Time-averaged velocity and scalar fields of the flow over and around a group of cylinders: a model experiment for canopy flows

Abstract

We conduct a well-controlled model experiment for a wide variety of canopy flows. Examples of these include engineering flows such as wind flow, dispersion of scalars through and over urban areas, and the convective heat transfer in many heat exchangers, as well as natural canopies such as flows through terrestrial or aquatic vegetation. We aim to shed the light on fundamental flow and transport phenomena common to these applications. Specifically, the characteristics of mean flow and scalar concentration characteristics of a turbulent boundary layer flow impinging on a canopy, which comprises a cluster of tall obstacles (this can also be interpreted as a porous obstruction). The cluster is created with a group of cylinders of diameter $d$ and height $h$ arranged in a circular patch of diameter $D$. The solidity of the patch/obstruction is defined by $\phi$ (the total planar area covered by cylinders), which is systematically varied ($0.098 \leq \phi \leq 1$) by increasing the number of cylinders in a patch ($N_c$). A point source is placed at ground level upstream of the patch and its transport over and around the patch is examined. Time-averaged velocity and scalar fields, obtained from simultaneous planar particle image velocimetry-planar laser-induced fluorescence (PIV-PLIF) measurements, reveal that the characteristics of wake and flow above porous patches are heavily influenced by $\phi$. In particular, we observe that the horizontal and vertical extent of the wake and scalar concentration downstream of the patches decreases and increases with $\phi$, respectively. Here, the recirculation bubble is shifted closer to the trailing edge (TE) of the patches as $\phi$ increases, limiting the flow from convecting downstream, decreasing the scalar concentration and virtually ‘extending’ the patch in the streamwise direction. As the bubble forms in the TE, vertical bleeding increases and hence the concentration increases above the patch where the cylinders appear to ‘extend’ vertically towards the freestream.

Impact Statement

Canopy flow covers extensive phenomena encountered in both nature and engineering applications. The characteristics of canopies (also referred to as porous obstructions) is known to affect the mixing, momentum and scalar (e.g. mass and heat) exchange processes in the flow. Examples include flow over and around urban areas, wind/tidal farms, patches of roughness in ship hulls, in-line heat-exchangers among various others. In almost all these applications, the patch of obstacles/roughness is typically characterised by its geometric properties, such as solidity (frontal and plan) or porosity, and it is important to understand the flow/scalar characteristics as well as transport processes for varied geometric parameters. For instance, an urban canopy with a given distribution of buildings and a pollutant source at ground level could result in different wake/dispersion characteristics depending on the frontal or plan solidity of distribution. This study attempts to carry out simplified experiments that will enable us to understand the intricacies of scalar-momentum transport over and around such canopies and allow us to build useful models for engineering design and/or prediction for engineering and environmental flows.

1. Introduction

Canopy flow refers to a scenario where a boundary layer develops over and around a porous surface/obstruction. Examples of such a phenomenon abound in both nature and engineering applications, including but not limited to flows past aquatic and terrestrial vegetations, atmospheric boundary layers developing over forests, clusters of tall buildings in cities, and wind/tidal farms, which consist of arrays of wind/tidal turbines. These obstructions are known to affect the scalar, momentum and energy transport process in the flow within the canopy and downstream of the canopies. Aquatic vegetation, for example, provides shielding (Kemp, Harper, & Crosa, 2000), affects the sediment formation and erosion (De Langre, 2008; Nepf, 2012a, 2012b; Tinoco & Coco, 2016), and also carbon sequestration in coastal ecosystems (Lei & Nepf, 2021). Urban and vegetation canopies induce coherent, multi-scale turbulent motions in the atmospheric boundary layer (Huang, Cassiani, & Albertson, 2009a, 2009b, 2011; Kanda, 2006; Katul, Kuhn, Schieldge, & Hsieh, 1997; Li & Bou-Zeid, 2011), which governs flow mixing, momentum, heat and mass exchange within the layer (Finnigan, 2000; Poggi, Porporato, Ridolfi, Albertson & Katul, 2004; Raupach & Thom, 1981), including pollutants and other chemical compounds (Aristodemou et al., 2018; Poggi, Katul, & Albertson, 2006). While the configuration and tuning of turbines in wind/tidal farms are vital in maximising power output (Myers & Bahaj, 2012; Vennell, 2011), they may also alter the meteorological pattern and ecosystem of the installation sites. For example, the installation of wind turbines has been shown to affect the temperature distribution and humidity of the surrounding air (Baidya Roy, Pacala, & Walko, 2004; Rajewski, Takle, Prueger, & Doorenbos, 2016), while wakes generated by tidal turbines can potentially modify the sedimentation process and bacterial population in estuarine waters (Ahmadian, Falconer, & Bockelmann-Evans, 2012). Considering its extensive industrial applications and environmental impacts, understanding the transport mechanism of the flow surrounding porous obstructions is, therefore, essential.

