3.1. General formulation

The flow around bluff bodies is complex. In order to obtain an analytical or semi-analytical description of the flow, a widely used approach consists in simplifying the governing equations using potential flow theory outside the wake while introducing free parameters such as base pressure to account for viscous and complex phenomena in the wake and near the solid structure (Parkinson & Jandali Reference Parkinson and Jandali1970). The model proceeds with the same idea. The system is separated into four regions delimiting four flow regimes, as shown in figure 7. The flow is assumed to be stationary, incompressible and inviscid everywhere except through the porous structure where viscous effects cannot be neglected.

Figure 7. Diagram of the model for a 3-D potential flow around and through a porous screen. The dashed lines are the separation streamlines used as a boundary between the regions. The dotted lines are the streamlines used in the model to calculate the velocities in regions I and II. The incoming flow is laminar and is extended over the entire height of the system. Here $\mathcal {C}_w$ denotes the section of the wake.

In region I we assume that the flow is potential and the velocity is denoted $\boldsymbol {v}_I(x,y,z)$. Region I is located upstream of the structure as well as downstream outside the wake zone contained by a well-defined streamtube attached to the contour of the porous surface as shown in figure 7. Therefore, the velocity derives from a velocity potential denoted $\phi _I(x,y,z)$ for region I. Using the method employed by Koo & James (Reference Koo and James1973), and initially suggested by Taylor (Reference Taylor1944), we calculate the flow by modelling the screen with a continuous source distribution with strength $\varOmega (x_s,y_s,z_s)$, where $(x_s,y_s,z_s)$ denotes a point on the surface. We obtain the potential flow in region I by superposing the resulting potential flow from the distribution of sources with the uniform laminar flow $\boldsymbol {v}_0$. The two streamfunctions $\psi _I$ and $\chi _I$ needed to describe general 3-D incompressible flows can then be deduced using the following relation (for a definition of 3-D streamfunctions, see Yih Reference Yih1957):

(3.1)\begin{equation} \boldsymbol{v}_I(x,y,z) = \boldsymbol{\nabla} \psi_I(x,y,z) \wedge \boldsymbol{\nabla} \chi_I(x,y,z). \end{equation}

Regions II and III are located downstream of the porous surface in the near wake. In these regions the flow can be rotational so that we cannot use anymore a velocity potential to describe the flow. In region II the pressure and the velocity are not constant since they are influenced by the surface. However, in region III the flow is sufficiently far from the screen so that the streamlines tend to be aligned with the uniform flow $\boldsymbol {v}_0$ as represented by the contour $\mathcal {C}_w$ in figure 7. Therefore, the pressure tends towards a base suction pressure $p_{III}$ that is a priori different (and lower) than the constant external pressure $p_0$. This region is mathematically at infinity (there are no finite separations between regions II and III), however, since the flow aligns rapidly with the uniform flow, we indicate a region III in the near wake in figure 7. In two dimensions, the approach of constant pressure along separating streamlines has been successfully used in free-streamline theory by Wu (Reference Wu1962), Parkinson & Jandali (Reference Parkinson and Jandali1970) and Roshko (Reference Roshko1954) to model the wake. In this model, since we consider three dimensions, we can not adopt free-streamline theory, but $p_{III}$ can be considered as having the same role as the constant pressure used in such theory. The flow in regions II and III is found with matching conditions as explained later.

In figure 7 we added a region IV that is located in the far wake where the mixing with the outer flow can not be ignored. In this region, the pressure should increase to reach again the pressure $p_0$ outside the wake. We assume that this region has little influence on the flow near the porous screen and on the aerodynamic forces, and therefore, it is not included in the model. Consequently, in our model the pressure in the far wake will remain equal to $p_{III}$.

3.2. Determination of the flow in region I

In region I the flow is potential, and we derive the velocity from the velocity potential. For simplicity, we set the reference frame so that the axis $(Oz)$ is aligned with the velocity $\boldsymbol {v}_0$ without loss of generality. Due to the linearity of the Laplacian, we first consider the potential flow $\phi (x,y,z)$ for the source distribution only, then we add the potential flow for the uniform flow. The velocity potential from the source distribution $\varOmega$ located on a general regular surface $\mathcal {S}_p$ is the solution of the equation

(3.2)\begin{equation} \Delta \phi(x,y,z) = \varOmega(x,y,z)\mathbb{1}_{\mathcal{S}_p}, \end{equation}

where $\mathds {1}_{\mathcal {S}_p}$ denotes the Dirac function associated to the surface $\mathcal {S}_p$. For a point source in three dimensions centred at the origin, the Green function of the Laplacian is

(3.3)\begin{equation} \varGamma(x,y,z) ={-} \frac{1}{4{\rm \pi}}\frac{1}{\sqrt{x^2+y^2+z^2}}. \end{equation}

Therefore, if we assume $\boldsymbol {\xi }: U \subset \mathbb {R}^2\to \mathbb {R}^3$ to be a surface patch of a general regular surface $\mathcal {S}_p$ with coordinates

(3.4)\begin{equation} \boldsymbol{\xi} =\begin{pmatrix} x_s(u,v)\\ y_s(u,v)\\ z_s(u,v) \end{pmatrix}, \end{equation}

parametrized by two parameters $u$ and $v$, with $(u,v)\in (U = [a,b]\times [c,d])$ with $(a,b,c,d)\in \mathbb {R}^4$, then the velocity potential $\phi (x,y,z)$ is expressed for all $(x,y,z)\in \mathbb {R}^3\setminus \mathcal {S}_p$ as (Pressley Reference Pressley2010)

(3.5)\begin{align} \phi(x,y,z) = \iint_{U} \varOmega(u,v)\varGamma(x-x_s(u,v),y-y_s(u,v), z-z_s(u,v)) \left\|\frac{\partial \boldsymbol \xi}{\partial u} \wedge \frac{\partial \boldsymbol \xi}{\partial v}\right\| \mathrm{d}u \,\mathrm{d}v, \end{align}

and the total velocity potential can be written as the sum

(3.6)\begin{equation} \phi_I(x,y,z) = {v_0} z + \phi(x,y,z). \end{equation}

We deduce the velocity in region I with $\boldsymbol {v_I}(x,y,z)=\boldsymbol {grad}(\phi _I(x,y,z))$. Note that the Green function can be changed without other modifications in the model to study the situation of a flow in a confined environment or near a wall.

3.3. Determination of the flow in regions II and III

In region II the flow can be rotational, and therefore, is not necessarily potential. The flow is obtained from the streamfunctions by considering, as done by Koo & James (Reference Koo and James1973), that the streamlines in region II have the same pattern as if they were obtained by the streamfunctions from the superposition of the distribution of sources and the uniform flow $\boldsymbol {v}_0$. This can be formulated in a general way by writing the two streamfunctions for the flow in region II as functions of the streamfunctions of region I. Let $\psi _{II}$ and $\chi _{II}$ be the streamfunctions in region II. As defined above, $\psi _I$ and $\chi _{I}$ are the streamfunctions deduced from the flow in region I, functions that can be considered in the whole space. Then, without loss of generality, we choose an analog formulation as the one proposed by Koo & James (Reference Koo and James1973) to describe the flow in region II, using the functions $f$ and $g$:

(3.7a,b)\begin{equation} \psi_{II}=f(\psi_I)\psi_I \quad {\rm and}\quad \chi_{II}=g(\chi_I)\chi_I . \end{equation}

This means that the velocity $\boldsymbol {v}_{II}$ in region II, and the velocity $\boldsymbol {v}_{I}$ obtained from the velocity potential $\phi _I(x,y,z)$, are co-linear at any point. We indeed obtain

(3.8)\begin{equation} {\boldsymbol{v}_{II}}(x,y,z) = \left(\frac{\mathrm{d}f}{\mathrm{d}\psi_I}+f(\psi_I)\right) \left(\frac{\mathrm{d}g}{\mathrm{d}\chi_I}+g(\chi_I)\right){\boldsymbol{v}_I}(x,y,z). \end{equation}

We define the attenuation function $E$ as

(3.9)\begin{equation} E(\psi_I,\chi_I)=\left(\frac{\mathrm{d}f}{\mathrm{d}\psi_I}+f(\psi_I)\right) \left(\frac{\mathrm{d}g}{\mathrm{d}\chi_I}+g(\chi_I)\right), \end{equation}

which is the crucial quantity to determine the flow in region II. Since $E$ is a function only of the streamfunctions, it is constant along a streamline and is therefore entirely defined by considering its value on the screen.

