4 results
Contrasting thrust generation mechanics and energetics of flapping foil locomotory states characterized by a unified $St$-$Re$ scaling
- Anil Das, Ratnesh K. Shukla, Raghuraman N. Govardhan
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- Journal:
- Journal of Fluid Mechanics / Volume 930 / 10 January 2022
- Published online by Cambridge University Press:
- 11 November 2021, A27
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Self-propelled flapping foils with distinct locomotion-enabling kinematic restraints exhibit a remarkably similar Strouhal number ($St$)-Reynolds number ($Re$) dependence. This similarity has been hypothesized to pervade diverse forms of oscillatory self-propulsion and undulatory biolocomotion; however, its genesis and implications on the energetic cost of locomotion remain elusive. Here, using high-resolution simulations of translationally free and restrained foils that self-propel as they are pitched, we demonstrate that a generality in the $St$-$Re$ relationship can emerge despite significant disparities in thrust generation mechanics and locomotory performance. Specifically, owing to a recoil reaction induced passive heave, the fluid's inertial response to the prescribed rotational pitch, the principal source of thrust in unidirectionally free and towed configurations, ceases to produce thrust in a bidirectionally free configuration. Rather, the thrust generated from the leading edge suction mechanics self-propels a bidirectionally free pitching foil. Owing to the foregoing distinction in the thrust generation mechanics, the $St$-$Re$ relationships for the bidirectionally and unidirectionally free/towed foils are dissimilar and pitching amplitude dependent, but specifically for large reduced frequencies, converge to a previously reported unified power law. Importantly, to propel at a given mean forward speed, the bidirectionally free foil must counteract the out-of-phase passive heave through a more intense rotational pitch, resulting in an appreciably higher power consumption over the range $10 \leq Re \leq 10^3$. We highlight the critical role of thrust in introducing an offset in the $St$-$Re$ relation, and through its amplification, being ultimately responsible for the considerable disparity in the locomotory performance of differentially constrained foils.
An asymptotic theory for the high-Reynolds-number flow past a shear-free circular cylinder
- Anuj Kumar, Nidhil M.A. Rehman, Pritam Giri, Ratnesh K. Shukla
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- Journal:
- Journal of Fluid Mechanics / Volume 920 / 10 August 2021
- Published online by Cambridge University Press:
- 16 June 2021, A44
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We present an asymptotic theory for analytical characterization of the high-Reynolds-number incompressible flow of a Newtonian fluid past a shear-free circular cylinder. The viscosity-induced modifications to this flow are localized and except in the neighbourhood of the rear stagnation point, behave like a linear perturbation of the inviscid flow. Our theory gives a highly accurate description of these modifications by including the contribution from the most significant viscous term in a correctional perturbation expansion about an inviscid base state. We derive the boundary layer equation for the flow and deduce a similarity transformation that leads to a set of infinite, shear-free-condition-incompatible, self-similar solutions. By suitably combining members from this set, we construct an all-boundary-condition-compatible solution to the boundary layer equation. We derive the governing equation for vorticity transport through the narrow wake region and determine its closed-form solution. The near and far-field forms of our wake solution are desirably consistent with the boundary layer solution and the well-known, self-similar planar wake solution, respectively. We analyse the flow in the rear stagnation region by formulating an elliptic partial integro-differential equation for the distortion streamfunction that specifically accounts for the fully nonlinear and inviscid dynamics of the viscous correctional terms. The drag force and its atypical logarithmic dependence on Reynolds number, deduced from our matched asymptotic analysis, are in remarkable agreement with the high-resolution simulation results. The logarithmic dependence gives rise to a critical Reynolds number below which the viscous correction term, counterintuitively, reduces the net dissipation in the flow field.
