The aim of this work is to experimentally examine flow over and near random porous media. Different porous materials were chosen to achieve porosity ranging from 0.95 to 0.99. In this study, we report the detailed velocity measurements of the flow over and near random porous material inside a rectangular duct using a planar particle image velocimetry (PIV) technique. By controlling the flow rate, two different Reynolds numbers were achieved. We determined the slip velocity at the interface between the porous media and free flow. Values of the slip velocity normalized either by the maximum flow velocity or by the shear rate at the interface and the screening distance K^{1/2} were found to depend on porosity. It was also shown that the depth of penetration inside the porous material was larger than the screening length using Brinkman’s prediction. Moreover, we examined a model for the laminar coupled flow over and inside porous media and analyzed the permeability of a random porous medium. This study provided detailed analysis of flow over and at the interface of various specific random porous media using the PIV technique. This analysis has the potential to serve as a first step toward using random porous media as a new passive technique to control the flow over smooth surfaces.

## I. INTRODUCTION

Soft porous media are used in a variety of materials, including filters and brushes, which interact with fluids. Understanding the velocity profile at the interfacial region between the porous layer and the free flow can help us to accurately predict the flow rate and momentum in a soft porous medium and can lead us to the design of advanced and efficient engineering and technological applications.

Coupling of free-fluid flow and a porous medium is very prominent both in nature and industry; thus, it has attracted significant attention from various fields of study. Darcy’s law^{1} facilitated breakthroughs in understanding and modeling the flow through porous media. The laminar flow in the free region can be modeled by the Navier-Stokes equation. Darcy’s equation coupled with the Navier-Stokes equation can be used to describe the flow within the porous medium and free-flow region; however, it would fail due to the jump in the velocity and shear stress at the interface. To extend Darcy’s equation and to contain these discrepancies, Brinkman’s equation^{2} was formulated

Here *μ* is the dynamic viscosity of working fluid, *μ*′ is the so-called effective viscosity, which takes into account the slip at the interface between the porous and the fluid. K is the permeability of the porous material. It depends on the fluid, the geometry, and the structure of the porous medium.

Many researchers have used Brinkman’s equation in combination with the Navier-Stokes equation as a method for coupling the two media. For instance, several numerical and analytical studies have studied flow over porous media.^{3–10} Researchers have investigated flow over sediment beds,^{11} coral reefs and submerged vegetation canopies,^{12} crop canopies and forests,^{13} endothelial glycocalyx of blood vessels,^{14} flow over carbon nanotubes (CNTs),^{15,16} polymer brushes,^{17,18} and flow in a channel bounded by one or two porous media.^{19} Coupled flows also occur in packed-bed heat exchangers, thermal insulation, geothermal engineering, and nuclear waste repositories.^{20} Studies of flow through porous media have also been conducted using a variety of models of porous media (as reviewed by Larson and Higdon^{21}). However, the detailed velocity profile measurements inside a soft random porous medium and its surface are still challenging; therefore, the experimental data concerning interfacial flow are very limited and restricted to simple ordered geometries, such as cylindrical arrays or structured geometries. To our knowledge, the slow flow of Newtonian fluids over random porous materials and their related slip velocity has never been addressed. Thus, the present study focuses on pressure-driven flow over random porous layers in low-Reynolds-number regimes, which is schematically shown in Fig. 1. The velocity profile, as shown in Fig. 1, is nearly parabolic in the open flow and almost constant inside the porous medium. In addition, there is an effective “slip” velocity at the interface, where several studies^{22–36} have been conducted to understand the effect of fluid flow and porous medium on the slip velocity.

One of the first papers on flow through and over a porous material was published by Beavers and Joseph.^{23} They used an experimental setup to understand flow at the interface of porous material in a two-dimensional Poiseuille flow. They suggested that there is a boundary layer region that forms inside the porous material which yields a velocity profile similar to Fig. 1 and it is due to the effects of viscous shear penetrating into the porous material; across this boundary region, the velocity reduces sharply from its value at the interface, u_{s}, to its Darcy value, u_{p}. They also indicated that the velocity near the surface of a permeable medium differs from its mean value within the porous material. To account for transport phenomena at the interface, they defined a new boundary condition that relates the exterior velocity and Darcy value to the shear rate at the permeable boundary and can be given by

where $\lambda =\alpha K$ is a parameter with a dimension of $L\u22121$ and depends on permeable surface properties. α is a dimensionless parameter that contains structural characteristics of the porous material and is called the slip coefficient.

