1. Introduction
Nowadays, horizontal flexible floating structures are becoming attractive and interesting, as are varieties of offshore and coastal structures in contrast to vertical designs that can be utilised in the sea environment for multi-use model applications such as breakwater [
1,
2] and wave energy converters. Recent studies have concentrated on using flexible floating or submerged structures in breakwaters and wave energy converters [
3,
4,
5]. The construction of these structures is economical when the water depth is significant and shielded from seismic activity, does not harm the marine environment or impede ocean currents, and can be removed or expanded effortlessly and, therefore, is environmentally sustainable.
Another impressive kind of flexible structure is the membrane structure, which has applications in wave energy conversion. This type of structure has become beautiful when using wave energy converters based on model tests, numerical tools, or theoretical approaches. For instance, the flexible-membrane wave energy converter was studied using the videogrammetric technique based on model tests [
6]. A comprehensive experiment was carried out to harness the power from an undulating membrane using linear electromagnetic generators [
7]. Further, the dynamic behaviour in terms of the membrane’s dynamic profile and hydrodynamic forces of an undulating tidal energy converter was assessed while subjected to wave and current forces based on a small prototype undulating membrane test [
8]. A thorough approach was suggested for analysing the dynamic behaviour of structural deformation, and the outcomes were compared with the existing published physical model test results [
9].
A rigorous review was carried out of the state-of-the-art on flexible membranes for harvesting wave energy, encompassing various computational modelling techniques. [
10]. The interaction of waves and flexible membranes connected with spring moorings in 3D was developed via theoretical tests and compared with experimental model tests with no current case [
11]. Through linear analysis utilising a straightforward theoretical model and model testing, a WEC (Wave Energy Converter) made up of two submerged, bottom-fixed, air-filled cylindrical chambers separated by a distance of roughly half a wavelength and capped with flexible membranes on top was examined along with an investigation into the stability of the flexible membranes [
12].
Theoretical models utilised submerged horizontal flexible membranes as breakwaters and anti-motion structures to mitigate the structural response of floating elastic structures. The orthogonal mode-coupling condition was applied to analyse the influence of a submerged flexible membrane on a floating plate in 3D [
3]. The relationship between floating membranes and ocean waves was studied utilising a hydro-viscoelastic model built on a monolithic finite element approach with linearised potential flow and viscoelastic membrane equations in 2D [
13]. A coupled mode system was developed to study the wave–current–seabed interaction on the shear currents in variable bathymetry regions [
14].
Introduced a novel numerical model capable of predicting the performance of floating-membrane solar islands based on linear potential flow theory principles. It was aimed at examining the various vertical motion modes displayed [
15]. A comprehensive approach was introduced to study the hydroelastic behaviour of membrane-based floating PV platforms using coupled dynamic equations [
16]. The theoretical solutions advanced for evaluating simplified membrane-type structures have been successfully applied in practice in the initial design analysis of membrane-type PV farms [
17]. The hydrodynamic behaviour of a floating PV farm was performed based on full-scale experiment tests and numerical simulation subjected to marine environmental conditions [
18].
Significant and influential works related to WEC are associated with floating structures connected with mooring lines, such as floating platforms supported by wind turbines especially based on experimental and theoretical ones discussed. For example, subjected to regular and irregular waves, the hydrodynamic responses of a floating spar wind turbine were obtained, where experimental techniques were employed to measure the four mooring lines, and the results of RAO were validated with numerical simulation [
19]. A spar buoy wind turbine’s model motion and mooring loads were analysed by conducting physical model tests [
20]. The dynamic properties of a spar platform that is linked with mooring lines were conducted based on the surrogate model generation approach [
21] under the action of the current. The dynamic response via hydrodynamic loads of a platform wind turbine and a WEC were calculated based on numerical simulation, and the results were compared with model test results [
22]. The hydrodynamic response of FOWT and the tensions between the two various mooring systems were analysed by applying the BEM (Boundary Element Method) theory, which indicated that the mooring system can significantly diminish the dynamic response of floating wind turbine platforms [
23]. The response is dynamic through peak movements, and RAO response under ranges of wave parameters was predicted based on experiments [
24]. The dynamic characteristics of a FOWT was examined through theoretical analysis and model testing in which the effects of second-order doubling and third-order tripling waves significantly influence both shear force at the tower top and tension in the mooring lines [
25].
