Numerical Simulation of Flow Field around Jacket Foundations on Flat-Bed and Equilibrium Scour Bathymetry

Abstract

In recent years, jacket foundations have been increasingly employed in offshore wind farms. Their complex design comprising piles and trusses poses challenges for conducting comprehensive flow field measurements using physical experiments. Consequently, the influence of the flow field on local scour around these foundations remains unclear. Therefore, numerical simulation methods are essential to depict the surrounding flow characteristics. This study utilizes large eddy simulation (LES) turbulence models within OpenFOAM to simulate the flow field around jacket foundations on flat-bed and equilibrium scour bathymetry. A flume experiment was conducted for numerical model establishment and validation. The close agreement between experimental and numerical results indicates that the LES model accurately reflects the flow patterns around the jacket foundation. Time-averaged and instantaneous flow characteristics, average kinetic energy (AKE), turbulence structure, and bed shear stress were analyzed. The results indicate that flow intensity is reduced due to the shielding effect and energy dissipation by the truss structure of the jacket foundation. Furthermore, the AKE of the flow upstream of the rear piles decreases by 18.9% in the flat-bed state and 28.0% in the equilibrium state, indicating more energy dissipation and less scour at the rear piles in the equilibrium state. The research findings offer valuable insights into the design and scour protection strategies for jacket foundations.

1. Introduction

Offshore wind power is recognized globally as a key source of clean, renewable energy due to its inherent cleanliness, abundant reserves, and ease of conversion [1]. Types of foundations for offshore wind turbines include gravity foundations, monopile foundations, pile group foundations, jacket foundations, and floating foundations [2]. Among these different foundation types, jacket foundations have gained widespread application in recent years due to their high structural strength, light weight, and ease of transportation and installation [3].

According to a study by Satari et al. [4], the installation of jacket foundations significantly affects the surrounding flow field, often resulting in local scour around the foundation. Yuan et al. [5] indicated that this scour can lead to uneven settlement, compromising the overall stability of the structure and the safe operation of the wind turbine. Additionally, based on the study by Zhu et al. [6], the complex truss structure of the upper part of the jacket foundation further complicates the development of local scour compared to simpler foundation types. Due to the different scour characteristics between monopile and jacket foundations, existing scour protection measures or countermeasures for monopile foundations may require improvement or adjustment before being applied to jacket foundations [7,8]. Therefore, studying the flow field around jacket foundations is crucial for understanding local scour, improving engineering design, and enhancing scour protection measures. This ensures safe operation and facilitates maintenance.

Over the past few decades, researchers have conducted extensive physical experiments and numerical simulations to study the flow characteristics around complex foundations for offshore wind turbines. Breusers and Raudkivi [9] found that, compared to monopile foundations, the key factors influencing scour around pile groups are the spacing between the piles and their arrangement. Currently, jacket foundations are increasingly adopted in offshore wind farms (OWFs) worldwide, yet research on the flow and local scour characteristics around such foundations remains limited. Moreover, the flow field and scour characteristics around jacket foundations under wave and current actions are not fully understood. Lian et al. [10] developed a three-dimensional numerical model to investigate the scour characteristics around large-diameter multi-pile foundations in deep water. They optimized the mesh partitioning and discussed the effects of jacket height, pile diameter, and pile spacing on the flow field and scour depth. Their study revealed that, as the height of the jacket piles increases, the scour depth first increases to a peak value and then decreases. The scour depths around the front row of piles decrease with an increase in the pile diameter, while they increase with an increase in the pile diameter around the rear row of piles. Jiang et al. [11] investigated the scour around jacket foundations on the cohesive seabed. Their experiments demonstrated that the scour depth around the foundation increases with larger pile diameters and higher wave–current velocities but decreases with greater water depth. They found a nearly linear relationship between the ratio of wave–current velocity to the critical initiation velocity of sediment and the maximum scour depth. Additionally, they observed that the maximum scour depth increases with both the ratio of wavelength to water depth and the increases in Froude and Reynolds numbers. Li et al. [12] systematically analyzed the effects of wave and current actions on the local scour depth and range around a jacket foundation for a wind farm project. Welzel et al. [13] simulated the scour problem around jacket foundations when waves and current flow directions were perpendicular. Their experiments revealed rapid scour depth development around the piles of the foundation due to the compression and intensified flow. They also found that scour under combined wave and current action is more severe than under steady flow conditions. Recently, Chen et al. [14] conducted a flume experiment to investigate scour characteristics around jacket foundations under different hydrodynamic conditions. The findings revealed that due to the resistance effect of the truss structure and flow disturbance at the piles, the scour extent around jacket foundations is less significant compared to monopile foundations under the same hydrodynamic conditions.

