Simulating forced time-periodic flows in industrial applications presents significant computational challenges, partly due to the need to overcome costly transients before achieving time-periodicity. Reduced-order modelling emerges as a promising method to speed-up computations. We extend upon the work of Lotz et al. (2024) where a time-periodic space–time model is introduced. We present a time-periodic reduced-order model that directly finds the time-periodic solution without requiring extensive time integration. The reduced-order model gives a reduction in variables in both space and time. Our approach involves a POD-Galerkin reduced-order model based on a time-periodic full-order model that employs isogeometric analysis, residual-based variational multiscale turbulence modelling and weak boundary conditions. The projection-based reduced-order model inherits these features. We evaluate the reduced-order model with numerical experiments on moving hydrofoils. The motion is known a priori and we restrict ourselves to two spatial dimensions. In these experiments we vary the Strouhal and Reynolds numbers, and the motion profile respectively. Reduced-order model solutions agree well with those of the full-order model. The errors over the entire time period of thrust and lift forces are less than 0.2%. This includes complex scenarios such as the transition from drag to thrust production with increasing Strouhal number. Our time-periodic reduced-order model offers speed-ups ranging from O(10²) to O(10³) compared to the full-order model, depending upon the basis size. This makes it an appealing solution for prescribed time-periodic problems, with potential for additional speedup through nonlinear reduction techniques such as hyper-reduction.

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The hydrodynamic influence of surface texture on static surfaces ranges from large drag penalties (roughness) to potential performance benefits (shark-like skin). Although it is of wide-ranging research interest, the impact of roughness on flapping systems has received limited attention. In this work, we explore the effect of roughness on the unsteady performance of a harmonically pitching foil through experiments using foils with different surface roughness, at a fixed Strouhal number and within the Reynolds number range of. The foils' surface roughness is altered by changing the distribution of spherical-cap-shaped elements over the propulsor area. We find that the addition of surface roughness does not improve the performance compared with a smooth surface over the range considered. The analysis of the flow fields shows near-identical wakes regardless of the foil's surface roughness. The performance reduction mainly occurs due to an increase in profile drag. However, we find that the drag penalty due to roughness is reduced from for a static foil to for a flapping foil at the same mean angle of attack, with the strongest decrease measured at the highest. Our findings highlight that the effect of roughness on dynamic systems is very different than that on static systems; thereby, it cannot be estimated by only using information obtained from static cases. This also indicates that the performance of unsteady, flapping systems is more robust to the changes in surface roughness.

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Bats fly using significantly different wing motions from other fliers, stemming from the complex interplay of their membrane wings’ motion and structural properties. Biological studies show that many bats fly at Strouhal numbers, the ratio of flapping to flight speed, 50–150% above the range typically associated with optimal locomotion. We use high-resolution fluid–structure interaction simulations of a bat wing to independently study the role of kinematics and material/structural properties in aerodynamic performance and show that peak propulsive and lift efficiencies for a bat-like wing motion require flapping 66% faster than for a symmetric motion, agreeing with the increased flapping frequency observed in zoological studies. In addition, we find that reduced membrane stiffness is associated with improved propulsive efficiency until the membrane flutters, but that incorporating microstructural anisotropy arising from biological fibre reinforcement enables a tenfold reduction of the flutter energy while maintaining high aerodynamic efficiency. Our results indicate that animals with specialized flapping motions may have correspondingly specialized flapping speeds, in contrast to arguments for a universally efficient Strouhal range. Additionally, our study demonstrates the significant role that the microstructural constitutive properties of the membrane wing of a bat can have in its propulsive performance.@en

