A Methodology for Studying the Hydroelastic Response of Submerged Flexible Vegetation

Abstract

Most fluvial vegetation canopies embody some degree of structural flexibility, resulting in hydroelastic phenomena linking plant deformation and fluid flow. Quantification of such coupled fluid-structure motion is critical in investigating key fluvial processes. The present work presents an experimental protocol that enables such quantification. High-speed particle image velocimetry data of unidirectional flow surrounding idealized, submerged canopy elements, consisting of polyurethane rubber models, together with the corresponding solid displacement field, were simultaneously obtained by combining fluorescent imaging and refractive index matching. The operating principle of the approach employed herein involves seeding the two phases with different tracers, facilitating independent interrogation of the dynamics of each. Time-resolved data allowed for observation of the dynamic link between a deformable object and the surrounding flow field.

Key Points

• Combining dual-phase tagging and index matching allows simultaneous probing of flow-induced flexible plant motion and wake dynamics

• Time-resolved data enables study of temporal link between plant model motion and surrounding flow fluctuations

• Selected results demonstrate protocol probing capability and potential for spatio-temporal analysis

1 Introduction

Investigations of the flow dynamics associated with vegetation, either experimental or numerical, have widely employed idealized rigid models as proxies for vegetation elements (Bai et al., 2015; Liu et al., 2008; Huai et al., 2015; Huang et al., 2011; Nepf, 1999; Stoesser et al., 2010; Tanino & Nepf, 2008). Though a significant simplification from reality, this approach has improved our understanding of how flow responds to the presence of plants, often arranged in complex multielement, canopy-like configurations (Nepf, 1999, 2012a,b). However, most plants, particularly in aquatic environments, embody some degree of structural flexibility, which determines the dynamic structural response of the submerged plant to the hydrodynamic forcing (Aberle & Järvelä, 2013; Gosselin, 2019; Rand, 1983; Whittaker et al., 2015). The swaying frequencies, for example, impacts flow separation and vortex shedding, resulting in hydroelastic mechanisms coupling the plant motion with the surrounding flow (Gosselin, 2019; Niklas, 1992). A growing number of studies addressing flexible canopies have recognized the importance of incorporating realistic plant flexibility in their modeling approach (Lei & Nepf, 2019; Sukhodolov et al., 2021; Zhang et al., 2021). Models that ignore this dynamic coupling fail to explain physical phenomena such as windthrow (Gardiner & Quine, 2000; Mayer, 1987). However, due to the inherent challenges associated with the simultaneous quantification of flow and motion of models that mimic real vegetation, most studies utilize experimental protocols that, at best, capture the independent physics of each component, that is, for either flow (Finnigan, 1979, 2000; Houseago et al., 2022; Jin et al., 2018; Raupach & Thom, 1981; Van Gardingen & Grace, 1991) or plant dynamics (Brüchert et al., 2003; Hassani et al., 2016; Jin et al., 2019; Niklas, 1992). As such, these approaches cannot reveal the instantaneous coupling between turbulent structures and plant deformation. Of the few experimental studies that have measured both quantities simultaneously, Okamoto and Nezu (2009) utilized a fluorescent imaging technique to capture both flow and canopy motion. The fluid flow was measured using a planar particle image velocimetry (PIV) approach. In addition, fluorescent markers were attached to the stem elements to quantify the plant movement through single-point tracking. An algorithm was employed to discriminate the stem markers from the flow tracers based on morphological attributes (Nezu & Azuma, 2004). A few other studies have employed digital image correlation (DIC) to quantify the deformation of crop canopies using in-field images of canopy motion (Doaré et al., 2004; Py et al., 2005, 2006). However, only single-point measurements of the flow producing the plant motion were made.

The present work presents an experimental protocol developed to study the coupled flow–structure interaction in submerged flexible canopies. This is accomplished by facilitating the simultaneous quantification of plant deformation and the flow driving it through the utilization of integrated high-speed PIV-DIC. The method interrogates the flow surrounding flexible plant models together with the corresponding plant model displacement field throughout the stem by combining fluorescent imaging and refractive index matching (RIM). The data acquired in this manner facilitates spatial and temporal correlations of fluid-solid fluctuations characterizing hydroelastic behavior.

