Chaotic advection,diffusion, and reactions in open flows

We review and generalize recent results on advection of particles in open time-periodic hydrodynamical flows. First, the problem of passive advection is considered, and its fractal and chaotic nature is pointed out. Next, we study the effect of weak molecular diffusion or randomness of the flow. Finally, we investigate the influence of passive advection on chemical or biological activity superimposed on open flows. The nondiffusive approach is shown to carry some features of a weak diffusion, due to the finiteness of the reaction range or reaction velocity. (c) 2000 American Institute of Physics.     

Population dynamics advected by chaotic flows: A discrete-time map approach

A discrete-time model of reacting evolving fields, transported by a bidimensional chaotic fluid flow, is studied. Our approach is based on the use of a Lagrangian scheme where fluid particles are advected by a two-dimensional symplectic map possibly yielding Lagrangian chaos. Each fluid particle carries concentrations of active substances which evolve according to its own reaction dynamics. This evolution is also modeled in terms of maps. Motivated by the question, of relevance in marine ecology, of how a localized distribution of nutrients or preys affects the spatial structure of predators transported by a fluid flow, we study a specific model in which the population dynamics is given by a logistic map with space-dependent coefficient, and advection is given by the standard map. Fractal and random patterns in the Eulerian spatial concentration of predators are obtained under different conditions. Exploiting the analogies of this coupled-map (advection plus reaction) system with a random map, some features of these patterns are discussed. (c) 2001 American Institute of Physics.     

Chaotic mixing induced transitions in reaction-diffusion systems

We study the evolution of a localized perturbation in a chemical system with multiple homogeneous steady states, in the presence of stirring by a fluid flow. Two distinct regimes are found as the rate of stirring is varied relative to the rate of the chemical reaction. When the stirring is fast localized perturbations decay towards a spatially homogeneous state. When the stirring is slow (or fast reaction) localized perturbations propagate by advection in form of a filament with a roughly constant width and exponentially increasing length. The width of the filament depends on the stirring rate and reaction rate but is independent of the initial perturbation. We investigate this problem numerically in both closed and open flow systems and explain the results using a one-dimensional "mean-strain" model for the transverse profile of the filament that captures the interplay between the propagation of the reaction-diffusion front and the stretching due to chaotic advection. (c) 2002 American Institute of Physics.     

A grid based particle method for solving partial differential equations on evolving surfaces and modeling high order geometrical motion

We develop numerical methods for solving partial differential equations (PDE) defined on an evolving interface represented by the grid based particle method (GBPM) recently proposed in [S. Leung, H.K. Zhao, A grid based particle method for moving interface problems, J. Comput. Phys. 228 (2009) 7706–7728]. In particular, we develop implicit time discretization methods for the advection–diffusion equation where the time step is restricted solely by the advection part of the equation. We also generalize the GBPM to solve high order geometrical flows including surface diffusion and Willmore-type flows. The resulting algorithm can be easily implemented since the method is based on meshless particles quasi-uniformly sampled on the interface. Furthermore, without any computational mesh or triangulation defined on the interface, we do not require remeshing or reparametrization in the case of highly distorted motion or when there are topological changes. As an interesting application, we study locally inextensible flows governed by energy minimization. We introduce tension force via a Lagrange multiplier determined by the solution to a Helmholtz equation defined on the evolving interface. Extensive numerical examples are also given to demonstrate the efficiency of the proposed approach.     

Can aerosols be trapped in open flows?

The fate of aerosols in open flows is relevant in a variety of physical contexts. Previous results are consistent with the assumption that such finite-size particles always escape in open chaotic advection. Here we show that a different behavior is possible. We analyze the dynamics of aerosols both in the absence and presence of gravitational effects, and both when the dynamics of the fluid particles is hyperbolic and nonhyperbolic. Permanent trapping of aerosols much heavier than the advecting fluid is shown to occur in all these cases. This phenomenon is determined by the occurrence of multiple vortices in the flow and is predicted to happen for realistic particle-fluid density ratios.     