Previous studies on porous obstructions (such as the examples given above) have typically involved some simplifications from real-life conditions. The obstruction may be simplified as an array of cylinders mounted on an isolated patch with an incoming flow upstream of the cylinders (illustrated in figure 1). Numerical simulations and experiments have been conducted on the flow surrounding such a cylindrical array (Ball, Stansby, & Allison, 1996; Chang & Constantinescu, 2015; Chang, Constantinescu, & Tsai, 2017; Chen, Ortiz, Zong, & Nepf, 2012; Nepf, 1999; Nepf, Mugnier, & Zavistoski, 1997; Nicolle & Eames, 2011; Rominger & Nepf, 2011; Tanino & Nepf, 2008; White & Nepf, 2003; Zong & Nepf, 2012), where the cylinders are as tall as, or protruding from, a free surface, as commonly observed in emergent aquatic vegetation. Here, the flow is dominated by a Kármán vortex street in the wall-normal axis trailing behind the cylinders and thus can be considered two-dimensional (2-D) (Nicolai, Taddei, Manes, & Ganapathisubramani, 2020; Taddei, Manes, & Ganapathisubramani, 2016). A more general approach is to fully submerge the cylinders in the flow such that the height $h \ll \delta _{bl}$, where $\delta _{bl}$ is the incoming boundary-layer thickness (figure 1b). Studies have been conducted for cylinders attached to a bottom wall (as in figure 1b, see also Chen, Jiang, & Nepf, 2013; Nicolai et al., 2020; Taddei et al., 2016) and/or suspended from a free surface (Plew, 2011; Tseung, Kikkert, & Plew, 2016; Zhou & Venayagamoorthy, 2019). In either case, the patch is considered as three-dimensional (3-D), as flow mixing occurs on the sides of, downstream of and above the body. Taddei et al. (2016), Zhou and Venayagamoorthy (2019), Nicolai et al. (2020) have identified three mechanisms in which the flow may escape (‘bleed’) from a fully submerged porous obstruction: (i) the lateral bleeding from the interior of the patch to the sides (spanwise direction, see figure 1a); (ii) the trailing edge (TE) bleeding, which is responsible in the formation of wake downstream of the body (figure 1a); and (iii) the vertical bleeding, in which the flow escapes from the interior to the top surface of the patch and towards the free stream (figure 1b).

Figure 1. Illustrations of (a) lateral and trailing edge bleeding, and (b) vertical bleeding of cylindrical arrays on a circular patch. Here, $U_{bl}$ is the incoming boundary-layer velocity, whose thickness is $\delta _{bl}$, and $h$ is the height of the cylinders.

It has been reported that the solidity of a porous obstruction is a critical parameter in determining drag (Chang et al., 2017; Nepf, 1999; Taddei et al., 2016; Zhou & Venayagamoorthy, 2019), the structure of wake formed by the obstruction (Ball et al., 1996; Chang & Constantinescu, 2015; Chang et al., 2017; Chen et al., 2012; Nicolai et al., 2020; Nicolle & Eames, 2011; Zhou & Venayagamoorthy, 2019; Zong & Nepf, 2012), scalar dispersion (Nepf et al., 1997; Tanino & Nepf, 2008; White & Nepf, 2003) and sediment disposition (Chang & Constantinescu, 2015; Chen et al., 2012). For the cylindrical array shown in figure 1, the solidity is measured in terms of the planar area covered by cylinders per total surface, $\phi \equiv N_c(d/D)^2$ (also commonly referred to as the solid volume fraction), where $N_c$ is the number of cylinders in a circular patch, $D$ is the diameter of the circular patch and $d$ is the diameter of the cylinder (see figure 2a,d for graphical descriptions of $D$ and $d$). It has been observed by Nicolle and Eames (2011), Taddei et al. (2016), Zhou and Venayagamoorthy (2019) that the total drag of the patch $C_D$ increases with the increasing $\phi$ then drops for a solid obstruction ($\phi = 1$, figure 2d). Higher patch solidity also yields shorter and taller wake (in streamwise and wall-normal direction, respectively, see Nicolai et al., 2020) in the trailing edge, and increases both lateral and vertical bleeding (Zhou and Venayagamoorthy (2019), see also figure 1). Further, it has been reported by Nicolle and Eames (2011) that at low $\phi (\phi < 0.05)$, each cylinder in an array behaves as a single obstruction and as the solidity increases to $\phi > 0.15$, the cylindrical array behaves closer to a solid obstruction whose size is equal to that of the patch ($\phi = 1$, figure 2d). Within the range of moderate solidity ($0.05 \leq \phi \leq 0.15$), suppression of vortex street formation downstream of 2-D porous patches has been observed. In its place is a steady wake (constant velocity) region, which is a consequence of flow bleeding. Zong and Nepf (2012) and Chen et al. (2012) showed that the streamwise length of this steady wake region scales on $C_D aD$, where $a$ is the frontal area per unit volume $a \equiv N_c d/(0.25{\rm \pi} D^2)$. Similarly, scaling by $C_D ab$ and $C_D ah$ have also been observed by Rominger and Nepf (2011) for a square patch and by Chen et al. (2013) for a fully submerged patch, respectively, where $b$ is the half-width of the patch.

Figure 2. Illustration of cylinder array test cases: (a) C39; (b) C64; (c) C133; and (d) the solid case (CS) in the $x$–$z$ plane (top view), $d$ is cylinder diameter and $D$ is the circular patch diameter. (e) Mean streamwise velocity $U^+$ ($\circ$) and variance of velocity fluctuation $\overline {u'u'}^+$ ($\bullet$, light grey) of the smooth-wall turbulent boundary-layer (TBL) case as functions of wall-normal location $y^+$ at $Re_{\theta } \approx {3000}$, where $\theta$ is the momentum thickness. Dashed line is ${1}/({0.41}) \log y^+ + {5.0}$, solid line is the DNS results of Sillero et al. (2014) at $Re_{\theta } = {4000}$, filtered according to the spatial resolution of the PIV measurements (Lee et al., 2016). Data are downsampled for clarity.

