On the Morison hydrodynamic forces on perforated flat plates in combined steady, low frequency and high frequency motion

Journal of Fluids and Structures 81 (2018) 514–527

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On the Morison hydrodynamic forces on perforated flat plates in combined steady, low frequency and high frequency motion H. Santo a, *, P.H. Taylor b,c , C.H.K. Williamson d a

Office of the Deputy President (Research and Technology), National University of Singapore, Singapore 119077, Singapore Faculty of Engineering and Mathematical Sciences, University of Western Australia, Crawley, WA 6009, Australia Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, UK d Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA b c

article

info

Article history: Received 1 March 2018 Received in revised form 11 May 2018 Accepted 24 May 2018

Keywords: Morison fluid loading Wave–current-structure blockage Porous block simulation Fluid–structure interaction

*

Corresponding author. E-mail address: [email protected] (H. Santo).

https://doi.org/10.1016/j.jfluidstructs.2018.05.013 0889-9746/© 2018 Elsevier Ltd. All rights reserved.

a b s t r a c t This paper aims to model the hydrodynamic forces on grids of perforated flat plates undergoing forced motions at three scales, namely steady (current), and combined low (wave) frequency and high (structural) frequency oscillatory motion. The intended application is the design and re-assessment of dynamically-responding offshore platforms. A recent set of experimental results by Santo et al. (2018c) is taken as the reference for comparison with the numerical predictions. A block of porous cells is used as a proxy to the grids of perforated plates in the numerical simulation, but with comparable resistance and added mass represented by equivalent Morison drag and inertia stresses both uniformly distributed over the porous cells. Both stresses are characterised by empirical force coefficients, F ′ and Cm , which correspond to Morison drag, Cd and inertia, Cm coefficients, respectively. Using these two adjustable empirical parameters, the simulated forces compare reasonably well with the measured hydrodynamic forces on the grids, both in terms of peak forces as well as the complete force–time histories for most of the flow conditions tested in the experiments. This is particularly true when the amplitude of wave velocity is larger than that of current velocity, a representation of large waves in a small current which is realistic for the harsh ocean environment. The porous block model is capable of capturing the global large-scale wake structures, which are responsible for the reduction in fluid flow velocity and associated forces on a structure. The simulated forces however only exhibit slight force asymmetry, unlike the measured forces, because the local fine-scale wake structures are not represented in the numerical modelling. For the scale of the experiments used for the comparison, the contribution from these small-scale wake structures to the global hydrodynamic forces can be quite significant, in particular when the amplitude of wave velocity is comparable to that of current velocity. Overall, this paper demonstrates the versatility of the porous block modelling approach in capturing most of the dominant flow physics and reproducing almost all the experimental results. The generality of the approach allows straightforward extension to wider range of flow conditions including three-dimensional flow. © 2018 Elsevier Ltd. All rights reserved.

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1. Introduction Individual members of a space-frame structure acting as an obstacle array help modify the local kinematics. Hence, the use of free-stream (upstream) flow velocity in the Morison equation (Morison et al., 1950) to estimate the hydrodynamic loading on the structure can be (overly) conservative. In offshore engineering, this effect is termed current blockage, as the original intent was to allow for flow and force reduction due to steady flow. An analytical model as an improvement to the Morison equation was first introduced by Taylor (1991). The model was subsequently included in the standard offshore industry practice, such as API (API RP 2A, 2000). Over the years, additional force reduction due to the contribution from waves has been demonstrated and established. The combined effect, wave–current blockage, can be accounted for either using analytical modelling, with some restrictions as the model is valid for regular waves only (Taylor et al., 2013; Santo et al., 2014b), or using numerical Computational Fluid Dynamics (CFD) modelling, which can cover a wider range of practical applications of ocean waves (Santo et al., 2014a, 2015). These models have been extensively validated with both small tests on moving grids and larger scale experiments on a jacket model in a large wave tank, but only for two scales of motion, i.e. waves and current, see Santo et al. (2014b, 2017, 2018b). When the space-frame structure in question moves due to the loads from waves and current, the effect of structural (dynamics) motion in modifying the flow fields needs to be considered. The Morison equation can be extended to include the structural motion, either based upon the concepts of independent flow (absolute velocity) and relative flow fields (relative velocity). Previous studies have been conducted to investigate the validity of the extended Morison equation, see for example Verley (1980), Williamson (1985), Shafiee-Far et al. (1996), Burrows et al. (1997) and Sumer and Fredsøe (2006), however most of the relevant work considered only a single oscillating cylinder in otherwise still water. Our intended application is to space-frame structures in offshore engineering. Santo et al. (2018c) conducted extensive small-scale laboratory experiments on grids of perforated flat plates driven at three scales of motions in otherwise still water. The three-scale motions represent steady (current), and combined low (wave) and high (structural) frequency regular oscillations. The total force exerted on the grids was then used to examine blockage effects. These grids are a highly idealised representation of space-frame (porous) structures, such as those in offshore industry (jackets, compliant towers and jack-up legs). The combined effect, now termed wave–current-structure blockage, can be modelled analytically with some success, but the application is limited to an asymptotic limit whereby the amplitude of wave velocity is much larger than the that of structural and current velocities. Therefore, it is the aim of this paper to model and reproduce the whole range of force measurements of Santo et al. (2018c) using numerical CFD modelling, in order to demonstrate the application of such numerical modelling as a more widely applicable approach for the simulation of hydrodynamic forces on space-frame offshore structures with arbitrary sized wave, current and structural velocities. The applicability of the proposed numerical modelling for this small-scale laboratory experiments tests the building block for the more realistic modelling of a scaled jacket model, which is allowed to respond dynamically when exposed to combined waves and in-line current in a large towing tank experiment, as described in Santo et al. (2018a). 2. Nomenclature u uc uw us aw as fw / Tw fs / Ts Cd Cm A Af V u

