Effects of blade flexural rigidity on drag force and mass transfer rates in model blades

We created a model physical system to investigate the role of blade rigidity in setting forces and rates of mass exchange for a blade exposed to unsteady flow in a vortex street. Using a combination of experimental and theoretical investigations, we find that, broadly, both the forces and the mass exchange are higher for more flexible blades. Below a critical value of the dimensionless blade rigidity, inertial forces from the rapidly deforming blades become significant, increasing both the time-averaged and instantaneous forces on the blade. Mass transfer is also affected by blade rigidity. As blades deform, they alter the relative fluid motion at the blade surface, affecting mass fluxes. We developed a novel experimental method that simulates nutrient uptake to a blade using the transport of a tracer compound into polyethylene models. Through these experiments and modeling, we demonstrate that increased blade flexibility is associated with increased mass transfer to blades. Kelp blades alter their morphology via the process of phenotypic plasticity, which is the process through which the manifested traits of an organism adapt to changes in environmental stimuli, such as changes in light or nutrient availability, changes in mechanical stresses, changes in temperature, and so forth. In kelp blades, the length, width, and thickness of the blades, as well as the overall characteristics of the shape (e.g., rectangular, tapered, undulatory, etc.) can change over the timescale of the kelp life cycle in response to hydrodynamic conditions (Druehl and Kemp 1982; Koehl et al. 2008). For kelp, two of the most important environmental constraints on blade development are nutrient flux and hydrodynamic drag. Blades must acquire nutrients directly from the surrounding water. Furthermore, in the coastal ocean, kelp blades can be exposed to strong currents and waves that can break blades or dislodge entire fronds from the substrate. For this reason, documented observations of kelp blade morphology are traditionally grouped into two categories: ‘‘exposed’’ and ‘‘sheltered,’’ which describe the blades’ general exposure to rapid flow and high drag forces from waves and currents, and to slow flow and reduced drag forces, respectively (Koehl et al. 2008). In sheltered environments, some blades have undulations, or ruffles, along the flow-parallel edges (Koehl and Alberte 1988; Koehl et al. 2008), and this morphology, which can promote flapping, has been suggested to increase flux at the blade surface at the expense of also increasing drag forces (Koehl and Alberte 1988). In sheltered environments, however, drag forces are generally lower and may not be an important factor. In the same study, the morphologies observed at exposed sites (e.g., thicker, narrower, more strap-like blades) exhibited both less flapping and lower drag forces per unit blade area. Gerard and Mann (1979) also observed clear differences in blade thickness between flow-exposed and flow-sheltered sites in Laminaria longicruris. The thickness of the flow-exposed blades was more than three times the thickness of the flowsheltered blades. The trend of thicker blades at exposed sites has also been documented in the following macroalgae genera: Agarum (Duggins et al. 2003), Durvillaea (Cheshire and Hallam 1989), Ecklonia (Wernberg and Thomsen 2005), Eisenia (Roberson and Coyer 2004), Gigartina (Jackelman and Bolton 1990), Laminaria (Parke 1948; Sjotun and Frederiksen 1995; Duggins et al. 2003), Macrocystis (Hurd et al. 1996), and Pachydictyon (Haring and Carpenter 2007). Moreover, the same behavior has been observed in seagrass blades of various species (Peralta et al. 2006). Changes in blade thickness (h) can have significant effects on the mechanical properties of the blade. The blade stiffness in tension is directly proportional to the blade cross-sectional area, and thus is linearly proportional to blade thickness. However, for a blade with a nominally rectangular cross section, the bending rigidity is proportional to the cube of the blade thickness. Thus, changes in thickness have a more pronounced effect on the structure’s resistance to bending than its strength under tension. The blades of Macrocystis not only have an increased thickness at exposed sites, but they also exhibit longitudinal corrugations (Fig. 1a). These corrugations are fundamentally distinct from the ruffles or undulations observed in species at sheltered sites. The peaks and troughs of the corrugations run parallel to the edges of the blades, whereas undulations or ruffles are oriented perpendicularly to the blade edges (see fig. 1b in Koehl et al. 2008). At flowsheltered sites, these corrugations are either entirely absent (Hurd and Pilditch 2011) or greatly reduced in amplitude (Hurd et al. 1996). When present, corrugations are regular in amplitude and wavelength, and generally extend almost the full blade length, but at points along the blade individual peaks end, split, or merge with other peaks (Fig. 1b). From previous studies of corrugated materials (Lau 1981; Briassoulis 1986), we propose that the primary function of blade corrugations is to increase blade flexural rigidity. This idea is explored in detail later in the paper. We anticipate from previous studies of flapping flags (Michelin and Smith 2009) that changes in rigidity will affect the degree of blade flapping, which in turn may affect the mean and peak drag on the blade. In addition, based on * Corresponding author: [email protected] Limnol. Oceanogr., 59(6), 2014, 2028–2041 E 2014, by the Association for the Sciences of Limnology and Oceanography, Inc. doi:10.4319/lo.2014.59.6.2028