Although the characteristics of the wake and drag of a porous obstruction shown in figure 1 have been discussed in detail for both 2-D and 3-D patches, an element that is still missing from these analyses is the intricacy between bleeding and scalar transport in 3-D porous patches. So far, discussions about scalar transport are limited to lateral and longitudinal dispersion in 2-D patches (Tanino & Nepf, 2008; White & Nepf, 2003), sediment deposition (Chen et al., 2012; Tinoco & Coco, 2016), qualitative flow visualisation using dye injection (Ball et al., 1996; Rominger & Nepf, 2011; Zong & Nepf, 2012), and non-simultaneous velocity and scalar measurements due to limitations of the experimental facilities (Nepf, 1999; Tanino & Nepf, 2008). Here, a 3-D porous patch serves as a more realistic model of urban landscapes, vegetation canopies and wind farms, which are typically subsumed within the atmospheric boundary layer. Inclusion of scalar measurements (in addition to the velocity components) has many potential benefits, for example, in air pollution studies, where one might consider cluster of densely populated urban landscape as a porous obstruction and the pollutant as scalar, or in studies about sediment disposition in ocean beds.

In this study, we aim to address the following question: given a point source of scalar located at the ground level just upstream of 3-D porous patches, what are the flow and scalar characteristics in the wake and above the patches for various solidities? To answer this question, we conduct a parametric study of turbulent boundary layers impinging on an array of cylinders mounted on a circular patch (see figure 1). A point source upstream of the patch is created by releasing a passive fluorescent dye. The solidity $\phi$ is varied by systematically increasing the number of cylinders in the patch, such that $0.098 \leq \phi \leq 1$. This covers both the medium and high solidity range described by Nicolle and Eames (2011). The axis system $(x,\, y,\, z)$ corresponds to the streamwise, wall-normal and spanwise direction, respectively. Simultaneous planar particle image velocimetry-planar laser-induced fluorescence (PIV-PLIF) measurements in the $x$–$y$ plane provide the instantaneous streamwise–wall-normal velocity components $u$ and $v$, respectively, and the scalar concentration field $c$, which can be decomposed further into their time-average and fluctuation about the mean

(1.1a–c)\begin{equation} u(x,y,t) = U(x,y) + u'(x,y,t); \quad v = V + v'; \quad c = C + c'.\end{equation}

It should be noted that due to the choice of the measurement plane, discussion about flow bleeding in the current study is limited only to the trailing edge and vertical bleeding (figure 1).

2. Experimental set-up

2.1 Test cases

Experiments are carried out in the closed loop water flume at the University of Southampton. The test section has the size of $6250\ {\rm mm} \times 1200\ {\rm mm}$ (length $\times$ width). In all measurements, the water level is kept constant at $600 \pm 1$ mm and the free stream velocity is set to $U_{\infty } \approx 0.45$ ms$^{-1}$. The free stream turbulence level is $\sim$1 %.

A circular patch with a diameter ($D$) of 100 mm is mounted on the floor of the tunnel at 5500 mm downstream of the test section entrance, which corresponds to the patch Reynolds number $Re_D \equiv U_{\infty }D/\nu \approx 4.5 \times 10^{4}$, where $\nu = 1.0035 \times 10^{-6}$ m$^2$ s$^{-1}$ is the kinematic viscosity of water at $20\,^{\circ }$C. The patch constitutes an array of rigid cylinders, each has the diameter of $d = 5$ mm and height $h = 30$ mm (figures 1b and 2a,d), which correspond to $d/D = 0.05$ and $h/D = 0.3$. The cylinders are arranged along concentric, evenly spaced circles whose axis is at the centre of the circular patch, such that the distance between two consecutive cylinders is constant within the patch (see figure 2a–c). The solidity of the patch is parametrically varied by gradually increasing the number of cylinders $N_c$ while keeping $D$ and $d$ constant. Table 1 and figure 2(a–c) show three different circular patches: C39, C64, C133, which correspond to $N_c = 39$, 64, 133, which cover a range of patch densities of $0.1 \lesssim \phi \lesssim 0.33 (2.5 \lesssim aD \lesssim 8.5)$. A solid obstruction case, i.e. a cylinder covering the entire circular patch (such that $N_c = 1$ and $\phi = 1$), is included in addition to the porous cases as case ‘CS’ in table 1 and figure 2(d). The solid cylinder in case CS has an equal height $h$ to the cylinder arrays in other test cases.

Table 1. List of cylinder array test cases. Here, $N_c$ is the number of cylinders in a patch, $d$ and $D$ are the diameters of the cylinders and the patch, respectively, $h$ is the height of the cylinders, $\phi$ is patch solidity, and $a$ is the frontal area per unit volume. The ‘PIV’ and ‘PLIF’ columns show the availability of PIV and PLIF images of each test case. The last three columns show the availability of those images in each field of view (FOV).