A dimensionless slip velocity was derived by James and Davis^{24} used dimensional analysis. Consequently, they derived a dimensionless slip velocity of the form $us/\gamma \u0307K$, where $\gamma \u0307$ is the shear rate at the interface and it is much larger than $K$. It should be noted that if the effective viscosity is chosen to be the same as the viscosity of the fluid in the flow, as was assumed by Brinkman, then $us/\gamma \u0307K$ and the slip coefficient α are unity. They investigated both plane shear and pressure-driven flows across a square array of rods with solid volume fractions ranging from 0.001 to 0.1.^{24,27,28} In the case of the pressure-driven flow, values of $us/\gamma \u0307K$ varied from 0.15 to 0.3, depending on the solid volume fraction, fraction of the channel filled by rods, and the number of rows of rods that create a particular filling fraction. Their study also demonstrated very little flow penetration and slip velocity values are at most a third of the values predicted by Brinkman’s equation when the rods are oriented across the flow.

In order to understand the flow over and through porous media, many experiments were conducted. The velocity measurements at the interface of a porous medium and in an open channel have been investigated using a laser Doppleranemometer by Gupte and Advani.^{1} They used glass fiber material to form porous media with three different solid volume fractions: 0.07, 0.14, and 0.2. Agelinchaab *et al.*^{32} conducted experiments using PIV and various combinations of rod diameter and rod spacing to achieve a solid volume fraction in the range of 0.01–0.5. They reported that the slip velocity in the open flow depends on the solid volume fraction, rod spacing, and fraction of the channel filled by rods. The flow field near the edge of the model porous medium consisting of an annular array of regular-spaced circular rods installed vertically on an acrylic disk to provide 0.025, 0.052, and 0.1 solid volume fractions has been examined by Shams *et al.*^{30} using PIV technique. They generated circular Couette flow between the outermost rods and a rotating outer cylinder and observed secondary motion inside the porous medium for solid volume fractions of 0.52 and 0.1. Next, Tachie *et al.*^{31} conducted velocity measurements, using PIV, in square arrays of rods with circular, triangular, and square sections. Tachie *et al.*^{17} also conducted PIV velocity measurements of circular Couette flow through a “brush” modeled by an array of uniformly spaced rods mounted perpendicular to the stationary inner cylinder for solid volume fractions of 0.025, 0.05, and 0.1. Arthur *et al.*^{37} investigated the velocity profiles of pressure-driven flow through porous media under three different boundary conditions using two-dimensional PIV measurements. They used a porous medium constructed of a square array of circular acrylic rods. By changing the volume fraction of porous media, different velocity profiles were obtained. They then conducted an experimental work aimed at studying the effects of the PIV interrogation area and overlaps, location of the interface, and depth ratios on the flow at the interface between a model porous medium and an overlying free flow.^{38} The modified Brinkman equation (MBE) has been studied as a tool to predict the macroscopic velocity profile across a Sierpinski carpet interface.^{39} It was found that the MBE can describe the macroscopic velocity at the interface of a Sierpinski carpet, even though it was originally derived for the much simpler case of brush configurations. Although the macroscopic formulations for the two flow cases have an identical form, the permeability term in the brush configuration represents the viscous drag alone, whereas in the Sierpinski carpet, it accounts for both the viscous drag and form drag. However, to our knowledge, no one has yet attempted to analyze pressure-driven flow over soft random porous media.

In summary, the foregoing review of the literatures indicates that despite significant progress and study which has been made to understand flow over and inside porous media, most of the experimental and theoretical studies in this area mainly focused on a structured porous media;^{17,19,22,24,27,28,30–32,37,38,40} thus, the detailed analysis of the flow over and at the fluid-porous interface of random porous media is still unknown. The porous media in most of these studies consisted of an annular array of regularly spaced circular rods. In this paper, we perform several experiments to study the characteristics of flow over random porous materials in which the porous media coated the lower surface of the channel. The void volume fraction (i.e., porosity) varied from 0.95 to 0.99. In all the experiments, the Reynolds number was kept low so that the flow was laminar. The PIV technique was used to obtain a detailed velocity profile over the porous media and at the interface between the porous media and the flow. The present study provides only two-dimensional whole-field velocity component measurements in two planes (where changes of velocity distributions are most dramatic) for various porous media. Using the obtained measurements, the slip velocity was determined, and its variation with porosity, Reynolds number, and height of model porous medium relative to the channel height was studied. A coupled flow model over and inside the soft porous media was examined via experimental measurements; the proper experiments to determine the Darcy permeability for random porous media were also conducted.

