Chaotic Advection

Mixing and transport processes are extremely important topics with regard to the water quality of environmental flows. Examples include transport and dispersion of contaminants, distribution of mine tailings in tailings ponds, growth and decay of micro-organisms in reservoirs or lakes, motions of suspended sediment in rivers, lakes and estuaries, distribution of heavy metal particles near outfalls, and so on.  Recent research has shown that mixing in non-turbulent flows can be greatly enhanced by complicated particle behaviour caused by chaotic advection.  Chaotic advection, firstly introduced by Hassan Aref in early 1980s, is a state-of-the-art concept derived from nonlinear dynamics and is widely used as an approach to investigate transport and mixing problems in fluid flows.

Consider a passive particle tracer in a two-dimensional flow field (u, v).  Here ‘passive’ means that the particle is inert and light, so its presence has no effect on the background fluid flow.  The motion of the individual particles can be described by the following system of ordinary differential equations

,

which are the so-called advection equations and provide a Lagrangian description of the fluid motion.  The above two-dimensional advection equations are integrable if the flow is steady, while for unsteady flow they may be non-integrable.  Integrable solutions of the advection equations lead to regular advection, and the non-integrable cases are characterized by chaotic advection which can be described as particle motion sensitive to initial conditions, i.e., initially nearby trajectories diverge at an exponential rate.  To produce chaotic advection, the Eulerian velocity field in the advection equations is not necessarily very complicated (turbulent); the Lagrangian particle trajectories can become chaotic even when the Eulerian velocity at any given fixed point in space is periodic in time, e.g. the blinking vortex (BV) flow proposed by Aref (Aref 1984, J. Fluid Mech. 143).  The BV flow is generated by a pair of point vortices separated a fixed distance of 2a apart in an ideal fluid.  The two point vortices blink on and off periodically with a constant period of T in an unbounded domain (Khakhar et al. 1986, J. Fluid Mech. 172).  Consequently, two distinct gyre-type flow patterns occur, each centred about whichever vortex is switched on.  The advected particle dynamics is then governed by a dimensionless flow strength μ and  change from regular to chaotic as μ increases from zero.

Since it was introduced in early 1980s, chaotic advection has been widely used by researchers in diverse areas of fluid mechanics to explain a variety of experimental or real world problems.  However the chaotic advection problems considered in most of the published work are restricted to use idealised analytical, numerical solutions or experimental data as input flow fields.  Few investigations have considered mixing and transport processes involving chaotic advection in natural shallow water bodies. 

We investigate wind-induced passive mixing and transport in shallow water basins.  A shallow water basin, like a lake or reservoir, can often be defined as an enclosed system, in which wind shear is normally the major source of momentum while other factors such as inflows and outflows can only locally modify the flow pattern.  Kranenburg (Kranenburg 1992, J of Hydraulic Research 30) studied the chaotic advection caused by oscillatory wind-induced circulation in a circular lake with the intention of applying the concept of chaotic advection to environmental hydraulics.  Using a simplified analytical model for the wind-induced hydrodynamics, Kranenburg demonstrated that inert particle motions became chaotic when the surface wind stress periodically and abruptly changed direction even though its magnitude was constant.

Kranenburg’s method is not general since an idealised analytical velocity field is not available for natural shallow water flows, where the domain geometries and flow patterns are usually very complicated.  Herein we examine the chaotic mixing of particles in shallow flows using numerical methods that are applicable to natural cases.  The wind-induced flow field is numerically predicted using an adaptive quadtree grid based shallow flow solver that can be refined according to the boundary geometry and local flow features.  Wind-induced mixing and transport of passive re-suspended material in a natural tailings pond is then considered.  And the wind field is assumed to be sinusoidally changing , to provide a more realistic representation for natural cases.

In certain cases, the particles themselves impose dynamical effects in addition to being advected by the background fluid flows.  Examples include the growth and decay of algae in lakes, and chemical reactions due to mine tailing re-suspension.  The dynamical behavour of active particles is significantly different from that of passive particle tracers in chaotic hydrodynamical flows.  Herein, the advected active particles are assumed to undergo certain chemical or biological changes but do not affect the background fluid flow, i.e., the advected substances are chemically or biologically active but hydrodynamically passive.  This topic has attracted considerable attentions, and the derived theories have been applied to a wide range of chemical and ecological problems.

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