Advection Of Fluid Particles Research Articles

Thermal fronts and attracting Lagrangian Coherent Structures in the north Bay of Bengal during December 2015–March 2016

We investigate two-dimensional advective stirring by satellite-derived ocean surface currents in the north Bay of Bengal during December 2015–March 2016 using the framework of Lagrangian Coherent Structures (LCS). The forward and backward finite-time Lyapunov exponent (fFTLE and bFTLE, respectively) fields indicate that freshwater parcels from the river run-off are subjected to intense stirring by the ocean surface currents, which contain both geostrophic and wind-forced Ekman currents. Large-scale thermal fronts present in the satellite-based sea surface temperature (SST) fields are then shown to closely resemble attracting LCS (aLCS) that govern horizontal advective stirring on subseasonal time scales. Statistical estimates of the angle between the thermal fronts (of all spatial scales present in the SST data) and the aLCS from the entire north Bay of Bengal show a preference to align with each other. Significantly, in the presence of sufficiently strong aLCS in their neighbourhood, thermal fronts are shown to have a strong preference to align with them. Finite-time advection of fluid particles over around a week to compute the bFTLE fields results in the strongest alignment between the thermal fronts and neighbouring aLCS. The robustness of our results are established by presenting analyses of (i) geostrophic currents from altimeter data, and (ii) a reanalysis-based fine-scale salinity field. Attracting LCS, identified as sufficiently strong ridges in the bFTLE fields, are argued to represent reasonable estimates of the location and orientation of fronts (thermal or salinity) formed by advective stirring over subseasonal time scales.

Advection of passive scalars induced by a bay-trapped nonstationary vortex

A simple model of fluid particle advection induced by the interaction of a point vortex and incident plane flow occurring near a curved boundary is analyzed. The use of the curved boundary in this case is aimed at mimicking the geometry of an isolated bay of a circular shape. An introduction of such a boundary to the model results in the appearance of retention zones, where the vortex can be permanently trapped being either stationary or periodically oscillating. When stationary, it induces a steady velocity field that in turn ensures regular advection of nearby fluid particles. When the vortex oscillates periodically, the induced velocity field turns unsteady leading to the manifestation of chaotic advection of fluid particles. We show that the size of the fluid region engaged into chaotic advection increases almost monotonically with the increased magnitude of the vortex oscillations provided the magnitude remains relatively small. The monotonicity is accounted for the fact that the frequency of the vortex oscillations incommensurable with the proper frequency of fluid particle rotations in the steady state. Another point of interest is that it is demonstrated that bounded regions, in which the vortex may be trapped, can appear even at a significant distance from the bay. Making use of a Lagrangian indicator, examples of fluid particle advection induced by the periodic motion of the vortex inside the bay are adduced.

Subgrid-scale stresses and scalar fluxes constructed by the multi-scale turnover Lagrangian map

The multi-scale turnover Lagrangian map (MTLM) [C. Rosales and C. Meneveau, “Anomalous scaling and intermittency in three-dimensional synthetic turbulence,” Phys. Rev. E 78, 016313 (2008)] uses nested multi-scale Lagrangian advection of fluid particles to distort a Gaussian velocity field and, as a result, generate non-Gaussian synthetic velocity fields. Passive scalar fields can be generated with the procedure when the fluid particles carry a scalar property [C. Rosales, “Synthetic three-dimensional turbulent passive scalar fields via the minimal Lagrangian map,” Phys. Fluids 23, 075106 (2011)]. The synthetic fields have been shown to possess highly realistic statistics characterizing small scale intermittency, geometrical structures, and vortex dynamics. In this paper, we present a study of the synthetic fields using the filtering approach. This approach, which has not been pursued so far, provides insights on the potential applications of the synthetic fields in large eddy simulations and subgrid-scale (SGS) modelling. The MTLM method is first generalized to model scalar fields produced by an imposed linear mean profile. We then calculate the subgrid-scale stress, SGS scalar flux, SGS scalar variance, as well as related quantities from the synthetic fields. Comparison with direct numerical simulations (DNSs) shows that the synthetic fields reproduce the probability distributions of the SGS energy and scalar dissipation rather well. Related geometrical statistics also display close agreement with DNS results. The synthetic fields slightly under-estimate the mean SGS energy dissipation and slightly over-predict the mean SGS scalar variance dissipation. In general, the synthetic fields tend to slightly under-estimate the probability of large fluctuations for most quantities we have examined. Small scale anisotropy in the scalar field originated from the imposed mean gradient is captured. The sensitivity of the synthetic fields on the input spectra is assessed by using truncated spectra or model spectra as the input. Analyses show that most of the SGS statistics agree well with those from MTLM fields with DNS spectra as the input. For the mean SGS energy dissipation, some significant deviation is observed. However, it is shown that the deviation can be parametrized by the input energy spectrum, which demonstrates the robustness of the MTLM procedure.

