Eulerian Picture Research Articles

Comparison of particle image velocimetry and the underlying agents dynamics in collectively moving self propelled particles

Collective migration of cells is a fundamental behavior in biology. For the quantitative understanding of collective cell migration, live-cell imaging techniques have been used using e.g., phase contrast or fluorescence images. Particle tracking velocimetry (PTV) is a common recipe to quantify cell motility with those image data. However, the precise tracking of cells is not always feasible. Particle image velocimetry (PIV) is an alternative to PTV, corresponding to Eulerian picture of fluid dynamics, which derives the average velocity vector of an aggregate of cells. However, the accuracy of PIV in capturing the underlying cell motility and what values of the parameters should be chosen is not necessarily well characterized, especially for cells that do not adhere to a viscous flow. Here, we investigate the accuracy of PIV by generating images of simulated cells by the Vicsek model using trajectory data of agents at different noise levels. It was found, using an alignment score, that the direction of the PIV vectors coincides with the direction of nearby agents with appropriate choices of PIV parameters. PIV is found to accurately measure the underlying motion of individual agents for a wide range of noise level, and its condition is addressed.

Stability Analysis of Fluid Flows Using Lagrangian Perturbation Theory (LPT): Application to the Plane Couette Flow

We present a new application of Lagrangian Perturbation Theory (LPT): the stability analysis of fluid flows. As a test case that demonstrates the framework we focus on the plane Couette flow. The incompressible Navier-Stokes equation is recast such that the particle position is the fundamental variable, expressed as a function of Lagrangian coordinates. The displacement due to the steady state flow is taken to be the zeroth order solution and the position is formally expanded in terms of a small parameter (generally, the strength of the initial perturbation). The resulting hierarchy of equations is solved analytically at first order. We find that we recover the standard result in the Eulerian frame: the plane Couette flow is asymptotically stable for all Reynolds numbers. However, it is also well established that experiments contradict this prediction. In the Eulerian picture, one of the proposed explanations is the phenomenon of `transient growth' which is related to the non-normal nature of the linear stability operator. The first order solution in the Lagrangian frame also shows this feature, albeit qualitatively. As a first step, and for the purposes of analytic manipulation, we consider only linear stability of 2D perturbations but the framework presented is general and can be extended to higher orders, other flows and/or 3D perturbations.

A Lagrangian formalism for thermal analysis of laminar convective heat transfer

Heat transfer in essence is the transport of thermal energy along certain paths in a similar way as fluid motion is the transport of fluid parcels along fluid paths. This similarity admits Lagrangian heat-transfer analyses by the geometry of such "thermal paths" analogous to well-known Lagrangian mixing analyses. Essential to Lagrangian heat-transfer formalisms is the reference state for the convective flux. Existing approaches admit only uniform references. However, for convective heat transfer, a case of great practical relevance, the conductive state that sets in for vanishing fluid motion is the more natural reference. This typically is an inhomogeneous state and thus beyond the existing formalism. The present study closes this gap by its generalisation to non-uniform references and thus substantially strengthens Lagrangian methods for thermal analyses. This ansatz is demonstrated by way of a 2D case study and offers new fundamental insight into thermal transport that is complementary to the Eulerian picture based on temperature and heat-transfer coefficients.

A generalised Lagrangian formalism for thermal analysis of laminar convective heat transfer

Heat transfer in essence is the transport of thermal energy along certain paths in a similar way as fluid motion is the transport of fluid parcels along fluid paths. This similarity admits Lagrangian heat-transfer analyses by the geometry of such “thermal paths” analogous to well-known Lagrangian mixing analyses. Essential to Lagrangian heat-transfer formalisms is the reference state for the convective flux. Existing approaches admit only uniform references. However, for convective heat transfer, a case of great practical relevance, the conductive state that sets in for vanishing fluid motion is the more natural reference. This typically is an inhomogeneous state and thus beyond the existing formalism. The present study closes this gap by its generalisation to non-uniform references and thus substantially strengthens Lagrangian methods for thermal analyses by (i) greatly extending their applicability, (ii) resolving the fundamental ambiguity concerning arbitrariness of the reference state that limits the original formalism, (iii) facilitating accessible physical interpretation of heat fluxes and thermal paths and (iv) enabling subtler distinction of (Lagrangian) heat-transfer phenomena. The generalised Lagrangian formalism is elaborated for laminar convective heat transfer, which can be done without loss of generality, and completed by a comprehensive geometrical framework for the composition and organisation of thermal paths. This ansatz is demonstrated by way of 2D (un)steady case studies and offers new fundamental insight into thermal transport that is complementary to the Eulerian picture based on temperature.

