3.1. Coral branch as a circular spring-mounted cylinder in flow

Following the identification of VIVs as the probable source of soft coral branch vibration, we modelled the capture of particles advected by the fluid flow. We first idealised the coral branch as a circular cylinder. From a geometrical perspective, the polyps covering a branch of A. bipinnata are small perturbations, of approximately 10 % of the branch diameter (Bayer Reference Bayer1961). From a fluid–structure interaction perspective, isolated circular cylinders are only prone to VIVs (Nakamura et al. Reference Nakamura, Hirata and Kashima1994), which are the flow-induced vibrations in which we are interested. Additionally, simplifying the coral branch to a circular cylinder brings our study back into the classical problem of particle capture by a circular collector in the field of particle filtering.

We simulated a fluid flow of upstream velocity $U_{0}$ around a fixed or vibrating cylinder of diameter $D$ to highlight the effect of vibrations on particle capture. We considered the water flow as being incompressible and two-dimensional. In addition, we assumed it was unidirectional because the Keulegan–Carpenter number evaluated in the stream-wise direction of a coral branch was sufficiently large $KC \sim 250$, as estimated in § 2, and we are interested in the vortex shedding regime. The flow is governed by the continuity and Navier–Stokes equations, written in an Arbitrary Lagrangian Eulerian (ALE) framework as

(3.1a)\begin{gather} \boldsymbol{\nabla} \cdot \boldsymbol{U}_{f} = 0, \end{gather}

(3.1b)\begin{gather}\frac{\partial{\boldsymbol{U}_{f}}}{\partial{t}} + \left[ (\boldsymbol{U}_{f} - \boldsymbol{V}) \cdot \boldsymbol{\nabla} \right] \boldsymbol{U}_{f} = \frac{1}{\rho_{f}} \left[ -\boldsymbol{\nabla} p + \mu_{f} \nabla^{2}\boldsymbol{U}_{f} \right], \end{gather}

where $\rho _{f}$, $\mu _{f}$, $\boldsymbol {U}_{f}$ and $p$ are, respectively, the fluid density, dynamic viscosity, velocity and pressure. The vector $\boldsymbol {V}$ is the velocity of the moving mesh. If the cylinder is fixed, we simply have $\boldsymbol {V} = \boldsymbol {0}$. The hydrodynamic load applied on the cylinder is expressed as

(3.2)\begin{equation} \boldsymbol{F}_{hydro} = \oint \left[{-}p{\boldsymbol{\mathsf{I}}} + \mu_{f} \left( \boldsymbol{\nabla}\boldsymbol{U}_{f} + \boldsymbol{\nabla}\boldsymbol{U}_{f}^{\textsf{T}} \right) \right]\mathrm{d}\boldsymbol{l}, \end{equation}

with $\mathrm {d}\boldsymbol {l}$ being the integration element around the cylinder, pointing outwards. Furthermore, we considered the cylinder as a spring-mounted oscillator free to move stream-wise and transverse to the main flow under the hydrodynamic load $\boldsymbol{F} _{hydro}$, as schematised in figure 4(a). Neglecting structural damping, the position of the cylinder centre $\boldsymbol {X}_{cyl}$ is governed by

(3.3)\begin{equation} m\frac{\mathrm{d} ^{2}\boldsymbol{X}_{cyl}}{\mathrm{d} t^{2}} + k\boldsymbol{X}_{cyl} = \boldsymbol{F}_{hydro}, \end{equation}

where $m = {\rm \pi}\rho _{s} D^{2}/4$ is the cylinder mass and $k$ the spring stiffness, both per unit length.

Figure 4.

(a) Schematics of particles advected by a flow around a cylinder free to oscillate in the transverse and stream-wise directions. The upstream fluid velocity is parallel to $\boldsymbol {e}_{x}$. Particles have a density $\rho _{p}$ and diameter $d_{p}$. The mass per unit length of the cylinder is $m$, and the spring stiffness per unit length $k$ is the same in both directions. (b) Drawing contrasting the size of particles simulated with the size of the cylinder, where $R = d_{p}/D$ is the diameter ratio.