This paper presents a general and robust method for the fluid-structure interaction of membranes and shells undergoing large displacement and large added-mass effects by coupling an immersed-boundary method with a shell finite-element model. The immersed boundary method can accurately simulate the fluid velocity and pressure induced by dynamic bodies undergoing large displacements using a computationally efficient pressure projection finite volume solver. The structural solver can be applied to bending and membrane-related problems, making our partitioned solver very general. We use a strongly-coupled algorithm that avoids the expensive computation of the inverse Jacobian within the root-finding iterations by constructing it from input-output pairs of the coupling variables from the previous time steps. Using two examples with large deformations and added mass contributions, we demonstrate that the resulting quasi-Newton scheme is stable, accurate, and computationally efficient.@en

In this work, we describe the impact of aspect ratio on the performance of optimally phased, identical flapping flippers in a tandem configuration. Three-dimensional simulations are performed for seven sets of single and tandem finite foils at a moderate Reynolds number, with thrust producing, heave-to-pitch coupled kinematics. Increasing slenderness (or) is found to improve thrust coefficients and thrust augmentation but the benefits level off towards higher values of. However, the propulsive efficiency shows no significant change with increasing, while the hind foil outperforms the single by a small margin. Further analysis of the spanwise development and propagation of vortical structures allows us to gain some insights into the mechanisms of these wake interactions and provide valuable information for the design of novel biomimetic propulsion systems.

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This study examines the effects of surface topography on the flow and performance of a self-propelled swimming (SPS) body. We consider a thin flat plate with an egg-carton roughness texture undergoing prescribed undulatory swimming kinematics at Strouhal number and tail amplitude to length ratio; we use plate Reynolds numbers, 12 and, and focus on. As the roughness wavelength is decreased, we find that the undulation wave speed must be increased to overcome the additional drag from the roughness and maintain SPS. Correspondingly, the extra wave speed raises the power required to maintain SPS, making the swimmer less efficient. To decouple the roughness and the kinematics, we compare the rough plates to equivalent smooth cases by matching the kinematic conditions. We find that all but the longest roughness wavelengths reduce the required swimming power and the unsteady amplitude of the forces when compared to a smooth plate undergoing identical kinematics. Additionally, roughness can enhance flow enstrophy by up to 116 % compared to the smooth cases without a corresponding spike in forces; this suggests that the increased mixing is not due to increased vorticity production at the wall. Instead, the enstrophy is found to peak strongly when the roughness wavelength is approximately twice the boundary layer thickness over the range, indicating the roughness induces large-scale secondary flow structures that extend to the edge of the boundary layer. This study reveals the nonlinear interaction between roughness and kinematics beyond a simple increase or decrease in drag, illustrating that roughness studies on static shapes do not transfer directly to unsteady swimmers.

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Accurate modelling of the performance of a yacht in varying environmental conditions can significantly improve a yachts performance. However, a racing yacht is a highly complex multiphysics system meaning that real-time performance prediction tools are always semi-empirical, leaving significant room for improvement. In this paper we first use unsupervised machine learning to analyse full-scale yacht performance data. The widely documented ORC VPP (ORC, 2015) and the commercial Windesign VPP are compared to the data across a range of wind conditions. The data is then used to train machine learning models. A number of machine learning regression algorithms are explored including Neural Networks, Random Forests and Support Vector Machines and improvements of 82% are obtained compared to the commercial tools. The use of physics-based learning models (Weymouth and Yue, 2013) is explored in order to reduce the amount of data required to achieve accurate predictions. It is found that machine learning models can outperform empirical models even when predicting performance in environmental conditions that have not been supplied to the model as part of the training dataset.

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Pressure projection is the single most computationally expensive step in an unsteady incompressible fluid simulation. This work demonstrates the ability of data-driven methods to accelerate the approximate solution of the Poisson equation at the heart of pressure projection. Geometric Multi-Grid methods are identified as linear convolutional encoder–decoder networks and a data-driven smoother is developed using automatic differentiation to optimize the velocity-divergence projection. The new method is found to accelerate classic Multi-Grid methods by a factor of two to three with no loss of accuracy on eleven 2D and 3D flow cases including cases with dynamic immersed solid boundaries. The optimal parameters are found to transfer nearly 100% effectiveness as the resolution is increased, providing a robust approach for accelerated pressure projection of unsteady flows.