Universality in active chaos

Many examples of chemical and biological processes take place in large-scale environmental flows. Such flows generate filamental patterns which are often fractal due to the presence of chaos in the underlying advection dynamics. In such processes, hydrodynamical stirring strongly couples into the reactivity of the advected species and might thus make the traditional treatment of the problem through partial differential equations difficult. Here we present a simple approach for the activity in inhomogeneously stirred flows. We show that the fractal patterns serving as skeletons and catalysts lead to a rate equation with a universal form that is independent of the flow, of the particle properties, and of the details of the active process. One aspect of the universality of our approach is that it also applies to reactions among particles of finite size (so-called inertial particles).     

A conservative scheme for the relativistic Vlasov–Maxwell system

A new scheme for numerical integration of the 1D2V relativistic Vlasov–Maxwell system is proposed. Assuming that all particles in a cell of the phase space move with the same velocity as that of the particle located at the center of the cell at the beginning of each time step, we successfully integrate the system with no artificial loss of particles. Furthermore, splitting the equations into advection and interaction parts, the method conserves the sum of the kinetic energy of particles and the electromagnetic energy. Three test problems, the gyration of particles, the Weibel instability, and the wakefield acceleration, are solved by using our scheme. We confirm that our scheme can reproduce analytical results of the problems. Though we deal with the 1D2V relativistic Vlasov–Maxwell system, our method can be applied to the 2D3V and 3D3V cases.     

Basin topology in dissipative chaotic scattering

Chaotic scattering in open Hamiltonian systems under weak dissipation is not only of fundamental interest but also important for problems of current concern such as the advection and transport of inertial particles in fluid flows. Previous work using discrete maps demonstrated that nonhyperbolic chaotic scattering is structurally unstable in the sense that the algebraic decay of scattering particles immediately becomes exponential in the presence of weak dissipation. Here we extend the result to continuous-time Hamiltonian systems by using the Henon-Heiles system as a prototype model. More importantly, we go beyond to investigate the basin structure of scattering dynamics. A surprising finding is that, in the common case where multiple destinations exist for scattering trajectories, Wada basin boundaries are common and they appear to be structurally stable under weak dissipation, even when other characteristics of the nonhyperbolic scattering dynamics are not. We provide numerical evidence and a geometric theory for the structural stability of the complex basin topology.     

Sharp interface tracking using the phase-field equation

A general interface tracking method based on the phase-field equation is presented. The zero phase-field contour is used to implicitly track the sharp interface on a fixed grid. The phase-field propagation equation is derived from an interface advection equation by expressing the interface normal and curvature in terms of a hyperbolic tangent phase-field profile across the interface. In addition to normal interface motion driven by a given interface speed or by interface curvature, interface advection by an arbitrary external velocity field is also considered. In the absence of curvature-driven interface motion, a previously developed counter term is used in the phase-field equation to cancel out such motion. Various modifications of the phase-field equation, including nonlinear preconditioning, are also investigated. The accuracy of the present method is demonstrated in several numerical examples for a variety of interface motions and shapes that include singularities, such as sharp corners and topology changes. Good convergence with respect to the grid spacing is obtained. Mass conservation is achieved without the use of separate re-initialization schemes or Lagrangian marker particles. Similarities with and differences to other interface tracking approaches are emphasized.     

Operator Splitting Implicit Integration Factor Methods for Stiff Reaction-Diffusion-Advection Systems

For reaction-diffusion-advection equations, the stiffness from the reaction and diffusion terms often requires very restricted time step size, while the nonlinear advection term may lead to a sharp gradient in localized spatial regions. It is challenging to design numerical methods that can efficiently handle both difficulties. For reaction-diffusion systems with both stiff reaction and diffusion terms, implicit integration factor (IIF) method and its higher dimensional analog compact IIF (cIIF) serve as an efficient class of time-stepping methods, and their second order version is linearly unconditionally stable. For nonlinear hyperbolic equations, weighted essentially non-oscillatory (WENO) methods are a class of schemes with a uniformly high-order of accuracy in smooth regions of the solution, which can also resolve the sharp gradient in an accurate and essentially non-oscillatory fashion. In this paper, we couple IIF/cIIF with WENO methods using the operator splitting approach to solve reaction-diffusion-advection equations. In particular, we apply the IIF/cIIF method to the stiff reaction and diffusion terms and the WENO method to the advection term in two different splitting sequences. Calculation of local truncation error and direct numerical simulations for both splitting approaches show the second order accuracy of the splitting method, and linear stability analysis and direct comparison with other approaches reveals excellent efficiency and stability properties. Applications of the splitting approach to two biological systems demonstrate that the overall method is accurate and efficient, and the splitting sequence consisting of two reaction-diffusion steps is more desirable than the one consisting of two advection steps, because CWC exhibits better accuracy and stability.     

