Experiments in Fluids

, Volume 53, Issue 1, pp 51–76

Quantifying the dynamics of flow within a permeable bed using time-resolved endoscopic particle imaging velocimetry (EPIV)

  • G. Blois
  • G. H. Sambrook Smith
  • J. L. Best
  • R. J. Hardy
  • J. R. Lead
Research article

DOI: 10.1007/s00348-011-1198-8

Cite this article as:
Blois, G., Sambrook Smith, G.H., Best, J.L. et al. Exp Fluids (2012) 53: 51. doi:10.1007/s00348-011-1198-8

Abstract

This paper presents results of an experimental study investigating the mean and temporal evolution of flow within the pore space of a packed bed overlain by a free-surface flow. Data were collected by an endoscopic PIV (EPIV) technique. EPIV allows the instantaneous velocity field within the pore space to be quantified at a high spatio-temporal resolution, thus permitting investigation of the structure of turbulent subsurface flow produced by a high Reynolds number freestream flow (Res in the range 9.8 × 103–9.7 × 104). Evolution of coherent flow structures within the pore space is shown to be driven by jet flow, with the interaction of this jet with the pore flow generating distinct coherent flow structures. The effects of freestream water depth, Reynolds and Froude numbers are investigated.

1 Introduction

Permeability is encountered in a large number of small- and large-scale physical systems, in natural (e.g. blood vessels, non-cohesive sediment deposits, forests) and engineering environments (e.g. heat exchangers, biological, chemical and nuclear reactors). It has been recognised that turbulence not only may occur in such systems, but more importantly may control flow mixing and thus govern the kinetics of the physical processes occurring within porous media (Nield and Bejan 2006). The presence of turbulence within porous media has been postulated since the early experimental work of Forchheimer (1901) who highlighted that, under certain flow conditions, the relationship between pressure drop and flow rate across a porous medium deviates from Darcy’s law. Although this could be easily interpreted as the onset of turbulence, the definition of the transition in flow regime in porous media remains controversial. Hlushkou and Tallarek (2006) pointed out that ‘the only experimental evidence of the presence of turbulence in a three-dimensional porous media is that of microscopic turbulence’ (where ‘microscopic’ refers to the scale of the individual pore space) and emphasised that, strictly speaking, non-Darcian macro-scale phenomena, such as given by the pressure drop law, cannot identify the transition from one regime to another. In fact, it was not until the visualisation study of Dybbs and Edwards (1984) that the first evidence for eddy formation within the voids of porous media was provided.

Understanding and predicting macro-scale processes (i.e. at the scale of the entire porous domain) requires robust theoretical as well as numerical models. At the same time, validation of such models requires knowledge of fluid dynamics at the micro-scale (i.e. pore space). However, while non-Darcian macro-scale phenomena have been largely investigated (see review in Nield and Bejan 2006), micro-scale turbulent phenomena still remain largely unquantified due to the technical difficulties associated with conducting direct measurements in such inaccessible environments. This lack of experimental data is especially true for ‘hybrid systems’, defined as fluid-saturated porous media that are bounded by a freestream flow (for instance, non-cohesive sediment beds overlain by water flows, or vegetation immersed within atmospheric boundary layers). In such systems, turbulence may penetrate from the freestream domain and extend into a significant portion of the porous domain before undergoing a transition to a laminar regime flow state (Goharzadeh et al. 2005).

Alluvial river beds represent one of the most important and least explored cases of such hybrid systems. Rivers are typically characterised by high Reynolds number flows, where turbulence penetrating across the free-flow-bed interface may become established within the pore spaces located at the top of the transitional layer. Pore space turbulence controls the dynamics of important physical phenomena, such as dissolved oxygen exchange, nutrients and contaminant transport, as well as playing a role in the mechanisms governing bed mobilization, which are still poorly understood. Rivers are extremely complex systems, and laboratory experiments conducted on simplified models are often the only way to validate theoretical and numerical subsurface flow models. A prerequisite to ensure that such physical models are representative of reality is the ability to establish high Reynolds number open-channel flows. However, none of the currently-available techniques meet the significant challenges required to fully characterise pore space turbulence produced within permeable beds overlain by a highly turbulent free-surface flow. This is especially true for models of coarse-grained rivers where, due to the high bed permeability and high pore flow velocity, the formation of complex coherent flow structures can be expected. The main motivation of the present research is thus to develop an experimental apparatus that allows high-resolution optical measurements (i.e. using Particle Imaging Velocimetry) to be conducted within the pore spaces of an idealised open-channel model, which is characterised by high Reynolds numbers and a high permeability, in order to overcome the technical and practical limitation of current techniques.

The absence, and/or the extremely difficulty, of physical and optical access are the main factors that make pore space measurements particularly challenging. Techniques that are minimally invasive, such as hot-wire anemometry (e.g. Mickley et al. 1965), micro-electrodes (e.g. Seguin et al. 1998) and ultrasonic Doppler velocity profiling (Pokrajac et al. 2007) have provided at-a-point data within opaque solid matrices. Optical access is the prerequisite for techniques such as LDV and PIV to be applied, and hence these techniques have been generally restricted to cases where the geometry of an opaque solid matrix is relatively simple, such as arrays of circular cylinders (e.g. Masuoka et al. 2002; White and Nepf 2007) or beds of uniform spheres packed in a regular arrangement (Pokrajac and Manes 2009), thus providing the optical access needed for illumination/imaging from the side.

Refractive index matching (RIM) techniques represent one of the most established tools to provide optical access to the interstices of more complex porous media. In the RIM approach, the solid matrix is manufactured in a transparent material and immersed in an appropriate fluid that, nominally, has an identical index of refraction (RI). However, maintaining a perfect matching is extremely challenging and thus the accuracy of optical measurement in RIM is unavoidably affected by the geometrical complexity of the solid phase. Specifically, the experimental error due to refraction of light crossing a fluid-solid interface is greater for more highly curved solid surfaces; additionally, the deviation of light occurs every time the light crosses the interface between the fluid and solid phases. Therefore, obtaining highly accurate optical measurements in complex geometries such as porous media is inherently challenging. Since the pioneering work of Jolls and Hanratty (1966), who provided qualitative visualisations of pore space flow using RIM, quantitative optical techniques such as LDV (e.g. Yevseyev et al. 1991) and PIV (e.g. Zachos et al. 1996; Khalili et al. 1999; Moroni and Cushman 2001; Goharzadeh et al. 2005; Hassan and Dominguez-Ontiveros 2008; Huang et al. 2008; Morad and Khalili 2009; Arthur et al. 2009) have also been successfully applied in the context of RIM porous media. However, additional technical limitations concerning use of RIM in porous media, such as limited flow seeding density and limited duration of the experiments, still remain unresolved (see discussion in Huang et al. 2008). Great progress in applying new technologies that need no physical or optical access to quantify flows in porous media has been made. For instance, magnetic resonance velocimetry (MRV) techniques have been used to resolve flow through opaque solid matrices (Elkins and Alley 2007). MRV techniques are currently under development, but a major limitation here is related to the strength and distortion of the signal, as well as the limited spatial and/or temporal resolution of this technique that restricts its use to relatively low Reynolds numbers (see discussion in Elkins and Alley 2007).

In the present paper, an alternative solution for providing optical access to the pore spaces of a submerged bed is outlined. We detail a high-resolution PIV technique based upon endoscopy. Endoscopic PIV (EPIV), where a camera endoscope (Tani et al. 2002; Heenan et al. 2003) and/or an illumination endoscope (Wernet 2000; Tani et al. 2002) are utilised, has not been widely used because of the practical difficulties in obtaining high-quality particle images. This is especially true for a full EPIV technique, for which both imaging and illumination are obtained by endoscopy (e.g. Dierksheide et al. 2002). Applications of EPIV within a porous media are particularly challenging due to the complexity of the solid matrix geometry and the constraint of the pore space. For example, Klar et al. (2004) used fiberscopes in order to track the 3D motion of tracer particles within the pores of a gravel bed and successfully obtained time series of the interstitial velocity components by spatially-averaging the instantaneous velocity data taken within the pore volume. However, fiberscopes, as compared with borescopes, have a low-light transmittance and hence poor image resolution and contrast. Klar et al. (2004) thus concluded that lack of illumination within the pore space was the greatest experimental limitation and resulted in low data quality, thus rendering resolution of the instantaneous flow structure particularly challenging.

