Dynamics of a shrouded cantilevered pipe subjected to internal and annular flows

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Abstract

The system considered consists of a hanging cantilevered pipe, surrounded at its upper part by a concentric rigid cylindrical tube, mounted in a tank full of water. Water enters the pipe at its upper and is discharged into the tank at its free end. This generates an upwards flow in the annulus between the pipe and the rigid tube, exiting the system at its upper end. The reverse flow configuration is also considered in which water enters the tank via the annulus and is aspirated by the pipe, exiting at its upper, clamped end. The free end of the discharging pipe could be plain or fitted with an end-piece which allows straight-through axial flow or forces the fluid to exit/enter radially. The system under study may be considered to be a highly idealized version of the operation of salt-mined caverns used for storage and subsequent retrieval of liquid and gaseous hydrocarbons. The problem is studied analytically, and the derivation of the pertinent linear equations of motion of the pipe is outlined. Extensive experiments conducted in a bench-top apparatus are also presented and compared with theoretical predictions. Experimental results and theoretical predictions show flutter for the pipe discharging fluid axially. Annular flow proves to have a strong destabilizing effect on the discharging pipe. The theoretical model predicts stability for the discharging pipe with radial exit flow and this is demonstrated experimentally for light-weight end-pieces. In the reverse flow configuration, flutter is observed in the experiments, while the theory predicts static divergence for low flow velocities and flutter for higher flow velocities.

Introduction

Solution mining is a process in which water-soluble minerals, such as ordinary salt, magnesium or potassium salts, are extracted from deep underground. After drilling to the salt deposits, fresh water is pumped down through very long pipes, dissolving the minerals; the brine is then pushed to the earth’s surface by pumping in more fresh water. The minerals are retrieved from the brine by natural drying at ground-level through recrystallization. The brine-filled caverns that remain underground are often used for storage of crude oil, natural gas and other hydrocarbons. The lighter-than-brine product is pumped in replacing some of the brine; it is later retrieved by pumping in brine or water.

A typical arrangement involves a long pipe (the “brine string”), 1 or 2 km long, extending from the surface of the earth, through the overburden, all the way to nearly the bottom of the salt-mined cavern. It is surrounded by a shorter larger-diameter outer rigid tube (the “casing”) cemented onto the overburden and caprock and extending a little into the cavern. Within the cavern most of the brine string is not surrounded by the casing. A schematic of the system is shown in Fig. 1.

Flow-induced vibration or fluid-elastic instability have been identified to be probable causes for the occasional breakage of brine strings; ‘probable’ because salt-mined caverns are not instrumented. Repair and replacement costs are very high. In order to undertake a systematic fundamental study of the dynamics of such systems, an extensive research program was initiated by the McGill Fluid-Structure Interactions research group. Several theoretical and experimental studies have been made on various aspects of the system. Although not pretending to model closely a salt-mined cavern or its operation and dynamics, the present paper draws its inspiration from that system. As already reported — see, e.g., Moditis, Païdoussis & Ratigan [2] — and as the present paper shows, the dynamics of the system is anything but simple.

In this paper, we consider two flow configurations, as sketched in Fig. 2: (a) the cantilevered pipe discharges fluid into the cavern, and the fluid exits the cavern via the annulus between the pipe and the coaxial rigid tube at the upper end of the pipe; (b) the reverse flow configuration, in which the fluid enters the cavern via the annulus and exits (or ‘is aspirated by’) the pipe. In the first case the fluid may be made to exit the pipe radially rather than axially, as shown in Fig. 3. For simplicity, the density of the fluid entering and exiting the system is considered to be the same.

It is well known that a cantilevered pipe conveying fluid, discharging it at the free end, loses stability at sufficiently high flow velocities by flutter [3], [4], [5], [6], [7]. The dynamics of the system, both linear and nonlinear, steady or pulsatile, with attached masses or springs, as well as many other variations, has been studied extensively by many researchers. If, however, the straight-through flow path at the free end of the pipe is blocked, forcing the fluid to exit radially (using an end-piece similar to the one considered in this paper), then flutter is totally suppressed, as shown by Rinaldi & Païdoussis [8]. Reviews of all these studies may be found in Ibrahim [9], [10] and Païdoussis [11].