2.2 Validation with smooth-wall turbulent boundary layer

A smooth-wall turbulent boundary layer (TBL) case is first tested to validate PIV measurements in the present study. The measurement is conducted in the same test section as other test cases by removing the circular patch. This flow has the Reynolds number (based on momentum thickness $\theta$) of $Re_{\theta } \equiv \theta U_{\infty }/\nu \approx 3000$ and friction Reynolds number $Re_{\tau } \equiv \delta _{bl} U_{\tau bl}/\nu \approx 1500$, where $\delta _{bl} = 82 \pm 3\,\%$ mm is the 99 % boundary-layer thickness, $U_{\tau bl} \equiv \sqrt {\tau _0/\rho } = 0.019 \pm 2\,\%$ ms$^{-1}$ is the friction velocity for this case, $\tau _0$ is the wall shear stress and $\rho = 998.12$ kg m$^{-3}$ is the density of water at 20 $^{\circ }$C. Figure 2(e) shows the profiles of mean streamwise velocity $U^+ \equiv U/U_{\tau bl}$ and the variance of velocity fluctuations $\overline {u'u'}^+ \equiv \overline {u'u'}/U_{\tau s}^2$ as functions of wall-normal location $y^+ \equiv y U_{\tau s}/\nu$, where subscript ‘$+$’ denotes viscous scaling and the viscous length scale is $\nu /U_{\tau s} = 0.053$ mm. The logarithmic region of the mean velocity collapses to the line $1/\kappa \log y^+ + A$, where $\kappa = 0.41$ is the von Kármán constant and $A = 5.0$ is the log-law shift (Schlichting & Gersten, 2017). The variance $\overline {u'u'}$ is in good agreement with that obtained from the direct numerical simulation (DNS) of Sillero, Jiménez, and Moser (2014) at $Re_{\theta } = 4000$. Note that the DNS results presented in figure 2(e) are filtered according to the spatial resolution of the PIV measurements (Lee, Kevin, Monty, & Hutchins, 2016), ${\rm \Delta} x^+ \times {\rm \Delta} y^+ = 15 \times 15$ in streamwise and wall-normal directions, respectively.

2.3 Particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF)

For simultaneous planar PIV and PLIF measurements, the measurement domain is located in the streamwise–wall-normal ($x$–$y$) plane, slicing the centreline of the circular patch (figure 3). The measurement domain is created by combining three successive experiments in field of view (FOV) 1, FOV 2 and FOV 3, which extends downstream of the circular patch. Dashed lines in figure 3 illustrate these FOVs and the overlaps between each FOV. It should be noted that although the PIV and PLIF measurements are conducted simultaneously in each FOV, these measurements are not conducted simultanenously between FOV 1, 2 and 3. The PIV and PLIF images are recorded first in FOV 1, then the cameras and optics are traversed downstream to form FOV 2, then FOV 3. Each FOV has the size of $253\ {\rm mm} \times 126\ {\rm mm} (x \times y)$, which approximately extends 2.5 times the patch diameter (2.5$D$) in streamwise and 4$h$ in height. When the three FOVs are stitched together, the total length in $x$ is 689 mm ($6.89D$). Measurements are conducted in all three FOVs for the cylindrical array cases (C39, C64 and C133), and only the first two FOVs for solid obstruction and the reference TBL cases (CS and TBL, see table 1).

Figure 3. Contours of instantaneous (a) streamwise velocity $u/U_{h/\textit{2}}$ and (b) scalar concentration $c/C_s$ (in logarithmic scale) of case C39, obtained from (a) PIV and (b) PLIF measurements, where $U_{h/2}$ is the incoming velocity at half-height of the cylinders. Dashed lines show the extents of FOV 1, 2 and 3. Black rectangles illustrate the cylinder array of height $h$ mounted on a circular patch. The origin $x = {0}$ is at the centre of the patch and $y = {0}$ is on the floor of the tunnel. Measurement data are not available in the hatched regions (between the cylinders). In panel (b), the blue arrow shows the location of the scalar point source.

In each FOV, two 12-bit 4 MP CMOS cameras acquire the PIV images, while PLIF images are recorded by a 5.5 MP 16-bit sCMOS camera, to maximise the signal-to-noise ratio of the PLIF images. The PIV cameras are equipped with wavelength filters to filter out the PLIF signal, and vice versa for the PLIF camera. For PLIF measurements, a point source is created upstream of the circular patch by injecting Rhodamine 6G dye solution to the flow at a constant rate. The dye has the Schmidt number ($Sc \equiv \nu /\mathfrak {D}$, $\mathfrak {D}$ is the mass diffusivity) of $2500 \pm 300$ (Vanderwel & Tavoularis, 2014), which corresponds to the Peclet number based on patch diameter $Pe_D \equiv Re_D Sc = 1.12 \times 10^{8}$. The dye is supplied by a thin tube embedded on the floor of the tunnel, located 125 mm upstream from the centre the patch ($x/D = 1.25$, see figure 3b). Scalar concentration fields presented in this study are normalised by the measured concentration at the point source, $C_s$. The complete description of the experimental set-up is given in appendix A. Further details on the PIV-PLIF measurements, calibration, post-processing and uncertainty analysis are given by Lim et al. (2021).

3. Results and discussions

3.1 Extent of the wake and scalar concentration

Figure 4(a–d) shows the contours of time-averaged streamwise velocity $U$ for all porous test cases and the solid case (CS). The contours are normalised by $U_{h/2}$, which is defined as the incoming, undisturbed streamwise velocity upstream of the patch ($x/D = -1.3$) at the half-height of the cylinder, $y/h = 0.5$. The figures show the formation of wake for all porous and solid cases in the form of velocity deficit downstream of the patches. The magnitude of the deficit is significant, which signals the presence of patch-sized flow phenomena (Taddei et al., 2016) as opposed to those of individual cylinders observed in patches with low solidity (Chang & Constantinescu, 2015; Chen et al., 2012; Nicolai et al., 2020; Taddei et al., 2016). In figure 4(b,c), the blue-shaded contours show $U < 0$, which correspond to the recirculation of the flow downstream of the patch. The streamwise extent of the flow recirculation is approximately equal to $D$ (figure 4b,c). The strength of the recirculating flow appears to be affected by the solidity of the patch: recirculation is not apparent in case C39 (case with the lowest solidity, figure 4a), a weak recirculation then appears in case C64 (figure 4b), which grows stronger in case C133 (figure 4c). It is also noted that the recirculation bubbles are shifted downstream as solidity decreases. In case C64 (figure 4b), the bubble is observed at $1 \lesssim x/D \lesssim 2$, while in case C133 (figure 4c) and the solid case CS (figure 4d), the bubbles form immediately at the trailing edge of the patches ($x/D = 0.5$). Similar phenomena have been observed in 2-D patches (Chang & Constantinescu, 2015), 3-D patches (Taddei et al., 2016; Zhou & Venayagamoorthy, 2019) and porous plates (Castro, 1971).

