1. Introduction
The interaction of a vortex ring with porous walls is an important branch of vortex–wall interactions. It is also a basic model for understanding the complex mechanism of flows interacting with porous media in various applications, such as filter membranes, energy absorbers, insulation and acoustic materials, and flow conditioning devices (Bradshaw Reference Bradshaw1965; Ali et al. Reference Ali, Rady, Atti and Ewais2020). The effects of the Reynolds number (
${{Re}}$
), porosity (
$\phi$
) and shape parameters of porous walls, such as the wire structure size and wall thickness, on the associated vortex dynamics have been the primary focuses of research. Adhikari & Lim (Reference Adhikari and Lim2009) reported that the critical
${{Re}}$
at which the flow passed through a porous wall to form a transmitted vortex ring depended on
$\phi$
. They ascribed the mechanism of vorticity transmission at a high
${{Re}}$
to vortex stretching and reconnection. Hrynuk, Luipen & Bohl (Reference Hrynuk, Van Luipen and Bohl2012) considered the effect of the wire structure size for a constant
$\phi$
based on flow visualisation. In the case of fine wire mesh, a transmitted vortex ring immediately formed downstream of the porous wall. Conversely, as the wire diameter increased, the vortex ring was disrupted and did not re-form because of the generation of fine-scale vortical structures. Naaktgeboren et al. (Reference Naaktgeboren, Krueger and Lage2012) employed particle image velocimetry (PIV) to measure the flow fields of a laminar vortex ring impinging on a porous wall. They revealed that, compared to
${{Re}}$
,
$\phi$
had a predominant effect on the kinetic energy loss. In particular, they proposed a model relating the losses in kinetic energy (
$\Delta E$
) and impulse (
$\Delta I$
) to elucidate their different dependence on
$\phi$
, and to clarify the effect of
$\phi$
on fluid dynamics. A wide range of parameters were examined numerically by Cheng, Lou & Lim (Reference Cheng, Lou and Lim2014). They concluded that vorticity transmission was enhanced by increasing
${{Re}}$
and
$\phi$
, while being impeded by reducing
$\phi$
and increasing the wall thickness. Furthermore, a larger wire structure size promoted vortex shedding from the wire mesh, and reduced the coherence of the transmitted ring. This flow behaviour was further investigated by Hrynuk, Stutz & Bohl (Reference Hrynuk, Stutz and Bohl2018). They found that the formation of a transmitted ring was a function of vortex shedding behaviour and not
$\phi$
, which could be characterised by an interaction Reynolds number (defined by the wire diameter and the convection velocity of the vortex ring). Recently, Jain et al. (Reference Jain, Sharma, Roy and Basu2023) focused on the interaction of an underwater vortex ring with an oil-impregnated porous wall, providing a specific example for the potential application of the vortex ring–porous wall interaction.
In addition to the case of a single vortex ring, the interaction of successive vortex rings generated by synthetic jets with porous walls has attracted attention because of their more practical applications in mixing and heat transfer. Xu et al. (Reference Xu, Wang, Feng, He and Wang2018) varied the hole diameter (
$d_h$
) while maintaining a constant
$\phi$
. They claimed that
$d_h$
significantly influenced the formation of transmitted vortex rings through a vorticity cancellation mechanism. Li, Xu & Wang (Reference Li, Xu and Wang2020) and Xu, Li & Wang (Reference Xu, Li and Wang2021) introduced several similarity parameters that incorporated
${{Re}}$
,
$d_h$
and
$\phi$
to characterise the losses in kinetic energy and momentum flux caused by vortex rings impacting a porous wall. Sun et al. (Reference Sun, Sun, Fan, Yu, Chen, Kong and Yu2023) used a novel similarity parameter that further considered the wall thickness for scaling related dynamic quantities.
The losses in kinetic energy and momentum/impulse constitute important measures of the fluid transport capability in the vortex ring–porous wall interaction. In this regard, we can recall the model proposed by Naaktgeboren et al. (Reference Naaktgeboren, Krueger and Lage2012), which considered the effects of the drag force from the porous wall and fluid deceleration on the energy loss process. However, their model related
$\Delta E$
and
$\Delta I$
only in terms of bulk flow parameters, while neglecting the key role of the vortex ring characteristics. Moreover, multiple approximation processes were adopted during the model derivation. For example, the total drag force was replaced by the time-averaged drag force; it was assumed that there was a linear relationship between the drag force and the flow velocity. This led to rather rough agreement between the model and the measured results. Therefore, the present study aims to construct a more accurate model to characterise the flow physics of vortex ring–porous wall interactions.