However, the above experimental studies mainly focus on the scour problem around jacket foundations, particularly focusing on the scour process and equilibrium scour depth around the foundation. Few studies have simulated the flow field around jacket foundations, and there is a lack of comprehensive analysis regarding flow characteristics around these structures and their influence on the local scour process. To better understand the flow characteristics around the jacket foundation during the initial and equilibrium stages of the scour process, flow fields around jacket foundations on flat-bed and equilibrium scour bathymetry were analyzed. This study employed the OpenFOAM platform and used the LES method to simulate the flow field around the jacket foundation under constant flow. A physical experiment was conducted in a current flume, and the experimental results were used for numerical modeling and validation. The close agreement between experimental and numerical results indicates that the LES model accurately reflects the flow patterns around the jacket foundation. Subsequently, the characteristics of the time-averaged flow field, turbulence field, instantaneous flow field, AKE, and bed shear stress were studied numerically. Their influence on the local scour around the foundation was analyzed and discussed to further explain the local scour mechanism around jacket foundations.

2. Flume Experiment and Methodology

To obtain the equilibrium scour hole topography around jacket foundations under steady flow, physical model tests were conducted. Figure 1 shows the experimental flume used in this study, which is 50 m long, 1 m wide, and 1.5 m high. A 2.5 m long, 1 m wide, and 0.3 m deep sand recess was designed at the middle section of the flume, 24 m from the flume inlet. The model was positioned at the center of the sand recess, with 0.34 m from the pile to the flume wall. A concrete slope with a degree of 1:10 and a height of 0.3 meters was placed 14 m from the flume inlet to ensure smooth water flow transition. An acoustic Doppler velocimetry (ADV) was deployed to measure the velocity distributions at various locations within the flume. After the experiments were completed, the output data from the ADV were filtered using WinADV software, V2.024,the United States [15]. Data with low correlation (minimum COR < 70) were removed. Figure 2 shows the schematic diagram of the jacket foundation model, and

Table 1. Dimensions of components of the jacket foundation model.

Component Number	C-01	C-02	C-03	C-04	C-05	C-06	C-07	C-08	C-09	C-10
Diameter (mm)	40	19	14–19	14	7.8	7.6	6.6	6.6	6.6	6.6
Length (mm)	360	17	33	288	163	98	101	81	71	59

lists the dimensions of the model components. The pile diameter (D) was 0.04 m, the spacing between piles was 0.24 m, and the water depth above the artificial deck was 0.3 m.

Once the scour reached equilibrium, the flume was slowly drained, and the topography of the equilibrium scour hole was scanned for numerical modeling. Approximately 150 photos of the sand bed around the jacket foundation model were taken from different angles and levels after each test. These photos were then used to establish a three-dimensional numerical model. For ease of scanning, the jacket foundation could be disassembled into an upper truss structure and piles. In this study, the clear spacing between the piles of the jacket foundation was six times the pile diameter, with the equivalent diameter-to-flume width ratio being 8%, which essentially negates the influence of boundary effects [16]. Figure 3a shows the particle size distribution of the sand used in the experiment. The sand used in the experiment had a median particle size (d₅₀) of 0.25 mm, a geometric standard deviation of 1.49, a measured void ratio of 0.47, and a relative weight of 2.65. Based on previous studies, the critical sediment initiation velocity in the experiment was calculated to be 0.231 m/s [17]. The average flow velocity, which was 0.9 times this critical velocity, was used in this study to ensure a clear water scour condition. Only clear-scour conditions were considered in this study, and no sand was involved in the water motion. The equilibrium scour topography obtained from the experiment is shown in Figure 3b.

