Finite-horizon, energy-efficient trajectories in unsteady flows

Intelligent mobile sensors, such as uninhabited aerial or underwater vehicles, are becoming prevalent in environmental sensing and monitoring applications. These active sensing platforms operate in unsteady fluid flows, including windy urban environments, hurricanes and ocean currents. Often constrained in their actuation capabilities, the dynamics of these mobile sensors depend strongly on the background flow, making their deployment and control particularly challenging. Therefore, efficient trajectory planning with partial knowledge about the background flow is essential for teams of mobile sensors to adaptively sense and monitor their environments. In this work, we investigate the use of finite-horizon model predictive control (MPC) for the energy-efficient trajectory planning of an active mobile sensor in an unsteady fluid flow field. We uncover connections between trajectories optimized over a finite-time horizon and finite-time Lyapunov exponents of the background flow, confirming that energy-efficient trajectories exploit invariant coherent structures in the flow. We demonstrate our findings on the unsteady double gyre vector field, which is a canonical model for chaotic mixing in the ocean. We present an exhaustive search through critical MPC parameters including the prediction horizon, maximum sensor actuation, and relative penalty on the accumulated state error and actuation effort. We find that even relatively short prediction horizons can often yield energy-efficient trajectories. We also explore these connections on a three-dimensional flow and ocean flow data from the Gulf of Mexico. These results are promising for the adaptive planning of energy-efficient trajectories for swarms of mobile sensors in distributed sensing and monitoring.

Application of Computer Graphics Flow Visualization Methods in Vortex Rope Investigations

Computer graphics visualization techniques for application on data from Computational Fluid Dynamics (CFD) simulations of the vortex rope, a phenomenon present in hydraulic turbines operating in off-design conditions, were devised. This included not only objects for visualization (what to visualize) but also methods of the visualization itself (how to do it). By means of advanced methods based particularly on volume rendering of Eulerian fields in combination with Lagrangian objects, various phenomena were captured, such as the motion of the vortex rope or the backflow zone. The data came from simulations using a scale-resolving hybrid turbulence model, the Stress-Blended Eddy Simulation. In such detailed simulations and other applications involving complex three-dimensional structures, proper visualization methods are needed to leverage the content captured in the resultant data.

Quantitative flow analysis of swimming dynamics with coherent Lagrangian vortices

Undulatory swimmers flex their bodies to displace water, and in turn, the flow feeds back into the dynamics of the swimmer. At moderate Reynolds number, the resulting flow structures are characterized by unsteady separation and alternating vortices in the wake. We use the flow field from simulations of a two-dimensional, incompressible viscous flow of an undulatory, self-propelled swimmer and detect the coherent Lagrangian vortices in the wake to dissect the driving momentum transfer mechanisms. The detected material vortex boundary encloses a Lagrangian control volume that serves to track back the vortex fluid and record its circulation and momentum history. We consider two swimming modes: the C-start escape and steady anguilliform swimming. The backward advection of the coherent Lagrangian vortices elucidates the geometry of the vorticity field and allows for monitoring the gain and decay of circulation and momentum transfer in the flow field. For steady swimming, momentum oscillations of the fish can largely be attributed to the momentum exchange with the vortex fluid. For the C-start, an additionally defined jet fluid region turns out to balance the high momentum change of the fish during the rapid start.

Separability of drag and thrust in undulatory animals and machines

For nearly a century, researchers have tried to understand the swimming of aquatic animals in terms of a balance between the forward thrust from swimming movements and drag on the body. Prior approaches have failed to provide a separation of these two forces for undulatory swimmers such as lamprey and eels, where most parts of the body are simultaneously generating drag and thrust. We nonetheless show that this separation is possible, and delineate its fundamental basis in undulatory swimmers. Our approach unifies a vast diversity of undulatory aquatic animals (anguilliform, sub-carangiform, gymnotiform, bal-istiform, rajiform) and provides design principles for highly agile bioinspired underwater vehicles. This approach has practical utility within biology as well as engineering. It is a predictive tool for use in understanding the role of the mechanics of movement in the evolutionary emergence of morphological features relating to locomotion. For example, we demonstrate that the drag-thrust separation framework helps to predict the observed height of the ribbon fin of electric knifefish, a diverse group of neotropical fish which are an important model system in sensory neurobiology. We also show how drag-thrust separation leads to models that can predict the swimming velocity of an organism or a robotic vehicle.

Flow structure and transport characteristics of feeding and exchange currents generated by upside-down Cassiopea jellyfish

Quantifying the flows generated by the pulsations of jellyfish bells is crucial for understanding the mechanics and efficiency of their swimming and feeding. Recent experimental and theoretical work has focused on the dynamics of vortices in the wakes of swimming jellyfish with relatively simple oral arms and tentacles. The significance of bell pulsations for generating feeding currents through elaborate oral arms and the consequences for particle capture are not as well understood. To isolate the generation of feeding currents from swimming, the pulsing kinematics and fluid flow around the benthic jellyfish Cassiopea spp. were investigated using a combination of videography, digital particle image velocimetry and direct numerical simulation. During the rapid contraction phase of the bell, fluid is pulled into a starting vortex ring that translates through the oral arms with peak velocities that can be of the order of 10 cm s(-1). Strong shear flows are also generated across the top of the oral arms throughout the entire pulse cycle. A coherent train of vortex rings is not observed, unlike in the case of swimming oblate medusae such as Aurelia aurita. The phase-averaged flow generated by bell pulsations is similar to a vertical jet, with induced flow velocities averaged over the cycle of the order of 1-10 mm s(-1). This introduces a strong near-horizontal entrainment of the fluid along the substrate and towards the oral arms. Continual flow along the substrate towards the jellyfish is reproduced by numerical simulations that model the oral arms as a porous Brinkman layer of finite thickness. This two-dimensional numerical model does not, however, capture the far-field flow above the medusa, suggesting that either the three-dimensionality or the complex structure of the oral arms helps to direct flow towards the central axis and up and away from the animal.