In addition to (3.8), the mass flow rate must be conserved when the fluid passes through the screen implying the continuity of the normal velocity at the screen between region I and region II:

(3.10)\begin{equation} {v_{I}}_n(x_s,y_s,z_s) = {v_{II}}_n(x_s,y_s,z_s). \end{equation}

At this point, our system thus contains two unknowns that are the attenuation function $E$ and the distribution of sources $\varOmega$. We have one equation (3.10), and another equation linking the velocities and pressures in the vicinity of the porous structure is required to close our system of equations. For this purpose, two streamlines are considered as shown in figure 7: $(AP)$ and $(PB)$ where the point $P$ is on the screen taken as a surface from a macroscopic point of view. Along each of these streamlines, Bernoulli's equation can be applied, and we thus obtain, for $(AP)$,

(3.11)\begin{equation} \tfrac{1}{2}\rho v_{I}^2(x_A,y_A,z_A)+p_{I}(x_A,y_A,z_A) = \tfrac{1}{2}\rho v_{I}^2(x_s,y_s,z_s)+p_{I}(x_s,y_s,z_s), \end{equation}

and, for $(PB)$,

(3.12)\begin{equation} \tfrac{1}{2}\rho v_{II}^2(x_s,y_s,z_s)+p_{II}(x_s,y_s,z_s) = \tfrac{1}{2}\rho v_{II}^2(x_B,y_B,z_B)+p_{II}(x_B,y_B,z_B). \end{equation}

We consider that the points $A$ and $B$ are far enough from the screen so that we can take constant values of the velocities and pressures (see Fail, Lawford & Eyre (Reference Fail, Lawford and Eyre1957) for flat plates normal to an air stream). Therefore, upstream we have $\boldsymbol {v}_{I}(x_A,y_A,z_A)= \boldsymbol {v}_0$ and $p_{I}(x_A,y_A,z_A)=p_0$; downstream we take the mean value of the velocity over a section of the wake orthogonal to the far-field stream direction ($\boldsymbol {v}_0$): $\boldsymbol {v}_{II}(x_B,y_B,z_B) = \overline {\lim _{z \rightarrow +\infty }\boldsymbol {v}_{II}(x,y,z)} = \overline {E(\psi _{I},\chi _{I})}\boldsymbol {v}_0$. In the rest of the paper, $\overline {E(\psi _{I},\chi _{I})}$ will be denoted as $\bar {E}$. Koo & James (Reference Koo and James1973) considered a far-downstream constant pressure $p_{0}$; however, it is known that the pressure in the wake is lower than the pressure outside the wake that contributes to aerodynamic forces. Steiros & Hultmark (Reference Steiros and Hultmark2018) therefore introduced a suction base pressure $p_{III}$ at the point $B$, which is assumed to be constant far enough from the screen in region III as explained above. Thus, we introduce a third free parameter $p_{III}$ that will be determined from conservation equations in § 3.5. Note that since we assume that the pressure $p_{III}$ is constant, the pressure will be discontinuous across the wake boundaries (as in the model of Koo & James (Reference Koo and James1973) in the 2-D case). By combining (3.11) and (3.12), decomposing the velocities according the tangential and normal components on the screen (respectively, ${v_I}_t$ and ${v_{I}}_n$) and by using (3.10), we obtain the pressure difference

(3.13)\begin{align} p_0 - p_{III} &= \tfrac{1}{2}\rho(1-E^2(\psi_{I},\chi_{I})){v_I^2}_t(x_s,y_s,z_s) + \tfrac{1}{2}\rho(\bar{E}^2-1) v_0^2 \nonumber\\ &\quad + p_I(x_s,y_s,z_s) - p_{II}(x_s,y_s,z_s). \end{align}

If we are able to determine the pressure differences $p_0 - p_{III}$ and $p_I(x_s,y_s,z_s) - p_{II}(x_s,y_s,z_s)$ independently from these equations, then we can use (3.10) to find the attenuation function $E$ and (3.13) to find the distribution of sources $\varOmega$. In order to determine the suction pressure $p_{III}$, we follow the method of Steiros & Hultmark (Reference Steiros and Hultmark2018) in § 3.5, using the conservation law of the momentum in the control volume $V_1$ shown in figure 7, and at the vicinity of the screen. Before addressing this problem, we focus in the following § 3.4 on the pressure jump $p_I(x_s,y_s,z_s) - p_{II}(x_s,y_s,z_s)$.

3.4. Pressure jump across the screen

A summary of the models for the relation between the pressure jump and screen porosity can be found in Xu et al. (Reference Xu, Patruno, Lo and de Miranda2020). This problem has largely been discussed in many papers, raising the difficulties of a general formulation. In their model, Steiros & Hultmark (Reference Steiros and Hultmark2018) consider two streamlines passing through the screen in a hole where the velocity and the pressure are assumed to be uniform. Immediately upstream after the acceleration of the fluid, the characteristic pressure of the flow is denoted $p_h$ and is assumed to correspond physically to the mean pressure inside the hole. As noted by several authors including Taylor & Davies (Reference Taylor and Davies1944) and Wieghardt (Reference Wieghardt1953), the characteristic velocity immediately upstream should be regarded as the mean velocity after contraction of the flow within the holes, ${v_h}_1$, denoting the average velocity through the screen expressed as

(3.14)\begin{equation} {v_h}_1 = \frac{{v_I}_n(x_s, y_s, z_s)}{(1-s)}, \end{equation}

The velocity accelerates to ${v_h}_1$ in the hole and the pressure decreases to $p_h$ so that at this point there are no losses. Along this first streamline, Bernoulli's equation leads to

(3.15)\begin{equation} p_{I}(x_s, y_s,z_s) + \tfrac{1}{2}\rho {v_I^2}(x_s, y_s,z_s) = p_h + \tfrac{1}{2}\rho {v_{h}^2}_1. \end{equation}

Immediately downstream, considering also a homogenized velocity, the flow enlarges and the velocity reaches a value equal to

(3.16)\begin{equation} {v_h}_2 = {v_{II}}_n(x_s, y_s, z_s), \end{equation}

taken as the characteristic velocity just after the hole. As discussed by Taylor & Davies (Reference Taylor and Davies1944) and Steiros & Hultmark (Reference Steiros and Hultmark2018), there are pressure losses in the pores, i.e. $p_{II}< p_{I}$. Taylor & Davies (Reference Taylor and Davies1944) consider that a fraction $\lambda$ of the pressure lost in acquiring the velocity $v_h$ is regained when the stream becomes uniform again behind the sheet. Steiros & Hultmark (Reference Steiros and Hultmark2018) consider that the pressure loss during the fluid acceleration is not recovered, i.e. all surplus kinetic energy due to fluid acceleration is lost and not reconverted into pressure (i.e. $\lambda =0$). In that case, Bernoulli's equation along this second streamline leads to

(3.17)\begin{equation} p_h + \tfrac{1}{2}\rho {v_{h}^2}_2 = p_{II}(x_s, y_s, z_s) + \tfrac{1}{2}\rho {v_{II}^2}(x_s, y_s,z_s). \end{equation}

We combine (3.8), (3.10) and (3.14)–(3.17) to obtain

(3.18)\begin{align} {p_{II}}(x_s,y_s,z_s)-{p_{I}}(x_s,y_s,z_s) &= \tfrac{1}{2}\rho {v_I}_t^2(x_s, y_s,z_s) (1-E^2(\psi_I,\chi_I)) \nonumber\\ &\quad + \tfrac{1}{2}\rho{v_I}_n^2(x_s, y_s,z_s)\theta(s), \end{align}

with

(3.19)\begin{equation} \theta(s) = 1-\frac{1}{(1-s)^2}. \end{equation}

Injecting the pressure difference $p_{I}$–$p_{II}$ obtained in (3.18) in (3.13), we have

(3.20)\begin{equation} p_0\unicode{x2013}p_{III} ={-}\tfrac{1}{2}\rho{v_I^2}_n(x_s,y_s,z_s)\theta(s) + \tfrac{1}{2}\rho(\bar{E}^2-1) v_0^2. \end{equation}

We assumed that $p_{III}$ is constant, therefore, the right-hand side of the above equation has to be constant, which leads to the condition

(3.21)\begin{equation} \boldsymbol{grad}_{\mathcal{S}_p}({v_I}_n^2(x_s,y_s,z_s)\theta(s))= \boldsymbol{0}, \end{equation}

with $\theta (s)$ that can vary for non-homogeneous porous surfaces. Therefore, under the assumptions made so far, we are looking for a source strength $\varOmega$ that satisfies this condition. It is possible to relax certain restrictions on the value of $\varOmega$ by considering a variable base suction pressure $p_{III}$ or a variable far wake velocity $\boldsymbol {v}_{II}(x_B,y_B,z_B) = E(\psi _{I},\chi _{I})\boldsymbol {v}_0$. In that case, the problem is far more difficult, and should be solved numerically. Such a resolution is beyond the scope of the present work.