3. Numerical Model and Computational Details

3.1. Numerical Modeling Method

The numerical model in this study employs LES. To filter out small-scale fluctuations, an additional stress term is introduced in the Navier–Stokes equations to dissipate the small-scale vortices and prevent excessive turbulent kinetic energy. After filtering, the three-dimensional Navier–Stokes equations for LES are as follows:

$${\displaystyle \frac{\partial \overline{u_i}}{\partial t}}+{\displaystyle \frac{\partial \overline{u_i{u}_j}}{\partial x_j}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \overline{p}}{\partial x_i}}+\gamma {\displaystyle \frac{{\partial }^2\overline{u_i}}{\partial x_j\partial x_j}}$$

(1)

$${\displaystyle \frac{\partial \overline{u_i}}{\partial x_i}}=0$$

(2)

where x_i represents the Cartesian coordinates (i = 1, 2, 3 correspond to the x, y, z directions, respectively); t denotes time in seconds; $\rho$ is the fluid density in kg/m³; $\overline{{\mu }_i}$ is the filtered velocity vector in m/s; and $\overline{p}$ is the filtered pressure in Pascals (Pa). Let ${\tau }_{ij}=\overline{{\mu }_i{\mu }_j}-\overline{{\mu }_i}\overline{{\mu }_j}$, which is termed the subgrid-scale (SGS) stress. This term is an unsolved term in the equations. Therefore, the above equations can be transformed into the following form:

$${\displaystyle \frac{\partial \overline{u_i}}{\partial t}}+\overline{u_j}{\displaystyle \frac{\partial \overline{u_i}}{\partial x_j}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \overline{p}}{\partial x_i}}+\gamma {\displaystyle \frac{{\partial }^2\overline{u_i}}{\partial x_j\partial x_j}}-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial (\rho \overline{u_i{u}_j}-\rho \overline{u_i}\overline{u_j})}{\partial x_j}}$$

(3)

Equation (3) is similar to the Reynolds-averaged Navier–Stokes (RANS) equation, where an unclosed subgrid-scale (SGS) stress term appears. This term cannot be directly calculated and requires simulation using an eddy-viscosity approach. Based on the concept of eddy viscosity, this study employs the dynamic one-equation eddy-viscosity model (DOEE) to solve the governing equations [18].

3.2. Numerical Modeling Conditions

The origin of the coordinate system in the numerical model is set at the center of the jacket foundation at the original bed level. The x-axis is aligned with the flow direction, the y-axis is transverse to the flume, and the z-axis is vertical. As shown in Figure 4a, the computational domain has a length (L) of 2.5 m, a width (B) of 1 m (equal to the width of the flume), and a height of 0.3 m, which corresponds to the water depth (h). The diameter of the jacket foundation piles is D, and the gaps between the piles are 7D.

Given the complexity of the jacket structure, this study employed non-uniform hexahedral unstructured grids. To accurately capture various vortex structures, the mesh near the jacket foundation was locally refined using density boxes. Additionally, to simulate the flow details near the wall and the bed shear stress, the mesh near the bed surface was divided into boundary layers. Since the LES model is sensitive to grid scale, the dimensionless distance y⁺ in the normal direction near the wall should be less than 1.5. After estimation and verification, the thickness of the first layer of the grid near the piles and the bottom surface was set to 0.1 mm. A total of 6,000,000 cells were used.

The boundary conditions of the computational domain were set as follows: The inlet boundary was specified as a velocity inlet, with the upstream measured flow velocity varying with water depth, as shown in Figure 4b. The outlet boundary used a free outflow condition. A slip boundary was employed for the water surface boundary in the modeling. The jacket foundation structure and the sidewalls were set with no-slip conditions. It should be highlighted that in the flume experiment, the bed sand is movable to allow the development of a scour hole around the foundation. However, this study aimed to investigate the flow field characteristics around the jacket foundation on both flat-bed and equilibrium scour bathymetry using numerical modeling. Therefore, the bed sands for both the flat-bed and equilibrium scour bathymetry were set to be immovable in the numerical model. The bed was set with a no-slip condition and included a rough wall boundary condition, with an equivalent sand roughness height of k_s = 2.5d₅₀.

This study utilized the OpenFOAM platform, V9, the United Kingdom, to develop a large time-step transient incompressible flow solver, pimpleFoam, for solving the numerical model. This solver utilized the PIMPLE algorithm to couple the pressure–velocity equations. Throughout the solution process, second-order accurate finite difference schemes were applied to address time, gradients, pressure terms, and convective terms. Discretization was managed using the geometric algebraic multigrid (GAMG) matrix solver for all discretized components. To ensure closure of the governing equations during computation, a dynamic k equation (DKE) subgrid-scale stress model was integrated. The time step ∆t was set in an adjustable mode to stabilize the calculations in the LES framework, with the maximum Courant number maxCo = 1.0. Initial computations began with a time step of 0.002 s.

