Wake Effects on Wave-Induced Loads Acting on Cylinders in a Tripod Configuration

Abstract

Comparative hydrodynamic loads caused by a focused wave acting on differently sized slender vertical cylinders placed in a wave canal were predicted at model scale using an unsteady Reynolds-averaged Navier–Stokes (URANS) solver and the Morison equation. This paper focused, first, on wake effects that need to be considered for the structural design of tripod configurations and, second, on the influence of these wake effects on wave-induced loads. For the Morison approach, flow velocities were gathered at incremental representative locations along the length of the cylinders, whereby the use of a central differencing approach obtained the associated flow accelerations. A first scenario dealt with a small- and a large-diameter cylinder, implemented individually at two immersion depths. A second scenario considered medium- and large-diameter cylinders arranged in a tripod configuration, where the upstream and downstream cylinders had the same diameter. For the single cylinder, impact loads obtained by both methods compared favorably to experimental measurements. For the tripod configuration, however, impact loads obtained by both methods differed due to the influence of wake effects. Thus, it was necessary to determine the influence of wake effects to assess the structural integrity of tripod configurations.

1. Introduction

Slender cylindrical structures are widely used not only for offshore structures and wind tower foundations, but also for so-called pump towers in liquefied natural gas (LNG) tanks. Such towers are subject to sloshing-induced hydrodynamic impact loads. To reliably assess these loads, computational fluid dynamics (CFD) simulations or experimental investigations are generally performed. Besides these relatively elaborate computations, using the well-known Morison equation [1] offers an efficient method to approximate hydrodynamic loads acting on cylindrical structures:

$$dF=\rho C_M\frac{dU}{dt}A·dS+\frac{1}{2}\rho C_{D}U\left|U\right|D·dS,$$

(1)

where dF represents the force per segment length, $\rho$ is the density of the fluid phase, C_M is the inertia coefficient, U is the flow velocity, t is time, $\frac{dU}{dt}$ is the acceleration of the flow, A is the cross-sectional area of the cylinder, dS is the cylinder’s segment length, C_D is the drag coefficient, and D is the cylinders diameter.

When using the Morison equation, the structure itself is not directly part of the numerical flow model, i.e., the presence of the structure does not affect the water particle kinematics. The geometry, flow velocity, flow acceleration, fluid density, and drag and inertia coefficients must be specified at the location of the missing structure and inserted into the Morison equation. Keulegan and Carpenter were amongst the first to perform experiments to determine drag and inertia coefficients [2]. They introduced the dimensionless Keulegan and Carpenter parameter K_C to relate the amplitude of the oscillating wave velocity, the wave period, and the cylinder’s diameter. Sarpkaya later introduced a new parameter β relating K_C and the Reynolds number Re [3].

As sloshing tests in a model tank provide mainly statistical data on sloshing-induced impact loads acting on pump towers [4,5,6,7], it is difficult to precisely understand all physical phenomena necessary to accurately predict these loads. Impact loads occur quickly and are susceptible to local flow conditions [8]. Therefore, we decided to model the effect of a predefined focused wave acting on a pump tower configuration at a specific location in a wave canal. This enabled us to better observe and pinpoint local phenomena.

The hydrodynamic loads acting on a tripod configuration consisting of three cylinders were considered, where two downstream cylinders were located in the wake of an upstream cylinder. The upstream cylinder generated a wake field and, depending on the flow’s Reynolds number and the distance between the upstream and downstream cylinders, this wake may influence the loads acting on the downstream cylinders. In the wake field, the fluid’s mean velocities were less than the free stream velocities [9], thereby reducing the drag forces acting on the downstream cylinders [10,11].

An increase of wave loads was observed for cylinders installed side by side [12]. These loads were higher for regular non-breaking waves than for irregular non-breaking waves that were generated from a JONSWAP spectrum. Experimental investigations yielded modified drag and inertia coefficients for cylinders that were installed either in tandem or side by side [13]. Without calculating loads, numerical simulations with two side-by-side circular cylinders of Li et al. [14] illustrated the influence of the gap between the cylinders for a free surface flow. Tong et al. [15] investigated a steady uniform flow around two staggered cylinders at a low Reynolds number of $Re=1\times {10}^3$. The presence of a second cylinder altered the pressure distribution and the location of stagnation points on the surface of both cylinders. A reciprocal influence between an upstream and a downstream cylinder was also observed when the spacing ratio $\frac{L}{D}$ (with L being the distance of the center between the cylinders) was less than 3.5 for $Re=5.8\times {10}^4$, equal to 3.5 for $Re=8.3\times {10}^4$, and slightly higher than 3.5 for $Re=1.1\times {10}^5$ [16] for cylinders being installed in tandem. Zdravkovich investigated the influence on the drag coefficient for different spacing ratios between two cylinders that were installed in tandem. He found that the influence of the downstream cylinder on the upstream cylinder is significant when the spacing ratio is less than 3.5 for subcritical Reynolds numbers [17].

This paper dealt with the investigation of hydrodynamic loads on single cylinders and cylinders arranged in a tripod configuration. The Morison equation was used to calculate the wave loads on the cylinders in undisturbed waves. These wave loads were compared to those acting on the cylinders in disturbed waves, where all cylinders were geometrically modeled. For the configuration consisting of large-sized cylinders, Reynolds numbers ranged between $Re=5.0\times {10}^{4 }$ and $Re=1.6\times {10}^5$; for the configuration consisting of medium-sized cylinders, the Reynolds numbers ranged between $Re=4.3\times {10}^4$ and $Re=1.3\times {10}^5$. The tripod configuration represented a simplified LNG pump tower structure with staggered and side by side arrangements of the cylinders.

Using a turbulence model, the validation study was performed for the case with a single cylinder and presented in our conference paper [11]. It was summarized in this paper and extended by an additional validation study, also using a laminar flow model.

2. Test Case Description

The experiments of hydrodynamic loads acting on the cylinders were performed in a wave canal of 0.65 m width and 16.77 m length [18]. The initial water level was 0.667 m. A hydraulically powered flat-type wave maker generated the focused waves. A vertical wall installed at a distance of 12.53 m from the wave maker limited the tank’s test length.

The first scenario comprised tests of two single steel cylinders situated vertically in the canal at a longitudinal distance of 11.62 m from the wave maker and 0.91 m from the rear wall. A six degrees-of-freedom load sensor was fixed to a flange mounted at the top of each cylinder. This load sensor was enclosed in a frame whose height position was adjustable. Both cylinders had a length of 0.80 m, but they were each fixed at two different height positions to enable investigating impact loads at different immersion depths. In their low positions, the lower end of the cylinders was situated 0.198 m above the tank’s bottom; in their high positions, 0.398 m above the tank’s bottom. The small cylinder’s diameter was $D_{small}=0.0889 \mathrm{m}$; the large cylinder’s diameter was $D_{large}=0.1524 \mathrm{m}$. The purpose of these first scenario tests was to compare experimental data with numerical predictions.