3.5. Drag coefficient

At this point, our system of equations is not closed. Therefore, in this section we find another equation by adapting the model of Steiros & Hultmark (Reference Steiros and Hultmark2018) in three dimensions with a screen of arbitrary shape and orientation in the flow. It uses two expressions of the aerodynamic forces that can be combined to give a final equation, closing eventually our system of equations.

A first expression is given by the momentum balance around the surface of the screen, using the drag coefficient as defined in (2.2), giving

(3.22)\begin{equation} C_D = \frac{1}{\dfrac{1}{2}\rho S_{p} v_0^2} \left(\iint_{\mathcal{S}_p}(p_I\unicode{x2013}p_{II})\boldsymbol{e}_{z}\boldsymbol{\cdot} \boldsymbol{n}_s\,\mathrm{d}S+\rho\iint_{\mathcal{S}_p} {v_I}_n \boldsymbol{e}_{z}\boldsymbol{\cdot} (\boldsymbol{v}_{I}\unicode{x2013}\boldsymbol{v}_{II})\,\mathrm{d}S\right), \end{equation}

where the normal vector $\boldsymbol {n}_s$ points in the direction of region II. The pressure difference $p_I\unicode{x2013}p_{II}$ in (3.22) is given by (3.18).

The second expression of the drag coefficient is provided by the momentum balance in a control volume $V_1$ around the screen as shown in figure 7. We consider that this volume $V_1$ is large enough so that the velocities at the surfaces $\mathcal {S}_x$ and $\mathcal {S}_y$ on the sides of the block parallel to the $z$ axis are equal to $\boldsymbol {v}_0 + \boldsymbol {v}_{\epsilon }$ where $v_{\epsilon } \ll v_0$. With this approximation, the projection of momentum balance equation onto the far-field stream direction ($\boldsymbol {v}_0$) gives an expression of the drag $F_D$,

(3.23)\begin{equation} F_D =\rho S_w (1-\bar{E}^2){v^2_0} + S_w (p_0\unicode{x2013}p_{III}) - \rho v_0 \iint_{\mathcal{S}_x + \mathcal{S}_y} \boldsymbol{v}_I \boldsymbol{\cdot} \boldsymbol{n}\,\mathrm{d}S, \end{equation}

where $S_w$ is the area of the section of the wake (orthogonal to the $z$ axis), and the bar over $E$ denotes the mean over the considered surface. The normal vector $\boldsymbol {n}$ points outwards from the control volume. Then, a mass balance in the same control volume gives

(3.24)\begin{equation} \rho v_{0} S_w = \rho \iint_{\mathcal{S}_x + \mathcal{S}_y} \boldsymbol{v}_I \boldsymbol{\cdot} \boldsymbol{n}\,\mathrm{d}S + \rho \bar{E}v_{0}S_w. \end{equation}

This equation allows us to find the value of the last term of the conservation of momentum equation (3.23). Moreover, the section of the wake $S_w$ is determined with a mass balance through the screen

(3.25)\begin{equation} \rho \iint_{\mathcal{S}_p} \boldsymbol{v}_I \boldsymbol{\cdot} \boldsymbol{n}\,\mathrm{d}S = \rho \bar{E}v_{{0}} S_w. \end{equation}

Equations (3.23)–(3.25) are now combined to obtain a second expression of the drag coefficient, which reads

(3.26)\begin{equation} C_D = \frac{1}{\dfrac{1}{2}\rho S_p v_0^2}(p_0\unicode{x2013}p_{III} + \rho v_0^2 (\bar{E}-\bar{E}^2))\frac{1}{v_0 \bar{E}}\iint_{\mathcal{S}_p}{v}_{I_n}\,\mathrm{d}S. \end{equation}

Equation (3.20) giving an expression of $p_0\unicode{x2013}p_{III}$ is used so that the second expression of the drag (3.26) is calculated further:

(3.27)\begin{equation} C_D = \frac{1}{S_p}(-(1-\bar{E})^2 - {v}_{I_n}^2\theta(s)) \frac{1}{v_0 \bar{E}}\iint_{\mathcal{S}_p}{v}_{I_n} \,\mathrm{d}S. \end{equation}

The final equation of the problem is obtained by equalizing the two expressions of the drag coefficient (3.22) and (3.27), yielding

(3.28)\begin{align} & \bar{E}\iint_{\mathcal{S}_p} ({\tilde{v}_{I_t}^2}(1-E^2) + {\tilde{v}_{I_n}^2}\theta(s)) \boldsymbol{n}_s\boldsymbol{\cdot}\boldsymbol{e}_z + 2\tilde{v}_{I_n}(1-E){\boldsymbol{\tilde{v}}_{I_t}}\boldsymbol{\cdot} \boldsymbol{e}_z \,\mathrm{d}S \nonumber\\ &\quad =(-(1-\bar{E})^2 - \tilde{v}_{I_n}^2\theta(s)) \iint_{\mathcal{S}_p}\tilde{v}_{I_n}\,\mathrm{d}S, \end{align}

with dimensionless velocities $\tilde {v}={v}/{v_0}$.

Finally, we give the explicit expression for $v_{I_n}$ and $E$. Finding $v_{I_n}$ leads to a boundary surface potential problem. An explicit expression of $v_{I_n}$ is found from the gradient of the velocity potential (3.6) having therefore an integral term. This integral term is known as a harmonic double-layer potential with density $\varOmega$ (Gunter Reference Gunter1967), which is defined on a subdomain of $\mathbb {R}^3\setminus \mathcal {S}_p$. This integral term becomes singular if it is evaluated on the surface $\mathcal {S}_p$, however, it can be continuously extended on the surface for each side and the value depends on the side by which we approach the surface. From the definitions of regions I and II, for $\boldsymbol {m}=(x,y,z)\in \mathcal {S}_p$, we have

(3.29)\begin{equation} v_{I_n}(\boldsymbol{m}) = v_0\boldsymbol{e_z}\boldsymbol{\cdot}\boldsymbol{n}_s + v_n^-(\boldsymbol{m}), \end{equation}

and

(3.30)\begin{equation} v_{{II}_n}(\boldsymbol{m}) = E(\psi_I,\chi_I)(v_0\boldsymbol{e_z}\boldsymbol{\cdot}\boldsymbol{n}_s + v_n^+(\boldsymbol{m})), \end{equation}

with

(3.31)\begin{equation} v_n^{{\pm}}(\boldsymbol{m}) = \lim_{\epsilon \to 0^+}\frac{\partial \phi}{\partial \boldsymbol{n}_s} (\boldsymbol{m} \pm \epsilon \boldsymbol{n}_s), \end{equation}

with $\phi$ defined in (3.5) and the directional outward normal derivative

(3.32)\begin{equation} \frac{\partial \phi}{\partial \boldsymbol{n}_s}(\boldsymbol{m}) = \boldsymbol{grad}(\phi(\boldsymbol{m}))\boldsymbol{\cdot} \boldsymbol{n}_s. \end{equation}

From a theorem that can be found, for instance, in Kress (Reference Kress1999, p. 80), the value of the limits above can be expressed using an improper integral. For $\boldsymbol {m}=(x,y,z)\in \mathcal {S}_p$, this expression reads

(3.33)\begin{equation} v_{n}^{{\pm}}(\boldsymbol{m}) = \iint_{U} \varOmega(u,v) \frac{\partial \varGamma}{\partial \boldsymbol{n}_s} (\boldsymbol{m}-\boldsymbol{m}_s(u,v)) \left\|\frac{\partial \boldsymbol \xi}{\partial u} \wedge \frac{\partial \boldsymbol \xi}{\partial v}\right\| \mathrm{d}u\,\mathrm{d}v \pm \frac{1}{2}\varOmega(\boldsymbol{m}), \end{equation}

with $\boldsymbol {m}_s(u,v)=(x_s(u,v),y_s(u,v),z_s(u,v))$. Finally, $E$ is found by applying the continuity equation (3.10) for the normal velocities across the surface, with the expressions given in (3.29) and (3.30). For $\boldsymbol {m}=(x,y,z)\in \mathcal {S}_p$, we have

(3.34)\begin{equation} E(\boldsymbol{m}) = \frac{v_0\boldsymbol{e_z}\boldsymbol{\cdot}\boldsymbol{n}_s + v_n^-(\boldsymbol{m})}{v_0\boldsymbol{e_z}\boldsymbol{\cdot}\boldsymbol{n}_s + v_n^+(\boldsymbol{m})}. \end{equation}

The system of equations is now closed and enables us to obtain the velocities and pressure at any location in the flow by determining the source strength $\varOmega$, solution of (3.28). In the next section we solve the problem analytically in a basic but very common geometry.

