Numerical Study of Hydrodynamics Characteristics of Cylinders with Intermittent Spanwise Arrangements

Abstract

Subsea pipelines with intermittent spanwise arrangements are commonly encountered in offshore engineering, yet their complex hydrodynamic interactions remain insufficiently understood. In this study, three-dimensional numerical simulations were conducted to investigate the hydrodynamics of intermittently spanning cylinders at a Reynolds number of 40,250. The hydrodynamic coefficients and flow fields of cylinders with different gap ratios e/D, total spanning ratios L/H, and individual spanning ratios l/D were investigated (where e is the gap height, D is the diameter of the cylinder, L is the total spanning length, H is the length of the cylinder, and l is the individual spanning length). Moreover, this work validates the applicability of existing hydrodynamic prediction formulas for spanning cylinders under complex spanning conditions, as proposed by previous researchers. Numerical results show that the existing formulas can accurately predict the drag coefficient ${\overline{C}}_D$ of spanning cylinders under different uniform l/D ratios, but it fails to provide reliable predictions for the lift coefficient ${\overline{C}}_L$. These findings provide critical insights for optimizing the design of subsea pipelines and marine structures with intermittent spanwise arrangements.

1. Introduction

Steady incoming flow past a smooth circular cylinder (referred to as flow past a circular hereafter) has been a classical problem in fluid mechanics for more than a century, given its fundamental significance and practical applications. Due to its complex hydrodynamic phenomena, such as flow separation, vortex shedding, and the evolution characteristics of the flow structure, a large number of experimental and numerical simulation studies have been conducted. And a wealth of data and results have recently been obtained [1,2]. It is well known that the benchmarking case is mainly controlled by the Reynolds number Re, defined as Re = UD/ν, where U is the characteristic velocity, D is the diameter of cylinder, and ν is the fluid kinematic viscosity. Despite its geometrically simple structure, many flow regimes are observed as increasing Re. For instance, two-dimensional vortex shedding begins to occur at Re ≈ 47, the wake transitions from two-dimensional to three-dimensional at Re ≈ 190, and the phenomenon of drag crisis caused by the transition of the boundary layer of the cylinder [1]. To date, there have been numerous investigations on the evolution characteristics of flow fields and the hydrodynamic characteristics of the fluid around a cylinder in an unbounded flow field at different Reynolds numbers [1,2,3,4].

Compared to the problem of flow past a circular cylinder in an unbounded flow field, the wake structure of a near-wall cylinder is more complex, and the interaction between the wall boundary layer and the wake behind a cylinder will profoundly alter the hydrodynamic characteristics of the cylinder. For more than half a century, extensive experiments and numerical simulations have been conducted on the forces and flow-field structure of the near-wall cylinder. Bearman and Zdravkovich [5] investigated the wake patterns of a cylinder placed above a flat plate at different heights, as well as the pressure distribution along the surface of the cylinder through wind tunnel experiments, with the inflow boundary layer thickness being 0.8D in the experiments. The results of the experiment indicated that in the case of a small gap ratio e/D (where e is the distance of the cylinder from the wall), the vortex shedding in the wake will be suppressed, while the presence of the wall significantly alters the symmetry of the pressure on the surface of the cylinder. Zdravkovich [6] measured drag force and lift force acting on a near-wall circular cylinder in subcritical flow. The experimental results show that when the cylinder is placed outside of the incoming boundary layer, the average drag coefficient ${\overline{C}}_D$ increases with the increase in the gap ratio e/D, and approaches ${\overline{C}}_D$ in an unbounded flow field when the gap ratio e/D is greater than 2. When the cylinder is placed within the boundary layer, the average drag coefficient ${\overline{C}}_D$ is mainly influenced by the thickness of the boundary layer, while the average lift coefficient ${\overline{C}}_L$ is primarily controlled by the gap ratio. Further investigations have found that the mean drag coefficient ${\overline{C}}_D$ and the mean lift coefficient ${\overline{C}}_L$ both strongly depend on the gap ratio e/D and are affected by the boundary layer thickness. Within the gap ratio range of e/D from 0.2 to 0.3, the shedding of the cylinder’s trailing vortex is suppressed; when the gap ratio of e/D is in the range of 0.5 to 0.6, the suppression of the vortices near the wall weakens, and the mean drag coefficient ${\overline{C}}_D$ reaches its maximum value, which is approximately 1.2 times of the mean drag coefficient of the cylinder in an unbounded flow field under the same conditions [6,7,8]. In addition, the wake structure behind the near-wall cylinder is asymmetric. Two peaks are observed in the variation in mean lift coefficient ${\overline{C}}_L$ as the gap ratio e/D changes. One peak occurs at e/D = 0, where the suction on the surface of the cylinder is dominant, and the other peak occurs at approximately e/D = 0.1, where the stagnation pressure is dominant [9].

