1 Introduction

In recent years, a new technology called piggyback pipeline has been gradually adopted to ensure the transmission of monitoring signals during the process of marine oil and gas exploitation. This technology is intended to enhance the performance and safety of submarine pipelines, which are usually strapped together at certain intervals and laid on the seabed as a bundle. One of the most popular configurations of pipeline bundles consists of a main pipeline and a small pipeline, as shown in Fig.1.This configuration is called the piggyback pipeline. Owing to its advantages over the traditional single pipeline,the piggyback pipeline has been used increasingly for economic and technical purposes. As a result of changes in geometric configuration, the application of the piggyback pipeline has induced numerous problems in fluid dynamics. Compared with the traditional single pipeline,the piggyback pipeline encounters complications in sediment transportation and flow structure because of the presence of the small pipeline. For instance, the interaction of complicated shedding vortices between the main and small pipelines can increase the scour depth in comparison with that of the traditional single pipeline. Therefore, research on the local scour around the piggyback pipeline, particularly on its optimum configuration design,is necessary and valuable. Studies on piggyback pipelines are also significant to protect them in complex marine environments.

Owing to a wide range of Engineering practices, the local scour and hydrodynamic loadings of the traditional single pipeline under both current and wave-current interaction have been extensively studied through numerical and experimental work. Reviews of this subject can be found in many published papers (Chiew, 1990; Brors,1999; Sumer and Fredsoe, 2001; Sumer et al., 2001a, b;Myrhaug et al., 2009; Zakeri et al., 2009; Zhou et al.,2014; Postacchini and Brocchini, 2015; Zhao et al., 2015).The scour around the submarine pipeline involves two stages: tunnel scour in the early stage and lee wake scour in the late stage. After a systematic study of the scour depth below the submarine pipelines in clear water, Dey and Singh (2007) found that a critical ratio between the seepage velocity and main flow velocity corresponds to the minimum scour depth. With the development of research technologies, the influences of multiple factors,such as current, wave, and sediment properties, on the scour around the pipeline have been investigated. Sudhan et al. (2002) conducted an experiment to analyze wave-induced pressures on a pipeline buried in a permeable seabed, and found that high-pressure values were recorded at the top and low-pressure values were recorded at the bottom. Zhang et al. (2015) established a 3D poro-elastoplastic soil model to study the local scour of submarine pipelines in which both oscillatory and residual mechanisms can be simulated at the same time. The researchers found that, as the seabed scour depth increases, the normalized pore pressures decrease rapidly in the upper part of seabed, and then change slightly in the lower part.Cheng et al. (2014) conducted an experiment on 3D local scour below a rigid pipeline subjected to both wave and wave-current interaction. The effects of flow incidence angle, flow ratio (steady current velocity versus combined wave/current velocity) and pipeline initial embedment depth on the free span expansion rate were investigated.To control the degree of the scour below the pipeline, a spoiler is proposed to be installed on the pipeline; many studies have been conducted to analyze the influence of the spoiler on the local scour of the single pipeline. The results showed that implementing the spoiler can enhance the scour depth in both steady currents and waves (Yang et al., 2012a; Zhu et al., 2013; Zhao et al., 2016).

Although numerous studies have been conducted to investigate the local scour and hydrodynamic characteristics of the submarine single pipeline, the process of local scour evolution and its mechanism around the piggyback pipeline are not sufficiently understood. To the best of our knowledge, limited simulations and few experiments have been published on the scour of the piggyback pipeline under steady current, let alone under wave-current interaction. Through a simulation conducted based on a finite element solver with a κ-ω turbulence closure, Zhao et al.(2007) found that the relative position of the small pipe to the main pipe has important effects on the vortex shedding characteristics and hydrodynamic loadings of the piggyback pipeline. Liang and Cheng (2008) employed a 2D numerical model to investigate the scour hole around the piggyback pipeline. In their study, the piggyback pipeline was analyzed using only an equivalent pipeline without considering the mutual influences between the main and small pipelines. Cheng et al. (2013) applied a 2D hybrid numerical model to analyze the hydrodynamic loadings at a submarine piggyback pipeline; results indicated that the force coefficients initially decrease and then remain constant when the gap ratio between the gap height beneath the pipe and the diameter of the main pipeline is beyond 0.5. Zang and Gao (2014) conducted a series of experiments on the vibration of piggyback pipelines induced by steady current close to a plane seabed with a hydro-elastic facility in a conventional water flume.The effects of the mass-damping parameter, diameter ratio,gap-to-diameter ratio, spacing-to-diameter ratio, and position angle on the vortex-induced vibration responses were studied. The results showed that the peak vibration amplitude of a near-wall piggyback pipeline is less than that of a wall-free single pipeline.