3.6. Application to an inclined rectangular screen in free flow

For a rectangular geometry and homogeneous solidity, it is possible to obtain without major difficulty an analytical solution of the equations of our model. We therefore apply the 3-D model to the simple case of a rectangular screen (centred at $z=0$ as shown in the figure 8) in a free laminar flow in order to find the flow and the drag coefficient as a function of the solidity. In that case, the velocity potential (3.6) becomes

(3.35)\begin{align} \phi_I(x,y,z) = v_0z + c - \frac{1}{4{\rm \pi}}\iint_{\mathcal{S}_p} \frac{\varOmega(u,v)\,\mathrm{d}u\,\mathrm{d}v}{\sqrt{(x-v)^2+ (y-u\sin{(\beta)})^2+(z-u\cos{(\beta)})^2}}. \end{align}

Figure 8. Diagram of the inclined rectangular porous screen in the $(Oyz)$ plane in a 3-D free flow with an angle $\beta$. The dashed lines are the separation streamlines used as a boundary between the regions of the model. The dotted line between points A and B is the streamline used in the model to calculate the velocities in the regions I and III. The incoming flow is laminar.

We then calculate the velocities at the screen. The objective of the calculus is to obtain $\varOmega$ as a function of the solidity, which will determine all the other variables of the problem.

The normal component of the velocity in region I at the arbitrary position $(w,t)$ on the surface is

(3.36)\begin{equation} {v_{I}}_n^{{\pm}}(w,t) = v_0\sin{(\beta)} + \lim_{\epsilon \to 0^\pm}\frac{1}{4{\rm \pi}} \iint_{\mathcal{S}_p} f_{\epsilon}(u,v) \varOmega(u,v)\,\mathrm{d}u\,\mathrm{d}v, \end{equation}

with

(3.37)\begin{equation} f_{\epsilon}(u,v) = \frac{\epsilon\cos{(\beta)}\sin{(\beta)}}{((t-v)^2+(w-u)^2+(2(w-u) + \epsilon) \epsilon\cos{(\beta)})^{{3}/{2}}}. \end{equation}

The details of the calculations are given in Appendix B. At the screen, depending on the direction from which we approach the screen ($\epsilon \to 0^{\pm }$), the magnitude of the normal component of the velocity is constant and is equal to

(3.38)$$\begin{gather} {v_{I}}_n(w,t) = {v}_n^{-}(w,t) = v_0\sin{(\beta)} - \tfrac{1}{2} \varOmega(w,t), \end{gather}$$

(3.39)$$\begin{gather}{v_{II}}_n(w,t) = E(\psi_I,\chi_I){v}_n^+(w,t) = E(\psi_I,\chi_I)(v_0\sin{(\beta)} + \tfrac{1}{2} \varOmega(w,t)). \end{gather}$$

At this point, for an homogeneous screen ($s$ constant on the surface), (3.21) leads to $\boldsymbol {grad}(\varOmega (w,t))=\boldsymbol {0}$, i.e. the source strength is a constant.

Thus, (3.10) leads to a constant attenuation function

(3.40)\begin{equation} E = \frac{v_0\sin{(\beta)} - \dfrac{1}{2} \varOmega}{v_0\sin{(\beta)} + \dfrac{1}{2} \varOmega}. \end{equation}

If we found constant normal velocity, it is not the case of the tangential velocity for which the magnitude varies on the surface of the screen. The tangential component of the velocity at the surface is

(3.41)\begin{align} {v_I}_t(x,w) &= (\varOmega^2 \mathcal{I}_x^2(x,w) + (\varOmega(\sin{(\beta)}\mathcal{I}_y(x,w)+\cos{(\beta)}\mathcal{I}_z(x,w)) \nonumber\\ &\quad + \cos{(\beta)}v_0)^2)^{{1}/{2}}, \end{align}

with $\mathcal {I}_x$, $\mathcal {I}_y$ and $\mathcal {I}_z$ the following surface integrals that are calculated in Appendix A:

(3.42)$$\begin{gather} \mathcal{I}_x(x,w) = \frac{1}{4{\rm \pi}}\iint_{\mathcal{S}_p} \frac{x-v}{((x-v)^2+(w-u)^2)^{{3}/{2}}} \mathrm{d}u \,\mathrm{d}v , \end{gather}$$

(3.43)$$\begin{gather}\mathcal{I}_y(x,w) = \frac{1}{4{\rm \pi}}\iint_{\mathcal{S}_p} \frac{(w-u)\sin{(\beta)}}{((x-v)^2+(w-u)^2)^{{3}/{2}}} \mathrm{d}u \,\mathrm{d}v, \end{gather}$$

(3.44)$$\begin{gather}\mathcal{I}_z(x,w) = \frac{1}{4{\rm \pi}}\iint_{\mathcal{S}_p} \frac{(w-u)\cos{(\beta)}}{((x-v)^2+(w-u)^2)^{{3}/{2}}} \mathrm{d}u \,\mathrm{d}v. \end{gather}$$

For the sake of simplicity, we approximate the magnitude of the tangential component of the velocity as its root mean square. We note that since in (3.28), $v_{I_t}$ appears both in linear and quadratic form, this approximation becomes exact for a surface normal to the mean flow direction ($\beta ={{\rm \pi} }/{2}$). We thus take

(3.45)\begin{equation} {v_I}_t = (\varOmega^2 \gamma_0 + v_0^2\cos^2{(\beta)})^{{1}/{2}}, \end{equation}

where $\gamma _0$ can be considered as a shape factor (see Appendix A). Equation (3.18) becomes

(3.46)\begin{equation} {p_{II}}-{p_{I}} = \tfrac{1}{2}\rho ({v_I}_t^2(1-E^2)+{v_I}_n^2\theta(s)). \end{equation}

Equation (3.13) leads to

(3.47)\begin{align} {p_{III}}-{p_{0}} &= \tfrac{1}{2}\rho (1-E^2)(v^2_0-{v_I}_t^2) + p_{II} - p_{I} \nonumber\\ &= \tfrac{1}{2}\rho((1-E^2)v^2_0 + {v_I}_n^2\theta(s)). \end{align}

Using (3.23)–(3.25), we obtain a first expression of the magnitude of the drag force $F_D$:

(3.48)\begin{equation} F_D = \rho v_0(1-E){v_n} {S_p} + \frac{1}{v_0}(p_0 - p_{III}) \frac{{v_n}}{E} {S_p}. \end{equation}

The second expression for the drag force $F_D$ is obtained with (3.22),

(3.49)\begin{equation} F_D = ({p_I}\unicode{x2013}{p_{II}})\sin{(\beta)}{S_p} + \rho {v_n} v_0\cos^2(\beta)(1-E){S_p}. \end{equation}

Note that, for a rectangular screen orthogonal to the free flow, the second term of (3.49) vanishes since $\beta ={{\rm \pi} }/{2}$ and we obtain a drag coefficient proportional to the pressure difference ${p_I}\unicode{x2013}{p_{II}}$. Now, by denoting $\omega = {\varOmega }/{v_0}$ and combining (3.48) and (3.49), we obtain the following equation that we have to solve to find the value of the source strength:

(3.50)\begin{equation} -\tfrac{1}{8} \omega^4\theta(s) + \omega^2 \sin^2{(\beta)}(8\gamma_0 + \theta(s) - 2) -4\omega\sin{(\beta)} - 2\sin^4{(\beta)}\theta(s) = 0. \end{equation}

This equation has been solved using Python 3.9.5 and the method fsolve from scipy.optimize. The solution for a square porous plate at normal incidence with $\beta ={{\rm \pi} }/{2}$ is plotted in figure 9(a). At low solidity, our model for a square plate is close to the prediction of Steiros & Hultmark (Reference Steiros and Hultmark2018) and Taylor & Davies (Reference Taylor and Davies1944). Above a solidity $s=0.4$ the drag coefficient becomes slightly different from the prediction of Steiros & Hultmark (Reference Steiros and Hultmark2018) and the difference increases with increasing solidity. Three-dimensional effects are therefore important at high solidity.