While advancements in flexible-membrane structure design have occurred, the effect of currents on a 3D analytical model associated with a horizontal floating flexible membrane (HFFM) connected with spring moorings remains absent. This model offers a valuable understanding of the intricate connection between currents, hydrodynamic behaviour, and structural stability of these membranes. Therefore, this work aims to develop a 3D mathematical model that operates under current loads as well as to provide its analytical solutions and provide significant insights into the present literature.
Section 2 presents the mathematical formulation of the referenced model under linearised water wave theory in three dimensions in the presence of current loads, in alignment with this study’s objective.
Section 3 utilises techniques involving Green’s function to obtain the dispersion relation in the presence of currents in different water depths. The derived dispersion relation is then utilised in the analytical solution by applying the velocity decomposition method. A brief description of the experiment is presented in
Section 4.
Section 5 compares the analytical results for reflection and transmission coefficients with experimental model results available in recent publications. Additionally, the effects of varying structural and environmental parameters on membrane deflection and vertical force are examined through numerical simulations. In addition, the impact of current loads and environmental depths on the phase and group velocities are studied.
Section 6 provides a concluding overview of this study’s findings and identifies promising avenues for further investigation.
2. Model Formulation
The mathematical modelling of the considered problem is based on small-amplitude linearised water wave theory and structural response in three dimensions, with the horizontal plane and the y-axis being the vertical downward positive direction in finite water depth. Under the assumption of linearised theory, the velocity of the water particle, displacement, and their derivatives are small quantities. Moreover, the current flow is assumed to be much smaller than wave components. The flexible membrane is modelled as a thin, inextensible sheet that acts under tension at its interface. The two-dimensional string equation governs the behaviour of the flexible membrane, assuming it is made of a material with constant and uniform density.
The floating flexible membrane of finite length
and finite width
occupies the region
on the mean free surface
over an impermeable sea bed
making an angle
θ along the direction of wave propagation in the positive direction of the
x- and z-axis referred to as the following current; hence, the components of the current are
, and the floating flexible membrane is connected with spring moorings with stiffness
for
j = 1, 2 at the edges of the horizontal flexible membrane (see
Figure 1).
Hence, the whole fluid domain can be designated as
Under the assumption of linear water wave theory, it is assumed that a progressive wave with angular frequency
is incident on the flexible membrane with the positive
x-axis. It is assumed that the fluid is inviscid and incompressible and that the motion is irrotational. The wave–current flow in a channel is restricted by vertical walls at
that results in the reflection of current flow by the side walls and could be confined to
. Therefore, the total velocity potential is defined as
such that
where
, and Re is the real part of the complex velocity potential where
. Further, the displacement of the floating flexible membrane is presumed to be of the form
.
is the complex spatial velocity potential that satisfies the 3D Laplace equation,
where
denotes the Laplacian operator.
The linearised kinematic and dynamic conditions at
in the presence of current speed
are read as
By combining the above dynamic and kinematic linearised conditions (3) and (4), the linear form of the free surface boundary condition can be obtained as
As the bottom is rigid, the water bottom boundary conditions in finite water depth (FWD) and infinite water depth (IWD) are given by
Now, the dynamic condition on the HFFM satisfies
where the hydrodynamic pressure
in the presence of current is given by
Combining Equations (7) and (8), the dynamic condition on the HFFM
in the presence of current is obtained as
where the subscripts ‘
x’, ‘
z’, and ‘
t’ refer to the partial derivatives with respect to
x,
z, and
t, respectively. Further,
and
are the tension acting on the flexible membrane in
x- and
z-directions.
Eliminating the deflection
from Equations (3) and (9), the membrane-covered boundary condition under the influence of the following current is determined as
Assuming that uniform tension acts on the membrane along
x- and
z-directions, that is,
, condition (8) can be re-written as
where
,
, and
σ,
ρm, and
d are the tension, density, and thickness of the HFFM.
