3.1. Downstream wake topology

Since the velocity deficit serves as a useful indicator of the downstream wake topology behind bluff bodies, we computed the velocity deficit, which is defined as the difference between the incoming mean flow and the mean longitudinal velocity around the porous cylinders, $(U_e- \langle u\rangle )/U_e$ . Figure 3 displays selected contour maps of the velocity deficit alongside a schematic representation of the porous cylinders at the origin. This facilitates in understanding the cross-section of the cylinder and the effect of permeability. For a baseline of comparison, a solid case is included to highlight the influence of permeability on the downstream wake topology. The coordinate system is normalized by the cylinder width, $D$ .

Figure 3.

Selected contour maps of the normalized velocity deficit, $(U_e - \langle u \rangle )/U_e$ , illustrating the wake topology behind the cylinders. Panels ( $a$ )–( $f$ ) depict the variation in velocity deficit behaviour with increasing Darcy number ( $ \textit{Da} $ ).

Figure 3 illustrates the evolution of the velocity deficit with increasing $ \textit{Da} $ , which highlights the main flow features influenced by the permeable nature of the cylinders. A representative subset of configurations (S, A1, B2, A4, B5, C5) is shown to span the tested $ \textit{Da} $ range and to sample both odd and even pore arrangements. The remaining cases exhibit similar trends and are omitted here for clarity. It should be noted that in figure 3 contour levels less than 0.5 were omitted to clearly identify the downstream wake topology imposed by the current porous cylinders. This means that at the edge of the coloured regions in the contour maps, the streamwise velocity has recovered 50 $\,\%$ of its undisturbed value upstream of the cylinders.

As shown in figure 3, the velocity deficit and its spatial evolution behind the cylinders are clearly dictated by the permeability and the characteristics of the bleedings through the porous structure. Even with the smallest permeability (i.e. $ \textit{Da}=2.44 \times 10^{-5}$ ) considered in this study (see figure 3 b), the downstream wake structure is significantly altered compared with the solid case shown in figure 3(a). Specifically, the overall shape of the wake becomes elongated in both longitudinal and lateral directions due to the presence of bleeding flows in these directions, featuring a smaller magnitude of velocity deficit near the porous-fluid interface compared with the solid case. With increasing $ \textit{Da} $ , the downstream wake topology is remarkably modified, as shown in figure 3(c–f). The lateral extent of the wake appears to decrease, and the magnitude of velocity deficit diminishes with increasing $ \textit{Da} $ . This observation is consistent with the findings of Nicolai et al. (Reference Nicolai, Taddei, Manes and Ganapathisubramani2020), reporting similar trends in flows through cylinder arrays at high Reynolds numbers ( $ \textit{Re} = 1.3\times 10^5$ ). This topological change in the wake is due to the fact that the higher permeability allows more fluid to pass through the porous cylinder, reducing the momentum deficit and resulting in a structural alteration of the downstream wake (Taddei et al. Reference Taddei, Manes and Ganapathisubramani2016).

Furthermore, the dependence of lateral wake extent on permeability observed here is in close agreement with previous studies of porous plates at lower Reynolds numbers. For example, Castro (Reference Castro1971) reported that increasing longitudinal bleeding through porous plates leads to a reduction in the lateral wake width, which is a consistent pattern with the present results. Similarly, Steiros, Bempedelis & Ding (Reference Steiros, Bempedelis and Ding2021) provided both theoretical and experimental analysis of wakes behind porous plates, showing that increased longitudinal bleeding reduces wake width by weakening the momentum deficit in the wake region. Their results indicated that the reduction in wake width results from attenuated velocity gradients across the shear layer, reducing downstream wake entrainment and suppressing lateral wake growth. This mechanism has also been supported in other studies involving porous geometries (Zong & Nepf Reference Zong and Nepf2012; Taddei et al. Reference Taddei, Manes and Ganapathisubramani2016; Ciuti et al. Reference Ciuti, Zampogna, Gallaire, Camarri and Ledda2021). The current results in figure 3 exhibit a similar structural trend, supporting the notion that enhanced permeability mitigates the momentum deficit and effectively reduces the lateral wake extent. The agreement between the present results and previous studies suggests that this trend is robust across different configurations of porous bluff-body and Reynolds number conditions.

It should be also noted that figure 3 captures the structural features of longitudinal bleeding flow immediately downstream of the porous cylinders depending on the pore size and the corresponding permeability. For the porous cylinders with a sufficiently small permeability, as shown in figure 3(b,c), the longitudinal bleeding appears to behave as a combined single jet. In contrast, as pore size increases, inherently leading to higher permeability, the longitudinal bleeding transitions into discrete jet structures emitted from each lattice pore. These discrete jets are demarcated by the internal boundaries of the coloured contours within the wake topology (see figure 3 d–f).

The observed transition in wake structure – from a single, combined jet to spatially discrete bleeding jets – can be further interpreted through the framework of homogenization theory (Mei & Auriault Reference Mei and Auriault1991; Zampogna & Bottaro Reference Zampogna and Bottaro2016). Classical homogenization assumes a strong scale separation between the microscopic porescale and the macroscopic domain size, quantified by the scale separation parameter $\epsilon =d_1/D$ , where $d_1$ is the unit cell size and $D$ is the cylinder width. When $\epsilon \ll 1$ , the flow through the porous medium can be approximated as spatially uniform, thereby justifying the emergence of a homogenized jet-like structure in the near wake, as illustrated in figure 3(b,c). However, in the present study, the scale separation parameter spans the range $0.05\lt \epsilon \lt 0.25$ , and for cases with larger pore size (i.e. $\epsilon \gt 0.1$ , see figure 3 d–f), the assumption of homogenization breaks down. The pores are macroscopically resolvable, and the flow exhibits discrete jet-like features aligned with the pore geometry. This breakdown highlights the limitation of homogenization theory in high-permeability regimes, where pore-resolved effects, such as localized bleeding and wake variability, become significant.

It is also worth noting that porescale inertial effects play a critical role in determining the flow characteristics particularly under the homogenization framework (Zampogna & Bottaro Reference Zampogna and Bottaro2016). These effects can be quantified using the microscopic-scale Reynolds number $ \textit{Re}_l=U^*l/\nu$ , where $U^*$ is the characteristic internal velocity through the porous matrix, and $\nu$ is the kinematic viscosity (Zampogna & Bottaro Reference Zampogna and Bottaro2016). However, in our current experimental set-up, the internal velocity cannot be captured using a standard PIV system. Accurate measurement would require advanced techniques such as refractive-index matching (Blois et al. Reference Blois, Bristow, Kim, Best and Christensen2020) or magnetic resonance velocimetry (Elkins & Alley Reference Elkins and Alley2007). In this regard, we are currently unable to systematically assess the inertial contribution at the pore level. Nonetheless, the scale-separation parameter $\epsilon$ remains a useful metric for evaluating the applicability of homogenization in porous bluff-body flows and provides an important physical basis for interpreting the observed transition from homogenized to discrete bleeding behaviour.

Figure 4.

(a) Schematic representation of the lateral wake extent, $W_L$ , for case A1. (b) Variation of the lateral wake extent, $W_L$ , as a function of $ \textit{Da} $ . The crosses bounding the symbols indicate the sensitivity of the wake extent to the threshold level (set at 50 $\%$ ) used to define the wake edge.