Figure 9. Theoretical prediction of the drag coefficient and velocities on the surface as a function of the solidity obtained with the 3-D model applied to a square homogeneous screen normal to the flow ($\beta = {{\rm \pi} }/{2}$), and comparison with different 2-D models. The full lines are obtained with the 3-D model developed in this paper using the same pressure jump law across the screen as Steiros & Hultmark (Reference Steiros and Hultmark2018) (but taking account of geometric 3-D effects). (a) Drag coefficient. (b) Normal velocity and quadratic mean of the tangential velocity as defined respectively in (3.38) and (3.45). (a) Theoretical drag coefficient. (b) Theoretical velocity at the screen.

At high solidity ($s\lesssim 1$), the curve converges towards the drag coefficient of a flat solid plate. This value is predicted to be $1.2$ in our model, which is close to the experimental measurement at the global Reynolds number $Re \approx 10^4$, giving approximately $1.05$ according to Blevins (Reference Blevins1992), $1.17$ according to the synthesis of Hoerner (Reference Hoerner1965) on various drag measurements and $0.939$ in pour experiments.

Our model takes into account the shape of the porous surface through the parameter $\gamma _0$. For a rectangular screen, the aspect ratio, taken into account in $\gamma _0$, has an influence on the result as detailed in Appendix A. According to Hoerner (Reference Hoerner1965), for a flat plate normal to the flow, the drag coefficient increases very slowly when the aspect ratio is reduced until a ratio of approximately $0.1$. Beyond this point the increase becomes more pronounced, until it reaches $C_D=2.0$ for an infinitely thin plate ($1.90$ according to Blevins Reference Blevins1992). In our model, we indeed observe an increase of the drag when the aspect ratio decreases (see Appendix A) from $C_D=1.2$ for a square plate (with $\gamma _0=0.0998$), 1.29 for a rectangle with aspect ratio 1/10 (with $\gamma _0=0884$) to $C_D=1.33$ for an infinitely thin plate. This value for an infinitely thin plate is lower than the experimental value, that is, $C_D \approx 2.0$ according to Blevins (Reference Blevins1992) and Hoerner (Reference Hoerner1965). This difference may be due, in this case of very high aspect ratio, to the vortex shedding that is not taken into account in the present wake model, as noted by Steiros & Hultmark (Reference Steiros and Hultmark2018).

In figure 9(b) we compare the normal and tangential velocities in the 2-D and 3-D cases. The tangential velocity is taken in both cases as the quadratic mean over the whole surface, in three dimensions it is defined in (3.45), the normal velocity is constant on the surface in both cases also, in three dimensions it is defined in (3.38). While the normal velocity is the same in two dimensions and three dimensions, we see an influence of 3-D effects on the tangential velocity.

While taking into account the 3-D effects improve the prediction of $C_D$ at high solidities, it does not improve the prediction at moderate and low solidities. The limitations of potential flow theory to describe the complex flow around bluff bodies are arising at high solidity, but should be negligible at low and moderate solidity. Here, we hypothesize that the main reason for the discrepancy between our experimental results (figure 3) and the theoretical prediction (figure 9a) lies in the pressure loss at the pore scale. Indeed, the relation used to estimate the pressure jump $p_{I}\unicode{x2013}p_{II}$ (3.18) is a strong assumption, and does not consider the pressure losses by viscous friction. In particular, it does not explicitly account for the viscosity, and can not account for the variations of the drag coefficient with the local Reynolds number observed in the experiments.

4.1. Pressure jump dependency on the local geometry of the pores and viscous effects

As explained above, in the equation of the pressure jump (3.18) with (3.19), as formulated by Steiros & Hultmark (Reference Steiros and Hultmark2018), we do not take into account the dependency of the pressure drop on the local Reynolds number $Re_d$, computed at the scale of the screen pores (and thus, much smaller than $Re$), the geometry of the holes and other possible dependency like the energy transfer between the material of the screen and the fluid. For instance, Ando et al. (Reference Ando, Nishikawa, Kaneda and Suga2022) showed that a layer of flexible fibres can have a higher permeability than the same layer of rigid fibres due to a flow-induced deformation. Moreover, as shown by Schubauer, Spangenberg & Klebanoff (Reference Schubauer, Spangenberg and Klebanoff1950), the angle of the screen relative to the laminar upstream free flow has an impact on the pressure drop. Kalugin et al. (Reference Kalugin, Epikhin, Chernukha and Kalugina2021) explained also that for inclined perforated plates, the structure of the flow in the holes depends on (1) the distance of the hole on the plate from the leading edge, and (2) the angle of attack of the plate. For a low angle of attack, the effective hole area can be significantly reduced due to the difficulty of the flow to deflect from its original direction mostly parallel to the surface of the plate. All these studies underline the current difficulty to obtain a general formulation of the pressure loss for an arbitrary porous screen. Therefore, in what follows, we adopt another method based on empirical laws in order to test whether this would be sufficient to estimate the drag accurately.

Numerous experimental investigations have shown that the pressure drop can be reasonably considered proportional to the square of the velocity normal to the screen at the vicinity of it through the resistance coefficient $k$, especially Taylor & Davies (Reference Taylor and Davies1944). More recently, Ito & Garry (Reference Ito and Garry1998) studied this problem in the 2-D case of a flow around and through a gauze for low resistance coefficient, while Eckert & Pflüger (Reference Eckert and Pflüger1942) studied the resistance coefficient for the case of a gauze spanning the entire section of a channel. In these cases, the pressure drop can be written as

(4.2)\begin{equation} \Delta p = \tfrac{1}{2} k \rho {v_{I}^2}_n(x_s,y_s,z_s). \end{equation}

The resistance coefficient $k$ depends on the geometry of the holes, the material of the screen and the Reynolds number based on the scale of the holes (sometimes $k$ is directly related to what is called the loss factor or friction factor). This has been mostly studied when the screen spans entirely a channel with normal incidence, and oblique incidence (Schubauer et al. Reference Schubauer, Spangenberg and Klebanoff1950; Reynolds Reference Reynolds1969). Note that in (3.18), $\theta (s)$ can be interpreted as a resistance coefficient, the intervention of the tangential velocity in (3.18) comes from the fact that in our case the fluid can pass around the porous structure.

To formulate the resistance coefficient dependency for porous screens, Pinker & Herbert (Reference Pinker and Herbert1967) has shown that the resistance coefficient $k$ can reasonably be considered as a product of a function of the solidity, $G(s)$, and a function $f$ of the local Reynolds number in a pore. In our definition of $Re_d$, we use the upstream velocity of the fluid far from the mesh; in order to take into account the changes in velocity close to the mesh, we define the local approach Reynolds number,

(4.3)\begin{equation} Re_n = Re_d \frac{v_{I_n}}{v_0}, \end{equation}

based on the scale of holes and the approach velocity that in our formulation corresponds to $v_{I_n}$. Among several fitted expressions for $G$ with respect to the solidity, Pinker & Herbert (Reference Pinker and Herbert1967) found that $G(s) = -\theta (s)$, which exhibits the best agreement with their data.