In the present study, the local scour around the piggyback pipeline under steady current has been systematically investigated through both numerical simulations and laboratory experiments. The effects of the prominent factors, such as inflow velocity, pipe diameter, and gap ratio on the evolution of the local scour, have been studied in the laboratory experiments. In addition to the corresponding experimental work, a sediment scour model has been developed based on a two-phase flow solver, which has been used to analyze the hydrodynamic characteristics of the piggyback pipeline during the evolution process of the scour hole under it. The air-water interface has been captured with the volume-of-fluid (VOF) method.Then, water-sediment interface behavior has been determined with the moving-mesh method (Zhang and Shi,2014). Furthermore, a theoretical formula to determine the local scour depth has been derived with consideration of the gap ratio, pipe diameter, flow velocity, and other factors. The feasibility of the theoretical formula in predicting the scour depth has also been calibrated by the experimental data and simulated results.

The rest of this paper is organized as follows. The theoretical analysis of the local scour around the piggyback pipeline is discussed in Section 2, where the theoretical formula to predict the maximum scour depth of the piggyback pipeline is derived in detail. Section 3 presents the details of the experimental setup and the corresponding results. Numerical methods and simulated results are reported in Section 4. The feasibility of the proposed formula to predict the scour depth is described in Section 5. Finally, Section 6 summarizes the findings of the study.

2 Theoretical Analysis

2.1 Description of Scour Phenomena

When water flows past a pipeline, the pressure difference between the stagnation pressure upstream and the low pressure in the separation zone induces a hydraulic gradient on the seabed. When this hydraulic gradient exceeds the flotation gradient of the sediment, piping may occur. The piping can substantially reduce the sediment transport threshold, and the upward flow of water associated with it can also offset the effective weight of the sediment. This phenomenon helps the upstream stagnation eddy to breach the sand barrier under the pipeline and causes the onset of scour (Chiew, 1990). With the sediment transport on the seabed under the submarine pipeline, the clearance between the seabed and pipeline grows and extends in both horizontal and vertical directions around the pipeline. Finally, a scour hole is formed.When the clearance is small, the current speed between seabed and pipeline could reach several times the upstream velocity, and the shear stress on the sediment bed significantly exceeds the stress threshold of sediment transport, thereby increasing the rate of sediment transport. When the evolution of the scour reaches a certain stage, the surface pattern of the scour hole does not change overall. Finally, the scour equilibrium is obtained.

2.2 Theoretical Analysis of Scour Equilibrium

When the scour is investigated in the clear water with-out sediment supply upstream, the sediment transport stops as the scour reaches the equilibrium state. The flow field and local scour hole around the piggyback pipeline are depicted in Fig.1.

Fig.1 Schematic of velocity distribution around piggyback pipeline.

Owing to the coexistence of the small pipe, the blocking area of the piggyback pipeline increases. This condition leads to an increment of flow velocity and sediment transport rate below the pipeline. Compared with the traditional single pipeline, the piggyback pipeline has a scour hole length that increases severely along the flow direction. Thus, the dynamic angle of repose under the piggyback pipeline is smaller than that for the single pipeline.

Based on the scour equilibrium model, the following formula for predicting the scour depth under the traditional single pipe was derived by Yang et al. (2012b):

where H is the scour depth; n is the index number of velocity distribution in the scour hole, D is the pipe diameter, u0 is the incoming velocity on the height of the pipe axes, ds is the sandy particle diameter, h is the clearance height between the stagnation point and sediment bed, ues is the erosion-stopping velocity, and β is the index number of velocity distribution. For the open channel flows, β is always chosen as 1/6, as suggested by Chen (1991).

In the work of Yang et al. (2012b), however, the stagnation height h in Eq. (1) is not explicitly given. Besides,compared with the scour under the single pipeline, the scour hole under the piggyback pipeline is also affected by two extra factors including gap ratio and the diameter of small pipe. The formula proposed by Yang et al. (2012b)is no longer suitable and accurate for predicting the scour depth of the piggyback pipeline. Thus, proposing an exact formula to predict the scour depth of the piggyback pipeline is necessary. In the rest of this section, the derivation of the formula to predict the scour depth of the piggyback pipeline is demonstrated in detail.