The use of endoscopes to gain optical access has enormous practical advantages in open-channel flow experiments when compared to other approaches, principally in the low complexity of the experimental set-up, which in turn translates into a reduction in costs. For instance, RIM and MRV techniques require complex and expensive facilities and often demand use of chemicals that may be dangerous to handle. Additionally, most of the suitable RIM working fluids have a high viscosity, which renders achieving high Reynolds number flows very challenging. Lower viscosity aqueous solutions (e.g. Sodium Iodide) are chemically unsteady in the presence of oxygen, thus introducing additional challenges when operating with free-surface flows.

The principal aim of the present paper is to address the technical limitations of previous EPIV techniques in order to provide high-resolution measurements of flow within the pore space of a porous medium immersed in an open channel. Specifically, we present details of a fully endoscopic PIV (EPIV) technique that is readily applicable in the challenging conditions of a packed bed underlying a free-surface flow at high Reynolds numbers. The specific objectives of the paper are to:
  1. 1.

    Provide the first quantitative characterisation of the mean and instantaneous flow within the pore space of a submerged permeable bed at a high spatio-temporal resolution.

     
  2. 2.

    Identify how the conditions of the freestream flow influence the mean pore flow.

     
  3. 3.

    Identify the coherent flow structures occurring within the pore space and elucidate their modes of inception and evolution.

     

2 Experimental set-up

2.1 Physical model

Experiments were conducted in a purpose-built recirculating water flume that housed a fixed packed bed and was instrumented with an EPIV system (Fig. 1a, b). The flume test section was 4.8 m long, had a zero slope and rectangular cross-section (width W = 0.35 m and height H = 0.60 m; Figs. 1, 2a); the cross-section was vertically elongated (aspect ratio H/W = 1.71) in order to maximise the thickness (hbed) of the permeable domain and the range of water depths (hw) that could be investigated. For this study, hbed was kept constant (hbed = 0.24 m) while hw was varied over the range hw/hbed = 0.25–1.
Fig. 1

Experimental set-up: a photograph showing the test section that contains the packed bed and is instrumented with EPIV. The camera endoscope (CE) is connected to the 4 Mpixel image-intensified camera (IIC) and the laser endoscope (LE) is connected to the articulated arm (AA). In the top left of the picture, the camera used to measure the free flow is also shown. b Schematic diagram of the longitudinal view (XY plane) of the flume test section (length L = 2.80 m, height H = 0.60 m, slope α = 0). Flow is from left to right; Qstream and Qbed refer to the mean flow discharge above and through the bed, respectively (U0 is the mean freestream flow velocity)

Fig. 2

Endoscope configurations: a transverse section (width = 0.36 m) showing the set-up of the endoscopes in the YZ plane: the camera endoscope is inserted horizontally from the wall, while the laser endoscope is inserted vertically through the floor; b details of the customised laser endoscope LE; photographs of the adjustable endoscope ports, CE c and LE d, respectively

The model built for these experiments is an idealised representation of a gravel-bed river characterised by high permeability. For the experiments reported herein, we used a simplified geometry comprising six layers of spheres (D = 0.04 m diameter) rigidly fixed in a cubic arrangement. In order to construct a solid matrix that could handle large flow dynamic loads, the spheres were connected through 0.6-m-long stainless steel threaded rods (2.8 mm diameter). The motivation for using a regular packing configuration in the present experiments is that the internal morphology of the porous bed is completely known. Consequently, this geometry can be readily replicated in numerical codes, thus allowing use of this experimental data set for validation of numerical schemes developed for porous media applications.

To obtain a high-quality uniform freestream flow, the fluid was conveyed from the inlet tank to the channel through a contraction section (the radius of curvature was 0.4 m for the lateral contractions and 1.2 for the ramp) that prevented any flow detachment at the inlet section. A honeycomb grid (0.05 m thick and 0.008 m mesh size) was inserted at the inlet section in order to destroy any large-scale turbulence produced in the inlet tank and straighten the flow. The permeable bed covered the entire width of the flume test section with a uniform thickness, but the ends of the bed were shaped with a gradually changing thickness to avoid sharp geometrical discontinuities (see Fig. 1b).

2.1.1 Boundary conditions

The total flow discharge entering the test section, Qt, splits into two parts (see Fig. 1b): Qstream that moves above the permeable bed and Qbed that represents the discharge globally flowing through the permeable bed. The Qbed/Qstream ratio is not known apriori as it is a function of the flow depth—bed thickness ratio hw/hbed, of Qtot and the characteristics of the bed such as the packing and permeability. Qbed was indirectly estimated by measuring Qt and Qstream and assuming that Qbed = QtQstream. Specifically, the discharge, Qt, was measured by an electromagnetic flow meter (Ultra Mag, Model MX06-W100) in the return discharge pipe. Qstream was obtained by measuring the velocity of the freestream flow along multiple vertical profiles and integrating the streamwise velocity components to compute the mean velocity U0. These profiles were measured using ultrasonic Doppler velocity profilers (UDVP; Best et al. 2001) in the arrangement shown in Fig. 2a. Specifically, the data were taken using four UDVP probes, equally spaced (0.10 m) in the spanwise direction (see Fig. 2a) in order to account for the lateral variability of flow. UDVP emits a straight beam and receives the Doppler signal (produced by dispersed particles) at different distances from the probe. This signal is then converted to obtain measurements of velocity. Specifically, the UDVP provides the velocity component directed along the UDVP beam at different locations along the beam. In order to use the UDVP for velocity profiling (i.e. obtain u(y)), the probes had to be directed at an angle with respect to the bulk flow and the raw data decomposed to extract the streamwise component (ux). The UDVP probes were mounted 0.15 m upstream of the EPIV pore space measurement volume and oriented at an angle, α = 30°, from the vertical (Fig. 1b). The data were then projected onto the vertical, making the assumption that the flow is self-similar within the 0.3 m of the beam range. In order to maximise the signal, 11 μm hollow glass spheres were added to the flow. For each experimental condition, UDVP data were collected at 5 Hz for a total sampling time of 120 s. The data were then filtered and averaged to obtain the four, time-averaged profiles, and these were then averaged to compute the mean velocity U0.

Qstream was calculated as Qstream = U0·hw·W.·U0 and was used to compute the freestream flow Reynolds number Res = U0·hw (where ν is the kinematic viscosity = 1.004 × 10−6 m2 s−1) and Froude number Frs = U0·(g·hw)−0.5 (where g is the acceleration due to gravity = 9.81 m s−2). UDVP can yield relatively accurate velocity measurements and is very convenient for rapid quantification of this type of flow boundary condition as it allows flow measurement at multiple locations simultaneously. However, due to the conical shape of the beam and the orientation of the probes, UDVP data may be biased near the freestream-bed interface. In order to validate the use of UDVP, we performed PIV measurements of the freestream flow along the channel centerline using a standard PIV system. The flow was illuminated in the XY plane (see Figs. 1, 2a) by a 50 mJ Nd:YAG laser (Litron Lasers) and was imaged by a 4 Mp camera (Redlake MotionPro Y5). The light was provided from the top through the water surface (see Fig. 1). At low discharges (Qt < 20 L s−1), the water surface is relatively flat and the laser light can be introduced from the top without the water surface affecting the quality of light. For higher flow discharges, a clear acrylic plate (0.10 m wide and 0.30 m long), rigidly positioned at the water surface, allowed a steady measurement light sheet to be used with low disturbance to the flow. Seeding particles, as well as image interrogation and validation schemes, were the same as those used for the EPIV and details can be found in the next section. PIV data showed that the free flow is uniform, thus confirming use of the UDVP approach (single profile). Vertical profiles were obtained by double-averaging the PIV data (in time and along X direction). A comparison between UDVP and PIV data (Fig. 3) shows a good agreement and confirms the UDVP could accurately measure the mean flow boundary condition.
Fig. 3

Evaluation of the boundary conditions. a Example of freestream flow fields measured using standard PIV; b Example of comparison between vertical profiles of the streamwise component of velocity obtained by PIV and UDVP (Qtot = 10 L s−1, hw = 0.18 m)

2.2 EPIV system

The 2D EPIV system developed for the present experiments consists of a digital video camera, a pulsed laser source and two borescopes (rigid endoscope) that provided optical access to the pore space (Fig. 1). The laser endoscope (LE) directed laser light into the pore, allowing an efficient illumination of the pore space measurement plane; the camera endoscope (CE) conveyed the light scattered by the seeding particles from the pore to the camera.

