^{1}, T. Séon

^{2,a)}, K. Wielage-Burchard

^{2}, D. M. Martinez

^{1}and I. A. Frigaard

^{2,3}

### Abstract

We study buoyant displacement flows with two miscible fluids of equal viscosity in ducts that are inclined at angles close to horizontal . As the imposed velocity is increased from zero, we change from an exchange flow dominated regime to a regime in which the front velocity increases linearly with . During this transition, we observed an interesting phenomenon in which the layer of displaced fluid remained at the top of the pipe (diameter ) during the entire duration of the experiment, apparently stationary for times (the stationary residual layer). Our investigation revealed that this flow marks the transition between flows with a back flow, counter to the imposed flow, and those that displace instantaneously. The same phenomena are observed in pipes (experiments) and in plane channels (two-dimensional numerical computations). A lubrication/thin-film model of the flows also shows the transition from back flow to instantaneous displacement. At long times, this model has a stationary residual layer solution of the type observed, which is found at a unique ratio of the axial viscous velocity to the imposed velocity. The prediction of the stationary residual layer from the lubrication model is compared with the transition in observed behavior in our pipe flow experiments and our 2D numerical displacements in the channel. Reasonable agreement is found for the pipe and excellent agreement for the channel. Physically, in either geometry at transition, the upper layer of the fluid is in a countercurrent motion with zero net volumetric flux; the velocity at the interface is positive, but the velocity of the interface is zero. This results from a delicate balance between buoyancy forces against the mean flow and viscous forces in the direction of the mean flow.

This research has been carried out at the University of British Columbia, supported financially by NSERC and Schlumberger through CRD Project No. 354717-07. T. Séon also acknowledges additional financial support from PIMS through a postdoctoral fellowship. The authors thank S. Gharib for assisting in running the experiment. We thank A. Wachs and D. Vola for their assistance in introducing PELICANS. Finally, we thank the reviewers for their helpful comments and for alerting us to Ref. 6.

I. INTRODUCTION

II. PIPE DISPLACEMENTS

A. Experimental setup and procedure

B. Experimental observations

C. Lubrication model

D. Experimental and theoretical comparison

III. PLANE CHANNEL GEOMETRY (2D)

A. Lubrication model

B. Numerical method and overview

C. Numerical results

IV. CONCLUSIONS

### Key Topics

- Viscosity
- 24.0
- Duct flows
- 23.0
- Lubrication
- 20.0
- Buoyancy driven flows
- 14.0
- Laminar flows
- 9.0

## Figures

(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.

(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.

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.

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.

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.

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.

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).

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).

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.

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.

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

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

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

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

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: .

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: .

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

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

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

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

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).

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).

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

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

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.

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.

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

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

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

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

(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.

(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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