3.3. Model Verification

Figure 5 shows the distribution of y⁺ around the jacket foundation on flat-bed and equilibrium scour bathymetry, respectively. It can be observed that y⁺ values are all less than 1.5, indicating that the first-layer boundary layer mesh thickness used in this study meets the accuracy requirements of LES.

Furthermore, as shown in Figure 6, the characteristic cross-section (y/D = 3.5) was selected for velocity verification. Velocity profiles of horizontal flow were compared between experimental results and numerical simulations at x/D = −8.5, −6.5, −5.5, −5, −4.5, −2.5, −2, −1.5, and −0.5. The results are shown in Figure 7, where profiles (a) to (e) depict velocity profiles on the upstream side of the foundation, indicating close agreement between experimental and numerical results with errors within 5%. Profiles (f) to (i) show the velocity between the front and rear piles of the jacket foundation, where slight differences up to 10% are observed due to the complex vortex structures behind the piles. Overall, the results demonstrate that the LES model established effectively reflects the flow patterns around the jacket foundation.

4. Results and Discussion

4.1. Time-Averaged Flow Field Characteristics

We selected the cross-section along the centerline of the front and rear rows of piles in the jacket foundation as the characteristic section (y/D = 3.5) for analysis. The blank areas in the section represent the pile structures, with the corresponding pile numbers attached. The water flow direction is from left to right. The pile diameter (D) was used as the basis for the non-dimensional distance parameter.

Figure 8a,b illustrate the time-averaged horizontal velocity distribution around the jacket foundation on flat-bed and equilibrium scour bathymetry, respectively. The time-averaged horizontal velocity shows significant changes at a distance of 2D to 3D from the front row pile (pile 2#), which is consistent with the experimental measurements by Istiarto and Graf [19]. Due to the contraction of the cross-section around the upper truss of the jacket foundation [20], the flow is compressed, and an accelerated jet is generated [21]. The peak value of the time-averaged horizontal velocity occurs at the junction of the front row pile and the truss, reaching 1.4 U₀ (U₀ is critical velocity). Due to the shielding effect from the front piles and obstruction and energy dissipation effects from the upper truss structure, the flow intensity on the upstream side of the rear piles is significantly reduced. As shown in Figure 8a, blue recirculation zones appear on the downstream side of each pile, indicating the formation of shear layers and wake vortices behind the piles. This is a critical factor in the rapid development of scour holes behind the piles and is similar to the reverse flow observed in the downstream and bed intersection areas of cylindrical structures found by Ozturk et al. [22]. As shown in Figure 8b, in the equilibrium state, the scale of the recirculation zones on the upstream side of each pile is consistent with the longitudinal distance of the scour hole, ranging between 2D and 2.5D. The flow velocity gradient is large in this region, indicating the extreme instability of the flow. In addition, inverted triangular recirculation zones appear on the downstream side of the front row piles, indicating the formation of wake vortices behind the piles.

Figure 8c,d show the time-averaged vertical velocity distribution around the jacket foundation in the flat-bed and equilibrium states, respectively. As illustrated in Figure 8c, there is a downflow between the front and rear piles, indicating that the flow descends after passing through the front piles and the upper truss structure. The maximum value of the downflow velocity was obtained behind the rear piles within the height range of 2.5 < z/D < 3.0, approximately 0.5 times the mean flow velocity.

As shown in Figure 8d, the downflow is prominent on the upstream side of each pile in the equilibrium state, with a maximum velocity being 1.7 times that of the flat-bed state. The significant downflow is enhanced by the scour hole, which increases the water depth upstream of the front pile and aids in the development of the downflow. This increased downflow can impact the bottom of the scour hole and lead to the initiation of sand sediment. The initiated sediment may be transported downstream by the horseshoe vortex and wake flow. Together with the recirculation shown in Figure 8b, this forms the primary vortex system of the horseshoe vortex in the scour hole. The stronger downflow is distributed in the range of 0 < z/D < 0.25 and vertically from 0.5 < z/D < 0. Due to the shielding effect, the range of downflow on the upstream side of the rear piles is significantly smaller than that of the front piles.

