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Stationary residual layers in buoyant Newtonian displacement flows
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View: Figures


Image of FIG. 1.
FIG. 1.

(a) Schematic view of the experimental setup. The shape of the interface is illustrative only. More realistic shapes are given in Fig. 3 where the interface shape was found to evolve both spatially and temporally. The arrows in the diagram represent the motion of the fronts. (b) Schematic views of the distribution of the two fluids in two perpendicular vertical planes of the pipe (diametrical and transversal). The notation is that used later in our models.

Image of FIG. 2.
FIG. 2.

Sequence of images showing the stationary upper layer. This sequence is obtained for 5, 25, 250, and 450 s after opening the gate valve. The field of view is and taken right below the gate valve. For this experiment, the pipe is tilted at 85° from vertical. The normalized density contract is , the viscosity is , and the mean flow velocity is . The figure below the sequence is a spatiotemporal diagram of the variation of the light intensity in the transverse dimension, averaged over 20 pixels along the pipe in the region marked on the pipe above, with a time step of . It shows the variation of the layer height with time.

Image of FIG. 3.
FIG. 3.

Four snapshots of video images taken at different mean flow rates and illustrating the different regimes. The heavy transparent fluid flows downward under the combined effects of buoyancy and pressure gradient . The light black fluid has different behaviors (flows upward or downward) depending on the control parameters values. These images were obtained at , , and . The mean flow velocities were: (a) , (b) , (c) , and (d) . The field of view is , and contains the gate valve (wide black stripe) and a pipe support (thin black stripe). The images are taken at: (a) 150 s, (b) 290 s, (c) 365 s, and (d) 75 s after opening the valve.

Image of FIG. 4.
FIG. 4.

Spatiotemporal diagrams of the variation of the light intensity along a line parallel to the pipe axis in the upper section of the pipe. The vertical scale is time ( and 500 s for both) and the horizontal scale is the distance along the pipe above the gate valve (see Fig. 3). The orientation of the axis is the same as in Fig. 3: downward. These diagrams correspond to the experiments: (a) Fig. 3(b) and (b) Fig. 3(c).

Image of FIG. 5.
FIG. 5.

Ultrasonic Doppler velocimeters profiles for the same series of experiments as Fig. 3: (a) [see Fig. 3(a)] sustained back flow regime, profiles averaged between 60 and 120 s; (b) [see Fig. 3(b)] stationary interface regime, profiles averaged between 240 and 300 s; and (c) instantaneous displacement regime, profiles averaged between 120 and 240 s. The vertical scale represents the distance from the upper wall ( measuring distance from the lower one) and the horizontal scale the corresponding value of the longitudinal flow velocity. The horizontal dashed line shows the position of the interface. The vertical dashed line shows the zero velocity. The oblique dashed line close to the lower wall has been added to guide the eye where the profiles are distorted by instrumental error.

Image of FIG. 6.
FIG. 6.

Contours of and the contour (bold black line). The intercept of and occurs at and

Image of FIG. 7.
FIG. 7.

Profiles of for , with . The broken line shows the theoretical stationary at . The inset shows the extension of the stationary frontal region.

Image of FIG. 8.
FIG. 8.

The experimental results in a pipe over the entire range of control parameters ( is in the range of , is in the range of , and is in the range of 83°–87°). The heavy line represents the prediction of the lubrication model for the stationary interface: .

Image of FIG. 9.
FIG. 9.

Contours of and the contour (bold black line), in a plane channel displacement. The intercept of and occurs at and .

Image of FIG. 10.
FIG. 10.

Sequence of concentration field evolution obtained for , , , and [, ]. The images are shown for , 25, 50, 100, 200, and 300 s (from top to bottom).

Image of FIG. 11.
FIG. 11.

Spatiotemporal diagram of the average concentration variations (white and black colors represent heavy and lighter fluids, respectively) along the channel for , , , and [, ]. Vertical scale: time; horizontal scale: distance along the channel. Dashed lines have slopes equal to velocities estimated for the front and back flows. The stationary slope (1) shows that the front velocity is constant. Dashed line (2) is the initial inertial velocity for the back flow, which is followed by a decreasing viscous velocity. Dashed line (3) is vertical, which implies that the lighter fluid velocity is zero (near-stationary).

Image of FIG. 12.
FIG. 12.

The velocity profiles corresponding to Fig. 10 for a channel flow at , 25, 50, 100, 200, and 300 s (from top to bottom).

Image of FIG. 13.
FIG. 13.

The velocity profile close to the pinned point (with the axial position ) corresponding to Figs. 10 and 12 for a channel flow at : illustrating the countercurrent inside the stationary (lighter/black) fluid. Dashed line represents the local height of the interface.

Image of FIG. 14.
FIG. 14.

Four possible conditions for a viscous buoyant channel flow in the presence of an imposed flow for , , and : (a) [, ]; (b) [, ]; (c) [, ]; (d) [, ].

Image of FIG. 15.
FIG. 15.

Classification of our simulation results in a channel. The heavy line represents the prediction of the lubrication model for the stationary interface: .

Image of FIG. 16.
FIG. 16.

(a) Schematic variation of the velocity as a function of distance from the gate valve (continuous line) in a viscous regime and for . The short dashed line represents the final viscous velocity . The dotted line marks the boundary between the transient inertial regime and the viscous regime. We also represent the case , using the long dashed line, to underline the stopping length condition. The arrows on the curve show the trend of the evolution of the velocity with time. (b) is plotted vs for two series of experiments at different angles : 83° (◻) and 85° (○), and same density contrast and viscosity (, ). The experiments plotted here are either in the temporary back flow regime or in the stationary interface regime, and represents the position where the front stops (maximal ). The dashed line is a guide for the eye to show the common linear curve.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Stationary residual layers in buoyant Newtonian displacement flows