Numerical Study of the Influence of Tidal Current on Submarine Pipeline Based on the SIFOM–FVCOM Coupling Model

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Article Numerical Study of the Inﬂuence of Tidal Current on Submarine Pipeline Based on the SIFOM–FVCOM Coupling Model

Enjin Zhao 1,2 , Lin Mu 1,2,* and Bing Shi 3

1 College of Marine Science and Technology, China University of Geosciences, Wuhan 430074, China; [email protected] 2 Shenzhen Research Institute, China University of Geosciences, Shenzhen 518057, China 3 College of Engineering, Ocean University of China, Qingdao 266100, China; [email protected] * Correspondence: [email protected]

Received: 29 October 2018; Accepted: 6 December 2018; Published: 10 December 2018

Abstract: The interaction between coastal ocean flows and the submarine pipeline involved with distinct physical phenomena occurring at a vast range of spatial and temporal scales has always been an important research subject. In this article, the hydrodynamic forces on the submarine pipeline and the characteristics of tidal flows around the pipeline are studied depending on a high-fidelity multi-physics modeling system (SIFOM–FVCOM), which is an integration of the Solver for Incompressible Flow on the Overset Meshes (SIFOM) and the Finite Volume Coastal Ocean Model (FVCOM). The interactions between coastal ocean flows and the submarine pipeline are numerically simulated in a channel flume, the results of which show that the hydrodynamic forces on the pipeline increase with the increase of tidal amplitude and the decrease of water depth. Additionally, when scour happens under the pipeline, the numerical simulation of the suspended pipeline is also carried out, showing that the maximum horizontal hydrodynamic forces on the pipeline reduce and the vertical hydrodynamic forces grow with the increase of the scour depth. According to the results of the simulations in this study, an empirical formula for estimating the hydrodynamic forces on the submarine pipeline caused by coastal ocean flows is given, which might be useful in engineering problems. The results of the study also reveal the basic features of flow structures around the submarine pipeline and its hydrodynamic forces caused by tidal flows, which contributes to the design of submarine pipelines.

Keywords: SIFOM–FVCOM; overset grid; submarine pipeline; tide; hydrodynamic force; ﬂow turbulence

1. Introduction For decades, much crude oil and natural gas have been found in the ocean, which are constantly mined and utilized with the development of marine science and technology. The exploited oil and gas are usually transported by submarine pipelines. Most of submarine pipelines are laid on the seabed and exposed to the various marine environments directly. The pipelines are strongly affected by forces from the complex marine environments such as unpredictable waves and currents. For example, due to the influence of the water flows, pipelines are prone to vibration damage or stress destruction in the deep sea, while pipelines are easily affected by coastal waves in shallow water. Pipeline accidents lead to not only huge economic losses but also serious marine environmental pollution because of crude oil leakage. In order to reduce accidents on submarine pipelines, it is necessary to carry out deep investigations on the interaction between tide-induced flows and submarine pipelines [1,2].

Water 2018, 10, 1814; doi:10.3390/w10121814 www.mdpi.com/journal/water Water 2018, 10, 1814 2 of 29

In the past, a series of studies about the submarine pipelines have been carried out experimentally and numerically. Through physical experiments, Kok [3] examined the locations of front stagnation and separation points on the pipeline as well as the magnitudes and directions of drag and lift forces, which gave a better understanding of various flow phenomena at the pipeline. Li et al. [4] experimentally studied the hydrodynamic force on a pipeline subjected to vertical seabed movement using an underwater shaking table, the results of which showed that the Reynolds number (Re) and Keulegan–Carpenter number (KC) are the two main parameters influencing the hydrodynamic force coefficients, known as the drag coefficient (CD), added mass coefficient (CA) and horizontal hydrodynamic force coefficient (CV). Besides, the investigation of the influence of close proximity of a submarine pipeline to the seabed on the vortex-induced vibrations (VIV) of the pipeline was conducted by Yang et al. [5]. In the experiment, visualization of the flow field with hydrogen bubbles indicated that the wake vortexes became much more irregular with the decrease of the gap between the pipe and boundary. Mattioli et al. [6,7] investigated the near-bed dynamics around a submarine pipeline laying on different types of seabed under currents and waves, which revealed the mechanics of flow–sediment–pipeline interactions based on dedicated laboratory experiments. Except for the experimental research, numerical simulations were also conducted to study the submarine pipeline. Pierro et al. [8] researched on the influence of boundary layer viscous dynamics on the lift and drag forces acting on a near-bed pipeline analytically and numerically. Investigation of high Reynolds number flows around a circular cylinder close to a flat seabed was conducted by Ong et al. [9] using a two-dimensional standard high Reynolds number κ-ε model. It was found that the time-averaged drag coefficient (CD) of the cylinder increased as the gap between pipeline and a seabed flat to pipeline diameter ratio (G/D) increased for small G/D. Cheng and Chew [10] studied the effect of a spoiler on flow around the pipeline and the seabed. An additional spoiler on top of a pipe would substantially increase the pressure on the upstream of the pipe and decrease the base pressure at the back of the pipe. Reviews of this subject also can be found in many published papers [11–20]. In these studies, hydrodynamic forces on pipelines were induced by viscosity, inertia, flow asymmetry and vortex shedding, while analytical expressions were proposed for the evaluation in some cases. Although many investigations have been conducted experimentally and numerically on hydrodynamics at submarine pipelines, the research focuses on the unidirectional steady flow in small scale flow field rather than realistic ocean environment. The past studies ignored the strong interaction between small- and large-scale phenomena. Spatially, the real sea area is much larger than the area near the submarine pipeline, indicating that the simulation accuracy of the large-scale model can hardly satisfy the simulation of a small-scale structure. In the meanwhile, although the simulation accuracy of small-scale models might meet with the simulation requirement of the submarine pipeline, they are not able to calculate the large-scale ocean model due to the limitation of calculation conditions. Temporally, the period of regular semi-diurnal tide is usually 12 h, leading to a large ratio of the time scale of ocean model against the small-scale model; therefore, the small-scale model cannot be used to simulate tidal flows accurately which oscillate in a much longer period. Since one of the major tasks in the design of submarine pipelines is the analysis of hydrodynamic stability, it is crucial to ensuring that during construction and operation stages, the pipelines are able to remain stable under the action of hydrodynamic forces induced by tidal currents and waves [21]. Due to the drawbacks of the existing references and the deficiency of hydrodynamic simulation for submarine pipelines, a multi-physics/multi-scale approach is indispensable for the accurate simulation of the interaction between flows and pipelines, which has become an important subject in the prediction of coastal ocean flows in recent years [22]. In this research, the Solver for Incompressible Flow on Overset Meshes (SIFOM) and the Unstructured Grid Finite Volume Coastal Ocean Model (FVCOM) [23] are interactively combined as one system to carry out numerical simulations of the interaction between flows and pipelines and study the hydrodynamic forces acting on pipelines, in which the SIFOM is used to calculate small-scale local hydrodynamics around the submarine pipeline, and the FVCOM is employed to capture large-scale coastal ocean tidal currents. SIFOM–FVCOM can solve the Water 2018, 10, 1814 3 of 29 incompatibility problems of different scale models according to the domain decomposition method. Besides, the data between SIFOM and FVCOM exchange in the nested meshes directly and march in time simultaneously, which omits the steps of data storage and readout between different models. The calculation time of the coupling model is shortened greatly which can provide predictive results quickly for the engineering practice. Besides, FVCOM as a marine prediction model has been used popularly to forecast the marine environment. So, this model also can predict the realistic coastal flow around the ocean and coastal engineering depending on the present marine conditions to achieve the purpose of the disaster prevention. The structure of this paper is organized as follows. Firstly, the modeling system is briefly introduced in Section2, while validations of the system are given in Section3. Secondly, systematic simulations of the interaction between flows and pipelines are carried out in a numerical channel, the results of which are analyzed with regard to flow structures and hydrodynamic forces in Section4. Meaningful conclusions of the study and some discussions are listed in Section5.

2. Solver for Incompressible Flow on Overset Meshes–Unstructured Grid Finite Volume Coastal Ocean Model (SIFOM–FVCOM) Modeling System In order to simulate the realistic multi-physics/multi-scale ocean phenomena, the domain decomposition approach is adopted in SIFOM–FVCOM, and the solution data exchange among different overset grids.

2.1. SIFOM Modeling In the SIFOM modeling, the Cartesian coordinate is used as a reference system and the mesh in this model is divided into rectangular grids for the calculation. The I-, J- and K-directions of the rectangular grids are corresponding to the x-, y- and z-directions of the Cartesian coordinate, respectively. The mass and momentum conservation equations are employed to govern three-dimensional ﬂow motion, and Chimera grids and the Schwarz alternative iteration are used in the modeling. The governing equations are as follows: ∇·u = 0 (1) where ∇ is the gradient operator; u is the velocity vector.

1 ut + ∇·uu = − ∇pd + ∇·((ν + νt)∇u) − g∇Hζ + g(α(T − T0) − β(C − C0))k (2) ρ0 where t is time; ρ0 is the reference density; pd is the dynamic pressure; ν is the fluid viscosity; νt is the turbulence viscosity; g is the gravity; ∇H is the horizontal gradient operator; ζ is the water surface elevation; α is the thermal expansion coefficient; T is the temperature; T0 is a reference temperature; β is a coefficient reflecting the expansion; C is the tracer concentration; C0 is a reference concentration; k is the unit vector in the vertical direction. A second-order accurate, implicit, finite-difference method is used on non-staggered, curvilinear grids. The QUICK scheme [24] is used to discretize convective terms. The odd-even decoupling of the pressure field is eliminated by a third-order, fourth-order difference artificial dissipation method [25].

2.2. FVCOM Modeling FVCOM is a geophysical ﬂuid dynamics model. The unstructured/triangle mesh is used in the horizontal plane to deal with complicate coastlines. In the vertical direction, in order to obtain a smooth representation of irregular bottom topography in the ocean, all equations are transformed into σ coordinate from z coordinate of the Cartesian coordinate. The coordinate transformation expression is written as follows: z − ζ z − ζ σ = = (3) H + ζ D Water 2018, 10, x FOR PEER REVIEW 4 of 29

zz−−ζζ Water 2018, 10, 1814 σ == 4 of(3) 29 HD+ζ where zz isis the the vertical vertical position position of of a certain a certain water water level, level,ζ is theζ is water the water surface surface elevation, elevation,H is the H bottom is the bottomdepth and depthD isand the D total is the water total columnwater column depth depth (Figure (Figure1). The 1). initial The initial water water surface surface is located is located at the at thez = 0z m= and0 m σandvaries σ varies from −from1 at the−1 bottomat the bottom to 0 at theto surface.0 at the Insurface. the vertical In the direction vertical ofdirectionσ coordinate, of σ coordinate,the calculation the domaincalculation is divided domain by is manydivided layers, by many which layers, is referred which to is as referredσ-grids. to as σ-grids.