2. Experimental set-up and methods
The experiments were carried out in a cubic plexiglass water tank measuring
$600$
mm with wall thickness
$20$
mm. As shown in figure 1(a), a piston–cylinder apparatus was used to generate the vortex ring, with a sinusoidal velocity programme for the piston. In each experiment, the servo motor operated within half a sinusoidal cycle at the set frequency. Moreover, an eccentric wheel drove the connecting rod to move from the same initial position, which was precisely determined through repeated pre-tests, ensuring that the piston maintained a forward stroke in the cylinder. Two frequencies,
$f = 0.5$
and
$1$
Hz, were selected in this study. The piston diameter was
$D_i = 29$
mm, and the jet exit diameter was
$D_0 = 15$
mm. Considering that the eccentricity of the eccentric disc (
$e = 5.5$
mm) was far smaller than the length of the connecting rod (
$l = 300$
mm), based on mass conservation of the incompressible flow, the instantaneous velocity at the jet exit
$u_0$
is calculated as (Xu et al. 2021)
\begin{equation} u_0(t) = u_p(t)\left ({D_i}/{D_0}\right )^2 \approx 2\unicode{x03C0} ef\left ({D_i}/{D_0}\right )^2\sin (2\unicode{x03C0} ft),\quad 0 \leqslant t \leqslant T_p, \end{equation}
where
$u_p$
is the piston velocity, and
$T_p\ (= 1/2f)$
is the motion time. This piston velocity programme produces a ratio of acceleration impulse to total impulse of
$0.5$
, at which no stopping vortex forms and interacts with the primary vortex ring (Das, Bansal & Manghnani Reference Das, Bansal and Manghnani2017). The time-averaged exit velocity
$U_0$
is further given by
\begin{equation} U_0 = \frac {1}{T_p}\int _{0}^{T_p}u_0(t)\,{\textrm d}t = 4ef\left ({D_i}/{D_0}\right )^2. \end{equation}
(a) Schematics of the vortex ring generator, experimental set-up and porous wall. The LIF visualisation (b) in the
$x$
–
$y$
plane of part of fluid passing through the porous wall (indicated by the white region at approximately
$x=0$
) and (c) in the
$y$
–
$z$
plane at
$x/D_0 = 0.67$
(indicated by the dashed white line in (b)) at the same instant for
${{Re}}_{\it \Gamma } = 700$
,
$d_{h}^{*} = 0.133$
.
With
$U_0$
, the vortex rings were generated at a constant stroke ratio
$L_0/D_0 = U_0T_p/D_0 = 2.7$
, ensuring that no trailing jet formed behind the primary vortex ring (Gharib, Rambod & Shariff Reference Gharib, Rambod and Shariff1998). For the two frequencies, the jet Reynolds numbers are
${{Re}} = U_0D_0/\nu = 617$
and
$1234$
, where
$\nu$
is the kinematic viscosity of water. Based on the circulations estimated from PIV measurements during vortex ring translation, the corresponding circulation Reynolds numbers are approximately
${{Re}}_{\it \Gamma } = {\it \Gamma }/\nu \approx 700$
and
$1800$
, respectively, indicating that the current vortex rings were laminar. Furthermore, following Saffman (Reference Saffman1970), the vortex ring geometry is defined using the integral parameters, resulting in the dimensionless core radius
$\varepsilon = a/R = 0.28$
and
$0.24$
, where
$a$
is the core radius calculated from the radial moments of the azimuthal vorticity (Archer, Thomas & Coleman Reference Archer, Thomas and Coleman2008), and
$R$
is the ring radius.
The porous wall, which was fabricated from aluminium plates via a high-precision computer numerical control machine, was placed vertically at the centre of the tank, with distance between its upstream surface and the jet exit
$H_0 = 49$
mm. The porous wall measured
$150$
mm (
$10D_0$
) in both length and width, with thickness
$2$
mm. In this study, four groups of circular holes with diameter
$d_h$
and pitch
$d_p$
were considered when maintaining a constant porosity
$\phi$
(
$= 0.906d_h^2/d_p^2$
) of
$0.75$
, where the normalised hole diameter was
$d_h^{*} = d_h/D_0 = 0.067$
,
$0.10$
,
$0.133$
and
$0.20$
. The centre of the porous wall was defined as the origin of the coordinates, with the
$x$
-axis along the jet centreline. Both Cartesian and polar coordinate systems were used, with the
$y$
- and
$z$
-axes, as well as the
$r$
- and
$\theta$
-directions, depicted in figure 1(a).
