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

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## Abstract

**:**

## 1. Introduction

_{D}), added mass coefficient (C

_{A}) and horizontal hydrodynamic force coefficient (C

_{V}). 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.

_{D}) 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 [11,12,13,14,15,16,17,18,19,20]. 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.

## 2. Solver for Incompressible Flow on Overset Meshes–Unstructured Grid Finite Volume Coastal Ocean Model (SIFOM–FVCOM) Modeling System

#### 2.1. SIFOM Modeling

**u**is the velocity vector.

_{0}is the reference density; p

_{d}is the dynamic pressure; ν is the fluid viscosity; ν

_{t}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; T

_{0}is a reference temperature; β is a coefficient reflecting the expansion; C is the tracer concentration; C

_{0}is a reference concentration;

**k**is the unit vector in the vertical direction.

#### 2.2. FVCOM Modeling

**V**is the depth-averaged velocity vector;

**τ**

_{b}is the shear stress on the seabed;

**τ**

_{s}is the shear stress on the water surface;

**G**includes Coriolis force and water friction force.

**e**is the strain rate;

**H**stands for the residual terms; κ and λ are coefficients.

#### 2.3. Coupling Strategy of SIFOM–FVCOM Modeling

_{SIFOM}and Ω

_{FVCOM}are the calculation domains of SIFOM and FVCOM, respectively. The boundaries of ∂Ω

_{SIFOM}and ∂Ω

_{FVCOM}are the interfaces of SIFOM and FVCOM, respectively. Ω

_{SIFOM}

_{-FVCOM}is the zone of SIFOM enclosed between the seabed and the red dashed line of ∂Ω

_{FVCOM}. In the overlap domain, SIFOM and FVCOM exchange solutions at the interfaces of ∂Ω

_{SIFOM}and ∂Ω

_{FVCOM}.

**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.

_{SIFOM}

_{-FVCOM}which is called the blanket zone. In this way, the structure of FVCOM does not need to be modified around the submarine pipeline. In the blanked zone, the solution values are redistributed and interpolated on the domain of FVCOM.

## 3. Modeling System Validation

#### 3.1. Validation on Flow Velocity and Vortex

#### 3.2. Validation on Force

^{6}. The velocity of incoming 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 Figure 6, indicating that the vortex behind the pipeline is shedding periodically.

_{p}= (p

_{c}− p

_{c}

_{∞})/(0.5ρU

^{2}

_{∞})), compared with the experimental data and simulation results. Here, p

_{c}is the static pressure at the peripheral angle θ of the cylinder, and θ is clockwise angle from the stagnation point. p

_{c}

_{∞}is the static pressure of the flow at infinity. The predictions of the skin friction coefficient (C

_{f}= τ/(0.5ρU

^{2}

_{∞})) are shown in Figure 7b. Here, τ is the tangential wall shear stress. In the figures, the simulation data are collected by the SIFOM–FVCOM system. The 3D LES and URANS results [9,31] are plotted for comparison. It is well known that the flow in the separation point region has a strong pressure gradient, and there is little difference between simulation and experimental results. In short, the simulation results agree well with the measurements around the cylinder.

## 4. Results and Discussions

#### 4.1. Hydrodynamics around Submarine Pipeline with No Scour

#### 4.1.1. The Mesh and Calculation Setting

#### 4.1.2. Effect of Pipeline Diameter

_{1}, t

_{2}and t

_{3}are depicted in Figure 11. At time t

_{1}, the flow velocity of the whole flow field is quite low (Figure 10a) since the flood tide has just begun. It is seen that there is one vortex behind the pipeline at the beginning of the development of flow field. The length of vortex is about 4 m and the shape of the vortex is flat. The vortex under tidal flow is much more than that under the wave, because the vortex formation is subject to several influence factors that the period of the tide is very long and the velocity changes slowly. Correspondingly, the period of vortex shedding is also relatively long and the swirl evolves slowly. Meanwhile, the seabed also affects the formation of the whirlpool to some extent. Then, the tide flow moves towards the right gradually, leading to a number of vortices behind or at the right side of the pipe at time t

_{2}(Figure 11b). At this time, both the velocity of the flow field and the forces on the pipeline reach the maximum. When the tide has reversed its direction afterwards at time t

_{3}, there are still some vortices at the left side of the pipe (Figure 11c). Interestingly, at this moment, the reversed current takes place underneath the top current, which remains flowing towards the right; while the middle and bottom layers of water move in opposite directions. The flow is very weak at this time and there is a larger whirlpool appearing at the intersection of the flow field.