To further examine the influence of permeability on the wake size, the wake extent is evaluated. In this study, the lateral wake extent is only considered since the longitudinal extents for the porous cylinders are beyond the present PIV FoV under 50 $\,\%$ of the cutoff defining the wake edge. The lateral wake extent $W_L$ , defined as the maximum lateral distance reached by the wake edge, is illustrated in figure 4(a). In figure 4(b), the measured $W_L$ for all porous cases is plotted against $ \textit{Da} $ , with uncertainty indicated by cross symbols. The uncertainty in $W_L$ was quantified by performing a sensitivity analysis, varying the threshold level used to define the wake edge by $50 \pm 10\,\%$ (Nicolai et al. Reference Nicolai, Taddei, Manes and Ganapathisubramani2020), resulting in a maximum variation of $\pm 2.46\,\%$ in $W_L$ .

As seen in figure 4(b), $W_L$ expands laterally up to approximately twice the cylinder width $D$ for the lowest $ \textit{Da} $ (i.e. case A1) and decreases progressively with increasing $ \textit{Da} $ . This reduction in $W_L$ with increasing $ \textit{Da} $ is attributed to the effect of lateral bleeding, a phenomenon previously shown to strongly influence the wake size in the $y$ -direction (Taddei et al. Reference Taddei, Manes and Ganapathisubramani2016). Nicolai et al. (Reference Nicolai, Taddei, Manes and Ganapathisubramani2020) also observed a decreasing trend in wake extent related to cylinder porosity $\varPhi$ , although the relation in their study was implicit. In contrast, the present observations in figure 4(b) reveal an explicit log–linear relationship between $W_L$ and $ \textit{Da} $ . This result further emphasize the role of permeability as an important control parameter for the wake structures induced by porous cylinders (Cummins et al. Reference Cummins, Viola, Mastropaolo and Nakayama2017; Ledda et al. Reference Ledda, Siconolfi, Viola, Gallaire and Camarri2018; Seol et al. Reference Seol, Hong and Kim2023).

Based on the observations in figures 3 and 4, several noteworthy points emerge. First, bleeding flows in both longitudinal and lateral directions significantly affect shear layer development and wake formation behind porous square cylinders. For example, it has been documented that longitudinal bleeding contributes to the weakening of shear layer intensity, while lateral bleeding causes the vertical displacement of the top and bottom shear layers away from the wake core (Taddei et al. Reference Taddei, Manes and Ganapathisubramani2016; Nicolai et al. Reference Nicolai, Taddei, Manes and Ganapathisubramani2020). In this study, figure 3 illustrates a substantial decrease in velocity deficit with increasing $ \textit{Da} $ , indicating a significant attenuation in shear layer intensity. Simultaneously, figure 4 demonstrates a decreasing pattern in lateral wake extent with increasing $ \textit{Da} $ , reflecting weakened lateral bleeding. These findings are consistent with recent observations from both qualitative and quantitative perspectives (Taddei et al. Reference Taddei, Manes and Ganapathisubramani2016; Zhou & Venayagamoorthy Reference Zhou and Venayagamoorthy2019; Nicolai et al. Reference Nicolai, Taddei, Manes and Ganapathisubramani2020), reaffirming that cylinder permeability and the associated bleeding characteristics predominantly control the flow structures downstream of the porous cylinders.

Second, discrete longitudinal bleeding jets observed in cases with larger pores and higher permeability (figure 3 d–f) evolve structurally, resembling multiple plane jets as they move downstream (Miller & Comings Reference Miller and Comings1960; Zhao & Wang Reference Zhao and Wang2018). This structural similarity suggests that the discrete jets behind the porous square cylinders can be classified into two distinct regions based on their streamwise location: merging and combined regions. In the merging region, as seen in figure 3(d), individual jets curve towards the symmetry plane (i.e. $y/D=0$ ) and begin to merge. As they move downstream, these jets interact further and coalesce into a single jet in the combined region. Notably, the pore configuration of the cylinder significantly influences local flow structures immediately behind the cylinder. For instance, in the case of cylinders with odd pore configuration (figure 3 d), the momentum of the longitudinal bleeding reaches its maximum at $y/D=0$ and decreases laterally due to the presence of a central pore along the lateral centreline. In contrast, cylinders with even pore configuration (figure 3 e) generate dual jet-like structure, with the highest momentum occurring along the off-centreline, due to the locally impermeable boundary at $y/D=0$ . These differences in the longitudinal bleeding patterns and their interactions lead to complex flow dynamics within the merging region before the jets fully combine into a single structure.

Lastly, drawing an analogy from multiple plane jets (Miller & Comings Reference Miller and Comings1960; Zhao & Wang Reference Zhao and Wang2018), the current observations of the permeability effect on longitudinal bleeding provide a basis for developing an analytical model linking the structural characteristics of bleeding jets to permeability. Specifically, the combined point on the centreline marks where the jets coalesce into a single entity. Beyond this point, the effects of the initial conditions have largely dissipated, and the influence of permeability, as a global factor, prevails in the downstream bleeding flow. In our previous work (Seol et al. Reference Seol, Kim and Kim2024), we successfully proposed an analytical model for downstream flow adjustment behind porous square cylinders in relation to permeability. We believe that the connection between the structural behaviour of the bleeding (i.e. the combined point) and permeability can be analytically approached. This relationship will be discussed in further detail in § 3.3.

3.2. Structural characteristics of longitudinal bleeding

As discussed in the previous section, bleeding flows play a pivotal role in shaping the wake structures downstream of porous square cylinders. The structural features of longitudinal bleeding are dictated by the permeability and pore configuration of the porous cylinders, which in turn exert a significant influence on the development of the downstream flow. To gain further insight into the fundamental physics governing these flows, it is essential to investigate how the structural characteristics of longitudinal bleeding evolve with permeability and pore configuration. In this section, we examine the downstream evolution of longitudinal bleeding jets, drawing on an analogy with multiple plane turbulent jets, with particular emphasis on the merging and coalescence behaviour of the jets (Miller & Comings Reference Miller and Comings1960; Zhao & Wang Reference Zhao and Wang2018).

Figure 5.

(a) Evolution of the longitudinal velocity profiles at multiple downstream positions ( $x/D = 1.5$ , 3, 4.5 and 6) for cases S, A1, B2, C4 and C5, corresponding to increasing $ \textit{Da} $ . (b) Schematic illustration of the development of longitudinal bleeding jets, interacting with surrounding shear layers. The diagram highlights the merging, combined, and wake regions for the odd pore configuration.

Figure 5(a) shows the lateral profiles of the mean longitudinal velocity, $\langle u \rangle /U_e$ , at several downstream positions for selected porous cases, emphasizing the structural evolution of bleeding flows. Four porous cases (A1, B2, C4 and C5) are selected as representative configurations to capture distinct structural patterns across pore arrangement and permeability. The solid square (case S) is included as a baseline without bleeding. These cases cover the tested $ \textit{Da} $ range and represent both odd and even pore configurations. Other tested configurations exhibit the same qualitative behaviour and are therefore omitted for clarity.

