1. Introduction
Aquatic vegetation provides essential ecosystem services (Barbier et al. Reference Barbier, Hacker, Kennedy, Koch, Stier and Silliman2011). It enhances biodiversity by providing habitat and shelter for fisheries (Costanza et al. Reference Costanza1997), and by supplying food for larger herbivorous animals such as dugongs and green turtles (Waycott, Longstaff & Mellors Reference Waycott, Longstaff and Mellors2005). Shoreline vegetation attenuates incoming waves, mitigating coastal erosion caused by wave impact (Koch et al. Reference Koch2009; Barbier et al. Reference Barbier, Hacker, Kennedy, Koch, Stier and Silliman2011). Furthermore, submerged macrophytes contribute to nutrient retention in local ecosystems (Barko & James Reference Barko and James1998) and act as carbon sinks, sequestering more carbon per hectare annually than tropical rainforests (Fourqurean et al. Reference Fourqurean2012). Given the critical role of aquatic vegetation in these processes, its protection and restoration have become a central focus in coastal management (Greiner et al. Reference Greiner, McGlathery, Gunnell and McKee2013). Understanding vegetation dynamics across various flow conditions is essential for predicting its coastal protection benefits, as well as its role in particle retention and carbon sequestration.
Numerous studies have examined wave attenuation by vegetation, using both laboratory experiments with model vegetation (Mendez & Losada Reference Mendez and Losada2004; Augustin, Irish & Lynett Reference Augustin, Irish and Lynett2009; Stratigaki et al. 2011; Xu & Lei Reference Xu and Lei2024b ), real vegetation (Maza, Lara & Losada Reference Maza, Lara and Losada2016; van Wesenbeeck et al. Reference van Wesenbeeck, Wolters, Antolínez, Kalloe, Hofland, de Boer, Çete and Bouma2022; Xie & Lei Reference Xie and Lei2025) and field observations (Bradley & Houser Reference Bradley and Houser2009; Paul & Amos Reference Paul and Amos2011; Infantes et al. Reference Infantes, Orfila, Simarro, Terrados, Luhar and Nepf2012). Some of these investigations have specifically investigated the role of vegetation flexibility in wave damping. For instance, Mullarney & Henderson (Reference Mullarney and Henderson2010) formulated an analytical model based on cantilever beam theory to simulate the motion of single-stemmed vegetation, such as sedges. Their findings indicated that wave dissipation by flexible vegetation was only 30 % of that observed for rigid vegetation. Similarly, Houser, Trimble & Morales (Reference Houser, Trimble and Morales2015) explored how blade flexibility influences the drag coefficient and, consistent with Mullarney & Henderson (Reference Mullarney and Henderson2010), found that increased flexibility led to a reduction in drag force.
Although waves and currents often coexist in coastal environments, relatively few studies have investigated how their combined effects influence the hydrodynamic response of flexible vegetation and its wave-damping capacity. Losada, Maza & Lara (Reference Losada, Maza and Lara2016) conducted laboratory experiments with real vegetation and observed that wave attenuation increased when the current opposed wave propagation, but decreased when the current aligned with it. They proposed a wave energy dissipation model incorporating vegetation deflection by replacing the full blade length,
$l$
, with the deflected height,
$l_d$
, which represents the vertical distance from the bed to the time-averaged blade tip position.
Another approach to characterising vegetation–flow interaction involves the concept of effective blade length, which quantifies the impact of reconfiguration on drag forces experienced by an individual blade in co-directional waves and currents. Originally introduced by Alben, Shelley & Zhang (Reference Alben, Shelley and Zhang2002), this concept has been developed and validated for various flow regimes, including pure currents (Luhar & Nepf Reference Luhar and Nepf2011, Reference Luhar and Nepf2013; Lei & Nepf Reference Lei and Nepf2021), pure regular waves (Luhar, Infantes & Nepf Reference Luhar, Infantes and Nepf2017; Lei & Nepf Reference Lei and Nepf2019b
), pure internal solitary waves (Sun, You & Lei Reference Sun, You and Lei2024), and co-directional waves and currents (Lei & Nepf Reference Lei and Nepf2019a
; Schaefer & Nepf Reference Schaefer and Nepf2024). The effective length
$l_{e}$
represents the equivalent length of a rigid blade that experiences the same drag as a flexible blade of actual length
$l$
. Typically, reconfiguration reduces drag force, resulting in
$l_e\lt l$
. This drag reduction occurs due to a decrease in the plant’s frontal area, streamlining of the blade, and, in wave environments, blade motion that lowers the relative velocity between the blade and the surrounding water.
Despite these advances, a comprehensive predictor for vegetation dynamics under orthogonal wave–current conditions, where the current is perpendicular to wave propagation, remains lacking. Our study addresses this knowledge gap by investigating the response of flexible vegetation under orthogonal wave–current conditions through controlled laboratory experiments. Specifically, we quantify how vegetation motion and drag forces are modified by wave–current interactions. By integrating force measurements, velocity analysis and video-based motion tracking with the Morison equation, we establish a framework for predicting hydrodynamic forces on flexible vegetation under orthogonal wave–current conditions. Finally, we evaluate whether scaling laws developed for pure currents, pure waves, and co-directional waves and currents can be adapted to the more complex orthogonal case.
2. Methods and data processing
2.1. Experimental set-up and flow conditions
The experimental tests were conducted in the wave–current basin at the Hydraulic Engineering Laboratory of the National University of Singapore. This basin, measuring
$33$
m in length,
$10$
m in width and
$0.9$
m in depth, is uniquely designed to generate orthogonal waves and currents. A
$19$
m
$\times$
$1.5$
m section was partitioned to create a reservoir that facilitates current recirculation. The current enters the main basin through a single opening with width of 2.8 m at a
$90^\circ$
angle to the waves and exits through an adjustable weir. Two
$75$
HP centrifugal pumps recirculate the flow from the basement tank to the roof tank (Lim & Madsen Reference Lim and Madsen2016). The current speed is controlled by adjusting the valve opening, weir height and water level in the roof tank. The wave propagation direction is defined as the
$x$
-direction, while the current flows along the
$y$
-direction (figure 1).
(a) Schematics of the truncated pyramid and instrument installation. (b) Sample GoPro frames showing stem motion in the
$x$
- and
$y$
-directions. (c) Illustration of the wave–current basin and the position of the truncated pyramid.
Waves were generated using a piston-type wave maker with 13 independently controlled paddles, each driven by an electric servo motor. A 1 : 10 sloped artificial gravel beach was installed to minimise wave reflection. Following the method of Goda & Suzuki (Reference Goda and Suzuki1976), wave reflection was measured using two wave gauges, yielding a reflection coefficient of 10 %. This study focused exclusively on regular waves, with a constant water depth of
$h = 0.3$
m measured from the top of a false bottom. The experiments covered five wave periods (
$T = 1.0, 1.2, 1.4, 1.6, 2.0$
s) and four wave amplitudes (
$a_w = 0.03, 0.04, 0.05, 0.06$
m) (see the detailed test cases and results in Appendix A). Orthogonal wave–current flows were classified into two categories: wave–current flows with high current speed (WCHC) and wave–current flows with low current speed (WCLC), corresponding to current speeds of
$u_c=0.16$
m s−1 and
$u_c=0.06$
m s−1, respectively. The flow conditions in this study closely resemble those observed in natural coastal environments.
To mimic individual elements of flexible vegetation, cylindrical rods were selected due to their uniform second moment of inertia, which suppresses torsional motion. As a result, torsion effects on vegetation rods were not considered in this study. Furthermore, in this study, the second moment of inertia is given as
$I=\pi d^4/64$
. Two EPDM-based materials with differing flexibilities were tested, denoted HF (high flexibility) and LF (low flexibility). The Young’s modulus (
$E$
) of each material was measured using a low-force tensile strength tester (Instron 5969). The material LF (
$E = 5.36$
MPa,
$\rho _v = 923$
kg m
$^{-3}$
) exhibited a higher rigidity, while material HF (
$E = 0.96$
MPa,
$\rho _v = 704$
kg m
$^{-3}$
) was more flexible. The diameter,
$d$
, was fixed as
$6$
mm, while the vegetation lengths (
$l$
) were 9, 14 and 19 cm, resulting in the bending stiffness,
$\textit{EI}$
, from
$6.1 \times 10^{-5}$
to
$3.4 \times 10^{-4}$
$\mathrm{N\,{m}^{2}}$
and buoyancy parameter,
$B$
, from 0.21 to 11.64. In most cases,
${\textit{Ca}}\gg B$
, indicating that hydrodynamic drag dominated over the restoring force due to buoyancy. Therefore, buoyancy effects were considered negligible in this study. For comparison, wooden cylinders with identical geometrical parameters were used as rigid references. In total, 121 experimental cases with replicates were conducted under pure wave (PW), WCHC and WCLC conditions, covering a range of current-to-wave velocity ratio (
$\beta$
) from 0.19 to 1.56,
${\textit{Ca}}$
from 0.3 to 32 and
$L$
from 1.1 to 10.