2 Experimental Method

The measurement protocol presented herein is based on an optical method that combines DIC, to quantify the plant deformation, and PIV, to resolve the surrounding flow field. A dual-signal visualization approach based on fluorescent imaging and accomplished via selective particle tagging and monochromatic light excitation was utilized to separate the fluid and solid phases. Unobstructed optical access was achieved by leveraging a fully controlled refractive index-matching environment (Blois et al., 2020). The particle-based signal approach allowed the use of well-established particle image algorithms independently tailored to reconstruct the instantaneous flow field in the fluid phase and the instantaneous solid model deformation.

The imaging principle followed a standard PIV (and/or DIC) approach. Here, a 2D planar PIV method (R. J. Adrian & Westerweel, 2011) in combination with a DIC method (Blaber et al., 2015; Chu et al., 1985; Palanca et al., 2016; Pan, 2011; Peters & Ranson, 1982) were used to resolve the flow around the flexible model and the model deformation, respectively. Figure 1a presents a 3D schematic view of the experimental setup and illustrates the arrangement of the laser sheet and the orientation of the camera used to image the flow-object system. In this case, the vegetation model is an idealized plant model featuring multiple branches. The model was secured at the center of a 11 × 15.7 × 0.5 cm removable PVC plate that was flush mounted into an inset in the floor of the test section.

Figure 1

(a) Schematic of the experimental setup. The bounding box represents a subset of the flume test section. The plant model is bent by the flow. The dots on the model represent the particle tracers in it. An instantaneous snapshot of the streamwise velocity around the model is shown to illustrate the turbulent flow generated by the interaction; (b) CAD model of the idealized branch and the approximate location of the laser sheet used in a, c, and d. (c) Raw PIV-DIC image; (d) Illustration of the image segmentation technique: the image portion colored in green corresponds to the fluid region illuminated by the laser lightsheet, while the magenta portion delineates the cross section of the solid model transected by the lightsheet.

An instantaneous snapshot of the streamwise velocity around the model is also shown in Figure 1a. The laser sheet is configured in the streamwise–wall-normal (x − y) plane to best observe the model bending owing to the mean flow. Use of particle tracers conveniently dispersed in the fluid and solid phases at high concentrations facilitated high-resolution reconstructions. All tracer particles were illuminated with the same pulsed laser light formed into a thin sheet that defined the streamwise–wall-normal (x − y) measurement plane that cut through a portion of the vegetation model. A high-speed digital camera captured the position of the tracer particles at two sequential times, defined by conveniently timed laser pulses, allowing the quantification of local instantaneous displacements with high spatial resolution. A raw PIV image of the model in the configuration depicted in Figure 1a, subject to a freestream velocity of U_∞ = 0.49 m/s, is shown in Figure 1c.

The operating principle of the technique presented herein involved optically separating the fluid and solid phases (i.e., image segmentation). This segmentation was accomplished by seeding the two phases with different tracers which facilitated the interrogation of each phase independently. More specifically, the fluid was seeded with 2-μm silver-coated hollow-glass spheres (Blois et al., 2020) while the solid phase was seeded with fluorescent particles with diameters in the range of 53–63 μm, with peak excitation and emission wavelengths of 577 and 606 nm, respectively. Figure 1d illustrates an example of signal separation to obtain a segmented image.

The fluorescent particles are dispersed within the plant model and the laser sheet cut through the centerline of the model. As such, the measurement plane for the flow and for object deformation coincided (Figure 1a). This required the use of transparent solid models and the minimization of light refraction effects at the interface, achieved through a refractive index matching (RIM) approach. The method is based on utilizing transparent materials to fabricate solid models and immersing the models within a fluid with a refractive index (RI) as close as possible to that of the solid material (Budwig, 1994). When RIM is achieved, the solid-fluid system is rendered an optical continuum.

The remainder of this section describes the RIM facility, the model fabrication protocol, and the image acquisition and processing schemes.