Figure 1. IllustrationIllustration of of the the orthogonal orthogonal Cartesian coordinate system.

FVCOM consistsconsists ofof anan external external mode mode and and an an internal internal mode. mode. Every Every mode mode includes includes continuity continuity and andmomentum momentum equations equations as the as governing the governing equations. equati Aons. second-order A second-order accurate accurate finite volume finite methodvolume methodis used inis theused model, in the which model, is ablewhich to is preserve able to conservationpreserve conservation of mass and of momentummass and momentum accurately. accurately.The governing The equations governing of equation external modes of external are averaged mode governing are averaged equations governing in the equationsσ-coordinate: in the σ-coordinate: (ζ)t + ∇H·(VD) = 0 (4) ζ + ∇ ⋅ = ( ) t H (V D ) 0 (4) τs − τb (VD)t + ∇H · (VVD) = −gD∇Hζ + ττ− + G (5) ()VVVDDgD+∇ ⋅ ( ) = − ∇ζ +sbρ0 + G tH H ρ (5) where V is the depth-averaged velocity vector; τb is the shear stress0 on the seabed; τs is the shear stress on the water surface; G includes Coriolis force and water friction force. where V is the depth-averaged velocity vector; τb is the shear stress on the seabed; τs is the shear In the internal mode, the three dimensional governing equations are used based on the stress on the water surface; G includes Coriolis force and water friction force. hydrostatic assumption: In the internal mode, the three dimensional governing equations are used based on the (ζ) + ∇H·(VD) + (ω) = 0 (6) hydrostatic assumption: t σ (VD)t + ∇H·(VVD) +ζ(Vω+)σ∇= ⋅ −gD∇+Hζω+ ∇=H·(κe) + (λVσ)σ/D + H ( ) t H ( V D ) ( ) σ 0 (6) gD R 0 0 (7) − ∇H D ρdσ + σρ∇H D ()+∇ ⋅ ( )() +ωζκλ = − ∇ρo + ∇ ⋅ ()()σ + + VVVVDDtHσσ gD H H eH Vσ D where V is the horizontal velocity; ω is the vertical velocity in σ coordinate; e is the strain rate; H stands gD 0 (7) for the residual terms; κ and λ are coefficients. −∇()(Ddρσ′) +∇ σρ D In the two modes of FVCOM, the time derivativeρ HH termsσ are discretized by the Runge–Kutta o method, and the convection terms are discretized using upwind schemes [26]. where V is the horizontal velocity; ω is the vertical velocity in σ coordinate; e is the strain rate; H stands2.3. Coupling for the Strategy residual of terms; SIFOM–FVCOM κ and λ areModeling coefficients. InFrom the thetwo above modes discussion of FVCOM, of thethe SIFOMtime derivati and FVCOM,ve terms itare demonstrates discretized thatby the two Runge–Kutta models are method,different and in computationalthe convection meshes,terms are governing discretized equations using upwind and numericalschemes [26]. algorithms. In order to simulate the multi-physics/multi-scale ocean flow, the SIFOM and FVCOM must be integrated using 2.3. Coupling Strategy of SIFOM–FVCOM Modeling a coupling strategy, which is presented in this section. FromThe sketch the above of the discussion devised domains of the SIFOM around and the FV submarineCOM, it pipelinedemonstrates is shown that intwo Figure models2. The are differentmodels of in SIFOM computational and FVCOM meshes, overlap governing around theequations submarine and pipeline. numericalΩSIFOM algorithms.and ΩFVCOM In orderare theto simulatecalculation the domains multi-physics/multi-scale of SIFOM and FVCOM, ocean flow, respectively. the SIFOM The and boundaries FVCOM of must∂ΩSIFOM be integratedand ∂ΩFVCOM using aare coupling the interfaces strategy, of SIFOMwhich is and presented FVCOM, in respectively.this section. ΩSIFOM-FVCOM is the zone of SIFOM enclosed between the seabed and the red dashed line of ∂ΩFVCOM. In the overlap domain, SIFOM and FVCOM exchange solutions at the interfaces of ∂ΩSIFOM and ∂ΩFVCOM. Water 2018, 10, x FOR PEER REVIEW 5 of 29

The sketch of the devised domains around the submarine pipeline is shown in Figure 2. The models of SIFOM and FVCOM overlap around the submarine pipeline. ΩSIFOM and ΩFVCOM are the calculation domains of SIFOM and FVCOM, respectively. The boundaries of ∂ΩSIFOM and ∂ΩFVCOM are the interfaces of SIFOM and FVCOM, respectively. ΩSIFOM-FVCOM is the zone of SIFOM enclosed ∂Ω Waterbetween2018 ,the10, 1814seabed and the red dashed line of FVCOM. In the overlap domain, SIFOM and FVCOM5 of 29 exchange solutions at the interfaces of ∂ΩSIFOM and ∂ΩFVCOM.

FigureFigure 2. 2.The The decomposition decomposition domains domains of of Solver Solver for for Incompressible Incompressible Flow Flow on theon the Overset Overset Meshes–Finite Meshes– VolumeFinite Volume Coastal OceanCoastal Model Ocean (SIFOM–FVCOM) Model (SIFOM–FVCOM) modeling modeling around thearound submarine the submarine pipeline. pipeline.

The velocity u of SIFOM are the same as the velocity u(V, ω) of FVCOM. The velocity values are exchanged from FVCOM to SIFOM at the interfaces. = ∂ Ω u SIFOM u FVCOM , on SIFOM (8) uSIFOM = uFVCOM, on ∂ΩSIFOM (8) Due to the change of water level and velocity, the pressures in a domain of SIFOM includes the staticDue pressure to the and change hydrodynamic of water level pressure. and velocity, the pressures in a domain of SIFOM includes the static pressure and hydrodynamic pressure. =−+ρζ pgzp0 ()d (9) p = ρ g(ζ − z) + p (9) where z is vertical position of a certain water level.0 d When the solutions of SIFOM exchange to FVCOM, similar conditions are adopted: where z is vertical position of a certain water level. = ∂Ω When the solutions of SIFOMu exchangeFVCOM u toSIFOM FVCOM,, on similarFVCOM conditions are adopted: (10) In these conditions, the solution exchange from the SIFOM to FVCOM is also conducted in the uFVCOM = uSIFOM, on ∂ΩFVCOM (10) domain of ΩSIFOM-FVCOM which is called the blanket zone. In this way, the structure of FVCOM does not needIn these to be conditions, modified around the solution the submarine exchange pipe fromline. the In SIFOM the blanked to FVCOM zone, is the also solution conducted values in theare redistributeddomain of ΩSIFOM and interpolated-FVCOM which on is the called domain the blanket of FVCOM. zone. In this way, the structure of FVCOM does not need to be modiﬁed around the submarine= pipeline.∂ Ω In the blanked zone, the solution values are u FVCOM u SIFOM , on SIFOM − FVCOM (11) redistributed and interpolated on the domain of FVCOM. The coupling solving process between SIFOM and FVCOM can be seen in the Appendix.

uFVCOM = uSIFOM, on ∂ΩSIFOM−FVCOM (11) 3. Modeling System Validation The coupling solving process between SIFOM and FVCOM can be seen in the AppendixA. Depending on the investigation data of DF1-1 submarine pipeline off the west coast of Dongfang,3. Modeling Hainan System Island, Validation the measured maximum near-bed current velocity varies between 0.40 m/s and 0.91 m/s and the submarine pipeline diameter changes from 0.30 m to 1.60 m [27]. It is noted that whenDepending the submarine on the investigation pipeline lays data on of the DF1-1 seab submarineed, due topipeline the influence off the of west the coastseabed of Dongfang,boundary layerHainan on Island,the tidal the flow, measured the veloci maximumty around near-bed the pipeline current is not velocity so much varies that betweenthe Reynolds 0.40 m/snumbers and are0.91 in m/s the and supercritical the submarine flow pipelineregime. diameterBased on changesthis prototype from 0.30 conditions, m to 1.60 in m order [27]. Itto is ensure noted thatthe whenSIFOM–FVCOM the submarine modeling pipeline system lays on works the seabed, appropriatel due to they, three influence flows of are the simulated seabed boundary and the layer results on arethe compared tidal flow, with the velocity existing around experimental the pipeline data. is not so much that the Reynolds numbers are in the supercritical flow regime. Based on this prototype conditions, in order to ensure the SIFOM–FVCOM modeling system works appropriately, three flows are simulated and the results are compared with existing experimental data.

3.1. Validation on Flow Velocity and Vortex The ﬁrst test is that a steady current passes a cylinder which is suspended above the ground. The ﬂow parameters employed in the computation are the same as those in the experiments by Jensen [28]. The total water depth is 0.6 m and Reynolds number is 7000 which is based on the incoming velocity of Water 2018, 10, x FOR PEER REVIEW 6 of 29

3.1. Validation on Flow Velocity and Vortex

WaterThe2018 ,first10, 1814 test is that a steady current passes a cylinder which is suspended above the ground.6 of 29 The flow parameters employed in the computation are the same as those in the experiments by Jensen [28]. The total water depth is 0.6m and Reynolds number is 7000 which is based on the incoming0.07 m/s andvelocity pipeline of 0.07 diameter m/s and of 0.1 pipeline m. The diamet ratio ofer the of gap 0.1 betweenm. The theratio bottom of the and gap pipeline between to the the − bottompipeline and diameter pipeline is 0.37.to the The pipeline cylinder diameter center wasis 0.37. placed The at cylinder (x, y) = (0center m, was0.513 placed m). In at the (x simulation,, y) = (0 m, −a0.513 rectangular m). In the ﬂow simulation, domain is a considered. rectangular Theflow length domain and is widthconsidered. of the The calculation length and domain width are of 10 the m calculationand 5 m, respectively. domain are SIFOM 10 m and encloses 5 m, the respectively region near. SIFOM the pipeline, encloses while the FVCOMregion near covers the the pipeline, region whileaway fromFVCOM it. There covers are the two region sets ofaway meshes from (G1 it. Ther ande G2) are in two SIFOM sets of and meshes one set (G1 of meshesand G2) in in FVCOM SIFOM and(Figure one3 ).set The of meshes grid layer in FVCOM is nested, (Figure and the 3). density The grid of thelayer grid is nested, is increasing and the from density the far of ﬁeld the grid to the is increasingpipeline and from seabed. the far field to the pipeline and seabed.