As evident in figure 5(a), the flow structure downstream of porous cylinders differs considerably from that of the solid cylinder due to the influence of bleeding flows. In the solid case, the centreline velocity reaches its minimum around $x/D=1.95$ (not shown for brevity), marking the onset of cylinder-scale vortex formation, followed by velocity recovery. In contrast, for porous cases, longitudinal bleedings shifts the minimum velocity point farther downstream, delaying the formation of large-scale vortices (Zong & Nepf Reference Zong and Nepf2012). In figure 5(a), the near-wake ( $x/D=1.5$ ) exhibits more pronounced longitudinal bleeding, with individual bleeding jets becoming increasingly distinct and aligned with the lattice pores as pore size and permeability increase. These velocity profiles in the near-wake quantitatively confirm that the formation of the bleeding jet structure is strongly influenced by the pore configuration whether it corresponds to an odd or even case, as noted in the wake topologies (see figure 3). Interestingly, for the cases with larger pores (i.e. C4 and C5), figure 5(a) illustrates the merging of multiple parallel jets, emitted from each lattice pore, as they propagate downstream. These jets eventually combine into a single structure, which spreads and decays farther downstream. In contrast, for smaller pore cases (i.e. A1 and B2), their velocity profiles at $x/D=1.5$ resemble a single jet, indicating rapid flow adjustment immediately behind the porous cylinder (Seol et al. Reference Seol, Kim and Kim2024).

The observations from figure 5(a) suggest a new conceptual model of bleeding flows behind porous square cylinders, analogous to the behaviour observed in parallel plane jets (Miller & Comings Reference Miller and Comings1960; Zhao & Wang Reference Zhao and Wang2018), although with distinct shear layer dynamics due to the influence of a strong external flow. Figure 5(b) illustrates this proposed model, which divides the longitudinal bleeding domain into three distinct flow regions based on their structural attributes. The first region, termed the merging region, situated near the jet exit where individual jets begin to interact and merge, leading to significant mixing and causing the flow to lose its individual jet characteristics. This is followed by the combined region, where the coalesced jets behave as a single, unified flow that progressively spreads and decays while interacting with the top and bottom shear layers induced by the cylinder. Finally, the wake region is located downstream of the combined region, where the combined jet fully dissipates and transitions into a wake characterized by the onset of wake recovery process. This conceptual framework captures the complex evolution of longitudinal bleedings behind porous square cylinders, offering a detailed understanding of how these jets shape the downstream flow structure under the influence of surrounding shear layers.

Figure 6.

Selected contour maps of the normalized mean longitudinal velocity, $\langle u \rangle /U_e$ , superimposed with streamlines for case B, where $ \textit{Da} $ varies while maintaining a constant porosity of $\varPhi =0.8$ . Red dashed circles indicate the main recirculation bubble at $y/D=0$ , while yellow dashed circles highlight the second recirculation bubble attached to the cylinder trailing edge.

To investigate the overall behaviour of longitudinal bleeding flows and their influence on downstream flow structures, contour maps of the mean longitudinal velocity ( $\langle u \rangle /U_e$ ) with streamlines are presented in figure 6. Here, $\langle \boldsymbol{\cdot }\rangle$ denotes the ensemble average of a quantity. The analysis focuses on porous cylinders from case B, where permeability varies with unit-cell scaling while maintaining a constant porosity of $\varPhi =0.8$ . For clarity, a side view of the cylinder cross-section is included at the origin to aid in understanding the structural features of the longitudinal bleeding flows.

Figure 6 reveals the presence of longitudinal bleeding flow along the symmetric plane ( $y/D=0$ ) for all porous cases due to the permeable nature of the cylinders. However, both the momentum and the longitudinal extent of this bleeding flow are strongly influenced by cylinder permeability ( $ \textit{Da} $ ), which in turn modifies the main recirculation bubble. As highlighted by the red dashed circles in figure 6, increasing permeability leads to a reduction in the size of the main recirculation bubble and causes it to shift farther downstream, consistent with the findings of Ledda et al. (Reference Ledda, Siconolfi, Viola, Gallaire and Camarri2018). When reaches its critical value ( $ \textit{Da}_c=2\times 10^{-4}$ ), as reported in our previous work (Seol et al. Reference Seol, Kim and Kim2024), the reverse flow in the downstream region vanishes. Instead, a steady wake region with positive and nearly constant velocity develops, as seen in figure 6(e) and 6(f) (Zong & Nepf Reference Zong and Nepf2012). Based on this structural behaviour, Seol et al. (Reference Seol, Kim and Kim2024) categorized flows into two regimes: high flow-blockage cases ( $ \textit{Da} \lt Da_c$ ) and low flow-blockage cases ( $ \textit{Da} \gt Da_c$ ).

Additionally, figure 6 captures a second pair of recirculation bubbles attached to the trailing edge of the porous cylinders, highlighted by the yellow dashed circles. These bubbles have been reported in past numerical and experimental studies (Fang et al. Reference Fang, Yang, Ma and Li2020; Seol et al. Reference Seol, Kim and Kim2024) and attributed to the shear layer interactions between the separated external flow and the longitudinal bleeding flow. The second recirculation bubble represents a low-pressure zone that envelops the longitudinal bleeding flow near the trailing edge. These second bubble pairs thus influence the structural development of the longitudinal bleeding, inducing local acceleration and deceleration as the flow convects downstream in the near wake. Consequently, the observations in figure 6 suggest that the coupled effects of the main and second recirculation bubbles play a role in governing the structural behaviour of longitudinal bleeding flows downstream of the porous cylinders.

Figure 7.

Selected contour maps of the normalized Reynolds shear stress, $-\langle u'v' \rangle /U_e^2$ , for the same cases shown in figure 6.

To further explore the structural evolution of longitudinal bleeding flows and their resulting shear layers as a function of permeability, contour maps of Reynolds shear stress, $-\langle u'v' \rangle$ , normalized by $U_e^2$ are presented in figure 7. Here, $u'$ and $v'$ are the fluctuating velocity components in the longitudinal and lateral directions, respectively. For consistency, the same porous cases as in figure 6 are considered. A schematic representation of the cylinder cross-section is included to facilitate understanding of the structural characteristics of the bleeding shear layers, which are influenced by pore size and the corresponding permeability. Here, the term bleeding shear layer refers to the shear layers generated by longitudinal bleeding jets immediately downstream of the porous cylinders. This terminology is introduced intentionally to avoid confusion with the shear layers originating from flow separation at the leading edge of the cylinders. For reference, the solid cylinder case is also included to highlight the modifications in shear layer structure induced by the permeable nature of the cylinders and the resulting bleeding flows.

Figure 7 clearly illustrates the impact of permeability on the intensity and development of the primary shear layers shed from the leading edge of the cylinders. As $ \textit{Da} $ increases, the magnitude of $-\langle u'v' \rangle$ diminishes, indicating suppression of the unsteady wake and a reduction in turbulent mixing intensity. This trend suggests that the presence of bleeding flows attenuates the shear layer development compared with the solid case. Furthermore, the merging point of the two primary shear layers shifts progressively downstream with increasing $ \textit{Da} $ , emphasizing the role of permeability in modulating the downstream flow dynamics.