Evaluation of mean energy fluxes in thermoacoustic oscillations of a gas in a tube

In a previous paper [N. Sugimoto and M. Yoshida, Phys. Fluids 19, 074101 (2007)], the marginal condition for the onset of thermoacoustic oscillations of a gas in a tube subjected to the parabolic temperature distribution was derived in the framework of the linear and first-order theory in the boundary-layer thickness. This paper examines the marginal oscillations from a viewpoint of mean energy fluxes averaged over one period of oscillations, aiming at understanding an action of the boundary layer under finite temperature gradient. Using the nonlinear energy equation, formulas for the acoustic energy flux and the convective heat flux (entropy flux times temperature) are derived in the main-flow region and in the boundary layer within the lowest, quadratic order in the pressure amplitude. These fluxes may be evaluated in terms of the linearized solutions and their axial distributions are displayed graphically. The boundary layer occupying nearly half of the side wall near the open end plays an active role to pump energy into the acoustic main-flow region, whereas the other half plays a passive role to dissipate energy. The acoustic energy flux in the main-flow region flows toward the closed end, while the flux in the boundary layer flows down the gradient of the pressure amplitude and toward the open end. The convective heat flux flows only in the boundary layer and directed down the temperature gradient. In the vicinity of the closed end, however, this heat flux flows against the temperature gradient. The heat flux through the side wall comes mostly into the boundary layer but goes out of it near the closed end. Such an Eulerian picture of the flow field is examined from a Lagrangian viewpoint in terms of thermodynamic relations for a gas particle fixed. While the particle in the main-flow region is subjected to the adiabatic change to yield no net power per one period of oscillations, the one in the boundary layer is subjected to a thermodynamic cycle of a prime mover or a heat pump, depending on whether the value of the product of the gradients of the logarithmic temperature and pressure amplitude is beyond a certain value. Discussions are also focused on the mechanisms to maintain the marginal state of oscillations and the efficiency as heat engines.

Hydrodynamic construction of the electromagnetic field

We present an alternative Eulerian hydrodynamic model for the electromagnetic field in which the discrete vector indices in Maxwell's equations are replaced by continuous angular freedoms, and develop the corresponding Lagrangian picture in which the fluid particles have rotational and translational freedoms. This enables us to extend to the electromagnetic field the exact method of state construction proposed previously for spin 0 systems, in which the time-dependent wavefunction is computed from a single-valued continuum of deterministic trajectories where two spacetime points are linked by at most a single orbit. The deduction of Maxwell's equations from continuum mechanics is achieved by generalizing the spin 0 theory to a general Riemannian manifold from which the electromagnetic construction is extracted as a special case. In particular, the flat-space Maxwell equations are represented as a curved-space Schrödinger equation for a massive system. The Lorentz covariance of the Eulerian field theory is obtained from the non-covariant Lagrangian-coordinate model as a kind of collective effect. The method makes manifest the electromagnetic analogue of the quantum potential that is tacit in Maxwell's equations. This implies a novel definition of the ‘classical limit’ of Maxwell's equations that differs from geometrical optics. It is shown that Maxwell's equations may be obtained by canonical quantization of the classical model. Using the classical trajectories a novel expression is derived for the propagator of the electromagnetic field in the Eulerian picture. The trajectory and propagator methods of solution are illustrated for the case of a light wave.

Computing the wavefunction from trajectories: particle and wave pictures in quantum mechanics and their relation

We describe the particle method in quantum mechanics which provides an exact scheme to calculate the time-dependent wavefunction from a single-valued continuum of determinstic trajectories where two spacetime points are linked by at most a single orbit. A natural language for the theory is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture and the particle theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle. The Euler–Lagrange equations imply a fourth-order nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates using as input the initial wavefunction. The admissible solutions are those consistent with quasi-potential flow. The effect of the superposition principle is represented via a nonclassical force on each particle. The wavefunction is computed via the standard map between the Lagrangian coordinates and the Eulerian fields, which provides the analogue in this model of Huygens’ principle in wave mechanics. The method is illustrated by calculating the time-dependence of a free Gaussian wavefunction. The Eulerian and Lagrangian pictures are complementary descriptions of a quantum process in that they have associated Hamiltonian formulations that are connected by a canonical transformation. The de Broglie–Bohm interpretation, which employs the same set of trajectories, should not be conflated with the Lagrangian version of the hydrodynamic interpretation. The theory implies that the mathematical results of the de Broglie–Bohm model may be regarded as statements about quantum mechanics itself rather than about its interpretation.