The CE was connected to a CMOS camera (2,352 × 1,728 pixels) with maximum frame rate of 250 fps. Due to the small optic aperture of the endoscopes, the amount of light transmitted by the CE is much lower compared to standard lenses (Dierksheide et al. 2002). To maximise the light energy transmitted to the camera sensor, an image-intensified camera (IIC, Redlake MotionPro X5) was used. In addition, the LE was connected to a double-pulsed laser (New Wave Nd:YAG, 532 nm) with output energy up to 120 mJ per pulse and repetition rate up to 15 Hz. The coupling of these two devices allowed us to solve issues encountered by previous studies related to an insufficient light signal (see discussion in Klar et al. 2004).

2.2.1 Camera endoscope

While fiberscopes used in previous similar studies (Klar et al. 2004) offer a higher flexibility in terms of probe positioning, their resolution is limited to the number of fibres in the bundle (~tens of thousands), which is typically orders of magnitude smaller than modern digital sensors (several millions). Due to this technical issue, image interrogation procedures for fiberscopic EPIV are usually restricted to particle tracking velocimetry with streak searching (PTV-SS, see Tani et al. 2002), thus limiting the seed particle concentration and consequently the ability to resolve complex instantaneous flow structures. The borescopes chosen for the present study offered a high light transmission quality and resolution, thus allowing application of conventional cross-correlation PIV. The CE used herein was 300 mm long with an 8 mm external diameter shaft borescope equipped with an imaging lens with variable focus adjustment. The choice of the CE diameter was a compromise between the requirement for a small diameter, in order to minimise flow perturbations in the pore space, but a large enough optic aperture to minimise loss of light. A rod lens (6 mm diameter) system was used in order to maximise light transmission, thus generating enhanced image resolution and brightness. The CE had a 0° view direction and a 67° angle of view and was connected to the camera through a 50 mm focal length Nikon lens to provide appropriate magnification of the pore space.

2.2.2 Laser endoscope

A 0° view angle customised laser endoscope (LE) was used to convey the laser beam into the pore space and generate the measurement plane (Fig. 2b). The LE consisted of a 270 mm long probe (12 mm external diameter) that terminated at an 8-mm diameter cylindrical lens (Fig. 2b). The LE produced a light sheet that was 0.2–0.5 mm thick with a lateral divergence angle of ~26° (Fig. 2b). The cylindrical lens was sealed within a waterproof casing, and the LE probe was connected to an optical chamber that housed a spherical lens with a fixed focal length of 350 mm (Fig. 2b). The axial position of the lens within the mounting chamber was adjustable to allow collimation of the beam focus in the center of the pore space. The optical connection between the laser source and the LE was provided by an articulated optical arm (AA) that facilitated accurate alignment of the laser into the LE (Figs. 1, 2).

2.2.3 Endoscope configuration

The pore flow measurements were obtained in a specific section (X = 0) located 1.9 m downstream of the channel inlet and 0.9 m upstream of the channel outlet. This location was chosen to minimise any inlet and outlet boundary effects. The endoscopes were introduced into the flume through customised endoscopic ports that provided appropriate protection for the endoscopes, minimised the influence of wall vibrations and provided precise adjustment during endoscope positioning. The CE was mounted perpendicular to the wall of the flume (Fig. 2a), thus imaging the pore space in the streamwise vertical (XY) plane (Fig. 4a). Six endoscopic ports were located in the flume sidewall in the vertical (Y) and were equally spaced (dY = 0.04 m) in order to allow quantification of flow at different depths within the bed (Fig. 2a, c). The pore specifically investigated herein is the one accessed by port B, which is the second pore space beneath the bed surface (Y = −0.08 m). The location of the endoscope relative to the pore is a crucial parameter, and its choice was dictated by the aim of maximising the spatial resolution and field of view. The downside of this criterion is the increase in image distortion and thus worsening of problems associated with endoscope imaging, as discussed below. The minimum distance between the end of the CE and the measurement plane was 0.04 m (Fig. 2a), thereby resulting in negligible flow perturbation, maximum image resolution and a more complete field of view in comparison with previous external PIV applications in packed beds of spheres. A mean magnification of 18 μm pixel−1 was found to be an acceptable compromise between resolution and accuracy.
Fig. 4

a Details of the pore space measurement volume (see also Fig. 1b) in the XY plane, showing the portion of the pore space illuminated by the laser sheet (divergence angle β = 26°) and the field of view imaged by the camera endoscope. An example of a raw EPIV image is also shown; b instantaneous flow field within the pore space at location B (see Fig. 2b); boundary conditions are hw = 0.24 m; Qt = 35 L s−1; the color-map refers to the swirling strength (i.e. imaginary portion of the complex eigenvalue of the local velocity tensor, see Adrian et al. 2000) expressed in s−1 and the streamlines are also illustrated (high resolution version of this figure is given as a supplementary material)

In order to avoid any disturbance to the freestream flow above the sediment bed, the laser endoscope (LE) shaft was inserted vertically through the floor of the flume via a customised port (Fig. 2b). The LE port allowed adjustable positioning of the LE in the vertical (Y) direction and in the horizontal plane (XZ), therefore permitting precise alignment of the light sheet in the pore space centerline. The end of the LE was positioned 0.05 m from the center of the pore space (Fig. 2a). This configuration thus provided the desired aim of negligible local flow perturbation and complete illumination of the whole pore space.

3 EPIV measurements and analysis

A single-exposure, double-frame, PIV approach was adopted for image acquisition. Figure 4a shows an example of EPIV image taken within the pore space and displays the typical circular field of view (FOV) obtained using endoscopes. Figure 4 shows the size of the image obtained through this configuration (FOV, about 25 mm diameter) relative to the size of the whole pore space. An example of instantaneous flow field obtained by applying the methodology described below is shown in Fig. 4b. Importantly, it can be seen (Fig. 4) that significant areas (those hidden by the spheres) are imaged using EPIV that would not be visualised by external PIV techniques. The downside of this FOV is that it included part of the background of solid spheres, thus introducing problems of background noise due to reflections. This problem was exacerbated by two factors: deposition of seed particles and deposition of air bubbles on the surface of the spheres, although both of these were mechanically removed by accessing the pore before each experiment (see next section). Reflection was particularly severe in conditions where the rate of bubble formation was higher than the duration of the experiments and in which bubbles reformed during image acquisition. High rates of bubble formation typically occurred when the water temperature was much lower than the ambient air temperature due to the release of dissolved gas from water. Warming the water before data collection was thus sufficient to resolve this problem.

3.1 Seeding particles

Neutrally buoyant, spherical, hollow glass particles (density, ρp = 1,050 kg m−3), with a nominal mean diameter of 11 μm were used to seed the flow. A uniform distribution of tracer particles was readily obtained due to the high degree of fluid turbulence both over and within the bed.

It is worth noting that use of a cross-correlation approach requires high seed particle concentrations in order to maximise the statistical significance of the correlation. However, due to the low velocity and stresses occurring in porous media, seed particles are easily deposited on the solid matrix surface, potentially causing high levels of reflection. This problem can be readily addressed in a cubically-packed bed by reaching into the pore space with a brush and mechanically removing the particles before starting image acquisition. Pore spaces in more dense packing arrangements are less accessible and would require disassembly of the bed, which is particularly challenging for random packing. The endoscopic approach based upon borescopes allowed us to optimise the seed particle concentration and maximise performance of the cross-correlation processing algorithms described below.

3.2 Image acquisition

A series of preliminary PIV measurements were performed over a wide range of acquisition parameters, such as the time interval between laser pulses (Δt), acquisition frequency and sample size, in order to determine the optimum values. Due to the occurrence of energetic episodes of pore flow that produced high in-plane accelerations (velocities higher than 5 standard deviations) over the image capture period (33.5 s), Δt was found to be a crucial parameter in order to optimise the accuracy of the measurements and capture the full velocity dynamic range. The 3-dimensionality of the flow caused areas of particle mismatch due to occasional spanwise accelerations that introduced significant out-of-plane motion. Preliminary data taken at a low sampling frequency (1 Hz) were used to determine the longest Δt for which the number of spurious vectors was a minimum. The values of Δt used resulted in a wide range of particle displacements (2–18), 80% being in the range of 5–15 pixels. Over the whole experimental campaign, Δt was varied over an order of magnitude (400–18,000 μs).