2.1 Flow Facility

Laboratory experiments presented herein were performed in the refractive-index-matched (RIM) flow facility at Notre Dame (Blois et al., 2020). The cross-section of the flume test section measures 112.5 × 112.5 mm and the length is 2.5 m. The incoming flow had a turbulent boundary layer of thickness δ = 58.4 mm and a freestream turbulence intensity of 2.4%. Other properties of the incoming flow are listed in Table 1 for the two vegetation models utilized. Further analysis of the incoming flow is provided in previous work (Gundersen et al., 2021).

Table 1. Characteristics of the Incoming Flow for the Two Flexible Vegetation Models

	Cylinder	Idealized branch
Free-stream velocity, U∞ (m/s)	0.98	0.49
δ99 (mm)	56.5	56.7
uτ (m/s)	0.038	0.020
ReD = U∞D/ν	4,450	1,340
Reδ = U∞δ99/ν	50,000	25,000
Reτ = uτδ99/ν	1,920	1,040

A sodium iodide (NaI) solution (approximately 63% by weight) was used as the working fluid in these experiments. An advantage of using this fluid is that it has a high density to kinematic viscosity ratio relative to other fluids commonly used for index matching. This characteristic allows for high-Re flow conditions to be achieved. Furthermore, the solution is nontoxic and relatively inexpensive compared to other options, so it can be used in a large-scale laboratory setting. Lastly, the working fluid, though prone to chemical instabilities in the presence of oxygen that can potentially compromise its optical properties, its integrity can be preserved over long periods of time by ensuring it remains contained within a closed-circuit system and isolated from the surrounding air. Additional details on the properties of the NaI solution and practical strategies for handling a large-scale experimental setup are outlined in Blois et al. (2020). The optical RI of the solid models was very close to that of the aqueous sodium iodide working fluid. By fine-tuning the fluid temperature to further optimize the match between the solid and fluid RIs, the model immersed in the working fluid optically “disappeared” and light was able to pass through the model with minimal reflection or refraction. This experimental protocol therefore enabled unobstructed and unaberrated optical access to the entire fluid domain as well as to the interior of the model, allowing full quantification of the fluid flow via PIV and object deformation via DIC, respectively.

2.2 Vegetation Models

The models in these experiments were fabricated from a clear polyurethane rubber to take advantage of its elastic nature. This material has an index of refraction of n = 1.48822 (at 20°C), a Young's Modulus of E = 3 MPa, and a density of ρ_s = 1040 kg/m³. The overall dimensions of the model were chosen based on multiple constraints. The length was selected to allow the model to be fully submerged within the incoming turbulent boundary layer. The diameter was chosen by imposing a length-to-diameter ratio of 10 which provided a model flexural rigidity within the range of most aquatic plants (Hurd & Pilditch, 2011; Miler et al., 2012; Rominger & Nepf, 2014).

In this study, two simplified geometries were utilized as proxies for an idealized plant: a cylinder and a branched stem that mimics a defoliated plant. The fabrication process closely followed the protocols detailed in Blois et al. (2020). First, a rigid opaque object (positive) of the desired shape was obtained from 3D printing. Second, a model mold (negative) was created by pouring a silicone rubber mixture (Mold Star™ 30) around the positive shape. A two-part (A and B) urethane rubber, Clear Flex^® 50, was then poured into the silicone mold. Previous to pouring, a suspension containing fluorescent particles was dispersed in part A via mixing. Part B was then added and thoroughly mixed with part A to maximize uniformity of particle dispersion in the curing process. The optimal particle concentration was refined over multiple iterations with the goal of satisfying upper and lower particle density constraints. That is, the particle concentration must be sufficient to capture “local” cylinder deformation but not overly saturated which could reduce light transmittance.

The cylinder model was L = 50 mm long and had a diameter of D = 5 mm. The geometry of the branched plant model was created in reference to a digital rendering of a defoliated sample of Prunus laurocerasus that was acquired via terrestrial laser scanning (TLS) by Boothroyd et al. (2016). The CAD model of the idealized branch is shown in Figure 1b along with an illustration of the approximate location of the laser sheet for the model configurations displayed in Figures 1 and 2a, 2c and 2d. The original geometry was modified to satisfy the constraints of the facility while attempting to maintain some of the morphological complexity. Specifically, the height was scaled down to H = 50 mm such that the deflected model was fully immersed in the turbulent boundary layer. This resulted in a scale factor of about 20. The diameter of the branches was dictated primarily by considerations relative to the model casting process. In order to appropriately limit the complexity of the fabrication, it was established that the diameter must be no smaller than D = 3 mm.