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Immersed boundary methods are extensively used for simulations of dynamic solid objects interacting with fluids due to their computational efficiency and modeling flexibility compared to body-fitted grid methods. However, thin geometries, such as shells and membranes, cause a violation of the boundary conditions across the surface for many immersed boundary projection algorithms. Using a one-dimensional analytical derivation and multi-dimensional numerical simulations, this manuscript shows that adjustment of the Poisson matrix itself is require to avoid large velocity, pressure, and force prediction errors when the pressure jump across the interface is substantial and that these errors increase with Reynolds number. A new minimal thickness modification is developed for the Boundary Data Immersion Method (BDIM-σ), which avoids these issues while still enabling the use of efficient projection algorithms for high-speed immersed surface simulations.

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The optimal stiffness for soft swimming robots depends on swimming speed, which means no single stiffness can maximise efficiency in all swimming conditions. Tunable-stiffness would produce an increased range of high-efficiency swimming speeds for robots with flexible propulsors and enable soft control surfaces for steering underwater vehicles. We propose and demonstrate a method for tunable soft robotic stiffness using inflatable rubber tubes to stiffen a silicone foil through pressure and second moment of area change. We achieved double the effective stiffness of the system for an input pressure change from 0 to 0.8 bar and 2 J energy input. We achieved a resonant amplitude gain of 5 to 7 times the input amplitude and tripled the high-gain frequency range compared to a foil with fixed stiffness. These results show that changing second moment of area is an energy effective approach to tunable-stiffness robots.

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The importance of the leading-edge sweep angle of propulsive surfaces used by unsteady swimming and flying animals has been an issue of debate for many years, spurring studies in biology, engineering, and robotics with mixed conclusions. In this work, we provide results from three-dimensional simulations on single-planform finite foils undergoing tail-like (pitch-heave) and flipper-like (twist-roll) kinematics for a range of sweep angles covering a substantial portion of animals while carefully controlling all other parameters. Our primary finding is the negligible 0.043 maximum correlation between the sweep angle and the propulsive force and power for both tail-like and flipper-like motions. This indicates that fish tails and mammal flukes with similar range and size can have a large range of potential sweep angles without significant negative propulsive impact. Although there is a slight benefit to avoiding large sweep angles, this is easily compensated by adjusting the fin's motion parameters such as flapping frequency, amplitude and maximum angle of attack to gain higher thrust and efficiency.

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This technical note presents a simplified method for quantifying the added mass of the soil that is mobilised as part of the failure mechanism around a foundation during rapid loading, and the resulting additional soil resistance. This note focuses on the solutions for an embedded plate anchor, which is a potential foundation system for offshore floating facilities. In current practice, only the shear strength of the soil surrounding the foundation is considered in calculations of the ultimate bearing capacity. However, the solutions presented in this technical note show that the added mass of the soil involved in the failure mechanism around the foundation can result in a significant increase in ultimate bearing capacity during extreme dynamic wave loading events. These lead to snatch loads transmitted to the anchoring and mooring system, which are high but brief. The technical note provides a general approach applicable to all foundation types, and illustrates the effect of the added mass term and the additional capacity for an embedded plate anchor with typical input conditions.