## II. EXPERIMENTAL SETUP, MATERIALS, AND MEASUREMENT TECHNIQUE

The experiments of this study were conducted in a three-dimensional channel where the soft porous media coated the bottom wall of the channel. Schematic diagrams of the channel and arrangements of the test models are shown in Fig. 2. This figure also describes the coordinate system used in this study where “x” represents the streamwise direction and “y” indicates the vertical direction. “z,” which indicates the spanwise direction, is perpendicular to the “x” and “y.” The following references were used in this study to describe the test section: x = 0 is located at 70 cm downstream from the inlet of the channel, in order to ensure that the flow over the porous media is fully developed, y = 0 at the fluid-porous media interface, and z = 0 at the mid-plane of the interface.

To obtain a void fraction in the desired range, three different random porous media were tested. The porous media in the experimental analysis of this work, manufactured by Mountain Mist, have been made of 95% polyester and 5% silk, which according to our analysis could provide a relatively smooth boundary surface. Three different porous media heights—h_{p} = 5.5 mm, 9 mm, and 11 mm—were also used to study how the fraction of the channel filled by the porous media modifies the flow and slip velocity. The porosity (void fraction) of the porous media was calculated using direct volume measurements, ε = V_{pore}/V_{total} = 1 − ρ/ρ_{solid}, where ρ and ρ_{solid} are the densities of a porous material and its solid phase, respectively. We have defined the solid phase density using the water volume displacement method^{41–43} and the working fluid similar to the experiments (i.e., water/glycerin solution consisting of 80% glycerin and 20% water by weight). First, a particular volume of glycerin-water solution was added to a graduated cylinder. Then, a sample of porous material was weighed with 0.0001 g accuracy and saturated in the glycerin-water solution. We then defined the volume of the material’s solid phase by the change in volume of the glycerin-water solution in the graduated cylinder and calculated ρ_{solid}. By measuring the mass and3t the volume of the porous layer, we then defined the density of the porous material and accordingly calculate the porosity. The porosity of our materials has been defined as 0.95, 0.98, and 0.99, respectively. Figure 3 shows a sample of porous media with a porosity of 0.95. The relevant properties of the porous media are summarized in Table I, where ε is the porosity, h_{p} is the thickness of the porous material coating the lower wall of the channel, and K is the permeability of the porous materials. It should be noted that all the reported formulas to determine the permeability of porous media are generally empirical formulas containing several fitting parameters. Therefore, their applicability is restricted to the physical conditions in which they have been expressed. In this study, we defined the permeability of the porous media experimentally using Darcy’s law.

h_{p} (mm)
. | ε . | K (mm^{2})
. |
---|---|---|

5.5 | 0.95 | 0.083 |

11 | 0.98 | 0.105 |

9 | 0.99 | 0.140 |

h_{p} (mm)
. | ε . | K (mm^{2})
. |
---|---|---|

5.5 | 0.95 | 0.083 |

11 | 0.98 | 0.105 |

9 | 0.99 | 0.140 |

The working fluid was water/glycerin solution consisting of 80% glycerin and 20% water by weight, which was well stirred to maintain transparency. The viscosity of the working fluid, which was measured before and after an experiment using a Malvern Rheometer with an uncertainty of ±2%, never changed by more than 1%. The working fluid has a viscosity and density of μ = 0.07 Pa⋅s and ρ = 1.21 g/cm^{3}, respectively, which enabled low-Reynolds number testing. Ensuring adequate control of the viscosity of the working fluid was difficult because the viscosity of aqueous solutions of glycerol is very sensitive to temperature changes. Since using a water jacket to control the temperature of the working fluid was impractical, the temperature of the working fluid was continuously monitored. The temperature of the working fluid tracked the room temperature, which varied by no more than 25 °C over the course of an experiment. The index of refraction of this fluid is 1.443 ± 0.002 at 20 °C.^{44} The rectangular duct had a cross section of 25 × 42 mm^{2} and a length of 90 cm. A peristaltic pump (Simplypump, Inc.) was used to pump the fluid through the system. The pump had a maximum head of 20 ft. A flowmeter (Hedland, Inc.; Range: 0.1–1.0 GPM) was used to measure the flow rate through the system. An interchangeable surface was used at the bottom wall of the channel and was covered with our sampled porous material. The Reynolds number was calculated from the bulk velocity and the hydraulic diameter of the free flow region. It was also verified by the numerical integration of the velocity profiles when the PIV data of the entire channel height were described to ensure that the fluid was laminar. We prepared the flow setup by draining the duct, replacing the sliding plate with the porous material, and finally calibrating it. Before each experiment, the test was run between 10 and 45 min to reach a steady-state condition.