When the mass conservation equation is applied to derive the predicting formula for scour depth, the volume of flow through the scour hole should be confirmed. When the flow approaches the piggyback pipeline, the flow is separated at a point and crosses the pipeline from the upside and downside. The downside flow is through the scour hole. The demarcation point of the flow is where the stagnation point locates. Thus, to calculate the downside flow, the stagnation point should be determined first. In view of the experimental work in the present study, the height of the stagnation point varies under the variation of the pipeline diameter, Reynolds number, and gap G between the small and main pipes. Owing to the minimum velocity at the stagnation point, the location of minimum velocity can be measured by using an acoustic Doppler velocimeter (ADV). The height of the stagnation point can be decided by adjusting the vertical position of the ADV. After the analysis of stagnation positions measured in the experiment (as shown in Section 3.3.3), the height of the stagnation point can be evaluated as

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In this case, D is the main pipe diameter and d is the small pipe diameter.

The velocity distribution from the seabed to the location of h in section A-A can be formulated as

where y is the vertical coordinate with respect to the horizontal seabed, and U(y) is the velocity at the location of y. Based on Eq. (2), the stagnation point velocity U0′at the location of h in section A-A can be expressed as

where u0 is the inflow velocity in the main pipe centerline(y=D/2).

According to the experimental data, when the evolution of the local scour approaches the state of equilibrium, the velocity along the vertical direction of the scour hole presents an exponential distribution. As depicted in Fig.1,the maximum velocity in the scour hole exists at 3H/4 in section B-B because of the influences of the seabed and the boundary of the pipeline (Yang et al., 2012b). With y representing the distance from the bottom of the scour hole to the 3H/4, the velocity of u(y) can be expressed as

Meanwhile, the velocity u′(y) from the wall of the pipeline to 3H/4 can be established as

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The relationship between the bottom velocity of the scour hole and the velocity at 3H/4 can be written as

where ub is the bottom velocity of scour hole, ub′ is the velocity at 3H/4, and y' is the vertical coordinate with respect to the wall of the pipeline.

The bottom velocity ub at the deepest point of the scour hole decreases with the increasing of the scour depth.When the forces that act on the sand particles attain balance and the sand transportation stops, erosion-stopping velocity ues exists. At equilibrium state, the bottom velocity is equal to the erosion-stopping velocity (ub=ues). Fig.1 also depicts the snapshot of the equilibrium state among drag force (FD), uplift force (FL), Frictional resistance of particle (Rf), and submerged weight (Ws). When the sum of the component of submerged weight and frictional resistance equates the drag force, the erosion-stopping velocity can be calculated using the following formula(Yang et al., 2012b):

where s is the specific gravity of sediment particles (s=ρs/ρ); ρ is water density; ρs is density of sediment; α is dynamic angle of repose; CL is the lift coefficient, which can be chosen as a constant of 0.178; and CD is the drag coefficient, which can be taken as a constant of 0.45 for the turbulence condition (Qian and Wan, 1991).

In all of the tests, the depth of the scour hole is measured by using a type of measuring pin with minimum precision of 0.1 mm, which was also previously used by Yang et al. (2012a, b) to measure the section profiles of the scour hole. The velocity profiles are measured using an ADV with maximum measuring range of 4 m s−1 and accuracy of ±1 mm s−1. The horizontal distance from the measurement point (1) of inflow velocity (u0) to the main pipe center is approximately 0.5 m, and the vertical distance from the seabed to the measurement point (1) is half of the main pipe diameter. When the scour approaches equilibrium, the bottom velocity (ub) is measured. The monitor (2) of the ADV is set in the deepest point of the scour hole without touching the seabed. This condition ensures that ADVs disturb the flow only slightly. The measuring probe and ADV are fixed on the instrumentcarrying car, which can move freely along the flume to collect the data. The sediment is measured before every experiment to ensure the same conditions in all experiments, such as median grain size, sand porosity, and sediment bulk density. If the sediment conditions change,the sediment is replaced by the initial backup sediment.

Substituting Eqs. (1)–(7) into Eq. (8) establishes the formula of the scour depth as

The formula parameters, such as n, in the preceding equations will be experimentally determined in Section 3.The difference in the scour depths predicted by Eqs. (1)and (9) under varying upstream velocities is demonstrated in Fig.2, where the main pipe diameter is 0.1 m and the gap ratio e0 (e0=G/D) is 0.25. This figure shows that the scour depth predicting Eq. (1) for the single pipeline is no longer applicable to the piggyback pipeline. This condition will be further confirmed by the experimental data presented in the following sections.