_{1}, t

_{2}and t

_{3}correspond to three moments of flow files in Figure 11a–c, respectively. The simulation time of tidal flow is four days, namely 96 h. f

_{x}and f

_{z}are the horizontal and vertical hydrodynamic forces on unit length of pipeline, respectively. The maximum horizontal and vertical hydrodynamic forces are 108.96 N/m and 29.42 N/m, respectively. The dotted line represents the simulation results of SIFOM–FVCOM, while the solid line is the filtering curve which is an average curve made with a filtering frequency of 0.30 Hz depending on the simulation results. There are two reasons for the addition of the filtering curve in every figure. Firstly, due to the turbulence of local flow around the pipeline, the simulation forces on the pipeline fluctuate to a certain extent in a short time. With the fluctuation of the tide, the forces on the pipeline also fluctuate regularly in a long time. In order to show the fluctuation regularity of the force in the long period, the smooth fitting filtering line is calculated. Secondly, due to the filtering curves are smooth and match the simulation results well, when the effects of different influence factors on the forces are analyzed, the filtering curves can be as the reference lines which can obviously demonstrate the force changes in different conditions. In order to compare the forces in the different cases conveniently, the predicted horizontal and vertical hydrodynamic forces are normalized with the hydrostatic force in this basic case. The formula of the hydrostatic force is written as: F

_{s}= ρgHS where ρ is water density, g is the gravity, H is the water depth, S is the section area of the pipeline (S = πD

^{2}/4). (In the rest of the paper, the predicted hydrodynamic forces are also normalized using this force). The non-dimensional horizontal and vertical forces are represented by fx* and fz*. Due to the oscillatory flow of the tide, the forces on the pipeline change periodically and symmetrically with a period of 12 h. At the beginning of the simulation, the unstable flow causes the force to fluctuate severely. When the flow field becomes stable after about 20 h, the change of force tends to become gentle gradually. However, because of the turbulence of flow, there is still little fluctuation of forces. Furthermore, the hydrodynamic forces on the pipeline at the flood tide are larger than that at the ebb tide. Because in the process of tide propagation, the higher the tidal elevates, the larger the bottom horizontal velocity grows. The difference of velocity leads to the change of the hydrodynamic forces. From the figure, it is seen that the horizontal force has two peaks that are about the same in magnitude but opposite in direction, while the vertical force is always in the upward direction. The peak values in the horizontal and vertical directions occur at the same time.

#### 4.1.3. Effect of Tidal Amplitude

#### 4.1.4. Effect of Water Depth

#### 4.1.5. Hydrodynamic Forces on the Submarine Pipeline

_{x,max}and F

_{z,max}) in every case are normalized by F

_{D}= F

_{x,max}/(ρgDA) and F

_{L}= F

_{z,max}/(ρgDA), respectively. In order to eliminate the effect of the constant value of the kinematic viscosity coefficient on the empirical coefficients of the formulas, the Reynolds number is redefined by the Re* = Re|ν|, where |ν| is the value of the kinematic viscosity coefficient. The formulas for the maximums of hydrodynamic forces are written as:

^{2}of F

_{D}and F

_{L}are 0.881 and 0.860, respectively. The Froude number and Reynolds number are calculated based on the water depth H, pipe diameter D, and velocity v*. The velocity is evaluated by v* = A(g/H)

^{1/2}. The modeling results and the extracted formulas are plotted in Figure 22.