In order to maximise the ability to track the evolution of the flow, the image acquisition rate was kept constant at 15 Hz, which was found sufficient to capture the evolution of large vortical structures for a reasonable number of sequential images, especially at low flow discharges. The downside of this approach was that it required longer samples to obtain the convergence of flow statistics. Image sequences consisting of a minimum of 1,500 image pairs were collected and subsequently processed to evaluate data convergence. Each final data set consisted of a minimum of 1,500 up to 2,500 image pairs, and these sample lengths were found satisfactory for low- and high-flow discharges, respectively.

3.3 Endoscope imaging

The main drawback of using EPIV is related to the high level of distortion induced by endoscopes. As discussed by Soloff et al. (1997), every time the image recording plane (camera sensor) and the measurement plane (laser sheet) are not parallel, the relationship between homologue points in the two spaces will not be linear. This means that the image will be distorted and, as a result, the magnification factor αm will not be uniform over the image (i.e. distortion). Additionally, spherical lenses project a spherical image onto a planar surface (sensor) or a planar surface onto a spherical surface. In either case, the two surfaces cannot be parallel and thus, as explained above, αm will be variable. The radius of such a surface coincides with the focal length of the lens multiplied by the magnification. This means that, for a fixed magnification, the larger the angle of view, the higher the distortion. For this reason, image distortion is greater for images acquired with endoscopes due to the short focal length of the endoscope entrance optic that is typically a few millimetres. The large angle-of-view of endoscopes introduces a distortion that is known as a fish-eye view and consists of the projection of a low radius spherical surface onto a plane. Figure 5a shows the effect of such distortion on a uniform Cartesian grid of white circles on a black background placed in the object plane. The high image distortion has three main consequences that are relevant for EPIV: defocusing, non-uniform magnification and out-of-plane motion.
Fig. 5

a Calibration image; b location of the centroids of the dots in the image space (co-ordinates are in pixels); cred dots represent the location of the centroid in the fluid space; black crosses represent the location of the dots obtained applying the polynomial function to the measured dots (coordinates are in [mm]; the point (0,0) refers to the center of the pore space); d magnification factor for the u component; e magnification factor for the v component

3.3.1 Defocusing

Since the image plane is projected onto a spherical surface, it is not possible to obtain uniformly-focused images over the whole FOV (Dierksheide et al. 2002). Out-of-focus particles will spread the scattered light over a higher area of the sensor and thus will be recorded with a lower signal intensity and contrast. As long as the signal is of sufficient intensity, this does not represent a limit for PIV, and PIV data can be recorded even though the accuracy may be affected due to a lower signal-to-noise ratio. In the present experiments, we sought to maximise the focus towards the borders of the FOV where the light scattered by the particles is weaker. This subsequently allowed us to improve the light signal at the sides and top of the image, although this partially sacrificed the bottom of the image where occasionally oversaturated particles occurred due to a higher light intensity nearer the LE.

3.3.2 Image calibration

As mentioned above, image distortion introduces a non-linear relationship between the location of a particle in the image plane (sensor) and in the fluid space (laser plane). This produces two problems for PIV: (1) a regular interrogation grid in the fluid space will correspond to a warped grid in the image space, and vice versa, and (2) the conversion of each in-plane particle displacement from the image space to the fluid space will depend on the location of the displacement through the local magnification. If not corrected, the distortion introduces systematic errors into both the location and magnitude of the velocity vectors. As discussed by Soloff et al. (1997), assuming that the out-of-plane motion does not contaminate the in-plane velocity component, the distortion can be corrected by experimentally determining a mapping function F (x, y, z0) such that given a point (X, Y) in the image space, its homologue (x, y) in the fluid space can be determined. Such a function allowed us to express the displacement in the image space (ΔX, ΔY) as a function of the displacements in the fluid space (Δx, Δy). A third-order polynomial function was used to map the effect of the spherical distortion.
$$ F\left( {x, \, y, \, z_{0} } \right) = a_{0} + a_{1} x + a_{2} y + a_{3} x^{2} + a_{4} y^{2} + a_{5} xy + a_{6} x^{3} + a_{7} y^{3} + a_{8} x^{2} y + a_{9} xy^{2} $$
(1)

The nine coefficients (a0a9) of the mapping function were determined by measuring, from a calibration image, the coordinate (X, Y) of a number of points for which their real-world coordinates (x, y) were known; for each homologue point, the Fi(X, Y) and Fi(x, y) were calculated from Eq. 1 and finally a least-squares method was used to define the F(X, Y) that best approximated the analytic function F(x, y). The set of coefficients that minimised the mean square error between F(X, Y) and F(x, y) was used.

The following procedure was employed:
  1. 1.

    A calibration target was placed within the pore space prior to experiments taking place and imaged through the endoscope placed in its final configuration. The target consisted of a 30 × 30 mm plate (5 mm thick) with a precise grid of circular dots (200 μm diameter) spaced 1 mm apart. The target was mounted on a specially-designed frame (35 mm thick) that allowed precise positioning of the target face in the center of the light sheet.

     
  2. 2.

    Illumination of the target was provided by a continuous white light source located above the bed and directed towards the pore space. The light intensity and angle were adjusted in order to maximise the illumination uniformity. Figure 5a shows an example of a calibration image taken when positioning the endoscope in port B.

     
  3. 3.

    Image processing based upon previous image normalisation and subsequent binarisation was applied to the calibration image. Automatic blob analysis algorithms were applied to determine the centroid locations of the dots in the image space (X, Y). The result of the centroid identifications is shown in Fig. 5b. This information was used to generate an approximate mapping function of the image distortion.

     
  4. 4.

    Figure 5c shows the application of the inverse mapping function to transfer the location of the dots from the image space onto the fluid space. The accuracy of the method depends on accuracy in the determination of the centroids of the dots and on the degree of the polynomial applied. The results showed that a third-order polynomial described the distortion with a satisfactory level of accuracy.

     

As discussed by Soloff et al. (1997), the mapping function could be used to dewarp the raw images or alternatively to dewarp the vector fields. The latter method was used herein to process the EPIV data as detailed below.

3.3.3 Out-of-plane motion

The interrogation windows in a distorted image lie on a surface in the fluid space that is not parallel to the light sheet. In the case of endoscope imaging, this surface is spherical and the angle between the light sheet and the surface varies with location. If the light sheet had a nominally zero thickness, images would only be recorded along the intersection between the light sheet and the image surface. In reality, the light sheet defines a volume of measurement. This means that if the camera axis is not perpendicular to the light sheet, out-of-plane motions are recorded. Consequently, the particle displacements recorded by the camera are the projection of the 3D motion within the light volume onto the image surface, and therefore inherently contain the w component that introduces an error. Depending on the location and the direction of the out-of-plane motion, this error can result in under- or over-estimation of the in-plane component and is proportional to both the angle between the camera axis and the laser sheet and the sheet thickness. In the case of a short focal lens system, this error is zero or negligible at the center of the image and increases radially away from it.

As discussed by Soloff et al. (1997), the calibration method illustrated above corrects only the in-plane components of velocity and is strictly valid only when the w component is zero or perspective angle is zero. Perspective errors due to the out-of-plane component are not negligible in the case examined here, as the ratio between the size of the FOV and the object distance is high. In the case of 2D EPIV, these errors were discussed by Reeves and Lawson (2004). Due to the lack of a second IIC camera, the EPIV system used here was necessarily restricted to a single camera, and thus the out-of plane component of motion could not be quantified. However, in order to minimise this error, the thickness of the laser sheet was reduced to just 0.2–0.5 mm and the vector reconstruction was limited to a smaller area of the FOV.

3.4 Image interrogation

As mentioned above, the images were not dewarped, i.e., image interrogation was performed on the original images, thus optimising the computation time while maintaining a high level of accuracy. An interrogation grid of equally-spaced points was defined in the sensor array within the region imaged by the camera endoscope. This entailed masking a circular region of approximately 25 mm diameter and applying a cross-correlation function to each image pair. A fast Fourier transform (FFT) method was applied in locating cross-correlation peaks (Raffel et al. 2007). Sub-pixel accuracy was obtained by a standard three-point Gaussian fitting technique.

A multi-pass procedure based upon secondary peak mapping (Hart 2000) coupled with an adaptive window displacement was used in order to minimise the interrogation window. This procedure allowed us to use the low noise vector maps gained on a 64 × 64 pixel grid to construct the final interrogation on a 32 × 32 pixel grid (we call this grid Xi). Such a grid corresponds to an average window size in the range of 0.5–1 mm (depending on local magnification). This allowed us to detect vortical structures with dimensions of 1.5–2 mm in diameter, which equates to eddies smaller than 1/20th of the void size.