4.2. Horseshoe Vortex System

Figure 9 shows the distribution of two-dimensional time-averaged streamlines on the characteristic cross-section (y/D = 3.5) of the jacket foundation in the flat-bed and equilibrium scour states. As shown in Figure 8a, two horseshoe vortices (HSVs), HV1 and HV2, develop upstream of the front pile. HV1 is located 1D upstream of the front pile, with the vortex core at (−4.85D, 0.06D). A smaller secondary vortex, HV2, develops upstream of HV1, with its core at (−5.30D, 0.04D). Additionally, a saddle point, S1, can be observed between HV1 and HV2. Another saddle point, S2, is present upstream of HV2. The upstream side of the rear piles also exhibits the aforementioned flow field structure, though with a relatively smaller range due to the shielding effect of the front piles.

Furthermore, three vortex structures develop downstream of the front piles in flat-bed conditions: a primary clockwise vortex, DV1; a secondary counterclockwise vortex, DV2; and a smaller counterclockwise vortex, DV3. DV1 is located downstream of the joint of the front pile and the truss structure, with a vertical scale of 1.00D and a streamwise scale of 1.95D. DV2 is located at the bottom downstream of the front pile, with a vertical scale of 0.50D and a streamwise scale of 1.90D. DV3 is the smallest vortex that develops downstream of the front pile, with a vertical scale of 0.10D and a streamwise scale of 0.35D. The flow rapidly transitions to a smooth phase behind the vortex, indicating that the vortex structures on the downstream side of the front piles are well developed. The downstream side of the rear piles also exhibits the three vortices mentioned above, with their streamwise scale being about 50% of those on the front piles, indicating a similar vortex system around the front and rear piles but with the vortex scale at the rear piles being weaker than that at the front.

The vortex system around the jacket foundation on equilibrium scour bathymetry Is presented in Figure 9b. There are four vortices inside the scour hole on the upstream side of the front piles in the equilibrium state, namely HV1, HV2, HV3, and JV. The largest vortex, HV1, forms near the front pile with both horizontal and vertical scales of 1.30D. Its location is in the interface between positive and negative flow velocities. This vortex induces nearby water movement, forming a coherent junction vortex (JV) at the bottom of the front pile, which causes localized flattening at the bottom of the scour hole. In addition, two secondary vortexes, HV2 and HV3, form upstream of HV1, along the scour hole slope. These four vortices are interconnected, and their distribution is similar to the “horseshoe vortex” observed upstream of a monopile. Due to the influence of the upflow, the inflow intensity to the rear piles is reduced, resulting in less developed vortices inside the scour hole.

4.3. Average Kinetic Energy

Figure 10 shows the distribution of average kinetic energy (AKE) at the characteristic cross-section of the jacket foundation on flat-bed and equilibrium scour bathymetry. It can be seen that as the flow approaches the front pile, the AKE gradually decreases in both cases. Additionally, due to the contraction of the flow cross-section within the upper truss structure, the flow around the truss structure experiences a sharp increase in AKE in both the flat-bed and equilibrium states. However, Figure 10a shows that the low AKE in the flat-bed state occurs mainly downstream of the piles. In contrast, Figure 10b shows that the low AKE in the equilibrium state occurs at the interface between the truss structure and the piles, as well as on the upstream slope of the scour holes around both the front and rear piles. The different locations of low AKE in the flat-bed and equilibrium states may result from the existence of scour hole and the upflow from the scour hole, which increases the AKE downstream of the piles.

Table 2. AKE comparison in different piles.

	Pile No.	1#	2#	3#	4#
State
Flat bed		0.74	0.74	0.60	0.60
Equilibrium		0.75	0.75	0.54	0.54

compares the AKE of the flow at a position 1D above the bed and 1D upstream of the piles of the jacket foundation in both flat-bed and equilibrium scour bathymetry conditions. It shows that the AKE of the flow upstream of the rear piles is lower than the flow upstream of the front piles, with a reduction of 18.9% in the flat-bed state and 28.0% in the equilibrium state. The AKE of the flow upstream of the rear piles drops from 0.60 to 0.54 from the flat-bed state to the equilibrium state. This suggests that in the equilibrium state, more kinetic energy is consumed before the flow reaches the rear piles of the jacket foundation, resulting in less significant scour at the rear piles. Additionally, the drop in the AKE during the scour process indicates that the local scour at the rear piles becomes less significant from the beginning of the scour process to the final equilibrium state. Furthermore, the AKE of the flow upstream of the front piles remains nearly the same in both the flat-bed and equilibrium states, suggesting that the development of the scour hole around the foundation does not affect the AKE of the flow upstream of the front piles, compared to the rear piles.