Normalising velocities with $U_{0}$ and distances with $D$, and scaling time by $D/U_{0}$ and pressure by $\rho _{f} U_{0}^{2}$, the governing equations of the fluid and cylinder interaction can be rewritten as

(3.4a)\begin{gather} \bar{{\boldsymbol{\nabla}}} \cdot \bar{{\boldsymbol{U}}}_{f} = 0, \end{gather}

(3.4b)\begin{gather}\frac{\partial{\bar{{\boldsymbol{U}}}_{f}}}{\partial{\bar{t}}} + \left[ (\bar{{\boldsymbol{U}}}_{f} - \bar{{\boldsymbol{V}}} ) \cdot \bar{{\boldsymbol{\nabla}}} \right] \bar{{\boldsymbol{U}}}_{f} = \left[ -\bar{{\boldsymbol{\nabla}}} \bar{p} + \frac{1}{{\textit{Re}}} \bar{\nabla}^{2}\bar{{\boldsymbol{U}}}_{f} \right], \end{gather}

(3.4c)\begin{gather}\bar{\boldsymbol{F}}_{hydro} = \oint \left[ -\bar{p}{\boldsymbol{\mathsf{I}}} + \frac{1}{\textit{Re}} \left( \bar{{\boldsymbol{\nabla}}}\bar{{\boldsymbol{U}}}_{f} + \bar{{\boldsymbol{\nabla}}}\bar{{\boldsymbol{U}}}_{f}^{\textsf{T}} \right) \right]\mathrm{d}\bar{{\boldsymbol{l}}}, \end{gather}

(3.4d)\begin{gather}\frac{\mathrm{d} ^{2}\bar{{\boldsymbol{X}}}_{cyl}}{\mathrm{d} \bar{t}^{2}} + \left( \frac{2{\rm \pi}}{U_{r}} \right)^{2} \bar{{\boldsymbol{X}}}_{cyl} = \frac{4/{\rm \pi}}{M} \bar{\boldsymbol{F}}_{hydro}, \end{gather}

with the bar $\bar {(.)}$ denoting dimensionless variables. The dynamical parameters present in the system of (3.4d) are the Reynolds number ${\textit{Re}}$, the reduced velocity $U_{r}$ and the mass number $M$, which are defined as

(3.5a–c)\begin{equation} {\textit{Re}} = \frac{\rho_{f} U_{0} D}{\mu_{f}},\quad U_{r} = \frac{2{\rm \pi} U_{0}}{D} \sqrt{\frac{m}{k}},\quad M = \frac{m}{{\rm \pi}\rho_{f} D^{2}/4}. \end{equation}

As mentioned in § 2, we supposed that soft coral branches have a density roughly the same as that of water $\rho _{s} \sim \rho _{f}$, thus we considered a constant cylinder mass number $M = 1$ throughout this study.

Figure 5 shows the computational domain. The dimensions were sufficiently large ($-40 \leq \bar {x} \leq 120$ and $-60 \leq \bar {y} \leq 60$) to avoid confinement effects (Persillon & Braza Reference Persillon and Braza1998). The domain was discretised with 96 000 nodes and 48 000 Taylor–Hood ($\mathcal {P}_{2}-\mathcal {P}_{1}$) triangular elements (Taylor & Hood Reference Taylor and Hood1973), so it follows that the velocity was third-order accurate and the pressure was second-order accurate. These elements were small in the wake and close to the cylinder to resolve the vortex shedding and the boundary layer. The grid in figure 5 was validated in Etienne & Pelletier (Reference Etienne and Pelletier2012) and yielded accurate results in good agreement with the existing data. The boundary conditions on the fluid velocity $\bar {{\boldsymbol {U}}}_{f}$ were a uniform Dirichlet at the entry ($\bar {x} = -40$), homogeneous Neumann at the exit ($\bar {x} = 120$), symmetry at the top and bottom boundaries ($\bar {y} = \pm 60$), and no-slip at the cylinder wall.