After each intermediate iteration, statistical validation criteria based upon the velocity range recorded from preliminary experiments and local validation algorithms based on neighbourhood interrogation methods (Westerweel 1994) were applied in order to detect potentially erroneous data. Application of these techniques resulted in 1–5% of the instantaneous vectors being deemed as erroneous, with these being mainly caused by a localised lack of seeding particles or low illumination. Missing data in the instantaneous flow fields were partially reconstructed using second-order correlation techniques (Hart 2000). The data-interpolation algorithms used were recursive methods based upon data validated with the highest confidence as input and finally applying 2D interpolation functions.

3.5 Flow field correction

The procedure used to reconstruct the flow field in the fluid space used the information gained from the mapping function X = F (x, y) and from the image interrogation (Xi, Yi, ui, vi). The first step was to define a regular grid of points in the fluid space (xg, yg), To this end, (xg, yg) was defined as a regular orthogonal grid of points spaced 0.2 mm apart. The corresponding grid in the image space (Xg, Yg) was then determined by applying the mapping function. We defined a 0.4-mm ring whose external border coincided with the border of the FOV and filtered out the grid points within it. A cubic interpolation scheme was used to interpolate the vectors (ui, vi) previously determined on (Xi, Yi) onto (Xg, Yg). The final step required correction of the magnitude of the vectors. This was simply determined by calculating the gradient of F (which corresponds to the magnification map) and multiplying the velocity components by the corresponding matrix (Soloff et al. 1997). In Fig. 5c and d, the magnification factor for the u and v components, respectively, is reported. As shown, the magnification factor, αm, is non-uniform over the image but is typically at a maximum in the image center (~70 pixels mm−1) and a minimum on the borders (~58 pixels mm−1).

Since this procedure involves a vector interpolation, a smoothing effect of the instantaneous flow field is unavoidable. Additional data smoothing was undertaken for representation of the instantaneous flow field, but only fully validated vectors were used for calculation of the mean flow fields.

An example of an instantaneous flow field obtained by applying the methodology described above is shown in Fig. 4b. The data were collected keeping the flow depth (hw = 0.24 m) and flow rate (Qt = 35 L s−1) constant and under a turbulent pore flow regime (Rep = 960). The image demonstrates the capability of the technique in capturing small-scale eddies (order of 1.5–2 mm), thus providing a useful tool to investigate relatively high Reynolds number flows.

4 Results

All experiments were conducted under steady flow boundary conditions. Data were collected at four different water depths hw (0.06, 0.12, 0.18 and 0.24 m), thus yielding a particle diameter: flow depth ratio, D/hw, in the range 0.17–0.67 and a bed thickness: flow depth ratio, hb/hw in the range 1.0–4.0. Flow discharge, Qt, was varied from 5 to 40 L s−1, yielding Res numbers in the range 1 × 104–1 × 105 and Frs numbers in the range 0.04–0.90. A summary of the experimental conditions examined in this paper are reported in Table 1.
Table 1

Summary of the experimental flow boundary conditions

Qt (L s−1)

hw (m)

U0 (m s−1)

Qstream (L s−1)

Qbed (L s−1)

Qb/Qs

Qb/Qt

Res

Frs

5

0.06

0.16

3.4

1.6

0.46

0.31

9.8E+03

0.21

10

0.06

0.30

6.2

3.8

0.61

0.38

1.8E+04

0.39

15

0.06

0.43

9.0

6.0

0.67

0.40

2.6E+04

0.56

20

0.06

0.56

11.8

8.2

0.70

0.41

3.4E+04

0.73

25

0.06

0.69

14.5

10.5

0.72

0.42

4.2E+04

0.90

5

0.12

0.10

4.3

0.7

0.16

0.14

1.2E+04

0.09

10

0.12

0.18

7.6

2.4

0.31

0.24

2.2E+04

0.17

15

0.12

0.26

11.0

4.0

0.37

0.27

3.1E+04

0.24

20

0.12

0.34

14.3

5.7

0.40

0.28

4.1E+04

0.31

25

0.12

0.42

17.7

7.3

0.42

0.29

5.0E+04

0.39

30

0.12

0.50

21.0

9.0

0.43

0.30

6.0E+04

0.46

35

0.12

0.58

24.3

10.7

0.44

0.30

7.0E+04

0.53

40

0.12

0.66

27.7

12.3

0.45

0.31

7.9E+04

0.61

5

0.18

0.08

4.9

0.1

0.01

0.01

1.4E+04

0.06

10

0.18

0.14

8.7

1.3

0.14

0.13

2.5E+04

0.10

15

0.18

0.20

12.5

2.5

0.20

0.16

3.6E+04

0.15

20

0.18

0.26

16.3

3.7

0.22

0.18

4.7E+04

0.20

25

0.18

0.32

20.1

4.9

0.24

0.20

5.7E+04

0.24

30

0.18

0.38

23.9

6.1

0.25

0.20

6.8E+04

0.29

35

0.18

0.44

27.7

7.3

0.26

0.21

7.9E+04

0.33

40

0.18

0.50

31.5

8.5

0.27

0.21

9.0E+04

0.38

5

0.24

0.06

5.0

0.0

0.01

0.01

1.5E+04

0.04

10

0.24

0.11

9.3

0.7

0.08

0.07

2.7E+04

0.07

15

0.24

0.16

13.4

1.6

0.12

0.11

3.8E+04

0.10

20

0.24

0.21

17.5

2.5

0.14

0.12

5.0E+04

0.14

25

0.24

0.26

21.6

3.4

0.15

0.13

6.2E+04

0.17

30

0.24

0.31

25.8

4.2

0.16

0.14

7.4E+04

0.20

35

0.24

0.36

29.9

5.1

0.17

0.15

8.5E+04

0.23

40

0.24

0.40

34.0

6.0

0.18

0.15

9.7E+04

0.26

4.1 Flow in a symmetrically and asymmetrically bounded permeable domain

Since the permeable domain investigated herein is asymmetrically bounded, underneath by an impermeable surface and above by a freestream flow, the flow may be expected to display an asymmetric behaviour (e.g. non-horizontal flow direction, asymmetric turbulence distribution). In order to test the sensitivity of our set-up to detecting elements of asymmetry present in the flow, we conducted two simple experiments in which the parameters Qt (10 L s−1) and hw (0.18 m) were kept constant, but the boundaries were changed. In the first experiment (Fig. 6a, b), we provided symmetrical boundaries by placing an impermeable wall on top of the bed. In the second experiment (Fig. 6c, d), the solid top was removed, thus allowing water to flow across the interface.
Fig. 6

Comparison between time-averaged flow fields within the pore space at location B with different boundaries; on the lefta, b the porous domain is symmetrically-bounded by two permeable walls on the top and bottom; a and b illustrate the u component and v component, respectively. On the right sidec, d the impermeable wall on the top is removed allowing full interaction between the free flow and subsurface flow, creating an asymmetrically-bounded condition; cu component; dv component. Flow conditions within the channel were kept constant: hw = 0.18 m; Qt = 10 L s−1; the color-map refers to the magnitude of the velocity component normalised by its maximum value

As expected (see Yevseyev et al. 1991), the u component in the symmetrical configuration (Fig. 6a) is characterised by a relatively symmetrical distribution of velocity (in the vertical), with higher u velocities along the plane y/D = −2 and lower momentum regions at the top and bottom of the pore. If we focus on the plane y/D = −2, the u component shows larger values at the entrance (x/D < −0.2) and exit (x/D > 0.2) of the pore and a region of lower velocity in the pore center (−0.2 < x/D < 0.2). As expected, horizontal profiles of u (along y/D = −2) show a wavy shape, with deceleration and accelerations that reflect longitudinal changes in the cross-sectional area of the pore. The distribution of the v component (Fig. 6b) shows a high level of symmetry in the vertical, with values of v close to zero at the center (y/D = −2) of the pore and progressively higher values towards both the top and bottom. Figure 6b shows a symmetrical vertical divergence of flow on the left of the image (x/D < 0) and a subsequent contraction at the very right of the image (x/D > 0.3). This vertical symmetry of the flow is not perfect, which is most likely due to the fact that the pore is not at the exact same distance from the top and bottom walls.