Resonance phenomena in a two-layer two-vortex shear flow.

The paper deals with a dynamical system governing the motion of two point vortices embedded in the bottom layer of a two-layer rotating flow experiencing linear deformation and their influence on fluid particle advection. The linear deformation consists of shear and rotational components. If the deformation is stationary, the vortices can move periodically in a bounded region. The vortex periodic motion induces stirring patterns of passive fluid particles in the both layers. We focus our attention on the upper layer where the bottom-layer singular point vortices induce a regular velocity field with no singularities. In the upper layer, we determine a steady-state regime featuring no closed fluid particle trajectories associated with the vortex motion. Thus, in the upper layer, the flow's streamlines look like there is only external linear deformation and no vortices. In this case, fluid particles move along trajectories of almost regular elliptic shapes. However, the system dynamics changes drastically if the underlying vortices cease to be stationary and instead start moving periodically generating a nonstationary perturbation for the fluid particle advection. Then, we demonstrate that this steady-state regime transits to a perturbed state with a rich phase portrait structure featuring both periodic and chaotic fluid particle trajectories. Thus, the perturbed state clearly manifests the impact of the underlying vortex motion. An analysis, based on comparing the eigenfrequencies of the steady-state fluid particle rotation with the ones of the vortex rotation, is carried out, and parameters ensuring effective fluid particle stirring are determined. The process of separatrix reconnection of close stability islands leading to an enhanced chaotic region is reported and analyzed.

Vortex dynamics of a fluid near a boundary with a circular cavity

The motion of a point vortex along a rectilinear boundary with a circle cavity, which models the coastline of a bay, and associated fluid particle advection are studied within a model of barotropic inviscid fluid. Using an analytical expression for the complex potential through which the velocity field is determined, we show that fluid particles start moving irregularly when the vortex is passing the cavity due to the nonstationarity of the velocity field generated by the vortex. Some of the fluid particles which were initially inside the vortex atmosphere leave it due to the irregularity and remain within the cavity vicinity. Depending on the initial position of the vortex and a parameter that determines the cavity size, the fraction of these fluid particles can differ significantly from fluid particles initially uniformly distributed within the vortex atmosphere. The escape of fluid particles from the vortex atmosphere is shown to be most efficient in the case of a relatively closed cavity under the condition that the initial vortex atmosphere area should be significantly smaller than the cavity area.

Interaction of a monopole vortex with an isolated topographic feature in a three-layer geophysical flow

Abstract. In the frame of a three-layer, quasi-geostrophic analytical model of an f-plane geophysical flow, the Lagrangian advection induced by the interaction of a monopole vortex with an isolated topographic feature is addressed. Two different cases when the monopole is located either within the upper or the middle layer are of our interest. In the bottom layer, there is a delta-function topographic feature, which generates a closed recirculation region in its vicinity due to the background flow. This recirculation region extends to the middle and upper layers, and it plays the role of a topographic vortex. The interaction between the monopole and the topographic vortex causes a complex, including chaotic, advection of fluid particles. We show that the model's parameters, namely the monopole and topographic vortices' strengths and initial positions, and the layers' depths and densities, are responsible for the diverse advection patterns. While the patterns are rather complicated, one can single out two major processes, which mostly govern the fluid particle advection. The first one is the variation in time of the system's phase space structure, so that within the closed region of the topographic vortex, there appear periodically unclosed particle pathways by which the particles leave the topographic vortex. The second one is chaotic advection that arises from the nonstationarity of the monopole–topography interaction.