In addition to the primary shear layers, figure 7 reveals the development of bleeding shear layers for the porous cylinders. These shear layers exhibit distinct structural behaviour: their growth remains constrained by the primary shear layers, confining them within the outer boundaries of the main flow structures. Initially, the bleeding shear layers expand near the cylinder trailing edge but subsequently diminish in both size and intensity as they convect downstream. At the location where the primary shear layers merge, the bleeding shear layers dissipate entirely, as observed in figure 7. Moreover, it should be noted that the structural characteristics of the bleeding shear layers are closely related to the pore size and resulting permeability. For cases with larger pores and higher permeability (figure 7 d–f), multiple discrete bleeding shear layers are evident, corresponding to the individual jets emitted from each lattice pore. These discrete layers undergo merging and interaction processes, starting immediately behind the porous cylinder, consistent with the behaviour illustrated in figure 5(b). In contrast, cases with smaller pores and lower permeability (figure 7 b,c) exhibit a continuous mixing layer formed by a combined jet that undergoes lateral spreading near the cylinder trailing edge. This distinction highlights the critical role of pore configuration and permeability in shaping the evolution of bleeding shear layers within the primary wake.

Figure 8.

Lateral profiles of the mean longitudinal velocity $\langle u \rangle /U_e$ (blue circles) and the Reynolds shear stress $\langle u'v' \rangle /U_e^2$ (red solid line) at several downstream positions ( $1 \leqslant x/D \leqslant 3.5$ ) for each porous cylinder configurations shown in figures 6 and 7.

To provide further insight into the structural evolution of longitudinal bleeding flows, figure 8 presents lateral profiles of the mean longitudinal velocity ( $\langle u \rangle$ , blue circles) and Reynolds shear stress ( $-\langle u'v' \rangle$ , red lines) at various downstream positions within the range $1\leqslant x/D \leqslant 3.5$ for the porous cylinder cases shown in figure 6 and 7. Both $\langle u \rangle$ and $-\langle u'v' \rangle$ are normalized by their respective scaling factors to highlight overall trends with respect to $ \textit{Da} $ and streamwise location. Within the bleeding shear layer, rapid velocity change induced by longitudinal bleeding gives rise to intense turbulence, as evidenced by the $-\langle u'v' \rangle$ profiles. The centres of both the primary and bleeding shear layers can be identified either by the inflection points in the velocity profiles or the local maxima in the Reynolds shear stress profiles, providing insights into the complex interaction between the longitudinal bleeding flows and the surrounding mixing layers.

The evolution of the bleeding shear layers, as depicted in figure 8, offers a detailed view of the structural change of longitudinal bleeding jets as they propagate downstream from the porous cylinders. Consistent with the velocity structures presented in figure 5(a), figure 8 reveals a distinct merging process of the bleeding shear layers, followed by gradual dissipation as the bleeding jets weaken. These structural changes can be described with respect to the flow regions identified in figure 5(b). Specifically, in the merging region, multiple bleeding shear layers originating from the individual pores of the porous cylinders are discernible. As these shear layers progress downstream into the combined region, they coalesce into a single pair of shear layers with opposing signs, symmetrically aligned along $y/D=0$ , indicating the formation of a combined bleeding jet. In the wake region, the bleeding shear layers fully dissipate as the primary shear layers merge, marking the final phase of the structural evolution.

The structural characteristics of longitudinal bleeding flows, as shown in figure 8, provide a framework for quantitatively assessing the extents of the merging and combined regions downstream of the porous cylinders, denoted as $L_m$ and $L_c$ , respectively (see figure 5 b). These extents can be determined by analysing the longitudinal velocity and Reynolds shear stress profiles along the centreline, offering a deeper understanding of the flow dynamics behind the porous cylinders. In the context of parallel plane jets, it is well established that the merging region ends at the combined point, where two jets coalesce into a single jet flow along the centreline (Zhao & Wang Reference Zhao and Wang2018). Applying this concept to the present study, $L_m$ can be defined as the streamwise distance from the cylinder trailing edge to the combined point in the flow past porous cylinders. Conversely, the merging and dissipation processes of the longitudinal bleeding jets, as observed in figure 8, suggest that the longitudinal extent of the combined region, $L_c$ , aligns with the merging point of the primary shear layers along the centreline. For instance, Zong & Nepf (Reference Zong and Nepf2012) noted that following the merging of the primary shear layers, the centreline velocity begins to recover. This observation implies that the streamwise location where the centreline velocity reaches its minimum corresponds to the point where the primary shear layers merge, thereby defining the extent of the combined region, $L_c$ .

Figure 9.

Profiles of the normalized mean longitudinal velocity, $\langle u \rangle /U_e$ , along the centreline for the same cases shown in figure 6. Red crosses denote the local minima of the profiles, indicating the combined region length, $L_c$ . (b) Variation of the measured $L_c$ as a function of $ \textit{Da} $ , for all possible porous cases. The crosses bounding the symbols indicate the sensitivity of the $L_c$ based on the uncertainty of the longitudinal velocity, $\delta \langle u \rangle / U_e$ .

Figure 9 illustrates the method used to determine the combined region length, $L_c$ , by analysing centreline profiles of the mean longitudinal velocity $\langle u \rangle$ , normalized by the free stream velocity $U_e$ , as a function of the streamwise position $x/D$ . The shaded region in figure 9(a) represents the physical domain occupied by the porous cylinder ( $0\lt x/D\lt 1$ ). To maintain consistency with figures 6–8, the same porous cases (B1–B5) with a fixed porosity $\varPhi =0.8$ are examined. For comparison, the solid cylinder case (S) is included to highlight the wake behaviour in the absence of bleeding effects. Unlike previous studies (Seol et al. Reference Seol, Hong and Kim2023, Reference Seol, Kim and Kim2024), the present analysis focuses exclusively on the longitudinal velocity at $y/D=0$ , avoiding spatial averaging across the lateral direction (i.e. $0 \leqslant y/d_1 \leqslant 1$ , where $d_1$ is the unit cell length, as depicted in figure 1 f).

As shown in figure 9(a), the centreline velocity downstream of the porous cylinders is strongly influenced by permeability. With increasing $ \textit{Da} $ , the magnitude of the negative velocity associated with the recirculation bubble decreases, and the bubble shifts farther downstream. This trend is consistent with observations by Ledda et al. (Reference Ledda, Siconolfi, Viola, Gallaire and Camarri2018). Furthermore, the velocity gradient in the near wake, reflecting the flow adjustment process, is progressively weakened as $ \textit{Da} $ increases in agreement with findings by Zong & Nepf (Reference Zong and Nepf2012). As $ \textit{Da} $ increases, the local minimum in the centreline velocity (marked by red crosses in figure 9 a) shifts downstream, leading to an elongation of the combined region. This behaviour indicates that higher permeability promotes stronger longitudinal bleeding flow, delaying velocity recovery farther downstream.

