Flow filling a curved pipe

A small scale experiment was designed to study the propagation of the front of a viscous ﬂuid ﬁlling a curved pipe. Several Newtonian ﬂuids with different viscosities and a non-Newtonian ﬂuid have been used. The experiments show that there exists a minimum speed for completely ﬁlling the pipe, which depends on the parameters of the experiment (di-ameter d and radius of curvature R of the pipe, kinematic viscosity n of the ﬂuid). Appropriate dimensionless numbers are introduced to characterize the ﬂow and optimal ﬁlling conditions.


Introduction
Prestressed concrete is commonly used in construction. To prevent the prestressed strands from any corrosion, a cement solution is generally injected in pipes of complex geometry. The pipe has to be completely filled in order to avoid weak zones in the structure. The aim of the present study is to observe experimentally the propagation of the front of a viscous fluid injected in a curved pipe, in order to understand the reasons for the appearance of unfilled zones. The solution used in construction is composed of cement, water, liquefier, and retarder. It is a non-Newtonian fluid with air bubbles and particles in suspension. As a first step, we mainly focus in this paper on the filling by Newtonian fluids.
Permanent flows in pipes have been widely studied ͓1͔, particularly in the case of curved pipes ͓2-4͔ and in engineering applications ͓5,6͔. However, pipe flows involving a propagating front or interface remain poorly understood. The problem of an airwater interface propagating in straight inclined pipes has been addressed only recently ͓7,8͔. As stated above, in industrial applications such as construction, the fluid is often mixed with air bubbles and the flow ought to be treated as a slug flow ͑see for example ͓9,10͔ for recent work on slug flows in pipes͒. But the emphasis of the present work is in the propagation of the interface which clearly separates the gas from the liquid.
The front propagation of a perfect fluid in a horizontal pipe was studied by Benjamin ͓11͔. See also the review by Simpson ͓12͔ on gravity currents for related studies and Asavenant and Vanden-Broeck ͓13͔. Benjamin analyzed the front in terms of a ''cavity flow'' displacing a fluid beneath it. The two-dimensional geometry is shown in Fig. 1͑a͒͑Benjamin also considered the case of a circular cross-section͒. Note that Benjamin studied only the case where the free surface detaches with an angle of 60 degrees. By applying conservation of mass, momentum, and energy, he found that there is a unique solution, characterized by where the meaning of the various symbols is shown in Fig. 1. In other words, the cavity fills half of the box. An experiment was suggested by Benjamin to realize closely this flow. Liquid initially fills a long rectangular box closed at both ends and fixed horizontally. One end is then opened, and under the action of gravity the liquid flows out freely from this end. It can be expected that, after the transient effects of starting have disappeared, the air-filled cavity replacing the volume of the ejected fluid will progress steadily along the box. Observed in a frame of reference travelling with the front of the cavity, the motion of the liquid will appear to be steady, as shown in Fig. 1͑a͒. If the effects of viscosity and surface tension are neglected, the velocity of the cavity relative to a stationary observer will be U, and, since hϭd/2, the liquid will discharge from the open end with the same velocity. In addition to the solution studied by Benjamin, there is also a one-parameter family of solutions in which the free surface detaches from the box tangentially ͓14͔. Such a solution is plotted in Fig. 1͑b͒.A t the exit of the box, the Froude number u(gh) Ϫ1/2 for these solutions is between 1 and ͱ2. The limit ͱ2 corresponds to Benjamin's solution, while the limit 1 corresponds to the vanishing of the cavity. Using conservation of mass, momentum and energy, one finds the relation Unfortunately, simple arguments based on conservation of mass, momentum and energy do not give as much information in the case of a curved channel. It is nevertheless important to keep in mind that the Froude number prevails to determine the size of the cavity and thus the ability to fill the pipe. This point will be discussed in the section where we consider various dimensionless numbers. Before that, we first describe the manner in which the experiments have been conducted. Then we present and discuss the experimental results.