Fluid particle advection in the vicinity of the Föppl vortex system

Fluid particle advection in the vicinity of the Föppl vortex system is considered. Due to periodic motion of vortices about the Föppl equilibrium, fluid particles within the vortex atmosphere, the fluid region with a velocity field being induced by the vortices, can move chaotic in the sense of exponential divergence of near trajectories. This chaotic motion leads to the vortex atmosphere particles to be carried away from the atmosphere to the exterior flow. In this Letter, the part of the carried away fluid particles is numerically assessed and the dynamics of the fluid release from the vortex atmosphere is demonstrated.

A topological approach to three dimensional laminar mixing

In recent years, topological concepts have yielded valuable insights into the long standing problem of laminar fluid mixing. Topologically complex stirring protocols are typically far superior to topologically simple protocols, guaranteeing chaotic advection of fluid particles and the associated exponential dilation of material elements. Furthermore, topological approaches to mixer design are typically intuitive and insensitive to precise geometry or fluid properties. However, results to date have been limited to two dimensional flows (for example, batch stirrers in food or polymer manufacturing) and quasi three dimensional protocols (for example, continuous flow micromixers). Motivated by a simple stretching and folding argument that works well in two dimensions, we propose a topological approach to fully three dimensional fluid mixing. A transition matrix is derived to describe the mapping induced by a three dimensional `braid' on area elements, and the associated Perron--Frobenius eigenvalue provides a prediction of the large time asymptotic area growth rate. We show that these theoretical predictions agree well with numerical data obtained from simulations in a prototype three dimensional mixing device. References M. Bestvina and M. Handel. Train tracks for surface homeomorphisms, Topology 34, 1995, 109--140. doi:10.1016/0040-9383(94)E0009-9 P. L. Boyland, H. Aref and M. A. Stremler. Topological fluid mechanics of stirring, J. Fluid Mech., 403, 2000, 277--304. doi:10.1017/S0022112099007107 J. Chen and M. A. Stremler. Topological chaos and mixing in three dimensional channel flow. {Physics of Fluids}, 21, 2009, 021701. doi:10.1063/1.3076247 M. D. Finn and J.--L. Thiffeault. Topological entropy of braids on the torus. SIAM J. Appl. Dyn. Sys., 6, 2007, 79--98. doi:10.1137/060659636 O. S. Galaktionov, P. D. Anderson, G. M. W. Peters and F. N. van de Vosse. An adaptive front tracking technique for three dimensional transient flows. Int. J. Numer. Meth. Fluids, 32, 2000, 201--217. E. Gouillart, N. Kuncio, O. Dauchot, B. Dubrulle, S. Roux and J.--L. Thiffeault. Walls inhibit chaotic mixing. Phys. Rev. Lett. 99, 2007, 114501. T. G. Kang, M. K. Singh, T. H. Kwon and P. D. Anderson. Chaotic mixing using periodic and aperiodic sequences of mixing protocols in a micromixer. Microfluid Nanofluid, 4, 2008, 589--99. doi:10.1007/s10404-007-0004-x J.--O. Moussafir. On computing the entropy of braids. Func. Anal. Other Math., 1, 2006, 37--46. doi:10.1007/s11853-007-0206-z H. A. Stone and S. Kim. Microfluidics: basic issues, applications and challenges. American Institute of Chemical Engineers Journal, 47, 2001, 1250--54. doi:10.1002/aic.690470602 M. A. Stremler and J. Chen. Generating topological chaos in lid driven cavity flow, Physics of Fluids, 19, 2007, 103602. doi:10.1063/102772881 M. A. Stremler, F. R. Haselton and H. Aref. Designing for chaos: applications of chaotic advection at the microscale. Phil. Trans. R. Soc. Lond. A, 362, 2004, 1019--36. J.-L. Thiffeault and M. D. Finn. Topology, braids and mixing in fluids. Phil. Trans. R. Soc. Lond. A, 364, 2006, 3251--66. doi:10.1098/rsta.2006.1899 C. R. Thomas. Problems of shear in biotechnology. Critical Reports on Applied Chemistry, 29, 1990, 23--93. S. O. Unverdi and G. Tryggvason. A front tracking method for viscous, incompressible, multi-fluid fluids. J. Comp. Phys., 100, 1992, 25--37. G. M. Whitesides and A. D. Stroock. Flexible methods for microfluidics. Physics Today, 54, 2001, 42--48.