The results from the second experiment (Fig. 6c, d) show that the asymmetric boundary drastically alters the pore flow, inducing regions of upwelling flow (Fig. 6d) and a consequent reduction in the average u component (Fig. 6c), with high values of u velocity concentrated at the top left of the image. Overall, the mean flow is horizontal, but with a pronounced asymmetry when compared to the case of the symmetrically-bounded flow (the high momentum region is shifted downward compared to the centerline y/D = −2).

The flow pattern found in the asymmetric boundary was also found for other flow conditions considered herein (specifically at high flow depths, hw/D) and seems to be associated with the horizontal deceleration of flow as it enters the bed. Similar effects have also been observed in past investigations through qualitative visualisations (Packman et al. 2004). However, the simple comparison of these asymmetrical and symmetrical boundaries clearly demonstrates the ability of the EPIV technique to quantify these effects, thus demonstrating its utility as a method to measure, at a high resolution, the turbulent structures within pore spaces beneath high Reynolds number flows that have hitherto not been possible. The following analysis examines how changing the flow boundary conditions affects these flow patterns.

4.2 Effects of freestream boundary conditions

The mean flow within the pore space of a packed bed of spheres enclosed in an impermeable duct has been represented as consisting of symmetric jet flows whose characteristics are a function of the bulk flow Re number (Yevseyev et al. 1991). In the case examined herein, the asymmetries of the flow domain (see above) may be expected to have an impact on the mean flow in the pore space. In order to investigate the effects of the flow boundary conditions, results obtained for two flow depths (hw = 0.12 and 0.24 m) at several different Res and Frs are detailed herein (Res and Frs conditions are reported in Table 1).

Figure 7 illustrates the time-averaged flow fields collected at hw = 0.12 m (hw/D = 3) at various flow discharges (Qt = 5–40 L s−1). The color-maps express the distribution of the u (Fig. 7A) and v (Fig. 7B) components of velocity for eight different cases (from a to i). Over the range of discharges reported herein, the general pattern of the u component within the pore is relatively consistent and possesses four principal characteristics: (1) the flow has one inlet point but different outlets (as represented by the arrows in Fig. 7B(d)); (2) regions of lower u are found at the top and/or bottom of the pore; (3) the highest values of u are found at the upstream pore entrance, and (4) the flow decelerates as it moves across the pore space and then accelerates as it reaches the downstream end of the pore. However, beyond this simple pattern, the results clearly show that both the magnitude of the u component and the degree of flow symmetry (with respect to the X axis) change with both Res and Frs. The contour plots of the u component reveal that the regions of higher horizontal momentum are consistently asymmetrical, but this asymmetry shifts either upward (y/D > −2) or downward (y/D < −2) depending on the flow conditions. Regions of high u values represent the pathway of the mean jet entering the pore. For slow flow conditions (Res < 4.1 × 104 and Frs < 0.31), this jet pathway is located at the top of the image (y/D > −2). However, as Res and Frs increase, the high momentum shifts downward, thus suggesting a clockwise ‘rotation’ of the mean jet, as represented by the large arrows in Fig. 7A.
Fig. 7

Time-averaged flow at hw = 0.12. Color-maps refer to: A streamwise velocity component in m s−1; B wall-normal velocity component in m s−1; scale bar ranges are set to optimise the visual interpretation; a depicts the free-surface elevation and the location of the pore space. Streamlines are superimposed

The v component is sensitive to this rotation, and analysis of the v distribution confirms the trend discussed above (as represented by the arrows in Fig. 7B(b–i)) and demonstrates that for slow free-flow conditions (Res < 4.1 × 104Frs < 0.31), the magnitude of v at the top of the pore is higher than at the base. At intermediate flows (Res = 5.0 × 104Frs = 0.39), the upward flow at the top of the pore is equivalent to the downward flow at its base (Fig. 7f) and the flow shows the highest level of symmetry. Finally, for faster free-flow conditions, the magnitude of the v component at the base of the pore is clearly higher than at the top.

Analysis of the entire data set confirms that, at a fixed hw and progressively increasing Res and Frs, the mean direction of the flow rotates clockwise. This trend can be observed by examining values of constant Res and Frs, where, as hw decreases, the mean direction of flow rotates clockwise. At high flow depths, such behaviour is very evident. In Fig. 8, the mean flow fields for hw = 0.24 m are reported, and the mean flow is seen to be consistently directed diagonally, as illustrated by the streamlines. However, two contrasting flow configurations can be distinguished:
Fig. 8

Time-averaged flow at hw = 0.24. Color-maps refer to: A streamwise velocity component in m s−1; B vertical velocity component in m s−1; scale bar ranges are set to optimise the visual interpretation; a depicts the free-surface elevation and the location of the pore space. Streamlines are superimposed

  1. 1.

    Res ≤ 6.2 × 104; Frs ≤ 0.17: at the bottom of the image, a region of flow upwelling is observed, as suggested by the streamlines and confirmed by the v component distribution (see Fig. 8B(b–e)). Therefore, the pore presents two different inlet points (see Fig. 8A(b–e)): one inlet point is located at the left side of the image while the other is located at the bottom, as represented by the arrows in Fig. 8A(a). This flow pattern suggests the superimposition of a horizontal and a vertical flow that results in a ‘combined’ flow, as witnessed by streamlines that form a junction point at the bottom left of the image (region of low momentum). High values of the u component are concentrated at the top left of the image. The intensity of these flows changes with the boundary conditions, thus resulting in a change of direction of the bulk pore space flow. The distribution of the vertical component (Fig. 8B) also shows that a second region of high positive v is located at the top of the image. The coexistence of two orthogonal directions of motion in the pore space suggests that the pressure gradient controlling the flow is not horizontal, in marked contrast to the case of a symmetrically-bounded domain.

     
  2. 2.

    Res > 6.2 × 104; Frs > 0.17. Observations made for the previous case (hw = 0.12 m) are also applicable for this range of boundary conditions. The flow has one inlet point but different outlets (see arrows in Fig. 8A(f–i)), where the fluid enters from the upstream pore space, flows horizontally and diverges towards the top and bottom, and displays a relatively symmetric pattern with respect to the X axis.

     
The pore Reynolds numbers, Rep = Up·D/ν (where Up is the spatial average computed from the distribution of the mean u component 〈U〉 measured within the pore), are a measure of the ratio between inertial and viscous forces within the pore space. For the cases investigated herein, Rep values are in a wide range (Rep = 124–5,900) including flows in both the transitional and turbulent regimes (Rep > 500, see Nield and Bejan 2006). Figure 9 illustrates the relationships between the boundary conditions, expressed as hw, Res, Frs and the parameter Rep and illustrates that for a fixed flow depth, Rep increases with Res. However, for similar Res, as the flow depth increases, Rep decreases dramatically. This may be due to the change in direction of the flow associated with the horizontal deceleration observed in Figs. 7 and 8. The relationship between Frs and Rep (Fig. 9b) shows that as Frs increases, Rep increases at the same rate for all the cases of flow depth examined herein, suggesting that a correlation may exist between these two parameters.
Fig. 9

Relationship between the pore space Reynolds number, Rep, at different flow depths hw and a freestream Reynolds number Res; b freestream Froude number Frs. The horizontal line at Rep = 500 indicates the critical value of Res that delimits the boundary between transitional and turbulent pore space flows

The distribution of turbulence within the pore space has been estimated from the EPIV data by computing the two-dimensional turbulent kinetic energy (TKE) for the cases of hw = 0.12 and 0.24 m (Fig. 10). The TKE is in the range 0.002–0.8 for hw = 0.12 m and 6 × 10−4– 0.01 for hw = 0.24 m. Over the range of discharges reported here, the TKE increases with Res and Frs and decreases with increasing flow depths. The distribution of TKE shows high values at the left side of the pore where the jet–pore interaction takes place. For hw = 0.12 m and Res > 5.0 × 104 (Fig. 10A(e–i)), the high levels of turbulence (TKE ~ 0.6) become concentrated at the bottom of the pore. This suggests that the base of the pore space may be characterised by the production of turbulent eddies, as confirmed by the observations discussed below in the analysis of the instantaneous flow.
Fig. 10

Turbulent kinetic energy (TKE = 0.5(〈u2〉 + 〈 v2 〉)) within the pore space Ahw = 0.12 m, experimental conditions are reported in Fig. 7; Bhw = 0.24 m, experimental conditions are reported in Fig. 8. Scale bar ranges are set to optimise the visual interpretation and differ between A and B