#### 4.2. Hydrodynamics around Submarine Pipeline with Scour

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgment

## Conflicts of Interest

## Appendix A

^{n}to the new time level t

^{n}

^{+1}. The process is as shown follows:

_{SIFOM}is a grid node of SIFOM; N

_{FVCOM}is a grid node of FVCOM.

_{SIFOM}is an interface node of SIFOM; ∂N

_{FVCOM}is an interface node of FVCOM;

_{SIFOM}is the host cell of the interface node of FVCOM in the SIFOM; H

_{FVCOM}is the host cell of the interface node of SIFOM is in the FVCOM.

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

**Figure 3.**The calculation meshes. (

**a**) The horizontal plane mesh at z = −0.513 m. (

**b**) The vertical plane mesh.

**Figure 4.**Simulation of the flow passing a cylinder. (

**a**) Computed flow field near the cylinder. (

**b**) Comparison of the horizontal velocity distribution. (

**c**) Comparison of the vertical velocity distribution. Circle-measurement [28], green dash line, modeling with ĸ-ε turbulence closure [29], blue solid line, SIFOM, red dash-dot line, SIFOM–FVCOM.

**Figure 5.**Flow passes a cubic cylinder at bottom of channel. (

**a**) Computed flow field near the square cylinder. (

**b**) Comparison of the velocity distribution. Circle-Measurement [30], dash-dot line, DNS modeling, solid line, SIFOM–FVCOM.

**Figure 7.**Comparison between simulation results and measurement data. (

**a**) Mean pressure distribution. (

**b**) Skin friction distribution.

**Figure 11.**Simulated 2D flow field at the pipe. The flow in (

**a**–

**c**) are in order of time t

_{1}, t

_{2}and t

_{3}.

**Figure 13.**The hydrodynamic force of submarine pipeline. (

**a**) Maximum force. (

**b**) Non-dimensional maximum force.

**Figure 15.**The maximum hydrodynamic forces at different pipelines. (

**a**) Maximum force. (

**b**) Non-dimensional maximum force.

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

**Figure 18.**The hydrodynamic force on pipeline with different tidal amplitudes. (

**a**) Maximum force. (

**b**) Non-dimensional maximum force.

**Figure 21.**The hydrodynamic force on pipeline with different water depths. (

**a**) Maximum force. (

**b**) Non-dimensional maximum force.

**Figure 22.**Computed hydrodynamic force. (

**a**) The coefficient of drag force. (

**b**) The coefficient of lift force. Spheres-computed hydrodynamic forces, surfaces, Equations (12) and (13).

**Figure 23.**Scour morphology around the submarine pipeline. (

**a**) Surface map of scour pit. (

**b**) The measurement data of scour hole.

**Figure 24.**The meshes and scour shape of SIFOM–FVCOM. (

**a**) Grids. (

**b**) The mesh closing of the pipeline.

**Figure 25.**The simulation results with h = 0.25 m in time of t = 39 h. (

**a**) Flow field. (

**b**) Hydrodynamic force.

**Figure 26.**The simulation results with h = 0.875 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 28.**The hydrodynamic forces on pipeline with different scour depths. (

**a**) Maximum force. (

**b**) Non-dimensional maximum force.

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**MDPI and ACS Style**

Zhao, E.; Mu, L.; Shi, B.
Numerical Study of the Influence of Tidal Current on Submarine Pipeline Based on the SIFOM–FVCOM Coupling Model. *Water* **2018**, *10*, 1814.
https://doi.org/10.3390/w10121814

**AMA Style**

Zhao E, Mu L, Shi B.
Numerical Study of the Influence of Tidal Current on Submarine Pipeline Based on the SIFOM–FVCOM Coupling Model. *Water*. 2018; 10(12):1814.
https://doi.org/10.3390/w10121814

**Chicago/Turabian Style**

Zhao, Enjin, Lin Mu, and Bing Shi.
2018. "Numerical Study of the Influence of Tidal Current on Submarine Pipeline Based on the SIFOM–FVCOM Coupling Model" *Water* 10, no. 12: 1814.
https://doi.org/10.3390/w10121814