The second scenario dealt with hydrodynamic loads acting on the cylinders in a tripod configuration, immersed to a depth of 0.469 m below the free surface. In this scenario, the investigated diameters were $D_{medium}=0.1230 \mathrm{m}$ and $D_{large}=0.1524 \mathrm{m}$. The longitudinal distance between the center of the downstream cylinders and the rear wall was 0.91 m. The longitudinal distance between the center of the downstream cylinders and the upstream cylinder was two times the diameter. The upstream cylinder was located on the centerline of the wave canal and the downstream cylinder was located one diameter transversally from this centerline. The upstream and downstream cylinders always had the same diameter.

3. Experimental Setup

An MC12 series AMTI multi-component transducer was used to measure hydrodynamic loads at a sampling frequency of 2000 Hz [18]. This transducer’s load capacity ranged between 100 N and 250 N for forces in the vertical direction, between 50 N and 125 N for forces in the longitudinal and transverse directions, between 50 Nm and 125 Nm for torques about the vertical axis, and between 100 Nm and 250 Nm for torques about the longitudinal and transverse directions. The load sensor was rigidly bolted on a movable support mounted above the wave canal. With this movable support, the sensor could be adjusted along the vertical axis to keep the single cylinder at the low and the high position. The cylinders were bolted through a fixation plate to the longitudinal center of the sensor [18].

To account for the effect of Newtonian viscosity, the non-breaking focused wave was generated on the basis of the Ricker spectrum with a peak period of $T_P$ = 2.617 s. The focal point of the focused wave was set at a longitudinal distance of x = 11.69 m from the wave maker [18]. The position of the focal point was thus close to the center of the cylinders in the first scenario; specifically, it was positioned at a distance of 11.62 m from the wave maker.

Table 1. Parameters of the wave for the first and second scenarios.

${\mathit{A}}_{\mathit{w}} \left(\mathbf{m}\right)$	m	x (m)	T	TP (s)
0.214	1.42	11.69	0.2285	2.617

lists the focused wave’s greatest height of $A_w$ = 0.214 m and the two parameters of m = 1.42 and T = 0.2285, which had to be specified to generate this wave in the first and second scenarios.

4. Numerical Method

An Eulerian approach defined our liquid–gaseous two-phase flow according to the volume of fluid (VoF) method:

$${\alpha }_i=\frac{V_i}{V},$$

(2)

where ${\alpha }_i$ is the volume fraction of phase i, $V_i$ is the volume of the respective phase i, and $V$ is the volume of the cell.

Velocities, pressures, etc., were calculated for each grid cell of the computational domain. An appropriate coordinate system was defined to identify the location and time assigned to each liquid and gaseous particle of the respective cell [19]. Energy, mass, and momentum had to be conserved. For the prevailing two-phase flow, exchange terms for energy, momentum, and mass accounted for the interactions of the two phases [20]. The term for the continuity equation is defined as follows:

$$\frac{\partial }{\partial t}\left({{\displaystyle \int }}_V^{ }\rho dV\right)+{{\displaystyle \oint }}_A^{ }\rho \mathit{v}·d\mathit{a}={{\displaystyle \int }}_V^{ }{{\displaystyle \sum }}_i{S}_{{\alpha }_i}·{\rho }_{i}dV,$$

(3)

where $\mathit{v}$ is the mass-averaged velocity, and a is the surface area. The product $S_{{\alpha }_i}·{\rho }_i$ accounts for the phase source term in the mass source term. The term for the momentum equation is defined as follows:

$$\frac{\partial }{\partial t}\left({{\displaystyle \int }}_V^{ }\rho \mathit{v}dV\right)+{{\displaystyle \oint }}_A^{ }\rho \mathit{v}\otimes \mathit{v}·d\mathit{a}=-{{\displaystyle \oint }}_A^{ }p\mathit{I}·d\mathit{a}+{{\displaystyle \oint }}_A^{ }\mathit{T}·d\mathit{a}+{{\displaystyle \int }}_V^{ }\rho \mathit{g}dV+{{\displaystyle \int }}_V^{ }{\mathit{f}}_{b}dV-{{\displaystyle \sum }}_i{\alpha }_i{\rho }_i{\mathit{v}}_{d,i}\otimes {\mathit{v}}_{d,i}·d\mathit{a},$$

(4)

where p is the pressure, $\mathit{I}$ is the unity tensor, $\mathit{T}$ is the viscous stress tensor, g is the gravity, ${\mathit{f}}_b$ is the vector of the body forces, and ${\mathit{v}}_{d,i}$ is the diffusion velocity. The term for the energy equation is defined as follows:

$$\frac{\partial }{\partial t}\left({{\displaystyle \int }}_V^{ }\rho EdV\right)+{{\displaystyle \oint }}_A^{ }\left[\rho \mathit{v}H+p+{{\displaystyle \sum }}_i{\alpha }_i{\rho }_i{H}_i{\mathit{v}}_{d,i}\right]·d\mathit{a}=-{{\displaystyle \oint }}_A^{ }{\dot{\mathit{q}}}^{″}·d\mathit{a}+{{\displaystyle \oint }}_A^{ }\mathit{T}\mathit{v}·d\mathit{a}+{{\displaystyle \int }}_V^{ }{\mathit{f}}_b\mathit{v}dV+{{\displaystyle \int }}_V^{ }{S}_{E}dV,$$

(5)

where E is the total energy, H is the total enthalpy, ${\dot{\mathit{q}}}^{″}$ is the heat flux vector, and $S_E$ is a user-defined energy source term.

The simulations for the first scenario were performed with a turbulence model as well as with a laminar model to investigate the necessity of using a turbulence model for the single cylinder case. The simulations for the second scenario were performed solely with the turbulence model because the wake generated by the upstream cylinder affected the downstream cylinders of the tripod configuration.