Chaotic advection and nonlinear resonances in an oceanic flow above submerged obstacle

The effect of an isolated submarine obstacle on the motion of fluid particles in a periodic external flow is studied within the framework of the barotropic, quasi-geostrophic approximation on f -plane. The concept of background currents advanced by Kozlov [1995. Background currents in geophysical hydrodynamics. Izvestia, Atmos. Oceanic Phys. 31 (2), 245–250] is used to construct a dynamically consistent stream function satisfying the potential vorticity conservation law. It is shown that a system of two topographic vortices revolving about a rotation center can form in a circular external flow. Unsteady periodic perturbations, associated with either variations in the background current or deviations of the external flow from circulation, are analyzed. Unsteadiness in the external flow essentially complicates the pattern of the motion of fluid particles. Vortex-type quasi-periodic structures, identified with nonlinear resonances that form in Lagrangian equations of fluid particle advection, are examined. They either surround the stationary configuration by a vortex chain—a ringlet-like structure [ Kennelly, M.A., Evans, R.H., Joyce, T.M., 1985. Small-scale cyclones on the periphery of Gulf Stream warm-core rings. J. Geophys. Res. 90(5), 8845–8857], or they form a complex-structure multivortex domain. Asymptotic estimates and numerical modeling are used to study the distribution and widths of the nonlinear resonance domains that appear under unsteady perturbations of different types. The onset of chaotic regimes owing to the overlapping of nonlinear resonance domains is analyzed. Transport fluxes determined by chaotic advection and barriers for transport (KAM-tori) and the conditions of their existence are studied. The relation of the rotation frequency of fluid particles on their initial position (when the dependence is calculated in the undisturbed system) is shown to completely determine the main features of the pattern of Lagrangian trajectories and chaotization effects. Because of nonlinear effects, the domain involved in quasi-periodic and chaotic motions can be much larger than the domain occupied by steady topographic vortices. The results of study by Sokolovskiy et al. [1988. On the influence of an isolated submerged obstacle on a barotropic tidal flow. Geophys. Astrophys. Fluid Dyn. 88(1), 1–30] concerning the due regard on the irrotational background component as the necessary factor for the transportation of fluid particles from the vortex domain to infinity are confirmed.

AN EXPERIMENTAL INVESTIGATION AND MODELLING OF THE DEFFUSION PHENOMENA UNDER WIND WAVES

In the aspect of water-environments, the studies on wind waves are very important regarding transport phenomena in atmosphere, lakes and sea. Dispersion phenomena in water under wind waves have been studied by many researchers, but they have taken notice of only surface layer beneath the free surface. The purpose of this study is to elucidate the diffusion in the surface layer and the lower layer as well. Visualization of dye diffusion and velocity field measurements by means of PIV and 2D-electro magnetic velocity meters were performed. The mechanisms of dye diffusion between the surface and the lower layers are different. In the surface layer, the diffusion can be estimated well by using Taylor's turbulent diffusion theory. In the lower layer, however, we found that the diffusion is like the chaotic mixing. To investigate qualitative feature of the chaotic mixing under wind waves, a mathematical model represented by the superposition of a periodic array of vortices and stationary water waves was proposed. The coexistence of free surface waves and large-scale vortices causes chaotic advection of fluid particles. This phenomenon will induce strong mixing of water in the lower layer even if the turbulent intensity in this layer is small.