Considering that the side walls of the channel are rigid, that yields
The HFFM is free to the walls of the channel
at
; thus, at the edges of the HFFM, the restoring forces in the spring must be equal to the transverse components of the tension, which gives
To mitigate structural damage during storms, the floating flexible membrane is secured with mooring lines to withstand the force of strong waves and currents, it is assumed that the floating flexible membrane is moored at
with springs of stiffness
,
to the bottom, which yields
The mathematical problem under consideration requires the application of continuity conditions to solve at hand; specifically, deflection and slope of deflection are assumed to be continuous at
along
, respectively, yielding (as in [
3])
Further, the continuity of pressure and velocity for
give
where
.
In the end, the far-field radiation condition for surface gravity wave interaction with an HFFM in 3D is of the form
where
and
m refers to the mode of oscillations, which is
as the incident wave contains many directions (harmonic incident wave propagating along the x-axis of the channel and also additional transverse components corresponding to
) in the channel;
is the incident wave amplitude; and
and
are the amplitudes of the waves associated with the
m-th modes of oscillation of reflected and transmitted waves, respectively. Further,
, where
, and
satisfies the gravity wave dispersion relation in the presence of currents as
.
NB: The formulation is notable for accommodating both following and opposing wave–current conditions; however, the results are presented on the following currents. This is beneficial because wave blocking, which could occur in opposing current scenarios, is a condition to be avoided.
5. Numerical Results and Discussion
This section compares the reflection and transmission coefficients calculated from the current analytical solutions with experimental data from [
11], which did not account for the current. Secondly, the deflection of floating flexible membranes on different design parameters is presented, and the results are analysed. Thirdly, the wave forces on the moored floating flexible membrane for different membrane widths, modes of oscillations, CSs, and membrane tension are analysed. Finally, the phase and group velocities on the effect of different CSs are also analysed. Here, the results for the opposite current case are avoided as not considered in the formulations due to the case of wave blocking.
Hereafter, all numerical computations were executed based on an analytical solution by considering fluid density and acceleration due to gravity , and the series is truncated to as the series attends convergence for unless mentioned otherwise. MATLAB R2016b, 64-bit (win64) was utilised to carry out computations, and all numerical computations from the analytical solution were performed on a desktop machine with Intel® core i7-4790 CPU with 3.60 GHz CPU (Intel (American multinational corporation and technology, Santa Clara, CA, USA)) and 16 GB of memory. On average, each case took roughly 8–10 min to complete the simulation.
5.1. Comparison Results and Analysis
To check the level of accuracy, in this subsection, the present analytical results of reflection and transmission coefficients on different spring stiffnesses are compared against the experimental model test results with no current [
11] versus wave period
.
In
Figure 2 and
Figure 3, for different spring stiffnesses, it is observed that the trends of
and
are similar, and almost all points of the
analytical results agree well with experimental data sets. The higher reflection in Cr from the analytical solution is observed for the lower wave period because of the CS in the present model, leading to elevated reflection upstream. A higher reflection coefficient implies that the HFFM breakwater offers better wave protection and reflection for regular waves with a shorter peak period in real physical applications.
In
Figure 2, when
T 2.4, the analytical result for
Cr converged with the experimental model data sets [
11], whilst the
Ct values were very close, but at a specific wave period, it also converged with the experimental model presented in reference [
11]. However, for the lower wave period, the discrepancy arises from the exceptionally low tension of the membrane materials used [
11]. This low tension suggests that the membrane’s tension reduced the efficiency of reflecting incident waves, particularly for higher wave periods, resulting in increased transmission.
Furthermore, in
Figure 3,
Cr from the model [
11] is 0.09 m, while that from the present model is 0.0668 m, which is 34% lower. Conversely,
Ct from the analytical solution converged for
(s), and the reason for the lower T is similar as is explained in the previous paragraph.