In actual marine engineering, local scour often occurs at multiple positions along the pipelines and cables. Furthermore, the actual seabed can be uneven, causing the submarine cables to form multiple intermittent spanning sections, with adjacent spanning being supported by the seabed. The spanning distribution of the cylinder will dramatically alter the hydrodynamic forces acting on them, and its force characteristics involve obvious three-dimensional features, which significantly affect the assessment of the on-bottom stability of the cables [10,11]. Griffiths et al. [12] conducted a parametric study through two-dimensional and three-dimensional numerical simulations to explore the hydrodynamic characteristics of non-uniformly spanning cylindrical structures under combined wave and current conditions, quantifying the variation in hydrodynamic loads along the axial direction of the cylinder. Sollund et al. [13] investigated the dynamic response of offshore pipelines in free spans, and derived a novel semi-analytical method for calculating the dynamic response of multi-span pipelines. In addition, Teng et al. [14] studied the effects of the single uniform free spans, multiple regularly spaced free spans, and seabed roughness elements by conducting a large number of experiments under waves. The main purpose aims to quantify the effects of gap height to pipeline diameter ratio and the span length to pipeline diameter ratio on the peak force coefficients. Tong et al. [15,16] used large eddy simulation to study the hydrodynamics of multiple spanning sections near the wall under unidirectional flow. The results show that even a small gap beneath the cylinder can significantly alter the fluid forces acting on the cylinder. Additionally, empirical prediction formulas for the hydrodynamics of spanning cylinders are proposed, as shown in Equations (1) and (2):

$${\overline{C}}_{D,p}={\displaystyle \frac{L}{H}}{\overline{C}}_D^S+{\displaystyle \frac{B}{H}}{\overline{C}}_D^{NS}$$

(1)

$${\overline{C}}_{L,p}={\displaystyle \frac{L}{H}}{\overline{C}}_L^S+{\displaystyle \frac{B}{H}}{\overline{C}}_L^{NS}$$

(2)

Among them, ${\overline{C}}_{D,p}$ and ${\overline{C}}_{L,p}$ are the drag coefficient and lift coefficient predicted by empirical formulas, respectively. The superscript S represents the hydrodynamic coefficient of the spanning cylinder, and the superscript NS represents the hydrodynamic coefficient of the non-spanning cylinder. L is the total length of the spanning cylinder, B is the total length of the non-spanning cylinder, and H = L + B is the total length of the cylinder. It is noteworthy that the variables ${\overline{C}}_D^S$ and ${\overline{C}}_L^S$ denote the time-averaged drag and lift coefficients for fully free-spanning cylinder, respectively, while ${\overline{C}}_D^{NS}$ and ${\overline{C}}_L^{NS}$ represent the time-averaged drag and lift coefficients for the fully blocked cylinder.

Although significant achievements have been made in the study of cylindrical flow around spanning structures considering the effects of spanning sections, it is undeniable that there are still certain limitations. Although the above study considered the effects of the total spanning ratio L/H and the gap ratio e/D on the hydrodynamic characteristics of the cylinder, it did not take into account the effect of the individual spanning length l/D on the hydrodynamic characteristics of the cylinder. In addition, the applicability of the existing formula under complex spanning conditions in a cylindrical structure also needs to be examined. Therefore, this work mainly focused on the effect of the single spanning length l/D on the hydrodynamic characteristics of cylinders, while also exploring the applicability of the empirical prediction formula proposed by Tong et al. [16] with different spanning lengths l/D.

With this intention, we have structured the current paper as follows: a short overview and details of the chosen numerical method are presented in Section 2. The main results and discussion are given in Section 3, which includes the hydrodynamic coefficients and flow structure. Finally, Section 4 gives the conclusion of the paper.

2. Numerical Model

Previous similar studies have shown that the large eddy simulation method (LES) has the capability to model high-Reynolds-number flow and resolve the 3D bluff body flow at a relatively lower computational cost Therefore, numerical simulations are performed using the LES model provided by STAR-CCM+ (17.04.008) for investigating the forces acting on the circular cylinder with intermittent spanwise arrangements. It uses a SIMPLEC algorithm for pressure–velocity coupling, which requires Rhie and Chow interpolation to avoid odd–even oscillations [17]. Moreover, the dynamic Smagorinsky model is employed in this research, which automatically updates the Smagorinsky constant over both time and the computational domain and outperforms the conventional Smagorinsky model [18,19].

2.1. Governing Equations

Within the framework of large eddy simulation, the continuity equation and momentum equation for three-dimensional incompressible flow are given as follows:

$${\displaystyle \frac{\partial {\overline{u}}_i}{\partial x_i}}=0$$

(3)

$${\displaystyle \frac{\partial {\overline{u}}_i}{\partial t}}+{\overline{u}}_j{\displaystyle \frac{\partial {\overline{u}}_i}{\partial x_j}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \overline{p}}{\partial x_i}}+{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial }{\partial x_j}}\left(\mu {\displaystyle \frac{\partial {\overline{u}}_i}{\partial x_j}}-{\tau }_{ij}\right)$$

(4)

In the equation, “-” indicates the variable after filtering by a grid filter with a filtering width of Δ, where Δ = V^1/3, and V represents the volume of the grid cell. (x₁, x₂, x₃) = (x, y, z) are the Cartesian coordinate components, u_i is the velocity component corresponding to the x_i direction, t is time, ρ is the fluid density, p is the pressure, and μ is the dynamic viscosity coefficient. τ_ij is the subgrid stress, which characterizes the effect of unresolved small-scale turbulent motions on large-scale flow during the grid filtering process, and can be defined as

$${\tau }_{ij}={\overline{u}}_i{\overline{u}}_j-\overline{u_i{u}_j}$$

(5)