Chaotic advection by two interacting finite-area vortices

This article deals with the advection of fluid particles in the velocity field of two identical vortices with various vorticity distributions. The two-dimensional velocity field is aperiodic in the range of parameters studied here, namely, the neighborhood of the critical distance for merger. Ideas and methods from the theory of transport in dynamical systems are used to describe and quantify particle advection. These methods are applied to the numerical representation of the velocity field, which is obtained by solving the Euler equations with the vortex-in-cell method. It is found that the strongest stirring of vortex fluid occurs slightly above the critical distance for merger. In this regime the fluid located between the vortices is subjected to intense stirring, and some vortex fluid may be entrained into the chaotic region depending on the smoothness of the vorticity distribution. Initial conditions below the critical distance lead to stirring of fluid mainly before merger. In this case the flow geometry is used to quantify the efficiency of merger, which is defined as the ratio of the circulation of the resultant vortex to the total circulation of the original vortices. It is found that the vortices with the smoothest vorticity profile have the lowest efficiency. Experimental visualizations in a two-dimensional rotating fluid confirm the intense stretching and folding of fluid elements that occurs before the vortices merge.

An analysis of a Stokes flow in an annular region

A study is made of the steady two-dimensional Stokes flow stirred by an infinitesimal rotating cylinder (a line rotlet) in the annular region between two fixed concentric cylindrical walls. It is shown that this simple flow exhibits a rich diversity of possible flow structures. Generally, one com- ponent of the flow is associated with a net flux through the annular region in either a clockwise or anticlockwise direction depending on whether the radial distance between the rotlet and the center axis of the bounding cylinders is greater or less than a unique value at which this flow component vanishes. The remaining components of the flow are either eddies at- tached to the boundary or free eddies in the interior of the flow. Owing to some of the dynamical features of this flow, the results obtained here may be applicable in engineering set- tings where mixing by the chaotic advection of fluid particles is desirable.

Tidal mixing and cross-frontal particle exchange over a finite amplitude asymmetric bank: A model study with application to Georges Bank

Tidal mixing, internal wave bores, and cross-bank particle transport over a finite amplitude asymmetric bank are examined using a two-dimensional primitive equation ocean model with Mellor and Yamada (level 2.5) turbulent closure. Driven by the surface M 2 tide, the model results show that tidal mixing exhibits temporal and spatial asymmetries on southern and northern flanks of the bank. It is strongest near the bottom around maximum on-bank tidal flow as a result of gravitational instability when denser water is advected upslope over lighter water, A sharp thermal depression occurs on the steep northern flank, which produces large current shear and strong tidal mixing throughout the upper 50 m of the water column. Dissipation also exhibits a strong tidal variation, with the largest values of order 10 5 W/kg) occurring near-bottom around maximum on- and off-bank tidal flow. Dissipation generally decreases upward, with a distinct phase lag in the vertical. Fluid particles are advected upslope near the bottom in the upper slope region (depth <150 m) on both flanks, with some particles moving across the tidal mixing fronts near the bottom. The near-bottom residual Lagrangian current is opposite in direction to the residual Eulerian current on the northern flank due to strong nonlinearity over the steep bottom slope. The mean upslope advection of fluid particles near the bottom on both flanks is consistent with model passive tracer experiments, suggesting that strong tidal forcing of a stratified fluid over the bank can provide one physical mechanism responsible for high concentrations of nutrients and hence phytoplankton at the fronts on Georges Bank. The model predictions of eddy viscosity and turbulent dissipation rates are in good agreement with estimates based on recent current and microstructure measurements made on Georges Bank.