As it was necessary to employ an unsteady approach to simulate the two-phase flow, the VoF method was combined with a URANS solver for the turbulence model and only with the Navier–Stokes (NS) equations for the laminar model. An averaged value of the velocity, the pressure, the viscous stress tensor, the total energy, and the heat flux was implemented in the URANS calculations. This resulted in a change of the terms for the conservation equations. The continuity equation was written as follows:

$$\frac{\partial }{\partial t}\left({{\displaystyle \int }}_V^{ }\rho dV\right)+{{\displaystyle \oint }}_A^{ }\rho \overline{\mathit{v}}·d\mathit{a}={{\displaystyle \int }}_V^{ }{{\displaystyle \sum }}_i{S}_{{\alpha }_i}·{\rho }_{i}dV,$$

(6)

where $\overline{\mathit{v}}$ is the averaged velocity. The momentum equation was changed to:

$$\frac{\partial }{\partial t}\left({{\displaystyle \int }}_V^{ }\rho \overline{\mathit{v}}dV\right)+{{\displaystyle \oint }}_A^{ }\rho \overline{\mathit{v}}\otimes \overline{\mathit{v}}·d\mathit{a}=-{{\displaystyle \oint }}_A^{ }\overline{p}\mathit{I}·d\mathit{a}+{{\displaystyle \oint }}_A^{ }\left(\overline{\mathit{T}}+{\mathit{T}}_{RANS}\right)·d\mathit{a}+{{\displaystyle \int }}_V^{ }\rho \mathit{g}dV+{{\displaystyle \int }}_V^{ }{\mathit{f}}_{b}dV-{{\displaystyle \sum }}_i{\alpha }_i{\rho }_i{\overline{\mathit{v}}}_{d,i}\otimes {\overline{\mathit{v}}}_{d,i}·d\mathit{a},$$

(7)

where $\overline{p}$ is the averaged pressure, $\overline{\mathit{T}}$ is the averaged viscous stress tensor, and ${\overline{\mathit{v}}}_{d,i}$ is the averaged diffusion velocity. A RANS stress tensor ${\mathit{T}}_{RANS}$ was added to the mean viscous stress tensor in the momentum equation:

$${\mathit{T}}_{RANS}=-\rho \left(\begin{array}{ccc}\overline{u\prime u\prime }& \overline{u\prime v\prime }& \overline{u\prime w\prime }\\ \overline{v\prime u\prime }& \overline{v\prime v\prime }& \overline{v\prime w\prime }\\ \overline{w\prime u\prime }& \overline{w\prime v\prime }& \overline{w\prime w\prime }\end{array}\right)+\frac{2}{3}\rho k\mathit{I},$$

(8)

where u′, v′, and w′ are the fluctuating components of the velocity along the x-, y-, and z-axis, and k is the turbulent kinetic energy. The RANS stress tensor was also implemented in the energy conservation equation:

$$\frac{\partial }{\partial t}\left({{\displaystyle \int }}_V^{ }\rho \overline{E}dV\right)+{{\displaystyle \oint }}_A^{ }\left[\rho \overline{\mathit{v}}H+\overline{p}+{{\displaystyle \sum }}_i{\alpha }_i{\rho }_i{H}_i{\overline{\mathit{v}}}_{d,i}\right]·d\mathit{a}=-{{\displaystyle \oint }}_A^{ }{\overline{\dot{\mathit{q}}}}^{″}·d\mathit{a}+{{\displaystyle \oint }}_A^{ }\left(\overline{\mathit{T}}+{\mathit{T}}_{RANS}\right)\overline{\mathit{v}}·d\mathit{a}+{{\displaystyle \int }}_V^{ }{\mathit{f}}_b\overline{\mathit{v}}dV+{{\displaystyle \int }}_V^{ }{S}_{E}dV,$$

(9)

where $\overline{E}$ is the averaged total energy and ${\overline{\dot{\mathit{q}}}}^{″}$ is the averaged heat flux vector.

The conservation equations of momentum and mass were solved consecutively with the segregated flow solver. The segregated flow solver relied on a semi-implicit method for pressure linked equations (SIMPLE) algorithm to couple the pressure and velocity fields. With the predictor-corrector approach, a pressure correction equation generated a velocity field that satisfied the continuity equation. This pressure correction equation consisted of parts of the momentum and continuity equations. A pressure gradient $p_f$ was computed and replaced the pressure in the momentum equations:

$$p_f=\frac{\overline{a_0}{p}_{f0}+\overline{a_1}{p}_{f1}}{\overline{a_0}+\overline{a_1}},$$

(10)

where $\overline{a_0}$ and $\overline{a_1}$ denote the average for each of the coefficients in the momentum equation for two adjacent cells 0 and 1, and $p_{f0}$ and $p_{f1}$ are face values that were interpolated from reconstructed gradients and cell values. The continuity equation was rewritten for the pressure–velocity coupling:

$${{\displaystyle \sum }}_f{\dot{(m}}_f{}^{\ast }+{\dot{m}}_f{}^{\prime })=0,$$

(11)

where ${\dot{m}}_f{}^{\ast }$ is the uncorrected face mass flux, which was obtained by solving the discrete momentum equations with an approximated pressure field $p^{\ast }$. As the continuity was not yet satisfied with the approximated pressure field, it needed the corrected mass term ${\dot{m}}_f{}^{\prime }$:

$${\dot{m}}_f{}^{\prime }=Q_f\left({\dot{p}}_0{}^{\prime }-{\dot{p}}_1{}^{\prime }\right)+\frac{{\dot{m}}_f{}^{\ast }}{{\rho }_f}{\left(\frac{\partial \rho }{\partial p}\right)}_T{p}_{upwind}^{\prime },$$

(12)

where ${\dot{p}}_0{}^{\prime }$ and ${\dot{p}}_1{}^{\prime }$ are the cell pressure corrections. The uncorrected mass flux ${\dot{m}}_f{}^{\ast }$ is then defined as follows:

$${\dot{m}}_f{}^{\ast }={\rho }_f\mathit{a}\left(\frac{v_0^{\ast }+v_1^{\ast }}{2}\right)-{\gamma }_f,$$

(13)

where $v_0^{\ast }$ and $v_1^{\ast }$ are the velocities after solving the discrete momentum equations with the approximated pressure field. The Rhie–Chow dissipation factor ${\gamma }_f$ prevents the development of physically unrealistic checker boarding of pressure caused by the linear interpolation from the cell centers to the faces of the cells:

$${\gamma }_f=Q_f\left(p_1-p_0-\overline{\nabla p_f^{\ast }}·ds\right),$$

(14)

where $\overline{\nabla p_f^{\ast }}$ is the volume-weighted average of the cell gradients of pressure and $Q_f$ defined as follows:

$$Q_f={\rho }_f\left(\frac{V_0+V_1}{\overline{a_0}+\overline{a_1}}\right)\mathit{\alpha }·\mathit{a},$$

(15)

where $p_{upwind}^{\prime }$ is defined as:

$$p_{upwind}^{\prime }=\left\{\begin{array}{c}{p}_0^{\prime } for {\dot{m}}_f{}^{\ast }>0\\ p_1^{\prime } for {\dot{m}}_f{}^{\ast }<0\end{array}\right.,$$

