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Spike formation theory in three-dimensional flow separation

We develop a frame-invariant theory of material spike formation during flow separation over a no-slip boundary in three-dimensional flows with arbitrary time dependence. Based on the exact evolution of the largest principal curvature on near-wall material surfaces, our theory identifies fixed and moving separation. Our approach is effective over short time intervals and admits an instantaneous limit. As a byproduct, we derive explicit formulas for the evolution of the Weingarten map and the principal curvatures of any surface advected by general three-dimensional flows. The material backbone we identify acts first as a precursor and later as the centrepiece of Lagrangian flow separation. We discover previously undetected spiking points and curves where the separation backbones connect to the boundary and provide wall-based analytical formulas for their locations. We illustrate our results on several steady and unsteady flows.

Spike formation theory in three-dimensional flow separation

S. Santhosh, H. Qin, B. F. Klose, G. B. Jacobs, Jérôme Vétel and M. Serra

J. Fluid Mech. 969, A25 (2023) [PDF]

Optimal Control of Short-Time Attractors in Active Fluid Flows

Objective Eulerian Coherent Structures (OECSs) and instantaneous Lyapunov exponents (iLEs) govern short-term material transport in fluid flows as Lagrangian Coherent Structures and the Finite-Time Lyapunov Exponent do over longer times. Attracting OECSs and iLEs reveal shorttime attractors and are computable from the Eulerian rate-of-strain tensor. Here we devise an optimal control strategy to create short-time attractors in viscosity-dominated active fluids. By modulating the active stress intensity, our framework achieves a target profile of the minimum eigenvalue of the rate-of-strain tensor, controlling the location and shape of short-time attractors. We use numerical simulations to show that our optimal control strategy effectively achieves desired short-time attractors while rejecting disturbances. Combining optimal control and recent advances on coherent structures, our work offers a new perspective to steer material transport in unsteady flows, with applications in synthetic active nematics and multicellular systems.

Optimal Control of Short-Time Attractors in Active Fluid Flows

C. Sinigaglia, F. Braghin, & M. Serra

(submitted) [PDF]

Optimal Locomotion for Limbless Crawlers

Limbless crawling is ubiquitous in biology, from cells to organisms. We develop and analyze a model for the dynamics of one-dimensional elastic crawlers, subject to active stress and deformation-dependent friction with the substrate. Solving an infinite-dimensional variational problem using perturbation theory, we find that the optimal active stress distribution that maximizes the crawler's center of mass displacement given a fixed amount of energy input is a traveling wave. This theoretical optimum corresponds to peristalsis-like extension-contraction waves observed in biological organisms, possibly explaining the prevalence of peristalsis as a convergent gait across species. Our theory elucidates key observations in biological systems connecting the anchoring phase of a crawler to the retrograde and prograde distinction seen in peristaltic waves among various organisms. Using our optimal gait solution, we derive a scaling relation between the crawling speed and body mass, explaining experiments on earthworms with three orders of magnitude body mass variations. Our results offer insights and tools for optimal bioinspired crawling robots design with finite battery capacity.

Detecting Lagrangian coherent structures from sparse and noisy trajectory data

Many complex flows such as those arising from ocean plastics in geophysics or moving cells in biology are characterized by sparse and noisy trajectory datasets. We introduce techniques for identifying Lagrangian Coherent Structures (LCSs) of hyperbolic and elliptic nature in such datasets. Hyperbolic LCSs, which represent surfaces with maximal attraction or repulsion over a finite amount of time, are computed through a regularized least-squares approximation of the flow map gradient. Elliptic LCSs, which identify regions of coherent motion such as vortices and jets, are extracted using DBSCAN - a popular data clustering algorithm - combined with a systematic approach to choose parameters. We deploy these methods on various benchmark analytical flows and real-life experimental datasets ranging from oceanography to biology and show that they yield accurate results, despite sparse and noisy data. We also provide a lightweight computational implementation of these techniques as a user-friendly and straightforward Python code.

Detecting Lagrangian coherent structures from sparse and noisy trajectory data

S. Molawi, M. Serra, E. Maiorino and L. Mahadevan

J. Fluid Mech. 948, A4. (2022) [PDF] Python code available here

A mechanochemical model recapitulates distinct vertebrate gastrulation modes

Gastrulation is a critical event in vertebrate morphogenesis driven by cellular processes, and characterized by coordinated multi-cellular movements that form the robust morphological structures. How these structures emerge in a developing organism and vary across vertebrates remains unclear. Inspired by experiments on the chick, we derive a theoretical framework that couples actomyosin activity to tissue flow, and provides a basis for the dynamics of gastrulation morphologies. Our model predicts the onset and development of observed experimental patterns of wild-type and perturbations of chick gastrulation as a spontaneous instability of a uniform state. Varying the initial conditions and a parameter in our model, allows us to recapitulate the phase space of gastrulation morphologies seen across vertebrates, consistent with experimental observations in the accompanying paper. All together, this suggests that early embryonic self-organization follows from a minimal predictive theory of active mechano-sensitive flows.