Figure 5.

Close-ups of the fluid mesh highlighting the densification in the wake (a) and around the boundary layer (b).

3.2. Particle dynamics

Next, we considered food particles as spheres of diameter $d_{p}$ and density $\rho _{p}$. Several governing equations were established to describe the motion of particles in non-uniform or random flow fields (Maxey & Riley Reference Maxey and Riley1983; Ounis & Ahmadi Reference Ounis and Ahmadi1990; Barton Reference Barton1995). They included terms displaying the influence of particles on the flow. However, in the present study, we assumed that the particles did not disturb the flow and could not actively change their trajectory, unlike a swimming plankton for instance. Then the suitable governing equation to consider was the Basset–Boussinesq–Oseen (BBO) equation (Clift, Grace & Weber Reference Clift, Grace and Weber1978; Barton Reference Barton1995). We neglected gravity effects, and the remaining forces applied on a particle that we kept were the drag $\boldsymbol {F}_{D}$, the pressure load $\boldsymbol {F}_{P}$ and the added mass force $\boldsymbol {F}_{A}$. The governing equations describing the particle trajectory are

(3.6a)\begin{gather} \frac{\mathrm{d} \boldsymbol{x}_{p}}{\mathrm{d} t} = \boldsymbol{u}_{p}, \end{gather}

(3.6b)\begin{gather}\frac{\mathrm{d} \boldsymbol{u}_{p}}{\mathrm{d} t} = \frac{1}{m_{p}} (\boldsymbol{F}_{D} + \boldsymbol{F}_{P} + \boldsymbol{F}_{A}), \end{gather}

where $m_{p} = \rho _{p}{\rm \pi} d _{p}^{3}/6$ is the mass of the particle, $\boldsymbol {x}_{p}$ is its position and $\boldsymbol {u}_{p}$ is its velocity.

The drag applied on a sphere is

(3.7)\begin{equation} \boldsymbol{F}_{D} ={-}\frac{1}{2} C_{D}^{sph} \rho_{f} {\rm \pi}\left( \frac{d_{p}}{2} \right)^{2} \left|\left| \boldsymbol{u}_{p} - \boldsymbol{U}_{f} \right|\right|\left( \boldsymbol{u}_{p} - \boldsymbol{U}_{f} \right), \end{equation}

which is the same expression if the flow were steady. Here $C_{D}^{sph}$ is the drag coefficient of a sphere and it is a function of the particle-based Reynolds number

(3.8)\begin{equation} {\textit{Re}}_{p} = \frac{\rho_{f} \left|\left| \boldsymbol{u}_{p} - \boldsymbol{U}_{f} \right|\right| d_{p}}{\mu_{f}}. \end{equation}

Because this latter is unknown a priori, we considered the Schiller–Nauman interpolation (Clift et al. Reference Clift, Grace and Weber1978)

(3.9)\begin{equation} C_{D}^{sph} = \frac{24}{{\textit{Re}}_{p}} (1 + {\textit{Re}}_{p}^{0.687}), \end{equation}

which is valid for ${\textit{Re}}_{p} < 800$. The pressure load is related to the pressure gradient through

(3.10)\begin{equation} \boldsymbol{F}_{P} ={-} \frac{4}{3} {\rm \pi}\left( \frac{d_{p}}{2} \right)^{3} \boldsymbol{\nabla} p. \end{equation}

It is also known as the Froude–Krylov force (Fredsøe & Sumer Reference Fredsøe and Sumer2006) owing to the difference in pressure of the global flow, which depicts the unsteadiness of the flow in the absence of the particle. If the fluid were stagnant, there would be no pressure gradient accelerating it and this force would simply vanish.