(a)

(b)

Figure 3. The calculation meshes. (a) The horizontal plane mesh at z = −0.513 m. (b) The vertical Figureplane mesh. 3. The calculation meshes. (a) The horizontal plane mesh at z = −0.513 m. (b) The vertical plane mesh. Due to the disturbance of the cylinder, the flow behind the cylinder is unsteady (Figure4). The meanDue to horizontal the disturbance and vertical of the velocitiescylinder, the of instantaneousflow behind the solutions cylinder are is unsteady presented (Figure in Figure 4).4 Theb,c, meanrespectively. horizontal It is seenand thatvertical there velocities is little difference of instantaneous between thesolutions results are modeled presented by SIFOM in Figure alone 4b,c, and respectively.SIFOM–FVCOM, It is seen which that is attributedthere is little to thedifference fact that between the top boundarythe results ofmodeled the computational by SIFOM alone domain and is SIFOM–FVCOM,considered a rigid which plane is using attributed SIFOM to alone,the fact whereas that the the top one boundary using SIFOM–FVCOM of the computational is treated domain as a isfree considered surface. Besides, a rigid plane the coupling using SIFOM strategy alone, between whereas SIFOM the andone FVCOMusing SIFOM–FVCOM might leads to is small treated errors. as aIn free short, surface. the results Besides, obtained the coupling with SIFOM–FVCOM strategy between system SIFOM agree and reasonably FVCOM with might computational leads to small and errors.experimental In short, results the in theresults literature. obtained with SIFOM–FVCOM system agree reasonably with computationalThe second and test experimental is a 3D flow results passes in a the square literature. cylinder at bottom of a channel. The length and widthThe of thesecond channel test areis a 54 3D m flow and 30passes m, respectively. a square cy Thelinder water at bottom depth is of 3.14 a channel. m and Reynolds The length number and widthis 21,400 of whichthe channel is based are on 54 the m incoming and 30 m, velocity respectively. of 0.02 m/sThe andwater the depth square is length3.14 m of and 1 m. Reynolds The flow numberfield around is 21,400 the squarewhich is cylinder based on is shownthe incoming in Figure velocity5a, which of 0.02 is simulated m/s and the by square the SIFOM–FVCOM length of 1 m. Thesystem. flow The field result around of SIFOM–FVCOM the square cylinder in Figure is shown5b is in an Figure average 5a, of which instantaneous is simulated solution by the over SIFOM– time. FVCOMThe comparison system. The of the result velocity of SIFOM–FVCOM distribution also in showsFigure that5b is the an SIFOM–FVCOMaverage of instantaneous system produces solution overresults time. that The agree comparison well with existingof the velocity data in distri the literature.bution also shows that the SIFOM–FVCOM system produces results that agree well with existing data in the literature.

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Water 2018, 10, x FOR PEER REVIEW 7 of 29

(a) Snapshot of the flow passing a cylinder.

(b) Mean horizontal velocity;

FigureFigure 4. 4. SimulationSimulation of of the the flow flow passing passing a cylinder. a cylinder. (a) Computed (a) Computed flow flow field field near near the cylinder. the cylinder. (b) Comparison(b) Comparison of the of thehorizontal horizontal velocity velocity distribution. distribution. (c) (cComparison) Comparison of ofthe the vertical vertical velocity velocity distribution.distribution. Circle-measurement Circle-measurement [28], [28], green green dash dash line, line, modeling modeling with with ĸ-kε-ε turbulenceturbulence closure closure [29], [29], blueblue solid solid line, line, SIFOM, SIFOM, red red dash-dot dash-dot line, line, SIFOM–FVCOM. SIFOM–FVCOM.

Water 2018 10 Water 2018, , ,10 1814, x FOR PEER REVIEW 88 of of 29 29

(a) Snapshot of the flow passing a cubic cylinder.

(b) Velocity distribution.

FigureFigure 5. 5.Flow Flow passes passes a cubica cubic cylinder cylinder at at bottom bottom of of channel. channel. (a )(a Computed) Computed ﬂow flow ﬁeld field near near the the square square cylinder.cylinder. (b )(b Comparison) Comparison of of the the velocity velocity distribution. distribution. Circle-Measurement Circle-Measurement [30 [30],], dash-dot dash-dot line, line, DNS DNS modeling,modeling, solid solid line, line, SIFOM–FVCOM. SIFOM–FVCOM.

3.2.3.2. Validation Validation on on Force Force InIn order order to to ensure ensure the the accuracy accuracy of of the the simulation, simulation, validationvalidation onon forceforce is also necessary, which is 6 isthe the case ofof thethe ﬂowflow crossingcrossing a a circular circular cylinder cylinder with with the the Re Re = 1.2= 1.2× ×10 10.6 The. The velocity velocity of of incoming incoming steady current is about 1.2 m/s. The water depth is about 10 m. The pipeline diameter is 1.0 m. The

Water 2018, 10, 1814 9 of 29 Water 2018, 10, x FOR PEER REVIEW 9 of 29 Water 2018, 10, x FOR PEER REVIEW 9 of 29 steady current is about 1.2 m/s. The water depth is about 10 m. The pipeline diameter is 1.0 m. The flow field and vorticity field around the pipeline simulated by SIFOM–FVCOM system are drawn in flow field and vorticity field around the pipeline simulated by SIFOM–FVCOM system are drawn in flowFigure field 6, indicating and vorticity that field the vortexaround behi the ndpipeline the pipeline simulated is shedding by SIFOM–FVCOM periodically. system are drawn in FigureFigure6 6,, indicating indicating that that the the vortex vortex behind behind the the pipeline pipeline is is shedding shedding periodically. periodically.

(a) (b) (a) (b)

Figure 6. Flow field.field. ( a) Velocity. ((bb)) Vorticity.Vorticity. Figure 6. Flow field. (a) Velocity. (b) Vorticity. 2 Figure7a displays the predicted mean pressure distribution ( Cp = (pc − pc∞)/(0.5ρU ∞)), Figure 7a displays the predicted mean pressure distribution (Cp = (pc−pc∞)/(0.5ρU2∞)), compared compared with the experimental data and simulation results. Here,p pc isc thec∞ static pressure2∞ at the with Figurethe experimental 7a displays datathe predicted and simulation mean pressureresults. Here, distribution pc is the (C static= (p −pressurep )/(0.5 ρatU the)), peripheralcompared peripheral angle θ of the cylinder, and θ is clockwise angle fromc the stagnation point. pc∞ is the static withangle the θ of experimental the cylinder, dataand θand is clockwise simulation angle results. from Here, the stagnation p is the staticpoint. pressure pc∞ is the atstatic the pressureperipheral of 2 pressure of the flow at infinity. The predictions of the skin friction coefficientc∞ (C = τ/(0.5ρU ∞)) are anglethe flow θ of at the infinity. cylinder, The and predictions θ is clockwise of the angle skin from friction the stagnation coefficient point. (Cf = pτ/(0.5 is ρthef U2 static∞)) are pressure shown ofin τ 2 showntheFigure flow in7b. at Figure Here, infinity. 7τb. is Here,Thethe tangential predictionsτ is the wall tangential of shearthe skin stress. wall friction shear In the stress.coefficient figures, In the (C fsimulation figures,= /(0.5ρ theU data∞ simulation)) are are shown collected data in areFigureby the collected SIFOM–FVCOM7b. Here, by theτ is SIFOM–FVCOM the system.tangential The wall 3D system. shearLES and Thestress. URANS 3D In LES the resultsand figures, URANS [9,31] the aresimulation results plotted [9,31 fordata] arecomparison. are plotted collected for It comparison.byis well the SIFOM–FVCOMknown It isthat well the known flow system. thatin the The the separation flow3D LES in the and point separation URANS region results point has a region [9,31]strong hasare pressure aplotted strong gradient, for pressure comparison. and gradient, there It andisis welllittle there knowndifference is little that differencebetween the flow simulation in between the separation simulationand experimental point and region experimental results. has a In strong short, results. pressure the Insimulation short, gradient, the results simulation and agreethere resultsiswell little with agreedifference the well measurements withbetween the measurementssimulation around the and cylinder. around experimental the cylinder. results. In short, the simulation results agree well with the measurements around the cylinder.

(a) (b) (a) (b) Figure 7. Comparison between simulation results and measurement data. (a) Mean pressure distribution.Figure 7. Comparison (b) Skin friction between distribution. simulation results and measurement data. (a) Mean pressure Figuredistribution. 7. Comparison (b) Skin friction between distribution. simulation results and measurement data. (a) Mean pressure 4. Resultsdistribution. and Discussions (b) Skin friction distribution. 4. Results and Discussions 4.1. Hydrodynamics around Submarine Pipeline with No Scour 4. Results and Discussions 4.1. HydrodynamicsIn this section, around the hydrodynamics Submarine Pipeline around with theNo Scour submarine pipeline with no scour is studied based4.1. Hydrodynamics on the SIFOM–FVCOM around Subm couplingarine Pipeline model. with The No Scour influence of different pipeline diameters, tidal In this section, the hydrodynamics around the submarine pipeline with no scour is studied amplitudes and water depths on the flow structure and hydrodynamic forces on the pipeline is also basedIn on this the section, SIFOM–FVCOM the hydrodynamics coupling model.around The the influencesubmarine of pipelinedifferent with pipeline no scour diameters, is studied tidal researched. According to the research results, empirical formulas of hydrodynamic forces under tidal basedamplitudes on the and SIFOM–FVCOM water depths on coupling the flow model. structure The and influence hydrodynamic of different forces pipeline on the diameters, pipeline is tidal also flows are proposed. amplitudesresearched. andAccording water depths to the onresearch the flow results, structure em piricaland hydrodynamic formulas of hydrodynamicforces on the pipeline forces isunder also researched.tidal flows are According proposed. to the research results, empirical formulas of hydrodynamic forces under tidal flows are proposed.

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4.1.1.4.1.1. The MeshMesh andand CalculationCalculation SettingSetting InIn orderorder toto understandunderstand thethe hydrodynamicshydrodynamics atat submarinesubmarine pipelinespipelines inin tidaltidal ﬂows,flows, aa modelingmodeling studystudy hashas beenbeen mademade forfor aa problemproblem depicteddepicted inin FigureFigure8 .8.

FigureFigure 8. 8. TheThe numerical numerical experimental experimental sketch sketch of of f ﬂowlow passing passing a submarinesubmarine pipeline. pipeline.