Fig.2 Comparison of scour depth predicted by Eqs. (1)and (9).

3 Laboratory Tests

3.1 Experiment Setup

All of the experimental work presented in this paper has been conducted in the River Dynamics Laboratory of Engineering College, Ocean University of China. The experimental arrangement is depicted in Fig.3. The test annular flume is 25 m long, 0.5 m wide and 0.6 m deep.The piggyback pipeline is laid and fixed in the flume.

Part of the flume around the piggyback pipe is filled with sand to imitate the seabed, and the height and length of the sandy bed are 0.15 and 5 m, respectively. The depth of the scour hole is assumed to not exceed the depth of the sand pit. White foam boards are set into the pipes so that the scour around the pipeline can be observed clearly.The span length of the main pipe and the small pipe is 0.5 m. Three different outer diameters of the main pipe are selected in the present study: 0.08, 0.10 and 0.12 m. According to the actual engineering design requirement, the ratio of small pipe diameter to main pipe diameter is chosen as 0.358, as reported by Yang et al. (2007). Thus, the diameters of the three small pipelines are 0.03, 0.035 and 0.04 m. Five gap ratios are considered: 0, 0.125, 0.25,0.375 and 0.5. To investigate the influences of current velocities on the local scour of the piggyback pipe, three different inflow velocities are selected: 0.24, 0.3 and 0.4 m s−1. The experimental water temperature is measured using a thermometer and is maintained at 15 ℃, and the corresponding kinematic viscosity ν isIn the experiment, the non-uniformity coefficientof the sand is 1.25, which indicates that the sand is finer,and the median grain size d50 is 0.3 mm. The porosity of the sand n0 is 0.37. The water depth hw is maintained at 0.4 m in the experiment section for all the tests.

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Fig.3 Schematic of experiment arrangement.

By applying the mass conservation law, the following integral-formula can be derived:

Table 1 summarizes all of the test runs and their corresponding parameters. The Reynolds numbers are calculated depending on the main pipe diameter and inflow velocity. The final results obtained from all the test runs have been used to analyze the influence of gap ratio e0,inflow velocity, and piggyback pipeline diameters on the dynamic repose angle α and scour depth H.

Table 1 Experimental parameters

Runs Main/Small pipe diameter (m)Gap ratio(e0=G/D)Reynolds number Run01 0.08/0.030 0/0.125/0.25/0.375/0.5 17143 Run02 0.08/0.030 0/0.125/0.25/0.375/0.5 21429 Run03 0.08/0.030 0/0.125/0.25/0.375/0.5 28571 Run04 0.10/0.035 0/0.125/0.25/0.375/0.5 21429 Run05 0.10/0.035 0/0.125/0.25/0.375/0.5 26786 Run06 0.10/0.035 0/0.125/0.25/0.375/0.5 35714 Run07 0.12/0.040 0/0.125/0.25/0.375/0.5 25714 Run08 0.12/0.040 0/0.125/0.25/0.375/0.5 32143 Run09 0.12/0.040 0/0.125/0.25/0.375/0.5 42857

3.2 Experimental Results

In this section, the experimental results are systematically demonstrated and discussed in detail. During the experiment, the lights were switched on to ensure the clarity of the photos. Thus, the picture shows the light reflected by the glass wall. Furthermore, the yellow line in the figures is the vertical ruler that is used to control the water level. The pipe is made of an organic glass tube,so the pipe wall is transparent. When the gap ratio (e0) is zero, a gap appears between the two pipes in the photos.In the photos, the pipe wall is emphasized by a black bold circle.

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The profiles of the local scour hole at the equilibrium state are depicted in Fig.4 for the case of D=0.01 m under various combinations of gap ratio and inflow velocity.The profile of the scour hole is significantly influenced by the two prominent influencing factors: inflow Re number,and the gap ratio between the main and small pipes.

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Fig.4 Experimental photos of scour around piggyback pipeline for D=0.10 m.

Since the scour around the normal pipe has been investigated adequately in previous studies, this paper mainly focuses on the piggyback pipeline. When designing the experiment, influencing factors, such as the gap ratio and small pipe diameter, which are closely related to the piggyback pipe on the scour, are considered primarily. Other influencing factors on the scour are also considered in the experiment setup, such as median grain size, water depth,and sand porosity. Other parameters may also be a function of the two prominent factors: inflow Re number and gap ratio. For example, the Re number is related to incoming flow velocity, pipe diameter, and coefficient of viscosity. As depicted in Fig.4, the higher inflow velocity can cause a deeper and wider scour hole, with the sandy bump moving further backward. A comparison of the results shown in Figs.4a and b with those in Figs.4c and d shows that a smaller gap ratio has the potential to induce a larger maximum scour depth.

