Fluid Body Interaction Research Articles

Numerical Analysis of the Dynamic Interaction between Two Closely Spaced Vertical-Axis Wind Turbines

To investigate the optimum layouts of small vertical-axis wind turbines, a two-dimensional analysis of dynamic fluid body interaction is performed via computational fluid dynamics for a rotor pair in various configurations. The rotational speed of each turbine rotor (diameter: D = 50 mm) varies based on the equation of motion. First, the dependence of rotor performance on the gap distance (gap) between two rotors is investigated. For parallel layouts, counter-down (CD) layouts with blades moving downwind in the gap region yield a higher mean power than counter-up (CU) layouts with blades moving upwind in the gap region. CD layouts with gap/D = 0.5–1.0 yield a maximum average power that is 23% higher than that of an isolated single rotor. Assuming isotropic bidirectional wind speed, co-rotating (CO) layouts with the same rotational direction are superior to the combination of CD and CU layouts regardless of the gap distance. For tandem layouts, the inverse-rotation (IR) configuration shows an earlier wake recovery than the CO configuration. For 16-wind-direction layouts, both the IR and CO configurations indicate similar power distribution at gap/D = 2.0. For the first time, this study demonstrates the phase synchronization of two rotors via numerical simulation.

Моделирование траектории судна на волнах на базе STAR-CCM+

The reliable ship motion trajectory signal provides a basis for six-degree-of-freedom test bench to simulate ship motion in waves. The characteristics of ship motion in waves were analyzed. Aiming at the problem of complex hull shape and large displacement at sea, the dynamic overlapping grid technique was used. Based on STAR-CCM+, the free surface was solved by CFD (Computational Fluid Dynamics) numerical simulation method. The numerical model of ship motion in waves was established, and the DFBI (Dynamic Fluid Body Interaction) method was used to simulate the ship motion in the waves. The ship’s motion trajectory was obtained in the regular and irregular heading waves, which provides important support for the six-degree-of-freedom test bench wave simulation system.

The effects of caudal fin deformation on the hydrodynamics of thunniform swimming under self-propulsion

To investigate the effects of the caudal fin deformation on the hydrodynamic performance of the self-propelled thunniform swimming, we perform fluid-body interaction simulations for a tuna-like swimmer with thunniform kinematics. The 3-D vortices are visualized to reveal the role of the leading-edge vortex (LEV) in the thrust generation. By comparing the swimming velocity of the swimmer with different caudal fin flexure amplitudes fa, it is shown that the acceleration in the starting stage of the swimmer increases with the increase of fa, but its cruising velocity decreases. The results indicate that the caudal fin deformation is beneficial to the fast start but not to the fast cruising of the swimmer. During the entire swimming process, the undulation amplitudes of the lateral velocity and the yawing angular velocity decrease as fa increases. It is found that the formation of an attached LEV on the caudal fin is responsible for generating the low-pressure region on the surface of the caudal fin, which contributes to the thrust. Furthermore, the caudal fin deformation can delay the LEV shedding from the caudal fin, extending the duration of the low pressure on the caudal fin, which will cause the caudal fin to generate a drag-type force over a time period in one swimming cycle and reduce the cruising speed of the swimmer.

Numerical Prediction of Hydrodynamic Performance of Planing Trimaran with a Wave-Piercing Bow

The central hull is the most important structure in the planing trimaran. In order to gain insight into the relationship between hydrodynamic performance and main hull shape, experimental tests and numerical simulations were carried out for volume Froude Number (FrΔ) ranging from 1.31 to 4.98. Dynamic sinkage and trim in the Dynamic Fluid Body Interaction (DFBI) six-degree-of-freedom model were considered. A validation study carried out by comparison of experimental test results with numerical results showed good consistency. To analyze the process of tunnel penetration and pressure change at the bottom of the boat, numerical simulation results for free surface, bottom streamline, and pressure distribution around the hull are given. A large triangular high-pressure area was observed in the front of the main hull for all volume Froude numbers. Consequently, the central drainage body, in reference to the profile of single planing craft with distinctive resistance performance, was redesigned into a wave-piercing shape. Total resistance, sinkage, and trim angle of the new model were then predicted by numerical method. The results show that the central drainage body has a significant impact on the hydrodynamic performance of the planing trimaran. Furthermore, the wave-piercing shaped main hull has a drag reduction effect.