Flow-field measurements were developed using a LaVision Flow Master particle image velocimetry (PIV) system. The flow was seeded with 3.2 *μ*m fluorescent particles (Thermo Scientific Fluoro-Max Red, 36-3) that experience a peak excitation at 542 nm and a peak emission at 612 nm. In this manner, the 532 nm Nd:YAG (Litron Nano L, 50-50) laser causes the particles to fluoresce. The images were then acquired with a 2560 × 2160 pixel^{2} Imager sCMOS camera with a full-scale resolution up to 50 Hz. A band-pass filter (BP532_10) was placed over a 60-mm-focal-length lens attached to the camera in order to minimize spurious reflections and enhance the particle signal-to-noise ratio. Homogeneity of the seeding distribution to reduce the error in velocity vectors was also checked. All timing and software were handled using the DaVis8 software package on a 2× quad-core XEON processor PC with 12 GB RAM.

Several measurements were taken in the x-z plane at various y locations and in the x-y plane at various z locations in the free flow region and on the top of the porous media to fix the position of the laser and camera in order to obtain the best data.

Figure 4(a) shows an example of a PIV image, the so-called test window, which was used to collect the PIV data. The flow in the fluid layer was observed in a window with a large field of view, i.e., 26 mm width and 22 mm height. The time interval between each pair of images was 3000 *μ*s. It should be noted that the upper part of Fig. 4(a) is the fluid region, whereas the lower part is the porous bed with a porosity of 95%. The tiny bright spots are tracer (seeding) particles. A typical velocity profile of the horizontal component of PIV result is plotted in Fig. 4(b). The maximum horizontal velocity component u_{max} appears at the upper part of the flow near the middle of the free surface. The horizontal velocity component is constant in the vicinity of the free surface and decays continuously until it reaches the porous bed.

To observe the flow near the interface better and to obtain more details of the slip velocity with a better relative uncertainty, a smaller test window was picked. Figure 4(c) is an example of a testing field with 18 mm width and 16 mm height. The time interval between each pair of images was 5000 *μ*s. The relative uncertainty of the velocity obtained from the PIV technique was determined to be approximately 5% in the large test widows [i.e., in Fig. 4(a)] and 3% in the small test windows [i.e., in Fig. 4(c)] based on the procedures outlined by Coleman and Steele.^{45}

## III. RESULTS AND DISCUSSION

Prior to conducting measurements of the flow over different soft porous media, we conducted several experiments in an empty channel (i.e., without porous media) at various Reynolds numbers and different axial locations downstream from the inlet of the test section of the channel to ensure the independency of the results at the interrogation region from the Reynolds number and to verify that the flow is fully developed in the test section.^{37} Selected velocity profiles at the channel mid-plane for different Reynolds numbers and a sample size of N = 300 are shown in Fig. 5. We also calculated analytical velocity profile of the laminar flow in an empty channel using MATLAB. As can be seen, these selected velocity profiles together with the analytical profile collapsed reasonably well onto each other, which confirms the accuracy of the PIV measurements and ascertains the development of the flow at the location of the porous media in the test section.

Next, we have to ensure that the flow becomes periodic over the porous media for the streamwise direction because the flow in the channel is three-dimensional when the soft porous media are installed in the channel. To achieve this, we performed experiments in the x-y plane at various y locations to determine the value of x/L for which the flow becomes fully developed. Note that in this study, L is the length of the channel, and W is the width of the channel.

Figure 6 shows the velocity profiles obtained along the x-direction at the interface and over the porous material where y/h_{p} = 0 (i.e., the interface), y/h_{p} = 0.25, and y/h_{p} = 0.5. The light sheet is located at z/W = 0. Note that the velocity profile in the x-direction at z/W = 0 and different y locations also shows how well the camera has been aligned during the measurements. In addition, the velocity profile u(x) at y/h_{p} = 0 and z/W = 0 shows the characteristics of the porous material. As can be seen in Fig. 6, the values of u(x) in the PIV field of view vary from 0 to 18 mm/s along the y axis. As one moves far from the interface to the open flow region, these values become more constant, which confirms that the flow is periodic. Furthermore, in Fig. 6, since the variation of u(x) at the location where y/h_{p} = 0.25 and y/h_{p} = 0.5 from the interface is not noticeable, we can conclude that the alignment uncertainty is negligible.

