2 results
The kinematic genesis of vortex formation due to finite rotation of a plate in still fluid
- M. Jimreeves David, Manikandan Mathur, R. N. Govardhan, J. H. Arakeri
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- Journal:
- Journal of Fluid Mechanics / Volume 839 / 25 March 2018
- Published online by Cambridge University Press:
- 02 February 2018, pp. 489-524
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We present a combined experimental and numerical study of an idealized model of the propulsive stroke of the turning manoeuvre in fish. Specifically, we use the framework of Lagrangian coherent structures (LCSs) to describe the kinematics of the flow that results from a thin plate performing a large angle rotation about its tip in still fluid. Temporally and spatially well-resolved velocity fields are obtained using a two-dimensional, incompressible finite-volume solver, and are validated by comparisons with experimentally measured velocity fields and alternate numerical simulations. We then implement the recently proposed variational theory of LCSs to extract the hyperbolic and elliptic LCSs in the numerically generated velocity fields. Detailed LCS analysis is performed for a plate motion profile described by $\dot{\unicode[STIX]{x1D703}}(t)=\unicode[STIX]{x1D6FA}_{max}\sin ^{2}(\unicode[STIX]{x1D714}t)$ during $0\leqslant t\leqslant t_{o}$ and zero otherwise. The stopping time $t_{o}$ is given by $t_{o}=\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714}=10~\text{s}$, the value of $\unicode[STIX]{x1D6FA}_{max}$ chosen to give a stopping angle of $\unicode[STIX]{x1D703}_{max}=90^{\circ }$, resulting in a Reynolds number $Re=c^{2}\unicode[STIX]{x1D6FA}_{max}/\unicode[STIX]{x1D708}=785.4$, where $c$ is the plate chord length and $\unicode[STIX]{x1D708}=10^{-6}~\text{m}^{2}~\text{s}^{-1}$ the kinematic viscosity of water. The flow comprises a starting and a stopping vortex, resulting in a pair of oppositely signed vortices of unequal strengths that move away from the plate in a direction closely aligned with the final plate orientation at $t/t_{o}\approx 2$. The hyperbolic LCSs are shown to encompass the fluid material that is advected away from the plate for $t>t_{o}$, henceforth referred to as the advected bulk. The starting and stopping vortices, identified using elliptic LCSs and hence more objective than Eulerian vortex detection methods, constitute only around two thirds of the advected bulk area. The advected bulk is traced back to $t=0$ to identify five distinct lobes of fluid that eventually form the advected bulk, and hence map the long-term fate of various regions in the fluid at $t=0$. The five different lobes of fluid are then shown to be delineated by repelling LCS boundaries at $t=0$. The linear momentum of the advected bulk region is shown to account for approximately half of the total impulse experienced by the plate in the direction of its final orientation, thus establishing its dynamical significance. We provide direct experimental evidence for the kinematic relevance of hyperbolic and elliptic LCSs using novel dye visualization experiments, and also show that attracting hyperbolic LCSs provide objective characterization of the spiral structures often observed in vortical flows. We conclude by showing that qualitatively similar LCSs persist for several other plate motion profiles and stopping angles as well.
Thrust generation from pitching foils with flexible trailing edge flaps
- M. Jimreeves David, R. N. Govardhan, J. H. Arakeri
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- Journal:
- Journal of Fluid Mechanics / Volume 828 / 10 October 2017
- Published online by Cambridge University Press:
- 31 August 2017, pp. 70-103
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In the present experimental study, we investigate thrust production from a pitching flexible foil in a uniform flow. The flexible foils studied comprise a rigid foil in the front (chord length $c_{R}$) that is pitched sinusoidally at a frequency $f$, with a flexible flap of length $c_{F}$ and flexural rigidity $EI$ attached to its trailing edge. We investigate thrust generation for a range of flexural rigidities ($EI$) and flap length to total chord ratio ($c_{F}/c$), with the mean thrust ($\overline{C_{T}}$) and the efficiency of thrust generation ($\unicode[STIX]{x1D702}$) being directly measured in each case. The thrust in the rigid foil cases, as expected, is found to be primarily due to the normal force on the rigid foil ($\overline{C_{TN}}$) with the chordwise or axial thrust contribution ($\overline{C_{TA}}$) being small and negative. In contrast, in the flexible foil cases, the axial contribution to thrust becomes important. We find that using a non-dimensional flexural rigidity parameter ($R^{\ast }$) defined as $R^{\ast }=EI/(0.5\unicode[STIX]{x1D70C}U^{2}c_{F}^{3})$ appears to combine the independent effects of variations in $EI$ and $c_{F}/c$ at a given value of the reduced frequency ($k=\unicode[STIX]{x03C0}fc/U$) for the range of $c_{F}/c$ values studied here ($U$ is free-stream velocity; $\unicode[STIX]{x1D70C}$ is fluid density). At $k\approx 6$, the peak mean thrust coefficient is found to be about 100 % higher than the rigid foil thrust, and occurs at $R^{\ast }$ value of approximately 8, while the peak efficiency is found to be approximately 300 % higher than the rigid foil efficiency and occurs at a distinctly different $R^{\ast }$ value of close to 0.01. Corresponding to these two optimal flexural rigidity parameter values, we find two distinct flap deflection shapes; the peak thrust corresponding to a mode 1 type simple bending of the flap with no inflection points, while the peak efficiency corresponds to a distinctly different deflection profile having an inflection point along the flap. The peak thrust condition is found to be close to the ‘resonance’ condition for the first mode natural frequency of the flexible flap in still water. In both these optimal cases, we find that it is the axial contribution to thrust that dominates ($\overline{C_{TA}}\gg \overline{C_{TN}}$), in contrast to the rigid foil case. Particle image velocimetry (PIV) measurements for the flexible cases show significant differences in the strength and arrangement of the wake vortices in these two cases.