Figure 2

Raw images of the four configurations of the branch models subjected to a freestream velocity of U_∞ = 0.49 m/s. (a) Streamwise-normal and (b) streamwise-aligned configurations. Pair of branches in tandem alignment in streamwise–normal configurations and spaced by (c) 1H and (d) 0.5H in the streamwise direction, where H = 50 mm is the height of the model.

Raw PIV images for each of the setups are shown in Figure 2. Experiments were performed for single plant model in two orientations (Figures 2a and 2b) as well as for two plants streamwise-aligned in a tandem configuration (Figures 2c and 2d). The two branches in the tandem configuration were oriented normal to the freestream flow, and spaced 1H (Figures 2c) and 0.5H (Figure 2d). The cylinder and plant models were subjected to a free stream velocity of approximately 1 and 0.5 m/s, respectively.

2.3 Image Acquisition and Processing Scheme

The tracer particles in both the fluid and solid phases were illuminated with a Northrop Grumman Patara, dual-cavity (each cavity emitted 50 mJ/pulse at 1 kHz) Nd:YLF laser formed into a thin light sheet (approximately 1 mm thickness). A four-megapixel, high-speed frame-straddle CMOS camera (Phantom v642) with an array size of 2,560 × 1,600 pixels was used to image the field of view (FOV). The resulting dimensions of the FOV in physical units was 106 × 66 mm. For each data set, 2,734 pairs of PIV images were collected. The camera captured image pairs at a sample frequency of 320 Hz to acquire time-resolved data and assess the ability of the technique to track the hydroelastic dynamics of the fluid-structure interaction process.

The camera captured both the light scattered by the silver-coated particles seeding the fluid flow as well as the light fluoresced by the particles seeded within the vegetation models. As such, prior to image interrogation, a robust image segmentation algorithm was used to separate the regions corresponding to the solid model doped with fluorescent particles from the surrounding flow whose motion was discerned from images of the scattered light from silver-coated tracer particles. The algorithm leveraged the fact that the fluorescent particles are larger and more concentrated compared to the small and relatively sparse tracer particles in the flow. First, a morphological opening operation with a disk-shaped structuring element was applied to the raw image. The opening algorithm is defined as the dilation of the erosion of a block of pixels (Haralick et al., 1987). The operation has the effect of removing small-scale features, which in this case are the small groups of pixels corresponding to the flow tracer particles, while retaining large-scale features, corresponding to the relatively larger clusters within the vegetation model. An image segmentation algorithm was then used to label and separate pixels within the model from the pixels in the mostly black space surrounding the model. The two regions were separated based on the pixel intensity with a threshold level determined with Otsu's method (Otsu, 1979). This segmentation approach allowed the flow and the cylinder motion to be processed independently. An example result of the segmentation technique is presented in Figure 1d. The region colored in green was used to process the flow data while the magenta region was used to determine the solid model displacement field.

The image regions corresponding to the fluid flow were processed using LaVision's DaVis software with a normalized multipass iterative cross-correlation scheme with decreasing interrogation window size, starting with two passes with a 64 × 64 pixel spot size, followed by three passes with a 16 × 16 pixel spot size. The final spatial resolution of the vector fields was approximately 0.6 mm. Finally, spurious vectors were identified and removed using a universal outlier detection scheme (Westerweel & Scarano, 2005). Holes were filled via linear interpolation. Fewer than 5% of vectors within the shear layer and in the wake of the object were a result of interpolation.