Laser-induced fluorescence (LIF) flow visualisation and time-resolved PIV were employed. For PIV, a
$1$
mm thick laser sheet generated by an
$8$
W continuous laser passed through the jet centreline and illuminated the upstream and downstream flow fields of the porous wall in the
$x$
–
$y$
plane. A high-speed CMOS camera (Photron Fastcam SA2) equipped with a Nikon lens (AF
$105$
mm/f
$2.8$
) captured the particle images at sampling frequencies
$180$
Hz and
$360$
Hz for two Reynolds numbers. The camera resolution was
$2048\times 960$
pixels, with magnification
$0.068$
mm pixel
$^{-1}$
, covering field range approximately
$139\ \text{mm}\times65\ \text{mm}$
(
$9.3D_0 \times 4.3D_0$
) in the axial and radial directions. The multipass iterative Lucas–Kanade algorithm was used to derive the velocity vector fields, and the final size of the interrogation window was
$32 \times 32$
pixels with a
$75$
$\%$
overlap. With the assumed
$0.1$
pixel accuracy in particle displacement, the velocity uncertainty was less than
$2$
$\%$
of the maximum jet velocity.
For LIF, Rhodamine 6G was used as the fluorescent dye to capture the flow evolution in the
$x$
–
$y$
plane and
$y$
–
$z$
planes downstream of the wall at
$x/D_0 = 0.67$
,
$1.33$
and
$3.33$
. Figures 1(b) and 1(c) present a pair of LIF images in the two planes at the same instant for
${{Re}}_{\it \Gamma } = 700$
and
$d_h^{*} = 0.133$
. The results show that after impingement, part of the fluid of the primary vortex ring passes through the porous wall to form jet-like structures. These structures rapidly roll up and present as layers of petal-shaped structures along the azimuthal direction in the
$y$
–
$z$
plane (figure 1
c). With further merging of petal-shaped structures, they evolve into a large-scale and coherent transmitted vortex ring, as indicated later.
3. Results and discussion
3.1. Flow structure evolution
Figure 2 shows the evolution of the LIF visualisation and azimuthal vorticity (
$\omega _{\theta }$
) field for the cases
$d_{h}^{*} = 0.067$
, 0.133 and
$0.20$
at
${{Re}}_{\it \Gamma }= 700$
. The evolution for
$d_{h}^{*} = 0.10$
is similar to that for
$d_{h}^{*} = 0.133$
, thus is not provided for the sake of brevity. The initial instant (
$t = 0$
) is defined as the moment when the core of the primary vortex ring reaches
$x/D_0 = - 2$
. When the primary vortex ring approaches the porous wall, the transmission of vorticity near the wall on the downstream side is more prominent for a larger
$d_{h}^{*}$
(
$tU_0/D_0 = 2.5$
). After impingement, the primary vortex ring spreads along the wall, and the wall shear layer upstream separates to form a secondary vortex ring, which pairs with the primary vortex ring. As
$d_{h}^{*}$
increases, the primary and secondary vortex rings exhibit a short spreading distance and a lower strength (
$tU_0/D_0 = 5.5$
and
$8.5$
). In particular, for the smallest
$d_{h}^{*}\ (= 0.067)$
, the stronger primary vortex ring can cause the wall boundary layer to separate again and thereby a tertiary vortex ring (
$tU_0/D_0 = 11.5$
in figure 2
a). On the downstream side, as
$d_{h}^{*}$
increases, more fluid from the incident vortex ring passes through the porous wall and rapidly reorganises into the transmitted vortex ring (
$tU_0/D_0 = 5.5$
and
$8.5$
). Consequently, the strength of the transmitted vortex ring increases with
$d_{h}^{*}$
. Furthermore, the transmitted vortex ring for
$d_{h}^{*} = 0.20$
becomes less coherent than that for
$d_{h}^{*} = 0.133$
(
$tU_0/D_0 = 11.5$
in figures 2
b,c), which is attributed to insufficient vorticity cancellation between jet structures in the case of a large hole diameter (Xu et al. Reference Xu, Wang, Feng, He and Wang2018).
