Advances in Acoustics and Vibration

Volume 2017, Article ID 8141523, 20 pages

https://doi.org/10.1155/2017/8141523

## A Note on an Analytic Solution for an Incompressible Fluid-Conveying Pipeline System

^{1}Department of Systems Engineering, Faculty of Engineering, University of Lagos, Akoka-Yaba, Lagos 23401, Nigeria^{2}Centre for Space Transport and Propulsion, National Space Research and Development Agency, Federal Ministry of Science and Technology, FCT, PMB 437, Abuja, Nigeria^{3}Department of Mechanical & Biomedical Engineering, College of Engineering, Bells University of Technology, Ota 234037, Ogun State, Nigeria

Correspondence should be addressed to Charles A. Osheku; moc.oohay@2002ukehsoselrahc

Received 13 July 2016; Accepted 21 September 2016; Published 6 March 2017

Academic Editor: Mohammad Tawfik

Copyright © 2017 Vincent O. S. Olunloyo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents an integral transform analytic solution to the equations governing a fluid-conveying pipeline segment where a gyroscopic or Coriolis force effect is taken into consideration. The mathematical model idealizes a segment of the pipeline as an elastic beam conveying an incompressible fluid. It is clearly shown that when such a system is supported at both ends and in a free motion, the Coriolis force dissipates no energy (or simply does not work) as it generates conjugate complex vibratory components for all flow velocities. It is demonstrated that the modal natural frequencies can be computed from the algebraic products of the complex frequency pairs. Clearly, the patterns of the characteristics of the system’s natural frequencies are seen partly when the real and imaginary components are plotted, as widely seen in the literature. Nonetheless, results from this study revealed that a continuity profile exists to connect the subcritical, critical, and postcritical vibratory behaviours when the absolute values are plotted for any velocity. In the meantime, the efficacy and versatility of this method against the usual assumed spatial or temporal modal solutions are demonstrated by confirming the predictions and validity of results of earlier workers such as Paidoussis, Ziegler, and others where pre- and postdivergence behaviours are exhibited.

#### 1. Introduction

Fluid-conveying pipes are parts of the most common engineering examples of slender systems interacting with axial flows; another good example is the deployment of flexible conduits in the oil and gas exploration and production industry. A compendium of other examples can be found in Paidoussis [1] work spanning over the last 50 years. The list of examples is not limited to the field of engineering but cuts across other areas of human endeavor such as the study of pulmonary and urinary tract systems or even haemodynamics within human physiology.

These problems have generated a lot of research interests over the years partly because some served as models for studying the stability of certain classes of dynamical systems leading to the development of novel numerical and analytical methods for solving such problems. It has also turned out, over the years, that several of these techniques have found wider applications in other areas of research that otherwise appeared unrelated and have in fact occasionally led to the development of unanticipated practical applications and devices. Thus for linear dynamics for axial flows along slender structures, the pipe conveying fluid is regarded as the main paradigm. It also serves as model for classical problems involving axial momentum transport or axially moving continua such as high speed magnetic and paper tapes [1].

However, a careful study of the development of this area of research revealed that most of the research interests were curiosity-driven as some of the interesting phenomena observed occurred at velocities and conditions outside practical engineering and operational working limits as at that time [2]. This however gradually changed with the study of high velocity flow within light-gauge piping used in rocket engines and stability problems experienced by oil pipelines at modest conveyance speeds. Nowadays, people are looking at areas of direct applications, for example, in the behaviour of aspirating pipes for ocean mining and LNG in-situ production. This concept is to be utilized for the proposed offshore mining of methane liquid-crystal deposits and carbon sequestration. Here, the interest is in flow-induced vibrations and instabilities that can arise at high flow rates.

Furthermore, with the advent of High Pressure-High Temperature (HP-HT) oil and gas exploration and production, the lengths of the flexible risers deployed surely qualify them as hoses or pipe strings whose vibration behaviour is of interest in the exploration field.

In the modeling of the mechanics of fluid-conveying pipes, one of the terms that has received considerable attention over the years is the Coriolis force that was assigned the role of energy absorption that counters the centrifugal effect that normally arose in free motions. Broadly speaking, such an energy absorption affects the stability in conservative and nonconservative systems, as was shown by Section of Elishakoff’s work (2005) [2]; Öz and Boyaci (2000) [3]; Szmidt and Przybyłowicz (2013) [4]; Askarian et al. (2014) [5]; Kuiper and Metrikine (2004) [6]; Chellapilla and Simha (2008) [7]. Other contributors that attempted to investigate and explain the behaviour included Leklong et al. (2008) [8]; Al-Hilli and Ntayeesh (2013) [9]; Guo et al. (2006) [10]; Zhang et al. (2000) [11]; Modarres-Sadeghi and Païdoussis (2009) [12] as well as Ibrahim (2010) [13].