Existence of a sharp transition in the peak propulsive efficiency of a low-$Re$ pitching foil
- Anil Das, Ratnesh K. Shukla, Raghuraman N. Govardhan
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- Journal:
- Journal of Fluid Mechanics / Volume 800 / 10 August 2016
- Published online by Cambridge University Press:
- 07 July 2016, pp. 307-326
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We perform a comprehensive characterization of the propulsive performance of a thrust generating pitching foil over a wide range of Reynolds ($10\leqslant Re\leqslant 2000$) and Strouhal ($St$) numbers using a high-resolution viscous vortex particle method. For a given $Re$, we show that the mean thrust coefficient $\overline{C_{T}}$ increases monotonically with $St$, exhibiting a sharp rise as the location of the inception of the wake asymmetry shifts towards the trailing edge. As a result, the propulsive efficiency too rises steeply before attaining a maximum and eventually declining at an asymptotic rate that is consistent with the inertial scalings of $St^{2}$ for $\overline{C_{T}}$ and $St^{3}$ for the mean power coefficient, with the latter scaling holding, quite remarkably, over the entire range of $Re$. We find the existence of a sharp increase in the peak propulsive efficiency ${\it\eta}_{max}$ (at a given $Re$) in the $Re$ range of 50 to approximately 1000, with ${\it\eta}_{max}$ increasing rapidly from about 1.7 % to the saturated asymptotic value of approximately $16\,\%$. The $St$ at which ${\it\eta}_{max}$ is attained decreases progressively with $Re$ towards an asymptotic limit of $0.45$ and always exceeds the one for transition from a reverse von Kármán to a deflected wake. Moreover, the drag-to-thrust transition occurs at a Strouhal number $St_{tr}$ that exceeds the one for von Kármán to reverse von Kármán transition. The $St_{tr}$ and the corresponding power coefficient $\overline{C_{p,}}_{tr}$ are found to be remarkably consistent with the simple scaling relationships $St_{tr}\sim Re^{-0.37}$ and $\overline{C_{p,}}_{tr}\sim Re^{-1.12}$ that are derived from a balance of the thrust generated from the pitching motion and the drag force arising out of viscous resistance to the foil motion. The fact that the peak propulsive efficiency degrades appreciably only below $Re\approx 10^{3}$ establishes a sharp lower threshold for energetically efficient thrust generation from a pitching foil. Our findings should be generalizable to other thrust-producing flapping foil configurations and should aid in establishing the link between wake patterns and energetic cost of thrust production in similar systems.
Minimum power consumption for drag reduction on a circular cylinder by tangential surface motion
- Ratnesh K. Shukla, Jaywant H. Arakeri
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- Journal:
- Journal of Fluid Mechanics / Volume 715 / 25 January 2013
- Published online by Cambridge University Press:
- 09 January 2013, pp. 597-641
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We investigate the effect of a prescribed tangential velocity on the drag force on a circular cylinder in a spanwise uniform cross flow. Using a combination of theoretical and numerical techniques we make an attempt at determining the optimal tangential velocity profiles which will reduce the drag force acting on the cylindrical body while minimizing the net power consumption characterized through a non-dimensional power loss coefficient (${C}_{\mathit{PL}} $). A striking conclusion of our analysis is that the tangential velocity associated with the potential flow, which completely suppresses the drag force, is not optimal for both small and large, but finite Reynolds number. When inertial effects are negligible ($\mathit{Re}\ll 1$), theoretical analysis based on two-dimensional Oseen equations gives us the optimal tangential velocity profile which leads to energetically efficient drag reduction. Furthermore, in the limit of zero Reynolds number ($\mathit{Re}\ensuremath{\rightarrow} 0$), minimum power loss is achieved for a tangential velocity profile corresponding to a shear-free perfect slip boundary. At finite $\mathit{Re}$, results from numerical simulations indicate that perfect slip is not optimum and a further reduction in drag can be achieved for reduced power consumption. A gradual increase in the strength of a tangential velocity which involves only the first reflectionally symmetric mode leads to a monotonic reduction in drag and eventual thrust production. Simulations reveal the existence of an optimal strength for which the power consumption attains a minima. At a Reynolds number of 100, minimum value of the power loss coefficient (${C}_{\mathit{PL}} = 0. 37$) is obtained when the maximum in tangential surface velocity is about one and a half times the free stream uniform velocity corresponding to a percentage drag reduction of approximately 77 %; ${C}_{\mathit{PL}} = 0. 42$ and $0. 50$ for perfect slip and potential flow cases, respectively. Our results suggest that potential flow tangential velocity enables energetically efficient propulsion at all Reynolds numbers but optimal drag reduction only for $\mathit{Re}\ensuremath{\rightarrow} \infty $. The two-dimensional strategy of reducing drag while minimizing net power consumption is shown to be effective in three dimensions via numerical simulation of flow past an infinite circular cylinder at a Reynolds number of 300. Finally a strategy of reducing drag, suitable for practical implementation and amenable to experimental testing, through piecewise constant tangential velocities distributed along the cylinder periphery is proposed and analysed.