Experimental Setup
The experimental setup is shown in Fig. 2. Gravity acts downwards. A transparent PVC pipe ͑''Tubclair''͒ of inner diameter d is curved with a radius R. Different diameters and radii were considered: 0.6рdр2c m ;5 рRр30 cm. The special case of a horizontal straight pipe was also investigated. The fluid was injected with a centrifugal pump at a constant mean flow velocity ͑noted U).
Several fluids were considered in order to investigate the effects of density , surface tension , and mainly kinematic viscosity ͑see Table 1͒. A series of runs has been performed with a non-Newtonian fluid ͑gel͒. Its viscosity has been measured with a plane-plane viscosimeter ͑see Fig. 3͒.
We assume that capillary effects are negligible, so that dӷL c where L c ϭ 1/2 (g) Ϫ1/2 is the capillary length (ϭ73 gs Ϫ2 for water and ϭ21 gs Ϫ2 for silicone͒. In the case of water L c ϭ0.26 cm and this assumption might be a shortcoming for small diameters. We nevertheless performed a few runs with water contaminated with wetting agents in order to diminish L c : the behavior and the results were the same within measurements errors. Moreover, the results presented in this paper are limited to dу1 cm.

Observations
For a very low flow rate ͑a typical mean flow velocity is U ϳ0.1 cm/s͒ and a sufficiently large pipe section ͑so that dӷL c ) the interface is more or less horizontal in the upflow part of the pipe. The flow is dominated by gravity. In the downflow part of the pipe the liquid is creeping down the pipe.
If U is slightly increased, the interface becomes more or less perpendicular to the pipe when rising. At the top, the interface deforms as shown in Fig. 2. The intersection between the interface and the upper part of the pipe ͑point C in Fig. 2͒ is a stagnation point.
If U is increased, the point C moves ͑interface in dotted line in Fig. 2͒. As soon as U is decreased back to zero, the pipe is emptying and the interface moves back to the top.
We consider that the pipe is completely filled when the point C has reached the point B ͑Fig. 2͒. We define U c as the critical speed beyond which the interface moves until the pipe is completely filled. The speed v is the mean velocity of the point C moving from A to B ͑Fig. 2͒:i fU is larger than U c , then v is greater than zero.
For the special case of the non-Newtonian fluid gel, the viscosity is smaller where the shear stress is larger, that is near the boundaries. Away from the boundaries, the behavior of the fluid is similar to a solid core pushed by the flow. In other words, the characteristic time of deformation of the interface is very long.