The combined Lagrangian advection method

We present and test a new hybrid numerical method for simulating layerwise-two-dimensional geophysical flows. The method radically extends the original Contour-Advective Semi-Lagrangian (CASL) algorithm [5] by combining three computational elements for the advection of general tracers (e.g. potential vorticity, water vapor, etc.): (1) a pseudo-spectral method for large scales, (2) Lagrangian contours for intermediate to small scales, and (3) Lagrangian particles for the representation of general forcing and dissipation. The pseudo-spectral method is both efficient and highly accurate at large scales, while contour advection is efficient and accurate at small scales, allowing one to simulate extremely fine-scale structure well below the basic grid scale used to represent the velocity field. The particles allow one to efficiently incorporate general forcing and dissipation.     

Double cascade turbulence and Richardson dispersion in a horizontal fluid flow induced by Faraday waves

We report the experimental observation of Richardson dispersion and a double cascade in a thin horizontal fluid flow induced by Faraday waves. The energy spectra and the mean spectral energy flux obtained from particle image velocimetry data suggest an inverse energy cascade with Kolmogorov type scaling E(k) ∝ k(γ), γ ≈ -5/3 and an E(k) ∝ k(γ), γ ≈ -3 enstrophy cascade. Particle transport is studied analyzing absolute and relative dispersion as well as the finite size Lyapunov exponent (FSLE) via the direct tracking of real particles and numerical advection of virtual particles. Richardson dispersion with <ΔR(2)(t)> ∝ t(3) is observed and is also reflected in the slopes of the FSLE (Λ ∝ ΔR(-2/3)) for virtual and real particles.     

Shear-flow suppression of hotspot growth

We consider the role of fluid shear in maintaining anomalous “hotspot” solutions reported for a chaotically stirred bistable chemical reaction model [S.M. Cox, Persistent localized states for a chaotically mixed bistable reaction, Phys. Rev. E 74 (2006) 056206]. In the well-mixed regime, the chemical concentration is governed by a single autonomous ordinary differential equation with two stable equilibria. Whether the reaction goes to extinction or to an excited state depends on whether the initial concentration lies below or above some unstable threshold. By contrast, when the concentration varies spatially, and the chemical is stirred, the interplay between advection, diffusion and reaction is much more complicated, and the fate of the reaction depends sensitively on the initial conditions. It has previously been shown that, if the stirring is temporally periodic, a localised hotspot may form, and so the system never becomes fully extinct, nor fully excited. In this work, we first demonstrate that hotspots are in some sense generic, in that they are easily found by trial and error, by choosing reaction parameter values between those giving rise to global extinct and excited states. We also show that hotspots may be associated with hyperbolic, elliptic or even parabolic orbits associated with the underlying stirring. We observe that fluid shear is an important mechanism in localising a hotspot, and derive a reduced ordinary differential equation model which can predict the fate of a chemical stripe in a shear flow accurately.     

Concurrent implicit spectral deferred correction scheme for low-Mach number combustion with detailed chemistry

We present a parallel multi-implicit time integration scheme for the advection-diffusion-reaction systems arising from the equations governing low-Mach number combustion with complex chemistry. Our strategy employs parallelisation across the method to accelerate the serial Multi-Implicit Spectral Deferred Correction (MISDC) scheme used to couple the advection, diffusion, and reaction processes. In our approach, the diffusion solves and the reaction solves are performed concurrently by different processors. Our analysis shows that the proposed parallel scheme is stable for stiff problems and that the sweeps converge to the fixed-point solution at a faster rate than with serial MISDC. We present numerical examples to demonstrate that the new algorithm is high-order accurate in time, and achieves a parallel speedup compared to serial MISDC.     