The Loop Current and adjacent rings delineated by Lagrangian analysis of the near-surface flow

Evolution of the Loop Current and the adjacent mesoscale rings was investigated over a 20-day period, starting June 1, 1998. Model sea-surface height and near-surface velocity fields from the Colorado University Princeton Ocean Model of the Gulf of Mexico were used along with drifter data in the study. A Lagrangian analysis provided detailed information about ring interaction and revealed features that were not apparent in the Eulerian picture. During the observation period, a new ring was formed in the meander of the Loop Current and a large anticyclonic ring to the west was cleaved by a cyclonic eddy. The coherent structures were identified in the Lagrangian framework by means of material curves (manifolds) that act as their boundaries. Concurrent drifter trajectories were in qualitative agreement with the results of the Lagrangian analysis.

Statistical analysis of the surface circulation in the Algerian Current using Lagrangian buoys

The Algerian Current (AC) is one of the most energetic flows in the Mediterranean basin. A characteristic picture of this current is formed by a series of mesoscale eddies at different scales. Here, statistical analysis of 15 surface Argos buoy tracks in 1996–1997 provides a complete Lagrangian view of the AC. The buoys, released upstream and across a coastal meander between 0°E and 1°E longitude, were followed for 3 months. They travelled eastward at an average speed of 14 cm/s and showed high energetic fluctuations related to several mesoscale eddies. The characteristic integral time and space scales are highly anisotropic. For the zonal component, these are about 4 days and 66 km, and for the meridional component, about 2 days and 26 km. Representative values of effective diffusivities from single dispersion statistics are within 0.7–1.3×10 8 and 1.5–6.0×10 7 cm 2/s for the zonal and meridional directions, respectively. A local analysis shows that mesoscale motions are particularly relevant in the region 1–3°E and 7–8°E, provided the considerably high values of eddy kinetic energy in comparison with the mean kinetic energy. Eddy–mean current interactions are evidenced by the significant changes of sign of horizontal covariance from west to east. Finally, an Eulerian picture of the AC is built, exhibiting similar trends than previous and recent field observations.

Laser-induced two-dimensional periodic and chaotic motions on molten tantalum and molybdenum surfaces

Abstract Two-dimensional flow patterns generated in high-power short-time laser–material interaction on the vapourizing surface of tantalum and molybdenum have been studied. The inhomogeneous pattern of turbulent cells was divided into homogeneous zones (basins) of dynamics which form characteristic streamline patterns. They were simulated by using the model of regular and chaotic flow of Boldrigini and Francheschini, based on the real-mode truncation of the Navier–Stokes equation in k space. The nine-mode model with three-mode forcing reproduced the characteristic 'footprint' cells, triangular cells, 'potato-like' cells and square cells for the Reynolds numbers Re = 80, 100 and 110, appearing in various basins with time delay with respect to the beginning of the laser pulse. All the basins show chaotic fluid motion in the Eulerian picture, and the motion changing from regular to chaotic in the Lagrangian picture.

A multiple shooting approach for the numerical treatment of stellar structure and evolution

AbstractWe present a new numerical method for solving the system of partial differential equations describing the structure and evolution of a spherically symmetric star. As usual, we employ the transversal method of lines in order to split the equations into a coupled spatial and temporal part. The novel features of the algorithm are the following: (a) Instead of using the Lagrangian picture we formulate the system of partial differential equations in the Eulerian picture. (b) We reformulate the equations of stellar structure as a multipoint boundary‐value problem. By means of this reformulation the rather clumsy iterative matching procedure of stellar atmosphere and interior is avoided. (c) The multipoint boundary‐value problem is solved by the multiple shooting method. This approach not only ensures a high accuracy of the stellar models calculated at each time step but also allows the free boundaries inside the star due to different energy transport mechanisms to be located exactly. (d) The time derivatives involved in the stellar‐structure equations are discretized implicitly to second order accuracy. Moreover, at each time step, the chemical abundances are determined by using a sophisticated update procedure. In this way, a high accuracy is achieved with respect to the integration in time.The algorithm has turned out to be exceedingly reliable and numerically accurate. This is shown by the evolution of a 1 M⊙ star up to the hydrogen‐shell burning phase. In this example, the virial theorem, the law of mass conservation, and the law of energy conservation is fulfilled to a hitherto unattainable degree of accuracy.Since the multiple shooting method, which is at the heart of our approach, is a perfect example of a parallel algorithm, the computational speed of the algorithm might be substantially improved provided easy‐to‐program, high‐performance parallel computers with sufficiently many processors become available in the near future.