Finally, although previous studies have ignored the added mass force (Haugen & Kragset Reference Haugen and Kragset2010; Krick & Ackerman Reference Krick and Ackerman2015), we deem it important to be included in the force balance. It is proportional to the particle acceleration relative to the fluid

(3.11)\begin{equation} \boldsymbol{F}_{A} ={-} C_{m} \rho_{f} \frac{4}{3} {\rm \pi}\left( \frac{d_{p}}{2} \right)^{3} \left( \frac{\mathrm{d} \boldsymbol{u}_{p}}{\mathrm{d} t} - \frac{\mathrm{d} \boldsymbol{U}_{f}}{\mathrm{d} t} \right). \end{equation}

Here $C_{m}$ is the mass coefficient of a sphere. We took it to be a constant equal to 1/2. Generally, it increases rapidly when the particle comes close to the wall (Brennen Reference Brennen1982), so our leading order simplification becomes invalid with proximity to the cylinder. An elaborate model would account not only for a varying mass coefficient, but also for the repulsive effect just prior to capture owing to confinement, for example, by including an additional force in the momentum equation (3.6b) (Béguin, Pelletier & Étienne Reference Béguin, Pelletier and Étienne2016). In all of our simulations, however, we kept the assumption $C_{m} = 1/2$.

Denoting the ratio of diameters as

(3.12)\begin{equation} R = \frac{d_{p}}{D}, \end{equation}

the particle-based Reynolds number is ${\textit{Re}}_{p} = {\textit{Re}} R \left |\left | \bar {{\boldsymbol {u}}}_{p} - \bar {{\boldsymbol {U}}}_{f} \right |\right |$, and equations (3.6) become

(3.13a)\begin{gather} \frac{\mathrm{d} \bar{{\boldsymbol{x}}}_{p}}{\mathrm{d} \bar{t}} = \bar{{\boldsymbol{u}}}_{p}, \end{gather}

(3.13b)\begin{gather}\frac{\mathrm{d} \bar{{\boldsymbol{u}}}_{p}}{\mathrm{d} \bar{t}} = \bar{{\boldsymbol{F}}}_{D} + \bar{{\boldsymbol{F}}}_{P} + \bar{{\boldsymbol{F}}}_{A}, \end{gather}

with

(3.14a)\begin{gather} \bar{{\boldsymbol{F}}}_{D} ={-} \frac{18}{\bar{\rho} R^{2}{\textit{Re}}} \left( 1 + {\textit{Re}}_{p}^{0.687} \right) \left( \bar{{\boldsymbol{u}}}_{p} - \bar{{\boldsymbol{U}}}_{f} \right), \end{gather}

(3.14b)\begin{gather}\bar{{\boldsymbol{F}}}_{P} ={-} \frac{1}{\bar{\rho}} \bar{{\boldsymbol{\nabla}}} \bar{p}, \end{gather}

(3.14c)\begin{gather}\bar{{\boldsymbol{F}}}_{A} ={-} \frac{C_{m}}{\bar{\rho}} \left( \frac{\mathrm{d} \bar{{\boldsymbol{u}}}_{p}}{\mathrm{d} \bar{t}} - \frac{\mathrm{d} \bar{{\boldsymbol{U}}}_{f}}{\mathrm{d} \bar{t}} \right), \end{gather}

where $\bar {\rho } = \rho _{p}/\rho _{f}$ is the ratio of densities. In this paper, $\bar {\rho } = 2$, which we considered as the upper bound of densities of the existing food particles. According to Espinosa-Gayosso et al. (Reference Espinosa-Gayosso, Ghisalberti, Ivey and Jones2015), a sediment-type particle that is 2.6 times heavier than water achieves a capture efficiency equal to that of a perfect fluid tracer if the Stokes number $Stk = \bar {\rho } R^{2} {\textit{Re}}/9$ is less than 0.1 at ${\textit{Re}} \sim 100$. For a typical simulation case at this Reynolds number, we have $Stk \sim 0.06$ (e.g. taking $R \sim 0.05$), therefore, the choice of $\bar {\rho } = 2$ is legitimate.