To demonstrate how various porosities and the thicknesses of the random porous layers modify the velocity profile from the corresponding one in an empty channel, we plotted selected velocity profiles in Fig. 7. These profiles were obtained at the mid-plane of the channel (i.e., z/W = 0) with and without the porous media. The profiles were obtained for h_{p} = 5.5 mm, 9 mm, and 11 mm as well as for porosities ε = 95%, ε = 99%, and ε = 98% and Reynolds number equal to 2.5. It can be seen that one of the effects of the porous layer is to shift the location of the maximum velocity towards the upper wall of the channel. In addition, the values of velocity at the interface (y/h_{p} = 0) for h_{p} = 5.5 mm and 11 mm are very different—they are around 2% and 10% of the maximum velocity, respectively.

In addition, Fig. 7 reveals that if the height of the pure fluid region is fixed, for the same porosity material, to increase the height of the porous medium will not cause any change in the velocity profile. This is also shown in Fig. 8 that the velocity profile in the porous region goes to zero before the 1/10 depth of the porous media. Thus, increasing the thickness of the porous medium will not impact the velocity on the free-flow region. Moreover, the structure and the pore spaces play a critical role on porosity and consequently on the velocity profile inside the porous medium.

Figure 8 shows the variation of the normalized average velocity in the smaller test window, obtained from the PIV, where y/h_{p} = 0 corresponds to the fluid-porous interface. Note that the average velocity, in these figures, indicates that the velocity has been averaged in the same vertical location. We checked the profiles for various Reynolds numbers, and there were no significant inertia effects. As can be observed from the plots, the velocity decreases from the value at the interface to a constant value inside the porous media. In addition, the slip velocity appears to be higher as the porosity increases. This is because the lower resistance in the porous material with higher porosity. The detailed normalized slip velocity has been presented in Table II.

Porosity . | h_{p}
. | . | . | u_{max}
. | u_{s}
. | . | . | . | $K$ . | . | . |
---|---|---|---|---|---|---|---|---|---|---|---|

ε (%) . | (mm) . | u_{ave}
. | Re . | (mm/s) . | (mm/s) . | $\gamma \u0307$ (1/s) . | u_{s}/u_{max}
. | u_{ave}/u_{max}
. | (mm) . | $us/\gamma \u0307K$ . | α . |

95.00 | 5.5 | 4.8 | 2.5 | 10.1 | 0.37 | 1.14 | 0.037 | 0.475 | 0.289 | 1.126 | 0.888 |

95.00 | 5.5 | 16.2 | 7.5 | 35.4 | 1.13 | 2.88 | 0.032 | 0.458 | 0.289 | 1.360 | 0.735 |

98.00 | 11 | 5.6 | 2.5 | 15.6 | 1.47 | 3.89 | 0.094 | 0.359 | 0.324 | 1.168 | 0.856 |

98.00 | 11 | 18.9 | 7.5 | 51.5 | 5.40 | 12.63 | 0.105 | 0.367 | 0.324 | 1.318 | 0.759 |

99.00 | 9 | 7.6 | 2.5 | 18.1 | 1.78 | 3.48 | 0.098 | 0.420 | 0.374 | 1.367 | 0.731 |

99.00 | 9 | 19 | 7.5 | 44.4 | 4.61 | 8.18 | 0.104 | 0.428 | 0.374 | 1.506 | 0.664 |

Porosity . | h_{p}
. | . | . | u_{max}
. | u_{s}
. | . | . | . | $K$ . | . | . |
---|---|---|---|---|---|---|---|---|---|---|---|

ε (%) . | (mm) . | u_{ave}
. | Re . | (mm/s) . | (mm/s) . | $\gamma \u0307$ (1/s) . | u_{s}/u_{max}
. | u_{ave}/u_{max}
. | (mm) . | $us/\gamma \u0307K$ . | α . |

95.00 | 5.5 | 4.8 | 2.5 | 10.1 | 0.37 | 1.14 | 0.037 | 0.475 | 0.289 | 1.126 | 0.888 |

95.00 | 5.5 | 16.2 | 7.5 | 35.4 | 1.13 | 2.88 | 0.032 | 0.458 | 0.289 | 1.360 | 0.735 |