2.2. Velocity measurement
Velocity measurements were conducted at mid-vegetation height using an acoustic Doppler velocimeter (ADV, Nortek) operating at a sampling rate of
$50$
Hz. The ADV was positioned at the centre of the truncated pyramid top in the
$x$
-direction and
$5$
cm from the edge in the
$y$
-direction. Phase (
$\varphi$
)-averaged velocity,
$u_x(\varphi )$
and
$u_y(\varphi )$
, was calculated using the method developed by Lei & Nepf (Reference Lei and Nepf2019b
). The amplitude of the horizontal wave orbital velocity (
$\tilde {u}_w$
) and current velocity (
$u_c$
) were derived from
$u(\varphi )$
, as defined in (2.1) and (2.2), respectively. The measured velocities were subsequently used in force prediction. The wave excursion,
$A_w$
, was calculated by
$A_w = \tilde {u}_wT/{2\pi }$
.
\begin{align} \tilde {u}_w &= \sqrt {2} \sqrt {\frac {1}{2\pi } \int _0^{2\pi } u_x^2 (\varphi )\, {\rm d}\varphi }, \end{align}
\begin{align} u_c &= \frac {1}{2\pi } \int _0^{2\pi } u_y (\varphi ) \, {\rm d}\varphi .\end{align}
2.3. Video processing and stem velocity
Simultaneously with force measurements, underwater videos were recorded using two GoPro cameras to capture the motion of flexible stems. The cameras were mounted on metal plates positioned 1 m downstream in both the
$x$
- and
$y$
-directions to minimise flow interference. Videos were recorded over 10 wave cycles at a frame rate (
$f$
) of 60 fps. An example of the video processing workflow is shown in figure 2.
Example of the video processing workflow. (a) Raw video frame with a red rectangle highlighting the cropped region. (b) Processed frame showing the stem divided into 10 vertical segments, with red-filled circles marking the centre of mass of each segment. (c) Time series of the tip displacement. (d) Vertical velocity profile of the stem, with the error bars representing the standard deviation over 30 wave cycles.
Video processing was performed using MATLAB’s Image Processing Toolbox. Each video was converted to greyscale frame by frame, with the stem identified as the largest connected cluster of black pixels, while background noise was removed. The stem velocity was tracked along the stem by dividing it into 10 vertical segments. The centre of mass of each segment was determined by averaging the positions of black pixels. The travel distance (
$\Delta x$
) of each segment was then computed as the displacement of the centre of mass relative to its initial position in the absence of waves and currents. This displacement was converted to metres using a pixel-to-metre scale, derived from the known length of the stem. The stem velocity,
$u_{s}$
, was obtained by computing its temporal gradient along travel distance and then phase-averaged. The stem acceleration
$\dot {u}_{s}$
was subsequently derived as the time derivative of velocity. The total centre-of-mass stem position was determined by averaging the positions of all black pixels, and the corresponding velocity
$u_{s,c}$
was calculated from its displacement. To visualise the range of motion, edge detection was applied to each selected frame and the resulting binary images were stacked over one wave cycle to create a composite image. The tip excursion for each case was determined by measuring the distance between the leftmost and rightmost positions of the tip’s centre. The uncertainty was evaluated by the standard deviation across 30 wave cycles. With an average scale of 0.3 mm pixel–1 and a time interval of
$1/60$
s between frames, the resolution of the stem velocity detected from the video is estimated to be 1.8 cm s−1.
2.4. Estimation of phase shift between stem and flow velocity
We observed a phase shift (
$\Delta \varphi$
) between the stem and flow velocities. This phase shift results from the dynamic balance between the hydrodynamic forces and the restoring force due to stem stiffness. Accounting for this phase shift is essential to predict the total force on individual flexible stems. Although velocity measurements and video recordings were not synchronised, the phase shift was visually estimated by determining the frame difference (
$\Delta n$
) between the maximum flow velocity and the maximum stem velocity. The frame corresponding to the surface displacement peak or trough was identified by locating the highest or lowest interaction point between the water and the wave gauge or the supporting rod of the ADV (figure 3
a), whichever provided a clearer reference. Since the surface displacement of the peak or trough corresponds to the maximum positive and negative flow velocity, this served as a reliable marker for determining the phase shift.
Example of phase shift estimation. (a) Frame 445, where the wave trough reaches the ADV rod, indicating the maximum negative wave velocity. (b) Corresponding frame 432, showing the maximum negative stem velocity.
The frame in which the stem velocity reached its maximum was extracted from the velocity time series obtained through video processing (figure 3 b). The phase shift for each case was calculated as the mean frame difference over 10 wave cycles. Additionally, since the ADV and wave gauge were positioned 10 cm upstream of the tracked stem, the phase difference due to this spatial offset was incorporated into the visually determined frame difference. Finally, the frame difference was converted to a phase shift using
\begin{equation} \Delta \varphi = 2\pi \frac { \Delta n}{T} \textit {f}, \end{equation}
where
$\Delta n$
is the frame difference and
$f$
is the frame rate of the camera.
2.5. Force prediction
The force prediction analysis was conducted to identify the key factors influencing total force on flexible stems, enabling a deeper understanding of the differences between flexible and rigid stem responses. It should be noted that only the force at
$x$
-direction was predicted and compared with the measured forces. The total force at
$x$
-direction (
$F_x$
), acting normally on an individual stem, comprises inertial and drag forces, as described by the Morison equation (Denny et al. Reference Denny, Gaylord, Helmuth and Daniel1998) and (2.4) in this study, where
$F_I$
, the inertial force, consists of the Froude–Krylov force (
$F_{\textit{VB}}$
, also known as the virtual buoyancy force) and the added mass force (
$F_{\textit{AM}}$
), while
$F_D$
is the drag force and
$F_{I,s}$
is the stem inertia caused by the acceleration of the moving stem,
\begin{equation} F_x = F_I + F_D - F_{I,b} = F_{\textit{VB}} + F_{\textit{AM}} + F_D - F_{I,s}. \end{equation}
Here,
$F_{\textit{VB}}$
is given by
\begin{equation} F_{\textit{VB}} (\varphi ) = \int _0^l \frac {1}{4} \rho \pi d^2 \dot {u}_x(\varphi ) \, {\rm d}z ,\end{equation}
where
$\dot {u}_x$
denotes the acceleration of the wave velocity. Additionally,
$F_{\textit{AM}}$
is trapezoidal integrated along the stem by
\begin{equation} F_{\textit{AM}} (\varphi ) = \int _{z_1}^{z_{10}} \frac {1}{4} \rho \pi d^2 (C_m - 1) \dot {u}_{r,x}(\varphi ,z_i) \, {\rm d}z, \end{equation}
where
$\dot {u}_{r,x}$
denotes the acceleration of the relative velocity in the
$x$
- direction and
$C_m$
is the inertia coefficient. The vertical coordinate
$z$
was discretised into 10 segments with an increment of
$10/l$
, from
$z_1 = l/20$
to
$z_{10} = l - l/20$
. The
$x$
-direction relative velocity at each segment,
$u_{r,x}(\varphi , z_i)$
, was defined as the difference between the flow velocity and the stem velocity. Because flow velocity varied little along the stem, it was assumed uniform and represented by the single-point measurement at mid-vegetation height,
$u_x(\varphi )$
. The stem velocity at
$z_i$
,
$u_{s,i}$
, was extracted from videos as described in § 2.3.
To align stem and flow velocity, the visually determined phase shift was applied. The ADV sampling rate was 50 Hz, while the video frame rate was 60 fps. To synchronise the datasets, the phase-averaged wave velocity was first interpolated to 60 Hz. The phase-averaged wave and stem velocity data were first arranged by phase (from peak to peak), and then
$u_{s,i}$
was shifted forward by the identified phase shift.
Here,
$u_{r,x} (\varphi ,z_i)$
is given by
\begin{equation} u_{r,x} (\varphi ,z_i) = {u}_x(\varphi ) - u_{s,i} (\varphi , z_i), \end{equation}
and
$F_D$
is calculated by
\begin{equation} F_{D} (\varphi ) = \int _{z_1}^{z_{10}} \frac {1}{2} \rho C_{\!D} \, d |u_r(\varphi ,z_i) |u_{r,x}(\varphi ,z_i) \, {\rm d}z ,\end{equation}
where
$u_{r} (\varphi ,z_i)$
is the total relative velocity, representing the combination of relative velocity in the
$x$
- and
$y$
-directions. In the
$x$
-direction,
$u_{r,x}$
is calculated by (2.7). In the
$y$
-direction, consistent with the uniform
$u_c$
assumption, only the mean velocity was considered, neglecting oscillatory components. Video analysis showed that the mean stem velocity
$u_{s,y}$
was negligible, falling below the resolution of video-derived velocity measurements. Therefore,
$u_{s,y}$
is assumed to be 0 and
$u_{r,y}$
is expressed as
\begin{equation} u_{r,y} = {u}_c - u_{s,y} = {u}_c. \end{equation}
Subequently,
$u_{r} (\varphi ,z_i)$
is given by
\begin{equation} u_{r} (\varphi ,z_i) = \sqrt {u^2_{r,x}(\varphi ,z_i)+u^2_{c}}. \end{equation}
The inertia coefficient
$C_{m}$
and drag coefficient
$C_{D}$
are obtained by interpolating the relationships versus
${\textit{KC}}$
number,
$KC=\tilde {u}_{w}T/d$
, from Keulegan & Carpenter (Reference Keulegan and Carpenter1958) (their figures 10 and 11), an approach validated for orthogonal wave–current conditions with direct force measurements by Xu & Lei (Reference Xu and Lei2024b
). Other studies have adopted the empirical relationships with
${Re}$
(Petrolo et al. Reference Petrolo, Ungarish, Chiapponi and Longo2022; Zhu et al. Reference Zhu, Chen, Ding, Jafari, Wang and Johnson2023; Zhang et al. Reference Zhang, Chen, Lei, Zhou, Yao and Stive2023).
Here,
$F_{I,s}$
is calculated by
\begin{equation} F_{I,s} (\varphi ) = \int _{z_1}^{z_{10}} \frac {1}{4} \rho _b \pi d^2 \dot {u}_{s,i} (\varphi ,z_i) \, {\rm d}z, \end{equation}
where
$\dot {u}_{s,i}(\varphi ,z_i)$
is the acceleration of the stem velocity in the
$x$
- direction.