Principal among these findings is that while the centrifugal force imparts energy to the system, the Coriolis force absorbs energy from the system, such that the balance between the two, in the absence of dissipation, gives rise to flutter. However, when confronted with a nonconservative system the effect of the Coriolis force can lead to destabilization. These conclusions were arrived at partly on experimental work as can be found in [14, 15]. However, in the 1950s, proofs were claimed of researchers and mathematicians’ findings showing that the Coriolis has negligible effects even on the molecular scale interaction.

Part of the interest in this present study rests on the fact that previous explanations in literature as to the effect of the Coriolis force on the natural frequency of the system derived from ad hoc heuristic arguments accompanied sometimes by authoritative and masterly analyses and interpretation of results of numerical and experiments. What is however missing is adequate proof based on the results of blind solutions of the unabridged governing differential equations for the linear problems as simple examples or justifications of such rationalizations.

A separate but issue related is that although the gyroscopic (Coriolis) forces do no work in the course of free motions, they nonetheless exert important influences on the overall dynamical behaviour of a pipeline system as pointed out in [1]. It would therefore be useful to know exactly what their influences are in such cases.

Another issue that has arisen over the years is that of the efficacy of the methodology. To be sure, several methods have been used to tackle the class of problems associated with the dynamics of fluid-conveying pipes but prominent amongst these is the original work of Gregory and Paidoussis and the sequel as presented in [1] where the use of an eigenfunction expansion in a modified Galerkin scheme was introduced. Part of the initial challenge was the absence of computers and the availability of validation modules to check the results of numerical work. With the development of the Finite element method more emphasis was placed on numerical schemes and laboratory experiments were framed up to confirm the predictions of these studies. Some of the other methods used for analyses over the years included the spectral method, for example, deployed by Lee and Park (2006) [16] or the differential transformation method recently applied by Qiao et al. (2006) [17].

Other recent works are those of Dodds Jr. and Runyan (1965) [18], who used flow visualization and velocity measurement to experimentally clarify the mechanism underlying the fluid-induced vibration in double T-junction of pipeline systems; Yamaguchi et al. (2016) [19], who used a solution method based on the Frobenius power series on a derived asymptotic model from the solutions of the Pridmore-Brown equation for the Fourier transform of the vibrational fluid pressure; Kutin and Bajsić (2014) [20], who use smart materials; and Jweeg and Ntayeesh (2015) [21] who made use of application of method of multiple scales to analyse approximately the gyroscopic system for a nonlinear fluid-conveying pipeline.

This paper further establishes the method of complex integral transforms (where the cosine and sine transforms are special cases) as one other effective method that can be used to tackle such problems within the context of linear theory for a start and is organized as follows. Section 1 introduces the problems under investigation. In Section 2, the analysis of the pipeline conveying an incompressible fluid with the governing partial differential equation and appropriate boundary conditions are presented. In Section 3, the complex and natural frequencies of the system are computed together with the relationship of the system’s natural frequency with the flow critical velocity. Section 4 analyses the dynamic responses for purely elastic pipe in three cases, namely, simple supports at both ends, cantilever pipe, and a clamped-pinned ends pipe. In Section 5, results are analysed and discussed and Section 6 concludes the paper, whilst references are listed in the final section and at the end found the Appendices A and B.

#### 2. Analysis of the Pipeline System Conveying an Incompressible Fluid

The homogeneous Partial Differential Equation (PDE) governing the flow-induced vibration of a pipeline conveying an incompressible fluid is given bywhere is the flexural rigidity of the pipe, and are the mass per unit length of pipe and fluid, respectively, flowing with a steady flow velocity , and is the lateral deflection of the pipe. The parameters and are the axial coordinate and time variables, respectively.

In literature of fluid structure interaction mechanics, where fluid-induced vibrations are studied, (1) is often nondimensionalized aswherewhereas, for the present study, the dimensionless form is given bywhereand the components involved are, respectively, the restoring, inertia, bending/centrifugal, and Coriolis force terms.

For comparison purposes, we present the following.

*Case 1. *We expect that if the pipeline segment is conveying a fluid and an eventual situation calls for the valve to be closed or the pump/compressor shut-off, an entrained fluid, be it hot, cold, pressurized, or otherwise, will be trapped in the pipe. When the flow velocity (2) based on the nondimensionalizing method in the literature becomesleading towhich shows that there is no fluid inside the pipe if the flow velocity is zero.* This is not to be so*. However, considering (4), based on this paper’s dimensionless method, with , it becomesleading toRevealing that, the system’s vibration, configuration, and response are strongly affected by the entrainment fluid in the pipe.

*Case 2. *If there is no fluid in the pipe, that is, , (2) and (4), respectively, becomeleading toThe literature method (10) is showing that when no fluid is present in the pipe, there is still a flow velocity. This is not possible and is likely a fundamental error in physics. However, (12) shows that the restoring and the inertial accelerations in dimensionless form are the balancing vectors.

