(a) Schematic view of experimental set-up (interface shape is illustrative only). (b) Change in iso-viscous displacement mixing with β for At = 0.0035 and kinematic viscosity . Mean flow speed is (see Fig. 3 ). The field of view is 1330 × 19 mm2 located 1740 mm downstream of the gate valve. The color bar at the top left of the figure shows the corresponding concentration value, C, with 0 referring to the pure displaced fluid and 1 to the pure displacing fluid.
Comparison of the experimental front velocity values for the exchange flow (solid data) with the data reported by Seon et al. 8 (open data). The solid line shows . 8 Squares correspond to At = 0.001 and circles to At = 0.01. Inclination angle, β varies between 20° and 70° in the experiments shown. The dashed lines show the limit of above which the exchange front velocity values deviate from , for corresponding At ( for At = 0.001, and At = 0.01, respectively).
Spatio-temporal diagrams of depth-averaged concentration field, for the same experiments as shown in Figure 1(b) : (a) β = 0°, ; (b) β = 30°, ; (c) β = 45°, ; (d) β = 60°, ; (e) β = 70°, ; (f) β = 85°, . The dashed line in (f) indicates the position of the displacing front and its slope is . The spatiotemporal diagram gives a front speed .
Ultrasound Doppler Velocimetry (UDV) measurements obtained for At = 0.004, and (a) β = 0°, , (b) β = 60°, , and (c) β = 85°, .
(a) Evolution of the depth-averaged concentration field, C with time, s, and streamwise location, , measured form the gate valve for the same experiment as in Figures 1(b) (β = 30°) and 3(b) . The dashed line shows C = 0.1 which is used for measuring the displacing front velocity, , consistently. (b) Evolution of the front velocity value, , with time for the same experiment. The inset shows the constant value of the front velocity when the flow is fully developed ( in this case with 2% standard deviation).
Change in displacing front velocity, with tilt angle, β, and imposed velocity, for and (a) At = 0.001, (b) At = 0.0035, and (c) At = 0.01. Different markers represent (▷), 10 (◀), 20 (▲), 40 (*), 60 (•), 80 (▼), and 100 mm/s (■). The dashed lines are guide to the eye plotted at
(a) Presentation of our results for the full range of experiments: normalized front velocity , plotted against Fr and Re cos β/Fr. The rectangular area (0 < Re cos β/Fr < 120, 0 < Fr < 6) indicated by dashed lines, locates the range of nearly horizontal experiments and is studied in full details in Ref. 4 . (b) Instantaneous displacement flows marked by the superposed circles plotted against Fr and Re cos β/Fr. The heavy line represents the prediction of the lubrication model for the stationary interface. The dashed lines Re cos β/Fr = 650 and Fr = 2 roughly indicate the boundary between instantaneous and non-instantaneous displacement. Normalized front velocity scale is limited to 2 in Figure 7(b) to better show the variations of the front velocity in the plane of Fr and Re cos β/Fr.
Collapse of depth-averaged concentration profiles with for and (a) β = 45°, , (b) β = 20°, , (c) β = 45°, , and (d) β = 85°, . In (a) At = 0.08 and in (b)–(d) At = 0.0035. The solid lines in (a) and (b) show the error-function fits with , respectively. The upper right insets show the qualitative flow pattern in each case. In (d) the lower left inset shows the collapse of the concentration profiles when is used instead of .
Variation of the diffusion coefficient, versus mean imposed flow velocity, for and (a) At = 0.0035 and (b) At = 0.01.
Contours of the normalized diffusion coefficient (a) and (b) versus Fr and Re cos β/Fr.
Reynolds number, Re, plotted against Fr and Re cos β/Fr. The illustrative curves correspond to different inclination angles, β, and are show Re = Re cr = 2100. Flows marked by the superposed triangles and/or squares satisfy the criterion Re > 2100. The symbols used are for β = 70° (△), β = 60° (▽), β = 45° (▷), β = 30° (◁), and β = 0° (□).
Classification of our results for the full range of experiments, presented in the (Fr, Re cos β/Fr)-plane: (i) instantaneous displacement flows are colored in blue and non-instantaneous flows in red; (ii) fully diffusive flows have no superposed symbol; (iii) non-diffusive flows are marked as viscous (superposed circles) or inertial (superposed squares). The heavy line represents the prediction of viscous backflows, from the lubrication model in Ref. 3 , (χ = χ c = 116.32). The thick broken line represents Re cos β/Fr = −50Fr + 500. The point of intersection of the two lines is Fr ≈ 4.62 and Re cos β/Fr ≈ 270.
(a) Comparison of the front velocity values obtained from experiments classified as viscous with the predictions V f (χ) from the lubrication model in Refs. 3 and 4 (thick solid line). Color values indicate Re cos β/Fr for each experiment. The data with the solid boundary are for near-horizontal β, taken from Ref. 4 . The thick white circle indicates the theoretical transition (χ = χ c ≈ 116.32). The thin solid line shows below which leading front velocities are not possible due to conservation of mass (b) comparison of the experimental and predicted values of , for intermittent flows with the predictions following (6) . The dashed line in Figure 14(b) indicates . The contour values in both figures show the corresponding Re cos β/Fr to each experiment.
(a) Comparison of the experimental macroscopic diffusion diffusion coefficient, , against the prediction of (7) . The broken line shows . Meanings of the symbols are the same as in Figure 10 . (b) Front velocity values, , plotted against the mean flow velocity, for all fully diffusive experiments shown in Figure 13 . The color values show the corresponding Re cos β/Fr. The solid line is the linear fit . The broken line represents .
Parameter range of our experimental study. Note that most of the experiments were run for At = 0.001, 0.0035, and 0.01.
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