2.6. Scaling relationships for effective length
2.6.1. Previous scaling framework for effective length
The scaling framework for effective length was developed to quantify the drag reduction due to plant reconfiguration in response to steady flow. The effective length was found to scale with two dimensionless parameters: the Cauchy number (
${\textit{Ca}}_c$
) and the buoyancy parameter (
$B$
). Here,
${\textit{Ca}}_c$
represents the ratio of hydrodynamic drag to the restoring force due to blade stiffness, while the buoyancy parameter quantifies the ratio of buoyancy to the restoring force (Luhar & Nepf Reference Luhar and Nepf2011):
\begin{align} {\textit{Ca}}_c &= \frac {1}{2}\frac { C_{\!D} \rho b u_c^2 l^3}{\textit{EI}}, \end{align}
\begin{align} B &= \frac {\Delta \rho g b { d}l^3}{\textit{EI}}, \end{align}
where
$\rho$
is the water density,
$b$
is the vegetation element (e.g. blade, stem) width,
$u_c$
is the steady current speed,
$l$
is the length,
$d$
is the thickness,
$\Delta \rho$
is the density difference between water and vegetation,
$E$
is the Young’s modulus, and
$I$
is the second moment of inertia. Previous studies indicate that
$B$
is small for common seagrass species such as Thalassia testudinum, Posidonia oceanica and Zostera marina (
$B \leqslant 1.4$
, see table 1 of Lei & Nepf (Reference Lei and Nepf2016)). Within this range,
$B$
has a negligible effect on the posture of the vegetation.
In oscillatory flow, the wave Cauchy number,
${\textit{Ca}}_{\textit{p}w}$
, is similarly expressed as
\begin{equation} {\textit{Ca}}_{\textit{p}w} = \frac {1}{2}\frac { C_{\!D} \rho b u_w^2 l^3}{\textit{EI}}, \end{equation}
where
$u_c$
is replaced by the wave velocity
$u_w$
. Zeller et al. (Reference Zeller, Weitzman, Abbett, Zarama, Fringer and Koseff2014) observed that blade motion also depends on the ratio of wave excursion (
$A_w$
) to blade excursion, which represents the horizontal distances travelled by the wave and the blade within one wave period, respectively. To generalise this relationship, Luhar & Nepf (Reference Luhar and Nepf2016) introduced another dimensionless parameter, the length ratio (
$L$
):
\begin{equation} L = \frac {l}{A_w} = \frac {2 \pi l}{u_w T}. \end{equation}
Lei & Nepf (Reference Lei and Nepf2019a
) extended the scaling relationships for pure currents and pure waves to co-directional wave–current flows, and proposed the following equations to estimate
$l_{e}$
under various flow conditions:
\begin{equation} \frac {l_e}{l} = \begin{cases} (1.09 \pm 0.07) ({\textit{Ca}}_{\textit{p}w} L)^{-0.25 \pm 0.02} & \text{for } u_c \lt 0.25 u_w, \\[3pt] 0.9 {\textit{Ca}}_{w\textit{c}}^{-1/3} & \text{for } 0.25 u_w \leqslant u_c \lt 2 u_w, \\[3pt] 0.9 {\textit{Ca}}_c^{-1/3} & \text{for } u_c \geqslant 2 u_w, \end{cases} \end{equation}
where the subscripts ‘
$pw$
’, ‘c’ and ‘
$wc$
’ denote the pure wave, pure current and co-directional wave–current condition, respectively. The modified
${\textit{Ca}}$
in co-directional wave–current flows,
${\textit{Ca}}_{w\textit{c}}$
, is given by
\begin{equation} {\textit{Ca}}_{w\textit{c}} = \frac {1}{2} \frac {C_{\!D} \rho b l^3}{\textit{EI}} \left ( u_c^2 + \frac {1}{2} u_w^2 \right ) ,\end{equation}
where
$ ( u_c^2 + 1/2 u_w^2 )$
is derived based on the equivalent time-averaged total drag force. It should be noted that the characteristic velocity scale
$u_w^2$
in (2.14) is defined based on peak drag, instead of time-mean equivalent drag (Lei & Nepf Reference Lei and Nepf2019a
). The characteristic velocity scale for
${\textit{Ca}}_{\textit{p}w}$
defined based on the time-mean equivalent drag should be
$1/2 u_w^2$
.
Luhar & Nepf (Reference Luhar and Nepf2016) noted that under inertia-dominated conditions, the force balance suggests that the effective length scales with
${\textit{CaL}/\textit{KC}}$
rather than
${\textit{Ca}}L$
, following a similar relation:
$l_e/l \sim {({\textit{CaL}/\textit{KC}})}^{-1/4}$
, where
$KC=u_wT/d$
is the Keulegan–Carpenter number (Keulegan & Carpenter Reference Keulegan and Carpenter1958) and
$d$
is the diameter of stems. Jacobsen et al. (Reference Jacobsen, Bakker, Uijttewaal and Uittenbogaard2019) also highlighted the significance of
${\textit{CaL}/\textit{KC}}$
, representing the inertia-to-stiffness ratio, in capturing the dynamics of flexible stems. In this study,
${\textit{CaL}/\textit{KC}}$
was adopted as a key dimensionless parameter, and its effectiveness in characterising both stem motion and hydrodynamic forces was further evaluated.
2.6.2. Hypotheses for scaling the effective length under orthogonal wave–current conditions
We proposed a scaling framework for orthogonal wave–current conditions by reformulating the drag force expression. The scaling relationships in (2.16) were derived from a drag-dominated force balance. Because the total velocity entering the drag term differs when waves and currents are orthogonal, the drag expression must be modified accordingly.
We hypothesise that the drag force can be decomposed into
$x$
- and
$y$
-components. We consider several measure-time-averaged, root-mean-square (r.m.s.) and maximum-computed either from the total drag magnitude or from the directional components. For the following analysis, we adopt: (1) the time-averaged magnitude of the total drag and (2) the r.m.s. magnitude of the decomposed drag (C10). We choose the r.m.s. of the decomposed components to their time-averaged values because it yields a more compact analytical form. We do not use the maximum drag; justification is provided in § 4.1. Complete derivations for all measures (time-averaged total drag, time-averaged decomposed drag, r.m.s. decomposed drag and maximum decomposed drag) are given in Appendix C. Because the drag expression is modified under orthogonal wave–current forcing, the definition of the Cauchy number
${\textit{Ca}}$
must be adjusted accordingly, analogous to the modification in (2.17).
In this study, the scaling relationships for effective length under orthogonal wave–current flows were evaluated based on the total drag and also considered separately in the
$x$
- and
$y$
-directions. The expression for the time-averaged magnitude of the total drag (C7) suggests that the appropriate characteristic velocity for
${\textit{Ca}}_{w\textit{c}}$
should be
$(\sqrt { ( 1/2 \tilde {u}_w^2 + u_c^2 )}$
). Thus, the definition of
${\textit{Ca}}_{w\textit{c}}$
aligns with that used for co-directional wave–current flows (2.17). Here,
$\boldsymbol{Ca}_{{w\textit{c}}_{{{\textit{rms}}}}}$
in each direction should be modified as
\begin{equation} \boldsymbol{Ca}_{{w\textit{c}}_{{{\textit{rms}}}}} = \frac {1}{2}\frac { \rho C_{D}{ d}l^3\tilde {u}_{w}^{2}}{\textit{EI}} \begin{bmatrix} \sqrt {\dfrac {4\beta ^{2}+3}{8}} \\[5pt] \beta \sqrt {\beta ^{2}+\dfrac {1}{2}} \end{bmatrix}, \end{equation}
where
$\beta =u_{c}/\tilde {u}_{w}$
is the current-to-wave velocity ratio.
In co-directional wave–current flows, stem motion is restricted by an additional current, resulting in a transition from wave-dominated to current-dominated scaling (2.16) as
$u_c$
increases. Under orthogonal wave–current flows, we hypothesise that the swaying motion in the
$x$
-direction remains insignificantly affected by the orthogonal current (verified in §§ 3.1 and 3.2) and the reconfiguration is still dependent on the wave excursion. Finally, the effective length, corresponding to the total drag, is assumed to scale with
${\textit{Ca}}_{w\textit{c}} L$
as
\begin{equation} \frac {l_e}{l} = 1.1 ({\textit{Ca}}_{w\textit{c}} L)^{-0.25}. \end{equation}
In the
$x$
- and
$y$
-directions, the effective lengths that correspond to r.m.s. drag are assumed to scale with
${\textit{Ca}}_{{w\textit{c}}_{{{\textit{rms}},}x}} L$
and
${\textit{Ca}}_{{w\textit{c}}_{{{\textit{rms}},}y}}$
, respectively, as given by
\begin{align} \frac {l_{e,x}}{l} &=1.1 ({\textit{Ca}}_{{w\textit{c}}_{{{\textit{rms}},}x}} L)^{-0.25},\nonumber\\ \frac {l_{e,y}}{l} &= 0.9 {\textit{Ca}}_{{w\textit{c}}_{{{\textit{rms}},}y}}^{-1/3}. \end{align}
2.6.3. Force measurement and measured effective length ratio
The total forces acting on the flexible stems were measured using submersible force transducers (FUTEK LSB210, with IP68 waterproof rating) with a measurement range from
$0.001$
to
$10$
N and a resolution of
$0.001$
N. Since these submersible sensors measure force in only one direction, two force transducers were installed perpendicular to each other to capture forces in both the
$x$
- and
$y$
-directions, with synchronised measurements. To facilitate accurate force measurements, a truncated pyramid (figure 1
a) with eight inclined faces (
$1\!:\!5$
slope) was designed as a platform for mounting the force transducers. The pyramid had a height of
$0.1$
m and a total length of
$1.5$
m. A customised metal holder securely attached each flexible stem to the force transducer using screws, ensuring rigid coupling between the stem and the transducer. Force measurements were conducted at a sampling rate of
$1200$
Hz over a
$2$
-minute duration, capturing
$60$
to
$100$
wave cycles depending on the wave period. Simultaneously, surface displacement and flow velocities were recorded using a wave gauge and an ADV. Experiments were conducted to quantify background forces from the set-up without vegetation under various wave conditions. The recorded forces were negligible (of the order of
$0.001$
N) compared with those acting on the stems. Thus, flow-induced forces on the set-up itself were considered insignificant and omitted from further analysis. The uncertainties for force measurement were determined from replicate tests.