ρ p

τ µ S F ′ Cm L

wf Vp

total velocity of the grid as a function of time amplitude of the mean velocity (current) amplitude of wave velocity amplitude of structural vibration velocity amplitude of wave oscillation amplitude of structural vibration frequency/period of the wave-like motion frequency/period of the structural motion Morison drag coefficient Morison inertia coefficient solid drag area of structural elements projected frontal area of structural elements volume of structural elements vector velocity density pressure shear stress dynamic viscosity embedded Morison stress field Forchheimer resistance coefficient equivalent of Morison inertia coefficient downstream width of the block frontal width of the block enclosed volume of porous block

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H. Santo et al. / Journal of Fluids and Structures 81 (2018) 514–527 Table 1 Dimension details for one grid. Parameter

Value

Width of the vertical strips (cm) Width of the horizontal strips (cm) Width of the square holes (cm) Solid area, A (each, cm2 ) Frontal area, Af (cm2 ) Plate thickness (cm) Blockage ratio (A/Af ) Porosity ratio (1 − A/Af )

0.887 0.864 4.936 261 870 0.16 0.30 0.70

Table 2 Experimental parameters with range of us /uw and uc /uw ; uw , us and uc in cm/s. Note that aw is the amplitude of wave oscillation, Tw is the wave oscillation period, as is the amplitude of structural oscillation, and Ts is the structural oscillation period. uc /uw

us /uw

[uw , uc , us ]

Wave oscillation (aw , Tw )

Structural oscillation (as , Ts ) for fs /fw = 2

fs /fw = 2.5

fs /fw = 3

1/4

1/4 1/3 1/4 1/3 1/4 1/3 1/4 1/3

[12, 3, 3] [12, 3, 4] [12, 4, 3] [12, 4, 4] [10, 5, 2.5] [10, 5, 3.3] [7.5, 7.5, 1.9] [7.5, 7.5, 2.5]

(8.6 cm, 4.5 s)

(1.07 cm, 2.25 s) (1.43 cm, 2.25 s) (1.07 cm, 2.25 s) (1.43 cm, 2.25 s) (0.72 cm, 1.8 s) (0.95 cm, 1.8 s) (0.36 cm, 1.2 s) (0.48 cm, 1.2 s)

(0.86 cm, 1.8 s) (1.15 cm, 1.8 s) (0.86 cm, 1.8 s) (1.15 cm, 1.8 s) (0.57 cm, 1.44 s) (0.76 cm, 1.44 s) (0.29 cm, 0.96 s) (0.38 cm, 0.96 s)

(0.72 cm, 1.5 s) (0.95 cm, 1.5 s) (0.72 cm, 1.5 s) (0.95 cm, 1.5 s) (0.48 cm, 1.2 s) (0.63 cm, 1.2 s) (0.24 cm, 0.8 s) (0.32 cm, 0.8 s)

1/3 1/2 1

(8.6 cm, 4.5 s) (5.7 cm, 3.6 s) (2.9 cm, 2.4 s)