Digital image correlation (DIC) is widely used as a noncontact technique for measuring material deformation (Chu et al., 1985; Peters & Ranson, 1982). The vector fields corresponding to the model displacement were processed using Ncorr, an open-source 2-D DIC Matlab package (Blaber et al., 2015). The algorithm developed in this work used a reference image to obtain the fluctuating component of solid motion via correlation with the subsequent images. This allowed the decoupling of the motion of the model from its pure deformation. The displacement and strain data were thus independently extracted from the transformation. The reference image in this case was a selected snapshot that featured the model in its average bent configuration. Due to symmetry, this occurred without any out-of-plane displacement, thus ensuring that the full particle distribution pattern of the reference image was visible. Ncorr uses the Reliability Guided DIC (RGDIC) method (Pan, 2009) to obtain displacement values across the domain. Both large and small (i.e., <1 pixel) displacements were found within the model due to the high dynamic range of its motion. The algorithm is suited to resolve the large range of displacements by iteratively transforming the subset points from the previous image and is able to accurately resolve subpixel displacements via biquintic B-spline interpolation that acts to mitigate signal noise (Schreier et al., 2000; Unser, 1999).

The fact that the fluorescent particles were homogeneously distributed throughout the model and had optimal brightness levels resulted in high correlation coefficients and relatively few spurious vectors. The deformation data were processed with a circular interrogation window with a radius of 50 pixels with subsets spaced by 10 pixels, resulting in a grid spacing of 0.45 mm.

3 Results

A representative example of the instantaneous distribution of swirling strength, λ_ci, around the cylinder model subjected to the oncoming turbulent boundary layer with a freestream velocity of U_∞ = 0.98 m/s is shown in Figure 3a. A similar result is shown in Figure 3b corresponding to a tandem branch configuration subject to a freestream velocity of U_∞ = 0.49 m/s. Swirling strength is a robust vortex identifier and defined as the imaginary component of the complex eigenvalue of the velocity gradient matrix, ∇u (R. Adrian et al., 2000; Zhou et al., 1999). Here, λ_ci was multiplied by the sign of the local vorticity, ω_z, to visualize the sense of rotation (Wu & Christensen, 2006). In Figure 3a, a region of high λ_ci magnitude and negative (clockwise) rotation can be seen emanating from the tip of the model. Such vortices are produced by separation at the tip and are consistent with the region of high spanwise vorticity recently shown by tomographic PIV measurements on rigid cantilevered cylinders by Crane et al. (2022). These measurements also reveal, along the entire span of the model, spanwise vortices of alternating rotational sense in the near-wake of the model. These vortices are a manifestation of the shear layer separating along the sides of the cylinder. As explained by Crane et al. (2022) for finite, rigid cylinders, a more complex three-dimensional (3D) von Kármán wake arises from the shear layer emanating from the sides. This wake is characterized by the alternate shedding of antisymmetric quasi-vertical vortices on opposite sides. Unlike infinite cylinders, where vortices are 2D and their axes tend to remain parallel to the cylinder axis, in a finite cylinder, these vortices arch. In particular, the opposite vortices (“legs”) pinch near the tip of the cylinder where they connect forming a full loop. An analysis of the stretching and tilting terms in the vorticity transport equation allowed Crane et al. (2022) to elucidate such arching which is the cause of the 3D flow structure. Compared to a 2D cylinder, the stronger velocity gradients, particularly along the axis, result in stronger stretching and tilting terms. Streamwise and wall-normal tilting components are particularly important at the tip and base of the cylinder, resulting in the wall-normal vorticity to reorient into the spanwise direction. This explains the concentration of spanwise vorticity in these two areas in a rigid cylinder. We speculate that in a flexible cylinder, the cylinder bending enhances the velocity gradient along the axis and thus the contribution of vortex tilting along the entire cylinder, thus increasing the intensity and number of the spanwise vortices compared to a fixed cylinder. These hypotheses can be tested by applying this technique in a stereo or tomographic fashion by measuring the three components of velocity and performing a similar 3D vorticity analysis as Crane et al. (2022).

Figure 3

Instantaneous snapshot of swirling strength, λ_ci, colored with the local sign of spanwise vorticity, ω_z, in the fluid phase with vectors displaying the fluctuating streamwise and wall-normal velocities, u′ and v′. The vectors are spaced at every other grid point and are up-scaled by a factor of 4. The model is colored based on the corresponding solid displacement field, Δx, relative to the previous image. (a) Cylinder model and (b) branch models in a tandem configuration spaced by distance equal to the branch height, H. The freestream velocity, U_∞, was 0.98 m/s in (a) and 0.49 m/s in (b).