Effective wake estimation of KCS hull at full-scale by GEOSIM method based on CFD

In this study, the effective wake factors of a self-propelled container ship were investigated by Telfer's GEOSIM method based on CFD (Computational Fluid Dynamics). For this purpose, the self-propulsion characteristics of KCS (KRISO Container Ship) at three different model scales and full-scale were determined at Fr = 0.26 by unsteady RANS (Reynolds Averaged Navier-Stokes) solver. All scales corresponded to a wide range of Reynolds numbers. Turbulent flow around the KCS hull has been investigated by the k-ε turbulence model. Heave and pitch motions of KCS hulls have been modeled with the DFBI (Dynamic Fluid Body Interaction) method and overset mesh technique. The rotational motion of KP505 propeller has been simulated using the RBM (Rigid Body Motion) approach. The result obtained by the extrapolation of the effective wake fraction with the GEOSIM method was compared with the full-scale ones of CFD and ITTC methods. The results by the GEOSIM method are more compatible with those of full-scale CFD than the results of 1978 ITTC method. In the wake region, where viscous effects are predominant, the change of effective wake fraction with Reynolds number has been successfully represented by the GEOSIM method, thus more robust and reliable results have been found.

Numerical simulation of water entry of an inclined cylinder

Water entry of cylindrical structures is a complex multiphase flow problem involving a three-phase medium interaction that occurs instantaneously. It is a nonlinear and unsteady process, which the physical understanding of mechanism in the aspect of evolution of cavity dynamics and multiphase flow field is yet to be fully understood. In the present work, a three-dimensional numerical model of six degrees of freedom (DOF) based on the shear stress transport (SST) k−ω eddy viscosity model was established to capture the complicated characteristics of the vertical water entry of an inclined cylinder at various entry velocities. A volume of fluid method for tracking air-water interface and a combined dynamic fluid body interaction and overlapping grid technology were used to simulate the complex motion. The numerical model with laminar and turbulent model was verified against the published experimental data of cavity evolution and motion characteristics. A parametric study was conducted to study the translational and rotational characteristics of the structure due to the effect of water entry velocity. The investigation revealed the secondary closure phenomenon during the water entry. The asymmetrical multiphase flow field showed that its dynamic characteristics change with space and time under the effect of cavity evolution.

Parametric analysis and optimization of a simple wind turbine in high speed railway tunnels

The harvesting of wind energy from the slipstream generated when a train passes through a tunnel is a new way to produce renewable energy. From the perspective of CFD (Computed fluid dynamics), a three-dimensional URANS (unsteady Reynolds-averaged Navier–Stokes) model was used to study the energy harvesting performance of wind turbines with different design parameters inside the tunnel during the entire process of a train passing through a tunnel. A higher value of offset distance of blades could promote the power efficiency of the wind turbine inside the wind tunnel. The maximum power efficiency with regard to overlap distance was found when an overlap distance is 0 m. The turbine with a negative twist angle yielded a better performance than the turbine with a positive twist angle. The optimal wind turbine can generate up to 157.9 W power when a train passes at 350 km/h. Given the tunnel distribution in China, 4.8 × 10 12 J energy will be recovered each day by this way, which is sufficient to supply the emergency power for railway lighting. It can also effectively ease the pressure of daily lighting power in railway tunnels. • Harvesting wind energy from railway tunnels is considered feasible. • DFBI (dynamic fluid body interaction) was used to simulate the response of the turbine. • The power efficiency of turbines with various design parameters was quantitatively evaluated. • The optimal wind turbine can generate up to 157.9 W power when a train passes at 350 km/h.

Numerical Prediction of the Vertical Responses of Planing Hulls in Regular Head Waves

The evaluation of the hydrodynamic performance of planing vessels has always been one of the most attractive study fields in the maritime agenda. Resistance and self-propulsion studies have been performed using experimental and numerical methods by researchers for a long time. As opposed to this, the seakeeping performance of planing hulls is assessed with 2D approximation methods, but limitedly, while the experimental campaign is not cost-effective for several reasons. With this motivation, pitch and heave transfer functions and accelerations were obtained for a monohedral hull and a warped hull using a state of art commercial Reynolds-averaged Navier–Stokes (RANS) solver, in this study. Moreover, 2-DOF (degree of freedom) dynamic fluid–body interaction (DFBI) equations were solved in a coupled manner with an overset mesh algorithm, to find the instantaneous motion of the body. After verification, obtained numerical results at three different Froude numbers and a sufficiently large wave frequency range were compared with the experiments. The results showed that the employed RANS method offers a very accurate prediction of vertical motions and accelerations for planing hulls.