98.00 | 11 | 5.6 | 2.5 | 15.6 | 1.47 | 3.89 | 0.094 | 0.359 | 0.324 | 1.168 | 0.856 |

98.00 | 11 | 18.9 | 7.5 | 51.5 | 5.40 | 12.63 | 0.105 | 0.367 | 0.324 | 1.318 | 0.759 |

99.00 | 9 | 7.6 | 2.5 | 18.1 | 1.78 | 3.48 | 0.098 | 0.420 | 0.374 | 1.367 | 0.731 |

99.00 | 9 | 19 | 7.5 | 44.4 | 4.61 | 8.18 | 0.104 | 0.428 | 0.374 | 1.506 | 0.664 |

After analyzing the velocity profiles, we obtained the values of the average velocity at the fluid-porous interface, i.e., the slip velocity, u_{s}. We also defined the shear rate $\gamma \u0307=\u2202u\u2202yy=0$ estimated by differentiating the least square curve fits to the area-averaged mean velocity in the interfacial region and evaluating it at y/h_{p} = 0. The relative uncertainty of shear rate is estimated to be 7%. Thereafter, we normalized u_{s} either by u_{max} as u_{s}/u_{max} or by the $\gamma \u0307K$ as of $us/\gamma \u0307K$. The typical values for u_{s}, maximum velocity u_{max}, shear rate $\gamma \u0307$, u_{s}/u_{max}, u_{ave}/u_{max}, $us/\gamma \u0307K$, and α are listed in Table II. $us/\gamma \u0307K$ is a more useful dimensionless slip parameter, and it depends on the local conditions rather than the far-field velocity profile u_{max}.^{24,32} This value is also the inverse of the slip coefficient as proposed by Beavers and Joseph^{23} and can be related to the Brinkman’s equation. As can be seen in Table II, no matter if we normalize the slip velocity by u_{max} or by $\gamma \u0307K$, by increasing the porosity of the random porous media, the normalized slip velocity increases. We estimated the relative uncertainty in $us/\gamma \u0307K$ to be 10% and the relative uncertainty of u_{s}/u_{max} and u_{ave}/u_{max} to be 8%, respectively.

We also focused on the slip coefficient α, which is the inverse of $us/\gamma \u0307K$ and has been determined in several studies using analytical, numerical, and experimental analyses for different types of porous media. For instance, the values of the slip coefficient α obtained by Beavers and Joseph^{23} for a rock sample were in the range of α = 0.1–4 (or $us/\gamma \u0307K$ = 0.25-10). A wide range of values α = 0.07-0.43 (or $us/\gamma \u0307K=$ 2.3-14.3) for flow through a glass fiber material in an open channel has also been reported by Gupte and Advani.^{1} The values of $us/\gamma \u0307K$ reported by Davis and James^{28} are in the range of 0.76–0.88 for flow through and over circular rods aligned along the flow. A set of values of $us/\gamma \u0307K$ from 0.97 to 2.11 for various combinations of rod diameter and rod spacing in which circular cylindrical rods were installed vertically on the bottom wall of the channel in regular square arrays were found by Agelinchaab *et al.*^{32} This value was around 1.0 for the rods mounted perpendicular to the walls of a cylinder in Couette device.^{17} Herein, we found that $us/\gamma \u0307K$ ranges from 1.126 to 1.506.

We further compared the variation of $us/\gamma \u0307K$ versus solid volume fraction in Fig. 9 to compare our work with the previous studies.^{17,28,32} It can be seen that the values of $us/\gamma \u0307K$ obtained in the present study are different for various porosities and nearly independent of the Reynolds number.

In addition, the slip velocity appears to be increasing as the porosity increases; however, the values are smaller than the values obtained by Agelinchaab^{32} for the structured rod-like porous model mounted vertically on the bottom wall of the channel. This is because our test samples have complex and random structures that lead to the lower permeability. Thus, it can be concluded that $us/\gamma \u0307K$ depends on both permeability and porosity of the porous layer.