The measured total forces on the flexible stems were phase-averaged,
$F(\varphi )$
, following the method outlined by Lei & Nepf (Reference Lei and Nepf2019b
). The measured r.m.s. force at
$x$
-direction (
$F_{{{\textit{rms}}},x}$
), as described in (2.21), was used for comparison with predicted forces (calculated from § 2.5) in § 3.4.2.
\begin{align} F_{{{\textit{rms}}},x} &= \sqrt {\frac {1}{2\pi } \int _0^{2\pi } F_x^2(\varphi ) \, {\rm d}\varphi },\nonumber\\ F_{{{\textit{rms}}},y} &= \sqrt {\frac {1}{2\pi } \int _0^{2\pi } F_y^2(\varphi )\, {\rm d}\varphi } .\end{align}
The time-averaged total force was calculated using
\begin{equation} F_{{\textit{mean},\textit{total}}} = {\frac {1}{2\pi } \int _0^{2\pi } \sqrt {F_x^2(\varphi ) + F_y^2(\varphi )}\, {\rm d}\varphi } .\end{equation}
These measured effective length ratios (
$l_e/l$
) were used to evaluate the validity of the scaling laws presented in (2.19) and (2.20). The effective length
$l_{e}$
is defined as the length of a rigid stem that experiences the same drag as a flexible stem of length
$l$
. Therefore, the measured
$l_e/l$
was determined by calculating the ratio of the measured force on a flexible stem to that on a rigid cylinder of identical geometry.
The measured
$l_e/l$
corresponds to time-average total force was defined by
\begin{equation} \frac {l_{e}}{l} = \frac {F_{{\textit{flexible}}, {\textit{mean},\textit{total}}}}{F_{{\boldsymbol{rigid},\textit{mean},\textit{total}}}}. \end{equation}
The measured effective length ratios corresponds to r.m.s. magnitude of decomposed forces were calculated by
\begin{align} \frac {l_{e,x}}{l} &= \frac {F_{{\textit{flexible}}, {{\textit{rms}}},x}}{F_{{\textit{rigid}}, {{\textit{rms}}},x}},\nonumber\\ \frac {l_{e,y}}{l} &= \frac {F_{{\textit{flexible}}, {mean},y}}{F_{{\textit{rigid}}, {mean},y}} .\end{align}
It should be noted that for evaluating the effective length in the
$y$
-direction, oscillation was neglected in accordance with the uniform
$u_c$
assumption and the mean force
$F_{{mean},y}$
, defined as the mean of
$F_y(\varphi )$
, was used. The force measurements for flexible stems can be found in Appendix A, while the detailed data for rigid references can be found on Zenodo (Xu & Lei Reference Xu and Lei2024a
).
3. Results
3.1. Stem motion
Examples of stem motion extracted from video recordings are presented in figure 4. Stem motion is influenced by material properties, stem length and flow conditions. Within each row, the panels are arranged in ascending order of
$ {\textit{CaL}}$
. For stems of the same length, HF stems exhibit greater tip excursion than LF ones. For stems made of the same material, tip excursion increases with stem length. Comparing the first row figure 4(a–e) with the second row figure 4(f–j), the
$ z$
-values under orthogonal wave–current flows are lower than those observed under pure wave conditions. In the wave propagation direction, the presence of an orthogonal current causes greater stem bending, particularly for HF stems. Under pure waves, the tip excursion follows a nearly two-dimensional trajectory, whereas under orthogonal wave–current conditions, the stem undergoes three-dimensional movements. In the current direction figure 4(k–o), the stem tends to bend more as
$ {\textit{CaL}}$
increases.
The asymmetric motion of the HF stems under pure waves, observed in figures 4(d) and 4(e), has also been reported in previous studies (Luhar & Nepf Reference Luhar and Nepf2016; Zhu et al. Reference Zhu, Zou, Huguenard and Fredriksson2020; Schaefer & Nepf Reference Schaefer and Nepf2024) and is attributed to stem movement in a spatially varying velocity field. Under pure waves, the stems exhibit a mean forward bending, but this asymmetric pattern was not observed under orthogonal wave–current conditions. This is attributed to the bending in the
$y$
-direction, which effectively increases the rigidity of the stem and constrains the motion in the
$x$
-direction.
3.2. Range of motion
The non-dimensional tip excursion of the stem is plotted as a function of
$ {\textit{CaL}}$
and
$ {\textit{CaL}/\textit{KC}}$
in figures 5(a) and 5(b), respectively. Tip excursion quantifies how the stem tip moves with the flow. The dimensionless parameter
${\textit{Ca}}L$
, adopted by Luhar & Nepf (Reference Luhar and Nepf2016), represents the ratio of drag force to blade stiffness, while
${\textit{CaL}/\textit{KC}}$
, introduced by Jacobsen et al. (Reference Jacobsen, Bakker, Uijttewaal and Uittenbogaard2019), represents the ratio of inertial force to blade stiffness. As expected, tip excursion increases with
$ {\textit{CaL}}$
and
${\textit{CaL}/\textit{KC}}$
. The presence of an orthogonal current does not significantly affect the horizontal range of stem motion, as tip excursions under PW, WCHS and WCLS follow similar trends with
$ {\textit{CaL}}$
and
$ {\textit{CaL}/\textit{KC}}$
. For stems with shorter lengths, higher elastic moduli and under mild waves (i.e.
$ {\textit{CaL}} \lt 1$
), their motion closely resembles that of nearly rigid stems, consistent with the findings of Luhar & Nepf (Reference Luhar and Nepf2016) and Jacobsen et al. (Reference Jacobsen, Bakker, Uijttewaal and Uittenbogaard2019).
In figure 5(a), data points tend to cluster based on stem length, particularly for
$ {\textit{CaL}} \gt 20$
, where tip excursion varies significantly despite similar
$ {\textit{CaL}}$
values. However, in figure 5(b), the clustering of data points was improved by plotting against
${\textit{CaL}/\textit{KC}}$
, suggesting that
$ {\textit{CaL}/\textit{KC}}$
better captures the range of stem motion than
$ {\textit{CaL}}$
alone. This finding aligns with Jacobsen et al. (Reference Jacobsen, Bakker, Uijttewaal and Uittenbogaard2019) and Leclercq & De Langre (Reference Leclercq and de Langre2018). Tip excursion reaches a value of approximately 2.0, consistent with Jacobsen et al. (Reference Jacobsen, Bakker, Uijttewaal and Uittenbogaard2019) within a similar
$ {\textit{CaL}}$
and
$ {\textit{CaL}/\textit{KC}}$
range. Leclercq & De Langre (Reference Leclercq and de Langre2018) also reported a comparable maximum value, though their
$ y$
-axis variable,
$ \Delta x/A_w$
, is twice the magnitude of the corresponding values in this study.
3.3. Phase shift
Stem velocity and flow velocity are not necessarily in phase, as theoretically described by Mullarney & Henderson (Reference Mullarney and Henderson2010) and observed in the field by Bradley & Houser (Reference Bradley and Houser2009). Accounting for this phase shift is essential for accurately simulating stem motion and predicting the forces acting on vegetation. For nearly rigid vegetation, the stem velocity leads the surrounding water velocity by 90
$^\circ$
. In contrast, highly flexible vegetation moves in phase with the water flow. The phase shift arises from the interplay between the restoring force due to stiffness and the hydrodynamic forces exerted by the water. For vegetation with intermediate flexibility, the phase shift falls within the range of 0
$^\circ$
to 90
$^\circ$
. Understanding this phase shift is crucial for capturing the dynamic response of flexible vegetation under wave–current interactions.
The phase shift, determined from video recordings, is presented in figure 6 as a function of
$ {\textit{CaL}}$
and
$ {\textit{CaL}/\textit{KC}}$
. The phase shift patterns are consistent across different flow conditions, and generally decrease with increasing
$ {\textit{CaL}}$
and
$ {\textit{CaL}/\textit{KC}}$
. When
$ {\textit{CaL}} \lt 10$
and
$ {\textit{CaL}/\textit{KC}}\lt 0.2$
, stems exhibit a phase shift close to 90
$^\circ$
, behaving similarly to rigid structures. Some values exceeding 90
$^\circ$
may be attributed to uncertainties in visually counting phase differences. In this study, stems moved nearly in phase with the water when
$ {\textit{CaL}} \gt 70$
and
$ {\textit{CaL}/\textit{KC}} \gt 1$
, consistent with the findings of Jacobsen et al. (Reference Jacobsen, Bakker, Uijttewaal and Uittenbogaard2019). Additionally, the presence of an orthogonal current did not affect the observed phase shift. The measured phase shift was subsequently applied to align the stem velocity with the flow velocity for accurate force prediction on the flexible stems.