Modeling and numerical simulation of ricochet and penetration of water entry bodies using an efficient free surface model

In this paper, the enhancement and application of the recently developed two-phase flow model for the simulation of ricochet and penetration of water entry bodies are presented. The simple two-phase three-equation model, which is a reduced version of the full two-fluid model for compressible multiphase flows, is extendedly applied. The numerical procedure is advanced by implementing the scheme on moving overset body-fitted grids to facilitate flow simulation of complex geometries and arbitrary motions of objects and improve computational productivity by employing their flexibility. A coupling algorithm for the fluid-body interaction and a moving overset grid modeling strategy are integrated into the flow solver. Validation with numerical grid refinement and convergence studies of water entry problems is addressed, and the capability and robustness of the present model in the accurate simulation of free surface and water-impact flows are demonstrated. The method was subsequently employed for the investigation of the ricochet of a circular cylinder off the water surface, through which the physical picture of the ricochet is more clearly observed.

Two-phase SPH model based on an improved Riemann solver for water entry problems

There are challenges in simulating two-phase flows with large density ratios using traditional SPH. Taking full advantages of Riemann solver in dealing with contact discontinuity problems and combining two-phase flow SPH method with Riemann solver, a new model of two-phase flow coupling with structure based on an improved Riemann solver is described for the water entry problems with the effect of air. The flow momentum equation of the Riemann form is improved to decrease the Riemann dissipation and simulate the two-phase flows with different viscosity ratios. One-sided Riemann problem considering the free-slip and no-slip conditions at a wall is used to deal with the fluid-body interaction. A switch-function-based Riemann solver dissipation is used to improve the instability of interface owing to the strong impact. In order to test the accuracy of the model in fluid-structure coupling problems, five cases, including dam break, sinking of a rectangle body, water entry of a wedge, flat plate slamming on water and water entry of a catamaran, are numerically simulated. The agreements between the present results and other reference results demonstrate that the proposed method is robust and stable to simulate the water entry problems with the air.

Optimal boundary control for steady motions of a self-propelled body in a Navier-Stokes liquid

Consider a rigid body 𝒮 ⊂ ℝ3 immersed in an infinitely extended Navier-Stokes liquid and the motion of the body-fluid interaction system described from a reference frame attached to 𝒮. We are interested in steady motions of this coupled system, where the region occupied by the fluid is the exterior domain Ω = ℝ3 \ 𝒮. This paper deals with the problem of using boundary controls v*, acting on the whole ∂Ω or just on a portion Γ of ∂Ω, to generate a self-propelled motion of 𝒮 with a target velocity V (x) := ξ + ω × x and to minimize the drag about 𝒮. Firstly, an appropriate drag functional is derived from the energy equation of the fluid and the problem is formulated as an optimal boundary control problem. Then the minimization problem is solved for localized controls, such that supp v*⊂ Γ, and for tangential controls, i.e, v*⋅ n|∂Ω = 0, where n is the outward unit normal to ∂Ω. We prove the existence of optimal solutions, justify the Gâteaux derivative of the control-to-state map, establish the well-posedness of the corresponding adjoint equations and, finally, derive the first order optimality conditions. The results are obtained under smallness restrictions on the objectives |ξ| and |ω| and on the boundary controls.

Numerical simulation of water entry of a wedge using a modified ghost-cell immersed boundary method

The hydrodynamic problems of water entry of a wedge are investigated by a modified ghost-cell immersed boundary method. The modifications correspond to a newly proposed scheme for dealing with fluid–body interaction, and incorporation of CIP (Constraint Interpolation Profile) method and THINC/SW (Tangent of Hyperbola for Interface Capturing with Slope Weighting) method. The modified ghost-cell immersed boundary method uses a compact interpolation structure and gives the fluid properties to the ghost cell to achieve a more accurate treatment of fluid–body interaction. It can preserve the sharpness of the immersed boundary. To validate the new method, simulations of water entry of symmetric and asymmetric wedges in a single degree of freedom and water entry of an asymmetric wedge in three degrees of freedom are carried out, respectively. The variations of pressure, vertical velocity, vertical acceleration, angular displacement, and angular acceleration against time are focused on. The evolution of the free surface and velocity distributions of fluid is also analyzed. The results of numerical simulations are compared with available experimental results and fairly good agreement is obtained, which indicates that the present model is reliable.