Figure 4. Contours of time-averaged streamwise velocity $U/U_{h/2}$ of cases: (a) C39; (b) C64; (c) C133 and (d) CS. Lines are the time-averaged spanwise vorticity $\varOmega _y D/U_{h/\textit{2}}$ with the contour levels of $-{8},\, -{7},\,\ldots,\, -{2}$ (solid line) and ${0.4},\, {0.8},\,\ldots,\, {1.6}$ (solid blue line).

Superimposed with the contours of $U$ in figure 4(a–d) are the contours of time-averaged spanwise vorticity $\varOmega _y \equiv \partial V/\partial x - \partial U/\partial y$ normalised by $U_{h/2}/D$. Black solid lines are the contours of $\varOmega _y < 0$ (clockwise) and the blue lines $\varOmega _y > 0$ (counter-clockwise). Negative $\varOmega _y$ show the development of shear layers forming above and downstream of the patches, which appear to be stronger (higher in magnitude) as solidity increases. The choice of using $\varOmega _y$ is based on the inclusion of both $\mathrm {d}U/\mathrm {d} y$ and $\mathrm {d}V/\mathrm{d}{\kern0.7pt}x$, which is especially important when the vertical bleeding becomes significant (closer to leading edge of the patch at high solidities). Previous work (figure 10a in Nicolai et al., 2020) has shown that the vorticity and the Reynolds shear stress have similar behaviour. Similar to the observation by Taddei et al. (2016), here we also observe the elevation of the shear layer further upward from the patches as solidity increases (compare, for example, figure 4c with figure 4a,b). The lift-off is attributed to vertical bleeding from the interior of the patches, which is noticeably absent in the solid case CS (figure 4d) where there is no vertical bleeding.

Figure 5(a–d) shows the contours of time-averaged scalar concentration $C$ (normalised by the concentration at the source $C_s$ and presented in logarithmic scale) for all porous test cases and the solid case. The dark-shaded contours in figure 5(a–c) illustrate how the scalar permeates through and downstream of the patches. With less obstructions to block the flow, high scalar concentration (purple-shaded contours) is observed downstream of the C39 and C64 patches in figure 5(a,b), respectively. As blockage ($\phi$) increases, TE bleeding decreases and consequently, the downstream concentration decays (see C133 patch, figure 5c). Instead, the scalar escapes the patch by means of vertical bleeding, as shown by higher scalar concentration above the patch ($y/h \geq 1$) in figure 5(c). As $\phi$ reaches 1 (solid case), TE and vertical bleeding are non-existent, as evidenced by $C \approx 0$ above the patch and low magnitude of $C$ downstream of the patch (figure 5d). Here, the observed concentration is mainly the result of flow mixing in the wake instead of the scalar permeating through the patch.

Figure 5. Contours of time-averaged scalar concentration $C/C_s$ (in logarithmic scale) of cases: (a) C39; (b) C64; (c) C133 and (d) CS. Measurement data are not available in the hatched regions.

Figures 4 and 5 show the dependency of wake and scalar concentration downstream of the porous bodies as a result of TE bleeding. We attempt to measure the extent of the wake by using a similar method to Nicolai et al. (2020). First, we define the velocity defect as ${\rm \Delta} U = U_{bl} - U$, where $U_{bl}(y)$ is the incoming boundary layer velocity profile upstream of the patch (at $x/D = -1.3$), and arbitrarily choose a threshold for the magnitude of ${\rm \Delta} U$ as a wake definition. Figure 6(a) shows the contours of ${\rm \Delta} U/U_{h/2}$ for all porous cases and the solid case at the chosen threshold of 0.3. With this threshold, we are able to define the length and height of the wake, $L_w$ and $H_w$, respectively (figure 6a). Here, $L_w$ is defined as the length of the wake measured from the centreline of the patch ($x = 0$) to the $x$-maxima of the chosen contour level, while $H_w$ is the height of the wake measured from the floor of the tunnel ($y = 0$) to the $y$-maxima of the same contour level. Figure 7(a,b) shows the normalised length and height of the wake, $L_w/D$ and $H_w/h$, respectively, as a function of solidity $\phi$. The figures show that $L_w$ decreases with increasing $\phi$, but $H_w$ increases (and possibly reaches a maxima at a certain value of $\phi$) and then decreases as $\phi \rightarrow 1$, where the obstacle is no longer porous and there is no flow escaping from the top of the patch. The wake ‘blow-up’ (i.e. taller wake) in higher $\phi$ has also been recorded by Taddei et al. (2016) and Nicolai et al. (2020). These tendencies of $L_w$ and $H_w$ are insensitive to the change in threshold, as illustrated by the hatched regions in figure 7(a,b), which correspond to the same quantities measured when the threshold for ${\rm \Delta} U/U_{h/2}$ is varied by $0.3 \pm 20\,\%$. As a comparison, $L_w$ and $H_w$ measured by Nicolai et al. (2020) are shown in white-filled circles in figure 7(a,b). The study involved similar arrays of cylindrical patches of various $\phi (0.018 \leq \phi \leq 1)$ that impinged a developing turbulent boundary layer at $Re_{\theta } \approx 60\ 000$ (20 times higher than that of the current study). It should be noted that the measured $L_w$ and $H_w$ are greater in magnitude as the wake is defined at a lower threshold of ${\rm \Delta} U/U_{h/2} = 0.2$ instead of 0.3 due to the increasing streamwise length of the FOV ($8D$ instead of $6.89D$). It should also be noted that the values of $H_w$ (white-filled circles in figure 7b) are scaled by a factor of $3.58 h/\delta _{bl}$, where 3.58 is the ratio of incoming boundary layer thickness to cylinder height $\delta _{bl}/h$ of Nicolai et al. (2020). However, the differences between studies do not change the observed trend in $L_w$ and $H_w$ as a function of $\phi$.

