User:MasterSmcd

Page of graduate students of the SMCD Master

We are graduate students in a Physics/Mechanics Master's program in Paris, France. In the course of a project, we plan to modify and complement the ChaoticMixing page. For this purpose, we will first modify the page in our user page, using it as a sandbox, and the copy back the modifications to the ChaoticMixing page, once a satisfactory new collaborative version exists.

Project
modify the page on chaotic mixing

Chaotic mixing


In chaos theory and fluid dynamics, chaotic mixing is a process by which flow tracers develop into complex fractals under the action of a fluid flow. The flow is characterized by an exponential growth of fluid filaments. Even very simple flows, such as the blinking vortex, or finitely resolved wind fields can generate exceptionally complex patterns from initially simple tracer fields.

The phenomenon is still not well understood and is the subject of much current research.

Fluid flows
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.

Shear
A flow promotes mixing by the separation of neighboring fluid particles. This separation occurs because of velocity gradients, a phenomenon called shearing. Let us $$\mathbf{X}_1$$ and $$\mathbf{X}_2$$ be two neighboring fluid particles, separated by $$d\mathbf{X}=\mathbf{X}_2-\mathbf{X}_1$$ at time $$t$$. When the particles are advected by a flow $$\mathbf{v}$$, at time $$t+dt$$ the separation between the particles is given by :



d\mathbf{X}(t + dt) = d\mathbf{X}(t) + dt (d\mathbf{X}\cdot\nabla)\mathbf{v} $$ hence

\frac{d(d\mathbf{X})}{dt}=(d\mathbf{X}\cdot\nabla)\mathbf{v} $$ 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.

Conditions for chaotic advection
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:



\frac{d x}{dt} = u(x,y,z,t) $$

\frac{d y}{dt} = v(x,y,z,t) $$

\frac{d z}{dt} = w(x,y,z,t) $$

where $$u$$, $$v$$ and $$w$$ are the components of the velocity field, which are assumed to be known from the solution of one of the standard dynamical equations governing fluid flow, typically the Navier-Stokes equation in one of its many forms. If the dynamical system governing trajectories is chaotic, the integration of a trajectory is extremely sensitive to initial conditions, and neighboring points separate exponentially with time. This phenomenon is called chaotic advection.

Dynamical systems and chaos theory state that at least 3 degrees of freedom are necessary for a dynamic system to be chaotic. 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.



That is the case for 2-D stationary flows in which there are only two degrees of freedom $$x$$ and $$y$$. For time-independent flows, Lagrangian trajectories of fluid particles coincide with the streamlines of the flow. In 2-D, streamlines are concentric closed curves that cross only at [[ stagnation point | stagnation points]. Thus, in a mixing, a blob of dye can only explore the region bounded by the most external and internal streamline on which it is lying at the initial time. Regarding practical application, this incomplete or ineffective mixing is not very satisfying.

For 2-D unstationary flow which is time-dependent, instantaneous closed streamlines and Lagrangian trajectories do not coincide any more. We consider an example of the first protocol proposed, the blinking vortex flow which is piecewise stationary, and constitutes a form of time dependency. In this type of flow, two circular rods are plunged in a circular domain, and the two rods are rotated alternately in opposite senses. The rotation of the rods is alternately switched on and off implying a blinking vortex flow which is periodic. Trajectories obtained are complicated and intersect themselves because particles can follow different streamlines which would be impossible in a stationary flow. They consist of a succession of arcs of circles matching streamlines. During the finite time of the simulation, the trajectory sweeps a large fraction of the fluid domain, like 3-D stationary flows, and contrary to the case of 2-D stationary flows. Hence, 3-D stationary flows are more favorable to mixing.