These results show that, despite a slight decrease in illumination energy in the shadow region beneath the cylinder, the data quality is relatively uniform throughout the FOV. Similar observations can be made by examining results in Figure 3b for the tandem-aligned branch models where the vortices shedding from the upstream plant advect downstream and eventually impinge onto the downstream plant, providing a unique data set to explore proximal plants turbulent interaction processes. Figure 3b confirms that the simultaneous deformation of proximal plants can be achieved and can be used to investigate their coupling mechanisms.

Figure 3 also shows the instantaneous solid model displacement maps relative to the previous image frame at the same instant in time, obtained through the use of DIC. The displacement magnitudes, Δx, increase with distance from the base in a manner qualitatively consistent with the expected displacement distribution of a cantilever beam that is fixed to the wall and subjected to a unidirectional force (Gere & Goodno, 2012), herein owing to the oncoming flow. While this force is unknown, it could be estimated by combining knowledge on the cylinder bending and constitutive properties of the material.

Access to time-resolved acquisition further enhances the probing capability of this measurement protocol. An animation illustrating how the dynamics of the cylinder and that of the flow can be simultaneously quantified in a time-resolved manner is available in Supporting Information S1. This animation highlights how the hydroelastic behavior and coupled dynamics can be studied in a quantitative fashion. This approach, for example, allowed for the accurate and simultaneous calculation of quantities such as power spectra of the cylinder vibrations from the DIC data and the vortex shedding frequency from the PIV flow field data. In addition, it also facilitated temporal correlations between the model and flow and/or between the two models.

Figure 4a illustrates time series of the x- and y- components of fluctuations for the flow and model at selected points in close proximity as identified in Figure 4b. Specifically, the fluid fluctuation, shown in blue, is sampled just above the tip of the model and the time derivative of the solid displacement, plotted in red, is sampled at a point within the model near the tip and along the model centerline. The velocities within the model were calculated as the product of the solid displacement relative to the previous image, Δx, Δy, and the sample frequency, f_s = 320 Hz. The time series, shown in Figure 4a, highlight that both velocities have periodic behaviors and closely follow one another. The periodicity is further examined in the power spectral densities (PSD) of each velocity component, S_uu and S_vv (Figure 4c). The PSDs were calculated using Welch's method (Welch, 1967) with 16 segments that overlap by 50%. As an example of quantitative information and insight that can be extracted from these data, a brief analysis of these results is provided. The PSDs for the horizontal and vertical components of both the flow and model motions display a dominant peak at f = 52 Hz, corresponding to a Strouhal number, St, of 0.26, where St = fD/U_∞. This frequency is associated with the shedding of the large-scale arched vortices, presumably similar to those showed by Crane et al. (2022). These structures form a 3D von Kármán wake and cause the model to oscillate with a strong periodicity. As previously discussed, the head of these vortices connecting the two legs is revealed in Figure 3a near the tip by concentrated flow rotation. The time series data indicates that, while the oscillations and flow fluctuations have a clear periodicity, their amplitude is not uniform. As suggested by a qualitative inspection of Figure 4a, the streamwise velocity fluctuations alternate events of low and high amplitude, with the latter corresponding to high-momentum streamwise flow events (u′ > 0). Similarly, the oscillation of the model experiences protracted periods of low and high amplitude. The data shows that high streamwise momentum flow events and high amplitude model oscillations occur at the same time. Furthermore, these protracted events alternate periodically, as indicated by the existence of secondary peaks in the S_uu spectra (Figure 4b). In this regard, the secondary peaks for the model motion, Δx, and the flow, u′, coincide, indicating the longitudinal swaying of the model is locked to and likely modulated by a multimodal vortex shedding activity. Moreover, Figure 4a suggests that u′ and Δx are out of phase, with the maximum positive bending in each cycle corresponding to the minimum velocity and vice versa. The vertical velocity PSD, S_vv, shows a significant secondary peak in the vertical motion of the model velocity at roughly half the dominant peak frequency. Such pronounced secondary peak does not appear in the fluid velocity spectrum near the model tip suggesting that the low frequency vertical motion of the model and the near tip are uncoupled.