In the near-bed portion of a flexible stem, phase shift is 90
$^\circ$
. The phase shift decreases to zero within the elastic boundary layer (Mullarney & Henderson Reference Mullarney and Henderson2010). Jacobsen et al. (Reference Jacobsen, Bakker, Uijttewaal and Uittenbogaard2019) observed a constant phase shift along the stem when
$ {\textit{CaL}} \lt 20$
and
$ {\textit{CaL}/\textit{KC}} \lt 0.5$
, with stem motion following a single-mode response. At higher
$ {\textit{CaL}}$
, inflection points appear along the stem, resembling motion mode 1, 2 and 3 defined in Mullarney & Henderson (Reference Mullarney and Henderson2010). Under these conditions, different points along the stems do not reach maximum bending simultaneously. However, in this study, we assume a uniform phase shift along the stem length. This assumption is justified for LF stems, where the measured phase shift averages 82
$^\circ$
, closely aligning with the 90
$^\circ$
shift characteristic of the near-bed rigid layer. For HF stems, the near-bed rigid layer is much thinner than the elastic boundary layer, as seen in figure 4. Above the near-bed boundary layer, inflection points were not observed and the stems remained straight, exhibiting the mode 0 behaviour defined by Mullarney & Henderson (Reference Mullarney and Henderson2010), where all points along the stem reach maximum bending simultaneously. Given these observations, assuming a uniform phase shift along the stem length is a reasonable simplification in this study.
Examples of stem motion under wave conditions of
$ T = 1.6 \, \text{s}$
and
$ H = 0.10 \, \text{m}$
. (a–e) Stem motion in the wave propagation direction under pure waves. (f–j) Stem motion in the wave propagation direction under WCHC conditions. (k–o) Stem motion in the current direction under WCHC conditions. Each column corresponds to stems with the same properties. Orange lines represent frames with positive velocity, while blue lines indicate frames with negative velocity. Black dashed ellipses show the wave horizontal excursion,
$ 2A_w$
, at the mid-vegetation height.
Tip excursion normalised by wave excursion as a function of (a)
$ {\textit{CaL}}$
and (b)
$ {\textit{CaL}/\textit{KC}}$
. Marker shape indicates stem length. PW, WCHC and WCLC are shown in blue, orange and green, respectively. Closed and open markers represent HF and LF stems, respectively. Standard deviations are less than 10 % and smaller than the symbol size, and thus not displayed.
Phase shift between the water particle and the stem as a function of (a)
${\textit{Ca}}L$
and (b)
${\textit{CaL}/\textit{KC}}$
. The uncertainties come from the standard deviation across 10 wave cycles.
3.4. Force prediction
3.4.1. Velocities and the predicted forces
Examples of velocities and their corresponding predicted forces are shown in figure 7. Relative velocity reduction is a key factor in the force reduction in flexible vegetation compared with rigid vegetation. Bradley & Houser (Reference Bradley and Houser2009) observed decreased wave energy attenuation by flexible vegetation due to its relative motion between the real seagrass blades and oscillatory flows. In figure 7(a), the flow velocity and stem velocity are in quadrature. The centre-of-mass stem velocity,
$u_{s,c}$
, was used to illustrate the differences between HF and LF stems. The maximum
$|u_{s,c}|$
of LF stems is 7.9 cm s−1 due to the relatively small range of motion. With minimal influence from
$u_{s,c}$
, the relative velocity
$u_r$
closely follows
$u_w$
in magnitude, with a slight phase shift.
Wave velocity, stem velocity and relative velocity for (a) LF and (c) HF stems. Drag force, inertial force and total force in
$x$
-direction for (b) LF and (d) HF stems. Both cases are under WCHC conditions with
$ T = 1.2 \, \text{s}$
and
$ H = 0.10 \, \text{m}$
.
In contrast, in figure 7(c),
$u_{s,c}$
and
$u_w$
are nearly in phase, with a phase difference as small as
$0.18\pi$
. The maximum
$|u_{s,c}|$
of HF stems reaches 10.9 cm s−1, reducing the magnitude of
$u_r$
to approximately half of
$|u_w|$
. In this scenario, the total force remains primarily governed by drag. Under these conditions, the measured
$F_{x,{{\textit{rms}}}}$
for LF and HF stems corresponds to 75 % and 35 % of
$F_{{\textit{rigid}},x,{{\textit{rms}}}}$
, respectively, highlighting the significant force reduction in flexible vegetation.
3.4.2. Force prediction with different assumptions
The comparison between predicted and measured forces is shown in figure 8. Overall, LF stems experience higher forces than HF stems due to their greater rigidity. A comparison of the left and right columns of figure 8 reveals that incorporating phase shift improves the accuracy of force predictions. Specifically, when the counted phase shift is included, the root-mean-squared error (RMSE) between predicted and measured forces decreases from 0.0060 N to 0.0038 N (see figures 8 a and 8 b).
Comparison between measured and predicted forces. The left column assumes water and stem velocity are in phase, while the right column incorporates the exact phase shift. (a,b) Forces under pure waves. (c,d) Forces under orthogonal wave–current flows using the original vegetation length
$l$
. (e,f) Forces under orthogonal wave–current flows with the deflected vegetation length
$l_d$
. (g,h) Forces under orthogonal wave–current flows considering solely
$l_d$
, treating flexible stems as rigid. Uncertainties arise from force measurement replicates.
Under orthogonal wave–current conditions (figures 8
c and 8
d), force predictions using in-phase assumption overestimated the measured forces, with an RMSE of 0.0035 N. Including the phase-shift effect improved the overestimation but yielded the same RMSE of 0.0035 N. This discrepancy arises because flexible stems bend in the current direction, reducing their frontal area normal to the horizontal wave velocity. To account for this effect, the deflected height,
$l_d$
, was measured from video recordings in
$y$
-direction. By replacing
$l$
with
$l_d$
in the force prediction equations, the corrected predicted forces (figure 8
f) align well with the measured values, reducing the RMSE from 0.0035 N to 0.0025 N.
With the inclusion of relative velocity, phase shift and deflection effects, our force prediction model demonstrates strong agreement with the measured values. Additionally, force predictions were compared using a simplified approach that considers only deflection while ignoring changes in relative velocity. This approach treats flexible vegetation as rigid stems of length of
$l_d$
. The results for LF and HF stems are shown in figures 8(g) and 8(h), respectively. Although this simplified approach works better for LF stems, generating an RMSE of 0.0031 N, it strongly overestimates the forces on HF stems, with an RMSE of 0.0054 N, resulting in an overestimation of wave attenuation by highly flexible vegetation such as seagrass. This is because less flexible vegetation behaves more like rigid stems, exhibiting relatively low stem velocity. In contrast, highly flexible vegetation responds more actively to oscillatory flows, making the difference between flow velocity and stem velocity more pronounced.
In summary, accurately predicting forces on flexible vegetation requires accounting for both relative velocity and frontal area reduction. Under pure wave conditions, the reduction in relative velocity plays a dominant role in force reduction compared with rigid vegetation. However, under orthogonal wave–current conditions, both reduced frontal area and relative velocity contribute to force reduction. Furthermore, the strong agreement between predictions and measurements suggests that drag and inertia coefficients can be reliably determined from their relationship with the
${\textit{KC}}$
number (Keulegan & Carpenter Reference Keulegan and Carpenter1958).
3.4.3. Inertia and drag
Inertial forces were found to contribute significantly to the total force acting on flexible vegetation within the tested
$\beta$
range from 0.19 to 1.56 in this study. Previous studies have assessed the relative importance of inertial forces using the
${\textit{KC}}$
number. However, since Keulegan & Carpenter (Reference Keulegan and Carpenter1958) originally developed this parameter for rigid cylindrical structures in oscillatory flows,
${\textit{KC}}$
is primarily applicable to rigid vegetation. In this study, we found the significance of inertial effects varies with flexibility. To capture this variability, we compare the ratio of predicted inertial force to total force as a function of
${\textit{CaL}/\textit{KC}}$
in figure 9.
Predicted inertial forces normalised by the total force under (a) pure wave and (b) orthogonal wave–current conditions. Circle, triangle and square markers represent stem length of 19, 14 and 9 cm, respectively. Uncertainties can come from measured
$\tilde {u}_w\text{sin}(\varphi )$
and video-derived stem velocity.
Inertial contributions increase with increasing
${\textit{CaL}/\textit{KC}}$
for both pure wave and orthogonal wave–current conditions. This suggests that inertial forces become more pronounced for shorter wave periods, higher stem flexibility and longer stems. For stems of the same length, HF stems experience a larger inertial impact than LF stems because the latter are more rigid and therefore subject to higher drag forces.
Under orthogonal wave–current conditions (figure 9
b), trends differ between the WCLC and WCHC cases. In WCHC, inertial forces increase gradually with
${\textit{CaL}/\textit{KC}}$
, reaching up to 29 %. In contrast, WCLC follows a pattern similar to that of pure waves, with inertial contributions increasing rapidly with
${\textit{CaL}/\textit{KC}}$
to 73 %. The presence of a strong orthogonal current (
$\beta$
from 0.65 to 1.56) appears to weaken the relative significance of inertia. This is because the increased total flow velocity enhances drag, while the virtual buoyancy force (
$F_{\textit{VB}}$
) and the added mass force (
$F_{\textit{AM}}$
) are not influenced.
Inertial forces consist of the
$F_{\textit{VB}}$
,
$F_{\textit{AM}}$
and stem inertia (
$F_{I,s}$
). The stem inertia, which scales with the acceleration of the stem itself, is relatively small in this study. The relative importance of
$F_{\textit{VB}}$
and
$F_{\textit{AM}}$
depends on the flow conditions. Here,
$F_{\textit{VB}}$
becomes more important when the wave period and wave height are smaller and the magnitude of acceleration is relatively large compared with the wave velocity. In contrast,
$F_{\textit{AM}}$
, which scales with relative acceleration, contributes more substantially when wave velocities are higher.
In general, inertial forces require particular attention in flow conditions characterised by shorter wave periods and highly flexible vegetation. Previous studies often neglected these effects, as they primarily examined thin rectangular plates (e.g. 0.4 mm by Luhar & Nepf (Reference Luhar and Nepf2016), and 0.1 mm by Lei & Nepf (Reference Lei and Nepf2019a ) and Schaefer & Nepf (Reference Schaefer and Nepf2024)), resulting in significantly lower mass compared with the cylindrical stems examined in this study. Consequently, inertial forces play a more prominent role here.