Featured in Science Advances

A mechanochemical model recapitulates distinct vertebrate gastrulation modes

M. Serra, Guillermo S. Nájera, Manli Chuai, A. Plum, S. Santhosh, Vamsi Spandan, Cornelis J. Weijer and L. Mahadevan

Science Advances 9.49 (2023) [PDF] Featured image in Science Advances, UCSD News

Reconstruction of distinct vertebrate gastrulation modes via modulation of key cell behaviors in the chick embryo

Manli Chuai*, Guillermo S. Nájera*, M. Serra, L. Mahadevan and Cornelis J. Weijer

Science Advances 9.1 (2023) [PDF]

Defect-mediated dynamics of coherent structures in active nematics

Active fluids, such as cytoskeletal filaments, bacterial colonies and epithelial cell layers, exhibit distinctive orientational coherence, often characterized by nematic order and topological defects. By contrast, little is known about positional coherence -- i.e., how a hidden dynamic skeleton organizes the underlying chaotic motion -- despite this being one of their most prominent and experimentally accessible features. Using a combination of dynamical systems theory, experiments on two-dimensional mixtures of microtubules and kinesin and hydrodynamic simulations, we characterize positional coherence in active nematics. These coherent structures can be identified in the framework of Lagrangian dynamics as moving attractors and repellers, which orchestrate complex motion. To understand the interaction of positional and orientational coherence on the dynamics of defects, we then analysed observations and simulations and see that +1/2 defects move and deform the attractors, thus functioning as control centers for collective motion. Additionally, we find that regions around isolated +1/2 defects undergo high bending and low stretching/shearing deformations, consistent with the local stress distribution. The stress is minimum at the defect, while high differential stress along the defect orientation induces folding. Our work offers a new perspective to describe self-organization in active fluids, with potential applications to multicellular systems.

Defect-mediated dynamics of coherent structures in active nematics

M. Serra, L. Lemma, L. Giomi, Z. Dogic and L. Mahadevan

Nature Physics 1-7 (2023) [PDF]

Dynamic morphoskeletons in development

Morphogenetic flows in developmental biology are characterized by the coordinated motion of thousands of cells that organize into tissues, naturally raising the question of how this collective organization arises. Using only the kinematics of tissue deformation, which naturally integrates local and global mechanisms along cell paths, we identify the dynamic morphoskeletons behind morphogenesis, i.e., the evolving centerpieces of multi-cellular trajectory patterns. These features are model and parameter-free, frame-invariant, robust to measurement errors, and can be computed from unfiltered cell velocity data. It reveals the spatial attractors and repellers of the embryo by quantifying its Lagrangian deformation, information that is inaccessible to simple trajectory inspection or Eulerian methods that are local and typically frame-dependent. Computing these dynamic morphoskeletons in wild-type and mutant chick and fly embryos, we find that they capture the early footprint of known morphogenetic features, reveal new ones, and quantitatively distinguish between different phenotypes.

Video Credit: C.J. Weijer Lab (Univ. of Dundee)

Dynamic morphoskeletons in development

M. Serra S. Streichan, M. Chuai, C.J. Weijer and L. Mahadevan

PNAS 117.21 (2020) [PDF]

Featured on Harvard SEAS News, SIAM NEWS

Search and rescue at sea aided by hidden flow structures

Every year hundreds of people die at sea because of vessel and airplane accidents. A key challenge in reducing the number of these fatalities is to make Search and Rescue (SAR) algorithms more efficient. Here we address this challenge by uncovering hidden TRansient Attracting Profiles (TRAPs) in ocean-surface velocity data. Computable from a single velocity-field snapshot, TRAPs act as short-term attractors for all floating objects. In three different ocean field experiments, we show that TRAPs computed from measured as well as modelled velocities attract deployed drifters and manikins emulating people fallen in the water. TRAPs, which remain hidden to prior flow diagnostics, thus provide critical information for hazard responses, such as SAR and oil spill containment, and hence have the potential to save lives and limit environmental disasters.