For the pressure drop, depending also on the inclination of the surface, we have to consider in fact the function $f(Re_n,\beta )$, as first proposed by Schubauer et al. (Reference Schubauer, Spangenberg and Klebanoff1950). Since the geometry of our porous screen is arbitrary, $\beta$ should be considered as a local characteristic of the inclination of the surface of the screen relative to the direction of the laminar free flow in the far upstream. Here $\beta = 0$ means that the surface is parallel to the flow $\boldsymbol {v}_0$. We thus propose the following relation for the pressure jump:

(4.4)\begin{align} {p_{II}}(x_s,y_s,z_s)-{p_{I}}(x_s,y_s,z_s) &= \tfrac{1}{2}\rho {v_I}_t^2(x_s, y_s,z_s) (1-E^2(\psi_I,\chi_I)) \nonumber\\ &\quad + \tfrac{1}{2}\rho{v_I}_n^2(x_s, y_s,z_s)\theta(s)f(Re_n,\beta). \end{align}

The expression of $f$ is thus expected to be found experimentally. As far as we know, there is no general physical formulation of the pressure drop through the holes at the microscopic level that can cover all types of screens, and there is no general demonstration of an analytical expression of $f$ for an arbitrary porous screen shape. Therefore, it is expected that some modifications are required for particular porous structures taking into account, for instance, the geometry of the holes. For screens composed of fibres, with an angle of attack $\beta$ with the upstream flow, and for $10^{-4} < Re_n < 10^4$, Brundrett (Reference Brundrett1993) gives the following empirical expression of $f$:

(4.5)\begin{align} f(Re_n,\beta) = \sin^2{(\beta)}\left(\frac{c_1}{Re_n\sin{(\beta)}}+ \frac{c_2}{\ln{(Re_n\sin{(\beta)}+1.25)}}+c_3\ln{(Re_n\sin{(\beta)})}\right). \end{align}

Here $c_1$, $c_2$ and $c_3$ are real constant. In this model (4.5), the first term corresponds to the laminar contribution, the second term to the turbulent friction and the last term is valid at large Reynolds numbers (Brundrett Reference Brundrett1993). Brundrett (Reference Brundrett1993) obtain a good fitting with his data for wire mesh screens by taking $c_1\approx 7.125$, $c_2\approx 0.88$, $c_3\approx 0.055$. Bailey et al. (Reference Bailey, Montero, Parra, Robertson, Baeza and Kamaruddin2003) found also a good fitting with their data by taking $c_1\approx 18$, $c_2\approx 0.75$ and $c_3\approx 0.055$. For the following sections of the paper, we take the geometric mean of the different values $c_1 \approx 11$, $c_2 \approx 0.8$ and $c_3 \approx 0.055$. For high $Re_n$, the function $f$ behaves as a logarithmic function of the Reynolds number $Re_n$, while for low $Re_n$, the variation of $f$ is much more pronounced (inverse function of the Reynolds number $Re_n$). We thus expect the viscous effects to be important for the typical flow speeds and mesh sizes considered in this work, and in applications such as fog harvesting and face masks.

We thus use this empirical expression, which is in principle only valid for fibrous screens, to account for local viscous effects in our experiments. The last term in the pressure jump (3.18) is thus multiplied by $f(\beta, Re_n)$. This is carried through all the calculation. In particular, our equation to determine the source strength (3.50) is modified such that

(4.6)\begin{align} & -\tfrac{1}{8} \omega^4\theta(s)f(Re_n,\beta) + \omega^2 \sin^2{(\beta)} (8\gamma_0 + \theta(s)f(Re_n,\beta) - 2) \nonumber\\ &\quad -4\omega\sin{(\beta)} - 2\sin^4{(\beta)}\theta(s)f(Re_n,\beta) = 0. \end{align}

This equation has been solved using Python 3.9.5 and the method fsolve from scipy.optimize. In the following, we keep and compare both formulations (3.50) and (4.6). For high solidity, and for a certain range of the two Reynolds numbers $Re$ and $Re_d$, the term $\theta (s)f(Re_n,\beta )$ should behave as the inertial term of the Darcy–Forchheimer equation for porous media. For porous screens composed of square fibre meshes, the inertial term of the Darcy–Forchheimer equation calculated with the method of Wang et al. (Reference Wang, Yang, Huang, Huang, Zhuan and Wu2021) has a value reasonably close to $\theta (s)$ when $s \approx 0.9$. For a very thin porous surface, as discussed by Teitel (Reference Teitel2010), the concept of permeability for porous media involved in the equation of Darcy, and Darcy–Forshheimer, may not always hold for the pressure loss through screens depending on the regime of the flow. According to Brundrett (Reference Brundrett1993) and Bailey et al. (Reference Bailey, Montero, Parra, Robertson, Baeza and Kamaruddin2003), the expression (4.5) seems to be valid over a larger flow regime.

4.2. Three-dimensional and viscous effects on the drag coefficient

As seen previously, our experiments suggest a strong effect of the local Reynolds number on the drag coefficient. We now compare in figure 10 our experimental results with the prediction of the 3-D model, including local viscous effects using the function $f(Re_n,\beta )$. For all mesh Reynolds numbers $Re_d$, the trend of the curve remains globally the same: there is an approximately linear increase of the drag coefficient at low solidity before the curve flattens and reaches a plateau, which is well represented by the 3-D model derived using the method proposed by Steiros & Hultmark (Reference Steiros and Hultmark2018), i.e. without $f(Re_n,\beta )$ (3.50). However, the slope of the initial linear part strongly depends on $Re_d$: it decreases with increasing mesh Reynolds number $Re_d$ for $Re_d<10^3$.

Figure 10. Drag coefficient as a function of the solidity for various square porous screens normal to the free flow with different $Re_d$, and comparison between the 3-D model. The full lines are obtained with the 3-D model developed in this paper, solution of (4.6). The black dash-dotted curve is the solution of (3.50), thus, without taking into account $Re_d$.

Using our model, (4.4) with the empirical formulation of Brundrett (Reference Brundrett1993), we obtain a good fitting with our data for solidity $s\leq 0.6$, for different Reynolds numbers. This shows the importance of both the 3-D effects and the viscous effects through the local Reynolds number $Re_d$. In particular, for a given solidity, the drag coefficient strongly decreases for increasing $Re_d$, as are plotted in figure 11.

Figure 11. Drag coefficient as a function of the Reynolds number $Re_d$ for three different narrow ranges of solidities: $s=0.1 \pm 0.02$, $s=0.28 \pm 0.04$, $s=0.585 \pm 0.025$. The bullets represent the experimental measurements, and the plain lines the results of our model ((4.6) taking into account the effect of mesh Reynolds number $Re_d$). The data points for the drag as well as the Reynolds number values are obtained for the velocity $5.0\leq v_0\leq 13\,{\rm m}\,{\rm s}^{-1}$. The dash-dotted lines represent the value of the drag coefficient for a solid plate (three dimensional) from our experiments and from Blevins (Reference Blevins1992).

We indeed observe a strong effect of the mesh Reynolds number, that also depends on the solidity. We observe a decrease of drag coefficient with increasing $Re_d$ that is well captured by the empirical formulation (4.6). This effect has also been observed for perforated plates by de Bray (Reference de Bray1957). At low $Re_d$, all curves tend towards the value of the drag coefficient of a flat solid plate, i.e. no fluid is passing through the screen and most of it is deviated around. We then note rapid variations for intermediate values of $Re_d$ ($5< Re_d<50$ depending on the solidity), to finally converge towards a constant value at high $Re_d$. This reduction of drag decreases with increasing solidity. As expected, for high solidity ($s=0.9$), there are no effects of $Re_d$, as the flow through the screen is weak. We obtain a similar drag coefficient using the values for the constants $c_1$, $c_2$ and $c_3$ from either Brundrett (Reference Brundrett1993) or Bailey et al. (Reference Bailey, Montero, Parra, Robertson, Baeza and Kamaruddin2003). Indeed, we see that for all solidity and moderate Reynolds numbers $Re_d={O}(200)$ that correspond to most of the screens we tested, the difference is small ($1\leq {C_D({\rm {Bailey}})}/{C_D({\rm {Brundrett}})}\leq 1.05$), and the difference is even lower with increasing Reynolds number $Re_d$. For low Reynolds numbers $Re_d={O}(5)$, the difference is however higher.

To further check the validity of our model, we made screens that have the same solidity but different hole size and number, keeping however $Re_d$ constant (see table 1 for $s=0.24$ (P14, P15 and P16) and $s=0.7$ (P8 and P25)), but still with a periodic distribution. As expected from our model, we do not observe any difference in the drag coefficient within the bounds of the measurement uncertainty, demonstrating that for these regular screens, the friction coefficient depends only on $s$ and $Re_d$.