3. Brief description of the experiments In the physical experiments, grids of perforated flat plates were towed and oscillated in otherwise still water in a computer-controlled XY towing tank in Cornell University. The tank is ∼ 6 m long, 1 m wide and 1 m deep. The water depth was set at 0.57 m. Previously, Santo et al. (2014b) conducted a series of tests involving two scales of forced motions using a combination of one to three grids, and recently Santo et al. (2018c) conducted another series of tests involving three scales of forced motions using just three grids. For each series of tests, the same set up was maintained, and this is illustrated in Fig. 1(a), which is taken from Fig. 2 in Santo et al. (2014b). The details of the grids are shown in Fig. 1(b) and Table 1. Each grid is 0.3 m wide and the submerged height to the water surface is 0.29 m. The total spacing of the three grids is 0.3 m along the direction of imposed motion. The motions of the grids were programmed and controlled through a LabView interface on a PC. The resultant total force on the grids were measured by a force transducer with a sampling rate of 1000 Hz. The force signals were subsequently low-pass filtered at 12× the frequency of the oscillatory wave motion as post-processing in MATLAB. For steady motion, the filtered forces are presented in the paper for comparison with the numerical predictions. For unsteady motion, the filtered forces are first averaged cycle-by-cycle over ∼ 10 cycles when the force signals are in steady-state (simply periodic — repeating every wave cycle or every two cycles), ignoring the starting and ending transients, before being used for comparison with numerical predictions. The experimental variability in the force signal cycle-by-cycle is discussed in Section 4. Three scales of imposed motion are considered, which represent steady current, wave and structural oscillation. The motion is defined as u = uc + uw cos(2π fw t) + us cos(2π fs t). We provide a nomenclature for the symbols in Section 2 of the paper. uw is the amplitude of the velocity component representing wave motion, uc is the steady velocity representing the current, and us is the amplitude of the high frequency component representing the structural vibration. Table 2 presents a list of the range of motions tested. The amplitude ratio of structural to wave velocity, us /uw , is varied from 1/4 − 1/3, and structural to wave frequency ratio, fs /fw is kept at 2, 2.5 and 3× so as to represent higher frequency structural vibrations. A range of amplitude ratio of current to wave velocity is considered, uc /uw = 1/4 − 1, hence examining a wide range of wave–current-structure interaction. Several motion combinations within a choice of uw /us and fs /fw ratios are made possible by introducing a discrete ±π/2 and π phase shift to us (or uw ), and the combinations are listed in Table 3 and plotted in Fig. 2, where F2 denotes fs /fw = 2 and so on. All the oscillations are sinusoidal in time, while the steady motion is simply constant in time. Throughout the paper, [uw , uc , us ] (in cm/s) naming convention as well as the labelling for each fs /fw (such as F2A etc.) for the relative phase between us and uw will be used to describe each case. Although the scale of our experiments is very small compared to those in the field for offshore structures, we have aimed to set the relative sizes of the imposed velocity components at appropriate values. The combinations of structural to wave frequency are chosen to be appropriate for dynamically sensitive offshore structures in deeper water, such as the first mode of deep-water jackets and the second mode of compliant towers. A 15 s wave and a 5 s structural period would be appropriate, and combining with a typical structural vibration of ∼ 1 member diameter, this results in a chosen combination of fs /fw and

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Fig. 1. (a) Schematic diagram of the XY towing tank. (b) Plan view of the different grid arrangement. (c) Elevation view of the grid.

Table 3 Motion combinations for each fs /fw . Note that F2 denotes fs /fw = 2 and so on. Label

Motion

F2A, F2.5A & F3A F2B & F3B F2C, F2.5C & F3C F2D & F3D

uw cos(2π fw t) + us cos(2π fs t) uw cos(2π fw t) − us cos(2π fs t) uw cos(2π fw t) − us sin(2π fs t) uw cos(2π fw t) + us sin(2π fs t)

us /uw . For large waves, the horizontal wave kinematics could be 8 m/s, so 4 − 8× the local current of 1 m/s. With the same current, smaller waves could have fluid velocities ∼ 1× the current, hence the combination of uc /uw = 1/4 − 1 is chosen. 4. Numerical modelling Here a brief description of numerical Computational Fluid Dynamics (CFD) modelling using open-source package R OpenFOAM⃝ (www.openfoam.org) is presented. Previous studies by Santo et al. (2014a, 2015, 2017) treated a space-frame structure as a porous block and found some success in reproducing the measured peak forces as well as complete force–time histories. In this paper, we apply a similar modelling approach by representing the grids of perforated flat plates as a porous