Evolution of LIF visualisation (above the symmetry axis) and the azimuthal vorticity
$\omega _{\theta }D_{0}/U_{0}$
field (below the symmetry axis) at
$tU_{0}/D_{0} = 2.5$
,
$5.5$
,
$8.5$
and
$11.5$
for
${{Re}}_{\it \Gamma } = 700$
: (a)
$d_{h}^{*} = 0.067$
, (b)
$d_{h}^{*} = 0.133$
and (c)
$d_{h}^{*} = 0.20$
. The initial instant
$t = 0$
corresponds to the moment when the core of the primary vortex ring reaches
$x/D_{0} = -2$
.
3.2. Impulse and kinetic energy
The fluid kinetic energy
$E$
and impulse
$I$
are crucial parameters for measuring the effect of the porous wall on the flow dynamics. For an axisymmetric flow, they are calculated as (Naaktgeboren et al. Reference Naaktgeboren, Krueger and Lage2012)
\begin{align} I &= \unicode{x03C0} \rho \iint \omega _{\theta } r^{2}\, \textrm{d}r\,\textrm{d}x, \end{align}
\begin{align} E &= \unicode{x03C0} \rho \iint \boldsymbol{u}\,{\boldsymbol\cdot}\, \boldsymbol{u}\,r\,\textrm{d}r\,\textrm{d}x, \end{align}
where
$\rho$
is the density of water. Figure 3 presents the evolution of the normalised impulse
${I}/{\rho U_{0} D_{0}^{3}}$
and kinetic energy
${E}/{\rho U_{0}^{2} D_{0}^{3}}$
in the upstream and downstream regions of the porous wall for all the cases examined in this study. The upstream and downstream flow quantities are uniformly denoted as
$({\cdot})_{\textit {up}}$
and
$({\cdot})_{\textit {do}}$
. The results represent the average values calculated by using the data above and below the jet centreline to reduce the influence of any slight asymmetry in the flow evolution.
Evolution of normalised (a,b) impulse
${I}/{\rho U_{0} D_{0}^{3}}$
and (c,d) kinetic energy
${E}/{\rho U_{0}^{2} D_{0}^{3}}$
in the upstream and downstream regions for all the cases: (a,c)
${{Re}}_{\it \Gamma}=\,$
700 and (b,d)
$Re_{\it \Gamma}=\,$
1800. The dashed black lines indicate the local mean values within the upstream and downstream constant stages, taking the case
$d_{h}^{*} = 0.20$
as an example.
In the initial stage, when the vortex ring translates from
$x/D_0 = - 2$
towards the wall, it shows approximately constant
$I_{\textit {up}}$
and
$E_{\textit {up}}$
values. However, after normalisation based on the slug model approximation, these values show distinct differences for two
${{Re}}_{\it \Gamma }$
. This is because the current vortex rings are generated by the orifice geometry. The orifice plate imposes a high turning angle at the exit, resulting in a non-negligible radial velocity component (Limbourg & Nedić Reference Limbourg and Nedic2021). Previous studies (Krieg & Mohseni Reference Krieg and Mohseni2013; Limbourg & Nedić Reference Limbourg and Nedic2021) suggest that the radial velocity can lead to a significant increase in
$\it \Gamma$
,
$I$
and
$E$
for orifices or conical nozzles compared to the equivalent slug model. Moreover, such differences enlarge as the Reynolds number increases (Rosenfeld, Katija & Dabiri Reference Rosenfeld, Katija and Dabiri2009). Therefore, the normalised values are larger for
${{Re}}_{\it \Gamma } = 1800$
.