Furthermore, we determined the depth of penetration δ in the vertical direction in which the local velocity u inside the porous media decays to 0.01(u_{s} − u_{p}). We compared δ with the screening distance obtained from Brinkman’s equation (i.e., K^{1/2}), which suggests that δ is of order K^{1/2}. Table III summarizes values of δ for some selected porosities and their comparison with K^{1/2}. It can be found that the penetration depth δ is larger than that suggested by Brinkman’s equation with $\mu \u2032/\mu $ = 1 and depends on both the porosity and permeability of the porous media. This result agrees with the prior results.^{1,32} We also found that the estimated relative uncertainty for δ is 5%.

h_{p} (mm)
. | Porosity ε . | $K$ (mm) . | δ (mm) . | $\delta /K$ . |
---|---|---|---|---|

5.5 | 0.95 | 0.289 | 0.72 | 2.49 |

11 | 0.98 | 0.324 | 0.85 | 2.62 |

9 | 0.99 | 0.374 | 1 | 2.67 |

h_{p} (mm)
. | Porosity ε . | $K$ (mm) . | δ (mm) . | $\delta /K$ . |
---|---|---|---|---|

5.5 | 0.95 | 0.289 | 0.72 | 2.49 |

11 | 0.98 | 0.324 | 0.85 | 2.62 |

9 | 0.99 | 0.374 | 1 | 2.67 |

To conclude this study, we then used the analytical prediction of the coupled flow over and inside the porous region to examine the flow over random porous media using the coupled Navier-Stokes and Brinkman’s equations and our experimental findings. The experimental data presented herein and from previous research^{32} show that the velocity profile inside and over the porous media, and consequently the shear stress in the system, depends on the value of K^{1/2}, the thickness of the porous layer, h_{p}, the height of the fluid flow, h_{f}, and the ratio of $\mu \u2032/\mu $; this result is consistent with the theoretical analysis.^{10,14–16} Our aim is to investigate the coupled velocity profile over and through a random porous medium once the channel geometrical features and the porous medium properties are known.

A general solution of Brinkman’s equation using coupled flow over and inside porous media in dimensionless form in a geometry shown in Fig. 1 can be given by^{10}

where

Here,

In addition, $\xfbs$ is the dimensionless interfacial velocity, *ξ* is the length ratio, β is the permeability parameter, and M is the viscosity ratio. It should be noted that for high porosity porous materials as reported by Ref. 10, M could be assumed to be 1, i.e., *μ*′ = *μ*. However, in the current manuscript, we defined *μ*′ precisely based on our experimental results from Table II and using Eq. (3) [i.e., $\alpha =(\mu /\mu \u2032)1/2$].

We then evaluated this model by comparing it with the experimental results of the current study for different porous media. The velocity measurements and model prediction are shown in Fig. 10. The value of β is determined to perform the best fit between the experimental and coupled analytical methods for each porous medium, that is, β = 18, β = 32, β = 19 for porosity of 0.95, 0.98, and 0.99, respectively. First, we defined the value of β for each porous media using the fitting of the interfacial velocity. This fitted value of β then has been used to perform a prediction of the remaining data set. As can be seen from the figure, the values of β are different depending on the porous material and the thickness of the permeable layer. Thus, one can define the velocity profile over and inside each porous medium using Eq. (4) when the porous medium property and the thickness of the layer are known.

Consequently, we calculated permeability K for each random porous medium from the fitted velocity profile and $K=hp2M\beta 2$ and then compared them with the experimentally measured values of the permeability of each porous medium using Darcy’s law. The details are reported in Table IV.

Porosity ε (%) . | h_{p} (mm)
. | K_fitted (mm^{2})
. | K_tested (mm^{2})
. |
---|---|---|---|

95 | 5.5 | 0.074 | 0.083 |

98 | 11 | 0.087 | 0.105 |

99 | 9 | 0.120 | 0.140 |

Porosity ε (%) . | h_{p} (mm)
. | K_fitted (mm^{2})
. | K_tested (mm^{2})
. |
---|---|---|---|

95 | 5.5 | 0.074 | 0.083 |

98 | 11 | 0.087 | 0.105 |

99 | 9 | 0.120 | 0.140 |

Figure 11 shows the ratio between the fitted and predicted permeability values for various porous media. The results show that the fitted and tested permeability are in good agreement. The discrepancy between the fitted and predicted values is expected due to deviations of the experiments from the model approximations and/or idealized flow conditions, such as flow steadiness, one-dimensionality, and smoothness of the porous surface. However, this approach shows its ability to directly link the soft porous media geometry to flow response using analytical predictions.