Conservative Formulation of Large Deformation Plasticity

We explain several ideas which may, either singly or in combination, help achieve high resolution in simulations of large-deformation plasticity. Because of the large deformations, we work in the Eulerian picture. The governing equations are written in a fully conservative form, which are correct for discontinuous as well as continuous solutions. Models of shear bands are discussed. These models describe the internal dynamics of a developed shear band in terms of time-asymptotic states; in other words, the smooth internal structure is replaced by a jump discontinuity, and the shear band evolution is determined by jump relations. This analysis is useful for high resolution numerical methods, including both shock capturing and shock tracking schemes, as well as for the understanding and validation of computations, independently of the underlying method. Preliminary computations, which illustrate the feasibility of these ideas, are presented.

General-relativistic approach to the nonlinear evolution of collisionless matter.

A new general-relativistic algorithm is developed to study the nonlinear evolution of scalar (density) perturbations of an irrotational collisionless fluid up to shell crossing, under the approximation of neglecting the interaction with tensor (gravitational-wave) perturbations. The dynamics of each fluid element is separately followed in its own inertial rest frame by a system of twelve coupled first-order ordinary differential equations, which can be further reduced to six under very general conditions. Initial conditions are obtained in a cosmological framework, from linear theory, in terms of a single gauge-invariant potential. Physical observables, which are expressed in the Lagrangian form at different times, can be traced back to the Eulerian picture by solving supplementary first-order differential equations for the relative position vectors of neighboring fluid elements. Similarly to the Zel'dovich approximation, in our approach the evolution of each fluid element is completely determined by the local initial conditions and can be independently followed up to the time when it enters a multistream region. Unlike the Zel'dovich approximation, however, our approach is correct also in three dimensions (except for the possible role of gravitational waves). The accuracy of our numerical procedure is tested by integrating the nonlinear evolution of a spherical perturbation in an otherwise spatially flat Friedmann-Robertson-Walker universe and comparing the results with the exact Tolman-Bondi solution for the same initial profile. An exact solution for the planar symmetric case is also given, which turns out to be locally identical to the Zel'dovich solution.

Eulerian variational principle for ideal hydrodynamics and two-fluid representation

The hydrodynamical equations for an ideal fluid with vorticity are derived, in the Eulerian picture, from a constrained variational principle, which, exploiting a two-fluid representation for the density field, both yields a consistent representation for the velocity field and allows one to develop a Hamiltonian formalism.

Lagrangian formulation of neoclassical transport theory

Neoclassical transport theory is developed in a Lagrangian formulation in contrast to the usual Eulerian development. The Lagrangian formulation is constructed from the three actions: magnetic moment, parallel invariant, and bounce-averaged poloidal flux. By averaging over the fast orbital time scales, an equation in the actions alone of the Fokker–Planck type is obtained. The coefficients give the rates of the elementary neoclassical scattering processes. This action space form of the kinetic equation contains in explicit form processes like banana diffusion and the kinematic Ware pinch. The associated fluxes can be computed by simple moments without having deviations from the local Maxwellian. All the trapped-particle contributions are of this explicit type. Another class of fluxes arise from perturbations to the Maxwellian and are termed implicit. The decomposition of the fluxes into explicit and implicit parts is a key feature of the Lagrangian formulation. These contributions correspond to distinct physical processes and have separate Onsager symmetry theorems (explicit and implicit) for their respective transport matrices. The theory does not depend on the details of the Fokker–Planck coefficients but only on some very general properties and is thus applicable—without modification of the formalism—to nonaxisymmetric and turbulent systems. This general formulation is the primary purpose of the work. To benchmark the theory, in the present paper, the tokamak transport coefficients (for the Lorentz gas) are computed and compared to the known Eulerian results, demonstrating the equivalence of the two formulations. The elementary processes responsible for the neoclassical pinch and bootstrap effects (somewhat obscured in the Eulerian picture) are identified and the physical basis for their Onsager symmetry relationship is clarified.

On the momentum of Eulerian phonons

For a one-dimensional dispersive medium the linear momentum of a phonon is discussed in both the Lagrangian and the Eulerian picture, i.e. with the use of substantial (material) and local coordinates, respectively. As phonons are usually considered as solutions of the linearized equations of motion, in the Eulerian picture the linear momentum of a phonon is only defined up to linear terms in the fields. To obtain results relevant towards higher order in the fields, one has to solve the nonlinear equations of motion. This is done to obtain expressions for the linear momentum up to terms quadratic in the fields.