Comparison of the pendulum motion of a buoy subjected to wave load with experiments

This paper describes a method for the coupled analysis between incompressible smoothed particle hydrodynamics (ISPH) and multibody dynamics (MBD). ISPH is a numerical technique used to calculate the motion of a fluid that is expressed as a system of particles. The analysis program for SPH-MBD coupled analysis is developed. The fluid force acting on a body should be calculated to solve the coupled problem. The method for fluid-body interaction at an acceleration level is proposed. The fluid force is considered as the external force and moment for a body. The motion of a body due to the fluid force and other forces is calculated by an implicit HHT-α integration technique in MBD. A pendulum model is used to verify the accuracy of the analysis program. An experimental setup is designed using a dimensional water tank. Four wave conditions are used. The wave height and the pendulum motion of a buoy are compared with experimental results. Simulation results reveal that period and wave height are consistent with experimental findings, and the pendulum motion of a buoy shows a maximum error of 12.2 %.

Channel Flow Past A Near-Wall Body

Summary Near-wall behaviour arising when a finite sized body moves in a channel flow is investigated for high flow rates. This is over the interactive-flow length scale that admits considerable upstream influence. The focus is first on quasi-steady two-dimensional flow past a thin body in the outer reaches of one of the viscous wall layers. The jump conditions near the front of the body play an important part in the whole solution which involves an unusual multi-structured flow due to the presence of the body: flows above, below, ahead of and behind the body interact fully. Analytical solutions are presented and the repercussions for shorter and longer bodies are then examined. Second, implications are followed through for the movement of a free body in a dynamic fluid–body interaction. Particular key findings are that instability persists for all body lengths, the growth rate decreases like the $1/4$ power of distance as the body approaches the wall, and lift production on the body is dominated by high pressures from an unexpected flow region emerging on the front of the body.

TRAJECTORY SIMULATION OF SELF-PROPELLED ELASTIC RODS IN FLUID

The research for flexible structures coupling with fluid simulates and promotes the development of soft robotics. A fast and accurate numerical method is significant for a real-time simulation of robots. This research provides theoretical and critical information for experiments to reduce the possibility of failure, by anticipating the path of the underwater soft robots and the possible requiring parameters for materials. This paper researches for bio-swimming elastic rods coupling with two-dimensional incompressible Newton fluid. First, we discretize elastic rods to extensive springs and rotational springs and stablish the kinetic equations based on energy reflecting the influence of internal force on swimming rods, and solve the governing equation by leap-frog algorithm. Second, semi-Lagrangian method is used to establish a fluid solver. Finally, the simplified coupling algorithm based on immersed-boundary method is raised, calling immersed-boundary method for momentum equations. These equations update the velocity of grid near the coupling interface directly to replace the function source force play in immersed-boundary method. Combining the fluid solver, rods solver and the coupling algorithm, a integral program solving the problem of the dynamics for immersed rods numerically. Simulating the rods swimming with a referenced pose of sine curvature. Comparing the simulating result to existed experiments of filament swimming and to the swimming trajectory of soft-body fish, we find the numerical result conform to experiment result, leading to the conclusion that this algorithm and dynamic model can simulate the trajectory of discrete elastic rods swimming underwater smoothly under the influence of both internal elastic force and coupling soft body-fluid interaction. Using this program, we test several critical parameters relating to swimming performance of rods, including iteration numbers, frequency and initial phases, finding that changing the initial phase of rods will alter onward direction of elastic rods. These results prove the feasibility and possibility of this algorithm and program being used for guiding development of real soft swimmers.