5.2. Effect of Current and Structural Parameters on and
Figure 4 plots the influence of u and no current on the versus wave period
T(s) with membrane tension, spring stiffness, and versus wave period. An elevation in current value corresponds to a heightened transmission coefficient (
Ct1). This is because of the higher deflection of the floating flexible membrane caused by a higher CS, thereby enabling the transmission of greater wave energy beneath the membrane. It is evident that in the absence of currents, no peak is observed in
Ct1, and the trend of
Ct1 is almost linear.
Figure 5 depicts the impact of the tensile force of the floating membrane on the
versus
T(s) with spring stiffness
,
, and
. It is observed that the reflection coefficient increases with an increase in tensional force on the membrane. However, this effect becomes reversed for a higher wave period. Due to the membrane’s higher tensile strength (
), it exhibits reduced deflection, resulting in decreased wave energy transmission and increased wave reflection.
Figure 6 simulates the effect of the non-dimensional water depth
on
versus
T(s) with spring stiffness
,
l/
h = 5, and
a/
h = 14. It is observed that the reflection coefficient decreases with an increase in tensional force. However, the observed effect is reversed for wave periods exceeding a specific threshold, provided all other parameters are held constant.
Figure 7 shows the impact of the membrane width a/
h on
Cr versus
T(s) with spring stiffness
and
. Due to the expanded membrane coverage on the water’s surface, the reflection coefficient drops and reaches its minimum value for a wider membrane. While the results for reflection coefficients across various membrane lengths are not presented here, it is important to note that the observations are similar to those depicted in
Figure 7.
5.3. Influence of Current and Structural Parameters on 3D Membrane Deflection
Figure 8 simulates the 3D deflection
of the HFFM for different CSs versus the length
and width
of the channel with membrane tension
with
. It is found that the deflection increases with an increase in the value of the CS. However, for lower values of CS, the variations among them are negligible, which is clear from
Figure 8.
Figure 9 simulates the 3D deflection
of the HFFM for different spring stiffnesses versus the length
and width
of the channel with membrane tension
and where
. An increase in spring stiffness results in a decrease in deflection. Despite their differences, they are essentially the same. It is clear from
Figure 9 that the effect of spring stiffness on the HFFM does not significantly affect the vibration of the membrane.
Figure 10 plots the effect of different membrane tensions on the 3D deflection of the HFFM versus channel length and width. It is seen that the observations are similar to those in
Figure 6; however, in this case, the deflection variations for different
are prominent.
In
Figure 11, the effect of different water depth
on the 3D deflection of the HFFM versus channel length
and width for
are plotted. It is found that the deflection decreases as the depth of water increases, which is because the wave force on the membrane becomes lower as the
increases.
5.4. Influence of Current and Structural Parameters on Vertical Force
Figure 12 shows the effect of CS on the non-dimensional vertical wave force
Fm for
versus non-dimensional wavelength
. As the values of CS increase, the vertical wave load on the membrane increases. Conversely, for an intermediate wavelength, the vertical wave load on the membrane increases with increasing CS.
Figure 13 shows the effect of membrane width on the non-dimensional vertical wave force where
and CS
versus non-dimensional wavelength
. In general, the wave force on the HFFM lessens as the width increases. However, for an intermediate value of wavelength, the effect is just reversed in nature.
In
Figure 14, the effects of zero, primary, and secondary modes of the HFFM on the vertical wave force where
and CS
versus
are plotted. It is observed that the vertical wave force increases with an increase in modes of oscillations. The tertiary mode exhibits a sharp peak in vertical load at a smaller wavelength. This phenomenon arises from generating small waves at the edges of the HFFM. These waves break before reaching a critical energy level.
Figure 15 plots the effects of different
on the non-dimensional vertical wave force where
versus
. It is found that the non-dimensional vertical wave force increases with an increase in the values of
, certain values of spring stiffness, and the primary mode of oscillation. However, for each value
, the vertical load attends to a minimum for particular values of
because the membrane tension increases the flexibility of the membrane, which becomes harder; as a result, the vertical force on the membrane increases.