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Fig. 2. Plots of all motion combinations in terms of velocity–time history for an example case of [12, 4, 4].

block with the same amount of resistance. Comparable work has been conducted on characterising resistance based on drag and Morison equations in related fields, see e.g. Kristiansen and Faltinsen (2012), Zhao et al. (2013) and Chen and Christensen (2016). We use a similar numerical setup to the two-dimensional forced motions described in Section 5.2 of Santo et al. (2014a). The flow is assumed to be in the x−direction with the plane of the grids in the y−direction. A porous block with uniformly distributed embedded Morison stresses (forces distributed over the block but using the local flow kinematics, so accounting for the global presence of the structure, see Eq. (2)) is used to model the grids of perforated flat plates. To account for the effect of the porous block, the conventional Navier–Stokes momentum equation within the block is modified by adding a sink term as follows:

∂ ∂ ∂p ∂τ (ρ u) + u (ρ u) = − +µ −S ∂t ∂x ∂x ∂x

(1)

where u is the fluid velocity, p is the fluid pressure, τ is the shear stress, µ is the dynamic viscosity. In the case of a simple homogeneous porous block, the sink term can be written as: S=

1 2

ρ F u|u| + Cm′

∂ρ u ∂t

(2)

′ where F is the Forchheimer resistance parameter and Cm is the equivalent of the local Morison inertia coefficient, Cm , but ′ here defined in the porous block context. Both F and Cm are related to the Morison drag and inertia coefficients through ′ Cd A/Af = FL and Cm = Cm V /VP , where A and Af are the solid drag area and the frontal area of the grids respectively, L is the downstream length of the porous block, V is the displaced volume of the elements in the grids, and VP is the volume of the porous block. The simulated force on the porous block is obtained during post-processing by volume integration using the local velocity and acceleration within each computational porous cell according to Eq. (2). A square porous block is chosen to represent grids of 1, 2 and 3 perforated plates in the 2D numerical simulation, with the length and width of 0.3 m, and thickness of 0.1 m (as required by OpenFOAM although there is no flow in the thirddimension). The best fit Cd for the asymptotic test cases (when uc /uw ≤ 1/3) described in Santo et al. (2018c) is found to be within the range of 2.4 − 2.6, which corresponds to F ∼ 7.5 m−1 in the numerical model. This value is slightly higher than expected for a rectangular bar in isolation. The best fit Cm for the same test cases in Santo et al. (2018c) is within the range ′ of 9.5 − 13, which corresponds to Cm ∼ 0.15. It is worth stressing that the numerical porous block does not include any representation of the details of the individual bars of the plates, hence small-scale vorticity/eddies which scale as the width of the individual bars are not present in the simulation. It is also worth stressing that in our porous body modelling we completely remove the effect of finite volume of solid, and replace the solid body with a distributed stress over the enclosed volume. Therefore, our model does not account for the physical volume of the structural elements within numerical computational cells (no ‘pore velocity representation’), nor local flow amplification due to the presence of other physical members. The reduced velocity obtainable from Eqs. (1) and (2) represents the mean velocity over the projected frontal area of the structure modified by global effect of the structure as obstacles (blockage). The pressure–velocity coupling is solved with the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) iterative algorithm, see Patankar (1980) and Patankar and Spalding (1972). In the fully unsteady flow simulation, an implicit Euler time stepping is used for the unsteady term (Ferziger and Perić, 2002). We use the k − ω Reynolds-Averaged Navier–Stokes

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Fig. 3. Comparison of simulated drag forces from 2D and 3D computational domains.

Fig. 4. Layout of the 2D computational domain, with wf = 0.3 m. x-axis is pointing to the right, y-axis is pointing up, and z-axis is pointing out of the figure.

