Added mass and wave radiation damping for flow-induced rotational vibrations of skinplates of hydraulic gates

One observed vibration mode for Tainter gate skinplates involves the bending of the skinplate about a horizontal nodal line. This vibration mode can be approximated as a streamwise rotational vibration about the horizontal nodal line. Such a streamwise rotational vibration of a Tainter gate skinplate must push away water from the portion of the skinplate rotating into the reservoir and draw water toward the gate over that portion of the skinplate receding from the reservoir. The induced pressure is termed the push-and-draw pressure. In the present paper, this push-and-draw pressure is analyzed using the potential theory developed for dissipative wave radiation problems. In the initial analysis, the usual circular-arc skinplate is replaced by a vertical, flat, rigid weir plate so that theoretical calculations can be undertaken. The theoretical push-and-draw pressure is used in the derivation of the non-dimensional equation of motion of the flow-induced rotational vibrations. Non-dimensionalization of the equation of motion permits the identification of the dimensionless equivalent added mass and the wave radiation damping coefficients. Free vibration tests of a vertical, flat, rigid weir plate model, both in air and in water, were performed to measure the equivalent added mass and the wave radiation damping coefficients. Experimental results compared favorably with the theoretical predictions, thus validating the theoretical analysis of the equivalent added mass and wave radiation damping coefficients as a prediction tool for flow-induced vibrations. Subsequently, the equation of motion of an inclined circular-arc skinplate was developed by incorporating a pressure correction coefficient, which permits empirical adaptation of the results from the hydrodynamic pressure analysis of the vertical, flat, rigid weir plate. Results from in-water free vibration tests on a 1/31-scale skinplate model of the Folsom Dam Tainter gate are used to demonstrate the utility of the equivalent added mass coefficient.

Finite Element Modeling of Shell Shape in the Freshwater Turtle Pseudemys concinna Reveals a Trade-Off between Mechanical Strength and Hydrodynamic Efficiency

Aquatic species can experience different selective pressures on morphology in different flow regimes. Species inhabiting lotic regimes often adapt to these conditions by evolving low-drag (i.e., streamlined) morphologies that reduce the likelihood of dislodgment or displacement. However, hydrodynamic factors are not the only selective pressures influencing organismal morphology and shapes well suited to flow conditions may compromise performance in other roles. We investigated the possibility of morphological trade-offs in the turtle Pseudemys concinna. Individuals living in lotic environments have flatter, more streamlined shells than those living in lentic environments; however, this flatter shape may also make the shells less capable of resisting predator-induced loads. We tested the idea that ‘‘lotic’’ shell shapes are weaker than ‘‘lentic’’ shell shapes, concomitantly examining effects of sex. Geometric morphometric data were used to transform an existing finite element shell model into a series of models corresponding to the shapes of individual turtles. Models were assigned identical material properties and loaded under identical conditions, and the stresses produced by a series of eight loads were extracted to describe the strength of the shells. ‘‘Lotic’’ shell shapes produced significantly higher stresses than ‘‘lentic’’ shell shapes, indicating that the former is weaker than the latter. Females had significantly stronger shell shapes than males, although these differences were less consistent than differences between flow regimes. We conclude that, despite the potential for many-to-one mapping of shell shape onto strength, P. concinna experiences a trade-off in shell shape between hydrodynamic and mechanical performance. This trade-off may be evident in many other turtle species or any other aquatic species that also depend on a shell for defense. However, evolution of body size may provide an avenue of escape from this trade-off in some cases, as changes in size can drastically affect mechanical performance while having little effect on hydrodynamic performance.