Lyapunov exponents
A trajectory in the fluid is determined by the following system of ordinary differential equations:



\frac{d \vec x}{dt} = \vec v(\vec x, t) $$

where $$\vec x$$ is the physical position, t is time, and $$\vec v$$ is the fluid velocity as a function of both position and time. If the flow is chaotic, then small initial errors in a trajectory will diverge exponentially. We are interested in calculating the stability—i.e., how fast do nearby trajectories diverge? Suppose we make a small perturbation, $$\delta \vec{x}$$, then using a Taylor expansion, we get:



\frac {\mathrm d} {\mathrm dt} (\vec{x}+\delta \vec{x}) \approx \vec{v} + \nabla \vec{v} \cdot \delta \vec{x} $$

The Jacobi matrix of the velocity field, $$\nabla \vec{v}$$, provides information about the local rate of divergence of nearby trajectories or the local rate of stretching of Lagrangian space. The rate of change of the error vectors is given approximately as:



\frac {\mathrm d} {\mathrm dt} \delta \vec x \approx \nabla \vec v \cdot \delta \vec x $$

We define the matrix H such that:



\frac {\mathrm d} {\mathrm dt} \boldsymbol{H} \equiv \nabla \vec{v} \cdot \boldsymbol{H}, \qquad \boldsymbol{H} (t=0)=\boldsymbol{I} $$

where I is the identity matrix. It follows that:



\delta \vec{x} (t) \approx \boldsymbol{H} \cdot \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{H^T} \cdot \boldsymbol{H} \cdot \delta \vec{x}_{0i} = h_i \cdot \delta \vec{x}_{0i} $$



\lambda_i(\vec{x},t) \equiv \frac {1} {2 t} \ln {h_i(\vec{x},t)} $$

where $$\lambda_i(\vec{x},t) \geq \lambda_{i+1}(\vec{x},t)$$ is the ith Lyapunov exponent of the system, while $$\delta \vec {x}_{0i}$$ is the ith principal component of the matrix H.

If we start with a set of orthonormal initial error vectors, $$\{\delta \vec x_{0i}\}$$ then the matrix H will map them to a set of final orthogonal error vectors of length $$\{\sqrt{h_i(\vec{x},t)}\}$$. The action of the system maps an infinitesimal sphere of inititial points to an ellipsoid whose major axis is given by the $$\sqrt{h_1(\vec{x},t)}$$ while the minor axis is given by $$\sqrt{h_N(\vec{x},t)}$$, where N is the number of dimensions.

This definition of Lyapunov exponents is both more elegant and more appropriate to real-world, continuous-time dynamical systems than the more usual definition based on discrete function maps. Chaos is defined as at least one positive Lyapunov exponent.

In a chaotic system, we call the Lyapunov exponent the asymptotic value of the greatest eigenvalue of H:



\lambda = \lim_{t \to \infty} \lambda_1(\vec{x},t) $$

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:
 * by following one single trajectory for very long times and computing $$ \lambda = \lim_{t \to \infty} \lambda_1(\vec{x},t)$$.


 * or by following an ensemble of trajectories for a given period of time, and computing the ensemble average: $$ <\lambda>_{trajectories} $$

The equivalence of the two methods is due to the ergodicity of the chaotic system.

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 \vec x| \approx |\delta \vec x_0| e^{\lambda_1 t}

Contour advection


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 over several days. The figure to the right shows the length of this contour as a function of time.



The link between exponential contour growth and positive Lyapunov exponents is easy to see. The rate of contour growth is given as:



\frac{\mathrm d L}{\mathrm d t} = \int | \nabla \vec v \cdot \mathrm d \vec s | $$

where $$\mathrm d \vec s$$ is the path and the integral is performed over the length of the contour. Contour growth rates will approximate the average of the large Lyapunov exponents:



L \approx L_0 \exp(\bar \lambda_1 t) $$

Poincaré sections
In the case of chaotic advection, a particle travels in 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. Here, we want to be able to distinguish regions of good and bad mixing. Poincaré sections are the most common method used.

For time-periodic flows, we can solve this problem by using Poincaré sections. The Poincaré map is defined as the transformation
 * $$\begin{align}

\boldsymbol{M} \colon \boldsymbol{X}(t=0)&\to \boldsymbol{x}(T,\boldsymbol{X}). \end{align}$$

$$\boldsymbol{M}$$ transforms a point-like particle into the position of the particle after a time-interval T. Especially, for a time-periodic flow with period T, applying the map several times to a particle gives the successive positions of the particle period after period. Then, a Poincaré section is built by starting from different initial conditions and plotting the corresponding iterates. This comes down to plotting the trajectories but only keeping one point every T. Hence, by this method, we stroboscope the Lagrangian trajectory every T.