TheThe influence influence factors factors include include pipeline pipeline diameter, diameter, tide tide amplitude amplitude and andwater water depth. depth. The Thecomputational computational domain domain is 50 is km 50 in km length, in length, 200 m 200 in m width, in width, with witha pipeline a pipeline laid laidon the on seabed. the seabed. The Theboundaries boundaries of the of computational the computational domain domain are controlled are controlled by the bytidal the elevation. tidal elevation. The mesh The of meshFVCOM of FVCOMcovers the covers entire the flow entire domain, flow and domain, SIFOM and contains SIFOM th containsree sets threeof grids, sets one of grids,(G3) of one which (G3) wraps of which the wrapspipeline, the with pipeline, the other with two the othersets of two grids sets (G1 of and grids G2 (G1) enclosing and G2) enclosingthe rest of thethe restnear-field of the near-fieldregion, as region,shown asin shownFigure 9 in Figure9 TheThe surfacesurface elevationelevation atat thethe boundariesboundaries ofof thethe flowflow domaindomain isisprescribed prescribedas asA Asin(sin(kkxx− − ωωtt),), inin whichwhichA Ais is the the amplitude amplitude of theof the tide, tide,k the k wavethe wave number, number, and ω andthe ω frequency. the frequency. Considering Considering the regular the semi-diurnalregular semi-diurnal tide, the tide, period the of period the tide of isthe set tide to beis 12set h,to whichbe 12 h, corresponds which corresponds to astronomical to astronomical tides. tides.The meshes include 1360 elements in the horizontal plane and 21 layers of grids in the vertical directionThe inmeshes the FVCOM. include As1360 for elements the SIFOM, in the horizontal pipeline diameter plane and has 21 some layers effects of grids on the in mesh the vertical which closesdirection to the in pipelinethe FVCOM. (namely, As gridfor the G3 inSIFOM, Figure the9), becausepipeline there diameter is a fluid has boundary some effects layer on around the mesh the pipeline.which closes If the to mesh the pipeline is so rough (namely, that the grid boundary G3 in Figure layer is9), ignored, because thethere detailed is a fluid flow boundary characteristics layer aroundaround thethe pipelinepipeline. are If the hard mesh to be is captured so rough and that the the accuracy boundary of thelayer results is ignored, may be the lower detailed by 20% flow to 40%characteristics than the actual around results. the pipeline In order are to ensurehard to the be accuracycaptured ofand the the simulation, accuracy theof the Y-plus results theory may [32 be] islower adopted by 20% when to the40% mesh than isthe set actual around results. the pipeline. In order When to ensure the pipeline the accuracy diameter of the is 0.3simulation, m, the layer the thicknessY-plus theory of mesh [32] is next adopted to the when pipeline the mesh is the is minimum set around of the 0.001 pipeline. m which When is the also pipeline satisfied diameter for the simulationsis 0.3 m, the of layer the larger thickness pipelines. of mesh Then, next the to layer the thicknesspipeline is of the mesh minimum increases of from 0.001 the m first which boundary is also layersatisfied of 0.001 for the m basedsimulations on the of increasing the larger ratio pipelines. of 1.12. Then, It can the not layer only thickness ensure the of calculationmesh increases accuracy, from butthe alsofirst reduceboundary the layer number of 0.001 of grids m resultingbased on inthe the incr reductioneasing ratio of the of computing1.12. It can time. not only ensure the calculationWith the accuracy, expansion but of also the reduce calculation the number range in of SIFOM, grids resulting compared in withthe reduction the flow field,of the the computing pipeline istime. rather smaller and the effect of the pipeline on the whole field flow is weak. Besides, with the increaseWith of the meshexpansion layer basedof the oncalculation the increasing range ratio in ofSIFOM, 1.12, thecompared effect of thewith pipeline the flow diameter field, the on thepipeline scale ofis rather grids also smaller can beand ignored the effect in the of the far flowpipeline field. on So, the the whole lengths field in flowx-direction is weak. of Besides, the grids with (G1 andthe increase G2) are theof the similar mesh in layer all cases based in on this the section. increasing In summary, ratio of 1.12, the totalthe effect number of the of gridspipeline inSIFOM diameter is 39,996,on the scale with of grid grids spacing also can refined be ignored as 0.001 in m the at far the flow wall field. of the So, pipe the and lengths the seabed. in x-direction of the grids (G1 and G2) are the similar in all cases in this section. In summary, the total number of grids in SIFOM is 39,996, with grid spacing refined as 0.001 m at the wall of the pipe and the seabed.

Water 2018, 10, x FOR PEER REVIEW 11 of 29 Water 20182018,, 1010,, 1814x FOR PEER REVIEW 1111 of of 29

(a) (a)

(b) (c) (b) (c)

Figure 9. The model setup. ( a)) Computational domain. ( (b)) Grids. Grids. ( (cc)) The The mesh mesh closing closing to to pipeline. pipeline. Figure 9. The model setup. (a) Computational domain. (b) Grids. (c) The mesh closing to pipeline. 4.1.2. Effect of Pipeline Diameter 4.1.2. Effect of Pipeline Diameter In this simulation, the influence inﬂuence of pipeline di diameterameter on hydrodynamic forces acting on the pipelineIn this is studied. simulation, The meshes the influence and calculation of pipeline settings diameter are represented on hydrodynamic in the Section forces 4.1.14.1.1. acting. The on tidal the amplitudepipeline is Astudied. is 1 m Theand mesheswater depth and calculation H isis 20 m. settingsThe The pipeline pipeline are represented diameter diameter DD in isis the designed designed Section to to4.1.1. 0.3 0.3 m, m,The 0.7 0.7 tidal m, m, 1.0amplitude m, 1.3 m A and is 1 1.7m and m, respectively. water depth H is 20 m. The pipeline diameter D is designed to 0.3 m, 0.7 m, 1.0 m,In 1.3 thethe m calculation calculationand 1.7 m, domain, respectively.domain, a gaugea gauge is set is above set above the pipeline the pipeline to monitor to monitor the free surfacethe free elevations, surface whichelevations,In indicates the whichcalculation that indicates the numericaldomain, that the a result gaugenumerical satisﬁes is set result theabove analyticalsatisfies the pipelinethe result analytical well to (Figuremonitor result 10 wellthe). Therefore, free(Figure surface 10). the Therefore,targetelevations, tide canthewhich target be simulatedindicates tide can accuratelythat be simulatedthe numerical in the accurately study. result insatisfies the study. the analytical result well (Figure 10). Therefore, the target tide can be simulated accurately in the study.

Figure 10. TheThe free free surface elevation comparison between numerical result and analytical solution. Figure 10. The free surface elevation comparison between numerical result and analytical solution.

Water 2018, 10, 1814 12 of 29

Water 2018, 10, x FOR PEER REVIEW 12 of 29 When the pipeline diameter is 1 m, the instantaneous flow structures at time instant t1, t2 and t3 are depictedWhen inthe Figure pipeline 11. diameter At time tis1, 1 the m, flowthe inst velocityantaneous of the flow whole structures flow field at time is quite instant low t1 (Figure, t2 and t103 a) sinceare thedepicted flood tidein Figure has just 11. begun.At time Itt1 is, the seen flow that velocity there is of one the vortex whole behindflow field the is pipeline quite low at the(Figure beginning 10a) ofsince the development the flood tide of has flow just field. begun. The lengthIt is seen of vortexthat there is about is one 4 mvortex and thebehind shape the of pipeline the vortex at the is flat. Thebeginning vortex under of the tidaldevelopment flow is much of flow more field. than The that length under of vortex the wave, is about because 4 m theand vortex the shape formation of the is subjectvortex to is severalflat. The influence vortex under factors tidal that flow the is much period more of the than tide that is under very longthe wave, and thebecause velocity the vortex changes formation is subject to several influence factors that the period of the tide is very long and the slowly. Correspondingly, the period of vortex shedding is also relatively long and the swirl evolves velocity changes slowly. Correspondingly, the period of vortex shedding is also relatively long and slowly. Meanwhile, the seabed also affects the formation of the whirlpool to some extent. Then, the the swirl evolves slowly. Meanwhile, the seabed also affects the formation of the whirlpool to some tide flow moves towards the right gradually, leading to a number of vortices behind or at the right extent. Then, the tide flow moves towards the right gradually, leading to a number of vortices side of the pipe at time t2 (Figure 11b). At this time, both the velocity of the flow field and the forces behind or at the right side of the pipe at time t2 (Figure 11b). At this time, both the velocity of the onflow the pipelinefield and reach the forces the maximum. on the pipeline When thereac tideh the has maximum. reversed itsWhen direction the tide afterwards has reversed at time its t3, there are still some vortices at the left side of the pipe (Figure 11c). Interestingly, at this moment, the direction afterwards at time t3, there are still some vortices at the left side of the pipe (Figure 11c). reversedInterestingly, current at takesthis moment, place underneath the reversed the curre top current,nt takes whichplace underneath remains flowing the top towards current, thewhich right; whileremains the middleflowing and towards bottom the layersright; while of water the movemiddle in and opposite bottom directions. layers of water The flow move is in very opposite weak at thisdirections. time and The there flow is a is larger very whirlpool weak at this appearing time and at thethere intersection is a larger of whirlpool the flow field.appearing at the intersection of the flow field.

(a)

(b)

(c)

Figure 11. Simulated 2D flow field at the pipe. The flow in (a–c) are in order of time t1, t2 and t3. Figure 11. Simulated 2D flow field at the pipe. The flow in (a–c) are in order of time t1, t2 and t3.

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Water 2018, 10, x FOR PEER REVIEW 13 of 29 When the pipeline diameter changes and the tidal free surface elevation above the pipeline reaches When the pipeline diameter changes and the tidal free surface elevation above the pipeline the maximum, the flow fields are shown in Figure 12. With the increase of pipeline diameter, the reaches the maximum, the flow fields are shown in Figure 12. With the increase of pipeline diameter, velocity above the pipeline increases and the swirls behind the pipeline become bigger and bigger, the velocity above the pipeline increases and the swirls behind the pipeline become bigger and while the vortices fall off the pipeline constantly with a flat and long shape. bigger, while the vortices fall off the pipeline constantly with a flat and long shape.

(a) D = 0.30 m

(b) D = 0.70 m

(d) D = 1.70 m

Figure 12. The ﬂow ﬁelds around different diameter pipelines in time of t = 39 h. Figure 12. The flow fields around different diameter pipelines in time of t = 39 h.