In contrast, interest in the dynamics of aspirating cantilevered pipes conveying fluid, i.e. with the fluid ingested at the free end, developed more recently. The first such study was probably by Païdoussis & Luu [12]. Many others followed, e.g. by Cui & Tani [13], Païdoussis [14], Kuiper & Metrikine [15], Païdoussis et al. [16], Kuiper et al. [17], Kuiper & Metrikine [18], Debut et al. [19], Giacobbi et al. [20], Butt [21] and Minas [22]. The dynamics of this deceptively simple system is anything but cut and dry; there have been many reversals of opinion as to whether the pipe flutters or not at high enough flow velocities. In this respect, this quandary is similar to that of the ‘reverse sprinkler’, i.e. a rotary sprinkler aspirating rather than discharging water [23]: does it rotate or not? The current thinking on the aspirating cantilever is that it does flutter, although in a very weak fashion as compared to the discharging cantilever; indeed, this flutter has aptly been described as anaemic.

Perhaps the first study on cantilevered pipes simultaneously subjected to internal and external axial flows was by Hannoyer & Païdoussis [24]. Another study, pertinent to the present one is by Païdoussis, Luu & Prabhakar [25] on the dynamics of a drill-string conveying fluid internally downwards to rotate the drill-bit, and with the mud and debris flowing up in the drilled-out annulus around the drill-string to the surface. A more recent study, very pertinent to the present paper, is by Moditis et al. [2].

The theoretical models are outlined in Section 2, and typical results for the dynamics of the systems considered are given in Section 3. The experiments are described in Section 4, and experimental observations are compared to theoretical predictions in Section 5.

Section snippets

The equation of motion for discharging cantilever

The system under consideration is shown in Fig. 3. It consists of a cantilevered flexible pipe, the upper portion of which is surrounded by a coaxial rigid tube, and the whole submerged in a fluid-filled tank. At the free end of the pipe a special end-piece is inserted which either (i) allows a straight-through axial flow or (ii) blocks the axial flow, forcing the fluid out radially through a number of holes; it is this latter flow configuration that is depicted in Fig. 3.

Water, or more

Axially discharging pipe, aspirating annulus

Typical Argand diagrams for a system with αD=0.397,αch=1.97,βi=0.0741,βo=0.471,γ=3.186,ε=28.46,cf=0.01593, χ=1.7,h=1.03,μe=0and rann=0.453,which are parameters corresponding to a pipe and annulus used in some of the experiments, are presented in Fig. 6 for Uo=0,Fig. 7 for Uo/Ui=0.05and Fig. 8 for Uo/Ui=0.40.

It is seen in Figs 6 and 7 that the

Experiments

Experiments were performed in an ad hoc bench-top apparatus in the Fluid-Structure Interactions Laboratory of McGill University. It consists of a cylindrical stainless-steel tank with four large symmetrically located viewing windows. At its upper end there are supports for the cantilevered pipe and the rigid tube around it. External and internal piping achieves the two flow arrangements of (a) discharging-pipe/aspirating-annulus and (b) discharging-annulus/aspirating-pipe.

A schematic of this

Comparison between experiment and theory

Theoretical and experimental values for the critical flow velocities for the pipe discharging fluid are presented in Table 2. It is seen that agreement is reasonable, but for Uo/Ui=0.05 and 0.40 theory overpredicts Ucr by 40% approximately.

Theoretical and experimental values for the pipe aspirating fluid and the annulus discharging fluid are presented in Table 3. Only material damping is taken into account in the calculation of the theoretical values, whereas ideally damping associated with the

Conclusion

In this paper, theoretical and experimental investigations are presented on the dynamics of the system under consideration: a hanging cantilevered pipe within a fluid-filled reservoir, the upper portion of the pipe being shrouded by a coaxial rigid tube, thus forming an annular passage. A special end-piece is inserted at the free-end of the pipe, via which the flow at the free end can be either axial or radial. The following flow configurations are considered: (a) axially discharging pipe,

CRediT authorship contribution statement

Sophie L. Minas: Writing - original draft. Michael P. Païdoussis: Funding acquisition, Supervision, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowlgedgments

The authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), the Solution Mining Research Institute (SMRI) and the Pipeline Research Council International (PRCI).

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