On optimal node spacing for immersed boundary–lattice Boltzmann method in 2D and 3D

A computational study on optimal spacing of Lagrangian nodes discretizing a rigid and immobile immersed body boundary in 2D and 3D is presented in order to show how the density of the Lagrangian points affects the numerical results of the Immersed Boundary–Lattice Boltzmann Method (IB–LBM). The study is based on the implicit velocity correction-based IB–LBM proposed by Wu and Shu (2009, 2010) that allows computing the fluid–body interaction force. However, the (original) method fails for densely spaced Lagrangian points due to ill-conditioned or even singular linear systems that arise from the derivation of the method. We propose a modification that improves the solvability of the linear systems and compare the performance of both methods using several benchmark problems. The results show how the spacing of the Lagrangian points affects the numerical results, mainly the permeability of the discretized body boundary in applications to fluid flows over rigid obstacles and blood flows in arteries in 2D and 3D.

A new constraint-based formulation for hydrodynamically resolved computational neuromechanics of swimming animals

Undulatory motion in aquatic animals is attained through waves of muscle contraction. Coordinated muscle contraction is produced by the orchestration of contralaterally anti-phased, caudally propagating waves of neural activity that deliver stimuli to skeletal muscles. This physical deformation generates muscle bending moments along the length of the body. These resultant moments in combination with hydrodynamic, inertial, and the body's constitutive forces (e.g., elasticity) determine the deformation kinematics for swimming. Hydrodyamically resolved simulations of neurally controlled locomotion can facilitate experimental neurobiologists in identifying and decoding activation patterns associated with distinct motor behaviors in swimming animals. When neurally activated, muscle stiffness and consequently the effective body stiffness dynamically increases. Computationally resolving large deformations for an effectively stiff, viscoelastic body immersed in fluid is expensive. For methods that explicitly couple the fluid–body interactions, the body's large elastic modulus imposes severe limitations on the time step size required to ensure numerical stability. Fully implicit methods are generally no more computationally efficient. When the effective body stiffness is sufficiently large, the realized deformation kinematics closely follow the time-varying preferred (or reference) configuration which is implied by the muscle dynamics. Rather than resolving the numerically stiff, elastic equations for the body, a fast and efficient, constraint-based self-propulsion formulation is employed to directly impose the preferred swimming kinematics. With this approach, the exploration of neuromechanical model for propulsion using fully resolved computational fluid dynamics becomes more tractable. The presented method is robust and may be employed to investigate the neuromechanics of motor control and locomotion. Two and three dimensional simulations (2D and 3D) demonstrating the neural activation patterns' effect on maneuverability, speed, and feedback driven obstacle navigation are presented. The phase lag between curvature and moment waves, a manifestation of neurally controlled propulsion, is reproduced, further signifying the robustness of the presented method.

Hydromorphological Classification Using Synchronous Pressure and Inertial Sensing

Classification of river morphology is often based on hydromorphological units (HMUs) identified from field measurements. Established survey methods rely on expert judgment or collection of field point measurements. When used for HMU classification, these methods can suffer from high errors due to the variations in the sampling environment, causing low repeatability. In order to expedite field data collection and increase HMU classification accuracy, we propose a multisensory device, the hydromast. Each hydromast provides a new source of data to classify HMUs. The modules are inexpensive and highly portable, consisting of a synchronous array of commodity pressure and inertial sensors. Rapid, local changes in the flow field are recorded with absolute and differential pressure sensors. At the same time, slower depth-integrated flow signals are obtained from a small damped cylindrical mast, driven by vortex-induced vibrations. In contrast to existing passive flow measurement technologies, the hydromast uses fluid–body interactions to provide flow measurements. This allows for minimal signal processing and simple feature extraction. An array of three hydromasts was used to collect ten samples in three river HMUs with shallow depths and highly turbulent flows with smooth and rough beds. We investigated classification accuracy using single, dual, and triple hydromast arrays with pressure, inertial, and combined features using linear regression, a genetic algorithm, and a neural network. Although limited in scope, the set of spot measurements covering three HMUs showed that a single multimodal sensor could deliver an overall classification accuracy of 89% of the HMUs, and an increase of up to 99% was achieved using a multimodal triple hydromast array. These preliminary results show promise in using hydromasts for rapid and robust HMU classification, providing a new way to collect and assess river survey data.