To quantify the influence of permeability on the combined region length $L_c$ , figure 9(b) presents $L_c$ as a function of $ \textit{Da} $ . The results reveal a clear log–linear relationship, demonstrating that $L_c$ increases systematically with increasing $ \textit{Da} $ . This strong correlation indicates that enhanced longitudinal bleeding flows with higher permeability are the dominant factor driving the expansion of the combined region. It is worth noting that for the cases with exceptionally high permeability (e.g. B5, C2–C5), $L_c$ exceeds $6D$ , surpassing the FoV of the current PIV measurements. While these cases could not be fully captured (see table 1), the trend remains consistent and robust. Based on these observations, figure 9 reaffirms that the combined region length $L_c$ is intrinsically linked to the strength of longitudinal bleeding flows induced by permeability. Furthermore, this elongation of the combined region has direct implications for the aerodynamic drag of the porous cylinders, following a consistent log–linear relation, as recently reported by (Seol et al. Reference Seol, Kim and Kim2024).

The merging region length, $L_m$ , is generally defined as the streamwise distance from the cylinder trailing edge to the combined point, where multiple bleeding jets coalesce into a single, unified jet along the centreline. To quantitatively assess $L_m$ , it is crucial to understand the overall merging behaviour of these jets in the near wake, where intense mixing of porescale jets occurs. Within this region, additional local pressure drops around the longitudinal bleeding flows lead to a complex diffusion process, distinct from the combined region. To simplify the analysis of these interactions, we introduce two primary assumptions.

First, the combined point is assumed to reside along the lateral centreline ( $y/D=0$ ), and thus, our evaluation in the following sections focuses solely on this axis. Second, the boundary layer assumption for turbulent plane jets (Bradbury Reference Bradbury1965) is applied:

(3.1) \begin{align} \langle u \rangle \frac {\partial \langle u \rangle }{\partial x}+ \langle v \rangle \frac {\partial \langle u \rangle }{\partial y} = -\frac {\partial \langle u'v' \rangle }{\partial y}. \end{align}

This assumption implies that at sufficient distances from the cylinder, where the jet flow becomes self-similar, advective transport of longitudinal momentum dominates over lateral turbulent and viscous transport, justifying the boundary layer approximation. The validity of this assumption in the wake region is supported by recent experimental findings (Nicolai et al. Reference Nicolai, Taddei, Manes and Ganapathisubramani2020), which demonstrated self-similar characteristics of the flow in the far-wake region of wall-mounted porous cylinders. Thus, the boundary layer assumption provides a practical framework for simplifying the analysis of the merging region by focusing on the lateral diffusion process, which predominantly governs the streamwise development of jet velocity profiles as the flow propagates downstream. Accordingly, the lateral gradient of Reynolds shear stress, $\partial \langle u'v' \rangle /\partial y$ , evaluated at $y/D=0$ , serves as a representation for the diffusion process in the wake and facilitates the determination of $L_m$ .

Figure 10.

Streamwise profiles of the lateral gradient of Reynolds shear stress, $\partial \langle u'v' \rangle / \partial y$ , and normalized longitudinal velocity, $\langle u \rangle /U_e$ , at the centreline ( $y/D=0$ ): (a) case B1, (b) case A3, (c) case B5 and (d) case B4. Red and black symbols represent $\partial \langle u'v' \rangle / \partial y$ and $\langle u \rangle /U_e$ , respectively. The vertical dashed line indicates the extent of the merging region, $L_m$ . The red shading denotes the uncertainty in $\partial \langle u'v' \rangle /\partial y$ , as described in Appendix A.

Figure 10 illustrates $\partial \langle u'v' \rangle /\partial y$ along the centreline (red symbols) for four representative cases, capturing the influence of pore configuration and permeability on the lateral diffusion process downstream of the porous cylinders. To complement this analysis, the centreline profiles of the mean longitudinal velocity $\langle u \rangle /U_e$ (black symbols), are included to provide insights into key structural behaviour such as the presence of the main recirculation bubbles or the steady wake region behind the cylinders. For consistency, the analysis mainly considers porous cylinders from case B, with the exception of case A3 (figure 10 b). This is because case B3 does not meet the criteria for both high permeability ( $ \textit{Da} \gt Da_c = 2.0\times 10^{-4}$ ) and an odd pore configuration. Instead, case A3, which satisfies both conditions, is included. Figure 10(a,b) corresponds to high flow-blockage cases ( $ \textit{Da}\lt Da_c$ ), while figure 10(c,d) represents low flow-blockage cases ( $ \textit{Da}\gt Da_c$ ). Figure 10(a,c) depicts cases with even pore configurations, where the momentum peaks of the longitudinal bleeding jets are positioned off-centre at half the unit cell length ( $d_1/2$ ). Conversely, figure 10(b,d) illustrates odd pore configurations, where the momentum peak aligns with the centreline due to the presence of a central pore.

It is important to note that the lateral gradient of Reynolds shear stress in figure 10 is intentionally presented in its dimensional form, as it is directly incorporated into the analytical model in (3.3) (§ 3.3). Since the model is formulated based on a dimensional representation of the lateral diffusion of longitudinal momentum, retaining the dimensional form of $\partial \langle u'v' \rangle /\partial y$ ensures a direct, one-to-one correspondence between the experimentally measured values and their role in determining the merging region length, $L_m$ , without the need for additional normalization.

In figure 10, the influence of pore configuration on $\partial \langle u'v' \rangle /\partial y$ profiles within the merging region is evident. For cylinders with even pore configurations (figures 10 a and 10 c), a negative gradient emerges immediately downstream of the cylinder, initially decreasing before increasing farther downstream. This negative gradient reflects inward lateral diffusion towards the centreline (Oskouie, Tachie & Wang Reference Oskouie, Tachie and Wang2020), indicating lateral energy transfer driven by off-centred bleeding jets. This behaviour arises from the momentum deficit between adjacent jets, with local maxima in velocity located at $y/d_1=\pm 1/2$ (see figure 8 f). As the jets diverge and subsequently converge in the merging region, inward momentum transfer is initially enhanced and then attenuates, as indicated by the $\partial \langle u'v' \rangle /\partial y$ profiles in figure 10(a) and 10(c). In contrast, for cylinders with odd pore configurations (figure 10 b and 10 d), the lateral gradient is positive immediately behind the cylinder and gradually decreases in magnitude farther downstream. This positive value in $\partial \langle u'v' \rangle /\partial y$ corresponds to outward momentum diffusion from the central bleeding jet at $y/D=0$ . As the longitudinal bleedings converge along the centreline, this outward momentum transfer weakens progressively.

Following the structural patterns dictated by pore configurations, the impact of permeability on the $\partial \langle u'v' \rangle /\partial y$ profiles becomes evident when comparing high flow-blockage cases (figure 10 a and 10 b) with low flow-blockage cases (figure 10 c and 10 d). In high flow-blockage cases, a distinct positive hump in $\partial \langle u'v' \rangle /\partial y$ appears downstream of a local minimum, followed by a subsequent negative hump farther downstream. These patterns arise from interactions between primary and second recirculation bubbles, as illustrated in figure 6. These low-pressure zones are strongly influenced by permeability (Fang et al. Reference Fang, Yang, Ma and Li2020; Seol et al. Reference Seol, Hong and Kim2023), driving complex energy transfer along the centreline. For even pore configuration (figure 10 a), recirculation bubbles change the direction of lateral momentum diffusion, as evidenced by the sign change in $\partial \langle u'v' \rangle /\partial y$ at $x/D \sim 1.5$ . For odd configuration (figure 10 b), reversed flows enhance outward momentum transfer from the central jet, as shown by the rapid increase following a positive local minimum at $x/D \sim 1.4$ . Conversely, for low flow-blockage cylinders (figure 10 c and 10 d), $\partial \langle u'v' \rangle /\partial y$ profiles approach zero farther downstream, corresponding to the development of a steady wake region.