(16)

In the URANS simulations, the shear stress transport (SST) k–ω model of Menter [21] was used to solve for turbulence. The term for the transport equations for the kinetic energy k was:

$$\frac{\partial }{\partial t}\left(\rho k\right)+\nabla \left(\rho k\overline{\mathit{v}}\right)=\nabla \left[\left(\mu +{\sigma }_k{\mu }_t\right)\nabla k\right]+P_k-\rho {\beta }^{\ast }{f}_{{\beta }^{\ast }}\left(\omega k-{\omega }_0{k}_0\right)+S_k,$$

(17)

where $\mu$ is the dynamic viscosity, ${\sigma }_k$ is a model coefficient, ${\mu }_t$ is the turbulent eddy viscosity, $P_k$ is a sum of production terms consisting of a turbulent, a buoyancy and a non-linear production term, ${\beta }^{\ast }$ is a model coefficient, $f_{{\beta }^{\ast }}$ is a free-shear modification factor, $\omega$ is the specific dissipation rate, ${\omega }_0$ and $k_0$ are ambient turbulence values to counteract the turbulence decay, and $S_k$ are the user-specified source terms. The term for the transport equations for the specific dissipation rate $\omega$ was:

$$\frac{\partial }{\partial t}\left(\rho \omega \right)+\nabla \left(\rho \omega \overline{\mathit{v}}\right)=\nabla \left[\left(\mu +{\sigma }_{\omega }{\mu }_t\right)\nabla \omega \right]+P_{\omega }-\rho \beta f_{\beta }\left({\omega }^2-{\omega }_0^2\right)+S_{\omega },$$

(18)

where $P_{\omega }$ is a sum of production terms consisting of a specific dissipation production and a cross-diffusion term, $\beta$ is a model coefficient, $f_{\beta }$ is a vortex-stretching modification factor, and $S_{\omega }$ is the user-specified source terms. The high-resolution interface capturing (HRIC) scheme [22] was used to discretize the transport equation for the volume fraction of water and air. The normalized variable diagram (NVD) [23] and normalized variables were used for the HRIC scheme. The normalized face value ${\xi }_f$ was calculated on the NVD using a combination of the upwind and downwind differencing schemes:

$${\xi }_f=\left\{\begin{array}{ccc}{\xi }_C& if& \hfill {\xi }_C<0\\ 2{\xi }_C& if& \hfill 0\le {\xi }_C<0.5\\ 1& if& \hfill 0.5\le {\xi }_C<1\\ {\xi }_C& if& \hfill 1\le {\xi }_C\end{array}\right.,$$

(19)

where ${\xi }_C$ is the normalized cell value. The normalized face value is improved by considering the local value of the Courant number (CFL) $Co$:

$$Co=\frac{\mathit{v}{\mathit{a}}_f}{V_{P_C}}\Delta t,$$

(20)

where ${\mathit{a}}_f$ is the face area, $\Delta t$ is the time step size, and $V_{P_C}$ is the control volume of the donor cell. A correction of the normalized face value ${\xi }_f^{\ast }$ was implemented to avoid stability problems during the simulations and to fulfill the convective boundedness criterion (CBC):

$${\xi }_f^{\ast }=\left\{\begin{array}{ccc}{\xi }_C& if& \hfill Co<C{o}_l\\ {\xi }_C+\left({\xi }_f-{\xi }_C\right)\frac{{Co}_u-Co}{{Co}_u-C{o}_l}& if& \hfill {Co}_l\le Co<{Co}_u\\ {\xi }_C& if& \hfill {Co}_u\le Co\end{array}\right.,$$

(21)

where $C{o}_l$ denotes the lower limit and $C{o}_u$ the upper limit for the CFL number. The HRIC scheme is used when CFL numbers are below $C{o}_l$,. A blending of the HRIC scheme and the first-order upwind scheme (FOU) is used when CFL numbers are between $C{o}_l$ and $C{o}_u$. Only the FOU is used if the CFL number is greater than $C{o}_u$.

5. Numerical Setup

To reduce the computational domain from 12.53 to 4.53 m, longitudinal and vertical velocities and wave heights were pre-calculated for a longitudinal distance of 8.00 m from the wave maker. By incorporating a re-derived Boussinesq-type model [24] and implementing an advanced shoaling enhancement operator, an accurate distribution of vertical velocities was obtained that always satisfied the kinematic bottom boundary condition. An advanced potential formulation improved the effectiveness of the prior models [25,26,27] by converting the velocity formulation and, thus, enhancing the relation between both formulations.

The liquid phase of the two-phase flow was defined as incompressible water with a density of 1000 kg/m³, and the ambient air was assumed to be compressible.

The simulations discretized time implicitly to second-order, and a second-order scheme together with the mid-point rule discretized space. Initially, a flat wave defined the VoF wave model with a water height at 0.667 m. The multi-phase VoF model was used with the high-resolution interface capturing scheme (HRIC). A CFL of 0.5 specified the lower limit for the HRIC scheme, and a CFL of 1.0 as the upper limit. Use of a sharpening factor of 0.1 for the HRIC scheme [7] ensured obtaining accurate results. The surface average CFL number was kept below 0.2 during an entire simulation.

Seven prism layers surrounded the cylindrical structures. The specified thickness of the first prism layer ensured that all y+ values in the URANS simulations were below 3.0 and, on this basis, the physical area of the viscous sublayer was defined. As initial conditions, the static temperature of 293.15 K, the hydrostatic pressure of the flat VoF wave, and the zero velocity were defined. For the URANS simulations, the initial condition for the turbulent kinetic energy $k$ was 0.001 J/kg and, for the specific dissipation rate $\omega$, it was 1.0 × 10⁻⁴ 1/s

The x-axis defined the length of the numerical canal; the y-axis, its width; and the z-axis, its height. The computational domain ranged from x = 0 m, defined as its inlet, to x = 4.53 m, defined as its back wall. The canal had a width of 0.65 m, ranging from z = −0.669 m to z = 0.52 m. The origin of the coordinate system was situated at the canal’s centerline and the intersection of the undisturbed water level with the inlet. The cylinders in the first scenario as well as the cylinders in the second scenario and all walls, except the top wall, were defined as walls with the no-slip condition. The top wall was defined as a pressure outlet; one part of the inlet, as the velocity inlet; and the other part, as a wall.

Table 2. Cylinder particulars describing the arrangement for the cylinders in the experimental setup and for the numerical setup (in brackets).