Measurements
The mean flow velocity U was obtained by measuring the flow rate in the steady-state regime, i.e., when the pipe was completely filled. The flow rate and v were estimated by measuring the filling time of a graded beaker and the travelling time of point C from A to B ͑Fig. 2͒ with a stopwatch. We may consider that these simple techniques do not lead to any bias error. By repeating the same run ͑at least five times͒ for different flow rates, we have estimated that the precision limit and thus the uncertainty was always better than 10 percent on U and v.
We measured v as a function of U for different experiments using various fluids, pipes, and curvatures ͑see Table 1͒. In Fig. 4, v is plotted versus U for several runs in four different configurations. The line vϭU delimits the region where physical solutions exist: v cannot be larger than U. In all our experiments v apparently varies linearly with U. We may thus determine the critical velocity U c ͑for vϭ0), the slope ␣ϭdv/dU and the interfacial velocity v(Uϭ0), by rms fitting a straight line into the data for each fluid, pipe diameter and radius of curvature.
Let us first consider the slope ␣ of the four straight lines in Fig.  4. The analysis presented in the Introduction considered a perfect fluid. It was assumed that the flow is identical in a reference frame moving with the front. In this framework, the filling of a pipe is similar to the emptying of the pipe. This would imply that ␣ is equal to one: adding ␦U to the mean entrance flow velocity would add the same value to the front propagation velocity. It is therefore not surprising that the slopes of the straight lines in Fig. 4 related to water are close to one. When the fluid is far from perfect  Diagram of the flow and notation. The interface is shown at two different times in solid and dotted lines. but still Newtonian ͑case of the silicone oil V100 in Fig. 4͒, ␣ is clearly smaller than one: when the viscosity is large, the deformation of the front strongly depends on U. For a low viscosity, the deformation of the interface takes place on a very short time scale and the propagation of the front apparently does not depend on the reference frame. In the other limit, that is for a very large viscosity, the deformation of the front takes place on a very long time scale and the slope is again close to one ͑this is the case of the non-Newtonian fluid͒.
The velocity v(0) corresponds to the draining of the pipe. The measurements have been performed by bringing U to zero after filling the pipe: the flow is then from right to left in Fig. 2. It is important to emphasize that v(0) is very small in the case of the non-Newtonian fluid. We have estimated that the draining velocity is about 1 cm per week. The deformation of the front is visible only when the fluid is sheared and U c is therefore also close to zero for the gel. The agreement in v(0) between the measured values and the values deduced from the fitted straight lines in Fig.  4 is rather good except in the case of water for Rϭ30 cm. The discrepancy is probably due to measurement errors but also to wetting effects. The contact between the fluid and the PVC is indeed not the same when filling or emptying the pipe. It was not our purpose to investigate wetting effects and we did not test other materials for the pipe. We therefore have to consider that wetting effects may induce a bias error over U c of about 10 percent in the worst case, i.e., an uncertainty of 20 percent in the water case.
The critical speed U c ͑mean entrance velocity for which v ϭ0) is plotted against the pipe diameter d in Fig. 5. U c clearly tends to increase with d. Considering gravitational effects, the larger is d, the more efficient are gravity effects to deform the interface. It is thus not surprising that inertial effects have to be increased to compensate for it. Gravity effects could also reasonably depend on the radius of curvature. The interface deforms more rapidly when the pipe is curved: for a same diameter, U c is therefore smaller when Rϭϱ. Otherwise, if R is large but not infinite, the travelling time of the front in the pipe is large and the mean entrance velocity has to be large for the front to propagate. In other words, for a very small radius, the front has not enough time to deform before the exit of the pipe. This would explain the fact that U c is larger for Rϭ30 cm than for Rϭ5 cm.
In addition, above ϭ1 cm 2 /s, viscous effects tend to maintain the shape of the interface and thus decrease U c . This is noticeable for the silicone V100 and clearly visible for the silicone V300. Again the case of the gel requires specific comments. U c is very close to zero because the front deforms only when the fluid is sheared as mentioned above. The graphical estimation of U c gives a nonzero value only for dϭ1.5 cm. This configuration will allow us to estimate dimensionless numbers for the gel in the next section.
The data in Figs 4 and 5 are available in tabular form in Table  2.

Interpretation in Terms of Dimensionless Numbers
We may consider various dimensionless numbers that could characterize the flow in our special configuration. Note that in our experiments both the Bond number (Boϭd 2 g/) and the Weber number (WeϭdU 2 /) are very large: capillary effects are a lot weaker than inertial and gravitational effects.
Considering inertial and viscous effects in the framework of a curved pipe, we introduce a modified Reynolds number Re, which partly takes into account centrifugal effects, and is in fact the classical Reynolds number plus the Dean number ͓15͔: To balance inertial and gravity effects, we build a modified Froude number Fr, such as: (3) The modification of the classical dimensionless numbers may be understood by considering that there is a need to add centrifugal effects to maintain the fluid against the concave side of the pipe. Note that ͑2͒ and ͑3͒ tend toward the classical Reynolds and Froude numbers in the limiting case of a straight pipe, i.e., when R tends toward infinity. Let us finally define K, which measures the balance between gravity and viscosity:
(4) Fig. 4 Speed of the front propagation v versus the mean entrance velocity U "dÄ1.5 cm; see list of symbols in Table 1…. The straight lines are fitted curves by quadratic means.