Due to the unknown value of subgrid stress term τ_ij, the group of equations is not closed. To close the group of equations, Smagorinsky introduced the Boussinesq hypothesis into large eddy simulation and proposed the relationship between subgrid stress and subgrid eddy viscosity as follows:

$${\tau }_{ij}-{\displaystyle \frac{1}{3}}{\tau }_{kk}{\delta }_{ij}=-2{\mu }_t{\overline{S}}_{ij}$$

(6)

where δ_ij is the Kronecker delta, μ_t is the subgrid-scale turbulent viscosity coefficient, and S_ij is the strain rate tensor, which is defined as

$$S_{ij}={\displaystyle \frac{(\partial u_i/\partial x_j+\partial u_j/\partial x_i)}{2}}$$

(7)

To solve the subgrid vortex viscosity coefficient μ_t, it is necessary to establish a turbulence model. The dynamic Smagorinsky model is employed, and the equation of the dynamic Smagorinsky turbulence model is

$${\mu }_t=\rho C_S^2{V}^{2/3}\left|\overline{S}\right|$$

(8)

where $\left|\overline{S}\right|=\sqrt{2{\overline{S}}_{ij}{\overline{S}}_{ij}}$, C_S is the coefficient of the Smagorinsky model. In the dynamic Smagorinsky turbulence model, C_S is required to be calculated through quadratic filtering, as follows:

$$C_S^2={\displaystyle \frac{〈L_{ij}{M}_{ij}〉}{〈M_{ij}{M}_{ij}〉}}$$

(9)

where < > indicates spatial averaging of physical quantities, and the calculation formulas of L_ij and M_ij are as follows:

$$L_{ij}=\widetilde{\overline{u_i}}\widetilde{\overline{u_j}}-\widetilde{\overline{u_i}\overline{u_j}}$$

(10)

$$M_{ij}=2\left({\tilde{\Delta }}^2\left|\tilde{\overline{S}}\right|\widetilde{\overline{S_{ij}}}-{\Delta }^2\widetilde{\left|\overline{S}\right|\overline{S_{ij}}}\right)$$

(11)

In the above formula, “~” represents the variable of the secondary filtering with a test filter of width 2Δ. For a detailed introduction of large eddy simulations, please refer to references [20,21].

2.2. Definition of Characteristic Coefficients

For convenience of subsequent descriptions, the characteristic parameters involved in this work are listed here. The drag coefficient C_D and lift coefficient C_L of the cylinder are defined as follows:

$$C_D={\displaystyle \frac{F_D}{0.5\rho U_0^{2}DH}},C_L={\displaystyle \frac{F_L}{0.5\rho U_0^{2}DH}}$$

(12)

The mean drag coefficient ${\overline{C}}_D$ and mean lift coefficient ${\overline{C}}_L$ are calculated using the following equations:

$${\overline{C}}_D={\displaystyle \frac{{\displaystyle \sum _{i=1}^N{C}_D\left(i\right)}}{N}},{\overline{C}}_L={\displaystyle \frac{{\displaystyle \sum _{i=1}^N{C}_L\left(i\right)}}{N}}$$

(13)

In addition, the standard deviation of the drag and lift coefficients is defined as follows:

$${\overline{C}}_{D-rms}=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^N\left(C_D\left(i\right)\right.}-{\left.{\overline{C}}_D\right)}^2}{N}}},{\overline{C}}_{L-rms}=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^N\left(C_L\left(i\right)\right.}-{\left.{\overline{C}}_L\right)}^2}{N}}}$$

(14)

The pressure coefficient C_p are determined from the time-averaged flow field, where C_p is calculated as

$$C_P={\displaystyle \frac{p-p_{\infty }}{0.5(\rho U_0^2)}}$$

(15)

The dimensionless vortex shedding frequency, Strouhal number St, defined as

$$St={\displaystyle \frac{fD}{U_0}}$$

(16)

where F_D and F_L are the drag force and lift force acting on the unit-length cylinder in the streamwise and cross-flow directions, respectively; U₀ represents the free stream velocity outside the boundary layer; A is the cross-sectional area of the cylinder; D is the diameter of the cylinder; p is the pressure distribution on the surface of cylinder; $p_{\infty }$ is the reference pressure at the inlet boundary, and f is the vortex-shedding frequency of the cylinder, which is typically obtained by performing fast Fourier transform (FFT) on the time history of the lift coefficient. It is note worthing that similar studies have indicated that the number of the statistical sample N has a critical impact on the accuracy and reliability of the statistical measures of the drag and lift coefficients. Therefore, the statistical period of more than 200 nondimensional time units (defined as $t^{*}=t{U}_0/D$) is adopted in this study (the section enclosed by the green dashed box in Figure 4).

2.3. Geometric Parameters

The hydrodynamic characteristics of cylinder with intermittent spanwise arrangements are mainly influenced by their substructure, which divides the cylinder into spanning and non-spanning sections through multiple blockage blocks, as shown in Figure 1. This work refers to the physical experiment conducted by Teng et al. [22] where the diameter of the cylinder is D, the total length is H, and the total length of the spanning section is L. The distance from the bottom boundary of calculation domain to the cylinder is e. The blocking structure at the lower part of the cylinder is simplified to a rectangular structure, with the blocking block parallel to the length of the cylinder being b and the width being 0.91D. In order to completely block the non-spanning section of the cylinder while accurately capturing the flow structure around the cylinder, a zero-thickness plate is set between the surface of the blocking rectangular structure and the bottom of the cylinder, with a height of e₀, where e₀ is taken as 0.01D. Therefore, the structural parameters that mainly affect the hydrodynamic coefficients of the cylinder with intermittent spanwise arrangements are the height of the spanning cylinder e/D, the total spanning length L/H, and the individual spanning length of a single spanning section l/D.

