Effects of inertia on the time-averaged propulsive performance of a pitching and heaving foil

The inertia may play an important role in the unsteady behavior of a flapping foil. Of particular relevance in forward flapping locomotion is how the foil inertia may affect its time-averaged propulsive performance. This is the question addressed by the present study from a general, nonlinear formulation of the unsteady interaction with the surrounding fluid of a thin, flexible, two-dimensional and non-uniform foil undergoing prescribed pitching and heaving motion of any amplitude about an arbitrary pivot axis. For a rigid foil it is shown that, although the unsteady inertial forces and moments may be much larger than the unsteady aerodynamic forces and moments if the fluid density is much smaller than the foil density, the inertia does not affect the cycle-averaged propulsive performance for harmonic pitching and heaving motion, independently of their amplitude. Inertia may affect only the cycle-averaged moment if the mean angle of attack is not zero, but without affecting the cycle-averaged power input, and therefore the propulsive efficiency. When a small flexural deflection of the foil is considered, although the cycle-averaged inertial thrust and lift also vanish for any amplitude of the pitching and heaving harmonic motion, the cycle-averaged power input does not. Thus, the foil inertia contributes through the moment and power to the timeaveraged propulsive performance for any flexural deflection of the foil, obtained her e analytically in terms of the trailing edge location for general pitching and heaving motion of any amplitude and phase and small flexural deflection amplitude. Simple analytical results are also provided for inertia dominated deflection, characterizing the conditions that minimize the power consumption. The results are valid for arbitrary chord-wise distribution of mass, thickness and (sufficiently large) stiffness of the foil.This research has been supported by the Junta de Andalucía, Spain through grant UMA18-FEDER-JA-047 Funding for open access charge: Universidad de Málaga / CBU

On the feasibility of a flexible foil with passive heave to extract energy from low wind speeds

We explore numerically and theoretically the capability of flexible foils elastically mounted to translational springs and dampers at the leading edge to extract energy from low-speed winds through its passive heave motion. Given the spring and foil stiffnesses, for each damper constant the theory (which is valid for high Reynolds numbers and small foil deflection amplitudes, i.e., in absence of separation) provides analytically a minimum wind velocity for flutter instability, above which energy can be harvested, that depends on the thickness-to-chord-length ratio of the foil. Simple analytical expressions for the flutter frequency are also provided. Minimum wind speeds and corresponding flutter frequencies are characterized for a carbon fiber foil as the spring stiffness and damper constant are varied, finding that energy can be extracted from wind speeds lower than in conventional wind turbines. These theoretical predictions are assessed from full numerical simulations at Reynolds numbers corresponding to these wind velocities and for chord lengths of the order of the meter (i.e. about 106 ) using appropriate turbulence models, which allow to compute the power extracted from the wind that the flutter stability analysis cannot provid

Flutter stability analysis of an elastically supported flexible foil. Application to the energy harvesting of a fully-passive flexible flapping-foil of small amplitude

The aerodynamic forces on an oscillating flexible foil are used to study the flutter instability when the flexible foil is elastically mounted to translational and torsional springs and dampers at an arbitrary pivot axis. The present linear theory, valid for small amplitudes of the heaving, pitching and flexural deflection motions, and therefore valid for sufficiently large stiffness ratios, characterizes analytically the onset of the flutter instability and the corresponding leading frequency in terms of the flow velocity and all the structural parameters of the system. The analysis may serve to guide the search for the parametric ranges of energy extraction by a fully-passive flexible flapping-foil hydrokinetic turbine, including the effect of some relevant nondimensional parameters which have not been considered before. The results for the rigid-foil case are validated with recent numerical simulations for a fully-passive flapping-foil turbine. As the stiffness of the foil decreases, the coupled-mode flutter instability of the elastically supported rigid foil may weaken and disappear, or become enhanced, depending on the remaining parameters, most particularly on the location of the centre of mass in relation to the pivot point, whose dependence is investigated for specific values of the rest of the nondimensional parameters.This research has been financed by the Ministerio de Ciencia e Innovación of Spain (PID2019-104938RB-I00). Funding for open access charge: Universidad de Málaga / CBU

Force and torque reactions on a pitching flexible aerofoil

Experimental measurements in a wind tunnel of the unsteady force and moment that a fluid exerts on flexible flapping aerofoils are not trivial because the forces and moments caused by the aerofoil's inertia and others structural tensions at the pivot axis have to be obtained separately and subtracted from the direct measurements with a force/torque sensor. Here we derive from the nonlinear beam equation general relations for the force and torque reactions at the leading edge of a pitching aerofoil in terms of the fluid force and moment on the aerofoil and its kinematics, involving geometric and structural parameters of the flexible aerofoil. These relations are validated by comparing high-resolution numerical simulations of the flow–structure interaction of a two-dimensional flexible aerofoil pitching about its leading edge with direct force and torque measurements in a wind tunnel.This research has been supported by the Junta de Andalucía, Spain (UMA18-FEDER-JA-047 and P18-FR-1532). Funding for open access charge: Universidad de Málaga