Water 2018, 10, x 1814 FOR PEER REVIEW 1414 of of 29 29

When the pipeline diameter is 1 m, the hydrodynamic forces on the pipeline are displayed in When the pipeline diameter is 1 m, the hydrodynamic forces on the pipeline are displayed in the the Figure 13. The time instants t1, t2 and t3 correspond to three moments of flow files in Figures 11a– Figure 13. The time instants t1, t2 and t3 correspond to three moments of flow files in Figure 11a–c, c, respectively. The simulation time of tidal flow is four days, namely 96 h. fx and fz are the horizontal respectively. The simulation time of tidal flow is four days, namely 96 h. f and f are the horizontal and and vertical hydrodynamic forces on unit length of pipeline, respectively.x Thez maximum horizontal vertical hydrodynamic forces on unit length of pipeline, respectively. The maximum horizontal and and vertical hydrodynamic forces are 108.96 N/m and 29.42 N/m, respectively. The dotted line vertical hydrodynamic forces are 108.96 N/m and 29.42 N/m, respectively. The dotted line represents represents the simulation results of SIFOM–FVCOM, while the solid line is the filtering curve which the simulation results of SIFOM–FVCOM, while the solid line is the filtering curve which is an average is an average curve made with a filtering frequency of 0.30 Hz depending on the simulation results. curve made with a filtering frequency of 0.30 Hz depending on the simulation results. There are two There are two reasons for the addition of the filtering curve in every figure. Firstly, due to the reasons for the addition of the filtering curve in every figure. Firstly, due to the turbulence of local turbulence of local flow around the pipeline, the simulation forces on the pipeline fluctuate to a flow around the pipeline, the simulation forces on the pipeline fluctuate to a certain extent in a short certain extent in a short time. With the fluctuation of the tide, the forces on the pipeline also fluctuate time. With the fluctuation of the tide, the forces on the pipeline also fluctuate regularly in a long time. regularly in a long time. In order to show the fluctuation regularity of the force in the long period, In order to show the fluctuation regularity of the force in the long period, the smooth fitting filtering the smooth fitting filtering line is calculated. Secondly, due to the filtering curves are smooth and line is calculated. Secondly, due to the filtering curves are smooth and match the simulation results match the simulation results well, when the effects of different influence factors on the forces are well, when the effects of different influence factors on the forces are analyzed, the filtering curves can analyzed, the filtering curves can be as the reference lines which can obviously demonstrate the force be as the reference lines which can obviously demonstrate the force changes in different conditions. In changes in different conditions. In order to compare the forces in the different cases conveniently, order to compare the forces in the different cases conveniently, the predicted horizontal and vertical the predicted horizontal and vertical hydrodynamic forces are normalized with the hydrostatic force hydrodynamic forces are normalized with the hydrostatic force in this basic case. The formula of the in this basic case. The formula of the hydrostatic force is written as: Fs = ρgHS where ρ is water hydrostatic force is written as: F = ρgHS where ρ is water density, g is the gravity, H is the water depth, density, g is the gravity, H is thes water depth, S is the section area of the pipeline (S = πD2/4). (In the S is the section area of the pipeline (S = πD2/4). (In the rest of the paper, the predicted hydrodynamic rest of the paper, the predicted hydrodynamic forces are also normalized using this force). The forces are also normalized using this force). The non-dimensional horizontal and vertical forces are non-dimensional horizontal and vertical forces are represented by fx* and fz*. Due to the oscillatory represented by fx* and fz*. Due to the oscillatory flow of the tide, the forces on the pipeline change flow of the tide, the forces on the pipeline change periodically and symmetrically with a period of 12 periodically and symmetrically with a period of 12 h. At the beginning of the simulation, the unstable h. At the beginning of the simulation, the unstable flow causes the force to fluctuate severely. When theflow flow causes field the becomes force to fluctuatestable after severely. about When20 hours, the flowthe change field becomes of force stable tends after to become about 20 gentle h, the gradually.change of forceHowever, tends tobecause become of gentle the turbulence gradually. of However, flow, there because is still of thelittle turbulence fluctuation of flow,of forces. there Furthermore,is still little fluctuation the hydrodynamic of forces. forces Furthermore, on the pipeline the hydrodynamic at the flood forcestide are on larger the pipeline than that at at the the flood ebb tide.tide areBecause larger in than the that process at the of ebb tide tide. propagation, Because in the the higher process the of tidetidal propagation, elevates, the the larger higher the thebottom tidal horizontalelevates, the velocity larger thegrows. bottom The horizontaldifference velocityof velocity grows. leads The to differencethe change of of velocity the hydrodynamic leads to the changeforces. Fromof the the hydrodynamic figure, it is forces.seen that From the the horizontal figure, it isforce seen has that two the horizontalpeaks that forceare about has two the peaks same that in magnitudeare about the but same opposite in magnitude in direction, but while opposite the vertical in direction, force whileis always the verticalin the upward force is direction. always in The the peakupward values direction. in the horizontal The peak values and vertical in the horizontaldirections occur and vertical at the same directions time. occur at the same time.

(a) (b)

Figure 13. The hydrodynamic force of submarine pipeline. (a) Maximum force. (b) Non-dimensional Figuremaximum 13. The force. hydrodynamic force of submarine pipeline. (a) Maximum force. (b) Non-dimensional maximum force. The simulation results and filtering curve of hydrodynamic forces with different pipeline diametersThe simulation are shown results in Figure and 14 .filtering All the hydrodynamiccurve of hydrodynamic forces on theforces submarine with different pipeline pipeline oscillate diametersintensely atare the shown beginning in Figure of simulation 14. All the due hydrodyn to the unsteadyamic forces tidal on flow. the Whensubmarine the tidal pipeline flow becomesoscillate intenselystable, with at the the beginning increase of of pipeline simulation diameter, due to thethe projectionunsteady tidal area flow. in the When force directionthe tidal flow also becomes enlarges, stable,leading with to a the larger increase force onof pipeline the pipeline. diameter, the projection area in the force direction also enlarges, leading to a larger force on the pipeline.

WaterWater 20182018,, 1010,, x 1814 FOR PEER REVIEW 1515 of of 29 29

(a) D = 0.30 m

(b) D = 0.70 m

(d) D = 1.70 m

Figure 14. The hydrodynamic forces on the pipeline with different pipeline diameters. Figure 14. The hydrodynamic forces on the pipeline with different pipeline diameters.

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DependingDepending on thethe simulationsimulation results, results, the the maximum maximum force force values values at different at different pipelines pipelines are drawn are drawnin Figure in Figure15, where 15, thewhere red the and red blue and lines blue are lines the ﬁttingare the curves. fitting curves. It can be It seencan be more seen clearly more thatclearly the thatmaximum the maximum force value force increases value increase as the pipes as the diameter pipe diameter increases. increases.

(a)

(b)

Figure 15. The maximum hydrodynamic forces at different pipelines. (a) Maximum force. Figure(b) Non-dimensional 15. The maximum maximum hydrodynamic force. forces at different pipelines. (a) Maximum force. (b) Non-dimensional maximum force. 4.1.3. Effect of Tidal Amplitude 4.1.3. InEffect this of section, Tidal theAmplitude influence of tidal amplitude on the hydrodynamic forces acting on the pipeline is analyzed.In this section, The meshes the influence and calculation of tidal settings amplitude are representedon the hydrodynamic in the Section forces 4.1.1 .acting The pipeline on the pipelinediameter isD analyzed.is 1 m and The water meshes depth andH is calculation 20 m. The tidalsettings amplitude are represented is set to 0.25 in the m, 0.5Section m, 1.0 4.1.1. m, 1.5 The m pipelineand 2.5 m,diameter respectively. D is 1 m and water depth H is 20 m. The tidal amplitude is set to 0.25 m, 0.5 m, 1.0 m, 1.5Figure m and 16 2.5 shows m, respectively. the flow fields around the pipeline when the crest of the tide is at the top of the pipeline.Figure When 16 shows the tidal the amplitudeflow fields isaround 0.25 m, the the pipeline maximum when velocity the crest around of the the tide pipeline is at the is top 0.35 of m/s. the pipeline.As the tidal When amplitude the tidal reaches amplitude 2.5 m,is 0.25 the maximumm, the maximum velocity velocity around around the pipeline the pipeline increases is 0.35 to about m/s. As1 m/s. the tidal The amplitude higher the reaches tidal water 2.5 m, elevation the maximum is, the largervelocity the around velocity the grows. pipeline Additionally, increases to there about are 1 m/s.whirlpools The higher formed the and tidal shed water behind elevation the pipeline is, the atlarger all flow the velocities.velocity grows. Additionally, there are whirlpools formed and shed behind the pipeline at all flow velocities.

Water 2018, 10, 1814 17 of 29 Water 2018, 10, x FOR PEER REVIEW 17 of 29

(a) A = 0.25 m

(b) A=0.50 m

(d) A = 2.50 m

Figure 16. The ﬂow ﬁelds around the pipeline under different tidal amplitudes in the time of t = 39 h. Figure 16. The flow fields around the pipeline under different tidal amplitudes in the time of t = 39 h.

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Water 2018, 10, x FOR PEER REVIEW 18 of 29 The simulation results of hydrodynamic forces are depicted in Figure 17. It is seen that the The simulation results of hydrodynamic forces are depicted in Figure 17. It is seen that the oscillation of force curves is severer with the tidal amplitude of 0.25 m than that of other curves oscillation of force curves is severer with the tidal amplitude of 0.25 m than that of other curves because larger tidal amplitude leads to larger velocity around the pipeline, thus a larger force. When the because larger tidal amplitude leads to larger velocity around the pipeline, thus a larger force. When hydrodynamic forces are more than the forces caused by ﬂow turbulence, the inﬂuence of turbulence the hydrodynamic forces are more than the forces caused by flow turbulence, the influence of on the pipeline appears weak. In addition, the performance of the hydrodynamic forces on the pipeline turbulence on the pipeline appears weak. In addition, the performance of the hydrodynamic forces is more obvious with the increase of tidal amplitude. on the pipeline is more obvious with the increase of tidal amplitude.

(a) A = 0.25 m (b) A = 0.50 m

Figure 17. The hydrodynamic force on pipeline with different tidal amplitudes. Figure 17. The hydrodynamic force on pipeline with different tidal amplitudes. In Figure 18, the maximum force values at different tidal amplitudes are displayed. As the tidal amplitudeIn Figure changes 18, the from maximum 0.5 m to force 1.5 m,values the slopeat different of the tidal ﬁtting amplitud curvees reaches are displayed. the largest As andthe tidal the increaseamplitude rate changes of the pipeline from 0.5 force m reachesto 1.5 m, its the fastest slope level. of the fitting curve reaches the largest and the increase rate of the pipeline force reaches its fastest level.