(RANS) turbulence model (Wilcox, 1988), where k is the turbulence kinetic energy, and ω is the specific dissipation rate, with the standard values of the closure coefficients and auxiliary relations. The initial values of k and ω are related to the turbulent kinetic energy, I, and the turbulent mixing length, Lt , respectively. In this work, I = 2.5% and Lt = 0.07 × wf are chosen as the initial estimates, where wf = 0.3 m is the frontal width of the grid. Rather than moving the porous block in the numerical simulation, we impose forced oscillation of the flow on a stationary porous block instead. The boundary condition for velocity at both inlet and outlet is the imposed velocity–time history used in the physical tests, and the inlet/outlet switches according to the direction of the net superimposed oscillating flow. A ∂ p/∂ n = 0 boundary condition for pressure is applied to both inlet and outlet. Preliminary assessment is performed to compare the simulated forces from two- and three-dimensional computational domains, with the same lateral and vertical widths as the actual XY towing tank. Fig. 3 provides the comparison in terms of the drag force. The drag force of the 2D simulation (dashed red line) is observed to be larger than that of the 3D simulation (solid grey line). This is mainly due to additional vertical flow divergence in the 3D simulation which slightly increases the otherwise reduced velocity in the porous block. The difference is expected because the geometric blockage ratio (wind tunnel blockage ratio) is different, defined as the ratio of the frontal area of the grids to cross sectional area of the tank. Since the cross-sectional area of the flow covered by the grids to the entire flow cross-sectional area (the physical wind tunnel blockage ratio) is ∼ 0.15, to obtain the same wind tunnel blockage ratio, the width of the 2D computational domain needs to be doubled, i.e. from 1 m to 2 m. We believe this wind tunnel effect is dominated by the area ratio, not how the area of the grids is arranged. The simulated drag force is also plotted in the same figure (solid black line), which now shows that the results of 3D and extended 2D simulations are quite similar. There is no noticeable difference in the inertia force so it is not shown on the same plot. 2D simulations are generally more efficient, and so to reduce computational effort, 2D simulation is preferred and will be used throughout the subsequent analysis. The layout of the 2D computational domain is shown in Fig. 4. The size of the computational cells is kept uniform at 5 × 10−3 m in both horizontal (x) and vertical direction (y). Preliminary studies have been conducted to assess the sensitivity of the numerical domain, porous block layout and mesh convergence. Simulation of three thin grids of block is carried out to investigate the influence of having just a single porous block. The influence of the chosen upstream and downstream distances from the porous block to the numerical boundary (inlet and outlet) is investigated by comparing to a domain with larger distances. A mesh convergence test is also performed by comparing the resultant force components with the same domain layout but with twice and half the number of

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Fig. 5. Force comparison due to steady flow of uc = 10 cm/s for different grid configurations and blockage ratio of 0.3. Solid lines are the measurements of Santo et al. (2014b), dashed lines are the numerical predictions. The F resistance parameter for 1, 2 and 3 grids is 1.75, 3.50 and 5.25 m−1 , respectively, ′ while the Cm parameter for 1, 2 and 3 grids is 0.02, 0.04 and 0.06, respectively.

computational cells. Taken overall, the simulated forces are found to be of negligible difference, confirming that the present configuration is robust. Instead of simulating forced oscillation at three scales, one can also embed the structural oscillation us into the sink term of Eqs. (1) and (2). This embedding process will avoid having to oscillate the computational mesh dynamically when solving for a more realistic 3D-type of problem involving for example a deepwater offshore jacket with free-surface variations (surface gravity waves). Thus, the simulation can still be performed in a numerical wave tank with a static porous tower representing a space-frame structure with an additional embedded time-varying (and depth-varying) local stresses representing the local oscillatory structural velocity and acceleration. This method of embedding has been recently demonstrated by comparison with experiments involving a realistic scaled jacket model (Santo et al., 2018a). 5. Results and discussions Here we present selected results for both steady and unsteady flow comparison. The steady tow motions in the physical experiments were preceded by a smooth ramp up in the beginning, followed by a steady tow, and lastly by a smooth ramp down towards the end. Thus, the force measurements would contain two distinct components, i.e. the force transients (starting and ending) as well as the steady mean force. It is of interest to reproduce numerically the two distinct components of the measured forces. Fig. 5 shows the force comparison due to steady flow between measurements (solid lines) and numerical predictions (dashed lines) for different grid configurations. For the numerical predictions, the F resistance parameters for 1, 2 and 3 grids ′ ′ are 1.75, 3.50 and 5.25 m−1 , respectively, and Cm are set at 0.02, 0.04 and 0.06, respectively, with both F and Cm increasing with the number of grids. Since the numerical porous block has a fixed aspect ratio, the numerical simulation is not able to distinguish between 2A and 2B grid configurations, of which the mean force of 2B is slightly larger than that of 2A due to slightly enhanced lateral flow mixing, given that the grid distance for 2B is twice the distance for 2A. Overall, reasonably good agreement in terms of mean forces can be obtained from the porous block simulation using a combination of F ′ and Cm . The force transients, however, are consistently under-predicted, due to the absence of local fine-scale wake structures in the numerical simulation. The initial force transient (when time < 10 s) in general represents a build-up of the global wake (steady flow blockage) as the otherwise stationary water begins to accelerate to flow past the grids in the opposite direction of the moving grid. Even so the local wake effect also appears to be important. Thus, the numerical modelling only picks up only a small portion of the initial transient. Likewise for the ending transient (when time > 40 s), the moving grid comes to rest eventually and the ambient water reverses direction and starts to flow forward across the grid in the downstream direction instead (hence negative force recorded). What the forward flow constitutes is actually the flow generated by the vorticity previously steadily shed in the upstream direction during the steady tow and as the grid stops, the vorticity is swept downstream across the grid creating the forward flow through the grid. Since in the numerical simulation only the large-scale vorticity is being shed (global wake structures which scale as the frontal width of the grid), the net effect is smaller. Moreover, when the grid stops moving, the global wake behind the grid also vanishes, hence the net force (ending force transient) on the grid is smaller. In the physical experiment, the small-scale vorticity (or local fine-scale wake structures which scale as individual bar width) is as intense as the large-scale vorticity, and the intensity increases as the number of grids increases. As a result, the induced forward flow contributes to a larger (negative) force, of which the magnitude is comparable to the total force on the grid because of the (small) scale of the experiment.