The variations of the maximum scour depths as a function of the gap ratios are plotted and compared in Fig.5 for the diameters 0.08, 0.10 and 0.12 m, where the influences of the inflow Re numbers are also considered. As shown in Figs.5a, b and c, the maximum scour depth is approximately a linear function of the gap ratios. The maximum scour depth decreases with the increase of the gap ratio,because if the gap increases, much water will passes the gap and the velocity under the pipeline decreases. However, the increase of the inflow Re numbers can enlarge the maximum scour depth gradually. This condition is due to the fact that a current with a higher Re number usually has a potential to cause higher shear stress on the sandy bed and an enhanced hydraulic gradient around the piggyback pipe. This condition will finally help the scour hole to reach an increased depth at the equilibrium state.

Fig.5 Maximum scour depth as function of gap ratios and inflow Reynolds numbers.

3.3 Formula Parameter Determination

To enable the application of Eq. (9) in predicting the maximum scour depth of the piggyback pipe under various inflow Re numbers and gap ratios, erosion-stopping velocity (ues) and angle of repose should be determined initially. In addition, the height of stagnation point and the index parameters of velocity profile around the pipe also have to be determined before the prediction.

3.3.1 Dynamic angle of repose

For the single pipeline (equivalently, gap ratio e0→∞for the piggyback pipeline), the dynamic angle of repose φ in Eq. (10) is only related to the Re number of the sand grain, which is defined as Re*=u*ds/v. u* is the friction velocity. The fitting curve is shown in Fig.6.

Fig.6 Dynamic angle of repose as a function of Re*.

Dynamic angle of repose as a function of the sandy particle Reynolds number can be expressed as

When the piggyback pipeline is laid on the seabed, the bed scour and sediment transport are more severe than that of the single pipe because of the reduction in the angle of repose. For the piggyback pipe, the reduction in the dynamic angle of repose is related to the gap ratio. After the analysis of experimental data, the dynamic angle of repose α for the piggyback pipe can be expressed by angle φ and correction coefficient k. The formula is written as α/φ = k = f(e0), which can be determined by experimental data. The corresponding fitting curve is shown in Fig.7.

Then, the relationship is expressed as

Fig.7 Correction coefficient k as a function of e0.

3.3.2 Index number

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where γ is the volume fraction of water in a cell and the volume fraction is defined by whether the control volume is filled by water or air, as follows:

Fig.8 Index number determination for u0=0.24 m s−1, D=0.12 m, and e0 =0.

3.3.3 Height of stagnation point

When the scour reaches the equilibrium state, the height of the stagnation point for the single pipe is approximately half of pipe diameter, which has been reported previously (Oner et al., 2010). With coexistence of the small pipe, the height of the stagnation point for the piggyback pipe is affected by both the small pipe diameter and the gap between the two pipes. Fig.9 plots the fitting relation curve between the height of the stagnation point h and gap ratio e0.

Fig.9 Fitting relation curve between height of stagnation point h and gap ratio e0.

The relationship shown in Fig.9 can be expressed as

Based on the parameters determined, Eq. (9) is used to predict the maximum scour depth of the piggyback pipe.Its feasibility is calibrated by both the experimental and numerical results presented in Section 4.5.

4 Numerical Simulation

Besides the experimental tests, the temporal evolution of the local scour around the piggyback pipe and the relevant hydrodynamic characteristics are also numerically investigated in the current study. In all of the simulations, the numerical settings are the same as those in the experiments. The total length of the computational domain is 5.0 m, the water depth is 0.4 m, and the finest mesh resolution around the pipe is approximately 0.005 m,which ensures that y+ is close to 65 in the computation.The value of y+ is in a reasonable range from 30 to 100 for the turbulence model (Von Kármán, 1934).

4.1 Flow Model

In the present paper, the flow field is numerically simulated by solving the governing equations for the two immiscible fluids, namely, Navier-Stokes equations for incompressible flows, which can be written in vector form as

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where U is the flow velocity, t is the time, p is the pressure, ρ is the density of the air–water mixture, ρref is the reference density, μeff=μl+μt is the effective viscosity that includes the laminar viscosity μl and turbulence viscosity μt, and g is the acceleration of gravity. The reference density is used to eliminate the hydrostatic pressure accumulation in the gas-phase region. The reference density is set as air density in the current study. The total pressure p=p*+ρref·r·g, where r is the vector that starts from the reference point to the locations where the total pressure values are calculated.