4. Discussion
4.1. Force metric selection for effective length analysis
The choice of force metric affects both the expression of the characteristic velocity scale in
${\textit{Ca}}$
and the resulting estimation of effective length. The concept of effective length was initially introduced for unidirectional flow, where a flexible stem of length
$l$
experiences the same drag force as a rigid stem of the same length (Luhar & Nepf Reference Luhar and Nepf2011). In oscillatory flows, different definitions of effective length have been proposed.
Luhar & Nepf (Reference Luhar and Nepf2016) extended the effective length framework to oscillatory flow using the r.m.s. force, which effectively represents the magnitude of force over one wave cycle. They also showed that the scaling law for pure waves remains valid when using the maximum force. Lei & Nepf (Reference Lei and Nepf2019a ) later calculated effective length based on time-mean force in co-directional wave–current flows. Caution is needed when using the maximum force within a wave cycle, as inertial forces can introduce a phase shift between the maximum total force and the maximum drag force (figure 7). When inertial forces are negligible, the maximum total force can be considered equivalent to the maximum drag force. Since inertial contributions in this study are non-negligible, the maximum total force cannot be used. Instead, we examine the effective length using both the time-mean total force and the r.m.s. force, as shown in figure 10.
(a) Effective length under pure waves versus
${\textit{Ca}}_{\textit{p}w}L$
. (b) Effective length from time-mean total force versus
${\textit{Ca}}_{w\textit{c}}L$
under orthogonal wave–current flows. (c) Effective length from r.m.s. force in the
$x$
-direction versus
${\textit{Ca}}_{{w\textit{c}}_{{{\textit{rms}}},x}}L$
. (d) Effective length from r.m.s. force in the
$y$
-direction versus
${\textit{Ca}}_{{w\textit{c}}_{{{\textit{rms}}},y}}$
. Red dashed lines represent fitted formulae with the adjusted
$r^2$
shown in the lower right corner of each panel, while black solid lines indicate the hypothesised scaling laws ((2.19) for panel b and (2.20) for panels c and d). Uncertainties of the measured force arise from replicate tests.
4.2. Comparison of scaling laws under pure waves
Figure 10(a) compares the normalised measured effective length, calculated from the r.m.s. force, as a function of
${\textit{Ca}}_{\textit{p}w}L$
for pure waves. All experimental cases in this study fall within the small deflection range (
$L \gt 1$
), where the effective length is expected to scale as
$l_{e}/l \sim {\textit{Ca}}_{\textit{p}w}L^{-0.25}$
(Luhar & Nepf Reference Luhar and Nepf2016; Lei & Nepf Reference Lei and Nepf2019b
). The power of
${\textit{Ca}}_{\textit{p}w}L$
, fitted as
$0.28 \pm 0.02$
, aligns closely with the theoretical value of 0.25. However, the scaling coefficient,
$1.49 \pm 0.08$
, is 35 % higher than 1.1, which was fitted by Lei & Nepf (Reference Lei and Nepf2019b
).
The scaling coefficient is affected by the aspect ratio of the stem cross-section. Previous studies modelled flexible vegetation using thin rectangular sheets with significantly smaller aspect ratio, such as 0.095 for foam blades and 0.02 for HDPE blades by Luhar & Nepf (Reference Luhar and Nepf2016), and 0.01 for LDPE blades by Lei & Nepf (Reference Lei and Nepf2019b ). In contrast, cylindrical silicone rubber stems with an aspect ratio of 1 were used in this study. The smaller aspect ratio will lead to significantly lower volume, leading to negligible inertial effects. However, the higher-volume cylindrical stems used in this study produced non-negligible inertial contributions. As shown in figure 9(a), inertial forces account for an average of 25 % of the total force, thus increasing the total force beyond the drag component alone.
Luhar & Nepf (Reference Luhar and Nepf2016) reported that the scaling for pure waves underestimated the effective lengths of HDPE stems, attributing this discrepancy to vortex shedding when the drag force and stem stiffness were of comparable magnitude, i.e.
${\textit{Ca}} \sim O(1)$
. A similar phenomenon may explain the higher measured effective lengths in this study. Here,
${\textit{Ca}}$
is of the order of
$O(1)$
for all LF stems and HF stems with a length of 9 cm. However, when
${\textit{Ca}} \gt 10$
(HF stems with lengths of 14 and 19 cm), the effective lengths converge to the expected scaling law (black solid line in figure 10
a).
4.3. Comparison of scaling laws under combined wave–current flows
Figure 10(b) compares the measured effective length, calculated from the time-mean total force, as a function of
${\textit{Ca}}_{w\textit{c}}L$
. The fitted scaling exponent (
$-0.24 \pm 0.02$
) agrees with the hypothesised value of –0.25 in (2.19). The fitted scaling coefficient (
$1.46 \pm 0.11$
) is larger than the expected value of 1.1. This finding is similar to pure wave results (figure 10
a) and is attributed to greater inertial contributions from the larger material aspect ratio used here. These results support using
${\textit{Ca}}_{w\textit{c}}L$
, with
$u_c^2+1/2u_w^2$
being the characteristic velocity scale in
${\textit{Ca}}_{w\textit{c}}$
, as the appropriate scaling parameter under orthogonal wave–current flows. This scaling differs from that used for co-directional wave–current flows, where
${\textit{Ca}}_{w\textit{c}}$
alone is applied when
$0.25 \lt \beta \lt 2$
in (2.16). This difference highlights a key physical insight, showing that with an orthogonal current within the
$\beta$
range from 0.19 to 1.56 in this study, the combined forcing is wave-dominated, whereas under co-directional conditions, it can be current-dominated.
The consistent alignment of data from under both WCHC and WCLC conditions demonstrates that the hypothesised scaling relationship (2.19) holds for a
$\beta$
range from 0.19 to 1.56. When
$\beta$
approaches 0 and current velocity becomes negligible, the characteristic velocity in
${\textit{Ca}}_{w\textit{c}}$
reduces to
$1/2 u_w^2$
, representing the time-mean magnitude of drag under pure wave conditions. Conversely, when
$\beta$
becomes large and the current velocity dominates, the characteristic velocity approaches
$u_c^2$
, which corresponds to the definition of
${\textit{Ca}}_c$
(2.12). We therefore recommend adopting the definition of
${\textit{Ca}}_{w\textit{c}}$
(2.17) for co-directional and orthogonal wave–current flows, as well as pure waves and pure currents, to provide a unified dimensionless scaling for effective length.
Figures 10(c) and 10(d) further evaluate the effective length in the orthogonal wave–current conditions along the
$x$
- and
$y$
-directions, respectively, where the effective lengths were derived from the r.m.s. forces in each direction. In the
$x$
-direction, the
$l_{e,x}$
exhibits a similar trend to figure 10(a), converging towards the solid line when
${\textit{Ca}}_{{w\textit{c}}_{{{\textit{rms}}},x}} \gt 10$
. In the
$y$
-direction, the current is expected to remain the dominant driver of the flexible stem motion, leading to a scaling parameter solely with
${\textit{Ca}}_{{w\textit{c}}_{{{\textit{rms}},}y}}$
, independent of
$L$
. However, the fitted power of
${\textit{Ca}}_{{w\textit{c}}_{{{\textit{rms}},y}}}$
is –0.23
$\pm$
0.04, larger than the expected (–0.33), indicating that forces in the current direction were notably influenced by the presence of orthogonal waves. The
$l_{e,y}/l$
at
${\textit{Ca}}_{{w\textit{c}}_{{{\textit{rms}}},y}}$
in the range of 0.3–2 fluctuates around 1, indicating behaviour similar to rigid stems. This result is consistent with Luhar & Nepf (Reference Luhar and Nepf2011), who found the stiff-to-flexible transition to occur when
${\textit{Ca}} \lt 2$
.
The effective length was also evaluated using
${\textit{Ca}}_{w\textit{c}}{\textit{L/KC}}$
. However, the data exhibited significant scatter, especially for the combined wave–current condition. As noted by Luhar & Nepf (Reference Luhar and Nepf2016), the effective length is expected to scale with
${\textit{CaL}/\textit{KC}}$
under inertia-dominated conditions. In contrast, for the experimental cases in this study, although inertial forces contributed to the total force, hydrodynamic drag remained the dominant component, explaining the absence of a clear scaling relationship with
${\textit{Ca}}_{w\textit{c}}{\textit{L/KC}}$
.
4.4. Application of effective length to real flexible vegetation
To estimate the effective length of real vegetation, we compiled data from prior studies on wave–current conditions, and on the mechanical and morphological properties of several species. Using these data, we calculated the corresponding
${\textit{Ca}}L$
values under natural conditions, with and without currents, and predicted the associated effective length ratios, as summarised in table 1.
Comparison between the physical parameters in natural environments and this study.
$^{*}$
de los Santos et al. (Reference de los Santos, Onoda, Vergara, Pérez-Lloréns, Bouma, La Nafie, Cambridge and Brun2016)
$^{**}$
de los Santos et al. (Reference de los Santos, Onoda, Vergara, Pérez-Lloréns, Bouma, La Nafie, Cambridge and Brun2016); Womersley (Reference Womersley1984)
$^{***}$
Zhang et al. (Reference Zhang, Chua and Cheong2015).
