phase) with ${\psi }_{\text{max}}=0.0060,0.0063,0.0065$. The black lines in all the graphs represent the numerical air-liquid interface.

Figure 14

(a) Probe locations for LDV measurements for the case $G=0.25,\text{Fr}=2.58$. The measured surface profile in the permanent regime is also plotted. (b)–(e) Temporal evolution of $V_{\theta }$ at four probe locations $(r,z)$, obtained by LDV measurements (red $+$ symbols) and simulation at $\text{Re}=2480$ (black solid lines).

Figure 15

Radial profile of $V_{\theta }$ at the three measured heights for the case $G=0.25,\text{Fr}=2.58$. Circles indicate measurements and solid lines numerical results at $\text{Re}=2480$. Steady velocity profiles at $\text{Re}=2140$ and 2910 obtained numerically are also plotted.

Figure 16

For the same parameters as in Fig. 12, the velocity field is computed by Sunfluidh with $128\times 64$ cells (multicolor) and $256\times 128$ cells (dashed black lines). The black lines in the two graphs represent the numerical air-liquid interface (indistinguishable for both mesh). (a) Iso-contours of $V_{\theta }$, with 21 equispaced contours between 0 and 1. The overshoot region (yellow area) is defined as in Fig. 8. (b) Streamlines of the meridional circulation consisting of isocontours of $\psi$ with 21 equispaced levels between 0 and ${\psi }_{\text{max}}=0.0065$ (red “$+$”) (the same maximum is found for both mesh) and one dashed contour at $\frac{1}{2}{\psi }_{\text{min}}$. The minimum, red and black “$\times$” located in the air phase, differs significantly as ${\psi }_{\text{min}}=-0.0034$ and ${\psi }_{\text{min}}=-0.0027$ for $128\times 64$ and $256\times 128$ grid, respectively. For Sunfluidh, air parameters and boundary conditions are as described in the caption of Fig. 8.

Figure 17

Effect of the Froude number on the azimuthal velocity in the permanent regime at Reynolds number $\text{Re}=2847$. (a), (b) Isocontours of $V_{\theta }$, with 21 equispaced contours between 0 and 1, together with the liquid-gas interface (bold line). The Froude number is (a) $\text{Fr}=1.8$ and (b) $\text{Fr}=5$. (c) Radial distributions of the azimuthal velocity $V_{\theta }$ at $z=h\left(r\right)/2$ for the wetted case $\text{Fr}=0.88$ (blue dotted) and three dewetted cases $\text{Fr}=1.8$ (red solid), 2.58 (black dashed), and 5 (magenta dot-dashed). The solid symbols indicate the positions $r=r_{\mathrm{w}}$ of the triple line at the disk, respectively, equal to 0.15, 0.29, and 0.4 for the three dewetted cases.