Similar to the study by Qu et al. (2017a, b), in the present model, the computational domain is represented by an unstructured mesh. The collocated finite volume method is applied to discretize Eqs. (13) and (14). To adopt a highorder advection scheme, the deferred correction proposed by Fergizer and Perić (2002) is applied to discretize the convective term, which combines the first- order upwind scheme and the second-order Gamma scheme (Jasak,1996). Based on the work of Fergizer and Perić (2002),the diffusion and pressure terms are approximated using central difference by using values at the auxiliary nodes,which are in the intersection of the cell-face normal and straight lines connecting the node of the control volume and the neighboring nodes. The momentum interpolation method proposed by Rhie and Chow (2012) is applied to interpolate the velocities on the faces of the control cell.Then, the velocity and pressure are coupled using pressure implicit split operator method proposed by Issa(1991).

The free surface between water and air is resolved by the VOF method, which is originally proposed by Hirt and Nichols (1981). The transportation equation of VOF can be expressed as

The index number n is determined by measuring the velocity distribution in the scour hole. The index number of velocity distribution in the scour hole changes unnoticeably in all of the experiment tests, and can be regarded as constant. As depicted in Fig.8, the approximate velocity distributions in the scour hole as a function of n are plotted and compared with the measured data for u0=0.24 m s−1, D=0.12 m, and e0=0. By gradually increasing the index number, n=1/3 is found to be the most suitable, and is applied in Eq. (9).

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The local density and laminar viscosity as functions of γ are computed as

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The convection equation of the volume fraction is calculated with an algebraic VOF method. The finite volume method is applied to numerically solve the convection Eq.(15), and the volume fraction at the cell face is interpolated with the switching technique for advection and capturing of surfaces (STACS) from Darwish and Moukalled(2006), which is a high-resolution scheme based on the normalized variable diagram concept (Leonard, 1991).For temporal discretization, the Crank-Nicolson differencing scheme is used to calculate the next time-step values by solving the discretized equation. Generally, predictions obtained by STACS are recognized as accurate with minimal diffusion.

The turbulent viscosity μt is calculated by the twoequation κ-ε turbulence model, which has the following control equations:

where κ is turbulent kinetic energy, and ε is turbulent energy dissipation rate. The feasibility of the κ-ε turbulence model in predicting the scour process under the pipeline has been calibrated by Zhao et al. (2010, 2015).

4.2 Scour Model

In the current numerical model, the bed load is considered in accordance with the experimental work. The Exner equation is used to update the bed elevation (García and Parker, 1999).

where xk+1 and xk are grid position vectors at time level k+1 and k, respectively, and Δt is the time step.

根据我国宪法以及相关税务法律法规的规定，我国公民具有纳税的义务，纳税人需要进行税务登记、按期申报税务、及时缴纳税务等。根据税法规定，不同行业其所实行的缴税政策是不尽相同的，因此其计算方式是十分复杂的。税务会计是具有专业知识的专门型人才，其可以依据税法规定的计税依据与煤炭企业财务会计所反映的依据确定所需缴纳税额。这样可以在很大程度上规范我国煤炭企业的纳税行为。

where τ is the bed shear stress calculated on the basis of the fluid flow model, and g is the gravitational acceleration.

where φ is the angle between the velocity vector and the steepest slope direction of the bed, β is the dynamic repose angle, and μs is the static friction coefficient.

where is the non-dimensional grain size.

The critical Shields number should be adjusted according to the local slope of the bed and local shear force direction. Depending on the approach suggested by Engelund and Fredsøe (1976), the critical Shields number is adjusted according to the following formula:

Once the Shields number exceeds the critical rate, the sediment transportation occurs. However, the critical Shields number for the sediment is always given for a flat bed. The widely used formula to calculate the critical Shields number is

According to the critical Shields number, the following bed-load transport rate formula proposed by Engelund and Fredsøe (1976) can be used for the current model:

The bed-load sediment transport rate at the slope is adjusted according to the following formula:

where τi is the shear stress at i direction, C is an empirical coefficient in the range of 1.5–2.3 (Brors, 1999), and hl is the sandy bed level.