Water 2018, 10, 1814x FOR PEER REVIEW 1919 of of 29

(a)

(b)

Figure 18. The hydrodynamic force on pipeline with different tidal amplitudes. (a) Maximum force. Figure(b) Non-dimensional 18. The hydrodynamic maximum force force. on pipeline with different tidal amplitudes. (a) Maximum force. (b) Non-dimensional maximum force. 4.1.4. Effect of Water Depth 4.1.4.In Effect this section,of Water the Depth influence of water depth on the hydrodynamic forces acting on the pipeline is researched.In this section, The meshes the influence and calculation of water settings depth are on represented the hydrodynamic in Section forces 4.1.1. acting The tidal on amplitudethe pipelineA isis 1.0researched. m and pipeline The meshes diameter andD is calculation 1.0 m. The watersettings depth are Hrepresentedis designed toin beSection 10 m, 154.1.1. m, 20The m, 30tidal m amplitudeand 50 m, respectively.A is 1.0 m and pipeline diameter D is 1.0 m. The water depth H is designed to be 10 m, 15 m, 20As m, can 30 m be and seen 50 in m, Figure respectively. 19, with the increase of water depth, the velocity of the flow field around the pipelineAs can decreases.be seen in DueFigure to the 19, obstruction with the increase of the pipeline, of water the depth, backflow the invelocity the front of ofthe the flow pipeline field aroundleads to the the pipeline formation decreases. of the whirlpool, Due to the and obstruction there are alsoof the some pipeline, vortexes the behindbackflow the in pipeline. the front of the pipelineThe simulationleads to the results formation of hydrodynamic of the whirlpool, forces on and the there submarine are also pipeline some at vortexes different waterbehind depth the pipeline.are displayed in Figure 20. Due to the turbulence around the pipeline, the simulation curve of force is not smooth.The simulation With the results increase of ofhydrodynamic water depth, forces the hydrodynamic on the submarine forces pipeline on the submarine at different pipeline water depthcontinue are todisplayed decrease; in however, Figure 20. the Due static to the pressure turbul onence the around pipeline the increases. pipeline, Therefore,the simulation for pipelines curve of forcelaid on is thenot deepsmooth. sea, With static the pressure increase is the of mainwater factor depth, which the hydrodynamic leads to damage forces of the on pipelines. the submarine pipeline continue to decrease; however, the static pressure on the pipeline increases. Therefore, for pipelines laid on the deep sea, static pressure is the main factor which leads to damage of the pipelines.

Water 2018, 10, 1814 20 of 29 Water 2018, 10, x FOR PEER REVIEW 20 of 29

(a) H = 10 m

(b) H = 15 m

(d) H = 50 m

Figure 19. The ﬂow ﬁelds around the pipeline in different water depths in the time of t = 39 h. Figure 19. The flow fields around the pipeline in different water depths in the time of t = 39 h.

Water 2018, 10, 1814 21 of 29 Water 2018, 10, x FOR PEER REVIEW 21 of 29

(a) H = 10 m

(b) H = 15 m

(d) H = 50 m

Figure 20. The hydrodynamic force on pipeline with different water depths. Figure 20. The hydrodynamic force on pipeline with different water depths.

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TheThe maximum maximum values values of of hy hydrodynamicdrodynamic forces forces at at different different water water depth depth are are shown shown in in Figure Figure 21. 21 . AsAs can can be seen, the inﬂuenceinfluence ofof thethe waterwater depthdepth onon the the hydrodynamic hydrodynamic forces forces is is quite quite weak weak for for cases cases of ofwater water depth depth exceeding exceeding 25 25 m. m.

(a)

(b)

Figure 21. The hydrodynamic force on pipeline with different water depths. (a) Maximum force. Figure(b) Non-dimensional 21. The hydrodynamic maximum force force. on pipeline with different water depths. (a) Maximum force. (b) Non-dimensional maximum force. 4.1.5. Hydrodynamic Forces on the Submarine Pipeline 4.1.5. Hydrodynamic Forces on the Submarine Pipeline The computed hydrodynamic forces on the pipe consist of the pressure and the shear stress, which are affectedThe computed by the flowhydrodynamic characteristics. forces All on the the influence pipe consist factors of (pipelinethe pressure diameter, and the tidal shear amplitude stress, whichand water are depth)affected induce by the the flow change characteristics. of flow characteristics All the influence which can factors be reflected (pipeline by thediameter, characteristic tidal amplitudecoefficients and of Froudewater depth) number induce and Reynoldsthe change number. of flow Forcharacteristics example, Reynolds which can number be reflected can be by used the characteristicto distinguish coefficients the laminar of and Froude turbulent number flow. and Besides, Reynolds the number. hydrodynamic For example, forces Reynolds are also related number to canthe be characteristic used to distinguish coefficients, the laminar for instance, and turbulen the resistancet flow. of Besides, the body the in hydrodynamic flow can be characterized forces are also by relatedReynolds to numberthe characteristic and the relative coefficients, magnitude for inst of inertiaance, the force resistance to the gravity of the can body be characterizedin flow can be by characterizedFroude number. by So,Reynolds the maximum number hydrodynamic and the relative forces magnitude are considered of inertia to force be a functionto the gravity of the Froudecan be characterizedand Reynolds by numbers. Froude Depending number. So, on thethe methodmaximum of the hydrodynamic polynomial fitting forces and are the considered discrete data to ofbe the a functionmaximum of forcesthe Froude in previous and Reynolds research, numbers. the two fitting Depending curves areon determined.the method of The the maximum polynomial horizontal fitting andand the vertical discrete forces data (Fx,max of theand maximumFz,max) in every forces case in areprevious normalized research, by F Dthe= Ftwox,max fitting/(ρgDA curves) and FareL = determined.Fz,max/(ρgDA The), respectively. maximum Inhorizontal order to eliminateand vertical the effectforces of(F thex,max constantand Fz,max value) in ofevery the kinematiccase are normalizedviscosity coefficient by FD = F onx,max the/(ρgDA empirical) and coefficientsFL = Fz,max/(ρgDA of the), formulas,respectively. the In Reynolds order to number eliminate is redefinedthe effect of by thethe constant Re* = Re| valueν|, whereof the |kinematicν| is the viscosity value of thecoeffici kinematicent on viscositythe empirical coefficient. coefficients The formulasof the formulas, for the themaximums Reynolds of number hydrodynamic is redefined forces areby writtenthe Re* as:= Re|ν|, where |ν| is the value of the kinematic viscosity coefficient. The formulas for the maximums of hydrodynamic forces are written as: −2 ∗2 −2 ∗2 −2 ∗ FD = −3.88 × 10 Re Fr + 1.11 × 10 Re + 5.39 × 10 Re Fr (12) =− ×−−−22*** + × 22 + × 2 FFrFrD 3.88 10 Re 1.11 10 Re 5.39 10 Re (12) −2 ∗2 −3 ∗2 −2 ∗ FL = −1.07 × 10 Re Fr + 2.9 × 10 Re + 1.45 × 10 Re Fr (13) =− ×−−−2*2 + × 3*2 + × 2* FFrFrL 1.07 10 Re 2.9 10 Re 1.45 10 Re (13)

Water 2018, 10, 1814 23 of 29 Water 2018, 10, x FOR PEER REVIEW 23 of 29

R2 F F The regression statistics index numbers R2 of FD andand FLL areare 0.881 0.881 and 0.860, respectively. The Froude number and and Reynolds Reynolds number number are are calculated calculated based based on on the the water water depth depth HH,, pipe pipe diameter diameter DD,, 1/2 and velocity v*. The velocityvelocity isis evaluatedevaluated bybyv v** = =A A(g(g//H))1/2. .The The modeling modeling results results and and the the extracted extracted formulasformulas are are plotted in Figure 2222..

(a)

(b)

Figure 22. Computed hydrodynamic force. (a) The coefficient of drag force. (b) The coefficient of lift Figureforce. Spheres-computed 22. Computed hydrodynamic hydrodynamic force. forces, (a) The surfaces, coefficient Equations of drag (12) force. and ( (13).b) The coefficient of lift force. Spheres-computed hydrodynamic forces, surfaces, Equations (12) and (13). 4.2. Hydrodynamics around Submarine Pipeline with Scour 4.2. HydrodynamicsDue to the influence around Submarine of flows on Pipeline the sediment with Scour of seabed, the scour around the pipeline on the seabedDue occurs to the easily influence which of mayflows induce on the suspension sediment of of seabed, the pipeline. the scour The Figurearound 23 thea showspipeline the on scour the seabedmorphology occurs around easily thewhich DF1-1 may submarine induce suspension pipeline which of the lies pipeline. off the The west Figure coast of23a Dongfang, shows the Hainan scour morphologyIsland. The data around of scour the DF1-1 hole in submarine Figure 23 bpipeline are measured which usinglies off a the dual-frequency west coast of side-scan Dongfang, sonar Hainan and Island.a swath The sounder data systemof scour [ 27hole]. in Figure 23b are measured using a dual-frequency side-scan sonar and a swath sounder system [27].

Water 2018, 10, x FOR PEER REVIEW 24 of 29 Water 2018,, 10,, 1814x FOR PEER REVIEW 2424 of of 29

(a) (b) (a) (b)

