## Abstract

In this paper, an advanced three-dimensional (3D) computational fluid dynamics (CFD) model is used to analyse the steady-state hydrodynamics of laminar flow through an extended partial blockage (PB) in a pressurised pipe. PB corresponds to one of the main faults affecting pipelines. In fact, it reduces its carrying capacity with economic consequences, and as it does not give rise to any external evidence, its detection can be very challenging. The performance of the model is evaluated by comparing the numerical results with the available experimental data from the literature. Subsequently, the velocity and pressure distributions are analysed, and the main features of the flow field are described in terms of both local and global dimensionless parameters. Furthermore, the behaviour of the discharge coefficient is also investigated. The obtained results confirm that steady-state measurements can identify the presence of PB and follow its evolution over time but cannot detect its location and size. On the other hand, the location and severity of PBs can be provided by means of transient tests.

## HIGHLIGHTS

Analysis of the steady-state laminar flow through an extended partial blockage.

Validation of a 3D CFD model of an extended partial blockage with experimental data from the literature.

Velocity and pressure fields characterisation around an extended partial blockage.

Characterisation and evaluation in time of an extended partial blockage.

Hydrodynamics of a flow through an extended partial blockage.

## INTRODUCTION

Pressurised pipe systems allow conveying fluids – such as water, gas, and oil – over long distances; in addition, the venous and arterial systems can also be assimilated into them. In pipe systems, partial blockages – hereafter referred to as PBs – may result from the deposition of sand and excess calcium, in the case of water pipes, and hydrates, paraffin and asphaltene, in the case of refined and crude oil pipes; plaques and clot may happen in the venous and arterial systems. PBs may also result from simultaneous processes with solid deposition triggered by the previous formation of other solids (i.e*.*, asphaltene or hydrates after wax deposition). PBs start from small growths in the roughness of inner pipe walls and, if not detected and addressed early on, they continue to grow and protrude transversely and longitudinally. Consequently, they may encroach on a significant part of the cross-sectional area of the pipe (Duan *et al.* 2015). In subsea pipelines, as an example, the cold temperature of the pipeline walls causes the wax particles in the oil to crystallise and accumulate over time on the inner pipe walls, eventually leading to a PB and potentially complete blockage, if left untreated. The circumferential shape of ‘natural’ PBs is quite common as the result of a radial protrusion. PBs may also be due to mechanical causes, such as collapsed pipes or negligently partially closed in-line valves. Due to such a variety of causes of formation, PBs can occur anywhere in the pipelines. PBs are such a widespread concern, that they are considered the most insidious fault to detect and manage, in both water and petroleum industries.

In gravity systems, PBs reduce the carrying capacity of the pipes, leading to a smaller discharge, whereas, in rising mains, they increase energy consumption and cause safety risks due to pressure build-up. This is also the case of plaques in the arterial system if left undetected and untreated. Accordingly, early detection is considered the best approach for maintenance purposes.

Since PBs do not provide external evidence (unlike, as an example, leaks), their detection and characterisation, i.e., the evaluation of their size and location is an arduous challenge. In addition, considering the fact that, in most cases, pipelines are buried, any intervention is highly expensive and time-consuming. Furthermore, the ‘nature’ of the PB – i.e., its consistency (hardness) – plays a crucial role in identifying the most appropriate rehabilitation technique. If the PB cannot be removed by local flushing/chemical injection, then the blocked pipeline section must be completely replaced. In pipelines conveying liquids, on which attention is focused in this paper, several methods have been proposed for detecting PBs: vibration analysis (Lile *et al.* 2012), pulse-echo methodology (Duan *et al.* 2015), acoustic reflectometry (Papadopoulou *et al.* 2008), pressure measurements in steady- (Yang *et al.* 2019), and unsteady-state flows (Meniconi *et al.* 2011; Duan *et al.* 2014; Louati *et al.* 2018). The last method is discussed here in more detail as connected to this paper.