Figure 16 depicts the effect of water depth on the non-dimensional vertical wave force versus non-dimensional wavelength
. It is found that as the water becomes deeper, the vertical wave load on the membrane usually becomes smaller. Further, at an intermediate wavelength, vertical loads reach a minimum.
5.5. Effect of CS on Phase and Group Velocities
This subsection examines the impact of current velocities and varying water depths on the phase and group velocities. From the dispersion relation (23), one can derive the phase velocity (
) and group velocity (
Vg) in the presence of CS by neglecting
[
3] in different water depths, as follows.
In the case of FWD, the phase and group velocities can be read as
where
In the case of IWD, Equation (38) yields
where
.
In the case of SWD, Equation (38) gives
Figure 17 compares the phase velocity
and group velocity
of waves over the floating flexible membrane versus wave number with CS,
. From
Figure 17a, in FWD, it is observed that as the CS increases, the values of
become higher, but the trend decreases in nature for higher values of wave number, while in the case of
(as shown in
Figure 17b), the peak values are the same with those of
; however, for smaller values of wave number, the
moves slower than those of
. This is because of the non-evanescent mode propagation of waves over the HFFM. Further, in the case of no current, both the phase and group velocity have attended lower values than those of any one of the current values.
Figure 18 shows the comparison of phase velocity
and group velocity
over a floating flexible membrane versus wave number where CS
. From
Figure 18a, it is observed that the variation
between finite and deep water depth for higher values of wave number is negligible. However, in deep water, the
moves are faster than that of finite depth for lower wave numbers with a certain value of CS due to less effect from currents. From
Figure 18b, the group velocity of waves in shallow water is noticeably faster than in the other two cases, demonstrating a significant difference in propagation speed. Further, the impact of shallow water currents is easily observed as their flow aligns with the path of advancing waves.
6. Conclusions
In this paper, the novel research contributions extend previous work [
11] by investigating the impact of CS on the dynamics of an HFFM using an analytical approach. Further, the effects of current on the hydrodynamic coefficients, membrane deflection, wave forces, and
and
for structural and spring mooring parameters are studied by analysing numerical results from the analytical solutions. It is observed that as the value of the CS rises, the deflection also increases, whereas it declines in deeper water. On the other hand, the spring stiffness has minimal effect on the vibrations of the flexible membrane. It is concluded that
The inclusion of current in the mathematical model and analytical solution is an original contribution of this work. It is observed that the comparison results of Crm and Ctm between analytical and experimental data sets support the present analytical solution.
Regarding the effect of springs and membrane tension, it is found that as the and membrane tension increase, the 3D deflection of the membrane decreases, whilst as the modes of oscillation increase, the deflection pattern of the HFFM increases significantly. The wave force analysis revealed that non-dimensional vertical wave loads become increasingly substantial for higher m and elevated . On the other hand, it is observed that the phase and group velocity increases for higher values of CS. Nevertheless, the impact of CS is more significant in shallow water regimes than that of DWD and FWD.
The current model has certain limitations: the applicable boundary conditions need to be linear and of the third order, in contrast to the floating flexible plate model, which operates with fifth-order conditions at the structural boundary. Additionally, the structural boundaries must remain constant with respect to the coordinates, necessitating a Fourier series of solutions to be obtained. As a result, the existing methodology is suitable for flexible or porous membrane structures with rectangular or circular shapes.
The analysis of the three-dimensional problem in the presence of oblique current flow and waves will be studied in future work. Further extensions of the present work include the following: (i) The integral form of Green’s function derived in this paper can be further simplified by using Cauchy integral theorem based on complex function theory to obtain an alternate expression in series form, and this approach can be helpful for demonstrating a realistic physical nature of a problem emerging in the field of Ocean Engineering and can be validated against the present developed solutions, (ii) the deflection results might be useful for comparisons with the independent experimental model test results based on the computer vision target tracking technique, and (iii) the present formulation and analysis can give us a better understanding of developing an articulated floating flexible membrane for applications to wave energy extraction systems by introducing or adding extra stiffness and lowering the current speed to mitigate the instability of a floating flexible membrane.