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We are using a separation of scales argument to separate the bulk wake structure, which is resolved by our simulations, from the numerically unresolved small-scale individual wakes behind each bar within each grid. In an effort to introduce a crude model for these bar wakes and the local eddy structure that will result, we examined the possibility of simulating such small-scale wake structures by injecting additional small obstacle-induced turbulence, following the blade-induced k − ϵ turbulence model of Nishino and Willden (2012). Apart from enhancing turbulent flow mixing inside the porous block, the overall effect on the force transients is not significant. It is thus apparent that the only way to capture the required force transients at the start of the motion would be by representing the individual bars in each grid in the numerical simulation. Figs. 6–9 show a series of comparisons of force–time history for all ranges of uc /uw between measurements (grey lines) ′ and numerical predictions (black lines). In all cases, Cm = 0.05 is used, and F is allowed to vary over the range 7.3 − 8 m−1 (Cd = 2.43 − 2.67, a 10% variation) depending on uc /uw , while within the same uc /uw , virtually the same F can be used ′ across all different ratios of fs /fw , us /uw and the motion combinations. It can be observed that with one fixed parameter Cm and one slightly adjustable parameter F , the numerical predictions of the force on the grids agree reasonably well with the measurements, in particular for the cases of uc /uw < 1. Better agreement can be obtained by searching for the best fit of ′ F and Cm for each case. However, this is not done as the aim of the paper is to demonstrate the applicability of the porous block method in capturing most of the dominant flow physics. The agreement becomes poorer for the cases of uc /uw = 1, in that most of the numerical predictions under-predict the (negative) force troughs, and there is a slight phase shift in some of the force time histories. When uc /uw = 1, the amplitude of wave and current velocity effectively cancels out in some parts of the wave oscillation cycle to form a flat motion plateau, and when superimposed with small us oscillation, there is very little reverse oscillatory motion. In the numerical simulation, the large-scale vorticity previously shed during the forward oscillatory motion contributes to small negative forces as vorticity is washed upstream during the small reverse oscillation. The measurements however exhibit larger negative forces which must be due to the contribution from the small-scale vorticity that are not represented in the numerical simulations. To indicate the amount of variability in the measured forces, thin grey lines are added to selected figures. These represent the mean signal ±2 standard deviations of the variation of the individual signals from the mean, taken over ∼ 10 cycles. Good repeatability is observed based on the tight band of the standard deviation. For the selected figures containing the measure of variability, the average root mean square error between the individual and the averaged force is ∼ 0.03 N, and the average normalised root mean square error, defined as the root mean square divided by the range of the measured force for each case, is ∼ 2%. Overall, the reasonable agreement demonstrates the versatility of the numerical porous block modelling approach in reproducing the measured forces by capturing most of the required flow physics, as well as the generality of the approach in terms of wider range of flow conditions (without being constrained to asymptotic flow regime only) as well as extension to 3D-type of problems. It is worth remarking that, in the absence of the analytical modelling which is limited to asymptotic flow regime (when uc /uw ≤ 1/3) as described in Santo et al. (2018c), and sophisticated numerical simulation resolving individual bars and grids, the numerical approximation using a porous block approach is the best available tool to model the complete hydrodynamic forces on obstacle arrays with arbitrary sized wave, current and structural velocities. We note also the asymptotic regime (uw /uc large), but with irregular waves, is appropriate for describing extreme wave loads on offshore platforms − the ultimate aim of this work. In the experiments modelled here, the free-surface motion induced by the grid motion is very small, typically ±2 mm. The grids were arranged so the undisturbed free-surface was across the middle of the line of holes in the plate. And even at maximum motion, the free-surface did not hit a line of horizontal bars. This explains why the free-surface motion was so small in all the experiments. The vertical bars penetrating the free-surface act as very poor wavemakers. Of course for real waves, there is substantial free-surface motion and vertical as well as horizontal kinematic components. Our previous work shows excellent agreement between forces on a jacket model in a large wave tank and the porous block predictions in a numerical wave tank (Santo et al., 2018b). 5.1. On the use of an approximate expanded form of the Morison relative-velocity term We have demonstrated the applicability of the porous block approach for simulating forces on obstacle grids in complex motion. In practical applications, the motion arises from three sources: incident waves (here modelled by uw ), a steady current (uc ), and the structural vibration (us ). The first two are incident onto the structure and independent of it. The structural motion depends on the unsteady hydrodynamic loading and the structural stiffness, and typically the hydrodynamic modelling will be based on the Morison with relative velocity formulation. Hence, to account for this in offshore engineering applications, it would be necessary to dynamically couple the hydrodynamic simulation to a structural dynamics code (for example OpenFOAM to USFOS, a frame analysis package widely used in the offshore industry). This would be possible in principle, but awkward and computationally demanding. In an effort to split the hydrodynamics from the structural motion simulations, we explore an approximation for the Morison-relative-velocity force equation, still within the porous block formulation.