4.4. Turbulent Flow Characteristics

Previous studies have indicated that the flow passing over the foundation increases turbulence intensity downstream [23]. Turbulence intensity is a primary parameter for quantifying turbulence, derived from the average kinetic energy of turbulent eddies [24]. The dimensionless formula for turbulence intensity is as follows:

$$TI=\sqrt{{\displaystyle \frac{{\left(u_{rms}^{\prime }\right)}^2+{\left(v_{rms}^{\prime }\right)}^2+{\left(w_{rms}^{\prime }\right)}^2}{3}}}/U_0$$

(4)

where $u_{rms}^{\prime }, v_{rms}^{\prime }, { \mathrm{a}\mathrm{n}\mathrm{d} w}_{rms}^{\prime }$ represent the root-mean-square of turbulent velocity fluctuations along the x, y, and z directions, respectively. Figure 11 illustrates the distribution of turbulence intensity at characteristic cross-sections of jacket foundations in the flat-bed and equilibrium states.

In the flat-bed state, minimal turbulence intensity is observed near the bed within 1D upstream of the front pile. Figure 11a shows that the turbulence intensity decreases from the front to the rear piles, but there is a slight increase near the bottom of the rear pile, which is consistent with the experimental findings of Behzad et al. [8]. In addition, downstream of the front pile, the distribution of maximum turbulence intensity exhibits a distinct ‘C’ shape. The development of turbulence and recirculation zones shows a significant correlation, initiating scour behind the pile. The maximum turbulence intensity for the front pile occurs in the range of −2 < x/D < 0, and for the rear pile, it occurs in the range of 4.8 < x/D < 6.6, with a vertical range in the z direction of 0 < z/D < 1.5.

As shown in Figure 11b, the distribution range of maximum turbulence intensity within the scour hole behind the front pile on equilibrium scour bathymetry is broad, indicating intense turbulence where the upflow from the scour hole meets the flow passing through the truss structure. Scour and turbulence diffusion are mutually influential physical processes. At the equilibrium stage, the scour hole reaches its maximum size, providing sufficient space for turbulent vortices to diffuse and for kinetic energy to dissipate, preventing further scour of the bed by the flow. Furthermore, the turbulence intensity and the area of high turbulence intensity between the front and rear piles of the jacket foundation in the equilibrium state are significantly greater than those in the flat-bed state. In contrast, the turbulence intensity downstream of the rear piles in the equilibrium state is significantly smaller than that in the initial flat-bed stage. This suggests that intense turbulence and energy dissipation occur between the front and rear piles, resulting in flow with reduced kinetic energy passing through the rear piles. In particular, in the equilibrium stage, the turbulence intensity reaches a high level, preventing scour at the rear piles of the jacket foundation.

4.5. Instantaneous Flow Field Characteristics around Jacket Foundations

In this section, the turbulent coherent structures in the instantaneous flow field around the Jacket foundations are visualized using the Q-criterion and compared with three-dimensional transient streamlines. The expression for Q is as follows:

$$Q={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial u_i}{\partial u_j}}\times {\displaystyle \frac{\partial u_j}{\partial x_i}}\right)$$

(5)

where u represents the velocity vector in m/s, and x represents the direction. Figure 12 shows the instantaneous vortex structures and three-dimensional streamlines around the jacket foundations at t = 30 s in the flat-bed and equilibrium states.

Figure 12a shows that in the flat-bed state, a horseshoe vortex with lower horizontal velocity forms upstream of each pile, extending to both sides like a necklace. This is the primary cause of the upstream recirculation around the pile in the initial stage. The convergence of streamlines results in higher velocities on both sides of the pile. Vortex shedding occurs behind each pile, leading to increased turbulence and reduced horizontal velocity. Figure 12c shows that the velocity on the vortex structure surface significantly decreases, indicating that the scour has reached an equilibrium state. At this stage, multiple necklace vortices can be observed within the scour hole. The shed vortex tubes downstream of the piles move upward along the slope of the scour hole.