*Case 3. *We now examine the governing equation (1) in its original form without nondimensionalizing, such that, when , it becomesleading toIn the absence of fluid in the pipe, that is, .leading toIt is confirmed that (13) is similar to (8), demonstrating that the natural frequency is dependent on the mass or density of fluid flowing in the pipeline.

Now, the following definitions hold for the Fourier complex integral transforms Wrede and Spiegel [22], Olayiwola [23], and Jeffrey [24]; namely, such that in this caseUsing (18) on (4), the following governing equation ensues in the transforms plane:subject to simply supported conditions at both ends; namely,In conjunction with the following conditions:Substituting (20) into (19) leads to the following ordinary differential equation (ode) in the transform plane:This is a nonhomogeneous second-order ordinary differential equation in time domain. We can now proceed to solve for the frequency and displacement responses.

#### 3. Complex and Natural Frequencies

In order to find the natural frequencies of the system we seek to solve the complimentary equation of the system in its Fourier complex transform plane by using the trial solution, namely,to obtain the following characteristic equation:The preceding equation can now be solved for the roots of s to obtain complex conjugate pairs of the forms:In order to isolate the effect of the Coriolis force, we introduce the expression into (25). On comparing these equations with the natural frequency, the relative frequencies are related as follows:whereMoreover, the product of the conjugate pairs gives the natural frequencies of the system; namely,that is,Although, from mathematical physics, the complex notation “” indicates that is acting perpendicularly to the natural frequency , nonetheless, an algebraic functional relation can be deduced as follows.

Substituting a square of (26) into (30) givesOn expanding, it yieldsBut for , or with , (24) tends to a limit; that is,The first observation to make from relations (18) and (22) is that the natural frequency is not in the same component with the frequency due to the Coriolis force.

Having noted this, it is also useful to conform with the general practice in the literature by expressing the result for the natural frequency, where possible, in a way that relates it to physical parameters or benchmarks associated with the flow. Thus from relation (19), it is straightforward to deduce the following.

(a) The critical axial velocity at which the natural frequency of the system is zero satisfies the relationand defines the condition for the onset of irregular oscillations.

(b) When there is no axial flow, that is, , the natural frequency of the system satisfies the relation

(c) The general relation for the magnitude of the natural frequency can be rearranged asThis is a simple expression that relates the natural frequency to the critical flow velocity. Equation (34) actually proves the existence of such relationship as it could be argued that the results of the experiment of Dodds Jr. and Runyan (1965) [18] provide indirect evidence of the existence of such a relationship for the eigenfrequency mode .

Furthermore, evaluation of the critical velocity and fundamental frequency can be carried out by substituting the appropriate eigenvalues into (24). Thus it can be asserted that, for this case, this method facilitates the derivation of explicit closed form expressions for possible design parameters such as the critical velocity. This method of solution also sets the stage for deriving equivalent results for and for the dynamic response when other effects such as damping for example are included.

#### 4. Dynamic Response Analysis for Purely Elastic Pipe

If we consider the dynamic response of the simple system of a purely elastic horizontal pipe with uniformly distributed loads of Newton per unit length, then (4) is transformed as follows:whereThe general solution for the deflection response of the system is hence given aswhereTherefore, in the Fourier plane, the dynamic response is obtained aswhereWe now examine the deflection responses for three cases of horizontal pipeline system with regard to solution (40) above.

*Case 1. * *Pipe with Simple Supports at Both Ends*. In this case, as shown in Figure 1, the initial configuration of the pipe before it is dynamically excited shows that deflection is symmetric about the middle of the beam (or the pipe).

For this case, when the system is at the static state, time , the flow velocity is also zero, though the pipe might have trapped some fluid within it. (See Appendix for further analysis.)

The deflection function is given by Nash [25]:where and .

Now, (37) reduces toso thatThe inverse Fourier transform is given asWith and using (45) and (40) the response becomesThat is,On enforcing the dynamic boundary conditions, namely, ; , the system’s dynamic response is given byorwhereTo enable us understand fully the characteristics of the natural frequencies of such a pipeline system, the three scenarios, namely, critical, subcritical, and postcritical flow points, are examined; that is, comparing (30) and (34), we deducewhereThe arguments for computing the residues in (40) are , whenNonetheless, a critical flow point is attained for any corresponding velocity, when (47) is zero; that is,For a subcritical flow , this corresponds to the characteristics of the real part of natural frequency against flow velocity as normally seen in literature; that is,and for any postcritical flow point, which corresponds to the characteristics of the imaginary part of natural frequency against flow velocity; that is,The point to note here is that, in actual practice, continuity must exist. As such the natural frequency cannot be zero perpetually for postcritical flow velocity. This necessitated the essence of the plot of absolute characteristics of the natural frequency for all regimes of flow, as demonstrated in this paper. These three scenarios are described graphically in Section 5.