Collision of dipolar vortices on a β plane

The interaction of two dipoles moving perpendicularly to the gradient of background vorticity is studied both numerically and experimentally. In the numerical computations the vorticity distribution is represented either by four point vortices (point-vortex model) or by thousands of them (vortex-in-cell method). The simplest model is used to study the dynamics and the advection of fluid particles in two kinds of interaction: coaxial couples of equal strength but different size, and equal parallel couples with a nonzero impact parameter (the distance between the dipoles’ axis). As a result of the interaction fluid masses are exchanged between the two dipoles and between each dipole and the ambient fluid. In the case of equal coaxial couples the amount of fluid exchanged depends on the gradient of ambient vorticity, with the largest mass exchange occurring always between the eastward traveling dipole and the ambient fluid. The collision of parallel couples with nonzero impact parameter leads to a large mass exchange, either because several interactions may occur or because when two independent couples arise, they have a nonuniform motion. Laboratory experiments in a rotating fluid (with a flat sloping bottom providing the β effect), confirm that an elastic interaction is a rare event. The unstable trajectory of the westward traveling dipole, as well as small perturbations unavoidable in the laboratory, invariably lead to collisions of nonaligned dipoles. The gross features of the vortex motion, as well as of the mass exchange, are well modeled using the point-vortex model, whereas the vortex-in-cell method reproduces many details of the vortex motion, the evolution of the vorticity field, and the exchange of mass.

Chaotic advection of fluid particles in a field of point vortices and vortex rings

Processes of formation of vortex rings and mushroom structures in the ocean, in geological structures in Earth’s mantle, and in various applications in chemical technology,1 provide conditions for advanced interest in the phenomena of internal mixing. In order to model such processes, an approach based on a direct study of the trajectories of motion of Lagrangian particles in a Eulerian velocity field currently attracts special attention. Although the ideas of such an approach may be found in the works of Maxwell2 and Reynolds,3 comparatively little has been done toward obtaining a clear view of what goes on, except in the simplest cases. To the more complicated cases belong the processes of chaotic advection of particles in bounded domains.4 This paper presents the results of a study of properties of chaotic particle advection of an ideal fluid in unbounded and bounded domains under free motion of point vortices or vortex rings. Data on the process of particle motion for particles that initially form the ‘‘atmosphere’’ of a vortex pair when it interacts with a single vortex, or with other noncollinear vortex pairs, are presented for a two-dimensional unbounded domain. Depending on the type of interaction—either ‘‘direct’’ or ‘‘exchange’’ collision or mutual trapping5–7—an estimate of the effectiveness of stirring of atmosphere particles and chaotization of their trajectories is given. The problem of advection of particles in a two-dimensional rectangular domain with a free vortex is solved. Although the locked trajectory of the vortex and the period of its revolution are explicitly calculated by means of Jacobi and Weierstrass elliptic functions, a complicated dependence of the streamfunction on time leads to the formation of ‘‘whorls’’ and ‘‘tendrils,’’ and to effective stirring of some domains. Poincaré sections of particles for characteristic values of the geometrical parameters of the domain and initial positions of the vortex are given. Various types of interaction such as ‘‘leap-frogging,’’ passing and mutual trapping,8 are studied for the axisymmetric motion of two coaxial vortex rings with the intensities of vorticity of the same or different signs. An analysis of chaotization of the motion for particles initially forming the atmosphere of the rings has been carried out for some cases of interaction.

Chaotic advection of fluid particles

Particle motion in a fluid can be chaotic even when the flow field is very simple from an eulerian point of view. This basic feature of fluid kinematics, known as chaotic advection, is reviewed and a number of applications are cited. The notion of a chaotic ‘kinematic template’ underlying dynamical processes is introduced and discussed. Some emerging directions of investigation for this application of chaos to fluid mechanics are indicated.

Mixing in laminar flows by chaotic advection

It is pointed out that mixing of clouds of fluid particles can take place by pure advection in laminar flows. This is illustrated in a simple model of flow consisting of the superposition of steady flow and a free oscillation mode in a shallow layer of water in a rectangular basin. Here the generation of regions of chaotic advection of fluid particles occurs via homoclinic bifurcation of the stable and unstable manifolds of an equilibrium point. Similar behaviours are found in plane reversible maps that model the essential geometric properties of the Poincare map associated with the set of ordinary differential equations describing advection in two-dimensional periodic flows with reflection symmetry and discretized time periodicity.