Based on mass balance principle of the sediment transportation, the temporal evolution of the bed level hl can be expressed using the following Exner equation:

Along with the sand transporting under the piggyback pipeline, the computational mesh has to be adapted tightly with the evolution of the sandy bed. In the present study,the automatic mesh deformation is calculated by applying a Laplacian smooth operator (Jasak and Tuković, 2007).The governing equation for the mesh-motion equation is

In the current study, the formulas of the sour model are computed simultaneously with the evolution of the flow field.

4.3 Mesh Deformation Solver

where m is the porosity factor of the sediment, and qi is the bed-load transport rate vector whose components are given by Eq. (25).

收集上海市浦东新区浦南医院肿瘤科2009年10月至2017年1月间收治的171例胰腺癌患者的临床资料。排除病理诊断为非胰腺导管腺癌、肿瘤分期为Ⅰ～Ⅱ期、合并重要脏器功能缺陷或其他部位恶性肿瘤、接受过抗肿瘤治疗、体能状况不佳或无法接受至少2周期化疗及临床病理资料不完整的患者，最终纳入94例晚期胰腺导管腺癌患者。本研究经浦南医院伦理委员会批准，所有研究对象签署知情同意书。

where ζ is the diffusion coefficient to control the mesh motion, and v is the mesh moving velocity. It can be chosen as a constant or a variable defined by other properties in the computational domain. Selecting the variable depends on the specific mesh-motion problem and needs objective judgment (Liu and Marcelo, 2008). Jasak and Tuković (2007) proposed several possibilities to set the ζ value based on distance (linear, quadratic, and experimental) from several boundary or mesh characteristics(orthogonality and skewness). In the present study, ζ is chosen depending on the mesh distance. The boundary condition for Eq. (28) is expressed by the control equations of the scour model. After each time step of bed elevation change, the bed grid velocity is known. When the grid motion velocity field is solved, the grid in the entire domain can be updated based on the following equation:

在大型灌区泵站改造技术方面，安排了泵站或泵变合一站的微机保护，水泵水轮机空化与振动监测，枢纽、灌区泵站CIMS以及大型水泵液压调节等用于泵站保护、监测、诊断的技术与设备；改进了泵站微机保护装置、水泵液压调节技术等；提高了我国大型灌区泵站的管理水平，保障了泵站的正常运行，为实现我国粮食增产提供了技术支撑。

For the bed-load transport, the onset of local scour is determined by the Shields number, which is defined as

4.4 Simulation Results

The profiles of the sandy bed at three instances for e0 =0 and u0 =0.4 m s−1 are shown in Fig.10. The depth of the scour hole gradually increases with time. At t=2500 s,only a small scour hole is formed below the pipe and a small sandy bump is formed behind the pipe, as shown in Fig.10a. With the development of the scour, the scour hole gradually deepens, and the sandy bump becomes increasingly large and moves backward at the same time,as indicated in Figs.10b and c. The velocity magnitude in the gap between the main pipe and sandy bed gradually decreases with the devolution of the scour depth. Furthermore, the velocity magnitude on the sandy bed becomes equal to the erosion-stopping velocity. Then, the equilibrium state is reached and the scouring stops.

Fig.10 Comparison of profiles of sandy bed and corresponding flow field around piggyback pipe at three instances for e0=0, and u0=0.4 m s−1.

To validate the computational capability of the numerical model in the present paper, the predicted profiles of the sandy bed under u0 = 0.4 m s−1 and different gap ratios are compared with the experimental data, as shown in Fig.11. Besides the experimental data in the current study,the numerical results obtained by Zhao and Cheng (2008)have also been used to verify the current numerical model.The current numerical results are in good agreement with the experimental data, and the computational capability of our model is better than that of the model presented by Zhao and Cheng (2008) on average. Both the numerical results and experimental data demonstrate that a smaller gap ratio could cause a larger maximum scour depth.

关于材料堆放，规程规定可燃类保温材料的库房应由不燃性材料搭设而成，并有专人看管。当材料露天堆放时，堆放场四周应由不燃性材料围挡；堆放区域禁火，其周围10m范围内及上空不得有明火作业；附近不得放置易燃、易爆等危险物品；应配备种类适宜的灭火器、砂箱或其他灭火器具；场内材料的存放量不应超过三天的工程需用量,并应采用不燃性材料完全覆盖。

The temporal evolution of the local scour depth under the piggyback pipe is depicted in Fig.12. The figure demonstrates that all of the scour could reach the equilibrium state after approximately 15000 seconds since the beginning of the simulation. Furthermore, the time duration for the scouring process to reach the equilibrium state increases with the decrease of the gap ratio.