As the vortex ring interacts with the porous wall,
$I_{\textit {up}}$
and
$E_{\textit {up}}$
rapidly decrease. During the vortex–wall interaction, both
$I_{\textit {do}}$
and
$E_{\textit {do}}$
continuously increase as the incident flow gradually penetrates the porous wall. After interaction,
$I_{\textit {up}}$
and
$E_{\textit {up}}$
of the residual flow upstream decay slowly due to viscous dissipation. Conversely, with the full formation of the transmitted vortex ring,
$I_{\textit {do}}$
and
$E_{\textit {do}}$
stabilise at a constant value. Furthermore, increasing
${{Re}}_{\it \Gamma }$
and
$d_{h}^{*}$
leads to a faster dissipation of
$I_{\textit {up}}$
and
$E_{\textit {up}}$
, as well as larger final magnitudes of
$I_{\textit {do}}$
and
$E_{\textit {do}}$
. In particular, the net effect of the porous wall on fluid transport can be characterised by the losses in kinetic energy and impulse as
\begin{align} \Delta I &= \overline {I}_{\textit {up}} - \overline {I}_{\textit {do}}, \end{align}
\begin{align} \Delta E &= \overline {E}_{\textit {up}} - \overline {E}_{\textit {do}}, \end{align}
where
$\overline {({\cdot})}$
represents the local mean values within the constant stage, as exemplified by the dashed black lines in figure 3.
3.3. A model relating
$\Delta E$
and
$\Delta I$
Considering the situation where the fluid of the primary vortex ring convects downstream as a vortex ring after passing through the porous wall, it is reasonable to construct a model based on the vortex ring characteristics to correlate
$\Delta E$
and
$\Delta I$
. Based on Lamb’s model (Lamb 1932) of a vortex ring whose cross-section has a much smaller core radius than the ring radius (
$a \ll R$
), for such a vortex ring with circulation
$\it \Gamma$
moving in an inviscid fluid of density
$\rho$
, its impulse and kinetic energy are expressed as (Sullivan et al. Reference Sullivan, Niemela, Hershberger, Bolster and Donnelly2008)
\begin{align} I &= \rho {\it \Gamma }\unicode{x03C0} R^{2}, \end{align}
\begin{align} E &= \frac {1}{2} \rho {\it \Gamma }^{2} R\left (\ln \frac {8R}{a} - \frac {7}{4}\right ). \end{align}
These are applicable to the primary and transmitted vortex rings during their free-travelling stage. Thus by using (3.5), the relative loss in impulse of the fluid passing through the porous wall,
$\Delta I^{*}$
, is given as
\begin{equation} \Delta I^{*} = {\Delta I}/{I_{\textit {up}}} = ({I_{\textit {up}} - I_{\textit {do}}})/{I_{\textit {up}}} = \big({{\it \Gamma }_{\textit {up}} R_{\textit {up}}^{2} - {\it \Gamma }_{\textit {do}} R_{\textit { do}}^{2}}\big)/{{\it \Gamma }_{\textit {up}} R_{\textit {up}}^{2}}. \end{equation}
For the current high porosity (
$\phi = 0.75$
), when a highly coherent transmitted vortex ring is formed, its diameter is similar to that of the primary vortex ring, i.e.
$R_{\textit {do}} \approx R_{\textit {up}}$
, which has also been verified in previous studies (Hrynuk et al. Reference Hrynuk, Van Luipen and Bohl2012; Naaktgeboren et al. Reference Naaktgeboren, Krueger and Lage2012; Xu et al. Reference Xu, Wang, Feng, He and Wang2018) with similar porosities (
$\phi = 0.64$
–
$0.79$
). Using this result, (3.7) gives
\begin{equation} \Delta I^{*} \approx ({\it \Gamma _{\textit {up}} - \it \Gamma _{\textit {do}}})/{\it \Gamma _{\textit {up}}}. \end{equation}
Likewise, the relative loss in kinetic energy,
$\Delta E^{*}$
, is given by
\begin{equation} \Delta E^{*} = {\Delta E}/{E_{\textit {up}}} = ({E_{\textit {up}} - E_{\textit {do}}})/{E_{\textit {up}}} \approx \big({{\it \Gamma }_{\textit {up}}^{2} - {\it \Gamma }_{\textit {do}}^{2}}\big)/{{\it \Gamma }_{\textit {up}}^{2}}. \end{equation}
Based on (3.8) and (3.9), one can derive a relationship between
$\Delta E^{*}$
and
$\Delta I^{*}$