Search and rescue at sea aided by hidden flow structures

M. Serra, P. Sathe, I. Rypina, A. Kirincich, S. Ross, P. Lermusiaux, A. Allen, T. Peacock and G. Haller

Nature Communications 11 2525 (2020). [PDF]

Featured on MIT News, ETH News, US NSF News, Scientific American, BBC Science, AMS

Science Journal for kids version

MURI project with: ETH Zurich, MIT, Berkeley, VTech, WHOI, & U.S. Coast Guard

PI: Prof. T. Peacock (MIT)

A drone-based video of the 2018 field experiment is available here

Video Credit: VTech

Vortex boundaries as barriers to diffusive vorticity transport in two-dimensional flows

We put forward the idea of defining vortex boundaries in planar flows as closed material barriers to the diffusive transport of vorticity. Such diffusive vortex boundaries minimize the leakage of vorticity from the fluid mass they enclose when compared to other nearby material curves. Building on recent results on passive diffusion barriers, we develop an algorithm for the automated identification of such structures from general, two-dimensional unsteady flow data. As examples, we identify vortex boundaries as vorticity diffusion barriers in two flows: an explicitly known laminar flow and a numerically generated turbulent Navier--Stokes flow.

Vortex boundaries as barriers to diffusive vorticity transport in two-dimensional flows

S. Katsanoulis, M. Farazmand, M. Serra and G. Haller

Phys. Rev. Fluid 5 (2020) 024701 [PDF]

The Kinematics of Lagrangian Flow Separation in External Aerodynamics

Kinematic aspects of flow separation in external aerodynamics are investigated in the Lagrangian frame. Specifically, the initial motion of upwelling fluid material from the wall is related to the long-term attracting manifolds in the flow field. While the short-time kinematics are governed by the formation of a material spike upstream of the zero-skin-friction point and ejection of particles in direction of the asymptotic separation line, the trajectories of the fluid tracers are guided by attracting ridges in the finite-time Lyapunov exponents once they leave the vicinity of the wall. The wall signature of this initial fluid upwelling event, the so-called spiking point [Serra, M., Vetel, J., Haller, G., "Exact theory of material spike formation in flow separation", J. Fluid Mech., Vol. 845, 2018], is computed from the curvature of advected material lines and, for the first time, from high-order numerical derivatives of the wall-normal velocity obtained from direct numerical simulations of a circular cylinder and a cambered NACA 65(1)-412 airfoil. As the spline-based boundary parametrization of the airfoil profile induces oscillations, the principle spiking point can be recovered robustly through appropriate filtering. The short-term kinematics correlate strongly with the scaling lengths in the boundary layer.

Video Credit: Bjoern Klose (San Diego State University)

The Kinematics of Lagrangian Flow Separation in External Aerodynamics

B. F. Klose, G. B. Jacobs and M. Serra

AIAA Journal (2020) [PDF]

Material spike formation in highly unsteady separated flows

We apply the frame-invariant theory of separation spike formation [Serra, M., Vetel, J., Haller, G., "Exact theory of material spike formation in flow separation", J. Fluid Mech., Vol. 845, 2018] to complex unsteady flows, including a turbulent separation bubble, an impinging jet, and flows around a freely moving cylinder and a freely rotating ellipse. We show how the theory captures the onset of material spike formation, without any assumption on the flow type (steady, periodic, unsteady) or separation type (on- or off-wall, fixed or moving boundaries). We uncover new phenomena, such as the transition from on-wall to off-wall separation, the merger of initially distinct spikes, and the presence of severe material spikes that remain hidden to previous approaches. Remarkably, even in steady flows around curved boundaries, we detect material spikes in the absence of flow reversal, the main ingredient to existing separation criteria. Together, our results unveil how an involved network of spikes arises, interacts and merges dynamically, leading to the final ejection of particles from the wall in highly transient flow separation processes.

Material spike formation in highly unsteady separated flows

M. Serra, Sean Crouzat, Gael Simon, Jérôme Vétel and George Haller

J. Fluid Mech. 883 A30 (2019) [PDF]

A Simple Variational formulation of the Incompressible Euler equations

In 1966, Arnold [1] showed that the Lagrangian flow of ideal incompressible fluids (described by Euler equations) coincide with the geodesic flow on the manifold of volume preserving diffeomorphisms of the fluid domain. Arnold's proof and the subsequent work on this topic rely heavily on the properties of Lie groups and Lie algebras which remain unfamiliar to most fluid dynamicists. In this note, we provide a simple derivation of Arnold's result which only uses the classical methods of calculus of variations. In particular, we show that the Lagrangian flow maps generated by the solutions of the incompressible Euler equations coincide with the stationary curves of an appropriate energy functional when the extremization is carried out over the set of volume-preserving diffeomorphisms.