3.3. Numerical solving

We solved the particle advection problem in two stages. First, we computed the FSI problem in an ALE framework using the in-house solver Cadyf (Etienne et al. Reference Etienne, Garon and Pelletier2009). This code was verified with the method of manufactured solutions (Hay et al. Reference Hay, Yu, Etienne, Garon and Pelletier2014; Yu et al. Reference Yu, Étienne, Hay and Pelletier2015) and for classical FSI problems such as airfoil plunging and pitching, and a flexible strip at the rear of a square cylinder (Cori et al. Reference Cori, Etienne, Garon and Pelletier2015). The code was also validated and produced dynamical results consistent with the data in the literature for several flow-induced vibrations including VIVs of circular (Etienne & Pelletier Reference Etienne and Pelletier2012) and square (Hay et al. Reference Hay, Etienne, Garon and Pelletier2015a) cylinders, and wake-induced vibrations of cylinders in tandem (Yu et al. Reference Yu, Étienne, Scolan, Hay, Fontaine and Pelletier2016).

Initially, the fluid was at rest, $\bar {{\boldsymbol {U}}}_{f} = \boldsymbol {0}$ and $\bar {p} = 0$, and ramped up to unity $\bar {{\boldsymbol {U}}}_{f} = \boldsymbol {e}_{x}$ in a few time steps to satisfy a divergence-free flow field. Also at the initial time, the cylinder stood at the position $\bar {{\boldsymbol {X}}}_{cyl} = \boldsymbol {0}$. Cadyf integrated the coupled system of equations (3.4) using hp-adaptive backward differential formula (BDF) methods (Hay et al. Reference Hay, Etienne, Pelletier and Garon2015b). The order and time step adjusted automatically so that the local truncation error remained smaller than a constant absolute tolerance equal to $10^{-5}$. We extracted the flow solution at each instant $\bar {t}_{n} = n/10$ starting from $\bar {t} = 400$, long after VIVs had reached their periodic limit-cycle.

Next, we exported these data into Paradvect (2020), a Python code we wrote to integrate the system of equations (3.13) in a Lagrangian framework using a forward Euler scheme. A code verification of Paradvect is available in the other supplementary material. To satisfy stability, the integration time step ${\rm \Delta} \bar {t}$ should be smaller than the characteristic time involved in the system $\tau = (\bar {\rho } + C_{m}) R^{2}{\textit{Re}}/9$. We compared $\tau$ with the time step $\bar {t}_{n+1} - \bar {t}_{n} = 1/10$ by which we extracted the flow solution from Cadyf: if $\tau > 1/10$, which is the case for large particles, we took ${\rm \Delta} \bar {t} = 1/10$, whereas if $\tau < 1/10$, which is the case for small particles, we linearly interpolated the flow solution in time. Linear interpolation is appropriate because the flow solution varies smoothly during a time step of $\sim$1/10 and does not fluctuate over the small timescale $\tau$.

Figure 4(b) juxtaposes the cylinder and the considered particles to give an idea on their relative size. The trajectory of every particle was integrated from a unique starting line $\bar {x}_{0} = 2$ upstream from the cylinder for several ordinates $\bar {y}_{0}$. The flow in these positions was horizontal and unperturbed. The local fluid velocity was then assigned as their initial velocity.

To calculate the total hydrodynamic force applied on the particle, we quadratically interpolated the fluid velocity and linearly interpolated the pressure at the particle centre. As this calculation requires knowledge of the finite element hosting the particle, we adopted the particle tracer algorithm proposed by Löhner & Ambrosiano (Reference Löhner and Ambrosiano1990). It searches recursively, neighbour to neighbour, the new host element in the vicinity of the known previous host element. This tracking technique is suitable, time-saving and easy to implement in particle-in-cell codes, such as ours, where physical particles do not jump over many elements in a single time step (Löhner Reference Löhner2008).

Finally, we assumed that the particle was captured as soon as it hit the edge of the cylinder. Because the particles were spherical and the cylinder was circular, the capture occurred when $\left |\left | \bar {{\boldsymbol {x}}}_{p}-\bar {{\boldsymbol {X}}}_{cyl} \right |\right |~\le (1+R)/2$. We refer to this capture condition as the solid contact criterion.