4.3. Flow visualization

We can gain some insights into those behaviours by plotting the streamlines and velocity magnitude of the flow around and through the screen with our model (figure 12). For a screen normal to the flow, we observe that as the solidity increases, a larger part of the flow is deviated around the screen and that the flow is strongly slowed down in front of the screen, which is consistent with the measured increase of drag coefficient (figure 12a–c).

Figure 12. Streamlines and velocity magnitude obtained with the 3-D model for a square screen normal to the flow at $v_0=2.0\,{\rm m}\,{\rm s}^{-1}$ for different solidities and the pressure jump defined in (3.18) (thus, without dependency on the mesh Reynolds number $Re_d$). Results are shown for (a) $s=0.3$ and $\beta = 90^{\circ }$; (b) $s=0.6$ and $\beta = 90^{\circ }$; (c) $s=0.9$ and $\beta = 90^{\circ }$.

This analysis is taken further by plotting in figure 13 both the PIV measurements (as described in § 2) in the symmetry plane of the screen, as well as the theoretical predictions using the pressure jump equation (3.18) for the sake of simplicity (thus, not accounting for the mesh Reynolds number $Re_d$ effect). As the solidity increases, we can observe a stronger attenuation of the velocity behind the screen, a larger deviation of the flow around the screen, as well as the apparition of a slower region upstream of the screen. We note that the theoretical prediction of the velocity magnitudes are in good agreement with the measured velocities even for the velocity attenuation downstream in the wake. We also observe that the attenuation of the velocity upstream is well captured by the model. However, for high solidity, the width of the wake appears larger in experiments than in the model, which might also be due to the presence of a thicker frame around the mesh in the experiments. Indeed, as observed in the streamlines in figure 4, there are vortices attached to the edges of the frame that may impact both the normal velocity and the shape of the wake.

Figure 13. Comparison between the 3-D theoretical model and our experiments for the prediction of the velocity field in the $(z,x)$ plane with $y=0$ for three different solidities (low, moderate and high). The theoretical velocity field has been obtained using the theoretical pressure jump law (3.18) and (3.19). Uncertainties on the velocities are estimated to be around $0.1\,{\rm m}\,{\rm s}^{-1}$. Results are shown for (a) P14, $s=0.24$, experimental; (b) P14, $s=0.24$, theoretical; (c) P6, $s=0.61$, experimental; (d) P6, $s=0.61$, theoretical; (e) P23, $s=0.82$, experimental; (f) P23, $s=0.82$, theoretical.

By mass conservation and since the normal velocity is assumed constant on the surface in this study, the proportion of the incoming fluid that goes through the screen is directly given by the dimensionless normal velocity ${v_n}/{v_0}$. Experimentally, we measure the velocity using a constant temperature anemometer from Dantec Dynamics (MiniCTA, with probe 55P11, tungsten wire with diameter $5\,\mathrm {\mu }$m and length 1.25 mm, precision of $0.01\,{\rm m}\,{\rm s}^{-1}$, minimum velocity of $0.20\,{\rm m}\,{\rm s}^{-1}$). In figure 14 we plot this ratio ${v_{I_n}}/{v_0}$ as a function of the solidity $s$ for different meshes, i.e. different $Re_d$ as presented in table 2. For a high mesh Reynolds number and low solidity, the model exhibits good agreements with the data. However, we observe a strong effect of $Re_d$ on the normal velocity: notably, for a small mesh Reynolds number $Re_d$, the normal velocity drops more rapidly than predicted by the model. The model also overpredicts the normal velocity for high solidity. For screens commonly used for fog harvesting ($Re_d = {O}(100)$), the normal velocity in the case where $Re_d$ is taken into account can be up to $21\,\%$ greater than the normal velocity in the case where $Re_d$ is not taken into account (according to our theoretical results, this is true for $0.25 < s < 0.75$), and is thus not negligible. These data are coherent with the PIV measurements.

Figure 14. Dimensionless normal velocity ${v_{I_n}}/{v_0}$ obtained with (4.6) for the solid lines, and with (3.50) for the dotted black line. The bullets represent the experimental measurements obtained using a hot wire anemometer.

Table 2. Reynolds number $Re_d$ calculated with the fibre diameter $d$ as characteristic size and with a velocity $v_0 = 2.84\,{\rm m}\,{\rm s}^{-1}$ and a kinematic viscosity $\nu = 15.6 \times 10^{-6}\,{\rm m}^2\,{\rm s}^{-1}$.

In figure 15 we plot the dimensionless tangential velocity distribution on the porous surface for different solidities $s$ and Reynolds number $Re_d$. We see that the tangential velocity is much lower than $v_0$ for a large part of the screen. We thus expect important differences in the streamlines that are deviated around the screen when the solidity or $Re_d$ varies.

Figure 15. Dimensionless tangential velocity magnitude ${|v_{I_t}|}/{v_0}$ on the screen surface obtained with the 3-D model for a square screen normal to a flow for different solidities and Reynolds numbers ((4.6) taking into account the effect of mesh Reynolds number $Re_d$). Results are shown for (a–c) $Re_d = 10$; (d–f) $Re_d = 100$; (a) $s=0.1$ and $Re_d = 10$; (b) $s=0.5$ and $Re_d = 10$; (c) $s=0.8$ and $Re_d = 10$; (d) $s=0.1$ and $Re_d = 100$; (e) $s=0.5$ and $Re_d = 100$; (f) $s=0.8$ and $Re_d = 100$.

In figure 16 we give a representation of the proportion of the fluid passing through the porous surface (and thus, the proportion of the fluid going around it) for different solidities and Reynolds numbers ($Re_d = 10$ and $Re_d = 100$). The contours of these sections $S_a$ represent the separation between the fluid going through the surface and the fluid going around the surface at $z=-\infty$, i.e. the section of the streamtube or separation surface. If we place a passive tracer inside these sections at $z=-\infty$ then the passive tracer will flow across the surface. We observe a clear effect of the 3-D nature of the flow on these separation surfaces. As measured in our experiments, the separation distance along a diagonal is much higher than along the transverse direction, and this effect is amplified as $s$ increases and $Re_d$ decreases. The fraction ${S_a}/{S_0}$ is always equal to ${v_n}/{v_0}$. However, in applications such as filtration or fog harvesting, where the main drop capture mechanism is inertial impact (Moncuquet et al. Reference Moncuquet, Mitranescu, Marchand, Ramananarivo and Duprat2022), the tangential velocity and the shape of the streamlines are expected to impact the amount of droplets that leave the streamlines and are collected on the fibres.

Figure 16. Upstream section $S_a$ of the streamtube containing all the fluid passing through the surface, at $z=-\infty$, seen from the front, for $Re_d = 10$ and $Re_d = 100$.

4.4. Drag coefficient for low angle of attack

We then vary the orientation angle of the screen in the flow $\beta$. When the orientation angle is increased away from the normal incidence, the deviation of the streamlines is less important and the velocity is slightly higher (figure 17a,b). For the extreme case of a solid plate in figure 17(c), two streamlines are deviated along the plate, the flow slows down first and increases again with a peak value above the plate as the fluid particle leaves it and is re-entrained in the surrounding flow. However, if the asymptotic behaviour of the model may provide some indications about the global flow and the aerodynamic forces, it is expected to be outside of the assumptions of the model as discussed in the next section.

Figure 17. Streamlines and velocity magnitude obtained with the 3-D model for a square screen inclined to a flow at $v_0=2.0\,{\rm m}\,{\rm s}^{-1}$ for different solidities and the pressure jump defined in (3.18) (thus, without dependency on the mesh Reynolds number $Re_d$). Results are shown for (a) $s=0.8$ and $\beta = 60^{\circ }$; (b) $s=0.8$ and $\beta = 30^{\circ }$; (c) $s=1$ and $\beta = 30^{\circ }$.

The experimental results are compared with our model for the two angles $\beta =65^\circ$ and $\beta =43^\circ$ on figure 18. As for panels at a normal incidence, the drag coefficient increases with increasing solidity. The drag coefficient decreases with decreasing angle of attack, i.e. as we get further away from the normal incidence. Our model is in good agreement with the experimental data at $\beta =65^\circ$ (figure 18a). However, it underestimates the drag at orientations further from the normal incidence ($\beta =43^\circ$ figure 18b). In the case of an inclined porous screen composed of fibres, the effective solidity should increase as the angle of attack decreases. The use of this effective solidity would result in shifting our data closer to the yellow curve ($87 \leq Re_d \leq 225$) in figure 18(b). We finally observe that both the slope of the linear part and the final value at $s=1$ depend on the angle and that the drag coefficient decreases with decreasing angle.