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Simulations of turbulent fluid flow around long cylindrical structures are computationally expensive because of the vast range of length scales, requiring simplifications such as dimensional reduction. Current dimensionality reduction techniques such as strip-theory and depth-averaged methods do not take into account the natural flow dissipation mechanism inherent in the small-scale three-dimensional (3-D) vortical structures. We propose a novel flow decomposition based on a local spanwise average of the flow, yielding the spanwise-averaged Navier–Stokes (SANS) equations. The SANS equations include closure terms accounting for the 3-D effects otherwise not considered in 2-D formulations. A supervised machine-learning (ML) model based on a deep convolutional neural network provides closure to the SANS system. A-priori results show up to 92% correlation between target and predicted closure terms; more than an order of magnitude better than the eddy viscosity model correlation. The trained ML model is also assessed for different Reynolds regimes and body shapes to the training case where, despite some discrepancies in the shear-layer region, high correlation values are still observed. The new SANS equations and ML closure model are also used for a-posteriori prediction. While we find evidence of known stability issues with long time ML predictions for dynamical systems, the closed SANS simulations are still capable of predicting wake metrics and induced forces with errors from 1-10%. This results in approximately an order of magnitude improvement over standard 2-D simulations while reducing the computational cost of 3-D simulations by 99.5%.

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The concept of added (virtual) mass is applied to a vast array of unsteady fluid-flow problems; however, its origins in potential-flow theory may limit its usefulness in separated flows. A robust framework for modeling instantaneous fluid forces is proposed, named Energized Mass. The energized-mass approach is tested experimentally by acquiring the fluid kinetic-energy history around an accelerating sphere at both subcritical and supercritical terminal velocities. By tracking the energized-mass volume, the force response is shown to be related to changes in shear-layer growth as a function of acceleration moduli and Reynolds number. The energized-mass framework is then used to develop a low-order force model, requiring only body geometry and kinematics as input. An analytical expression for the instantaneous force on a sphere due to energized-mass growth is derived based on shear-layer mass flux arguments. Instantaneous forces determined experimentally, and modeled using the energized-mass approach, show strong agreement with direct force measurements. The results of this investigation thus demonstrate that the energized-mass framework provides a viable low-order modeling approach, and in tandem, can provide new insights into the origin of forces on accelerating bodies.

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Elasticity has been linked to the remarkable propulsive efficiency of pulse-jet animals such as the squid and jellyfish, but reports that quantify the underlying dynamics or demonstrate its application in robotic systems are rare. This work identifies the pulse-jet propulsion mode used by these animals as a coupled mass-spring-mass oscillator, enabling the design of a flexible self-propelled robot. We use this system to experimentally demonstrate that resonance greatly benefits pulse-jet swimming speed and efficiency, and the robot's optimal cost of transport is found to match that of the most efficient biological swimmers in nature, such as the jellyfish Aurelia aurita. The robot also exhibits a preferred Strouhal number for efficient swimming, thereby bridging the gap between pulse-jet propulsion and established findings in efficient fish swimming. Extensions of the current robotic framework to larger amplitude oscillations could combine resonance effects with optimal vortex formation to further increase propulsive performance and potentially outperform biological swimmers altogether.

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Flapping flight and swimming are increasingly studied due to both their intrinsic scientific richness and their applicability to novel robotic systems. Strip theory is often applied to flapping wings, but such modeling is only rigorously applicable in the limit of infinite aspect ratio (AR) where the geometry and kinematics are effectively uniform. This work compares the flow features and forces of strip theory and three-dimensional flapping foils, maintaining similitude in the rolling and twisting kinematics while varying the foil AR. We find the key influence of finite AR and spanwise varying kinematics is the generation of a time-periodic spanwise flow which stabilizes the vortex structures and enhances the dynamics at the foil root. An aspect-ratio correction for flapping foils is developed analogous to Prandtl finite wing theory, enabling future use of strip theory in analysis and design of finite aspect ratio flapping foils.