Based on these observations, the extent of the merging region, $L_{m}$ , can be experimentally quantified according to the structural patterns of lateral diffusion in the bleeding flow behind the porous cylinders. Despite various structural scenarios arising from differences in cylinder permeability and pore configurations, it is consistently observed that the merging region extends up to the combined point, where the bleeding jets coalesce into a single entity, indicating the stoppage of lateral momentum transfer at this point (Oskouie et al. Reference Oskouie, Tachie and Wang2020). In light of this, $L_{m}$ can be defined as the distance from the trailing edge to the streamwise location where the lateral gradient change diminishes to $|\partial ^2 \langle u'v' \rangle /\partial x\partial y| \lt 0.1$ for low flow-blockage cylinders, independent of pore configuration. For high flow-blockage cylinders, however, $L_{m}$ for even and odd pore configurations can be defined differently: for the even case, it is the distance from the trailing edge to the point where $\partial \langle u'v' \rangle /\partial y \approx 0$ ; for the odd case, it is the distance to the first local minimum in $\partial \langle u'v' \rangle /\partial y$ as indicated in figure 10(a) and 10(b), respectively.

Consequently, from the analysis of the merging and combined regions and their corresponding structural features shown in figure 10, both permeability and pore configuration are found to play critical roles in shaping the structural patterns within the merging region. In contrast, the downstream flow structures in the combined region are predominantly governed by permeability. This distinction highlights why the lateral diffusion term, $\partial \langle u'v' \rangle / \partial y$ , used to evaluate $L_m$ , depends on both the permeability and pore configuration of the porous square cylinders. As the longitudinal bleeding flow transitions through the merging region, losing its individual jet characteristics and evolving into a single, unified jet within the combined region, the flow behaviour becomes increasingly dependent on cylinder permeability. This dependency on permeability is reflected in the trend of $\partial \langle u'v' \rangle / \partial y$ observed in figure 10. Specifically, for low flow-blockage cases, the $\partial \langle u'v' \rangle / \partial y$ profile stabilizes and becomes constant farther downstream, indicating a steady wake (Zong & Nepf Reference Zong and Nepf2012). Conversely, high flow-blockage cases exhibit a more complex $\partial \langle u'v' \rangle / \partial y$ profile, even within the combined region, due to the presence of downstream recirculation bubbles (Ledda et al. Reference Ledda, Siconolfi, Viola, Gallaire and Camarri2018). These results emphasize the complex interplay between permeability and pore configuration in lateral momentum transfer and the structural evolution of longitudinal bleeding flows downstream of porous square cylinders.

3.3. Analytical approach for longitudinal bleeding

The results presented in § 3.2 suggest a promising approach for developing an analytical model to estimate the extent of the merging region, $L_m$ . This approach integrates the momentum equation incorporating the DBF model (Yu et al. Reference Yu, Zeng, Lee, Bai and Low2010; Anirudh & Dhinakaran Reference Anirudh and Dhinakaran2018) with the boundary layer assumption (Bradbury Reference Bradbury1965). The DBF model, embedded within the momentum equation, has been demonstrated to effectively capture flow dynamics both within and around porous media across a range of Reynolds numbers and flow conditions, including bluff body flows to turbulent boundary layers (Kuwata & Suga Reference Kuwata and Suga2017; Rosti, Brandt & Pinelli Reference Rosti, Brandt and Pinelli2018; Chavarin et al. Reference Chavarin, Efstathiou, Vijay and Luhar2020; Seol et al. Reference Seol, Kim and Kim2024).

In this context, the DBF model characterizes the flow structure within the merging region by accounting for viscous, inertial and porous effects. The boundary layer assumption, on the other hand, provides a robust framework for describing the structural behaviour of bleeding jets in the combined region, where streamwise gradients are negligible compared with lateral gradients. This dual approach establishes a connection between the merging and combined regions, with $L_m$ defined as the streamwise position of the combined point, which serves as the boundary between these two regions. Consequently, integrating the DBF model with the boundary layer assumption offers a systematic framework for analytically determining the combined point and, in turn, facilitates the development of an analytical model for $L_m$ .

To develop an analytical model for the longitudinal extent of the merging region ( $L_m$ ), which corresponds to the streamwise location of the combined point, two separate governing equations are considered for the merging and combined regions:

(3.2) \begin{align} \frac {1}{\varPhi ^2} \left (\langle u \rangle \frac {\partial \langle u \rangle }{\partial x} + \langle v \rangle \frac {\partial \langle u \rangle }{\partial y} \right ) & = -\frac {1}{\rho } \frac {\partial p}{\partial x} + \underbrace {\frac {\mu }{\rho \varPhi } \left (\frac {\partial ^2 \langle u \rangle }{\partial x^2} + \frac {\partial ^2 \langle u \rangle }{\partial y^2} \right )}_{\text{Brinkman term}} - \underbrace {\frac {\mu }{\rho K}\langle u \rangle }_{\text{Darcy term}} \nonumber \\ - \underbrace {\frac {F}{\sqrt {K}} \langle u \rangle \left (\langle u \rangle ^2+ \langle v \rangle ^2 \right )^{1/2}}_{\text{Forchheimer term}}, \\[-12pt] \nonumber \end{align}

(3.3) \begin{align} \langle u \rangle \frac {\partial \langle u \rangle }{\partial x}+ \langle v \rangle \frac {\partial \langle u \rangle }{\partial y} & = -\frac {\partial \langle u'v' \rangle }{\partial y} . \\[9pt] \nonumber \end{align}

Equation (3.2) represents the longitudinal momentum equation integrated with the DBF model, where $p$ is the intrinsic average pressure, $\mu$ is the fluid dynamic viscosity, $\varPhi$ is the porosity, $F(=1.75/\sqrt {150\varPhi ^3})$ is the inertial factor (Vafai Reference Vafai1984) and $(u^2+v^2)^{1/2}$ is the resultant velocity. Equation (3.3) is the boundary layer equation, where $u'$ and $v'$ denote the fluctuating velocity components in the longitudinal and lateral directions, respectively.

In the momentum equation, the Brinkman term (also known as the Reynolds stress term) is considered negligible compared with the remaining terms within the merging region, as a part of the downstream flow adjustment (Rominger & Nepf Reference Rominger and Nepf2011). Therefore, it is omitted from the governing equation. Additionally, under high-Reynolds-number conditions, the Darcy term is small compared with the Forchheimer term and can also be neglected in (3.2) (Seol et al. Reference Seol, Kim and Kim2024).