Nonlocal electron heat flux revisited

A known nonlocal model of electron heat flux, applying for (scale length/thermal ion-electron mean-free path) of order Z)1/2(e*/T)312, ionization number Z, large, and e*~ 6.5 T (the energy of electrons carrying most of the flux), is reconsidered. The large e*/T ratio simplifies the complete formalism. A simple flux formula, exact for both smooth and steep profiles, is given. Thermoelectric effects and other models are discussed

Nonlocal electron heat relaxation in a plasma shock at arbitrary ionization number

A recently obtained nonlocal expression for the electron heat flux valid for arbitrary ionization numbers Z is used to study the structure of a plane shock wave in a fully ionized plasma. Nonlocal effects are only important in the foot of the electronic preheating region, where the electron temperature gradient is the steepest. The results are quantified as a function of a characteristic Knudsen number of that region. This work also generalizes to arbitrary values of Z previous results on plasma shock wave structure

Unsteady Propulsion of a Two-Dimensional Flapping Thin Airfoil in a Pulsating Stream.

The cruising velocity of animals, or robotic vehicles, that use flapping wings or fins to propel themselves is not constant but oscillates around a mean value with an amplitude usually much smaller than the mean, and a frequency that typically doubles the flapping frequency. Quantifying the effect that these velocity fluctuations may have on the propulsion of a flapping and oscillating airfoil is of great relevance to properly modeling the self-propelled performance of these animals or robotic vehicles. This is the objective of the present work, where the force and moment that an oscillating stream exerts on a two-dimensional pitching and heaving airfoil are obtained analytically using the vortical impulse theory in the linear potential flow limit. The thrust force of the flapping airfoil in a pulsating stream in this limit is obtained here for the first time. The lift force and moment derived here contain new terms in relation to the pioneering work by Greenberg (1947), which are shown quantitatively unimportant. The theoretical results obtained here are compared with existing computational data for flapping foils immersed in a stream with velocity oscillating sinusoidally about a mean value.The authors acknowledge support from the Advanced Grant of the European Research Council GRIFFIN, Action 788247, and from the Junta de Andalucía, Spain, Grant UMA18-FEDER-JA-047. Ernesto Sanchez-Laulhe also acknowledges his predoctoral contract at the University of Malaga

Efficient self-propelled locomotion by an elastically supported rigid foil actuated by a torque

A new theoretical model is presented for an aquatic vehicle self-propelled by a rigid foil undergoing pitching oscillations generated by a torque of small amplitude applied at an arbitrary pivot axis at which the foil is elastically supported to allow for passive heaving motion. The model is based on 2D linear potential-flow theory coupled with the self-propelled dynamics of the semi-passive flapping foil elastically mounted on the vehicle hull through translational and torsional springs and dampers. It is governed by just three ordinary differential equations, whose numerical solutions are assessed with full viscous numerical simulations of the self-propelled foil. Analytical approximate solutions for the combined effect of all the relevant non-dimensional parameters on the swimming velocity and efficiency are also obtained by taking advantage of the small-amplitude of the applied torque. Thus, simple power laws for the velocity and efficiency dependencies on Lighthill number and torque intensity are obtained. It is found that the swimming velocity and transport efficiency can be greatly enhanced by selecting appropriately the non-dimensional constants of the translational and torsional springs, which are mapped for typical values of the remaining parameters in aquatic locomotion. These resonant values serve to select optimal frequencies of the forcing torque for given structural and geometric parameters. Thus, the present model and analysis provide a useful guide for the design of an efficient flapping-foil underwater vehicle.This research has been supported by the Junta de Andalucía, Spain, through the project grants UMA18-FEDER-JA-047 and P18-FR-1532. The computations were performed in the Picasso Supercomputer at the University of Málaga, a node of the Spanish Supercomputing Network. // Funding for open access charge: Universidad de Málaga / CBU

Analytical results for the propulsion performance of a flexible foil with prescribed pitching and heaving motions and passive small deflection

The propulsive performance of a flexible foil with prescribed pitching and heaving motions about any pivot point location and passive chordwise flexural deflection is analysed within the framework of the linear potential flow theory and the Euler–Bernoulli beam equation using a quartic approximation for the deflection. The amplitude of the flexural component of the deflection and its phase, the thrust force, input power and propulsive efficiency are computed analytically in terms of the stiffness and mass ratio of the plate, frequency, pivot point location and remaining kinematic parameters. It is found that the maximum flexural deflection amplitude, thrust and input power are related to the first fluid–structure natural frequency of the system, corresponding to the deflection approximation considered. The same relation is observed for the propulsive efficiency when an offset drag is included in the analytical expressions. These results, which are valid for small amplitude and sufficiently large stiffness of the foil, are compared favourably with previous related results when the foil pivots about the leading edge. The configurations generating maximum thrust and efficiency enhancement by flexibility are analysed in relation to those of an otherwise identical rigid foil.Funding for open access charge: Universidad de Málaga. This research has been supported by grants from the Ministerio de Ciencia e Innovación of Spain (DPI2016-76151-C2-1-R and PID2019-104938RB-I00) and from the Junta de Andalucía, Spain (UMA18-FEDER-JA-047 and P18-FR-1532)