Figure 12b shows that in the flat-bed state, the streamlines on the upstream side of the piles descend before reaching the piles and then evolve into two flow patterns. One portion of the upper streamlines closely follows the piles to the downstream side, generating recirculation. The other portion of the streamlines near the bed is driven by the horseshoe vortex, moving downstream from both sides of the piles. This explains the initial scour observed on both sides of the piles during the early stages of scour around jacket foundations.

As shown in Figure 12d, the main vortex in front of the piles in the equilibrium state increases steadily along with the development of the scour hole. Compared to the flat-bed conditions, the streamlines diffuse after passing the front piles when the scour hole develops, indicating decreased flow intensity at the rear piles. Consequently, the scour process at the rear piles becomes insignificant.

4.6. Distribution of Bed Shear Stress around Jacket Foundations

The distribution of bed shear stress is closely related to the vortex characteristics around the foundation. Figure 13 illustrates the average x-direction bed shear stress distribution around the jacket foundations in the flat-bed and equilibrium states, with a critical bed shear stress ${\tau }_c$ of 0.18 Pa [25,26]. As shown in Figure 13a, in the flat-bed conditions, a fan-shaped distribution of negative shear stress appears on the upstream side of each pile, with a maximum value of up to 1.2 ${\tau }_c$. As the flow passes the piles, it is compressed on both sides, generating higher positive shear stress at 60° positions on either side of the piles, where local scour first occurs. Comparing the maximum positive shear stress on either side of each pile reveals that the stress on the side closer to the central axis is slightly higher than that on the outer side. This Is due to the combined effects of accelerated flow between the jacket foundation piles and disturbances caused by the truss. Negative shear stress also appears behind each pile due to recirculation. The rear piles experience a shielding effect from the front piles, resulting in smaller ranges of both positive and negative shear stress compared to the front piles, which significantly reduces the initial scour depth around the rear piles.

As shown in Figure 13b, the positive shear stress on both sides of each pile in equilibrium conditions is close to zero. A crescent-shaped negative shear stress caused by the horseshoe vortex system appears on the upstream side at the bottom of the scour hole around the piles, with magnitudes less than ${\tau }_c$. At this stage, the local scour around the foundation reaches an equilibrium state. Sand dunes are present behind the piles and downstream of the jacket foundation during the equilibrium state. Positive shear stress may occur in the top area of the scour hole slope downstream of the piles due to water flow turbulence, potentially causing some sediment transportation. However, this has an insignificant impact on the equilibrium scour depth around the jacket foundations.

5. Conclusions

This study investigated the numerical simulation results of the flow field around jacket foundations under steady flow conditions on flat-bed and equilibrium scour bathymetry during the scour process. The vortex system, AKE, the time-averaged flow field, turbulence characteristics, instantaneous flow field, and bed shear stress in flat-bed and equilibrium scour conditions were analyzed. The main conclusions are as follows:

1. Numerical modeling shows that the flow velocity and the extent of downflow on the upstream side of the rear piles are significantly reduced due to the shielding effect of the front row piles and the obstruction and energy dissipation effects of the upper truss structure.

2. The vortex system structure at the rear piles of the jacket foundation is similar to that of the front piles, but the scale of the vortices at the rear piles is approximately half of those at the front pile.

3. The AKE of the flow upstream of the rear piles decreases significantly from 0.60 to 0.54, with reductions of 18.9% in the flat-bed state and 28.0% in the equilibrium state. This indicates more energy dissipation in the equilibrium state and results in less significant scour at the rear piles over time. In contrast, the AKE of the flow upstream of the front piles remains nearly constant, showing that the front piles are less affected by the scour process.

4. The turbulence intensity and the area of high turbulence intensity between the front and rear piles are significantly greater under equilibrium conditions than those under flat-bed conditions. This indicates that intense turbulence and energy dissipation occur between the piles, reducing the kinetic energy of the flow passing through the rear piles.

5. The maximum bed shear stress around the jacket foundation occurs at the 60° position on both sides of the piles, where scour initiates at the foundation. The rear piles experience reduced shear stress magnitude and extent due to the shielding effect of the front piles.