Figure 11 illustrates the distribution of Reynolds stresses for these two flow depths. For the case hw = 0.12 m, the Reynolds stresses are particularly high and concentrated at both the top and bottom of the pore, displaying a relatively symmetric pattern (Fig. 11A). As the flow depth increases (hw = 0.24), this symmetry is dramatically lost (Fig. 11B) and the majority of the pore space exhibits negative stresses, thus providing additional confirmation of the asymmetry associated with the free-surface flow examined herein.
Fig. 11

Distribution of the Reynolds stresses (ρu′v′) within the pore space Ahw = 0.12 m, experimental conditions are reported in Fig. 7; Bhw = 0.24 m, experimental conditions are reported in Fig. 8. Scale bar ranges are set to optimise the visual interpretation. Data are expressed in [Pa]

4.3 Instantaneous flow structures

Analysis of the instantaneous vector fields reveals the presence of distinct vortical structures (Figs. 12, 13, 14) with a wide range of sizes (1–20 mm in diameter) and strengths (swirling strength λci = 0.1–10 s−1). For slow pore space flow velocities (Up ~ 0.02–0.05 m s−1), the coherent flow structures observed within the pore space are predominantly large structures (order of the pore size) that move downstream at an average convection velocity Upc = 0.70–0.80 Up. However, as pore space flow velocity increased (Up ~ 0.05–0.15 m s−1), smaller structures, characterised by higher swirling strength and faster evolution, populated the pore. For these cases, the mean convection velocity Upc could not be measured due to limitations in the laser pulse rate (15 Hz). For the low Upc cases, the evolution of slow-moving coherent structures can be readily tracked over the whole FOV and resolved in several frames, although the small structures (1–4 mm) present in the flow can be observed for only a few images due to their shorter lifetime.
Fig. 12

Sequence of instantaneous flow fields showing flow evolution processes typical of quasi-horizontal flow driven by a downstream jet leading to formation of a vortical pathway; the color-map refers to the swirling strength; streamlines are superimposed. (hw = 0.24 m; Qt = 20). Time delay between sequential images shown is Dt = 0.2 s (high resolution version of this figure is given as a supplementary material)

Fig. 13

Sequence of instantaneous flow fields showing a high velocity front moving downstream and formation of a vortical pathway; the color-map refers to the streamwise velocity component expressed in m s−1; streamlines are superimposed. (hw = 0.24 m; Qt = 20). Time delay between sequential images shown is Dt = 0.2 s (high resolution version of this figure is given as a supplementary material)

Fig. 14

Sequence of instantaneous flow fields showing flow evolution processes typical of quasi-horizontal flow driven by a downstream jet leading to formation of a vortical pathway; the color-map refers to the swirling strength expressed in s−1; streamlines are superimposed. (hw = 0.24 m; Qt = 35). Time delay between sequential images shown is Dt = 0.07 s, total duration of the event t = 0.6 s (high resolution version of this figure is given as a supplementary material)

Four common examples of the temporal evolution of coherent flow structures are presented herein in order to show how, similar to the mean flow, the mechanisms governing the pore space flow dynamics are dependent on the free-flow conditions.

4.3.1 Single jet

This sequence (Fig. 12) illustrates a typical mechanism of eddy evolution showing a quasi-horizontal jet (represented by the arrow in Fig. 12a) and multiple eddies (labelled AE) that move downstream. The mean flow distribution for this condition is reported in Fig. 9e and shows that this is one of the cases defined as a ‘combined flow’ as it possesses a mean diagonal motion. The initial condition (Fig. 12a) shows three eddies (A and C anticlockwise and B clockwise). Vortex B is larger and moves at a slower rate (upc ~ 0.009 m s−1) that is equal to 80% of the local average velocity and can be tracked over the entire sequence (1.8 s). Vortex C is pushed downstream at a faster rate (uc ~ 0.025 m s−1) and is consistent with the mean flow velocity distribution that shows high values of the local u component (u > 0.025 m s−1) above the centerline (y/D > −2). In Fig. 10c, vortices A and C merge and form vortex D. In Fig. 12e, the topology of the flow is more symmetric, showing two counter-rotating eddies of similar dimension that are moving downstream. It can be seen that while the trajectory of vortex D is horizontal, vortex B advects diagonally and is pushed slightly upward. This is consistent with the streamlines revealed by the mean flow field (Fig. 8e). An animation of the mechanism described above is shown in a more complete form in Online Resource 1.

4.3.2 Jet-to-jet interaction

A different type of interaction is shown in Fig. 13 for the same flow boundary conditions as Fig. 12 (Rep = 640), which shows nine steps in a typical pattern of pore flow evolution. The sequence illustrates a group of eddies (labelled A, B and C) that advect downstream and whose movement appears to be controlled by the interaction between a quasi-horizontal downstream jet (labelled jet J3, Fig. 13a) and a vertical jet (jet J2). The entire sequence is characterised by a region of consistent upward-moving flow (v ~ 0.009 m s−1) concentrated at the bottom of the pore, which is consistent with the structure of the mean flow discussed above (Fig. 8e). Examining the entire data set collected for this condition, the flow upwelling at the bottom of the pore occasionally penetrates further into the pore and creates a vertical path that extends across the entire pore space. The result of this motion is that the flow accelerates at the top of the pore (as represented by the arrow J1), thus generating clockwise-rotating vortices at the center of the pore. One of these cases is illustrated in Fig. 13 that shows a vortex (A) generated by J1 (v ~ 0.01 m s−1). At the bottom of the pore, a constant flow upwelling (J3, v ~ 0.008 m s−1) is visible. The evolution of vortex A is controlled by a jet (J2, u ~ 0.02 m s−1) entering the pore at the top left of the field of view. Jet J2 is directed diagonally (as represented by the arrow) and forms a velocity front, with high velocities being present on the left side of the front and low velocities on the right side. The front is initially straight, perpendicular to the direction of J2 and moves in the direction of J2. The motion of this front pushes vortex A downstream, with the interaction between the front and J3 subsequently inducing the formation of new vortical structures (B and C). Fig. 13d shows that the flow upwelling contrasts with the motion of the front, inducing a deflection of the front (Fig. 13e). This mechanism generates a local instability of the velocity front in which the streamlines bend and reveal the formation of clockwise-rotating eddies elongated in the direction of the jet. The motion of these structures is not perfectly horizontal but possesses a small vertical component (positive) that pushes the structures slightly upward. This observation is consistent with the mean flow field in which the streamlines show a diagonal motion as discussed above. This event can be tracked for about 2 s (~30 frames), and Online Resource 2 shows a more complete sequence of this event.

4.3.3 High Re numbers

Figure 14 illustrates an example of flow evolution for the case of faster pore flow (Up = 0.024 m s−1, Rep = 960) and reveals that the flow is populated by eddies of different size. The large eddies are in the range of 8–11 mm diameter, with the small ones, as revealed by plots of the swirling strength (see Adrian et al. 2000), being in the range 2–4 mm. The formation of large eddies (vortices labelled as A, B, C, D, E) is triggered by the two diagonal jets (J1 and J2, Fig. 14). The motion of these large-scale eddies can be tracked over the entire duration of the sequence (0.6 s), while the smaller eddies are visible for only 2–3 images (~0.2 s). Vortices A and D are only partially visible due to the limitation of the FOV, but their size presumably extends up to the solid surface of the permeable matrix. Vortex D is found in the majority of the data collected herein, and its diameter is consistently large (5–15 mm); vortex A is smaller (2–10 mm) and occurs more occasionally.

When compared to the case of combined flow illustrated above, the mean pore space flow velocities in Fig. 14 are comparable (~0.02 m s−1). However, the eddies generated in the case of combined flow appear to have a larger size but lower strength, suggesting less deep penetration of the small-scale turbulence into the bed from the free flow. This suggests that the cases characterised by jet-to-jet interaction may have a role in preventing turbulence penetrating the bed.

5 Discussion

The present results show how the characteristics of the time-averaged flow within the pore space are a function of the free-flow boundary conditions Rep = f (hw/hbed, Res, Frs). For example, at a fixed flow depth, hw, for higher values of the freestream velocity (i.e. higher Res), the mass and momentum transport through the bed are likely to be enhanced. This observation is confirmed by the EPIV results that show an increase in velocity magnitude and turbulence intensity within the pore space with increasing Res. Conversely, with a fixed Res, but increasing flow depths, hw, the pore space velocity Up dramatically decreases (by up to one order of magnitude). For example, at Res = 3 × 105, Rep is found in the range Rep = 360–5,600 and principally caused by the enhancement of the vertical component of flow, which implies low horizontal momentum transfer through the bed (Up/U0 decreases). Thus, although a correlation between Res and Up may exist, the present data provide no evidence of a strict relationship.

