In chaos theory and fluid dynamics, chaotic mixing is a processby which flow tracers develop into complex fractals under the actionof a fluid flow.The flow is characterized by an exponential growth of fluid filaments.[1] [2] Even very simple flows, such as the blinking vortex,or finitely resolved wind fields can generate exceptionally complexpatterns from initially simple tracer fields.[3]
The phenomenon is still not well understood and is the subjectof much current research.
Two basic mechanisms are responsible for fluid mixing: diffusion and advection. In liquids, molecular diffusion alone is hardly efficient for mixing. Advection, that is the transport of matter by a flow, is required for better mixing.
The fluid flow obeys fundamental equations of fluid dynamics (such as the conservation of mass and the conservation of momentum) called Navier–Stokes equations. These equations are written for the Eulerian velocity field rather than for the Lagrangian position of fluid particles. Lagrangian trajectories are then obtained by integrating the flow. Studying the effect of advection on fluid mixing amounts to describing how different Lagrangian fluid particles explore the fluid domain and separate from each other.
A fluid flow can be considered as a dynamical system, that is a set of ordinary differential equations that determines the evolution of a Lagrangian trajectory. These equations are called advection equations:
| d\vecx | |
| dt |
=\vecv(\vecx,t)
where
\vecv=(u,v,w)
\vecx=(x,y,z)
Dynamical systems and chaos theory state that at least 3 degrees of freedom are necessary for a dynamic system to be chaotic. Three-dimensional flows have three degrees of freedom corresponding to the three coordinates, and usually result in chaotic advection, except when the flow has symmetries that reduce the number of degrees of freedom. In flows with less than 3 degrees of freedom, Lagrangian trajectories are confined to closed tubes, and shear-induced mixing can only proceed within these tubes.
This is the case for 2-D stationary flows in which there are only two degrees of freedom
x
y
For 2-D unstationary (time-dependent) flows, instantaneous closed streamlines and Lagrangian trajectories do not coincide any more. Hence, Lagrangian trajectories explore a larger volume of the volume, resulting in better mixing. Chaotic advection is observed for most 2-D unstationary flows. A famous example is the blinking vortex flow introduced by Aref, where two fixed rod-like agitators are alternately rotated inside the fluid. Switching periodically the active (rotating) agitator introduces a time-dependency in the flow, which results in chaotic advection. Lagrangian trajectories can therefore escape from closed streamlines, and visit a large fraction of the fluid domain.
A flow promotes mixing by separating neighboring fluid particles. This separation occurs because of velocity gradients, a phenomenon called shearing. Let
\vec{x}1
\vec{x}2
\delta\vec{x}=\vec{x}2-\vec{x}1
\vec{v}
t+\deltat
| d | |
| dt |
(\vec{x}+\delta\vec{x}) ≈ \vec{v}+\nabla\vec{v} ⋅ \delta\vec{x}
\deltax(t+\deltat) ≈ \deltax(t)+\deltat(\deltax ⋅ \nabla)\vecv
| d | |
| dt |
\delta\vecx ≈ \nabla\vecv ⋅ \delta\vecx
The rate of growth of the separation is therefore given by the gradient of the velocity field in the direction of the separation. The plane shear flow is a simple example of large-scale stationary flow that deforms fluid elements because of a uniform shear.
If the flow is chaotic, then small initial errors,
\delta\vecx
\nabla\vec{v}
We define the matrix H such that:
| d | |
| dt |
\boldsymbol{H}\equiv\nabla\vec{v} ⋅ \boldsymbol{H}, \boldsymbol{H}(t=0)=\boldsymbol{I}
where I is the identity matrix. It follows that:
\delta\vec{x}(t) ≈ \boldsymbol{H} ⋅ \delta\vec{x}0
The finite-time Lyapunov exponents are defined as the time average of the logarithms of the lengths of the principal components of the vector H over a time t:
\boldsymbol{HT} ⋅ \boldsymbol{H} ⋅ \delta\vec{x}0i=hi ⋅ \delta\vec{x}0i
λi(\vec{x},t)\equiv
| 1 | |
| 2t |
ln{hi(\vec{x},t)}
where
λi(\vec{x},t)\geqλi+1(\vec{x},t)
\delta\vec{x}0i
If we start with a set of orthonormal initial error vectors,
\{\delta\vecx0i\}
\left\{\sqrt{hi(\vec{x},t)}\right\}
\sqrt{h1(\vec{x},t)}
\sqrt{hN(\vec{x},t)}
This definition of Lyapunov exponents is both more elegant and more appropriateto real-world, continuous-time dynamical systems than the more usual definition basedon discrete function maps.Chaos is defined as the existence of at least one positive Lyapunov exponent.
In a chaotic system, we call the Lyapunov exponent the asymptotic value of the greatest eigenvalue of H:
λ=\limtλ1(\vec{x},t).
If there is any significant difference between the Lyapunov exponents then as an error vector evolves forward in time, any displacement in the direction of largest growth will tend to be magnified. Thus:
|\delta\vecx| ≈ |\delta\vecx0|
| λ1t | |
| e |
.