4.1. Definition

To assess the filtering ability of the cylinder, we calculated the rate at which it intercepts particles. We defined the capture rate $\dot {N}$ as the number of particles that the cylinder captures per unit time. As represented in figure 6, the particle that would be ultimately captured necessarily entered through an opening that we called the capture window. Thereby, the capture rate also equals the flux of particles through this capture window

(4.1)\begin{equation} \dot{N} = C_{0}U_{0}w, \end{equation}

where $C_{0}$ is the particle concentration per unit length, which we assumed to be constant and uniform, and $w$ is the size of the capture window.

Figure 6.

Schematics of particle interception by a fixed cylinder. Particles are launched from a distance $x_{0}$ upstream from the cylinder. Because the blue particle starts near the symmetry line, it is captured, whereas the red particle starts from a higher $y$-position and succeeds in crossing over the cylinder and escaping capture. The grey particle is the farthest particle from the symmetry line that the cylinder intercepts. Its trajectory starts from the ordinate $y^+$, and defines the upper border of the capture domain. The size of the capture window is $w = y^+ - y^{-}$.

4.2. Calculation strategy

4.2.1. Automated dichotomy

Owing to the definition (4.1), we determined the capture rate through the calculation of the size of the capture window. The upper and lower boundaries of the capture region were the trajectories of the farthest captured particles, as shown in figure 6. Thus, it suffices to calculate the initial ordinates of these border particles, denoted $y^+$ and $y^{-}$, because

(4.2)\begin{equation} w = y^+{-} y^{-}. \end{equation}

First, we considered a captured particle (in blue, figure 6) released from ($x_{0}, y^{c}$), and a non-captured particle (in red) released from ($x_{0}, y^{nc}$). Their initial positions were necessarily bounds of either border, say the upper border ($y^{c} < y^+ < y^{nc}$). Next, we released a third particle between the two previous ones, from the middle of their initial ordinates $y^{m} = (y^{c} + y^{nc})/2$. If it escaped capture, then the upper border $y^+$ was necessarily between the initial ordinates of this new particle and the former captured one ($y^{c} < y^+ < y^{m}$). The code Paradvect (2020) repeated this dichotomic process until it reached a resolution of one thousandth of the particle diameter $| y^{c} - y^{m} | < d_{p}/1000$.

4.2.2. Temporal decomposition

When the cylinder vibrates, the borders of the capture region vary in time $y^{\pm } = y^{\pm }(t)$, and the capture strip is no longer straight, as seen in figure 7. A particle would be captured if it is launched at time $t$ between $y^+(t)$ and $y^{-}(t)$. The instantaneous capture window $w(t) = y^+(t) - y^{-}(t)$ must be periodic, thus we propose the following ansatz

(4.3)\begin{equation} w(t) = \langle w \rangle + w_{a} \sin\left( \frac{2{\rm \pi} t}{T} + \varphi \right). \end{equation}

The period $T$ is not equal to the vibration period, but half of it. Indeed, as illustrated in figure 7, the cylinder describes a lemniscate, so it captures particles in the same way whether during the upper or the lower loop. Therefore, the capture rate is determined by the stream-wise – not transverse – oscillations, which complete a cycle in half a period.

Figure 7.

Distortion of the capture domain owing to vibration. The initial ordinates of the grey border particles are now varying in time $y^{\pm } = y^{\pm }(t)$. The instantaneous capture window, through which the particles launched at time $t$ would be captured, is $w(t) = y^+(t) - y^{-}(t)$. At instants $t$ between $t_{1}$ and $t_{2} > t_{1}$, the blue particles are launched between $y^{-}(t)$ and $y^+(t)$, so they are captured. The red particles are launched outside of these limits, so they escape capture. A portion of particle trajectories is shown by the solid line. The cylinder draws a lemniscate trajectory, and $Y_{max}$ and $X_{max}$ are the transverse and stream-wise amplitudes of vibration. The motion of the cylinder is symmetric with respect to the horizontal line passing through the node of the lemniscate, hence the capture is periodic and goes in harmony with the stream-wise oscillations, which complete a cycle in half a period.