## IV. CONCLUSION

Coupled flows over and inside permeable media occur in a variety of natural phenomena. Apparently, knowledge of flow over soft porous media can play a significant role for the accurate prediction of flow rate and momentum transfer and the design of efficient engineering systems.^{10} In this study, we investigated the very first detailed analysis of pressure-driven flow over different random porous media. We found that the values of the slip velocity normalized by either the maximum flow velocity or by the shear rate at the interface and the screening distance K^{1/2} (i.e., u_{s}/u_{max} and $us/\gamma \u0307K$, respectively) depend on the porosity of the porous media. Moreover, it was observed that the depth of penetration inside the porous media δ is higher than the screening distance K^{1/2} (as has been defined in the Brinkman equation) and depends on both the porosity and the permeability of the porous media. We also compared the slip coefficient α as well as the experimentally measured slip velocity normalized by both shear rate ($\gamma \u0307$) and the screening distance ($K$) for our random porous media with the values reported in the previous literatures for the structured porous model.^{17,28,32} We found that the values of $us/\gamma \u0307K$ obtained in the present study are different for various porosities and nearly independent of the Reynolds number and it depends on both permeability and porosity of the porous layer.

Furthermore, we have examined the coupled flow model over and inside porous media using a coupling between the Navier-Stokes and Brinkman’s equations via our experimental study in this work, which, to our knowledge, has not been attempted in previous studies. We then characterized and compared the permeability of our random porous media, defined experimentally using Darcy’s law, with the analytical predictions. Comparison of the results showed good agreement between the experimental data and analytical predictions for permeability of random porous media. It is also expected that the model can be used to determine the velocity fields in the system as well as the shear stress, penetration length, and friction factor over soft porous media by noting the properties and geometrical features of porous material; this can lead to the design of efficient and advanced engineering systems.

There were several limitations to this study: (1) We assumed the porous-fluid interface was smooth, and we considered only the average of the uneven surfaces; (2) the experimental setup had a cross-sectional aspect ratio (2:1) where the channel considered to be two-dimensional in analytical purposes; (3) we neglected the impact of refractive index matching of the porous layer (the refractive index matching of the working fluid and the porous materials must be exactly the same in order to analyze the velocity through random media accurately). Our recent calculations and experimental measurements show that the refractive index of our porous medium with a 95% porosity at 5893 Å and 25 °C is 1.517 ± 0.005 in the longitudinal direction. This value could be different in the cross section of the porous layer. However, the refractive index matching of the aqueous solution of water and glycerin is 1.443 ± 0.002.^{44} This is the subject of our current investigation; (4) we obtained only two-dimensional whole-field velocity component measurements in two planes (where changes of velocity distributions were most dramatic), and considering three-dimensional velocity measurements might have yielded more precise results; and (5) because of the randomness in the porous media, the porosity and permeability of the media might have varied in different cross-section, a possibility that was neglected in our analysis. In our analytical analysis, potential limitations included: The model approximations and/or idealized flow conditions, such as flow steadiness, one-dimensionality, and smoothness of the porous surface.

Despite these limitations, this study provides the detailed analysis of flow over and at the interface of highly compressible and random soft porous media using PIV. Our work provides a step forward in understanding and modeling flows over and near the interface of the random soft porous media. The lubrication analysis due to the existence of random soft porous media has been discussed in our previous studies.^{36,46,47} It should be noted that although there is still controversy on the type of boundary conditions that are more proper to model Stokes-Darcy flows,^{48} the outcomes of this research could be used to shed new light on the characterization of dynamical behaviors controlled by porous media at the porous-fluid interfaces, including slip phenomena controlled by porous surfaces [e.g., Slippery Liquid-Infused Porous (SLIP) surfaces], and drag reducing surfaces.^{10}

This work also yields critical insights into understanding the flow inside the systems of different geometries when they are covered with a porous medium with specific permeability, porosity, and porous layer thickness. Our findings also help environmental researchers understand laminar flow transport over submerged aquatic vegetation.

Future studies shall focus on developing mathematical models and experimentally validating them for laminar flows of complex fluids over soft porous media and the related applications.^{49–51} In addition, the development of experimentally validated analytical models of more realistic porous media configurations—for example, highly flexible and heterogeneous porous layers by coupling the bending of the soft porous layers with flow-field dynamics—is needed. We expect that the model will reveal the relation between the physical properties of the permeable layer and its topology with regards to the fluid passing over it and can be used in designing advanced engineering systems.

## ACKNOWLEDGMENTS

This research was supported partially by the Army Research Office (ARO) under Award No. W911NF-17-1-0406 and partially by the National Science Foundation (NSF)–CBET Fluid Dynamics Program under Award No. 1706766.