Figure 6. Contours of time-averaged (a) defect velocity ${\rm \Delta} U/U_{h/\textit{2}} = {0.3}$, (b) wall-normal velocity $V/U_{h/\textit{2}} = {0.1}$ at $x/D \leq {0.5}$ and (c) scalar concentration $C/C_s = {3} \times {10}^{-{4}}$. Legends in panel (b) show the colour scheme for each case. Hatched regions illustrate the location of the patch. In panels (a,c), the horizontal and vertical extent of the wake are defined by $L_w$ and $H_w$, respectively, while $L_c$ and $H_c$ are the horizontal and vertical extent of the scalar dispersion. In panel (b), the grey solid line is the wall-normal velocity ($V/U_{h/\textit{2}} = -{0.07}$) of case CS.

Figure 7. (a) Normalised length of wake $L_w/D$ ($\bullet$) and scalar dispersion $L_c/D$ ($\blacklozenge$) (as illustrated in figure 6) of all test cases as a function of solidity $\phi$. (b) Height of wake $H_w/h$ ($\bullet$) and scalar dispersion $H_c/h$ ($\blacklozenge$). Hatched regions illustrate the sensitivity to the threshold level defining the length/height ($\pm {20}\,\%$). The same quantities measured by Nicolai et al. (2020) at $Re_{\theta } \approx {60\ 000}$ are shown for comparison. Legends show the colour scheme for each case.

Similarly, a threshold can also be applied to the scalar concentration field $C$ to measure the extent of scalar dispersion downstream of the patch. Figure 6(c) shows the contours of $C/C_s$ for all porous cases and the solid case at the chosen threshold of $3 \times 10^{-4}$. The length and height of scalar dispersion $L_c$ and $H_c$, respectively, are measured from the centreline of the patch $x = 0$ to the $x$-maxima of the chosen contour level and from the floor of the tunnel $y = 0$ to the $y$-maxima, respectively. Filled diamonds in figure 7(a,b) show the measured $L_c$ and $H_c$ as a function of $\phi$. Similar to those of the wake extent, $L_c$ and $H_c$ decreases and increases, respectively, with increasing $\phi$. It should be noted that although scalar concentration data are available for case CS, the magnitude of $C$ downstream of the patch is lower than that of the chosen threshold (see figure 5d) and thus we are unable to measure $L_c$ and $H_c$ for this case.

Figure 8(a–d) shows the coloured contours of $C$ at the trailing edge for all porous and solid cases, with black contour lines superimposed in the figures showing $U < 0$, corresponding to flow recirculation. For case C64, figure 8(b) shows a high scalar concentration (purple-shaded region) at the trailing edge and up to $x/D \approx 1$, immediately followed by the emergence of a recirculation bubble. In other words, the scalar is trapped between the porous patch upstream and the recirculation bubble that prevents the scalar from convecting downstream. By contrast, case C39 (figure 8a) that has lower solidity shows no recirculation bubble, and thus the scalar is freely transported downstream of the patch. Case C133 (figure 8c), however, shows the recirculation bubble being shifted to the trailing edge. The scalar in this case is largely prevented from escaping towards the trailing edge (this explains the decrease in $L_w$ and $L_c$ as solidity increases, see figure 7a). Conservation laws dictate two possible outlets left for the flow: lateral and vertical bleeding. The higher magnitude of $C$ above the patch, upstream of the bubble ($y/h \geq 1$, $0 \leq x/D \leq 0.5$) confirms increasing vertical bleeding in higher solidity, while the increase of lateral bleeding with $\phi$ has been confirmed by Zhou and Venayagamoorthy (2019). In some sense, the flow lingers downstream of the porous obstruction (characterised by high magnitude of $C$ in this region in case C64), it is as if the obstruction is ‘extended’ in the streamwise direction. The length of this extension seems to be inversely proportional to $\phi$, as it is longer than the FOV for case C39 (or perhaps infinitely long), approximately equal to $1.25D$ for case C64 (figure 8b) and 0 for case C133 (figure 8c). Possibly, the extension length is akin to the length of the steady wake region observed in 2-D patches (Ball et al., 1996; Chen et al., 2012; Zong & Nepf, 2012). When the obstruction is impermeable (figure 8d), the mechanism differs. It acts as a solid body, shedding a pair of shear layers at the trailing edge (see contours of $\varOmega _y$ at $0.5 \leq x/D \leq 1.5$ in figure 4d).

Figure 8. Contours of scalar concentration $\log _{10}(C/C_s)$ for cases: (a) C39; (b) C64; (c) C133 and (d) CS. Black solid lines are negative streamwise velocity, in panel (b) $U/U_{h/\textit{2}} = -{0.06},\, -{0.04},\, -{0.02}$ and in panels (c,d) $-{0.4},\, -{0.36},\,\ldots -{0.04}$. Gray solid lines are the normalised vorticity in $y$ axis $\varOmega _y D/U_{h/\textit{2}} = -{8},\, -{7},\,\ldots,\, -{4}$.