Detection methods by using pressure unsteady-state measurements, the so-called transient test-based techniques (TTBTs), are based on the properties of the pressure waves generated by a manoeuvre (Meniconi *et al.* 2018; Keramat *et al.* 2019). Such pressure waves travel along the pipeline, back and forward, reflected by any anomaly (e.g., leaks and PBs), as well as boundaries. The location of any anomaly, and then a PB, can be determined on the basis of the timing of the reflected pressure wave measured at suitable sections. Accordingly, this technique is also called time-domain reflectometry. The performance of TTBTs depends significantly on the modalities of transient tests. In particular, the characteristics of the generated pressure wave – the sharper, the better (Brunone *et al.* 2021) – the accuracy of the pressure transducers, the frequency of sampling – the higher, the better – and the steadiness of the pre-transient flow conditions play an important role (Meniconi *et al.* 2016). Precisely, as shown in Brunone *et al.* (2021), for a given measurement section, the smaller the value of the pre-transient Reynolds number, the easier the evaluation of the reflected pressure wave. Accordingly, as a pre-transient condition, it is appropriate to set a very small value of the Reynolds number corresponding to a small value of the mean velocity, for a given liquid and internal pipe diameter. Such a circumstance makes laminar flow very attractive as an appropriate pre-transient condition.

Within TTBTs, according to the mentioned mechanisms of interaction between pressure waves and PBs, a distinction can be made between ‘discrete’ and ‘extended’ PBs in terms of their length (Brunone *et al.* 2008). Precisely, at a partially closed in-line valve or local diameter reduction, as examples of discrete PBs, the incoming pressure wave gives rise to a single reflected pressure wave. On the other hand, in the case of a small bore pipe, as an example of the extended PBs, the transient response is bell-shaped (Meniconi *et al.* 2012). In such a latter feature, the pressure rise is due to the interaction of the incoming pressure wave with the shrinkage of the opening area, whereas the successive pressure drop is generated by the interaction with the expansion of the opening area. An alternative terminology defines ‘discrete’ and ‘extended’ PBs as ‘sharp-edged’ and ‘thick-walled’ (or ‘short tubes’) orifices, respectively.

This paper aims to accurately simulate the pressure and velocity distribution around an extended PB by means of a three-dimensional (3D) computational fluid dynamics (CFD) model. The performance of the model is evaluated on the basis of the experimental data available in the literature. As pointed out above, attention is focused on laminar flow conditions. The rationale for this approach is that the use of one-dimensional (1D) model results is quite inadequate since only the mean values of the pressure and velocity are provided. As a consequence, the 1D approach prevents the identification of possible complex structures (i.e., eddies) downstream of the PB that qualify the flow field and influence the interaction mechanisms with the pressure waves. On the other hand, the 3D CFD model allows capturing in any detail the pressure and velocity distribution around the PB, both upstream and downstream. With respect to Martins *et al.* (2021) – where the PB length is assumed as negligible and then a sharp-edged orifice is considered – in this paper, a much more complex flow field is explored, which indeed resembles real extended PBs.

This paper is organised as follows. The quantities that characterise the hydrodynamic behaviour of extended PBs, as well as experimental data used to validate the performance of the 3D CFD model, are reported in Section 2, along with a description of the 3D CFD model, with attention to the setup and convergence assessment phases. In Section 3, firstly, the 3D CFD model results are compared with the selected experimental data. Subsequently, the numerical model is used to highlight the main features of the flow through extended PBs for different flow conditions and geometric characteristics. In such a context, the weakness of the approach based on steady-state pressure measurements is pointed out. Finally, Section 4 summarises the main conclusions about the 3D CFD model results.

## METHODS

### Flow through extended partial blockages

As mentioned, in this section, attention is focused on the analysis of the available experimental results concerning extended PBs in laminar conditions. Subsequently, such results are used for evaluating the performance of the 3D CFD model.

*et al.*(2006), any feature, , characterising the steady-state hydrodynamics of the flow through extended PBs can be expressed by means of the following dimensionless relationship (Figure 1):where is the length-to-diameter ratio,with

*L*is the PB length measured in the flow direction, and

*d*is the orifice's inner diameter (aperture). The diameter ratio, , is given by:with

*D*representing the pipe's inner diameter. The Reynolds number describes the flow in the pipe:with

*U*being the mean flow velocity in the pipe, and being the fluid kinematic viscosity; alternatively, the orifice Reynolds number, (=, with being the mean velocity at the orifice) is also used. The Euler number,takes into account the ‘net pressure drop’ (Figure 1) through the orifice (Hasegawa

*et al.*1997), , with being the fluid density. The discharge coefficient, , describes globally the flow through PBs:whereas the global local energy dissipation can be evaluated by considering the local head loss coefficient:that is linked to by the following relationship:

According to the value of , orifices are classified into: (i) sharp-edged: ≤ 0.125; (ii) thick-walled: 0.125 < < 2; and (iii) short tubes: ≥ 2 (Tu *et al.* 2006; Jankowski *et al.* 2008).