2.4. Calculation Domain and Boundary Conditions

As is well known, for flow past circular flow problems with Reynolds numbers less than 190, two-dimensional numerical models can provide accurate simulations. However, when the Reynolds number exceeds 190, significant discrepancies arise between the results obtained from two-dimensional and three-dimensional numerical models. Therefore, based on the relevant studies by previous scholars and the research objectives of this paper, the computational domain of this study was established [23,24]. The computational domain, with dimensions of 30D × 10D × 8.7D, is schematically represented in Figure 2. The origin of the Cartesian coordinates is positioned at the center of the circular cross-section of the cylinder, located at the z = 0 plane. The inlet boundary is positioned 10D upstream from the cylinder’s central axis, while the outlet boundary is located 20D downstream from the cylinder’s center. the lower boundary is e away from the bottom of the cylinder, and the upper boundary is 10D away from the lower boundary. The length of the cylinder is 8.7D.

The solution conditions are as follows. Boundary conditions at the entrance:

$$u=u(y), v=0, w=0, \partial p/\partial n=0$$

(17)

Boundary conditions for export:

$$\partial u_i/\partial \mathbf{n}=0, p=0$$

(18)

Upper boundary condition:

$$v=0, \partial u_i/\partial \mathbf{n}=0, \partial p/\partial \mathbf{n}=0$$

(19)

Lower boundary condition:

$$u=v=w=0, \partial p/\partial \mathbf{n}=0$$

(20)

Surface conditions of cylindrical and blocking structures:

$$u=v=w=0, \partial p/\partial \mathbf{n}=0$$

(21)

It is worth noting that to accurately model the boundary layer thickness at the inlet, the velocity profile is defined as follows:

$$u(y)=\min \left\{{\displaystyle \frac{u_{*}}{\kappa }}\ln ({\displaystyle \frac{y}{z_w}}), U_0\right\}$$

(22)

where u_* is the friction velocity, defined as u_* = κU₀/ln(δ/z_w), κ is the Karman constant, taken to be 0.41, U₀ is the flow velocity outside the boundary layer, δ is the thickness of the boundary layer, and the seabed roughness length z_w = d₅₀/12, where d₅₀ represents the median particle size of the seabed sediment, taken as z_w = 10⁻⁶ m in this study.

2.5. Convergence and Accuracy of the Numerical Model

Firstly, the grid convergence check of the numerical model is given and the grid division schematic is shown in Figure 3. Due to the complex fluid flow near the lower wall of the computational domain, the cylinder, and the blocking structure, local refinement of the grid is carried out in these locations. The expansion rate from the first layer grid thickness near the cylinder surface to the surrounding grid transition does not exceed 1.1.

The main parameters for grid refinement are as follows: (1) the number of grid nodes around the cylinder N_C, which is the number of grid nodes around the cylinder; (2) the thickness of the first layer of grid on the cylinder surface Δ₁; (3) the number of grid nodes along the length of the cylinder N_Z, which is the number of nodes along the longitudinal direction of the cylinder. The grid convergence check is conducted under the conditions of Reynolds number Re = 40,250, boundary layer thickness δ/D = 3, gap height e/D = 0.27, the total spanning ratio L/H = 0.44, and single spanning length l/D = 0.77. The node parameters for the grid convergence check are shown in

Table 1. Convergence check of spanwise mesh.

NZ	NC	Δ1/D	Number of Mesh (Million)	Δt (s)	${\overline{\mathit{C}}}_{\mathit{D}}$	${\overline{\mathit{C}}}_{\mathit{L}}$	CL-rms
180	160	0.0004	4.29	0.002	0.845	0.261	0.019
270	160	0.0004	5.81	0.002	0.850	0.266	0.016
360	160	0.0004	7.52	0.002	0.851	0.267	0.017

,

Table 2. Convergence check of circumferential mesh.

NZ	NC	Δ1/D	Number of Mesh (Million)	Δt (s)	${\overline{\mathit{C}}}_{\mathit{D}}$	${\overline{\mathit{C}}}_{\mathit{L}}$	CL-rms
270	120	0.0004	4.38	0.002	0.846	0.259	0.016
270	160	0.0004	5.81	0.002	0.850	0.266	0.016
270	200	0.0004	7.18	0.002	0.850	0.268	0.016

and

Table 3. Convergence check of first-layer mesh.

NZ	NC	Δ1/D	Number of Mesh (Million)	Δt (s)	${\overline{\mathit{C}}}_{\mathit{D}}$	${\overline{\mathit{C}}}_{\mathit{L}}$	CL-rms
270	160	0.0002	6.04	0.002	0.851	0.266	0.016
270	160	0.0004	5.81	0.002	0.850	0.266	0.016

. The time history of the hydrodynamic coefficients C_D and C_L are shown in Figure 4, and the dimensionless sampling time of the hydrodynamic coefficients tU₀/D is between 100 and 330.