Variational Lagrangian formulation of the Euler equations for incompressible flow: A simple derivation

Mohammad Farazmand and M. Serra

arXiv:1807.02726 (2018) [PDF]

Exact Theory of Material Spike Formation in Flow Separation

We develop a frame-invariant theory of material spike formation during flow separation over a no-slip boundary in two-dimensional flows with arbitrary time dependence. Based on topological properties of material lines, our theory identifies both fixed and moving flow separation, is effective also over short-time intervals, and admits a rigorous instantaneous limit. The material backbone we identify acts as the first precursor, and the latter centerpiece, of unsteady Lagrangian flow separation. We also discover a previously undetected spiking point where the backbone of separation connects to the boundary, and derive wall-based analytical formulae for its location. Finally, our theory explains the perception of off-wall separation in unsteady flows and provides conditions under which such a perception is justified.

Exact Theory of Material Spike Formation in Flow Separation

M. Serra, Jérôme Vétel and George Haller

J. Fluid Mech. 845 (2018) 51-92. [PDF]

Efficient Computation of Null Geodesic with Applications to Coherent Vortex Detection

Recent results suggest that boundaries of coherent fluid vortices (elliptic coherent structures) can be identified as closed null geodesics of appropriate Lorentzian metrics defined on the flow domain. Here we derive a fully automated method for computing such null geodesics based on the geometry of geodesic flows and basic topological properties of closed planar curves.

Efficient Computation of Null Geodesics with Applications to Coherent Vortex Detection

M. Serra and George Haller

Proc. of the Royal Soc. A (2017) 473 20160807. [PDF]

Featured on the journal's cover page

The polar vortices play a crucial role in the formation of the ozone hole and can cause severe weather anomalies. Their boundaries, known as the vortex ‘edges’, are typically identi- fied via methods that are either frame-dependent or return non-material structures, and hence are unsuitable for assessing material transport barriers. Using two-dimensional velocity data on isentropic surfaces in the northern hemisphere, we show that elliptic Lagrangian Coherent Structures (LCSs) identify the correct outermost material surface dividing the coherent vortex core from the surrounding incoherent surf zone. Despite the purely kinematic construction of LCSs, we find a remarkable contrast in temperature and ozone concentration across the identified vortex boundary. We also show that potential vorticity-based methods, despite their simplicity, misidentify the correct extent of the vortex edge.

Uncovering the Edge of the Polar Vortex

M. Serra, Pratik Sathe, Francisco Beron-Vera and George Haller

J. Atm. Sci. 74 (11) 3871–3885, (2017). [PDF]

Featured on Forbes - (31-01-19)

Uncovering the Edge of the Polar Vortex

Lagrangian prediction from Eulerian vortices

We propose an objective non-dimensional metric to assess the short-term persistence of Eulerian vortices identified by elliptic objective Eulerian Coherent Structures (OECSs). Elliptic OECSs, equipped with this metric, give a continuously updated picture of the coherent vortices in the flow, distinguishing them in a quantitative fashion. Our method offers a kinematical (model-independent) tool for detecting the instantaneous signature of long lived Lagrangian vortices and could forecast flow topological changes like vortex disruption and formation.

Forecasting Long-Lived Lagrangian Vortices from their Objective Eulerian Footprints

M. Serra and George Haller

J. Fluid Mech. 813 (2017) 436-457. [PDF]

Objective Eulerian Coherent Structures

We define objective Eulerian Coherent Structures (OECSs) in two-dimensional, non-autonomous dynamical systems as instantaneously most influential material curves. Specifically, OECSs are stationary curves of the averaged instantaneous material stretching-rate or material shearing-rate functionals. From these objective (frame-invariant) variational principles, we obtain explicit differential equations for hyperbolic, elliptic and parabolic OECSs. As illustration, we compute OECSs in an unsteady ocean velocity data set. In comparison to structures suggested by other common Eulerian diagnostic tools, we find OECSs to be the correct short-term cores of observed trajectory deformation patterns.

Objective Eulerian coherent structures

M. Serra and George Haller

Chaos 26 (2016) 053110. [PDF] Editor's pick

Dependent modal space control

We propose a control logic, called Dependent Modal Space Control (DMSC), for vibration reduction in flexible structures. The well-known independent modal space control (IMSC) allows to change the frequencies and damping ratios of the controlled system, leaving the mode shapes unaltered. The DMSC, instead, allows to change also the closed loop mode shapes. We illustrate numerically and experimentally the advantages of the DMSC over the IMSC on a cantilevered beam.

Dependent modal space control: Experimental test rig

M. Serra, Francesco Ripamonti and Ferruccio Resta

J. Vibration and Control (2015): 1077546315616699. [PDF]

Dependent modal space control

M. Serra, Francesco Ripamonti and Ferruccio Resta

Smart Materials and Structures 22.10 (2013): 105004. [PDF]