	Cylinder Size	Length (m)	Diameter (m)	Longitudinal × Transversal × Vertical Position (m)
				Single Cylinder		Upstream	Downstream
				Low Position	High Position	Low Position	Low Position
Scenario 1	Small	0.8000	0.0889	11.62 (3.62) × 0 × −0.469—0.331 (−0.469—0.52)	11.62 (3.62) × 0 × −0.269—0.531 (−0.269—0.52)
	Large	0.8000	0.1524	11.62 (3.62) × 0 × −0.469—0.331 (−0.469—0.52)	11.62 (3.62) × 0 × −0.269—0.531 (−0.269—0.52)
Scenario 2	Medium	0.8000	0.1230			(3.374) × (0) × (−0.469—0.52)	(3.62) × (0.123) × (−0.469—0.52)
	Large	0.8000	0.1524			(3.3152) × (0) × (−0.469—0.52)	(3.62) × (0.1524) × (−0.469—0.52)

lists particulars describing the arrangement for the cylinders in the experimental setup, with the corresponding particulars describing the arrangement for the numerical setup in brackets. No brackets indicate that the arrangements in both setups were the same.

For 434 heights along the z-axis, two distinct files stored the local vertical and horizontal velocities and the wave elevation at the inlet. This length extended from z = −0.667 m to z = 0.199 m, and a distance of 0.002 m separated these incremental heights from each other. A total of 434 blocks were created to allocate the data to the velocity inlet. The first block had a height of 0.001 m and comprised data for the first point. The following blocks had a height of 0.002 m and consisted of data of the remaining 433 points. The rest of the inlet, extending from z = 0.2 m to z = 0.52 m, was defined as a wall with the no-slip condition.

To validate the numerical method, predicted impact forces acting on the single large cylinder were compared with experimentally measured impact loads. Some simulations implemented the laminar flow model and others the turbulent flow model. Two sets of three successively finer grids were generated on which to compute the simulations. The refinement ratio of $\sqrt{2}$ was applied for the spatial as well as the time step sizes to ensure obtaining the same Courant number for each grid.

Table 3. Number of cells (CVs) and time steps sizes of grids used to validate the numerical method. Data from [11].

	Coarse Grid (1,085,131 CVs)		Medium Grid (2,829,244 CVs)		Fine Grid (7,519,511 CVs)
	Time Step (Start) (s)	Time Step (s)	Time Step (Start) (s)	Time Step (s)	Time Step (Start) (s)	Time Step (s)
Set 1	8 × 10−3	2 × 10−3	5.657 × 10−3	1.414 × 10−3	4 × 10−3	1 × 10−3
Set 2	5.657 × 10−3	1.414 × 10−3	4 × 10−3	1 × 10−3	2.828 × 10−3	7.071 × 10−4

lists the number of cells and the associated time step sizes of the coarse, medium, and fine grid.

The cells of all grids were hexahedral, and each cell was aligned at the origin of its coordinate system located at the intersection between the gaseous and the liquid phase. As a result, all cells contained either 100% of the gaseous phase or 100% of the liquid phase at the beginning of the simulation. Thus, at the start no unintentional mixing of the individual phases occurred.

Grids were refined in the vicinity of the free surface, ranging from z = −0.10 to 0.20 m, and at the cylinder to dissolve the wave shape and the higher velocities more accurately and to prevent unwanted vortices forming at the cylinders. Cell lengths were 0.0125 m for the coarse grid, 0.00884 m for the medium grid, and 0.00625 m for the fine grid. For the remaining area, the cell length was 0.025 m for the coarse grid, 0.01768 m for the medium grid, and 0.0125 m for the fine grid. The cell length in the vicinity of the cylinder was 0.00625 m for the coarse grid, 0.00442 m for the medium grid, and 0.003125 m for the fine grid. The total thickness of the prism layer was 0.00532 m for the coarse grid, 0.00376 m for the medium grid, and 0.00266 m for the fine grid, whereas the thickness of the first prism layer was 0.031 × 10⁻³ m for all grids. Figure 1 shows a side view and Figure 2 a top view of the grids (at z = 0 m) surrounding the large cylinder.

The simulations for the second scenario were obtained only with the implemented turbulent model. The time step size and basic grid structure were based on results from our previous validation study [11] and adapted for the tripod configuration. A prism layer thickness of 0.00266 m was defined for the large-sized cylinders and 0.00215 m for the medium-sized cylinders. The thickness of the first prism layer was 0.031 × 10⁻³ m for the large cylinders and 0.03 × 10⁻³ m for the medium-sized cylinders, thereby obtaining the same low wall y+ number. Seven prism layers were considered. The cells surrounding the upstream and downstream cylinders were 0.003125 m long, which enabled to accurately capture the turbulent flow in the area between the cylinders and, therefore, to account for the vortex-induced interaction between the cylinders. To evaluate the effectiveness of the Morison equation, the cylinders were removed from the numerical canal, and the velocities and accelerations were gathered at the positions of the removed cylinders. Figure 3 depicts a top view of grid refinement in the vicinity of the three large cylinders at the height of z = 0 m.

Table 4. Overview of the numerical cases.

	Flow Model	Cylinder Sizes	Quantity of Simulations	Cylinder Immersion during Validation Study (m)	Cylinder Immersion after Validation Study (m)
				Low Position	High Position	Low Position
Scenario 1	Laminar	Large	6 (3 each with Set 1 and Set 2)	−0.469
	Turbulent	Small	4 (2 each with Morison and URANS for both positions)		−0.269	−0.469
	Turbulent	Large	6 (3 each with Set 1 and Set 2) for low position; 4 (2 each with Morison and URANS for both positions)	−0.469	−0.269	−0.469
Scenario 2	Turbulent	Medium	6 (2 each with Morison and URANS for all three cylinders)			−0.469
	Turbulent	Large	6 (2 each with Morison and URANS for all three cylinders)			−0.469

lists particulars for the numerical cases.

6. Validation

Two simulations were conducted per grid with the turbulence model and two simulations per grid with the laminar model. Set 1 comprised simulations with the larger time steps and set two simulations with the finer time step, see Table 3. Thus, the CFL number differed only between the two sets and remained the same within each set. The least squares method [28] was used to obtain grid and time step independent solutions for a constant CFL number [29].

The two top graphs of Figure 4 plot time histories of the longitudinal load, based on the laminar flow model, acting on the large cylinder obtained on the three grids of set 1 and set 2. The black line depicts experimentally measured loads; the yellow line, loads computed on the coarse grid; the red line, loads computed on the medium grid; the blue line, loads computed on the fine grid. To better distinguish differences between peak load predictions, the two bottom graphs of Figure 4 present zoomed views of these time histories representing time intervals within the small rectangular box in the top graphs of Figure 4. As seen, for both sets the predicted load obtained on the coarse grids came closest to the measured load. Furthermore, with increasing grid fineness computed loads increased and deviated more from the measured loads.