A self-organizing Lagrangian particle method for adaptive-resolution advection–diffusion simulations

We present a novel adaptive-resolution particle method for continuous parabolic problems. In this method, particles self-organize in order to adapt to local resolution requirements. This is achieved by pseudo forces that are designed so as to guarantee that the solution is always well sampled and that no holes or clusters develop in the particle distribution. The particle sizes are locally adapted to the length scale of the solution. Differential operators are consistently evaluated on the evolving set of irregularly distributed particles of varying sizes using discretization-corrected operators. The method does not rely on any global transforms or mapping functions. After presenting the method and its error analysis, we demonstrate its capabilities and limitations on a set of two- and three-dimensional benchmark problems. These include advection–diffusion, the Burgers equation, the Buckley–Leverett five-spot problem, and curvature-driven level-set surface refinement.     

Phototactic clustering of swimming microorganisms in a turbulent velocity field

We study the distribution of swimming microorganisms advected by a two-dimensional smooth turbulent flow and attracted towards a light source through phototaxis. It is shown that particles aggregate along a dynamical attractor with fractal measure whose dimension depends on the strength of the phototaxis. Using an effective diffusion approximation for the flow, we derive an analytic expression for the increase in light exposure over the aggregate and by extension an accurate prediction for the fractal dimension based on the properties of the advection and the statistics of the attracting field.     

Complete condensation in a zero range process on scale-free networks

We study a zero range process on scale-free networks in order to investigate how network structure influences particle dynamics. The zero range process is defined with the rate p(n) = n(delta) at which particles hop out of nodes with n particles. We show analytically that a complete condensation occurs when delta < or = delta(c) triple bond 1/(gamma-1) where gamma is the degree distribution exponent of the underlying networks. In the complete condensation, those nodes whose degree is higher than a threshold are occupied by macroscopic numbers of particles, while the other nodes are occupied by negligible numbers of particles. We also show numerically that the relaxation time follows a power-law scaling tau approximately L(z) with the network size L and a dynamic exponent z in the condensed phase.     

Population genetics in compressible flows

We study competition between two biological species advected by a compressible velocity field. Individuals are treated as discrete Lagrangian particles that reproduce or die in a density-dependent fashion. In the absence of a velocity field and fitness advantage, number fluctuations lead to a coarsening dynamics typical of the stochastic Fisher equation. We investigate three examples of compressible advecting fields: a shell model of turbulence, a sinusoidal velocity field and a linear velocity sink. In all cases, advection leads to a striking drop in the fixation time, as well as a large reduction in the global carrying capacity. We find localization on convergence zones, and very rapid extinction compared to well-mixed populations. For a linear velocity sink, one finds a bimodal distribution of fixation times. The long-lived states in this case are demixed configurations with a single interface, whose location depends on the fitness advantage.     

Steady state in two-dimensional diffusion-controlled reactions

We investigate the conditions under which a steady state can be reached in a two-dimensional diffusion-controlled trapping reaction. If there is no interaction between trap and diffusing particles, the reaction rate decreases monotonically to zero. Here we show that a logarithmic attractive potential between trap and diffusing particles leads to a finite steady-state reaction rate. A steady state can also be reached if the diffusing particles move under the action of a uniform external field. More unexpectedly, a steady-state rate can be obtained in the absence of any “assisting field” if the trap grows due to the absorption of the diffusing particles. The reaction rates are calculated in all cases.     

Where do inertial particles go in fluid flows?

We derive a general reduced-order equation for the asymptotic motion of finite-size particles in unsteady fluid flows. Our inertial equation is a small perturbation of passive fluid advection on a globally attracting slow manifold. Among other things, the inertial equation implies that particle clustering locations in two-dimensional steady flows can be described rigorously by the Q parameter, i.e., by one-half of the squared difference of the vorticity and the rate of strain. Use of the inertial equation also enables us to solve the numerically ill-posed problem of source inversion, i.e., locating initial positions from a current particle distribution. We illustrate these results on inertial particle motion in the Jung-Tél-Ziemniak model of vortex shedding behind a cylinder in crossflow.