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Fig. 6. Force–time history comparison due to unsteady flow of uc /uw = 1/4 for (a) [12, 3, 4] and (b) [12, 3, 3] between measurement (grey lines) and ′ numerical predictions (red lines). For all cases, F = 7.3 m−1 and Cm = 0.05 is used. The thin grey lines represent error bars (±2 standard deviation for the variation of the mean of the measured force).

Following Haritos (2007) and Merz et al. (2009), an approximation for the Morison with relative velocity formulation in the drag force is possible for the case when us is small compared to uwc = uw + uc . This has the form: FD =

= ≈ ≈

1 2 1 2 1 2 1 2

⏐ ( )⏐ ρ Cd A uwc − us ⏐uwc − us ⏐ ( ) ρ Cd A uwc |uwc | − 2|uwc |us + u2s ( ) ρ Cd A uwc |uwc | − 2|uwc |us ρ Cd Auwc |uwc | − ρ Cd A|uwc |us

This approximation can be implemented into the porous block model via the Morison stress term written in terms of local disturbed velocity, u, as: SD =

≈

1 2 1 2

ρ F (u − us )|u − us | ρ F u|u| − ρ F |u|us

(3)

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Fig. 7. Force–time history comparison due to unsteady flow of uc /uw = 1/3 for (a) [12, 4, 4] and (b) [12, 4, 3] between measurement (grey lines) and ′ numerical predictions (red lines). For all cases, F = 7.5 m−1 and Cm = 0.05 is used.

where us is the imposed motion of the structural vibration. The first term in the expanded form of the drag force in Eq. (3) is the applied hydrodynamic force on a structure due to forced oscillations at two scales, namely due to regular oscillations and steady flow. This serves as an input to the forcing term in a structural dynamics code representing hydrodynamic force assuming the structure is very rigidly supported (statically-responding structure). The second term is the product of the disturbed flow kinematics with the structural oscillation, which constitutes to an additional applied force reduction due to structural dynamics and can be regarded as an additional damping from the relative velocity contribution as discussed in Santo et al. (2018a). This second term is now de-coupled and can be included into a structural dynamics code as a slight modification to the equation of motion. Therefore, this term needs to be stored for post-processing using the same structural code and forcing input to solve for the structural dynamics, and this is applicable for any space-frame structures allowed to freely respond dynamically. In this paper, we simulate forced oscillations at two scales and obtain the simulated force on the porous block. Because in the experiments the structural oscillation was forced (or imposed), the disturbed flow kinematics is subsequently post-processed according to Eq. (3) together with the imposed structural oscillation to produce the force prediction for all cases. The predicted force resulting from the Haritos and Merz simplification is then compared with the simulated force from forced oscillations at three scales for a range of flow cases as shown in Fig. 10. In general the agreement is reasonably good, which demonstrates the suitability of the approximate expanded form for use in engineering applications.