The upper graph of Figure 5 plots time step refinement ratio versus grid spacing refinement ratio for simulations based on the laminar model. This grid spacing refinement ratio x_ij is defined as follows:

$$x_{ij}=\frac{Spacin{g}_{gri{d}_i\_se{t}_j}}{Spacin{g}_{gri{d}_1\_se{t}_1}},$$

(22)

where indexes i = 1, 2, 3 refer to the coarse, the medium, and fine grids, respectively, and indexes j = 1 and 2 refer to sets 1 and 2, respectively. The time step refinement ratio y_ij is defined as follows:

$$y_{ij}=\frac{t_{gri{d}_i\_se{t}_j}}{t_{gri{d}_1\_se{t}_1}},$$

(23)

where t represents the time step, indexes i = 1, 2, 3 refer to the coarse, the medium, and fine grids, respectively, and indexes j = 1 and 2 refer to sets 1 and 2, respectively. The denominator of this ratio is always the largest time step, i.e., the time step of grid 1 in set 1.

The number of control volumes remained the same for set 1 and set 2. Thus, the grid spacing differed only between the coarse, the medium, and the fine grid. Recall that the refinement factor of $\sqrt{2}$ was the same for the grid spacing and the time step size.

The lower graph of Figure 5 plots the grid spacing and time step independent longitudinal impact load versus the arithmetic mean X_ij of x_ij and y_ij defined as follows:

$$X_{ij}=\frac{x_{ij}+y_{ij}}{2},$$

(24)

The cyan line represents the time step and grid independent solution for the impact load in set 1; the green line, the time step and grid independent solution for the impact load in set 2. Sets 1 and 2 yielded the impact loads of 29.57 N and 29.62 N, respectively, and the experimentally measured impact load was about 28.55 N. Thus, the numerically computed impact load deviated by about 3.6% from the measured impact. Recall that these load computations were based on the laminar flow model.

The two top graphs of Figure 6 plot time histories of the longitudinal load, here based on the turbulent flow model, acting on the large cylinder obtained on the three grids of set 1 and set 2. As in Figure 4, the black line depicts experimentally measured loads; the yellow line, loads computed on the coarse grid; the red line, loads computed on the medium grid; the blue line, loads computed on the fine grid. Here, too, the two bottom graphs of Figure 6 present zoomed views of these time histories representing time intervals within the small rectangular box in the top graphs of Figure 6. As seen, the predicted impact load obtained on the finest grid in set 2 compared best to the experimentally measured impact load.

Sets 1 and 2 yielded the discretization independent impact loads of 28.54 N and 28.52 N, respectively, see Figure 7. The favorable agreement between the measured impact load of 28.55 N and the discretization independent impact loads in both sets indicated that the computations were able to yield reliably predictions.

Table 5. Discretization-independent predictions and impact loads from the coarse, medium, and fine grids and their deviations from experimental measurements (in brackets).

	Coarse Grid Impact (N) and Deviation from Experiments (%)		Medium Grid Impact (N) and Deviation from Experiments (%)		Fine Grid Impact (N) and Deviation from Experiments (%)		Discretization Independent Impact (N) and Deviation from Experiments (%)
	Laminar	(SST) k–ω	Laminar	(SST) k–ω	Laminar	(SST) k–ω	Laminar	(SST) k–ω
Set 1	28.41 (0.49)	27.39 (4.06)	28.73 (0.63)	27.65 (3.15)	28.98 (1.51)	27.99 (1.96)	29.57 (3.57)	28.54 (0.035)
Set 2	28.51 (0.14)	27.49 (3.71)	28.88 (1.16)	27.76 (2.78)	29.18 (2.21)	28.14 (1.44)	29.62 (3.75)	28.52 (0.11)

summarizes these results and lists the calculated and measured value for the impact loads and the deviation between both. Recall that these load computations were based on the turbulent flow model.

These uncertainty studies demonstrated that loads based on the turbulent flow model yielded more accurate and higher predictions than those based on the laminar flow model. Therefore, subsequent computations were carried out with the turbulence model on the finest grid with the time step for set 2. Of course, the grid was adapted for the differently sized cylinders and the tripod configuration accordingly.

7. Comparison of Load Coefficients for Single Cylinder and Cylinders in the Tripod Configuration Considering the Influence of Wake Effects

In the first scenario, longitudinal flow velocities were pre-calculated at 210 points along the z-axis located at the single cylinder’s longitudinal position of x = 3.62 m. These probe points, separated by 0.0032 m from each other, extended from z = −0.4674 m to z = 0.2014 m. The length per segment dS, therefore, was 0.0032 m for a cylinder immersed to a depth of z = −0.4690 m.

However, for a cylinder immersed to a depth of z = −0.2690 m, the pre-calculation of flow velocities started at the 64th probe point located at z = −0.2658 m. To avoid repeating the simulation with different positions of the probe points, the segment length dS was increased to 0.0064 m. Thus, flow velocities were obtained at every second probe point for this cylinder. Figure 8 shows the arrangement and location of the single cylinder in this first scenario and the 210 probe points at which the longitudinal velocities were pre-calculated. For illustrative purposes, only 22 of the 210 probes are shown in Figure 8.

In the second scenario, velocities were obtained at 80 probe points along the z-axis located at the three cylinders’ longitudinal positions, i.e., at a total of 240 probe points. These probe points were separated by a distance of 0.0086 m from each other and extended from z = −0.4647 m to z = 0.2147 m. The cylinders were immersed to a depth of z = −0.4690 m. Figure 9 shows the arrangements of the three cylinders for the tripod configuration in the second scenario and the 240 probe points at which longitudinal velocities were pre-calculated. For illustrative purposes, only 27 of the 240 probes are shown in Figure 9.

Figure 10 plots time histories of the temporal waveform at the longitudinal positions of the single cylinders and of the downstream cylinders, which is located at x = 3.62 m, of the medium-sized upstream cylinder, which is located at x = 3.374 m and of the large-sized upstream cylinder, which is located at x = 3.3152 m.

Figure 11a plots time histories of pre-calculated flow velocities at the probe points in the first scenario for the single cylinder immersed to a depth of z = −0.4690 m; and Figure 11b for the single cylinder immersed to a depth of z = −0.2690 m. A color bar indicates the vertical position of the probe points.