As shown in Table 1, the number of radial grid nodes N_Z was selected as 180, 270, and 360 for the check. By comparing the hydrodynamic coefficients ${\overline{C}}_D$, ${\overline{C}}_L$ and C_L-rms, it can be observed that the variation in hydrodynamic coefficients is relatively low when the radial grid nodes number N_Z is 270 and 360. Therefore, N_Z being 270 already meets the convergence requirements.

The circumferential grid parameters for the cylinder are shown in Table 2, with the number of circumferential grid nodes chosen as 120, 160, and 200. By comparing the hydrodynamic coefficients ${\overline{C}}_D$, ${\overline{C}}_L$ and C_L-rms calculated from the three kinds of grids, it can be observed that the variation in hydrodynamic coefficients is relatively low when the number of circumferential grid nodes changes from 160 to 200. Therefore, the number of circumferential grid nodes N_C is taken as 160.

Table 3 presents the convergence check of the first-layer mesh on the surface of cylinder. The dimensionless height Δ₁ of the first-layer mesh is set at two values, 0.0002D and 0.0004D. It can be observed that reducing Δ₁ from 0.0004D to 0.0002D has little effect on the numerical simulation results, which proves that Δ₁ of 0.0004D already meets the convergence requirements.

In order to ensure the accuracy of numerical simulations, in addition to grid convergence check, it is also necessary to conduct time independence check to eliminate the influence of the time step Δt on the numerical results. The results of the time independence check are shown in

Table 4. Convergence check of time step.

NZ	NC	Δ1/D	Number of Mesh (Million)	Δt (s)	${\overline{\mathit{C}}}_{\mathit{D}}$	${\overline{\mathit{C}}}_{\mathit{L}}$	CL-rms
270	160	0.0004	5.81	0.002	0.850	0.266	0.016
270	160	0.0004	5.81	0.003	0.849	0.268	0.016
270	160	0.0004	6.04	0.005	0.847	0.267	0.015

.

From Table 4, it is evident that when Δt decreased from 0.003 s to 0.002 s, it minimally affected the numerical results for the hydrodynamic coefficients ${\overline{C}}_D$, ${\overline{C}}_L$ and C_L-rms. Therefore, Δt = 0.003 s has already met the requirement for time independence.

According to the physical experiments on cylinder with intermittent spanwise arrangements conducted by Teng et al. [22] the accuracy of this numerical model has been validated. The calculated parameters are as follows: Reynolds number Re = 40,250, boundary layer thickness δ/D = 3, height ratio e/D = 0.27, length of a single blocking section b/D = 0.97, and spanning ratio L/H being 0, 0.30, 0.44, 0.72, and 1. The results of comparison are shown in Figure 5, and it can be observed that the numerical model’s results agree well with the experimental results from Teng et al. [22] illustrating the accuracy of the numerical model.

3. Results and Discussions

3.1. Computational Cases

As previously mentioned, under the given Reynolds number and boundary layer thickness, the primary parameters influencing the hydrodynamic coefficients of the cylinder with intermittent spanning arrangements are the gap ratio e/D, the total spanning ratio L/H, and the length of individual spanning l/D, where e/D takes values of 0.1, 0.2, and 0.3; the total spanning ratios take values of 0, 0.25, 0.50, 0.5, and 1; and the lengths of the single spanning l/D take values of 0, 8.7, and the intermediate value between the two values. Specific details can be seen in

Table 5. Computational cases.

e/D	L/H	l/D
0.1, 0.2, 0.3	0	0
	0.25	0.1, 0.5, 1, 1.5, 2, 2.5, 3, 4, 5
	0.5	0.1, 0.5, 1, 1.5, 2, 2.5, 3, 4, 5
	0.75	0.1, 0.5, 1, 1.5, 2, 2.5, 3, 4, 5
	1	8.7

. It should be noted that, due to the significant computational costs, additional cases were added or removed during the research process based on partial results obtained. These adjustments will be reflected in subsequent figures and analyses. For all computational cases in this study, the Reynolds number Re = 42,500, and the boundary layer thickness δ/D = 3.

3.2. Hydrodynamic Coefficients

For the cylinder with intermittent spanwise arrangements, Tong et al. [16] proposed the predictive formulas for hydrodynamic coefficients (1) and (2). To quantify the difference between the numerical results in this work and the predicted results established by Tong et al. [16] the parameters of relative errors η_D and η_L are used for representation, with the calculation formulas as shown in Equations (23) and (24):

$${\eta }_D={\displaystyle \frac{{\overline{C}}_D-{\overline{C}}_{D,p}}{{\overline{C}}_{D,p}}}$$

(23)

$${\eta }_L={\displaystyle \frac{{\overline{C}}_L-{\overline{C}}_{L,p}}{{\overline{C}}_{L,p}}}$$

(24)

where η_D and η_L are the parameters of relative errors between the mean drag coefficient ${\overline{C}}_D$ obtained from numerical simulations and the corresponding empirical formula predictions for the lift coefficient ${\overline{C}}_L$. ${\overline{C}}_{D,p}$ and ${\overline{C}}_{L,p}$ are the time-averaged drag coefficients and time-averaged lift coefficients predicted by Equations (1) and (2), respectively.