Fig. 5
Critical speed versus the pipe diameter "see list of symbols in Table 1… Transactions of the ASME Note that K does not depend on the mean flow velocity U.Reand Fr are computed for UϭU c and plotted against K in Figs. 6 and 7.
We may consider two kinds of estimation of the dimensionless numbers for the gel. First we may use the graphical estimation of U c from Fig. 4 for dϭ1.5 cm: U c ϳ1.2 cm/s, leading to ϳ200 cm 2 /s from Fig. 3 ͑assuming that ␥ ϳ2U c /d). This allows us to plot the squares in Figs. 6 and 7. Second, we may estimate from our measurement that U c ϳv(0) is nonzero but very small ͑six orders of magnitude less than the other values of velocity͒. In that case, the squares would be far off the bottom right-hand corners of Figs. 6 and 7. In any case, the gel is possibly following the trend of Newtonian fluids with very large viscosity.
In Fig. 6, the data can be separated into two regions corresponding to partially ͑below the data͒ and completely ͑above the data͒ filled pipes. In a situation where the flow is characterized by a point belonging to the border line between the two regions, the filling would be effective over an infinite time, using an infinite amount of fluid. In the frame of a linear approximation, the data follow approximatively the (Ϫ1) slope, showing that Fr(U ϭU c ) is constant from Eq. ͑4͒.
Another way to represent the same results is given in Fig. 7. One sees that Fr is of the same order of magnitude for all experiments while K varies over four orders of magnitude, but Fr is not rigorously constant. In the range of diameters we investigated, we can say roughly that Fr characterizes the ability to fill the pipes  ͑Frϭ1 corresponds to the theoretical limit ͑1͒ where the flow detaches tangentially at the exit of a horizontal box, as suggested above in the introduction͒. Setting a flow rate leading to Frу1 would certainly fill any straight pipe. A careful examination of Fig. 7 leads to several remarks. For Newtonian fluids, the dotted symbols, which correspond to straight pipes, lie below the points corresponding to curved pipes. The latter are gathered with no obvious dependence on the radius of curvature R. Therefore the modified Froude number, which takes into account centrifugal effects, can predict the effective filling of the curved pipes, but is of less interest in the limit of straight pipes. Each group of points corresponds to a different fluid, i.e., viscosity. The tendency for Fr(UϭU c ) to increase with d is clearly visible in each group of points. For Kу10 Ϫ2 , Fr(U ϭU c ) is apparently decreasing with K and the flow would eventually tend to follow a non Newtonian behavior.
In almost all our experiments, Re is larger than 10 ͑see Fig. 6͒ indicating that viscous effects are smaller than inertial ones. Meanwhile, the data on Fr indicate that both inertial and gravitational effects are comparable.

Conclusion
In order to understand the failure in filling curved pipes, we performed a series of experiments using various fluids. We measured the velocity of propagation of the front v and we deduced from its linear dependence with the mean flow velocity U a critical value U c , below which there is no longer a complete filling of the pipe. This critical velocity U c is increasing with d for each fluid.
Results are presented in a synthetic way as relations between dimensionless numbers. We have shown, in the scope of our work, that a modified Froude number determines the ability of the flow to fill a curved pipe. In all our experiments and for Newtonian fluids of different viscosities, the filling was complete for Fr у0.6Ϯ0.2.
We also performed experiments with a non-Newtonian fluid whose rheological behavior is similar to the one of cement suspensions. The front deforms very slowly when the shear is weak because this increases the viscosity of the fluid. In that case, the complete filling may be obtained for Fr very small. Following ͓16,17͔, a more extensive study would be needed in order to understand the detail of the non-Newtonian effects.
For industrial applications we advise considering pipes of small diameters and injecting the solution at a relatively low flow rate in order to increase its effective viscosity. However, a future study of interest will be to repeat the present experiments in the context of slug flows.  Table 1.…