We should mention that the dependency on time is also true for the fixed cylinder, albeit with a much smaller fluctuation. For the fixed cylinder, the variation of $w(t)$ arises from the periodic vortex shedding, which affects the streamlines in the front, and the period $T$ in the ansatz (4.3) is equal to the vortex shedding period.

We calculated the size of the capture window at instants $t = 0$, $T/4$, and $T/2$, and determined the mean and fluctuating components $\langle w \rangle$ and $w_{a}$ in (4.3) from the ensuing formulae

(4.4a)\begin{gather} \langle w \rangle = \frac{w(0) + w(T/2)}{2}, \end{gather}

(4.4b)\begin{gather}w_{a} = \sqrt{ \left[ w(0) - \langle w \rangle \right]^{2} + \left[ w(T/4) - \langle w \rangle \right]^{2} }. \end{gather}

4.3. Dimensional analysis

The capture rate is a function of the fluid flow, cylinder and particle properties

(4.5)\begin{align} \left.\begin{gathered} \dot{N}\ [\mathrm{s^{{-}1}}],\quad U_{0}\ [\mathrm{m\ s^{{-}1}}],\quad \rho_{f}\ [\mathrm{kg\ m^{{-}3}}],\quad \mu_{f}\ [\mathrm{kg\ m^{{-}1}s^{{-}1}}],\quad D\ [\mathrm{m}],\quad k\ [\mathrm{N\ m^{{-}2}}], \\ m\ [\mathrm{kg\ m^{{-}1}}],\quad d_{p}\ [\mathrm{m}],\quad \rho_{p}\ [\mathrm{kg\ m^{{-}3}}],\quad C_{0}\ [\mathrm{m^{{-}2}}]. \end{gathered}\right\} \end{align}

For convenience, we discarded the variation in time $t$ owing to the decomposition in (4.3). The Buckingham $\varPi$ theorem (Buckingham Reference Buckingham1914) states that a relation exists between $10 - 3 = 7$ independent dimensionless variables, which we choose to be

(4.6)\begin{equation} \left.\begin{gathered} \eta = \frac{\dot{N}}{C_{0}U_{0}D},\quad {\textit{Re}} = \frac{\rho_{f} U_{0} D}{\mu_{f}},\quad R = \frac{d_{p}}{D},\quad U_{r} = \frac{2{\rm \pi} U_{0}}{D} \sqrt{\frac{m}{k}}, \\ \bar{\rho} = \displaystyle\frac{\rho_{p}}{\rho_{f}},\quad M = \frac{m}{\rho_{f} D^{2}},\quad \bar{C}_{0} = C_{0}D^{2}. \end{gathered}\right\} \end{equation}

In this way, the dimensionless capture rate $\eta$ is also the capture efficiency defined by Weber & Paddock (Reference Weber and Paddock1983), Palmer et al. (Reference Palmer, Nepf, Pettersson and Ackerman2004), and Espinosa-Gayosso et al. (Reference Espinosa-Gayosso, Ghisalberti, Ivey and Jones2012, Reference Espinosa-Gayosso, Ghisalberti, Ivey and Jones2013), which itself, from the definition (4.1), equals the dimensionless size of the capture window

(4.7)\begin{equation} \eta = \frac{\dot{N}}{C_{0}U_{0}D} = \frac{w}{D} = \bar{w}. \end{equation}

According to (4.3), it follows that

(4.8)\begin{equation} \eta = \langle \eta \rangle + \eta_{a} \sin\left( \frac{2{\rm \pi} t}{T} + \varphi \right), \end{equation}

with $\langle \eta \rangle = \langle w \rangle /D$ and $\eta _{a} = w_{a}/D$. For all simulations in the present work, we kept a constant and uniform particle concentration $\bar {C}_{0} =$ const. In addition, as we assumed in § 3, we chose $M = 1$ and $\bar {\rho } = 2$. Henceforth, the remaining variables that determine the capture rate are the Reynolds number ${\textit{Re}}$, the diameter ratio $R$ and the reduced velocity $U_{r}$.