To quantify force on the spanning sections and non-spanning sections, the drag force coefficient and lift force coefficient of non-span sections and a single non-span sections in Formulas (25) and (26) are calculated as follows:

$$C_{D,S}={\displaystyle \frac{F_{D,S}}{0.5\rho U_0^{2}DL}},C_{L,S}={\displaystyle \frac{F_{L,S}}{0.5\rho U_0^{2}DL}}$$

(25)

$$C_{D,NS}={\displaystyle \frac{F_{D,NS}}{0.5\rho U_0^{2}DB}},C_{L,NS}={\displaystyle \frac{F_{L,NS}}{0.5\rho U_0^{2}DB}}$$

(26)

where C_D,S and C_L,S represent the drag coefficient and lift coefficient of the spanning section, respectively. C_D,NS and C_L,NS represent the drag coefficient and lift coefficient of the non-spanning section, respectively. Similarly to Equation (12), F_D,S and F_L,S are the drag force and lift force acting on spanning section of cylinder, while F_D,NS and F_L,NS are the drag force and lift force acting on non-spanning section of cylinder, respectively.

3.2.1. Mean Drag Coefficient and Mean Lift Coefficient

The parameter of relative error η_D of the drag coefficient for multiple spanning cylinders varies with l/D is shown in Figure 6. It can be observed that when l/D ≤ 1, the relative error of the drag coefficient fluctuates dramatically with l/D; when l/D > 1, as l/D increases, the overall trend of η_D decreases gradually. In addition, for all cases in this work, it can be observed that the value of η_D is less than 5%, and under higher l/D, η_D even remains below 3%. It indicates that the empirical force prediction formula proposed by Tong et al. [16] can effectively predict the average drag coefficient ${\overline{C}}_D$ for multiple spanning cylinders.

Figure 7 shows the variation in the relative error parameter of the mean lift coefficient η_L with l/D under different L/H conditions. It can be seen that when L/H = 0.25, η_L decreases gradually, then it increases at L/H = 0.5, and tends to stabilize. In most l/D cases, η_L is less than 5%. When L/H = 0.5 and 0.75, the results indicate that there is a significant deviation between the mean lift coefficients obtained through numerical simulations and the predicted results from the empirical formula proposed by Tong et al. [16]. Specifically, as l/D increases, the parameter of relative error η_L shows a decreasing trend. Furthermore, the results in Figure 7 demonstrate that the empirical prediction formula established by Tong et al. [16] tends to overestimate the mean lift coefficient, especially for the cases with large L/H. When L/H = 0.5, the values of η_L are approximately confined within the range of −25% to 0%. However, when L/H = 0.75, the values of η_L exhibit significant fluctuations within the range of −60% to 0%. This suggests that under larger L/D conditions, the overestimation of the existing formula becomes more marked, leading to greater deviation from the actual results. The above results indicate that the empirical prediction formula from Tong et al. [16] is effective in predicting the lift coefficient for small L/H ratio cylinders, but it cannot accurately predict the forces on spanning cylinders with L/H ≥ 0.5; therefore, further study on the lift coefficients of multiple spanning cylinders with L/H ≥ 0.5 is necessary.

The lift coefficients of cylinders with a spanning ratio L/H = 0.5 is shown in Figure 8, where e/D ranges from 0.1 to 0.3. It is noted that ${\overline{C}}_{L,S},$ ${\overline{C}}_{L,NS}$ and ${\overline{C}}_L$ represent the time-averaged lift coefficients of the spanning cylindrical section, the non-spanning cylindrical section, and the entire cylinder, respectively. It is not difficult to observe that, under different e/D and l/D conditions, the time-averaged lift coefficient ${\overline{C}}_{L,NS}$ of the non-spanning section of cylinder is always greater than that of the spanning section. Meanwhile, the lift coefficient ${\overline{C}}_L$ of the entire cylinder, serving as the average of the two parts, lies between the lift coefficient curves of the spanning and non-spanning section. This is primarily attributed to the fact that, beneath the non-spanning section of the cylinder, the flow is obstructed, resulting in higher pressure values. In contrast, beneath the spanning section of the cylinder, the flow is not impeded, leading to negative pressure values. In addition, it can be seen that when L/H = 0.5, the mean lift coefficient ${\overline{C}}_{L,S}$ of all cylinders in the spanning section for e/D values of 0.1, 0.2, and 0.3 gradually decreases with the increase in l/D, stabilizing at approximately l/D > 2. This indicates that for l/D < 2, there is a significant interaction between the fluid in the spanning section and the non-spanning section. However, for l/D > 2, the influence of the spanning section on the non-spanning section weakens, and ultimately, the spanning section approaches a fully spanning condition. Additionally, it can be observed that the mean lift coefficient ${\overline{C}}_{L,NS}$ of the non-spanning section also gradually increases with the increase in l/D; similarly, the value tends to stabilize at l/D > 2.