Figure 8(a–c) shows the contours of $\varOmega _y < 0$ as solid grey lines, representing the shear layer forming above the porous patches. In case C39 (figure 8a), the shear layer shields a region of (relatively) high scalar concentration (coloured contours in figure 8a) below the layer. As solidity increases in case C64 (figure 8b), higher $C$ is observed downstream of the patch (purple-shaded contours), which coincides with elevation of the shear layer compared with that of C39 in figure 8(a). In the case with the highest solidity (C133 in figure 8c), recirculation occurs on the trailing edge followed by higher $C$ above the patch compared with that downstream of the patch. Here the shear layer is elevated further upward, shielding the observed high $C$ above the patch ($0.25 \lesssim x/D \lesssim 0.5$). Here, there seems to be an ‘artificial’ upward extension of the porous obstruction as $\phi$ increases, which accompanies the previously discussed streamwise extension at the TE of the obstruction.

3.2 Trailing edge and vertical bleeding

We further quantify the velocity of TE bleeding $U_{bleed}$. Here, $U_{bleed}$ is defined as $U$ downstream of the patch ($x \geq D/2$) integrated over $y$. This definition can be extended further to the scalar concentration at the trailing edge $C_{te}$, such that

(3.1a,b)\begin{equation} U_{bleed}(x) = \frac{1}{h} \int_{0}^{h} U(x,y) \, \mathrm{d} y; \quad C_{te}(x) = \frac{1}{h} \int_{0}^{h} C(x,y) \, \mathrm{d} y \quad \mathrm{for} \ x \geq D/2. \end{equation}

Figure 9(a) shows the magnitude of $U_{bleed}$ at the trailing edge, $x = D/2$, as a function of $\phi$. White-filled circles are $U_{bleed}$ measured by Nicolai et al. (2020) for similar cylindrical arrangements at $Re_{\theta } \approx 60\ 000$. Despite the difference in the order of $Re$, both Nicolai et al. (2020) and the present study show that $U_{bleed}$ decreases with $\phi$ at the trailing edge. Figure 9(b,c) further shows the defects $1-U_{bleed}/U_{h/2}$ and $C_{te}/C_s$ as functions of $x$ downstream of the patch ($x \geq D/2$). Both figures show the decay of wake and scalar concentration with $x$, with (generally) faster decay as $\phi$ increases in the porous cases. In particular, $U_{bleed}$ in figure 9(b) shows that the solid case CS recovers the fastest to the upstream condition (i.e. the incoming flow, dashed line is the smooth-wall case TBL), followed by the highest solidity case C133. We note that the decay of the wake for the porous and solid cases (figure 9b) and the decay of $C$ for the porous cases (figure 9c) seem to follow a certain power law, possibly similar to that of Wygnanski, Champagne, and Marasli (1986) for various solid and porous wake generators, which is left for future works. The exception to this is the scalar concentration downstream of case CS (black solid line in figure 9c), which is approximately constant in $x$ and much lower in magnitude compared with the porous cases.

Figure 9. (a) Integrated velocity bleeding $U_{bleed}/U_{h/2}$ at the trailing edge $x = D/\textit{2}$. The same quantity measured by Nicolai et al. (2020) is shown for comparison. (b) $\textit{1}-(U_{bleed}/U_{h/\textit{2}})$ as a function of $x$ downstream of the patch. (c) Integrated scalar concentration $C_{te}/C_s$ as a function of $x$. (d) Integrated vertical bleeding $V_{bleed}/U_{h/\textit{2}}$ above the patch ($y/h > \textit{1}$) as a function of $y$. (e) Integrated scalar concentration above the patch $C_v/C_s$. In panels (b,d), the dashed line is the reference smooth-wall case TBL. In panel (d), hatched region shows $V_{bleed} \pm {{\rm 1}}\,\%$ of $U_{h/\textit{2}}$. Legends show the colour scheme for each case.

A closer look into figure 9(c) reveals that the integrated scalar concentration of case C64 peaks at $x/D \approx 1$, and the magnitude of this peak is much higher than that of cases with lower and higher $\phi$ (C39 and C133). This spike corresponds to the high scalar concentration (purple-shaded region) at the trailing edge and up to $x/D \approx 1$, which is immediately followed by the emergence of a recirculation bubble (figure 8b) and thus can be translated into the streamwise extension of the porous obstruction discussed in § 3.1.

Contours of positive (upward) wall-normal velocity $V > 0$ above the patches are shown in solid lines in figure 6(b) for the same magnitude $V/U_{h/2} = 0.1$. The contours reveal a very different pattern of flow above the patches for porous cases compared with the solid case CS (solid black line). In case CS, flow separation is characterised by high magnitude of $V$ observed in the leading edge of the patch ($x/D = -0.5$). Compared with the solid case, the magnitude of $V$ in the leading edge is lower for all porous cases, as the flow is able to penetrate the patches instead of being diverted upward (see Taddei et al. (2016) for similar 3-D patches and Zhou and Venayagamoorthy (2019) for suspended patches, note that the sign of $V$ is inverted in the latter). What is only observed in the porous patches, however, is the presence of vertical bleeding above the patches, characterised by the contours of $V > 0$ at $-0.5 \leq x/D \leq 0.5$ and $y/h \geq 1$. The magnitude and extent of this region appear to be dependent upon solidity. For the same magnitude of $V/U_{h/2}$, the contours for porous cases in figure 6(b) show that $V$ is extended upward (in $y$) as $\phi$ increases.