When waves approach the shoreline at an oblique angle, they generate longshore currents. However, relatively few field studies have quantified longshore current velocities. Galvin (Reference Galvin1967) summarised early measurements, reporting a mean velocity of 0.08 m s−1. Sherman (Reference Sherman1988) recorded longshore currents at five field sites in Sandy Hook, New Jersey, with a mean of 0.23
$\pm$
0.29 m s−1. On the central Atlantic coast of Florida, Burnette & Dally (Reference Burnette and Dally2017) reported a 10-year dataset showing strong seasonal and climatic variability, with monthly averaged velocities ranging from 0 to 0.15 m s−1 and 2-hour averaged velocities generally below 0.4 m s−1. Almar et al. (Reference Almar, Larnier, Castelle, Scott and Floc’h2016) conducted experiments at a 10-m depth off the Gulf of Guinea, West Africa, reporting surf-zone longshore currents of 0.4–0.8 m s−1. Field measurements by Li et al. (Reference Li, Shi, Li, Wang, Lv, Wang and Tian2020) in the southern East China Sea during summer to autumn revealed a significant influence of tides and seasonality, with residual currents ranging from 0.1 to 0.5 m s−1. In summary, longshore current velocities vary widely depending on location, season and climate, with typical values ranging from 0 to 0.8 m s−1. Nearshore short waves typically exhibit wave periods of
$1{-}5$
s and wave heights of
$3{-}20$
cm (Wiberg et al. Reference Wiberg, Taube, Ferguson, Kremer and Reidenbach2019; Lee et al. Reference Lee, Tay, Ooi and Friess2021). Given a nearshore wave velocity range of 0.3–1.3 m s−1, the corresponding
$\beta$
range in natural coastal environments typically spans from 0 to approximately 2.7.
Seagrasses exhibit a wide range of
$E$
, from 0.1 to 2.4 GPa across different species (Lei & Nepf Reference Lei and Nepf2016; de los Santos et al. Reference de los Santos, Onoda, Vergara, Pérez-Lloréns, Bouma, La Nafie, Cambridge and Brun2016; Vettori & Marjoribanks Reference Vettori and Marjoribanks2021). de los Santos et al. (Reference de los Santos, Onoda, Vergara, Pérez-Lloréns, Bouma, La Nafie, Cambridge and Brun2016) conducted a comprehensive analysis of the mechanical and morphological properties of 22 species of tropical and temperate seagrasses. Globally, seagrasses exhibit three distinct growth forms. The form commonly modelled by thin sheets consists of strap-like leaves arising directly from the rhizome, such as Zostera marina, Thalassia hemprichii, Posidonia oceanica. Another type consists of short strap-like leaves arising from a vertical stem, predominantly found in southern and western Australia, e.g. Amphibolis antarctica (commonly known as wire weed). de los Santos et al. (Reference de los Santos, Onoda, Vergara, Pérez-Lloréns, Bouma, La Nafie, Cambridge and Brun2016) reported that the mechanical properties of these stems differ significantly from their leaves, exhibiting greater strength, stiffness and extensibility, making them more resilient under varying hydrodynamic forces. The third growth form consists of plants with paired oval leaves attached to the rhizome via a petiole. These species are typically short and exhibit morphologies distinct from the flexible cylinder model used in this study.
The vegetation model used in this study can be considered a simplification of Amphibolis antarctica, representing only the stem and ignoring the leaves. Additionally, the model is also representative of mangrove pneumatophore roots. Mangrove pneumatophore roots have been reported to exhibit a cone-shaped appearance with diameters ranging from 2 to 15 mm and an
$E$
of approximately 0.8 GPa (Zhang, Chua & Cheong Reference Zhang, Chua and Cheong2015), resulting in a bending stiffness (
$\textit{EI}$
) from
$6.3 \times 10^{-4}$
to
$3 \times 10^{-2}\,\mathrm{ N {m}^{2}}$
, which closely resembles the LF stems (
$EI=3.4 \times 10^{-4}\,\mathrm{N {m}^{2}}$
) in this study. The
$\textit{EI}$
of stems of Amphibolis antarctica were calculated to range from
$1.25\times 10^{-4}$
to
$2.6\times 10^{-4}\,\mathrm{ N {m}^{2}}$
, also comparable to the LF stems and slightly larger than that of the HF stems (
$EI=6.1 \times 10^{-5}\,\mathrm{N {m}^{2}}$
) tested in this study.
To isolate the influence of longshore currents, the wave velocity
$u_w$
was fixed to 0.3 m s−1, while the current velocity varied from 0 to 0.8 m s−1. Mangrove pneumatophore roots were simplified as cylindrical structures with diameters ranging from 2 to 15 mm for the calculations. For strap-like seagrass,
$l_e/l$
was calculated using the scaling law for pure waves ((2.16) when
$u_c = 0$
m s−1), based on a fitted range of
${\textit{Ca}}_{w\textit{c}}L$
from
$O(10^{-1})$
to
$O(10^5)$
. For Amphibolis antarctica stems and pneumatophore roots,
$l_e/l$
was predicted using the fitted scaling shown in figure 10(b), with a fitting range of
${\textit{Ca}}_{w\textit{c}}L$
from
$O(10^0)$
to
$O(10^2)$
.
This study examined
${\textit{Ca}}_{w\textit{c}}L$
values ranging from
$O(10^0)$
to
$O(10^2)$
, due to the fixed morphology and flexibility of the stem material. However, natural vegetation spans a much broader range of
${\textit{Ca}}_{w\textit{c}}L$
values. When orthogonal longshore currents are considered, the predicted
$l_e/l$
values become markedly smaller. Strap-like seagrasses can experience
${\textit{Ca}}_{w\textit{c}}L$
from
$O(10^1)$
to
$O(10^6)$
, with
$l_e/l$
generally below 0.46. Notably, when
${\textit{Ca}}_{w\textit{c}}L \gt 10^3$
, the effective length no longer follows the scaling law but instead tends to stabilise around 0.1 (see figure 5 of Lei & Nepf Reference Lei and Nepf2019b
), suggesting that the scaling framework (2.16) may not be applicable beyond this range. Amphibolis antarctica stems exhibit
${\textit{Ca}}_{w\textit{c}}L$
from
$O(10^{1})$
to
$O(10^4)$
with the predicted
$l_e/l$
values ranging from 0.29 to 1.3. For mangrove pneumatophore roots,
${\textit{Ca}}_{w\textit{c}}L$
spans from
$O(10^{-5})$
to
$O(10^2)$
, with predicted
$l_e/l$
exceeding 0.44. The predicted
$l_e/l$
values are much greater than 1 when
${\textit{Ca}}_{w\textit{c}}L$
$\ll$
$O(10^0)$
, which is not physically realistic. When the stems are relatively short and stiff (
${\textit{Ca}}_{w\textit{c}}L \lt 1$
), although the scaling law ((2.16) when
$u_c = 0$
m s−1) may yield effective lengths greater than 1, the actual effective length should be around 1 due to their near-rigid behaviour. While both this study and Luhar & Nepf (Reference Luhar and Nepf2016) observed
$l_e/l$
can slightly exceed 1 due to vortex shedding, flexible stems are unlikely to experience forces several times larger than those acting on rigid stems.
The scaling framework of effective length physically links the hydrodynamic forces in flexible structures to the corresponding forces on rigid structures with the same morphology. By knowing the morphological and mechanical properties of the structures, this approach provides a practical way to estimate forces (Sun et al. Reference Sun, You and Lei2024). Moreover, it can be extended beyond force prediction for vegetation alone, serving as a foundation for other physical models, such as wave energy decay models, to quantify wave energy dissipation by flexible vegetation (Luhar et al. Reference Luhar, Infantes and Nepf2017; Lei & Nepf Reference Lei and Nepf2019b ; Beth Schaefer & Nepf Reference Beth Schaefer and Nepf2022).
5. Conclusion
This study addresses a research gap by investigating the reconfiguration and hydrodynamic forces of flexible stems under orthogonal wave–current conditions. We extended and validated existing scaling laws for effective length to orthogonal conditions by modifying the definitions of
${\textit{Ca}}$
, and we introduced a new method to predict the total force on flexible stems. A total of 121 experimental cases were conducted in a wave–current basin, varying stem elasticity, stem length, wave periods, wave heights and current speeds. Stem motion was analysed using video recordings, from which the centre of mass velocities were extracted to calculate the relative velocity between the flow and the stems. The results showed that the phase shift between flow velocity and stem velocity decreases from 90
$^\circ$
to 0
$^\circ$
as
${\textit{Ca}}L$
increases. The close agreement between the measured and predicted forces confirms that the extracted stem velocity provides a reliable basis for computing the relative velocity. By incorporating stem velocity, phase shift and deflection in height, the predicted forces aligned well with the experimental measurements.
Inertial effects contributed significantly to the total force, especially for more flexible stems, attributed to the larger material aspect ratio used here relative to previous research. Consistent with our hypotheses, under orthogonal wave–current flows, the effective length associated with the total force scales as
${\textit{Ca}}_{w\textit{c}}L ^{-0.25}$
. For cylindrical stems, the scaling coefficient exceeded values reported previously, reflecting the additional inertial contribution in this study. In the decomposed directions, the effective length in the wave-propagation direction scales with
${\textit{Ca}}_{{w\textit{c}}_{{{\textit{rms}},}x}}L$
, whereas in the current direction, it scales with
${\textit{Ca}}_{{w\textit{c}}_{{{\textit{rms}},}y}}$
. Although the non-dimensional parameter
${\textit{CaL}/\textit{KC}}$
better describes tip motion, it showed limited correlation with effective length because it primarily reflects inertial forces rather than drag-induced reconfiguration. By integrating previous studies on effective length under unidirectional currents, pure waves and co-directional wave–current flows with the present orthogonal case, we expand the effective-length framework, making it more comprehensive and applicable to complex real-world conditions.