Figure 23. Scour morphology around the submarine pipeline. ( (aa)) Surface Surface map map of scour pit. ( (b)) The Figure 23. Scour morphology around the submarine pipeline. (a) Surface map of scour pit. (b) The measurement data of scour hole. measurement data of scour hole. Depending on the measurement, the formula of fitting curve of scour hole is evaluated as: Depending on the measurement, the formula of fitting curve of scour hole is evaluated as: Depending on the measurement, the formula of fitting curve of scour hole is evaluated as: ( 00.11.11||xx|−| −h−h 0.0.1111||x||≤≤ h z z== 0.11| x | h 0.11| x | h (13)(14) z =  00 0.0.1111||x||>> h (13)  0 0.11| x |> h in which the deepest point of the scour hole is defineddefined as the originaloriginal point, namely x = 0; h is the in which the deepest point of the scour hole is defined as the original point, namely x = 0; h is the depth of the scour hole. depth of the scour hole. In this section, three different cases are simulatedsimulated and analyzed. Based on the fitting fitting Equation In this section, three different cases are simulated and analyzed. Based on the fitting Equation (14) of scour hole, the scour hole of simulation model is established. The depth ofof scour holehole isis 0.25 m, (14) of scour hole, the scour hole of simulation model is established. The depth of scour hole is 0.25 m, 0.875 m and 1.5 m, respectively, and the width of the scour hole is is 4.56 m, 15.91 15.91 m m and and 27.27 27.27 m, m, 0.875 m and 1.5 m, respectively, and the width of the scour hole is 4.56 m, 15.91 m and 27.27 m, respectively. The The water water depth depth is is 20 20 m. m. The The tidal tidal wa waveve amplitude amplitude is is 1 1m m and and th thee pipeline pipeline diameter diameter is is1 respectively. The water depth is 20 m. The tidal wave amplitude is 1 m and the pipeline diameter is 1 m.1 m. The The coordinate coordinate of of the the pipeline pipeline center center is is (x (x, ,zz) )= = (0 (0 m, m, 0.5 0.5 m). m). The The pipeline pipeline is is taken taken as as a a rigid rigid body body m. The coordinate of the pipeline center is (x, z) = (0 m, 0.5 m). The pipeline is taken as a rigid body and the scour hole maintains a steadily evolving topography. topography. In the model, there are three structure and the scour hole maintains a steadily evolving topography. In the model, there are three structure grids (G1,(G1, G2G2 and and G3) G3) of of SIFOM SIFOM and and one one unstructured unstructured grid ofgrid FVCOM. of FVCOM. The meshes The meshes of SIFOM–FVCOM of SIFOM– grids (G1, G2 and G3) of SIFOM and one unstructured grid of FVCOM. The meshes of SIFOM– FVCOMare shown are in shown the Figure in the 24 Figure, in which 24, in the which pipeline the ispipeline constructed is constructed by O-mesh by (G3)O-mesh and (G3) the meshand the of FVCOM are shown in the Figure 24, in which the pipeline is constructed by O-mesh (G3) and the meshFVCOM of FVCOM is similar is to similar that in to Section that in 4.1 Section. The resolution4.1. The resolution of mesh closingof mesh to closing the pipeline to the andpipeline seabed and is mesh of FVCOM is similar to that in Section 4.1. The resolution of mesh closing to the pipeline and seabed0.001 m. is 0.001 m. seabed is 0.001 m.

(a) (b) (a) (b) Figure 24. The meshes and scour shape of SIFOM–FVCOM. (a) Grids. (b) The mesh closing of Figurethe pipeline. 24. The meshes and scour shape of SIFOM–FVCOM. (a) Grids. (b) The mesh closing of the Figure 24. The meshes and scour shape of SIFOM–FVCOM. (a) Grids. (b) The mesh closing of the pipeline. Whenpipeline. the wave crest of tide is at the top of the pipeline, the velocity of flow field reaches the maximumWhen andthe wave the flow crest fields of tide are is shown at the intop Figures of the 25pipeline,a–27a. Thethe velocity flow velocity of flow above field thereaches pipeline the When the wave crest of tide is at the top of the pipeline, the velocity of flow field reaches the maximumdecreases withand the increaseflow fields of theare scourshown depth, in Figures since 25a–27a. when the The depth flow of velocity scour hole above increases, the pipeline more maximum and the flow fields are shown in Figures 25a–27a. The flow velocity above the pipeline decreases with the increase of the scour depth, since when the depth of scour hole increases, more decreases with the increase of the scour depth, since when the depth of scour hole increases, more

Water 2018, 10, x FOR PEER REVIEW 25 of 29

Water 2018, 10, 1814 25 of 29 Waterwater 2018 passes, 10, x FOR the PEER scour REVIEW hole below the pipeline which induces the volume of water above25 of the29 Water 2018, 10, x FOR PEER REVIEW 25 of 29 pipeline to decrease. Besides, vortex shedding can be found behind the pipeline in every case. In water passes the scour hole below the pipeline which induces the volume of water above the waterFigures passes 26a and the scour27a, it hole is seen below that the the pipeline backflow which leads induces to the theformation volume of of whirlpools water above at thethe pipelineupward pipelinewater passes to decrease. the scour Besides, hole belowvortex theshedding pipeline can which be found induces behind the the volume pipeline of waterin every above case. the In toslope decrease. of scour Besides, hole as vortex a result shedding of the obstruction can be found of the behind pipeline. the pipeline in every case. In Figures 26a Figurespipeline 26a to decrease.and 27a, itBesides, is seen thatvortex the shedding backflow canleads be to found the formation behind the of whirlpoolspipeline in atevery the upwardcase. In and 27Undera, it is seenthe effect that the of backﬂowthe tidal leadsflow, to hydrodynamic the formation offorces whirlpools on the at pipeline the upward change slope cyclically of scour slopeFigures of 26ascour and hole 27a, as it a isresult seen of that the the obstruction backflow of leads the pipeline.to the formation of whirlpools at the upward hole(Figures as a result25b–27b). of the The obstruction vertical forces of the stay pipeline. negative since the average velocity above the pipeline is slopeUnder of scour the hole effect as aof result the oftidal the flow,obstruction hydrodynamic of the pipeline. forces on the pipeline change cyclically smallerUnder than the that effect under of the the pipeline, tidal ﬂow, while hydrodynamic the vertical downward forces on force the pipelineon the pipeline change greater cyclically than (FiguresUnder 25b–27b). the effect The ofvertical the tidal forces flow, stay hydrodynamicnegative since theforces average on the velocity pipeline above change the pipeline cyclically is (Figuresthe upward 25b– force27b). according The vertical to the forces Bernoulli stay negative equation. since the average velocity above the pipeline smaller(Figures than 25b–27b). that under The verticalthe pipeline, forces while stay negatithe vertveical since downward the average force velocity on the pipelineabove the greater pipeline than is is smaller than that under the pipeline, while the vertical downward force on the pipeline greater than thesmaller upward than force that underaccording the pipeline,to the Bernoulli while the equation. vertical downward force on the pipeline greater than the upward force according to the Bernoulli equation. the upward force according to the Bernoulli equation.

(a) (b)

(a) (b) Figure 25. The simulation(a) results with h = 0.25 m in time of t = 39 h. (a) Flow(b field.) (b) Hydrodynamic force. Figure 25. 25. TheThe simulation simulation results results with hwith = 0.25h m= in0.25 time mof t in = 39 time h. (a of) Flowt = field. 39 h. (b) Hydrodynamic (a) Flow ﬁeld. force.(Figureb) Hydrodynamic 25. The simulation force. results with h = 0.25 m in time of t = 39 h. (a) Flow field. (b) Hydrodynamic force.

(a) (b)

(a) (b) Figure 26. The simulation(a) results with h = 0.875 m in time of t = 39(b )h. (a) Flow field. (b) Figure 26. The simulation results with h = 0.875 m in time of t = 39 h. (a) Flow ﬁeld. Hydrodynamic force. Figure(b) Hydrodynamic 26. The simulation force. results with h = 0.875 m in time of t = 39 h. (a) Flow field. (b) HydrodynamicFigure 26. The force. simulation results with h = 0.875 m in time of t = 39 h. (a) Flow field. (b) Hydrodynamic force.

(a) (b)

Figure 27. The simulation(a) results with h = 1.5 m in time of t = 39(b) h. (a) Flow ﬁeld. (bFigure) Hydrodynamic 27. The simulation force.(a) results with h = 1.5 m in time of t = 39 h. (a) Flow field. (b)) Hydrodynamic force.

Figure 27. The simulation results with h = 1.5 m in time of t = 39 h. (a) Flow field. (b) Hydrodynamic force. Figure 27. The simulation results with h = 1.5 m in time of t = 39 h. (a) Flow field. (b) Hydrodynamic force.

Water 2018, 10, 1814 26 of 29 Water 2018, 10, x FOR PEER REVIEW 26 of 29

With the increase of scour depth, the maximum horizontal hydrodynamic forces reduce and the vertical hydrodynamic forces grow. The trend of hydrodynamichydrodynamic forces at different scour depths is illustrated inin FigureFigure 2828..

(a)

(b)

Figure 28. The hydrodynamic forces on pipeline with different scour depths. (a) Maximum force. (Figureb) Non-dimensional 28. The hydrodynamic maximum forces force. on pipeline with different scour depths. (a) Maximum force. (b) Non-dimensional maximum force. 5. Conclusions 5. ConclusionsUsing the SIFOM–FVCOM system, a modeling study is carried out on hydrodynamics at submarineUsing pipelinesthe SIFOM–FVCOM under the forcesystem, of tidala modeling currents. study The is maincarried conclusions out on hydrodynamics are summarized at below.submarine Through pipelines the under comparison the force between of tidal the curr numericalents. The main results conclusions and experimental are summarized measurements, below. theThrough simulation the comparison accuracy of SIFOM–FVCOMbetween the numerical modeling results is verified. and experimental Depending on measurements, this modeling, the influencesimulation of accuracy pipeline diameter,of SIFOM–FVCOM tidal amplitude modeling and wateris verified. depth onDepending the hydrodynamic on this modeling, forces acting the oninfluence the pipeline of pipeline is investigated. diameter, tidal It is concludedamplitude thatand withwater the depth increase on the of hydrodynamic pipeline diameter forces and acting tidal amplitude,on the pipeline the hydrodynamicis investigated. forcesIt is concluded on the pipeline that with increase. the increase However, of pipeline the hydrodynamic diameter and forcetidal decreasesamplitude, with the thehydrodynamic water depth forces increasing. on the Besides,pipeline flowincrease. structures However, around the the hydrodynamic pipelines are force also revealeddecreases under with the different water conditions.depth increasing. Since the Besides, period flow of tide structures is very longaround and the the pipelines velocity changesare also slowly,revealed the under falling different vortex behind conditions. the pipeline Since the appears period flat of and tide long, is very while long the periodand the of velocity vortex shedding changes isslowly, relatively the longfalling and vortex the swirl behind evolves the pipeline slowly as appears well. In flat the and meantime, long, while the seabed the period also affectsof vortex the formationshedding is of relatively the whirlpool long to and some the extent. swirl evolves Furthermore, slowly the as maximum well. In the corresponding meantime, the hydrodynamic seabed also forcesaffects onthe the formation pipeline areof the empirically whirlpool formulated to some extent. as a function Furthermore, of the Froude the maximum and Reynolds corresponding numbers. Accordinghydrodynamic to the forces real caseson the in pipeline the submarine are empirica pipelinelly engineering,formulated as the a hydrodynamicsfunction of the Froude around and the suspendedReynolds numbers. submarine According pipeline is to researched. the real Whencases thein the seabed submarine scour appears pipeline around engineering, the pipeline, the thehydrodynamics pipeline is in around the suspended the suspended state. Thesubmarine results showpipeline that is with researched. the increase When of the scour seabed depth, scour the hydrodynamicappears around forces the pipeline, on the submarine the pipeline pipeline is in the decrease. suspended state. The results show that with the increase of scour depth, the hydrodynamic forces on the submarine pipeline decrease. This paper demonstrates the capability of the SIFOM–FVCOM system in direct numerical simulations of high-fidelity multi-physics phenomena at vastly distinct scales, which is more

Water 2018, 10, 1814 27 of 29

This paper demonstrates the capability of the SIFOM–FVCOM system in direct numerical simulations of high-ﬁdelity multi-physics phenomena at vastly distinct scales, which is more advantageous than existing models in some extent. The results presented in this paper might be useful and advisable in the mechanism of the ﬂows and the research of pipeline in the real marine environment from an engineering point of view.