Figure 12 and Figure 13 plot time histories of pre-calculated flow velocities for the tripod configuration considered in the second scenario. The time histories in Figure 12 refer to the configuration consisting of the large-diameter cylinders; the time histories in Figure 13, to the configuration consisting of the medium-diameter cylinders. Graph (a) of Figure 12 and Figure 13 refers to flow velocities at the upstream cylinder; Graph (b) refers to flow velocities at the starboard downstream cylinder; Graph (c) refers to flow velocities at the portside downstream cylinder.

The times between maximum and minimum flow velocity in Figure 11, Figure 12 and Figure 13 were multiplied by a factor of two, and these times were treated as a wave period of the focused wave representing the period of a sloshing motion acting on a single vertical cylinder or on a typical tripod configuration of a pump tower. A previous investigation of ours demonstrated that this approach yielded practically relevant estimates [11].

To establish load coefficients, the wave period needed to be specified as this enabled determining Sarpkaya’s beta factor $\beta$ [3]:

$$\beta =\frac{D^2}{\vartheta T_m},$$

(25)

where $\vartheta$ is the kinematic viscosity, and T_m is the wave period. Sarpkaya defined drag coefficients C_D and inertia coefficients C_M as a function of the Keulegan and Carpenter parameter K_C [2]:

$$K_C=\frac{U_m{T}_m}{D},$$

(26)

where U_m is the oscillating flow’s maximum velocity at the point where the load is acting, that is, at the probe points in our case.

In the first scenario, the wave period 1.87 s and the maximum velocity were the same for all simulations because the location of the probe points along the single cylinder did not change. In the second scenario, the longitudinal position of the upstream cylinder differed from that of the downstream cylinders and, thus, the wave period was modified. While the wave period for the downstream cylinders remained at about 1.87 s in all simulations, the modified wave period was 2.32 s for the large-diameter upstream cylinder and 2.29 s for the medium-diameter upstream cylinder.

In the first scenario, the $\beta$ parameter turned out to be 12,383 for the large-diameter cylinder and 4214 for the small-diameter cylinder, and the K_C parameter was 13.7 for the large-diameter cylinder and 23.4 for the small-diameter cylinder.

In the second scenario’s configuration with the medium-diameter cylinders, $\beta$ and K_C were 6587 and 19.6, respectively, for the upstream cylinder, and 8066 and 16.7, respectively, for the downstream cylinders.

In the second scenario’s configuration with the large-diameter cylinders, $\beta$ and K_C were 9981 and 15.9, respectively, for the upstream cylinder and 12,383 and 13.7, respectively, for the downstream cylinders.

Sarpkaya performed experimental tests for $\beta$ values of 4480, 6555, 8370, and 11,525 [3,30]. From his tests for $\beta =4480$, drag and inertia coefficients were extracted for the small-diameter cylinder in the first scenario; from his tests for $\beta =6555$, drag and inertia coefficients were extracted for the upstream cylinder in the configuration with the medium-diameter cylinders; from his tests for $\beta =8370$, drag and inertia coefficients were extracted for the downstream cylinders in the configuration with the medium-diameter cylinders, and from his tests for $\beta =11,525$, drag and inertia coefficients were extracted for the large cylinder in the first scenario and the upstream and downstream cylinders in the configuration with the large-diameter cylinders.

Table 6. Wave periods, β and K_C values, and extracted drag and inertia coefficients for the first and second scenarios.

Scenario	Cylinder Size	Cylinder Position	${\mathit{T}}_{\mathit{m}} \left(\mathbf{s}\right)$	$\mathit{\beta }$	KC (-)	CD (-)	CM (-)
1	Small		1.87	4214	23.4	0.66	1.7
	Large		1.87	12,383	13.7	0.68	1.77
2	Medium	Upstream	2.29	6587	19.6	0.684	1.714
		Downstream	1.87	8066	16.7	0.65	1.79
	Large	Upstream	2.32	9981	15.9	0.71	1.68
		Downstream	1.87	12,383	13.7	0.68	1.77

lists these extracted drag and inertia coefficients as well as the associated wave periods and ß and K_C values.

To compare the predicted hydrodynamic loads against experimental measurements, the coefficient c_x was introduced, which expresses the normalized loads as follows [7]:

$$c_x=\frac{F}{\rho ·\left(2{\pi }^2·\frac{1}{T_P{}^2}·A_w{}^2\right)·A_{Area}},$$

(27)

where F represents the hydrodynamic load, T_P the peak period, A_w the maximum wave height of the respective focused wave, and A_Area the surface area of the cylinder.

Figure 14 presents comparative time histories of load coefficient c_x for the low- and high-positioned cylinder in the first scenario. A black line identifies c_x values based on experimentally measured loads; a dark red line, c_x values based on URANS computed loads; an indigo line, c_x values based on loads calculated with the Morison equation. Figure 14a,c of this figure refer to loads acting on the small-diameter cylinder in its low and high positions, respectively; Figure 14b,d of this figure refer to loads acting on the large-diameter cylinder in its low and high positions, respectively.

Generally, in all four cases the computed and calculated loads compared favorably to measurements, especially in the vicinity of the impacting load peaks. However, after impact, predicted loads differed somewhat from measurements although the URANS computed loads acting on the large-diameter cylinder agreed fairly well with measurements. Nevertheless, for the small-diameter cylinder, the Morison equation yielded loads that closely matched the URANS computed loads even after impact. After impact, not only the URANS simulations, but also the Morison equation based solutions had difficulties reproducing the temporal progression. This could have been due to the reduced stiffness of the small-diameter cylinder.

Figure 15 presents the comparative frequency spectra of load coefficient c_x, derived via a fast Fourier transform (FFT), for the low- and high-positioned cylinder in the first scenario. Again, a black line identifies c_x values based on experimentally measured loads; a dark red line, c_x values based on URANS computed loads; an indigo line, c_x values based on loads calculated with the Morison equation. Figure 15a,c of this figure refer to loads acting on the small-diameter cylinder in its low and high positions, respectively; Figure 15b,d of this figure refer to loads acting on the large-diameter cylinder in its low and high positions, respectively.

The URANS computed load coefficients and experimental measurements compared favorably for the large-diameter cylinder in its low position. There were two visible peaks at about 0.5 Hz and 3 Hz. Although load coefficients based on the Morison equation were somewhat lower than measured coefficients, they had two peaks at these same frequencies. The same tendency was observed for the small-diameter cylinder in its low position at the first peak. Four peaks characterized the spectrum of the load coefficient based on experimental measurements. Load coefficients from URANS computations and from the Morison equation yielded three visible peaks. The second and third peaks were at the same frequency as the third and fourth peaks of the coefficients based on experiments although the experimentally based peaks were slightly larger. The number of peaks and their occurrence frequency were nearly the same for the small-diameter cylinder in its high position. The URANS based coefficients were higher at the first peaks. Load coefficients based on experiments and on the Morison equation matched favorably at the first peak. Coefficients for the large-diameter cylinder in its high position compared favorably to both coefficients based on predictions and measurements. The frequency of the first peak shifted slightly to a higher value for the large-diameter cylinder in its low position.