Man-made flows from a fish’s perspective: autonomous classification of turbulent fishway flows with field data collected using an artificial lateral line

The lateral line system provides fish with advanced mechanoreception over a wide range of flow conditions. Inspired by the abilities of their biological counterparts, artificial lateral lines have been developed and tested exclusively under laboratory settings. Motivated by the lack of flow measurements taken in the field which consider fluid-body interactions, we built a fish-shaped lateral line probe. The device is outfitted with 11 high-speed (2.5 kHz) time-synchronized pressure transducers, and designed to capture and classify flows in fish passage structures. A total of 252 field measurements, each with a sample size of 132 000 discrete sensor readings were recorded in the slots and across the pools of vertical slot fishways. These data were used to estimate the time-averaged flow velocity (R2 = 0.952), which represents the most common metric to assess fishway flows. The significant contribution of this work is the creation and application of hydrodynamic signatures generated by the spatial distribution of pressure fluctuations on the fish-shaped body. The signatures are based on the collection of the pressure fluctuations’ probability distributions, and it is shown that they can be used to automatically classify distinct flow regions within the pools of three different vertical slot fishways. For the first time, field data from operational fishway measurements are sampled and classified using an artificial lateral line, providing a completely new source of bioinspired flow information.

Horizontal locomotion of a vertically flapping oblate spheroid

We consider the self-induced motions of three-dimensional oblate spheroids of density $\unicode[STIX]{x1D70C}_{s}$ with varying aspect ratios $AR=b/c\leqslant 1$, where $b$ and $c$ are the spheroids’ centre-pole radius and centre-equator radius, respectively. Vertical motion is imposed on the spheroids such that $y_{s}(t)=A\sin (2\unicode[STIX]{x03C0}ft)$ in a fluid of density $\unicode[STIX]{x1D70C}$ and kinematic viscosity $\unicode[STIX]{x1D708}$. As in strictly two-dimensional flows, above a critical value $Re_{C}$ of the flapping Reynolds number $Re_{A}=2Afc/\unicode[STIX]{x1D708}$, the spheroid ultimately propels itself horizontally as a result of fluid–body interactions. For $Re_{A}$ sufficiently above $Re_{C}$, the spheroid rapidly settles into a terminal state of constant, unidirectional velocity, consistent with the prediction of Deng et al. (Phys. Rev. E, vol. 94, 2016, 033107) that, at sufficiently high $Re_{A}$, such oscillating spheroids manifest $m=1$ asymmetric flow, with characteristic vortical structures conducive to providing unidirectional thrust if the spheroid is free to move horizontally. The speed $U$ of propagation increases linearly with the flapping frequency, resulting in a constant Strouhal number $St(AR)=2Af/U$, characterising the locomotive performance of the oblate spheroid, somewhat larger than the equivalent $St$ for two-dimensional spheroids, demonstrating that the three-dimensional flow is less efficient at driving locomotion. $St$ decreases with increasing aspect ratio for both two-dimensional and three-dimensional flows, although the relative disparity (and hence relative inefficiency of three-dimensional motion) decreases. For flows with $Re_{A}\gtrsim Re_{C}$, we observe two distinct types of inherently three-dimensional motion for different aspect ratios. The first, associated with a disk of aspect ratio $AR=0.1$ at $Re_{A}=45$, consists of a ‘stair-step’ trajectory. This trajectory can be understood through consideration of relatively high azimuthal wavenumber instabilities of interacting vortex rings, characterised by in-phase vortical structures above and below an oscillating spheroid, recently calculated using Floquet analysis by Deng et al. (Phys. Rev. E, vol. 94, 2016, 033107). Such ‘in-phase’ instabilities arise in a relatively narrow band of $Re_{A}\gtrsim Re_{C}$, which band shifts to higher Reynolds number as the aspect ratio increases. (Indeed, for horizontally fixed spheroids with aspect ratio $AR=0.2$, Floquet analysis actually predicts stability at $Re_{A}=45$.) For such a spheroid ($AR=0.2$, $Re_{A}=45$, with sufficiently small mass ratio $m_{s}/m_{f}=\unicode[STIX]{x1D70C}_{s}V_{s}/(\unicode[STIX]{x1D70C}V_{s})$, where $V_{s}$ is the volume of the spheroid) which is free to move horizontally, the second type of three-dimensional motion is observed, initially taking the form of a ‘snaking’ trajectory with long quasi-periodic sweeping oscillations before locking into an approximately elliptical ‘orbit’, apparently manifesting a three-dimensional generalisation of the $QP_{H}$ quasi-periodic symmetry breaking discussed for sufficiently high aspect ratio two-dimensional elliptical foils in Deng & Caulfield (J. Fluid Mech., vol. 787, 2016, pp. 16–49).