Similarly, Figure 9 shows the dependence of the mean lift coefficients ${\overline{C}}_L$, ${\overline{C}}_{L,S}$ and ${\overline{C}}_{L,NS}$ of the cylinder on l/D when L/H = 0.75, and e/D varies from 0.1 to 0.3. It can be observed that the variation in the mean lift coefficient ${\overline{C}}_{L,NS}$ for the non-spanning section follows the same trend as that when L/H = 0.5, gradually increasing with the increase in l/D and eventually stabilizing. For the spanning section of cylinder, when e/D is 0.1, the variation of ${\overline{C}}_{L,S}$ aligns with the trend observed at L/H = 0.5, where it first decreases gradually, then increases slightly when l/D > 2.5, and finally stabilizes. When e/D is 0.2 and 0.3, the variation of ${\overline{C}}_{L,S}$ is not significant when l/D < 1.5, and for l/D > 1.5, the trend is consistent with that at L/H = 0.5, which also gradually decreases and then rebounds after reaching a certain value of l/D, ultimately stabilizing. Compared to the cylinder with L/H = 0.50, the cylinder with L/H = 0.75, which has an increased spanning length, exhibits a lift coefficient of the entire cylinder and the spanning section that become closer in value, with a difference of approximately 0.1. This also implies that the blockage effect relatively weakens, particularly under larger e/D conditions, where the suppression effect near the wall diminishes, making this trend more marked.

To study the reasons for the variation of ${\overline{C}}_{L,NS}$ with l/D, the analysis of ${\overline{C}}_{L,NS}$ under all working conditions was conducted and shown in Figure 10. It can be seen that when the gap ratio e/D is fixed, the variation curves of ${\overline{C}}_{L,NS}$ with b/D for different L/H values are basically overlapping. This indicates that the spanning ratio L/H has little influence on the ${\overline{C}}_{L,NS}$ of the non-spanning section of the cylinder, and ${\overline{C}}_{L,NS}$ is mainly affected by the non-spanning section length b/D and the gap ratio e/D.

3.2.2. Root Mean Square of Lift Coefficient and Root Mean Square of Drag Coefficient

This part focuses on the root mean square of lift coefficient C_L-rms and drag coefficient C_D-rms for multiple sections of cylinders. The analysis is conducted for the conditions of L/H = 0.5, e/D = 0.3, and l/D = 1. The results of C_L-rms and C_D-rms for each spanning section and non-spanning section are shown in Figure 11 and Figure 12. In the figures, S1–S5 represent the different spanning sections of the cylinder, while NS1-NS4 represent the different non-spanning sections; A represents the entire cylinder. It can be observed that the root mean square lift coefficient C_L-rms of the spanning sections of cylinder is significantly greater than that of the non-spanning sections, while the root mean square drag coefficient C_D-rms for both spanning and non-spanning sections is basically the same, indicating that the impact of the fluid in the gap on the drag coefficient is minimal. Additionally, it can be seen that the root mean square lift coefficient C_L-rms and the drag coefficient C_D-rms for each section of the cylinder, both spanning and non-spanning sections, are larger than the overall values of the cylinder. This is primarily due to the significant phase difference in vortex shedding across different cross-sections of the cylinder, which leads to mutual cancelation effects in the hydrodynamic coefficients during the integration process over the entire cylinder.

3.2.3. Pressure Coefficient

It is well known that the forces acting on a cylinder are primarily influenced by the pressure distribution on its surface. The pressure coefficient C_p of a cylinder is discussed here. Figure 13 shows the distribution of the pressure coefficient C_p at the bottom of the cylinder when L/H = 0.5, e/D = 0.2, and l/D = 2. It can be observed that the middle part of the flow-approaching side of the non-spanning section of the cylinder exhibits higher pressure, with a region near the spanning section where the C_p gradually decreases toward the spanning section. In this study, the region where the pressure coefficient gradually decreases in the non-spanning span section is referred to as the transition section. In the spanning section, small negative pressure zones exist near both ends adjacent to the non-spanning sections, while a larger negative pressure zone is present closer to the mid-span region. This paper refers to these two negative pressure zones as low-pressure section A and low-pressure section B, respectively. Specifically, the low-pressure section A is primarily attributed to the deflection of fluid and vortices toward the non-spanning sections as flow passes the spanning section of the cylinder, whereas the low-pressure section B arises purely from the low-pressure region formed by vortices shed behind the spanning cylinder, analogous to the pressure distribution observed in an isolated cylinder.

In order to investigate the variation in the lift coefficient ${\overline{C}}_{L,NS}$ of the non-spanning cylindrical sections with l/D, the pressure coefficient C_p of non-spanning cylindrical sections was analyzed under the conditions of e/D = 0.3, L/H = 0.5, and l/D = 0.5, 2, and 3. As shown in Figure 14, it can be observed that when e/D = 0.5, the high-pressure section in the middle of the non-spanning cylindrical sections is relatively low, while the transitional sections occupies a larger proportion. As l/D increases, although the range of the transitional sections does not show significant changes, its relative proportion gradually decreases, which leads to the gradual increase in the ${\overline{C}}_{L,NS}$ of the non-spanning cylindrical sections with the increase in l/D.

In addition, in order to study the variation in the lift coefficient ${\overline{C}}_{L,S}$ of the cylindrical spanning section with l/D, for the cases of e/D = 0.2, L/H = 0.5, and l/D = 0.5, 2, 3, as well as e/D = 0.3, L/H = 0.5, and l/D = 0.5, 1.5, 3, the distribution of the bottom pressure coefficient C_p of the cylindrical spanning section was shown in Figure 15 and Figure 16. It can be seen that when L/H = 0.5 and e/D = 0.2, as l/D increases from 0.5 to 2, the low-pressure section of the spanning cylinder gradually expands, resulting in a decrease in ${\overline{C}}_{L,S}$; when l/D increases from 2 to 3, the range of low-pressure section B no longer expands, which leads to a decrease in the proportion of low-pressure section B, thus ${\overline{C}}_{L,S}$ increases; when L/H = 0.75 and e/D = 0.3, as l/D increases from 0.5 to 1.5, the C_p of low-pressure section A decreases and the range of low-pressure section B is relatively small, resulting in little influence on ${\overline{C}}_{L,S}$ for the cylindrical spanning sections; as l/D increases from 1.5 to 3, the range of low-pressure section B expands, ultimately leading to a decrease in ${\overline{C}}_{L,S}$.