Let us consider an example. The figure presented here depicts the Poincaré section obtained when one applies an eight-like movement to the mixing rod. We can see that some trajectories span a large region: these are the chaotic or mixing regions, 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. Elliptic islands have to be avoided for mixing, for two reasons :
 * Fluid particles are unable to cross the boundaries of the islands (except by slow diffusion, which is negligible), thus there is segregation.
 * The mixing in these regions is not efficient because the trajectories are closed and therefore not chaotic.

If one wants to avoid non-chaotic islands, it is useful to try and understand the physical origins of these regions. Generally, changing the geometry of the flow can induce a change in the chaotic and regular regions. In the figure-eight flow for instance, when the rod is too small, a particle initially close to the rod doesn't see the eight-like movement of the rod. To this particle, it seems that the rod is making circles. This explains why some particles have an elliptic movement, and can't go out of their loop. If we use a larger rod, it might be possible to extract a particle from one loop to another, which allows to get rid of the elliptic islands and span the whole fluid domain, leading to a good mixing.

Poincaré sections give the possibility to analyze the mixing quality by distinguishing chaotic and elliptic regions. However, this is somehow a crude measure of the mixing process, since we cannot infer the stretching properties from this mapping method. Nevertheless, this technique is the most useful one to study the mixing of periodic flows and can be extended to a 3D-domain.

Fractal dimension
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 very long time integration, becomes fractal. Fractals composed of a single curve are infinitely long and when formed iteratively, have an exponential growth rate, just like an advected 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 function of time, measured in four different ways. A good method of measuring the fractal dimension of an advected contour is the uncertainty exponent.



Evolution of tracer concentration fields in chaotic advection
Concentration field is a molecular or mass density distribution of a particle dispersed into a fluid in its whole domain. As an example of a concentration field, we can consider a dye poured into water, or any transparent fluid. In chaotic mixing, the molecules of dye disperse in the fluid thanks to the stretching by advection and diffusion. Our objective is to study how the dye concentration evolves during the mixing procedure. An example is shown on the figure "Mixing patterns obtained from the chaotic advection of a blob of dye", which is on the right. The concentration field evolves according to the advection-diffusion equation. That means there will be a competetion between advection which decrease the width of filament, due to the stretching, and the diffusion which tends to increase the width of filament while decreasing the concetration fluctuation. The advection-diffusion equation, called also Convection–diffusion equation, is deduced by combining diffusion equation and advection equation to describe the particle concentration field in a moving fluid. It expresses the relationship between the sum of the diffusion and advection, and the gradient of concentration field, which is linked by Diffusion coefficient. In the beginning of mixing process, the advection term dominates the evoluation of the concentration field of particles, and decreases the width of filament until it reaches Batchelor scale, where the diffusion begins to play a significant role. The figure (c) and (f) gives an image of Batchelor scale. From (c) to (f), there is no change of filament length scale, but only the concentration. It is the smallest lengthscale that can be observed in the concentration pattern, since diffusion smears out any finer detail. It is defined as the square root of the ratio between Diffusion coefficient and Lyapunov Exponent. This scale measures the balance between the stretching and diffusion effect on the evolution of our concentration field.
 * Advection-diffusion equation
 * Batchelor scale

w_B = \sqrt{\frac{D}{\lambda}} $$ Where: \lambda $$ is the Lyapunov Exponent When most of filaments reach the Batchelor scale, the diffusion begins to decrease significantly the contrast of concentration between filament and the other part of the fluid domain. The resolution of the advection–diffusion equation shows that after mixing time tm, the decrease of concentration is exponantial. The rapid decay of concentration field is given by the following formula:
 * D is the Diffusion coefficient
 * Rapid decay of concentration fluctuations

c(0, t) \simeq \sqrt{\frac{\lambda s_0^2}{2\pi D}} \exp{-\lambda t} $$ Where: \lambda $$ is the Lyapunov Exponent s_0 $$ is the initial length scale of concentration
 * c(0,t) is the concentration field at the centre.
 * D is the Diffusion coefficient

History of chaotic advection
The birth of the theory of chaotic advection is usually traced back to a 1984 paper by Hassan Aref. In this work, Aref studied the mixing induced by two vortices switched alternately in 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 and Michel Hénon 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. Recent work has for example employed topological methods to characterize the stretching of fluid particles. Other recent directions of research concern the study of chaotic advection in complex flows, such as granular flows.