Similar to the analysis for the TE bleeding $U_{bleed}$, we quantify the magnitude of vertical bleeding as the integrated $V$ above the patch ($-D/2 \leq x \leq D/2$ and $y \geq h$), and this can be further extended to include the scalar concentration,

(3.2a,b)\begin{equation} V_{bleed}(y) = \frac{1}{D} \int_{{-}D/2}^{D/2} U(x,y) \, \mathrm{d}\kern0.7pt x; \quad C_{v}(y) = \frac{1}{D} \int_{{-}D/2}^{D/2} C(x,y) \, \mathrm{d}\kern0.7pt x \quad \mathrm{for} \ y \geq h. \end{equation}

Figure 9(d,e) shows the integrated vertical bleeding $V_{bleed}$ and scalar concentration $C_v$ as functions of $y$ above the patch. Both figures show higher magnitude of $V_{bleed}$ and $C_v$ closer to the patch, which decay further towards the free stream. Dependency on $\phi$ is apparent on both. In figure 9(d), $V_{bleed}$ generally increases with $\phi$, with a significantly smaller magnitude (within $\pm 1\,\%$ of the incoming flow $U_{h/2}$, see hatched region in figure 9d) for CS and TBL (black solid and dashed lines in figure 9d, respectively). For case CS, small $V_{bleed}$ is the result of averaging between $V > 0$ in the leading edge and $V < 0$ above the patch (see black and grey contour lines in figure 6b), while for case TBL ($V = 0$), this can be attributed to the error of the measurements. The integrated scalar concentration in figure 9(e) is 0 for the solid case CS (since there is no vertical bleeding, see black solid line in figure 9e), while for the porous cases, it increases with $\phi$ and decays to 0 further from the patch.

In addition to increasing vertical bleeding with $\phi$, we also observed that the maxima of $C_v$ in figure 9(e) occurs at the top surface of the patch $y/h = 1$ for all porous cases, but the maxima of $V_{bleed}$ in figure 9(d) occurs further from the patch (for example, it occurs at $y/h \approx 1.5$ for case C133 in figure 9d). This perhaps can be explained by the upward extension of the porous patches (figure 8c). Here, the high scalar concentration (purple-shaded contours in figure 8c) occurs exactly at the top surface of the patch (corresponding to maximum $C_v$) and the porous obstruction is artificially extended upward (corresponding to the maximum $V_{bleed}$ further from the top surface).

4. Conclusions and recommendations

We conducted planar PIV-PLIF measurements of the flow surrounding 3-D, circular porous patches of various solidity $\phi (0.098 \leq \phi \leq 1)$ to study the characteristics of flow in the wake of and above the patches. Porosity is generated by mounting rigid cylinders of height $h$ and diameter $d$ on the patch (whose diameter is $D$) and the variation in $\phi$ is achieved by systematically increasing the number of cylinders in a patch. The patches are fully submerged within the incoming flow, with a fluorescent dye injected from the floor of the tunnel, creating a point source upstream of the patches. Simultaneous PIV-PLIF measurements in the $x$–$y$ plane provide streamwise and wall-normal velocity components, as well as the concentration of the fluorescent tracer in the FOV.

Examination of the time-averaged velocity and scalar fields suggests that the trailing edge (longitudinal) and vertical flow bleeding from the obstructions decreases and increases with increasing $\phi$, respectively, as previously reported by Taddei et al. (2016), Zhou and Venayagamoorthy (2019), Nicolai et al. (2020). In addition to that, we observe that $\phi$ affects the scalar concentration in the wake and above the patch. In particular, the present results suggest that $\phi$ determines the following components.

1. (i) The horizontal extent of the wake ($L_w$) and scalar concentration ($L_c$) downstream of the porous patches, with $L_w$ and $L_c$ decreasing with $\phi$ (figure 7a). The decrease is attributed to the shift of recirculation bubbles towards the trailing edge of the patches as $\phi$ increases. High scalar concentration is observed downstream of the patch as the flow is trapped between patches and the recirculation bubble, acting as an elongation of the obstructions in streamwise direction.

2. (ii) The vertical extent of the wake ($H_w$) and scalar concentration ($H_c$) downstream of the porous patches, with $H_w$ and $H_c$ increasing with $\phi$ (figure 7b). As $\phi$ increases further, a recirculation bubble occurs at the trailing edge of the patch and the flow is trapped in the interior of the patch. The flow largely escapes by means of vertical (and lateral) bleeding. High concentration of scalar, elevation of shear layer and $V$ above the patch suggest that the flow is trapped between the top surface of the patch, from which the flow bleeds, and the shear layer. Here we observe an artificial extension of the obstacle further upward (i.e. the patch is ‘taller’ from the perspective of the flow) as $\phi$ increases.

As a final remark, the authors would like to note that this study is the first step towards understanding both momentum and scalar transport of the flow surrounding 3-D porous obstructions and building useful models for engineering design, prediction for meteorological or oceanographic flows, etc. The results presented in this study are limited to the time-averaged velocity and scalar components. However, we note that intermittency (i.e. discrepancies between the time-averaged and instantaneous flow fields) has been reported by Cassiani, Katul, and Albertson (2008) and Taddei et al. (2016). In the future, analysis about the fluctuating components ($u'$, $v'$ and $c'$) should be included to obtain the full picture of the flow characteristics. The decay of wake and scalar concentration in the trailing edge, shown in figure 7(d,e), indicate a possible wake model for both porous and solid cases, which should also be investigated in the future.