Figure 4

(a) Segment of a time series of the fluctuating streamwise velocity, u′ (top) and fluctuating wall-normal velocity, v′ (bottom). The blue curves correspond to the flow velocity near the tip of the model and the red curves correspond to the velocity of a selected point within the model. These sample locations are illustrated in (b). (c) Power spectral densities of the data in (a). The data set used for frequency analysis consisted of 2,734 images sampled at f_s = 320 Hz and the freestream velocity was U_∞ = 0.98 m/s.

Figures 5a and 5c show time series and power spectral densities, respectively, corresponding to the time derivative of the model displacement in the y-direction at various points within the model as illustrated in Figure 5b. The model displacement fields (bottom plot in Figure 5a) were calculated using DIC and are shown in the blue curves, with progressively darker colors approaching higher elevations. As expected, the amplitude of oscillation is larger for points at higher elevations. These points also correspond to higher power spectral density values. As seen in Figure 5c, both the dominant and the secondary spectral peaks, previously observed at the tip (see Figure 4c), are present in all the PSDs, indicating that all the points of the stem experience the same dynamics. It is worth emphasizing that this example of application, demonstrates the high sensitivity of the technique presented herein to reveal details of the structural dynamics that could not be captured by previous approaches.

Figure 5

(a) Full time series (top) and a zoomed-in segment of the time series (bottom) of the wall-normal component of the model displacement, Δy, at selected points within the cylindrical model. (b) Locations of the sample locations in the reference image. (c) Power spectral density of Δy. The data were acquired via DIC. The data set consisted of 2,734 images sampled at f_s = 320 Hz and the freestream velocity was U_∞ = 0.98 m/s.

These results may be extended to other experiments by acknowledging some of the key parameters in evaluating these flow-plant interactions. The density of the working fluid in this study is approximately 1.8 times that of water and has roughly the same kinematic viscosity, ν. Therefore, the models used herein would likely display a more deflected posture than if they were submerged in water subject to a uniform flow at the same Re, noting that the drag force on a vegetation element is proportional to the density of the fluid. Additionally, the fact that the dynamic viscosity, μ, of the NaI solution is larger than that of water implies that the viscous dissipation rate in the wake of the model would likely be lower in water. Similarly, because viscosity acts to dampen the model vibrations, one may find higher-energy oscillations of the model in an aquatic scenario. The mass ratio, , gives insight into the effect of buoyancy forces on a vegetation element. The mass ratio in these experiments, , closely matches that of many freshwater plants (Westlake, 1965), but is slightly lower than the mass ratio of marine species (Gaylord & Denny, 1997; Luhar & Nepf, 2011).

4 Conclusions

The selected results presented herein demonstrate an experimental protocol that allows the simultaneous quantification of both vegetation model movement and the surrounding flow field. This is accomplished by embedding fluorescent particles in a transparent flexible vegetation model and different tracer particles in the flow field, facilitating image segmentation. DIC was used to interrogate the region corresponding to the model and thus resolve the model deformation, while PIV analysis was used to quantify the surrounding flow field. Additionally, the utilization of a RIM environment allowed unaberrated dissection of the model via a thin laser light sheet and thus quantification of the internal deformation of the model rather. If the model were opaque, the flow tracer particles would not have been observable in the area under the cylinder due to obstructed illumination. Moreover, index-matching minimized light refraction at the solid-fluid interface, which also ensured that the measurement region remained planar. While the use of a RIM environment is ideal, a similar approach could also, at least in principle, be achieved in conventional water flumes using transparent objects with square or rectangular cross sections to limit light refraction.

High-speed PIV data facilitated observation of the dynamic link between a deformable object and the surrounding flow, thus giving the ability to probe the fluid-structure coupling. This experimental method could be utilized to aid in the study of other hydro-elastic flow-structure interactions.

Acknowledgments

The authors are grateful to Notre Dame Research and Notre Dame International for providing funding to support this research titled “Developing an Integrated Structural-Flow Tracking Experimental Protocol to Study the Hydro-Elastic Response of Fluvial Vegetation.”

Open Research

Data Availability Statement

The data associated with the article has been uploaded into a repository belonging to the University of Notre Dame (CurateND) and available at this link https://curate.nd.edu/show/6108v982x4s.