The Lyapunov exponent of a flow is a unique quantity, that characterizes the asymptotic separation of fluid particles in a given flow. It is often used as a measure of the efficiency of mixing, since it measures how fast trajectories separate from each other because of chaotic advection. The Lyapunov exponent can be computed by different methods:
λ=\limtλ1(\vec{x},t)
<λ>trajectories
The equivalence of the two methods is due to the ergodicity of the chaotic system.
The following, exact equation can be derived from an advection-diffusion equation (see below), with a diffusion term (D=0) of zero:
| d\nablaq | |
| dt |
=-\nablaq ⋅ \nabla\vecv
\boldsymbol{H}\prime
| d\boldsymbol{H\prime | |
\boldsymbol{H\prime}=\boldsymbol{H}-1
\lbrace
| \prime\rbrace | |
| h | |
| i |
\boldsymbol{H}\prime
| \prime=1/h | |
| h | |
| i |
\lbrace
| \prime\rbrace | |
| h | |
| i |
Contour advection is another useful method for characterizing chaotic mixing.In chaotic flows, advected contours will grow exponentially over time.The figure above shows the frame-by-frame evolution of a contour advected overseveral days. The figure to the right shows the length of this contouras a function of time.
The link between exponential contour growth and positive Lyapunov exponents iseasy to see. The rate of contour growth is given as:
| dL | |
| dt |
=\int|\nabla\vecv ⋅ d\vecs|
where
d\vecs
L ≈ L0\exp(\barλ1t)
In chaotic advection, a fluid particle travels within a large region, and encounters other particles that were initially far from it. One can then consider that a particle is mixed with particles that travel within the same region. However, the region covered by a trajectory does not always span the whole fluid domain. Poincaré sections are used to distinguish regions of good and bad mixing.
The Poincaré map is defined as the transformation
\begin{align} \boldsymbol{M}\colon\vec{xi}(ti)&\to\vec{x}i+1(ti+1=ti+T,\vec{xi}). \end{align}
\boldsymbol{M}
As an example, the figure presented here (left part) depicts the Poincaré section obtained when one applies periodically a figure-eight-like movement to a circular mixing rod. Some trajectories span a large region: this is the chaotic or mixing region, where good mixing occurs. However, there are also two "holes": in these regions, the trajectories are closed. These are called elliptic islands, as the trajectories inside are elliptic-like curves. These regions are not mixed with the remainder of the fluid. For mixing applications, elliptic islands have to be avoided for two reasons :
Avoiding non-chaotic islands requires understanding the physical origin of these regions. Generally speaking, changing the geometry of the flow can modify the presence or absence of islands. In the figure-eight flow for instance, for a very thin rod, the influence of the rod is not felt far from its location, and almost circular trajectories exist within the loops of the figure-eight. With a larger rod (right part of the figure), particles can escape from these loops and islands do not exist any more, resulting in better mixing.
With a Poincaré section, the mixing quality of a flow can be analyzed by distinguishing between chaotic and elliptic regions. This is a crude measure of the mixing process, however, since the stretching properties cannot be inferred from this mapping method. Nevertheless, this technique is very useful for studying the mixing of periodic flows and can be extended to a 3-D domain.
Through a continual process of stretching and folding, much like in a "baker's map,"tracers advected in chaotic flows will develop into complex fractals.The fractal dimension of a single contour will be between 1 and 2.Exponential growth ensures that the contour, in the limit of verylong time integration, becomes fractal.Fractals composed of a single curve are infinitely long and whenformed iteratively, have an exponential growth rate, just like anadvected contour.The Koch Snowflake, for instance, grows at a rate of 4/3 per iteration.
The figure below shows the fractal dimension of an advected contour as a functionof time, measured in four different ways. A good method of measuring thefractal dimension of an advected contour is the uncertainty exponent.
In fluid mixing, one often wishes to homogenize a species, that can be characterized by its concentration field q. Often, the species can be considered as a passive tracer that does not modify the flow. The species can be for example a dye to be mixed.The evolution of a concentration field
q
.
| \partialq | |
| \partialt |
=\nabla ⋅ D\nablaq-\vec{v} ⋅ \nablaq.
\vec{v}
wB=\sqrt{
| D | |
| λ |
λ
When most dye filaments reach the Batchelor scale, diffusion begins to decrease significantly the contrast of concentration between the filament and the surrounding domain. The time at which a filament reaches the Batchelor scale is therefore called its mixing time. The resolution of the advection–diffusion equation shows that after the mixing time of a filament, the decrease of the concentration fluctuation due to diffusion is exponential, resulting in fast homogenization with the surrounding fluid.
The birth of the theory of chaotic advection is usually traced back to a 1984 paper[7] by Hassan Aref. In this work, Aref studied the mixing induced by two vortices switched alternately on and off inside an inviscid fluid. This seminal work had been made possible by earlier developments in the fields of dynamical Systems and fluid mechanics in the previous decades. Vladimir Arnold[8] and Michel Hénon[9] had already noticed that the trajectories advected by area-preserving three-dimensional flows could be chaotic. However, the practical interest of chaotic advection for fluid mixing applications remained unnoticed until the work of Aref in the 80's. Since then, the whole toolkit of dynamical systems and chaos theory has been used to characterize fluid mixing by chaotic advection.[1] Recent work has for example employed topological methods to characterize the stretching of fluid particles.[10] Other recent directions of research concern the study of chaotic advection in complex flows, such as granular flows.[11]