In a qualitative sense, there appears more of a correlation between Frs and pore space flow. The relationship between Frs and Rep has been detailed for a range of ratios of flow depth: bed thickness, hw/hbed = 0.25–1. Although the present data provide no evidence for a quantitative relationship between the two, a qualitative analysis based upon the classification of pore flow patterns suggests the existence of ranges of Frs for which the flow is largely either horizontal or diagonal.

These results suggest that higher values of Frs (Frs ≥ 0.15) produce more horizontal flow, while at lower values of Frs, the direction of the bulk pore space flow rotates anticlockwise. For Frs ≥ 0.15, the pore space flow was found to be in the turbulent regime (Rep > 500) and coherent flow structures were observed forming and evolving, thus confirming that turbulence penetrated into the bed from the free flow. At decreasing Frs, the horizontal jet in the center of the pore tends to be progressively replaced by a vertical jet originating from the bottom of the pore, and in this range of Frs, the flow switches to a transitional flow regime (250 < Rep < 500). Such vertical motions may play a role in reducing Up and thus preventing turbulence developing within the pore.

Frs may also have a direct control on the pore flow direction through the pressure gradient. We speculate that such a correlation may appear more evident in the proximity of topography (e.g. bedforms, particle clusters) where the local Fr number can vary drastically over short distances and especially where the near-bed pressure gradients may be enhanced by low pressure associated with flow separation. Although an upward motion may appear counter-intuitive in a flat bed, previous research has also found evidence of such upward-moving fluid flow (Packman et al. 2004). Packman et al. (2004) demonstrated the existence of vertical and diagonal paths of subsurface flow within a planar permeable bed overlain by a free-surface flow, and explained this phenomenon by local non-uniformities in the bed geometry that may produce local diagonal pressure gradients. The ranges of Frs given herein cannot be considered universal, as they are likely due to the specific geometry of the domain used (e.g. total channel length, flow depth: bed thickness ratio, packing type and density). Nevertheless, they provide the basis for further investigations in which different experimental configurations can be examined and quantitative relationships derived.

The instantaneous flow fields shown herein also demonstrate that pore space flow is unsteady and turbulent. The production of pore space turbulence was found to be driven by either single or multiple pulsating jets. The intensity, periodicity and duration of these jets were extremely variable. The duration of these vortical structures within the pore space ranged from 0.1 to 1 s, depending on the pore flow velocity, with the lifetime of each vortex likely a function of the strength and duration of the initial jet flow from which the vortices were generated.

The turbulent events occurring in the free flow (e.g. sweeps and ejections, large-scale motions; Adrian 2007) are presumably the trigger of the unsteady phenomena observed within the pore space. Previous CFD simulations (Stoesser and Rodi 2007) have shown that flow within pore spaces is originally driven by the advection of large vortical structures in the freestream flow, which, as they form and translate downstream, generate alternate low- and high-pressure regions near the bed. We speculate herein that the resulting complex and unsteady pressure gradients generated by these freestream coherent flow structures may then propagate in three dimensions through the bed, thus producing fluctuating local pressure gradients between the interconnected pores. Local pressure gradients are directly responsible for the interstitial flow observed herein and can be considered the main mechanism of energy transfer from the freestream flow to the subsurface flow. Energy is injected into the pore by jets and then dissipated by the observed turbulent events and fluid mixing. These strong injections/ejections of fluid into a porous bed, as observed in the numerical simulations of Stoesser and Rodi (2007), are confirmed by the present results in the form of local jets that are found to be the main driver of a number of complex flow patterns.

6 Conclusions

A unique facility for quantifying flow within the pore spaces of a permeable bed in an open channel has been developed. The facility is instrumented with an endoscopic PIV (EPIV) system that allows high-resolution measurements, characterised by a field of view larger than previous external PIV applications and seeding concentrations higher than RIM methods (see discussion in Huang et al. 2008). This EPIV technique allows quantitative study of both the mean and instantaneous structure of flow within a pore space at high Reynolds numbers, which are challenging for both MRV and RIM techniques. Use of a pulsed double-headed laser, coupled with an image-intensified camera, has overcome issues of poor illumination and resolution noted with other endoscopic techniques and permits the flow structure to be quantified over a large range of fluid velocities. This approach is particularly suited to the investigation of permeable beds immersed in open-channel flows that replicate flow in a natural gravel-bed river, which are challenging for many low-viscosity RIM techniques, and thus allows quantification of hyporheic flow processes at a wide range of flow Reynolds numbers.

The development and application of this new methodology permits the following key conclusions:
  1. 1.

    EPIV based upon use of borescopes allows the instantaneous velocity fields to be resolved with a resolution sufficient to characterise the structure of the pore space flow.

     
  2. 2.

    When compared to the case of a symmetrically-bounded domain, the mean pore space flow patterns in an asymmetrically-bounded domain show clear asymmetries in both the u and v components of velocity as well as the turbulent statistics.

     
  3. 3.

    The nature of the mean pore space flow is dependent on the free-flow boundary conditions. Res appears to control the magnitude of the mean pore space velocity and turbulence, while Frs controls the mean direction of flow.

     
  4. 4.

    The instantaneous structure of flow within the pore space is dominated by either single or multiple jet flows entering the pore, which generate turbulent eddies within the pore space.

     
  5. 5.

    Jet flows can be horizontal, vertical or a combination of the two, with the mechanism governing the temporal evolution of flow within the pore space being a function of the intensity, direction and duration of such jets.

     
  6. 6.

    High levels of turbulence (RMS values higher than 10%) in the pore space are associated with the formation of coherent flow structures at different scales that can be tracked in their formation and advection along specific vortical pathways.

     

Characterisation of the pore space flow in the present study using this new EPIV technique has revealed several distinct forms of coherent flow structure and mechanisms of turbulence production/dissipation, thus allowing new insights into the nature of flow within porous media. Future work will aim to develop a technique capable of providing PIV data simultaneously from the free flow and the pore space, thus allowing the link between freestream turbulent structures and pore space turbulent events to be elucidated. These ongoing experiments will thus allow full quantification of the precise characteristics of the freestream that are responsible for dictating flow structure within the pore space. The present work represents the essential first step in a wider project investigating the role of pore space turbulence on the mechanisms governing nanoparticle transport and fluid exchange in the hyporheic zone. This novel EPIV technique now presents a methodology with which to investigate the influence of grain and bedform roughness, particle packing and grain shape on the nature of hyporheic zone flow and turbulence, and obtain data sets that can be used to test and validate numerical simulations of pore flow.

Acknowledgments

We thank the UK Natural Environment Research Council for funding this work (NE/E006884/1). All experiments were undertaken in the Ven Te Chow Hydrosystems Laboratory, University of Illinois, and we thank Professor Marcelo Garcia for allowing access to this facility, and Professor Kenneth Christensen for providing part of the PIV equipment. Stephan Kallweit of Intelligent Laser Applications (ILA GmbH) supplied the endoscopes and much useful advice on their application, for which we are very grateful.

Supplementary material

Online Resource 1 Sequence of images of instantaneous flow fields showing the evolution of flow driven by a downstream jet leading to formation of a vortical pathway. Supplementary material 1 (AVI 22777 kb)

Online Resource 2 Sequence of images of instantaneous flow fields showing the evolution of flow triggered by a spanwise jet leading to formation of a large vortical structure, whose evolution is driven by fluid motion that is first downstream and then upstream. Supplementary material 2 (AVI 6537 kb)

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High resolution of figure 14 (JPEG 4.57 MB)

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • G. Blois
    • 1
    • 2
  • G. H. Sambrook Smith
    • 1
  • J. L. Best
    • 3
  • R. J. Hardy
    • 4
  • J. R. Lead
    • 1
  1. 1.School of Geography, Earth and Environmental SciencesUniversity of BirminghamBirminghamUK
  2. 2.Department of Mechanical Science and EngineeringUniversity of IllinoisUrbanaUSA
  3. 3.Departments of Geology, Geography, Mechanical Science and Engineering, and Ven Te Chow Hydrosystems LaboratoryUniversity of IllinoisUrbanaUSA
  4. 4.Department of Geography, Science LaboratoriesDurham UniversityDurhamUK

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