In Martins *et al.* (2021), for sharp-edged orifices, the performance of the 3D CFD model was tested based on laboratory data provided by Zampaglione (1969), Johansen (1997), Jankowski *et al.* (2008), and Alvi *et al.* (1978). On the other hand, as mentioned, this paper focuses on thick-walled orifices and short tubes (both assimilable to extended PBs).

As a necessary premise, it is the case to point out the quite small number of laboratory tests available in the literature for laminar flow conditions. Moreover, only global quantities – i.e., and – and not pressure or velocity distributions – are provided. Accordingly, there are considered data from the experiments executed by Hasegawa *et al.* (1997), Kiljanski (1993), and Sahin & Ceyhan (1996), whose main characteristics are synthesised in Table 1.

. | Diameter ratio . | Length-to-diameter ratio . | Orifice Reynolds number . |
---|---|---|---|

Hasegawa et al. (1997) | |||

Kiljanski (1993) | |||

Sahin & Ceyhan (1996) |

. | Diameter ratio . | Length-to-diameter ratio . | Orifice Reynolds number . |
---|---|---|---|

Hasegawa et al. (1997) | |||

Kiljanski (1993) | |||

Sahin & Ceyhan (1996) |

### CFD model

In this section, the governing equations and open-source software OpenFOAM (OF) 3D CFD model are described.

#### Governing equations

*t*is the time, and is the velocity components in the spatial coordinate (with = 1, 2, 3). The conservation of momentum principle gives rise to the Cauchy equations:where is the Cauchy stress tensor, and is the mass forces per unit mass acting on the fluid.

*p*is the thermodynamic pressure, is the Kronecker function, is the dynamic viscosity, and (=− 2/3) is the second viscosity coefficient. Inserting Equation (11) – i.e., the stress–strain law – in the momentum equation, yields the general form of the N-S equations for an isotropic and Newtonian fluid:

To close the problem, initial and boundary conditions are needed to be specified in space and time (Denton & Dawes 1998).

#### Model setup

3D CFD simulations, carried out using OF, involve three key stages: pre-processing, simulation, and post-processing.

The pre-processing stage includes the fluid domain definition, mesh generation and model setup, with the definition of boundaries and solver parameters.

The mesh used in the simulations considers the parameters, , , and proposed by Martins *et al.* (2014), with = 0.57, = 0.009, and = 0.0005, where the subscripts a, c, and r indicate the axial, circumferential, and radial direction, respectively. Since high-velocity gradients are expected, meshes herein present a much larger number of cells with respect to Martins *et al.* (2014), with mesh refinements not only at the PB but also in the zone near it. This is a necessary premise for the following simulation in transient conditions. In fact, even though the upstream flow is laminar, downstream of the PB the flow is much more complex, with recirculation and consequently large circumferential eddies (Zampaglione 1969; Martins *et al.* 2021). This points out the necessity for extra mesh refinements to capture the interaction with the travelling pressure waves, such as those generated for the PB detection. The assumed boundaries conditions are defined in Figure 2: the constant pressure at the inlet, the no-slip condition at pipe walls, and the velocity distribution at the outlet. The local head loss completes (Figure 2) at a distance downstream of the PB equal to the length of the expansion cone, , where the flow reattaches the pipe wall. The flow region downstream of the PB (coloured in dark grey in Figure 2), surrounded by swirls, is called ‘*corrente viva*’. The area of the ‘*corrente viva*’, , increases with the distance from the orifice; at , it is , with *A* being the pipe cross-sectional area. The pre-processing stage divides the fluid domain into smaller, non-overlapping subdomains generating a mesh of cells in which Equations (9) and (10) will be numerically solved in the next stage. In the second stage, in OF, the N-S equation and continuity equations are solved by the pressure-based solver rhoSimpleFoam, within the finite volume method. RhoSimpleFoam considers the relationship between pressure and velocity adjustments to carry out the mass conservation to obtain the pressure field. The convergence of the numerical solution can be assessed by progressively tracking the imbalances (residuals). This imbalance measures the overall conservation of the flow properties. The solution is assumed to have converged when the residual values drop below . Finally, in the third stage – the post-processing stage, the obtained results, validated by considering the available experimental data, allow analysing the flow upstream and downstream of the PB in detail.