A comparison of the scour depth between the piggyback pipeline and single pipe is shown in Fig.13. The scour depth increases with the increasing Re number.Compared with the single pipe, the maximum scour depth of the piggyback pipe is larger, and it increases significantly with the decrease of e0. When e0 is equal to zero and the Re number is approximately 36000, the maximum scour depth of the piggyback pipe increases by 50%compared with that of the single pipe. In summary, the scour depth under the piggyback pipeline is always larger than that of the single pipe under the same conditions.The scour under the piggyback pipeline is also more complex than that of the single pipe. Thus, in-depth investigation is important.

Fig.11 Profiles of sandy bed at equilibrium under u0=0.4 m s−1 with different gap ratios.

Fig.12 Temporal evolution of scour depth.

Fig.13 Comparison between piggyback pipeline and single pipe.

The vortex patterns around the piggyback pipe under different e0 with D = 0.1 m and u0 = 0.3 m s−1 are presented in Fig.14. When e0 = 0, large vortices are formed in the wake region and the mutual interactions between the shed vortices from the small and main pipes are strong, as shown in Fig.14a. The intensity of the mutual interactions between these two pipes gradually weakens and when e0 is increasing, as shown in Figs.14b, c, d and e. This condition also directly causes a decrease in the size of the vortices behind the piggyback pipe. Usually, when the shed vortices become larger, the pressure in the wake region could decrease, thereby intensifying the hydraulic gradient around the piggyback pipe. Fig.14 shows that the weaker the intensity is, the lower the maximum scour depth could reach; this condition is verified by the results presented in Fig.12. Thus, the mutual interaction of the shed vortices between the small and main pipes has a significant influence on the development of the scour hole.

Fig.15 shows the temporal evolution of the force coefficients of the main and small pipes with u0 = 0.3 m s−1 and e0 = 0.25, where the force coefficients are computed by their own diameters. This figure demonstrates that the frequencies of both the lift and drag coefficients of the small pipe are nearly three times larger than those of the main pipe.

Fig.14 Vortex pattern around pipeline with D = 0.1 m, u0 = 0.3 m s−1 under different gap ratios.

Fig.15 Temporal evolution of force coefficients of main and small pipes at u0 = 0.3 m s−1 and e0 = 0.25.

4.5 Formula Verification

The feasibility of the proposed Eq. (9) to predict the maximum scour depth of the piggyback pipeline under steady current has been calibrated by both experimental data and simulation results. The comparison of the predicted maximum scour depth by Eq. (9), numerical results,and experimental data are plotted in Fig.16, where h1 represents the ratios of H/D calculated by Eq. (9), and h2 represents the corresponding experimental data and numerical results. As shown in Fig.16, the proposed formula works effectively to provide an accurate prediction of the maximum scour depth.

Fig.16 Comparison of maximum sour depth.

5 Conclusions

In this study, the local scour around the piggyback pipe under steady current has been investigated through both numerical simulation and experimental work. The main conclusions are summarized as follows:

1) The experimental data show that the local scour depth of the piggyback pipeline is significantly influenced by two prominent factors: inflow Re number and the gap ratio between the main and small pipes. As the inflow Re number increases, both the maximum scour depth and the length of the scour hole increase at the same time.Meanwhile, an increment in the gap ratio results in a reduction of the maximum scour depth of the piggyback pipeline.

2) The computational capability of the current numerical model in predicting the temporal evolution of the local scour of the piggyback pipeline under steady current has been calibrated through comparison with experimental data. Analysis of the simulation results demonstrate that the gap ratio has a significant influence on the intensity of the mutual interaction between the vortex shedding from the main and small pipes. The larger the gap ratio is, the weaker the intensity of the mutual interaction becomes.The numerical results also show that the time duration for the scour process to reach the equilibrium state increases with a decrease in the gap ratios. Furthermore, the force coefficients of the small pipe exhibit three times higher frequency than those of the main pipe.

3) An empirical Eq. (9) to predict the maximum scour depth of the piggyback pipeline has been proposed and its feasibility has been calibrated by both experimental and simulation results. The application of this formula is instructive in the future design and application of the piggyback pipeline.

Acknowledgements

This study is financially supported by the National Key Research and Development Program of China (No. 2017 YFC1404700), the National Natural Science Foundation of China (Nos. 51279189, 51239001 and 51509023) and the China Scholarship Council. The authors are grateful to the anonymous reviewers for their valuable suggestions to improve the quality of this manuscript.

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