as
\begin{equation} \Delta E^{*} = 2\,\Delta I^{*} - \Delta I^{*2}. \end{equation}
Figure 4(a) shows a comparison between the model (3.10) (indicated by the dashed line) and the measured data from the current and previous experimental studies (Naaktgeboren et al. Reference Naaktgeboren, Krueger and Lage2012; Li et al. Reference Li, Xu and Wang2020; Xu et al. Reference Xu, Li and Wang2021; Sun et al. Reference Sun, Sun, Fan, Yu, Chen, Kong and Yu2023). The error bars of the current results are determined by the standard deviation of the quantities within the constant stage in figure 3. The results indicate that for large
$\Delta I^{*}$
values (
$\gtrsim\! 0.7$
), the measured data from different studies are in good agreement with this model (see the inset in figure 4
a). However, as
$\Delta I^{*}$
and
$\Delta E^{*}$
decrease, most of the measured data evidently deviate from the model predictions. This is because small impulse and kinetic energy losses occur in cases with relatively large
${{Re}}$
,
$\phi$
or
$d_h$
. In these cases, more fluid upstream convects through the porous wall to form small-scale vortical structures, which are superimposed on the transmitted vortex ring, increasing the flow instability and thereby reducing its coherence (Hrynuk et al. Reference Hrynuk, Van Luipen and Bohl2012; Cheng et al. Reference Cheng, Lou and Lim2014). Figure 4(b) provides such an example based on the LIF visualisation for
$d_{h}^{*} = 0.20$
at
${{Re}}_{\it \Gamma } = 1800$
. The primary vortex ring is laminar before interacting with the wall (
$tU_0/D_0 = 1.0$
). However, the flow experiences significant distortion immediately downstream of the wall (
$tU_0/D_0 = 4.0$
), eventually leading to a less coherent transmitted vortex ring (
$tU_0/D_0 = 8.0$
). Therefore, it is necessary to consider the state change in the vortex rings upstream and downstream of the wall for model construction.
(a) Comparison between the model relating the losses in kinetic energy
$\Delta E^{*}$
and impulse
$\Delta I^{*}$
, and measured data from current and other studies (Naaktgeboren et al. Reference Naaktgeboren, Krueger and Lage2012; Li et al. Reference Li, Xu and Wang2020; Xu et al. Reference Xu, Li and Wang2021; Sun et al. Reference Sun, Sun, Fan, Yu, Chen, Kong and Yu2023). The dashed line indicates the model
$\Delta E^{*} = 2\,\Delta I^{*} - \Delta I^{*2}$
; the dash-dotted line indicates the model
$\Delta E^{*} = 1 - k(1 - \Delta I^{*})^2$
with
$\varepsilon = 0.24$
; the shaded region indicates the error band with
$\varepsilon = 0.20$
–
$0.28$
. (b) Evolution of LIF visualisation at
$tU_0/D_0 = 1.0$
,
$4.0$
and
$8.0$
for
${{Re}}_{\it \Gamma } = 1800$
,
$d_{h}^{*} = 0.20$
.
Since the orifice-to-wall distance is typically small in related studies, the primary vortex ring can maintain better coherence before interacting with the wall even at high
${{Re}}$
(Adhikari & Lim Reference Adhikari and Lim2009; Cheng et al. Reference Cheng, Lou and Lim2014). Thus we next consider a situation where the primary vortex ring is laminar, while the transmitted vortex ring loses coherence and becomes turbulent. In this situation, the impulse and kinetic energy of the upstream and downstream vortex rings are expressed as
\begin{align} I_{\textit {up}} &= k_1 \rho {\it \Gamma }_{\textit {up}} R_{\textit {up}}^{2}, \quad I_{\textit {do}} = k_2\rho {\it \Gamma }_{\textit {do}} R_{\textit {do}}^{2}, \end{align}
\begin{align} E_{\textit {up}} &= k_3 \rho {\it \Gamma }_{\textit {up}}^{2} R_{\textit {up}}, \quad E_{\textit {do}} = k_4 \rho {\it \Gamma }_{\textit {do}}^{2} R_{\textit {do}}, \end{align}
where
$k_1 = \unicode{x03C0}$
,
$k_3 = ({1}/{2}) (\ln ({8R}/{a}) - ({7}/{4}) )$
for the laminar state, and
$k_2 = 2.8773$
,
$k_4 = 0.5081$