Author Contributions: E.Z. carried out the simulations, analyzed the data and wrote the manuscript. L.M. and B.S. critically reviewed the manuscript. Funding: This research was funded by the National Key Research and Development Program of China (Grant No. 2017YFC1404700), the Guangdong Special Fund for Marine Economy Development (Grant No. GDME-2018E001), the Discipline Layout Project for Basic Research of Shenzhen Science and Technology Innovation Committee (Grant No.20170418), and the National Natural Science Foundation of China (No. 41806010). Acknowledgments: The authors are grateful to H.S. Tang of City College of New York for help with program development and computation setup and to C.S. Chen for his input on FVCOM. Our gratitude goes to the editors and anonymous reviewers whose comments and suggestions expressively contributed to the improvement of this paper. Conﬂicts of Interest: The authors declare no conﬂicts of interest.

Appendix A The Schwarz alternative iteration is used for solution exchange at model interfaces when marching from time level tn to the new time level tn+1. The process is as shown follows: I: Initialization at m = 0.

(u, p , T)m = (u, p , T)n , ∀N d NSIFOM d NSIFOM SIFOM

(u, ω, ζ, T)m = (V, ω, ζ, T)n , ∀N NFVCOM NFVCOM FVCOM where m is the iteration index; NSIFOM is a grid node of SIFOM; NFVCOM is a grid node of FVCOM. II: The solution exchange at model interfaces by interpolation “⇒”.

+ (u, p , T)m 1 ⇐ (V, ω, ζ, T)m , ∀∂N , ∃H h ∂NSIFOM HFVCOM SIFOM FVCOM

+ (V, ω, T)m 1 ⇐ (u, T)m , ∀∂N , ∃H ∂NFVCOM HSIFOM FVCOM SIFOM where ∂NSIFOM is an interface node of SIFOM; ∂NFVCOM is an interface node of FVCOM; III: solution of SIFOM and FVCOM. + Solution of SIFOM (u, p , T)m 1 , ∀N \∂N d NSIFOM SIFOM SIFOM + Solution of FVCOM (V, ω, η, T)m 1 , ∀N \∂N NFVCOM FVCOM FVCOM IV: convergence check. Determine whether to continue iterating through the convergence check.

+ k(u, p , T)m 1 − (u, p , T)m k < e d ∂NSIFOM d ∂NSIFOM

+ k(V, ω, ζ, T)m 1 − (V, ω, ζ, T)m k < e ∂NFVCOM ∂NFVCOM where e is a give convergence error; HSIFOM is the host cell of the interface node of FVCOM in the SIFOM; HFVCOM is the host cell of the interface node of SIFOM is in the FVCOM. V: solution at time level n + 1.

+ (u, p , T)n 1 = (u, p , T)m , ∀N d NSIFOM d NSIFOM SIFOM

+ (V, ω, ζ, T)n 1 = (V, ω, ζ, T)m , ∀N NFVCOM NFVCOM FVCOM Water 2018, 10, 1814 28 of 29

References

1. Shankar, N.J.; Cheong, H.F.; Subbiah, K. Forces on a smooth submarine pipeline in random waves—A comparative-study. Coast. Eng. 1987, 11, 189–218. [CrossRef] 2. Al-Dakkan, K.; Alsaif, K. On the design of a suspension system for oil and gas transporting pipelines below ocean surface. Arab. J. Sci. Eng. 2012, 37, 2017–2033. [CrossRef] 3. Kok, N.J. Lift and Drag Forces on a Submarine Pipeline in Steady Flow. Ph.D. Thesis, University of Cape Town, Cape Town, South Africa, 1988. 4. Li, X.; Li, M.G.; Zhou, J. Experimental study of the hydrodynamic force on a pipeline subjected to vertical seabed movement. Ocean Eng. 2013, 72, 66–76. [CrossRef] 5. Yang, B.; Gao, F.P.; Wu, Y.X.; Li, D.H. Experimental study on vortex-induced vibrations of submarine pipeline near seabed boundary in ocean currents. China Ocean Eng. 2006, 20, 113–121. 6. Mattioli, M.; Alsina, J.M.; Mancinelli, A.; Miozzi, M.; Brocchini, M. Experimental investigation of the nearbed dynamics around a submarine pipeline laying on different types of seabed: The interaction between turbulent structures and particles. Adv. Water Res. 2012, 48, 31–46. [CrossRef] 7. Mattioli, M.; Mancinelli, A.; Brocchini, M. Experimental investigation of the wave-induced flow around a surface-touching cylinder. J. Fluid Struct. 2013, 37, 62–87. [CrossRef] 8. Pierro, A.; Tinti, E.; Lenci, S.; Brocchini, M.; Colicchio, G. Investigation of the dynamic loads on a vertically-oscillating circular cylinder close to the sea bed: The role of viscosity. J. Offshore Mech. Arct. Eng. 2017, 139, 061101. [CrossRef] 9. Ong, M.C.; Utnes, T.; Holmedal, L.E.; Myrhaug, D.; Pettersen, B. Numerical simulation of flow around a smooth circular cylinder at very high Reynolds numbers. Mar. Struct. 2009, 22, 142–153. [CrossRef] 10. Cheng, L.; Chew, L.W. Modelling of flow around a near-bed pipeline with a spoiler. Ocean. Eng. 2003, 30, 1595–1611. [CrossRef] 11. Chiew, Y.M. Mechanics of local scour around submarine pipelines. J. Hydraul. Eng. 1990, 116, 515–529. [CrossRef] 12. Brors, B. Numerical modeling of flow and scour at pipelines. J. Hydraul. Eng. 1999, 125, 511–523. [CrossRef] 13. Sumer, B.M.; Fredsoe, J. Wave scour around a large vertical circular cylinder. J. Waterw. Port Coast. Ocean Eng. 2001, 127, 125–134. [CrossRef] 14. Myrhaug, D.; Ong, M.C.; Føien, H.; Gjengedal, C.; Leira, B.J. Scour below pipelines and around vertical piles due to second-order random waves plus a current. Ocean Eng. 2009, 36, 605–616. [CrossRef] 15. Zhou, X.L.; Wang, J.H.; Zhang, J.; Jeng, D.S. Wave and current induced seabed response around a submarine pipeline in an anisotropic seabed. Ocean Eng. 2014, 75, 112–127. [CrossRef] 16. Postacchini, M.; Brocchini, M. Scour depth under pipelines placed on weakly cohesive soils. Appl. Ocean Res. 2015, 52, 73–79. [CrossRef] 17. Zhao, M.; Vaidya, S.; Zhang, Q.; Cheng, L. Local scour around two pipelines in tandem in steady current. Coast. Eng. 2015, 98, 1–15. [CrossRef] 18. Lam, K.Y.; Wang, Q.X.; Zong, Z.A. Nonlinear fluid-structure interaction analysis of a near-bed submarine pipeline in a current. J. Fluid Struct. 2002, 16, 1177–1191. [CrossRef] 19. Lin, Z.B.; Guo, Y.K.; Jeng, D.S.; Liao, C.C.; Rey, N. An integrated numerical model for wave-soil-pipeline interactions. Coast. Eng. 2016, 108, 25–35. [CrossRef] 20. Zhao, E.J.; Shi, B.; Qu, K.; Dong, W.B.; Zhang, J. Experimental and Numerical Investigation of Local Scour around Submarine Piggyback Pipeline under Steady Currents. J. Ocean Univ. China 2018, 17, 244–256. [CrossRef] 21. Sabag, S.R.; Edge, B.L.; Soedigdo, I. Wake II model for hydrodynamic forces on marine pipelines including waves and currents. Ocean Eng. 2000, 27, 1295–1319. [CrossRef] 22. Fringer, O.B.; Gerritsen, M.; Street, R.L. An unstructured-grid, finite-volume, nonhydrostatic, parallel coastal ocean simulator. Ocean Model. 2006, 14, 139–173. [CrossRef] 23. Tang, H.S.; Qu, K.; Wu, X.G. An overset grid method for integration of fully 3D fluid dynamics and geophysics fluid dynamics models to simulate multiphysics coastal ocean flows. J. Comput. Phys. 2014, 273, 548–571. [CrossRef] 24. Leonard, B.P. A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Comput. Methods Appl. Mech. Eng. 1979, 19, 59–98. [CrossRef] Water 2018, 10, 1814 29 of 29

25. Tang, H.S.; Keen, T.R.; Khanbilvardi, R. A model-coupling framework for nearshore waves, currents, sediment transport, and seabed morphology. Commun. Nonlinear Sci. Numer. Simul. 2009, 14, 2935–2947. [CrossRef] 26. Chen, C.S.; Liu, H.D.; Beardsley, R.C. An unstructured, finite-volume, three-dimensional, primitive equation ocean model: Application to coastal ocean and estuaries. J. Atmos. Ocean. Technol. 2003, 20, 159–186. [CrossRef] 27. Xu, J.S.; Pu, J.J.; Li, G.X. Field Observations of Seabed Scours around a Submarine Pipeline on Cohesive Bed. In Advances in Computational Environment Science; Lee, G., Ed.; Springer: Berlin, Germany, 2012; Volume 142, pp. 23–33. 28. Jensen, B.L. Large-scale vortices in the wake of a cylinder placed near a wall. In Proceedings of the 2nd International Conference on Laser Anemometry-Advances and Applications, Strathclyde, UK, 21–23 September 1987; pp. 153–163. 29. Liang, D.F.; Cheng, L. Numerical modeling of flow and scour below a pipeline in currents. Part I. Flow simulation. Coast. Eng. 2005, 52, 25–42. [CrossRef] 30. Lyn, D.; Rodi, W. The flapping shear layer formed by flow separation from the forward corner of a square cylinder. J. Fluid Mech. 1994, 267, 353–376. [CrossRef] 31. Catalano, P.; Wang, M.; Iaccarino, G.; Moin, P. Numerical simulation of the flow around a circular cylinder at high Reynolds number. Int. J. Heat Fluid Flow 2003, 24, 463–469. [CrossRef] 32. White, F.M. Fluid Mechanics, 7th ed.; McGraw Hill Press: New York, NY, USA, 2009.

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