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Propulsive flapping foils are widely studied in the development of swimming and flying animal-like autonomous systems. Numerical studies in this topic are mainly two-dimensional (2-D) studies, as they are quicker and cheaper, but this inhibits the three-dimensional (3-D) evolution of the shed vortices from leading and trailing edges. In this work, we examine the similarities and differences between 2-D and 3-D simulations through a case study in order to evaluate the efficacy and limitations of using 2-D simulations to describe a 3-D system. We simulate an infinite-span NACA0016 foil in both two and three dimensions at a Reynolds number of 5300 and an angle of attack of 10^°. The foil is subject to prescribed heaving and pitching kinematics with varying trailing-edge deflection amplitude. Our primary finding is that the flow and forces are effectively 2-D at intermediate amplitude-based Strouhal numbers (, where is the free-stream velocity and is the flapping frequency), for heaving, for pitching and for coupled motion, while 3-D effects dominate outside of these ranges. These 2-D regions begin when the fluid energy induced by the flapping motion overcomes the 3-D vortex shedding found on a stationary foil, and the flow reverts back to 3-D when the strength of the shed vortices overwhelms the stabilising influence of viscous dissipation. These results indicate that 3-D to 2-D transitions or vice versa are a balance between the strength and stability of leading/trailing-edge vortices and the flapping energy. However, 2-D simulations can still be used for flapping flight/swimming studies provided that the flapping amplitude/frequency is within a given range.

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The connected component labeling technique (CCL), which labels regions of connected Eulerian field data, will inaccurately identify closely spaced components when applied to the volume-of-fluid function. We present two modifications to the CCL that improve its robustness and accuracy. This Informed Component Labeling algorithm (ICL) incorporates the normal and uses multilevel thresholding to improve and refine connectivity decisions for components with spacing just larger than the grid size. Through detailed verification and validation using synthetic volume fraction data, we show that the ICL algorithm removes the bias to larger components, provide guidelines for its use, and estimate its error bounds for the smallest components. The ICL produces zero standard deviation in the number of components identified for those with radius larger than twice the grid size and can reduce it by ∼ 38% for smaller components. The modifications that comprise the ICL can be applied to any existing CCL algorithm with a known increase in computational cost. It enables robust identification of connected components for accurate transfer of information in mixed Eulerian-Lagrangian methods and statistical analysis that use the volume-of-fluid function.

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Symmetric flapping foils are known to produce deflected jets at high frequency-amplitude combinations even at a zero mean angle of attack. This reduces the frequency range of useful propulsive configurations without side force. In this study, we numerically analyse the interaction of these deflected jets for tandem flapping foils undergoing coupled heave-to-pitch motion in a two-dimensional domain. The impact of the flapping Strouhal number, foil spacing and phasing on wake interaction is investigated. Our primary finding is that the back foil is capable of cancelling the wake deflection and mean side force of the front foil, even when located up to five chord lengths downstream. This is achieved by attracting the incoming dipoles and disturbing their cohesion within the limits of the back foil's range of flapping motion. We also show that the impact on cycle-averaged thrust varies from high augmentation to drag generation depending on the wake patterns downstream of the back foil. These findings provide new insights towards the design of biomimetic tandem propulsors, as they expand their working envelope and ability to rapidly increase or decrease the forward speed by manipulating the size of the shed vortices.

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The Energized-Mass approach [1] offers a simple and robust framework for modeling forces in separated flow, requiring only kinematics, Reynolds-number, and geometry-based inputs. To this end, an energized-mass-based force model has been developed to describe the force response of a generic gust-body interaction, and is tested against numerical simulations of an accelerating sphere at moderate-to-high sub-critical Reynolds numbers. An analytic expression for the instantaneous force due to energized-mass growth is derived based on shear-layer mass flux arguments. The forming vortex ring is controlled by the Reynolds-number-dependent boundary layer thickness, which in turn, governs the energized-mass accumulated by the body. Entrainment effects are increasingly significant at greater Reynolds number as vortex formation transitions from laminar to turbulent, promoting mass flux beyond the predictions of shear-layer feeding arguments. Model predictions of instantaneous energized mass and resultant forces exhibit strong agreement with simulated results, suggesting energized-mass growth in sub-critical regimes is well-described by laminar-boundary-layer feeding. Thus, the current study provides a framework for future low-order force models, using the kinematics, Reynolds number and geometry to describe energized-mass time evolution.

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