for the turbulent state (Akhmetov Reference Akhmetov2009). These allow (3.8) and (3.9) to be given as
\begin{align} \Delta I^{*} &\approx ({k_1 {\it \Gamma }_{\textit {up}} - k_2 {\it \Gamma }_{\textit {do}}})/{k_1 {\it \Gamma }_{\textit {up}}}, \end{align}
\begin{align} \Delta E^{*} &\approx \big({k_3 {\it \Gamma }_{\textit {up}}^{2} - k_4 {\it \Gamma }_{\textit {do}}^{2}}\big)/{k_3 {\it \Gamma }_{\textit {up}}^{2}}. \end{align}
Based on (3.13) and (3.14), the relationship between
$\Delta E^{*}$
and
$\Delta I^{*}$
can be re-derived as
\begin{equation} \Delta E^{*} = 1 - k(1 - \Delta I^{*})^{2}, \end{equation}
where
$k = {k_1^{2} k_4}/{k_2^{2} k_3} = {1.2115}/({\ln ({8}/{\varepsilon }) - ({7}/{4})})$
is a parameter dependent on the dimensionless core radius
$\varepsilon$
. Typically,
$k$
can be estimated within a range
$\varepsilon = 0.20$
–
$0.28$
as suggested by the current results and previous studies that employed a piston–cylinder apparatus to generate a relatively thin vortex ring (Chu, Wang & Chang Reference Chu, Wang and Chang1995; Jha & Govardhan Reference Jha and Govardhan2015; Das et al. Reference Das, Bansal and Manghnani2017). Substituting
$k$
into (3.15) gives rise to the model with an error band, as shown by the shaded region in figure 4(a). Specifically, the dash-dotted line represents the expression with the median value
$\varepsilon = 0.24$
, which shows excellent agreement with the measured data of a total of 76 cases.
Relative error between the measured and predicted kinetic energy losses for (a) the previous model of Naaktgeboren et al. (Reference Naaktgeboren, Krueger and Lage2012),
$RE_{pre}$
, and (b) the current model,
$RE_{cur}$
. The dashed red lines indicate the mean relative error (
$\textit{MRE}$
).
To assess the prediction accuracy of the model, figure 5 compares the relative error between the measured and predicted kinetic energy losses for the previous model of Naaktgeboren et al. (Reference Naaktgeboren, Krueger and Lage2012),
$RE_{pre}$
, and the current model,
$RE_{cur}$
. They are calculated as
\begin{equation} RE_{pre} = \frac {|\Delta E - \Delta E_{model,pre}|}{\Delta E}, \end{equation}
\begin{equation} RE_{cur} = \frac {|\Delta E^{*} - \Delta E^{*}_{model,cur}|}{\Delta E^{*}}, \end{equation}
where
$\Delta E_{model,pre} = \Delta I\,({U_r}/{\phi }) (1 - ({\Delta I}/{2I_{up})})$
, with
$U_r$
being the convection velocity of the upstream vortex ring as proposed by Naaktgeboren et al. (Reference Naaktgeboren, Krueger and Lage2012), and
$\Delta E^{*}_{model,cur} = 1 - k(1 - \Delta I^{*})^2$
, with
$\varepsilon = 0.24$
corresponding to (3.15). Note that the data of Sun et al. (Reference Sun, Sun, Fan, Yu, Chen, Kong and Yu2023) cited in figure 4(a) are not presented since
$U_r$
in the expression of
$\Delta E_{model,pre}$
cannot be directly obtained from their paper. As shown by the dashed red lines, compared with the model of Naaktgeboren et al. (Reference Naaktgeboren, Krueger and Lage2012), which yields mean relative error (
$\textit{MRE}$
)
$41.9$
$\%$
for all the tested cases, the model (3.15) significantly decreases the
$\textit{MRE}$
to
$3.7$
$\%$
. Moreover, even for the model (3.10), the
$\textit{MRE}$
of
$8.4$
$\%$
is still much smaller than the value obtained from their model. This indicates that the prediction accuracy has been significantly improved by the newly defined model. Such results are related to the fact that the current model derivation relies only on the flow state and characteristics of the upstream and downstream vortex rings rather than the specific details of the vortex–wall interaction, thereby avoiding many approximation processes used during the model derivation in Naaktgeboren et al. (Reference Naaktgeboren, Krueger and Lage2012).