## RESULTS AND DISCUSSION

### CFD results vs. selected experimental data

*et al.*(1997).

The above positive results in terms of the performance of the 3D CFD model authorise exploring both the global (e.g., in terms of *k* values) and local (i.e., in terms of pressure and velocity distributions) behaviour of extended PBs.

### Simulated scenarios

3D CFD simulations concerned three values of (=0.214, 0.425, and 0.697) for both thick-walled ( = 0.125, 0.5, and 1) and short-tube orifices ( = 2, and 5). In terms of , the numerical experiments range between 1 and 350; for each , the maximum value of is smaller than the critical value, , i.e., the value above which pulsating phenomena occur (Zampaglione 1969). Precisely, it is 110 < < 120 for = 0.214; 250 < < 300 for = 0.425; and 350 < < 400 for = 0.697. The main characteristics of the numerical simulations are reported in Table 2.

Diameter to ratio . | Length-to-diameter ratio . | Reynolds number . |
---|---|---|

Diameter to ratio . | Length-to-diameter ratio . | Reynolds number . |
---|---|---|

### The global behaviour of extended PBs

From this plot, it emerges the indelible mark of PBs in terms of the local head loss. However, the steady-state measurements do not allow identifying nor the length, , nor the severity, , of the PB. In fact, for a given , if an unexpected is measured between two sections, it is not possible characterising the PB. In fact, a given value of *k* may be due to different values of and . In other words, steady-state measurements allow identifying the existence of a PB, following its evolution in time, but not detecting its location and size. Such a result implies the need of using TTBTs for PBs identification and characterisation.

### Flow field analysis

In Figure 8, (=0.5) and (=100) are kept constant, whereas the value of varies (=0.214, 0.425, and 0.697). In Figure 9, constant values of (=0.425) and (=100) are assumed, for different values of (=0.5, 1, and 2).

_{,}and . A further confirmation is offered by curves in Figures 11–13, where the behaviour along the pipe axis of the following global parameters is reported:the kinetic energy correction factor, as the velocity axial component,

*u*, is predominant, with being the dimensionless axial coordinate,the power pressure, and the pressure distribution variance,where is the mean pressure at the upstream section.

These plots confirm the above-mentioned results concerning the velocity and pressure distributions along the pipe axis. Precisely, in the considered cases, the behaviour of , , and does not depend on the values of the parameters characterising the extended PB.

*et al.*(2008), the behaviour vs. is different for sharp-edged orifices with respect to thick-walled ones and short tubes. Precisely, for a discrete PB, due to the effect of the vena contracta, achieves a maximum value and then it slightly decreases. On the other hand, for an extended PB, increases and then it reaches an asymptotic value. The maximum values of vs. are reported in Figure 15.

## CONCLUSIONS

In this paper, the results of a three-dimensional (3D) CFD model describing the steady-state hydrodynamics of the laminar flow at an extended PB are presented.

As a necessary premise, in the first part of the paper, the results of the 3D CFD model are compared with the available experimental data. For all considered cases, there is a good fit between the numerical results and experimental data. Subsequently, the velocity and pressure distributions, provided by the 3D CFD model for a wide range of values of the Reynolds number, , length-to-diameter ratio, , and diameter ratio, , are presented. Such results have been compared by considering both local and global parameters. In such a context, the different behaviour of sharp-edged orifices (discrete PBs) and thick-walled orifices and short tubes (extended PBs) is highlighted. The discharge coefficient, , as an example, increases rapidly with for discrete PBs, whereas the maximum value is reached asymptotically for extended PBs. Such a behaviour has clear implications on the transient response of PBs and the related analysis performed within TTBTs.

## ACKNOWLEDGEMENT

The authors are grateful for the support of Fundação para a Ciência e Tecnologia, research projects: Flow Dynamics in Storage Tanks (CPCA/A2/2568/2020) and IMiST – Improving Mixing in Storage Tanks for Safer Water Supply (PTDC/ECIEGC/32102/2017). The authors also acknowledge the support of the Università degli Studi di Perugia by the Fondo di Ricerca di Ateneo (Edizione 2021).

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.