Thrust and drag scaling of a rigid low-aspect-ratio pitching plate
Introduction
There has been an considerable amount of studies on the energetics of swimming over the past decades, parts of which having recently been reviewed in Godoy-Diana and Thiria (2018). It appears that Lighthill’s celebrated elongated-body theory (Lighthill, 1960) remains the key model for thrust prediction when addressing swimming or flying bodies, or when considering artificial systems for locomotion like oscillating foils, with relatively low aspect ratios (span to chord). The pressure field over the moving body or object characterizes the inertial fluid effects and for oscillatory motions, the tail-beat frequency as well as its amplitude has been widely used to characterize aquatic locomotion (see for instance Triantafyllou et al., 2000, Anderson et al., 1998, Saadat et al., 2017 and references therein). For a freely swimming body, the thrust force is balanced by the resistive drag. Form drag depends on the swimming body’s shape and at the same time the body’s surface induces significant viscous forces, unless the Reynolds number range is sufficiently high such that viscous forces can be neglected.
The importance of viscous drag has been a matter of discussion and it has been recognized that the motion of finite-aspect swimming bodies or objects may induce a drag increase, due to what is known as the “Bone–Lighthill boundary-layer thinning hypothesis” (Lighthill, 1971). This hypothesis has recently been readdressed for flapping plates and a longitudinal drag formula depending on the plate’s aspect ratio, the wall-normal velocity induced by the plate’s movement and of course the Reynolds number has been proposed (Ehrenstein and Eloy, 2013, Ehrenstein et al., 2014). This friction law and in particular the finite-size scaling with the span of the foil is retrieved when modeling an actuated elastic swimmer (Piñeirua et al., 2017). Some discussion on the interaction between the surface of the body of swimming fishes and the induced boundary-layer flow is provided in Lauder et al. (2016). Other observations however, for instance the measurements for the boundary layer on the body surface of trout swimming at high turbulent Reynolds numbers (Yanase and Saarenrinne, 2015), do not support the boundary-layer thinning hypothesis, which is attributed to an energy-efficient swimming strategy in a turbulent environment.
The influence of the swimming object’s aspect ratio on the inertial pressure force is still not entirely elucidated. Scaling laws for propulsion for archetype geometries and motions, such as heaving, pitching or undulatory foils, often consider added-mass forces per unit span, which apply to rather large aspect ratio geometries, assuming a quasi two-dimensional setting along the foil’s centerline. On the contrary, when addressing elongated bodies, the reactive term during the swimming motion is known to be proportional to the (small) width of the body. Reliable scaling laws have for instance been reported in Floryan et al. (2017), for a large aspect ratio (plate’s span to plate’s length ) of . The thrust-performance and the wake structure for rigid rectangular pitching plates have been reported in Buchholz and Smits (2008), providing evidence for quasi two-dimensional structures when , whereas the aspect ratio affects the propulsive performance for narrower plates. Whatever aspect ratio is considered for three-dimensional oscillating foils or bodies in longitudinal motion, flow structures will evolve through the span, the manifestation being the generation of a pair of counter-rotating streamwise vortices at the lateral edges. These kind of structures are known as trailing vortices in wing theory, being responsible for what is often called induced drag. A vortex-induced drag model taking into account these streamwise vortex structures in undulatory swimming has been proposed in Raspa et al. (2014). This model based on the vortex circulation is similar to the vortex drag analysis in Aider et al. (2010). Quite interesting, in this latter investigation it is mentioned, that an alternative interpretation would be to consider the pressure deficit due to the high transverse velocity between the vortex cores as responsible for the drag.
The aim of the present work is to characterize the influence of finite-size effects on the forces and drag for an oscillating archetype geometry, by computing the three-dimensional flow field induced by the motion. The numerical investigation is performed for a rigid pitching plate, considering different aspect ratios in the range of . The plate has vanishing thickness in this numerical solution procedure and hence form drag due to body shape is absent. The flow structure along the plate as well as in its the very vicinity is numerically captured and the instantaneous as well as time-averaged propulsive and resistive forces can reliably be computed. The paper is organized as follows. In Section 2, the numerical solution procedure is briefly outlined and the flow configuration and pitching parameters are addressed in Section 3, together with some illustration of the three-dimensional flow structure. The forces and drag analysis is provided in Section 4. The validity of the finite-aspect ratio viscous drag formula, derived for uniform motions, is examined for the pitching motion. The pressure force across the plate’s span is analyzed and a scaling is derived for the pressure deficit. Finally, a scaling, function of the aspect ratio , is proposed for the pressure force, which takes into account the pressure deficit associated with the transverse flow. Some final discussion of the results is provides in Section 5.
Section snippets
Numerical solution procedure
A multi-domain approach has been used for the solution of the Navier–Stokes system for the velocity field and the pressure in the presence of the pitching plate. This approach has been already used for flapping plate computations in Ehrenstein et al. (2014). Also, the numerical approach, which will be briefly outlined hereafter, has been validated in Moubogha Moubogha et al. (2017) through comparisons with experimental measurements, for a pitching plate
Flow configuration and pitching parameters
All the following computations have been performed for a Reynolds number , the incoming uniform flow velocity, and the plate’s length being the reference velocity and the reference length, respectively. The motion of the pitching plate with vanishing thickness is given by (4) with the pitching function (5) and for convenience, we set in the following and hence the trailing edge . It is recalled that the pitch-pivot point is at the distance from the leading edge,
Forces and drag induced by the pitching plate
Before addressing the forces and drag computations, the main parameters which enter into the analysis are briefly summarized and a brief nomenclature of the drag force quantities is provided.
As mentioned before, the aspect ratio and the Strouhal number are In the forthcoming analysis, the dimensionless time-averaged wall-normal velocity is one of the key quantities, that is according to (12)) (taking ) where
Concluding discussion
The precise force balance for a pitching plate has been extracted from three-dimensional numerical simulation results, for plates with different and relatively small aspect ratios (span to length) and for different Strouhal number (with the frequency, the trailing edge peak-to-peak amplitude and the incoming flow velocity). The Strouhal numbers considered are in the range which for instance is typical for a large number of fish (Eloy, 2012).
A theoretical skin
Acknowledgment
This work was granted access to the HPC resources of IDRIS-France under the allocation A0042A01741 made by GENCI (Grand Equipement National de Calcul Intensif).
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