For both (3.2) and (3.3), the analysis focuses along the lateral centre, $y/D=0$ , reducing the problem to a one-dimensional approach since the combined point resides along this axis. We assume that $\langle u \rangle = \langle u \rangle _{y/D=0} = U_o$ , $\langle v \rangle = \langle v \rangle _{y/D=0} \approx 0$ and $\partial \langle u'v' \rangle /\partial y = \partial \langle u'v' \rangle _{y/D=0}/\partial y$ . Furthermore, the pressure gradient in (3.2) is scaled as $\partial p/\partial x = \Delta p/D$ , reflecting the pressure difference between the upstream and downstream regions of the porous square cylinders. Based on these considerations, the governing equations simplify to

(3.4) \begin{align} \frac {1}{\varPhi ^2} U_o \frac {\partial U_o}{\partial x} \sim -\frac {1}{\rho } \frac {\Delta p}{D} - \frac {F}{\sqrt {K}}U_o^2, \end{align}

(3.5) \begin{align} U_o \frac {\partial U_o}{\partial x} \sim -\frac {\partial \langle u'v' \rangle }{\partial y}\bigg \rvert _{y/D=0} \\[-2pt] \nonumber \end{align}.

It is worth noting that in the context of canonical turbulent twin jets, the combined point is often identified at the location where the mean centreline velocity, $U_o$ , reaches its maximum (Nasr & Lai Reference Nasr and Lai1997; Durve et al. Reference Durve, Patwardhan, Banarjee, Padmakumar and Vaidyanathan2012). However, this approach is not suitable for the present study due to the distinct structural characteristics of longitudinal bleedings, which vary with pore configuration. Instead, (3.5) indicates that the local maximum in $U_o$ corresponds to $\partial U_o/\partial x=0$ , and thus to $\partial \langle u'v' \rangle /\partial y\rvert _{y/D=0}$ . This relationship provides a more fundamental and robust method for determining the combined point, based on the structural characteristics of lateral diffusion, and thereby facilitates the determination of $L_m$ .

For the current flow configuration, it is also important to emphasize that the primary contribution to cylinder drag arises from pressure drag rather than friction drag. This observation allows the pressure term in (3.4) to be reformulated as $\Delta p \approx \rho C_{\kern-1pt D} U_e^2$ , based on the explicit relationship between the cylinder drag and the pressure drop across the structured porous media (Seol et al. Reference Seol, Hong and Kim2023). Substituting this expression into (3.4), the equation can be rewritten as

(3.6) \begin{align} U_o(x) \frac {\partial U_o(x)}{\partial x} \sim -\varPhi ^2 \left [ \frac {C_{\kern-1pt D} U_e^2}{D} + \frac {F}{\sqrt {K}}U_o^2(x) \right ]\! .\end{align}

This reformulation of the momentum equation reduces it to a first-order ordinary differential equation, which can be solved explicitly using separation of variables and integration by substitution over the range $D \leqslant x \leqslant L_m + D$ . The resulting solution is expressed as

(3.7) \begin{align} L_m \sim \frac {\sqrt {K}}{F\varPhi ^2} \ln {\left [ \frac {\varPhi ^2 \big\{C_{\kern-1pt D} U_e^2/D + F U_o^2(D)/\sqrt {K} \big\}}{\varPhi ^2 \big\{C_{\kern-1pt D} U_e^2/D + F U_o^2(L_m+D)/\sqrt {K} \big\}} \right ]} .\end{align}

To further evaluate $L_m$ , appropriate boundary conditions must be applied at $x = D$ and $x = L_m+D$ , corresponding to the cylinder trailing edge and the streamwise position of the combined point, respectively. At $x=D$ , the centreline velocity, $U_o(x=D)$ , represents the exit longitudinal velocity, $U_{o, \textit{exit}}$ , immediately behind the porous medium. In general, this exit velocity is substantially smaller than $U_e$ , leading to the approximation $U_{o, \textit{exit}}^2/U_e^2 \approx 0$ , consistent with the findings of Zong & Nepf (Reference Zong and Nepf2012) and Nicolai et al. (Reference Nicolai, Taddei, Manes and Ganapathisubramani2020). At $x=L_m+D$ , the combined point characterizes the boundary between the merging and combined regions. At this position, the momentum (3.7) transitions to being equivalent to the boundary layer (3.5), as described by

(3.8) \begin{align} U_o(L_m+D) \frac {\partial U_o(L_m+D)}{\partial x} \sim -\varPhi ^2 \left [ \frac {C_{\kern-1pt D} U_e^2}{D} + \frac {F}{\sqrt {K}}U_o^2(L_m+D) \right ] \nonumber \\ = -\frac {\partial \langle u'v' \rangle (L_m+D,0)}{\partial y} .\end{align}

In this study, the lateral diffusion term, $\partial \langle u'v' \rangle /\partial y$ , at the combined point is empirically determined, as illustrated in figure 10. It is observed that for most porous cylinder cases, the magnitude of this term spans the range $0 \lt \partial \langle u'v' \rangle /\partial y \lt 100$ , with typical values around $\partial \langle u'v' \rangle /\partial y \approx O(10)$ at the combined point. This experimental observation allows for a simplification in the present analytical model, setting $\varPhi ^2 [ ({C_{\kern-1pt D} U_e^2}/{D} )+ ({F}/{\sqrt {K}})U_o^2(L_m+D) ] = \partial \langle u'v' \rangle /\partial y \approx 10$ . By substituting this result into (3.8), (3.7) can now be further simplified to

(3.9) \begin{align} L_m \sim \frac {\sqrt {K}}{F \varPhi ^2} \ln {\left [ \frac {\varPhi ^2 U_e^2 C_{\kern-1pt D}}{10D} \right ]}. \end{align}

To finalize the analytical model for the longitudinal extent of the merging region, $L_m$ , it is essential to determine the drag coefficient, $C_{\kern-1pt D}$ , of the porous cylinder. In general, the drag coefficient of a 2-D porous cylinder is inversely correlated with the non-dimensional permeability, $ \textit{Da} $ (Cummins et al. Reference Cummins, Viola, Mastropaolo and Nakayama2017; Ledda et al. Reference Ledda, Siconolfi, Viola, Gallaire and Camarri2018). However, recent experimental observations by Seol et al. (Reference Seol, Kim and Kim2024) have shown that $C_{\kern-1pt D}$ can be more effectively characterized by the non-dimensional flow-blockage factor, $1/(2\sqrt {Da})$ , which represents the ratio of two key length scales: the cylinder half-width, $D/2$ , and the drag length scale, $\sqrt {K}$ . This finding aligns well with earlier studies on flow past 2-D porous bluff bodies (Chen et al. Reference Chen, Ortiz, Zong and Nepf2012; Zong & Nepf Reference Zong and Nepf2012). Seol et al. (Reference Seol, Kim and Kim2024) explicitly demonstrated that $C_{\kern-1pt D}$ is logarithmically proportional to $1/(2\sqrt {Da})$ across a broad range of cylinder permeabilities. This explicit relationship allows the drag coefficient to be expressed as $C_{\kern-1pt D} \sim \log (1/(2\sqrt {Da}))$ for the present analysis. Additionally, the inertial factor, $F$ , in (3.9) is approximated as $F=1.75/\sqrt {150\varPhi ^3} \approx 1/(7\sqrt {\varPhi ^3})$ (Vafai Reference Vafai1984). Substituting these relationships into (3.9), the final expression for $L_m$ is given by

(3.10) \begin{align} L_m \sim \frac {7\sqrt {K}}{\sqrt {\varPhi }} \ln {\left [ \frac {\varPhi ^2 U_e^2}{10D} \log \left ( \frac {1}{2\sqrt {Da}} \right ) \right ]}.\end{align}

Figure 11.