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Fig. 8. Force–time history comparison due to unsteady flow of uc /uw = 1/2 for (a) [10, 5, 3.3] and (b) [10, 5, 2.5] between measurement (grey lines) and ′ numerical predictions (red lines). For all cases, F = 7.75 m−1 and Cm = 0.05 is used.

6. Conclusions This paper has presented a numerical modelling method to simulate hydrodynamic forces on grids of perforated flat plates. Forced motions at three scales are simulated to represent steady (current) and combined low (wave) and high (structural) frequency regular oscillations, the aim being to investigate fluid–structure interaction in the context of wave–current-structure blockage. A porous block with comparable amount of resistance and added mass, represented by F ′ and Cm , respectively, is used as a proxy to the physical grids. This is based on the assumption of separation of length scale effect, in which only the large-scale wake flow structures, which scale as the frontal width of the grids, are assumed to be more dominant in capturing the blockage effect and thus more important than the small-scale individual obstacle wake structures. Hence, using the porous block model, only large-scale wake flow structures are resolved numerically, while the small-scale individual obstacle wake features are not represented. However, it is observed that for the scale of the experiments considered in this comparison, the contribution from the small-scale structures to the global hydrodynamic forces can be quite significant, which weakens the validity of the separation of length scale assumption. ′ Overall, with only two adjustable empirical parameters of F and Cm , reasonably good agreement in terms of complete force time histories between measurements and numerical predictions can be obtained for most of the flow conditions tested in the experiments. This is particularly true for the cases when uc /uw < 1, which represents large waves in a small current - a harsh offshore environmental setting. This demonstrates the applicability of the porous block method in capturing most of the dominant flow physics. The significance of the small-scale wake structures becomes more apparent when uc /uw → 1, in that the small scale vorticity contributes to larger (negative) force troughs, while the force peaks can be reasonably well

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Fig. 9. Force–time history comparison due to unsteady flow of uc /uw = 1 for (a) [7.5, 7.5, 2.5] and (b) [7.5, 7.5, 1.9] between measurement (grey lines) and ′ numerical predictions (red lines). For all cases, F = 8 m−1 and Cm = 0.05 is used. The thin grey lines represent error bars (±2 standard deviation for the variation of the mean of the measured force).

reproduced numerically. As a result, the numerical results tend to under-predict the force troughs when the adjustable parameters are chosen to fit the force peaks. For forced motions at three scales, the use of Morison with relative velocity formulation implies a coupling between structural velocity and the fluid velocity. Nevertheless, the approximate expanded form following Haritos (2007) and Merz et al. (2009) can be used so long as the structural velocity is small relative to the ambient fluid velocity. The adequacy of the expanded form is investigated, by using the simulated forces from forced motions at two scales which is then post-processed together with the disturbed flow kinematics to yield the predicted force. Comparison between the predicted force and the simulated force from forced motions at three scales show reasonably good agreement, demonstrating the adequacy of the expanded term for all cases. In reality, the predicted force can be obtained by time-marching an equation of motion for the structural dynamics with the additional damping term from the Morison relative-velocity contribution. This approach is potentially useful as it allows a de-coupling between a fluid loading solver and a structural dynamics solver to solve for any realistic dynamically-responding structures under combined waves and current loading, as demonstrated in Santo et al. (2018a). It is worth remarking that, in the absence of the analytical modelling as described in Santo et al. (2018c) which is limited to uc /uw ≪ 1, as well as sophisticated numerical simulation resolving individual bars and grids, the proposed numerical approximation using a porous block approach is the best available tool to model the hydrodynamic forces on obstacle arrays, which in this case subjected to forced motions at three scales: steady and combined low frequency and high frequency regular oscillations.

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Fig. 10. Force–time history comparison between forced oscillations at three scales (grey lines) and forced oscillations at two scales combined with the expanded form of the Morison relative-velocity term (red lines) for a range of cases. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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