3.3. Characteristics of Flow Field

Figure 17 shows the time-averaged streamline cross-section when the gap ratio e/D = 0.3 for a completely spanning cylinder without blockage. It can be found that when the fluid flows past the cylinder, due to the presence of viscosity and reverse pressure gradient, the boundary layer separates from the surface of the cylinder, forming two vortices behind the cylinder. To facilitate a clear and intuitive comparison of the blockage effect on the flow field, Figure 18 depicts the time-averaged streamline results for a completely blocked cylinder at e/D = 0.3. It can be observed that the distribution of time-averaged streamlines at this time is significantly different from that of a completely spanning cylinder, with a larger-scale vortex forming behind the cylinder, while two smaller-scale vortices form upstream and downstream of the cylinder blockage.

Figure 19 shows the time-averaged streamlines of a cylinder with l/D = 1, e/D = 0.3, and L/H = 0.5. It can be seen that when the fluid flows past the non-spanning section, due to the obstruction of the blockage at the lower part of the cylinder, the fluid is deflected towards the two sides of the spanning section. This results in a transitional area at the bottom of the non-spanning section where the pressure gradually decreases on both sides, and a low-pressure section A on both sides of the spanning section. When the fluid exits the gap, it is deflected towards the upper side and the blockage section of the cylinder, forming a three-dimensional vortex. The time-averaged streamline diagram for a cylinder with l/D = 1, e/D = 0.3, and L/H = 0.75 is shown in Figure 20. It can be found that the vortex is concentrated behind the cylinder, extending less towards the non-spanning section, and the rear vortex is more similar to a two-dimensional vortex. Figure 21 shows the time-averaged streamlines for l/D = 0.1, e/D = 0.3, and L/H = 0.75. It can be observed that after the fluid exits the gap, it flows perpendicular to the direction of the cylinder, forming two vortices behind, with larger scales in the downstream direction of the vortices.

In addition, the results presented by the streamline in Figure 19 are consistent with the results of the pressure distribution described above. For a fixed e/D and L/H, as b/D increases, the proportion of flow deflected from the spanning section to the non-spanning section decreases. This consequently diminishes the relative proportion of the transitional section, ultimately leading to a decrease in ${\overline{C}}_{L,NS}$. When L/H = 0.5 and e/D = 0.2, as l/D increases from 0.5 to 2, the gap beneath the cylinder expands. This allows more complete development of vortices behind the cylinder, leading to an expansion of the low-pressure section B and consequently a decrease in ${\overline{C}}_{L,S}$. As l/D further increases from 2 to 3, the extent of low-pressure section B ceases to expand. This reduction in the relative proportion of low-pressure section B ultimately results in an increase in ${\overline{C}}_{L,S}$.

4. Conclusions

The large eddy simulation method is used in this work to study the force characteristics and flow field distribution of circular cylinder with intermittent spanwise arrangements. The applicability of the hydrodynamic coefficient prediction formula for spanning cylinders proposed by Tong et al. [16] under the condition of a continuously varying spanning length l/D is explored. The main conclusions of this work are as follows:

The effect of the individual spanning length l/D on the time-averaged drag coefficient ${\overline{C}}_D$ of uniformly spanning cylinders is minimal. The empirical formula proposed by Tong et al. [16] can accurately predict the drag coefficient ${\overline{C}}_D$ of spanning cylinders under different uniform l/D ratios. For the lift coefficient, when the spanning ratio is small, such as L/H = 0.25, the effect of the single spanning length l/D on the lift coefficient ${\overline{C}}_L$ of a uniform spanning cylinder is negligible; the empirical formula by Tong et al. [15,16] can also predict the hydrodynamic coefficients for cylinders well. However, when L/H ≥ 0.5, the effect of the single spanning length l/D on the lift coefficient ${\overline{C}}_L$ of a cylinder is significant, and the previous empirical formulas cannot provide accurate predictions.

In addition, further research demonstrates that the total spanning ratio L/H has little influence on the ${\overline{C}}_{L,NS}$ of the non-spanning section of cylinder, and ${\overline{C}}_{L,NS}$ is mainly affected by the non-spanning section length b/D and the gap ratio e/D. Specifically, the lift coefficient ${\overline{C}}_{L,NS}$ of the non-spanning sections of the cylinder gradually increases with the increase in the single non-spanning length b/D. Flow visualization results show that it is mainly due to the fluid being deflected to the sides of the non-spanning sections after passing through the gap of the spanning section, creating a transitional region on both sides of the non-spanning section where the pressure gradually decreases. As b/D increases, the proportion of the transitional region decreases, resulting in an increase in ${\overline{C}}_{L,NS}$ for the non-spanning sections of the cylinder.