A numerical study of the effects of bell pulsation dynamics and oral arms on the exchange currents generated by the upside-down jellyfish Cassiopea xamachana

Mathematical and experimental studies of the flows generated by jellyfish have focused primarily on mechanisms of swimming. More recent work has also considered the fluid dynamics of feeding from currents generated during swimming. Here we capitalize on the benthic lifestyle of the upside-down jellyfish (Cassiopea xamachana) to explore the fluid dynamics of feeding uncoupled from swimming. A two-dimensional mathematical model is developed to capture the fundamental characteristics of the motion of the unique concave bell shape. Given the prominence of the oral arms, this structure is included and modeled as a porous layer that perturbs the flow generated by bell contractions. The immersed boundary method is used to solve the fluid–structure interaction problem. Velocity fields obtained from live organisms using digital particle image velocimetry were used to validate the numerical simulations. Parameter sweeps were used to numerically explore the effects of changes in pulse dynamics and the properties of the oral arms independently. Numerical experiments allow the opportunity to examine physical effects and limits within and beyond the biologically relevant range to develop a better understanding of the system. The presence of the prominent oral arm structures in the field of flow increased the flux of new fluid from along the substrate to the bell. The numerical simulations also showed that the presence of pauses between bell expansion and the next contraction alters the flow of the fluid over the bell and through the oral arms.

Vortexlet models of flapping flexible wings show tuning for force production and control

Insect wings are compliant structures that experience deformations during flight. Such deformations have recently been shown to substantially affect induced flows, with appreciable consequences to flight forces. However, there are open questions related to the aerodynamic mechanisms underlying the performance benefits of wing deformation, as well as the extent to which such deformations are determined by the boundary conditions governing wing actuation together with mechanical properties of the wing itself. Here we explore aerodynamic performance parameters of compliant wings under periodic oscillations, subject to changes in phase between wing elevation and pitch, and magnitude and spatial pattern of wing flexural stiffness. We use a combination of computational structural mechanics models and a 2D computational fluid dynamics approach to ask how aerodynamic force production and control potential are affected by pitch/elevation phase and variations in wing flexural stiffness. Our results show that lift and thrust forces are highly sensitive to flexural stiffness distributions, with performance optima that lie in different phase regions. These results suggest a control strategy for both flying animals and engineering applications of micro-air vehicles.

Fluid transport and coherent structures of translating and flapping wings

The Lagrangian coherent structures (LCSs) of simple wing cross sections in various low Reynolds number motions are extracted from high-fidelity numerical simulation data and examined in detail. The entrainment process in the wake of a translating ellipse is revealed by studying the relationship between attracting structures in the wake and upstream repelling structures, with the help of blocks of tracer particles. It is shown that a series of slender lobes in the repelling LCS project upstream from the front of the ellipse and "pull" fluid into the wake. Each lobe is paired with a corresponding wake vortex, into which the constituent fluid particles are folded. Flexible and rigid foils in flapping motion are studied, and the resulting differences in coherent structures are used to elucidate their differences in force generation. The clarity with which these flow structures are revealed, compared to the vorticity or velocity fields, provides new insight into the vortex shedding mechanisms that play an important role in unsteady aerodynamics.

Fast computation of finite-time Lyapunov exponent fields for unsteady flows

This paper presents new efficient methods for computing finite-time Lyapunov exponent (FTLE) fields in unsteady flows. The methods approximate the particle flow map, eliminating redundant particle integrations in neighboring flow map calculations. Two classes of flow map approximations are investigated based on composition of intermediate flow maps; unidirectional approximation constructs a time-T map by composing a number of smaller time-h maps, while bidirectional approximation constructs a flow map by composing both positive- and negative-time maps. The unidirectional method is shown to be fast and accurate, although it is memory intensive. The bidirectional method is also fast and uses significantly less memory; however, it is prone to error which is large in regions where the opposite-time FTLE field is large, rendering it unusable. The algorithms are implemented and compared on three example fluid flows: a double gyre, a low Reynolds number pitching flat plate, and an unsteady ABC flow.

Using Lagrangian coherent structures to analyze fluid mixing by cilia

Motivated by the desire to understand the fluid flow within the airway surface liquid of the lung, we consider the flow generated by a computational model of a motile, internally actuated cilium. The cilium, along with a mucus layer modeled by linear elastic elements, is coupled to a viscous, incompressible fluid. The evolution of this coupled system is captured using an immersed boundary method. The Eulerian velocity field computed on a grid is used to compute finite-time Lyapunov exponent fields, whose maximal ridges identify Lagrangian coherent structures (LCSs). The computed LCS uncovers a barrier that separates a recirculation region of fluid that remains near the beating cilium from fluid that is advected downstream. Moreover, periodic stretching and folding of this region gives rise to complex mixing. Flow structures around a cilium propelling a mucus layer are compared to flow structures around a cilium with no mucus load.