Figure 18. Comparison between the 3-D model and our experiments for the prediction of the drag coefficient at two angles of inclination for various porous square screens: (a) $\beta = 65^\circ$ (b) $\beta = 43^\circ$ using (3.50). The black dash-dotted line is the theoretical result obtained with (4.6).

In addition, our model predicts a maximum of the drag coefficient at high solidity, as shown in figure 3 (or 10) and more pronounced in figure 18. Such a non-monotonic behaviour of the drag coefficient of permeable shells with solidity has been shown and demonstrated in the viscous regime for Stokes flow by Ledda et al. (Reference Ledda, Boujo, Camarri, Gallaire and Zampogna2021). In our case, at high $Re_L$ and a square screen normal to the flow, several of the porous screens have indeed (on average) a higher drag coefficient than a solid screen (e.g. for P3, P5, P8, P9, P22, P23, P24 and P25). However, due to the uncertainty and the interference drag with the frame used for the support of the screens (see Appendix C), the difference is not significant enough to draw a clear conclusion. We do not observe such a maximum in our measurements for inclined plates at an angle $43^{\circ }$. However, for an angle of $65^{\circ }$, the drag coefficient of the screen $P23$ is on average higher than the solid screen. It is also possible that in our experiments we were outside the flow regime for such a non-monotonic behaviour in the drag coefficient.

5.1. Vena contracta

In our model, as done also by Steiros & Hultmark (Reference Steiros and Hultmark2018), we neglected the possible contraction of the flow within the holes. Considering the vena contracta within the pores, as shown in figure 19, the characteristic velocity immediately upstream given by (3.14), i.e. the mean velocity after contraction of the flow within the holes ${v_h}_1$, can be expressed as

(5.1)\begin{equation} {v_h}_1 = \frac{{v_I}_n(x_s, y_s, z_s)}{C(1-s)}, \end{equation}

where $C$ is the contraction coefficient. This implies that the coefficient $\theta (s)$ given in (3.19) is

(5.2)\begin{equation} \theta(s) = 1-\frac{1}{C^2(1-s)^2}. \end{equation}

As far as we know, there are no measurements of vena contracta for screens constituted of fibres and especially in the 3-D case of free flow. Simmons & Cowdrey (Reference Simmons and Cowdrey1945) performed measurements of the velocity profile behind a porous screen made of a square mesh of woven material spanning a section of a channel and suggest an estimate of the vena contracta assuming a uniform velocity ${v_h}_1 = v_0$ in the holes. In these experiments the screens are made of circular rods with diameters ranging from 0.112 to 0.373 mm arranged in a square mesh. The velocity used was from $2.44$ to $10.36\,{\rm m}\,{\rm s}^{-1}$. The local Reynolds number $Re_d$ thus varies from $18$ to $248$, which is the same order of most of the porous screens used in our experiments. For a solidity $s\approx 0.5$, the coefficient $C$ should be between $0.9$ and $1.0$. Note that with the formulation (5.2), if $C \neq 1$, the limit $s=0$ leads to a non-zero pressure difference, suggesting that $C$ may depend on the solidity $s$ at least for low solidities. We note that if the contraction of the flow is not neglected in $\theta (s)$, i.e. $C\neq 1$, there is an increase of the drag coefficient compared with the curve plotted in figure 9(a). We would thus still overestimate the drag compared with our data in figure 3.

Figure 19. Diagram of the flow at the scale of the pores in case of parallel fibres (left), and corresponding imaginary surface as used in the model (right).

5.2. Asymptotic behaviour at large solidities

Although the model is built for porous surfaces, it is interesting to explore its asymptotic behaviour when the solidity tends to $1$. In this limit, one can question whether a porous description of the surface is still valid.

When the screen tends to a solidity equal to one, we can obtain an expression of the drag coefficient. Taking (4.6), dividing by $\theta (s)$, which tends to $+\infty$ when $s \to 1$, we have

(5.3)\begin{equation} (\omega^2-4\sin^2{(\beta)})^2=0. \end{equation}

Therefore (excluding the case when $\omega$ is negative, which would not correspond to the type of flow we study in this paper), we obtain

(5.4)\begin{equation} \omega = 2\sin{(\beta)}, \end{equation}

which gives ${v_I}_n=0$ as expected. Introducing this value into the expression of the drag coefficient gives

(5.5)\begin{equation} C_D = 2\sin^3{(\beta)}(1-4\gamma_0). \end{equation}

Note that the solid drag coefficient does not depend on the assumption of the pressure loss across the screen through the expression of $\theta (s)f(Re_n,\beta )$ (in (4.4)) as expected.

We measured the drag coefficient as a function of the angle of attack for a square plate at solidity $1$. We plotted the result in figure 20, and as we can see, there is a gap between the prediction (5.5) and the experimental value of the drag coefficient especially at a low angle of attack.

Figure 20. Comparison between the drag coefficient prediction of the 3-D model and the experimental measurement for a square plate in a free flow for different angles of inclination. Data are added from Blevins (Reference Blevins1992), Okamoto & Azuma (Reference Okamoto and Azuma2011) and Torres & Mueller (Reference Torres and Mueller2004).

Several explanations for this gap can be listed. First, for the solid plate, assuming a no-slip boundary condition, the tangential component of the velocity on the surface is equal to zero and increases gradually in the boundary layer. Here, this component of the velocity is not equal to zero but takes a value that corresponds to the conservation of the pressure head along the streamlines in region I. Moreover, it is clearly possible that the pressure difference $p_I\unicode{x2013}p_{II}$ is not always well determined through the assumptions of the model, especially for low angle of attacks (or particular shapes) where detachment of the boundary layer and the formation of separation bubbles can occur that strongly affects the pressure distribution around the surface (Crompton & Barrett Reference Crompton and Barrett2000).

5.3. Vortex shedding

Our model is only valid for a steady wake, i.e. without vortex shedding. This could be taken into account, but is beyond the scope of the current study. However, we believe vortex shedding has little influence on our experimental results. For instance, it has been shown that vortex shedding was suppressed for the flow past a porous cylinder (Ledda et al. Reference Ledda, Siconolfi, Viola, Gallaire and Camarri2018). In fact, in our flow field measurements, we do not see any vortex shedding close to the plate. To evaluate the influence of the vortex shedding on our measurements, we used two splitter plates of different length located in the wake as done by Steiros & Hultmark (Reference Steiros and Hultmark2018). We note that this method was proposed for a 2-D flow, and might not apply directly to our 3-D configuration. In fact, we are not aware of experiments to suppress vortex shedding in three dimensions but we believe that splitter plates can still give pertinent information on vortex shedding. Following Apelt & West (Reference Apelt and West1974), for Reynolds numbers in the range $10^4< Re<5\times 10^4$, which corresponds to our case, the use of a splitter plate three times longer than the plate already suppresses the vortex shedding. In our experiments the splitter plate has the same height as the solid plate on which drag is measured but with length 11 and 30 cm and thickness of 3 mm. Figure 21 shows the measured drag force for a solid plate (solidity equal to $1$, square of $11.0\times 11.0\,{\rm cm}^2$, 3 mm thickness) with and without a splitter plate. We see a reduction of the drag coefficient of $0.07$ with a splitter plate three times longer than the plate. This difference is expected to be less and less significant with decreasing solidity.

Figure 21. Comparison of the drag force $F_D$ with and without a splitter plate for a solid plate (without frame support) of dimensions $11.0\times 11.0\,{\rm cm}^2$ and thickness 3 mm. The fitting curves are obtained using a quadratic law.

While these results are not sufficient to rule out completely the effect of vortex shedding, we note that vortex shedding increases the drag significantly and reduces the base pressure, as seen in drag measurements in two dimensions by Steiros & Hultmark (Reference Steiros and Hultmark2018); these effects cannot explain the difference between the theory and the experimental data since suppressing vortex shedding could lead to a lower value of the drag coefficient compared with our measurements, thus, the difference between the theoretical model and the data would be more important.