Variation of the merging region length, $L_m$ , as a function of $ \textit{Da} $ . Here, $L_{m,\textit{fit}}$ is obtained from the analytical model (3.10), while $L_{m,exp}$ is determined experimentally. The blue shaded area represents the 95 $\%$ confidence interval for (3.10). Panel (a) highlights the influence of porosity, considering values of $\varPhi =0.7$ , 0.8 and 0.9. Panel (b) investigates the effect of the lateral diffusion term, $\partial \langle u'v' \rangle /\partial y$ , at the combined point, with values of 1, 10 and 100 used in (3.10).

The longitudinal extent of the merging region, $L_m$ , normalized by the cylinder width, $D$ , is presented in figure 11 as a function of $ \textit{Da} $ using both the analytical model (3.10) and experimental data. In both figure 11(a) and 11(b), the black solid line represents $L_{m,\textit{fit}}$ , which is calculated from the analytical model (3.10). For consistency, the porosity is set to $\varPhi =0.8$ in (3.10), reflecting the midrange value of $\varPhi$ for the porous cylinders used in this study. The blue shaded region denotes the 95 $\,\%$ confidence interval for (3.10). Using this reference line from the analytical model, the effect of porosity ( $\varPhi$ ) and the lateral diffusion ( $\partial \langle u'v'\rangle /\partial y$ ) at the combined point are examined in figure 11(a) and 11(b), respectively.

In figure 11(a), $\varPhi =0.7$ and $\varPhi =0.9$ are applied to (3.10) to define the upper and lower bounds, represented by the red dashed and dashed–dot lines. In contrast, figure 11(b) explores the variability in $\partial \langle u'v' \rangle /\partial y$ , where values of 1 and 100 are substituted for the nominal value of 10 in the denominator of (3.10), reflecting the observed experimental fluctuation range. These substitutions result in the red dashed line (upper boundary, $\partial \langle u'v' \rangle /\partial y=1$ ) and the red dashed–dot line (lower boundary, $\partial \langle u'v'\rangle /\partial y=100$ ) for the analytical model in figure 11(b). Experimental measurements of the merging region length, $L_{m,exp}$ , are included in both figure 11(a) and 11(b) for comparison. Specifically, $L_{m,exp}$ is defined as the distance from the cylinder trailing edge to the combined point, which is identified based on the lateral diffusion characteristics at $y/D=0$ , as influenced by flow blockage and pore configurations, as illustrated in figure 10.

Figure 11 demonstrates strong agreement between the experimental data and the analytical model (black solid line) across the measured range of $ \textit{Da} $ , revealing a log–log relationship between $L_m$ and $ \textit{Da} $ . Notably, $L_m$ increases systematically with increasing $ \textit{Da} $ , indicating that the combined point shifts farther downstream as the momentum of the bleeding flow increases with $ \textit{Da} $ . This trend is consistent with the past observations related to structural modification in the downstream flow. Examples include the relocation of the downstream recirculation bubble with increasing longitudinal bleeding momentum (Ledda et al. Reference Ledda, Siconolfi, Viola, Gallaire and Camarri2018) and the corresponding enhancement of flow adjustment length (Seol et al. Reference Seol, Kim and Kim2024). Since longitudinal bleeding momentum is inherently linked to the permeability of the porous cylinder, permeability emerges as a critical parameter in determining the extent of the merging region and the resulting flow structures. Although porosity ( $\varPhi$ ) appears in (3.10) and may influence $L_m$ , figure 11(a) reveals that the effect of permeability dominates over porosity. This result is consistent with Seol et al. (Reference Seol, Kim and Kim2024), which emphasized the crucial role of permeability in controlling wake dynamics behind porous square cylinders.

Some experimental data points deviate from the model predictions (black solid line), primarily due to underestimation or overestimation of the lateral diffusion term, $\partial \langle u'v'\rangle /\partial y$ , at the combined point. As shown in figure 11(b), varying this term by two orders of magnitude introduces upper and lower prediction bounds, illustrating its sensitivity. This result suggests that the magnitude of $\partial \langle u'v' \rangle /\partial y$ plays a role in determining $L_m$ . Nonetheless, all data points in figure 11(b) fall within the predicted boundaries, supporting the robustness of the proposed model and its applicability to the experimental results.

These findings suggest that the DBF model, incorporated into the momentum equation, effectively captures the volume-averaged resistance of the porous medium and its impact on the near-wake region. It is important to note that the DBF model used herein is not intended to resolve porescale details near the fluid-porous interface. Instead, it serves as a macroscale closure, consistent with prior studies that have shown its validity in turbulent flows (Seol et al. Reference Seol, Kim and Kim2024; Hao & García-Mayoral Reference Hao and García-Mayoral2025).

While figure 11 supports the validity of the analytical model, it is equally important to clarify that the jet-based interpretation adopted in this study is not proposed as a replacement for classical porous-media models such as the DBF equation (Yu et al. Reference Yu, Zeng, Lee, Bai and Low2010), particularly in regimes where strong scale separation exists and homogenization theory is valid (Zampogna & Bottaro Reference Zampogna and Bottaro2016). Our objective is not to claim the superiority of the jet-based framework, but to highlight its complementary role, especially in high-permeability cases where scale separation weakens and porescale effects become non-negligible.

In this context, figure 11 illustrates that the proposed analytical model plays as a bridge between pore-resolved jet dynamics and volume-averaged porous media models. It combines the DBF model for near-wake momentum loss with a boundary-layer approximation in the far wake. As a result, the present framework should not be viewed as a purely jet-based model, but rather as a hybrid approach that accounts for both macroscale permeability effects and lateral momentum diffusion in predicting the merging point, $L_m$ , in high-Reynolds-number flows past porous bluff bodies.

Lastly, it is worth noting that the jet-merging framework developed here is intended for porous bodies with periodic, isotropic lattices in a high- $ \textit{Re} $ external flow, in which coherent porescale jets emerge at the trailing edge. Its validity is bounded by the scale separation $\epsilon =d_1/D$ (Zampogna & Bottaro Reference Zampogna and Bottaro2016) and the permeability $ \textit{Da} $ : in the present configurations $0.05 \lt \epsilon \lt 0.25$ and $2.4\times 10^{-5} \lt Da \lt 2.9\times 10^{-3}$ . In this regime, discrete jets are macroscopically resolvable when $\epsilon \gt 0.1$ and interact to form a merged structure downstream. In contrast, for $\epsilon \lt 0.1$ the outflow becomes effectively homogenized and can be adequately described by the DBF formulation alone. The interpretation presumes geometric regularity at the trailing edge (even/odd pore configurations corresponding to dual/triple-jet analogies) and is not claimed to extend to random foams or strongly anisotropic/fibrous media. At a qualitative level, increasing $ \textit{Re} $ is likely to enhance shear-layer inertia and tends to increase the merging length ( $L_m$ ), while lower $ \textit{Re} $ may promote homogenization and weaken the plane-jet analogy.