Figure 16 presents comparative time histories of load coefficients for the medium-diameter cylinders in their tripod configuration, i.e., in the second scenario. The dark red line identifies the URANS computed coefficients and the indigo line, the coefficients calculated with the Morison equation. These identifications apply also to the Figure 17, Figure 18 and Figure 19. Graphs (a) of these figures refer to coefficients for the upstream cylinder; Graphs (b) refer to coefficients for the downstream starboard cylinder; Graphs (c) refer to coefficients for the downstream portside cylinder. For the upstream cylinder, the impact from the Morison equation was slightly higher than the impact from for URANS computation; however, for both downstream cylinders, the impact from the Morison equation was slightly lower than the impact from the URANS computation. For the upstream cylinder, the temporal progression of load coefficients obtained from both methods agreed better than for the two downstream cylinders. The wake effect from the upstream cylinder could not be captured by the Morison equation because the upstream cylinder was not accounted for in these computations.

Figure 17 presents comparative frequency spectra of load coefficient c_x, derived via a fast Fourier transform (FFT), for medium-diameter cylinders in their tripod configuration, i.e., in the second scenario. The first peak of the coefficients occurred at nearly the same frequency for all cylinders, and this peak was slightly higher on the upstream cylinder for the calculations with the Morison equation. The opposite was noticeable for the downstream cylinders. For the upstream cylinder, the two methods yielded coefficients that agreed more closely with each other than for the two downstream cylinders. The greater difference for the two downstream cylinders occurred because the wake and the associated flow interaction between the two transverse downstream cylinders could not be considered by the Morison equation.

Figure 18 presents comparative time histories of load coefficients for the large-diameter cylinders in their tripod configuration, i.e., in the second scenario. For the upstream cylinder, the impact predicted by both methods matched well. However, this was not so with the load coefficients for the medium-diameter upstream cylinder, see Figure 16. The Morison calculated load coefficients for the downstream cylinders were lower than the URANS coefficients. The temporal progression of coefficients from these two methods matched more closely for the upstream cylinder, which was also so with the load coefficients for the medium-diameter upstream cylinder.

Figure 19 presents comparative frequency spectra of load coefficient c_x, derived via a fast Fourier transform (FFT), for the large-diameter cylinders in their tripod configuration, i.e., in the second scenario. For the upstream cylinder, the first peak from the two methods matched favorably. For the downstream cylinders, coefficients from the URANS computations exceeded those from the Morrison equation. This same tendency was also found for the medium-diameter cylinders; see Figure 17. This trend was expected as the missing upstream cylinder in the calculations with the Morison equation was unable to generate a wake on the downstream cylinders. Again, the flow interaction between the two downstream cylinders could not be captured by the Morison equation, because they were not accounted for in these simulations.

Sarpkaya [3,30] investigated drag and inertia coefficients for single cylinders without wake effects and flow interactions between cylinders being installed in a transverse row. Hildebrandt et al. [13] demonstrated that a wake field and flow interaction between cylinders in tandem and in a transverse row affected drag and inertia coefficients. In a previous investigation of ours [11], this led us to consider using the Morison equation for two cylinders in tandem, where an upstream cylinder created a wake field at the location of the downstream cylinder. As only the upstream cylinder was implemented, the downstream cylinder was considered missing. Then the velocities at the downstream cylinder, caused by the upstream cylinder’s wake field, were used to calculate the loads acting on the downstream cylinder via the Morison equation. It turned out that this method provided favorable results to efficiently assess impact loads on differently sized downstream cylinders.

Figure 20 presents URANS predicted velocities of the flow field surrounding the implemented tripod configuration made up of the medium-diameter cylinders, taken after a simulation time of 12.0002 s. Streamlines represent these flow velocities located on 37 vertical planes. As seen, flow velocities at the sides of all three cylinders were higher than free flow velocities, and they were accelerated in the gap between the downstream cylinders. The contraction of the streamlines in the gap between the downstream cylinders indicates an acceleration of the flow at this location. This was expected because the fluid flow had to pass through a reduced cross-section.

These URANS simulations nicely captured the wake generated behind the upstream cylinder as it influenced the flow past the downstream cylinders. For comparison, Figure 21 presents the simulated streamlines without the presence of the cylinders, taken at the same time. As seen, the flow field was undisturbed as no cylinders were implemented. Of course, the method via the Morison equation could not account for such a phenomenon. The accelerated flow velocities in the gap between the downstream cylinders caused a certain amount of suction, and the influence of this suction was largely responsible for the deviation between hydrodynamic loads acting on the downstream cylinders obtained from URANS simulations and from calculations using the Morison equation.

8. Conclusions and Outlook

An unsteady URANS solver and the Morison equation were used to predict hydrodynamic loads from a single focused regular wave acting on slender vertical cylinders. The first scenario dealt with single cylinders; the second scenario dealt with three cylinders in a tripod configuration similar to an LNG pump tower.

First, for an individually mounted large-diameter cylinder, preliminary comparative URANS simulations were performed using a laminar and a turbulent flow model. As more accurate results were obtained with the turbulent flow model, this model was chosen for all subsequent URANS simulations.

Experimentally measured loads on two single cylinders of different diameters and different immersion depths were compared to loads obtained from the URANS solver and from the Morison equation. The first load peak was captured more accurately on the large-sized cylinder than on the small-sized cylinder. After the load peak, the URANS-based temporal progression of the load acting on only the large-diameter cylinder compared favorably to experimental measurements. The reduced stiffness of the small-diameter cylinder affected load progression after the load peak.

For the tripod configuration, peak loads on the upstream cylinder obtained from the URANS solver were slightly less than peak loads from the Morison equation. The opposite was the case for peak loads on the downstream cylinders.

Of course, only the URANS simulations were able to capture the wake generated behind the upstream cylinder and the flow interaction between the two downstream cylinders of the tripod configuration. Nevertheless, using the Morison equation yielded loads within an acceptable range of accuracy. However, this accuracy depended solely on the choice of appropriate drag and inertia coefficients.

We demonstrated that a dedicated investigation had to be performed to assess hydrodynamic loads acting on single cylinders in a tripod configuration. This paper dealt with wake effects on wave induced loads acting on cylinders of different configurations. Such loads may be useful to assess the structural integrity of LNG pump towers installed in partially filled tanks.