Notably, the current model implies that the correlation between
$\Delta E^{*}$
and
$\Delta I^{*}$
seems to be independent of the geometry parameters of the porous wall. This should be the reason why this model could be applicable to cases of various porous wall geometries in different studies, such as the woven wire mesh with square holes in Naaktgeboren et al. (Reference Naaktgeboren, Krueger and Lage2012), and the perforated sheet with circular holes in this study, Li et al. (Reference Li, Xu and Wang2020) and Xu et al. (Reference Xu, Li and Wang2021), as well as the honeycomb with hexagonal holes in Sun et al. (Reference Sun, Sun, Fan, Yu, Chen, Kong and Yu2023). Additionally, Hrynuk et al. (Reference Hrynuk, Stutz and Bohl2018) excluded the effect of
$\phi$
when defining an interaction Reynolds number to describe the vortex ring–porous wall interaction. The present results supply a plausible theoretical explanation for this finding. The above analysis suggests that the current model has captured the dominant flow physics of the vortex ring–porous wall interaction.
A further discussion of the current model is presented below. The expression directly reveals the inherent nonlinear correlation between
$\Delta E^{*}$
and
$\Delta I^{*}$
, and reflects their magnitude relationship
$\Delta E^{*} \geqslant \Delta I^{*}$
. Furthermore, when
$\Delta I^{*}$
approaches 1, model (3.15) approaches (3.10), and both predicted
$\Delta E^{*}$
values tend to 1 (figure 4
a), suggesting a reduced effect of
$k$
. This is because
$\Delta I^{*} = \Delta E^{*} = 1$
represents the extreme case where the vortex ring impinges onto a solid wall. For the other extreme case, where the vortex ring freely develops in the absence of the porous wall, no flow state change occurs for the upstream and downstream vortex rings. In this case,
$k_1 = k_2$
and
$k_3 = k_4$
correspond to
$k = 1$
, enabling (3.15) to be transformed into (3.10), which covers this extreme case at
$\Delta I^{*} = \Delta E^{*} = 0$
. These findings confirm the validity of this new model. Finally, it should be noted that the current model is derived based on Lamb’s vortex ring model. Better agreement between the model predictions and the measured data might be attained when considering other theoretical models of vortex rings. Nevertheless, this lies beyond the scope of this study.
4. Conclusions
In this study, a vortex ring impinging onto a porous wall at various Reynolds numbers (
${{Re}}_{\it \Gamma } = 700$
and
$1800$
) and hole diameters (
$d_{h}^{*} = 0.067$
,
$0.10$
,
$0.133$
and
$0.20$
) is investigated via laser-induced fluorescence and particle image velocimetry at a constant porosity (
$\phi = 0.75$
). The evolution of the flow structures, as well as the kinetic energy and impulse in the upstream and downstream regions of the porous wall, is examined. To characterise fluid transport through a porous wall, starting from Lamb’s vortex ring model, a new model is derived theoretically, based on vortex ring characteristics to correlate the losses in kinetic energy
$\Delta E^{*}$
and impulse
$\Delta I^{*}$
. This new model comprehensively considers the flow transition from the upstream laminar state to the downstream turbulent state caused by the porous wall disturbance, and is expressed as
$\Delta E^{*} = 1 - k(1 - \Delta I^{*})^2$
, where
$k$
is a parameter dependent on the dimensionless core radius
$\varepsilon$
, with
$k = 1$
when no flow state change occurs. Using this new model, the correlation between
$\Delta E^{*}$
and
$\Delta I^{*}$
shows excellent agreement with the measured data of more than 70 cases from current and previous experiments, thereby achieving significantly improved accuracy compared with the model proposed by Naaktgeboren et al. (Reference Naaktgeboren, Krueger and Lage2012). Furthermore, this model could be applicable to various porous wall geometries, indicating that it captures the dominant flow physics of the vortex ring–porous wall interaction.
The current model can function as a simplified method for predicting
$\Delta E^{*}$
of the fluid passing through a porous wall based on the vortex ring characteristics estimated from flow visualisation results, without the need for quantitative measurements. Although the model is a posteriori predictive model, it could provide an important reference for establishing priori similarity parameters to characterise the vortex ring–porous wall interaction, as investigated in Li et al. (Reference Li, Xu and Wang2020). Furthermore, it may potentially contribute to the modelling of the interaction between vortex flows and porous media. For example, it can be used as a physical constraint condition incorporated into the numerical solution process or to validate computation results, thereby improving the accuracy of relevant numerical simulations.
Funding
This work was supported by the National Natural Science Foundation of China (nos 12322212, 12472281, 11902019) and the Fundamental Research Funds for the Central Universities (no. YWF-1113).
Declaration of interests
The authors report no conflict of interest.