This study provides valuable datasets on hydrodynamic forces and stem motion under orthogonal wave–current conditions, which remain under-studied. However, the experiments employed simple cylindrical flexible stems and did not include torsional effects. Future work should explore stems with more complex morphologies that better represent natural vegetation. In addition, more research is needed on the interactions between stems within vegetated meadows under oblique wave–current conditions to better quantify their influence on hydrodynamic forces and wave attenuation.
Acknowledgements
The authors thank Mr Shaja Khan for his assistance and the Central Workshop of NUS for manufacturing the load cell holder and the truncated pyramid. Artificial intelligence tools (ChatGPT, OpenAI) were used solely to improve the clarity and grammar of the text. No AI tools were used for data analysis, interpretation or generation of scientific content.
Funding
This study is supported by the National University of Singapore (NUS) start-up grant under A-0009542-02-00 and by the MOE Tier1 grant under A-8001202-00-00.
Declaration of interests
The authors report no conflict of interest.
Data availability statement
The data supporting the findings of this study have been uploaded to figshare by Xu & Lei (Reference Xu and Lei2025).
Appendix A. Table of experimental cases
Experimental cases under pure waves with
$\beta = 0$
.
Experimental cases under WCHC conditions with current velocity
$u_c = 0.16$
m s–1.
Experimental cases under WCLC conditions with current velocity
$u_c = 0.06$
m s−1.
Appendix B. Evaluation of linear stem velocity assumption
Stem velocities were obtained by dividing the stem into 10 segments and differentiating the centre-of-mass displacement of each segment over time. However, this segmentation and integration process is time-consuming and impractical for field applications, because complete stem motion, particularly near the base, is often not fully captured. To address this, we evaluated the linear velocity assumption, in which a linear velocity profile along the stem was assumed. Specifically,
$u_s$
was interpolated based on the assumptions that
$u_s=0$
at
$z=0$
m (stem base) and
$u_s = 2u_{s,c}$
at
$z=l$
, where
$u_{s,c}$
represents the velocity at the total centre of mass of the stem (
$z=l/2$
). Figure 11 compares the vertical velocity profiles derived from full-stem tracking against those obtained using the linear assumption. The comparison indicates that the linear assumption underestimates velocities near the tip and overestimates them near the base. The maximum deviation is approximately 40
$\%$
at the tip, as shown in figure 11(d).
Comparison of along-stem velocities from full-stem tracking and linear assumption under WCHC conditions (
$T = 1.6\,\text{s}$
,
$H = 0.10\,\text{m}$
). Panels (a–d) correspond to panels (g–j) in figure 4.
Comparison of force prediction on stems between along-stem velocities from linear assumptions and full-stem tracking under (a) PW conditions and (b) WCHC and WCLC conditions.
To assess the impact of these velocity discrepancies on force prediction, force predictions were calculated separately using velocities derived from the linear assumption and full-stem tracking methods with the original stem length
$l$
, and compared in figure 12. For pure wave conditions (figure 12
a) and orthogonal wave–current flows (figure 12
b), the resulting RMSE values are 0.0019 N and 0.0023 N, respectively. These results indicate that, within the
${\textit{Ca}}L$
range examined, the linear-velocity assumption produces acceptable force predictions. However, its applicability at higher
${\textit{Ca}}L$
should be evaluated before broader applications.
It should be noted that neither approach can perfectly recover the true forces because the stems oscillate about their neutral configurations and the flow field varies spatially. Future work should incorporate the effects of spatiotemporal flow variation during stem motion.
Appendix C. Derivation of drag forces under orthogonal wave–current flows
The drag force is originally expressed as
\begin{equation} F_D=\frac {1}{2}\rho C_{\!D} d l |u_r|u_r, \end{equation}
where
$u_r$
is the relative velocity between the flow and the object (Denny et al. Reference Denny, Gaylord, Helmuth and Daniel1998). For rigid vegetation,
$u_r$
equals to flow velocity. Therefore, under pure waves and pure currents,
$u_r$
is equal to wave velocity (
$u_w$
) and current velocity (
$u_c$
), respectively. However, under orthogonal wave–current flows,
$u_r$
is equal to the total velocity (
$u_{w\textit{c}}$
), which is the combination of wave and current velocities.
This study focuses on the water column from the bed to the stem height, where the depth-averaged velocity is close to the value of the mid-stem height. For simplicity, we assume
$u_w$
is uniform along the stem height and is approximated by its value at mid-stem height. Under the assumption of linear wave theory, the horizontal wave velocity is expressed as
\begin{equation} u_w (\varphi ,z) = a_w \omega \frac {\cosh ( k(z+h))}{\sinh ( kh)} \sin (\varphi ) = \tilde {u}_w (z) \sin ( \varphi ), \end{equation}
where the
$\omega = 2\pi /T$
is the radian frequency,
$\varphi$
is the wave phase and
$k$
is the wavenumber. The amplitude of the horizontal wave orbital velocity at this height is given by
$\tilde {u}_w = a_w \omega \cosh ( k(l/2))/\sinh ( kh)$
. In the
$y$
-direction, a steady current velocity
$u_c$
is assumed. Then,
$u_{w\textit{c}}$
is described by
\begin{equation} u_{w\textit{c}} (\varphi ) = \sqrt { (\tilde {u}_w\sin ( \varphi ))^2 + u_c^2 }. \end{equation}
C.1. Derivations of total drag
Substituting the total velocity (C3) into (C1) yields the total drag force under orthogonal wave–current conditions,
\begin{equation} F_{D}(\varphi ) = \frac {1}{2} \rho C_{\!D} d l u_{w\textit{c}}^2(\varphi ) = \frac {1}{2} \rho C_{\!D} d l \left(\tilde {u}_{w}^{2}\sin ^{2}{\varphi } + u_c^2\right)\!. \end{equation}
The total drag force force, which is in the same direction with
$u_{w\textit{c}}(\varphi )$
, can be decomposed in the
$x$
- and
$y$
-directions. The decomposed drag,
$\boldsymbol {F}_D(\varphi )$
, at each direction can be expressed as
\begin{equation} \boldsymbol {F}_D(\varphi ) = \frac {1}{2} \rho C_{\!D} { d}l \sqrt {u_c^2 + \tilde {u}_w^2 \sin ^2 \varphi } \begin{bmatrix} u_w \sin \varphi \\ u_c \end{bmatrix}, \end{equation}
where the underline denotes a vector. Let
$\beta =u_{c}/\tilde {u}_{w}$
be the current-to-wave velocity ratio and substitute into (C5), then
$\boldsymbol {F}_D(\varphi )$
can be expressed as
\begin{equation} \boldsymbol {F}_D(\varphi ) = \frac {1}{2}\rho C_{D}{ d}l\tilde {u}_{w}^{2}\sqrt {\beta ^{2}+\sin ^{2}{\varphi }} \begin{bmatrix} \sin \varphi \\ \beta \end{bmatrix}. \end{equation}
C.2. Derivations of time-averaged drag
The time-averaged magnitude of the total drag force is expressed as
\begin{equation} F_{D_{{mean}}} = \frac {1}{2\pi } \int _0^{2\pi } F_{D}(\varphi )\,{\rm d}\varphi = \frac {1}{2} \rho C_{\!D} d l \left ( \frac {1}{2} \tilde {u}_w^2 + u_c^2 \right ). \end{equation}
The decomposed components of the time-averaged total drag (C7) in the
$x$
- and
$y$
-directions can be obtained by integrating (C6) over one wave period from 0 to
$2\pi$
, and dividing by
$2\pi$
, yielding the following expressions:
\begin{equation} \boldsymbol {F}_{D_{{mean}}} = \frac {1}{2}\rho C_{D}{ d}l\tilde {u}_{w}^{2} \begin{bmatrix} \dfrac {\beta +(\beta ^{2}+1)\arctan {\dfrac {1}{\beta }}}{\pi }\\[8pt] \beta ^{2}E\left(x\left| -\dfrac {1}{\beta ^{2}}\right)\right|^{2\pi }_{0} \end{bmatrix}, \end{equation}
where
$E (x \mid -({1}/{\beta ^2}) )$
denotes the elliptic integral of the second kind, which can be calculated using (C9). The elliptic integral arises naturally from the integration of
$(\beta ^{2}+\sin ^{2}{\varphi })^{1/2}$
over a wave cycle for which no elementary solution exists. However, the elliptic integral of the second kind can be calculated numerically using the MATLAB function ellipticE (available in the Symbolic Math Toolbox).
\begin{equation} E\left (x \mid -\frac {1}{\beta ^2}\right ) = \int _0^x \sqrt {1 + \frac {\sin ^2 \varphi }{\beta ^2}} \, {\rm d}\varphi. \end{equation}
C.3. Derivation of r.m.s. drag and maximum drag
The r.m.s. magnitude of drag force that decomposed in the
$x$
- and
$y$
-direction can be expressed as
\begin{equation} \boldsymbol {F}_{D_{{{\textit{rms}}}}} = \sqrt {\frac {1}{2\pi } \int _0^{2\pi } \left (\boldsymbol {F}^2_{D}(\varphi ) \right )\, {\rm d}\varphi } = \frac {1}{2}\rho C_{D}{ d}l\tilde {u}_{w}^{2} \begin{bmatrix} \sqrt {\dfrac {4\beta ^{2}+3}{8}} \\ \beta \sqrt {\beta ^{2}+\dfrac {1}{2}} \end{bmatrix}, \end{equation}
where boldface denotes a vector form and
$\boldsymbol {F}_D(\varphi )$
is the decomposed drag force (C6).
The maximum drag corresponds to maximum
$\tilde {u}_w\sin {\varphi }$
and can be expressed as
\begin{equation} \boldsymbol {F}_{D{{max}}} = \frac {1}{2}\rho C_{D}{ d}l\tilde {u}_{w}^{2}\sqrt {\beta ^{2}+1} \begin{bmatrix} 1 \\ \beta \end{bmatrix}. \end{equation}
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