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.

 

Collaborator:  Dr. M. Farazmand

 

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.

 

 

 

 

 

 

 

 

 

 

Collaborators: Prof. J. Vétel and Prof. G. Haller

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

 

Collaborator: Prof. G. Haller

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.

Collaborators: P. Sathe, Prof. F. Beron-Vera Prof. G. Haller                             

Uncovering the Edge of the Polar Vortex

Uncovering the hidden skeleton of environmental flows for hazards prediction and response

The goal of this project is to employ an integrated theoretical, computational and observational approach to develop, implement and utilize cutting-edge methods with data-driven modeling for the purpose of uncovering, quantifying and predicting key transport processes and structures during regional flow-based hazards in the ocean and atmosphere.


Joint with: MIT, UC Berkeley, Virginia Tech, WHOI &  U.S. Coast Guard

PIs and collaborators: Prof. T. Peacock (MIT) & Prof. P. Lermusiaux (MIT) 
                                    
Prof. G. Haller & P. Sathe

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.

Collaborator: Prof. G. Haller

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.

Collaborator: Prof. G. Haller

Hyperbolic Attracting OECSs

Parabolic  OECSs

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.

Collaborators: Dr. F. Ripamonti and Prof. F. Resta