Advection along streamlines

As already mentioned the velocity field of incompressible two-dimensional flows can be represented by a scalar streamfunction, ip(x,y) as 

The rate of change of the time-independent streamfunction along the path of a fluid element, given by its Lagrangian derivative, vanishes 

An important characteristic of advective transport is relative dispersion that describes how the separation of nearby fluid elements changes in time. The separation dpt) between two fluid elements that are initially close to each other, 5(0) C L, can be decomposed into tangential and normal components relative to the streamlines. For particles moving on closed streamlines the normal component oscillates periodically in time but the dominant asymptotic behavior comes from the tangential component dt that increases by the same amount after each cycle due to the small difference in the periods, T(ip), of the motion on nearby closed periodic orbits. Therefore 

This linear growth also holds for the length of material lines, e.g. representing a boundary between two fluid regions with different properties. As we will see later, time-dependent flows allow for much faster, accelerating growth of the length of material lines. 

Typically, the flow field also contains stagnation points where v(x ) = 0. The stagnation points can be classified according to the geometrical structure of the trajectories in their neighborhood. The trajectory of a fluid element around a stagnation point can be written as r(t) = x + S(t) and approximating the velocity field by a Taylor 

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Fluid elements are advected along closed circular streamlines, but the separatrices connecting the hyperbolic points inhibit long range advective transport from one cell to another (see Fig. 2.4). Thus... 

Although dispersion can be described by the same law as diffusion, its nature is different. Dispersion is the result of the velocity shear, that is, of the velocity difference between adjacent streamlines in an advective flow. Due to turbulent exchange perpendicular to the direction of flow, water parcels continuously change the streamline along which they move. Since these streamlines move at different speeds, each water parcel has its own individual history of speed and thus its individual mean velocity. 

The fate of the pollutant moving in the aquifer along the streamlines is determined by the advection-dispersion equation, Eq. 25-10 or 25-18. For Pe 1, that is, for locations x dis / if, the concentration cloud can be envisioned to originate from an infinitely short input atx = 0of total mass (a so-called5 input) that by dispersion is turned into a normal distribution function along the x-axis with growing standard deviation. Since the arrival of the main pollution cloud at some distance x is determined... 

Fig. 7.8 Poincare sections after 2000 cycles. Initially nine marker points were placed along the y axis and six along the x axis. The dimensionless amplitude was 0.5, as in Fig. 7.7. The parameter was the dimensionless period (a) 0.05 (h) 0.10 (c) 0.125 (d) 0.15 (e) 0.20 (f) 0.35 (g) 0.50 (h) 1.0 (i) 1.5. For the smallest values of the time period we see that the virtual marker points fall on smooth curves. The general shape of these curves would he the streamlines of two fixed continuously operating agitators. As the time period increases the virtual marker particles fall erratically and the regions indicate chaotic flow. With increasing time periods larger and larger areas become chaotic. [Reprinted by permission from H. Aref, Stirring Chaotic Advection, J. Fluid Meek, 143, 1-21 (1984).]...

The advection problem is thus described by a periodically driven non-autonomous Hamiltonian dynamical system. In such case, besides the two spatial dimensions an additional variable is needed to complete the phase space description, which is conveniently taken to be the cyclic temporal coordinate, r = t mod T, representing the phase of the periodic time-dependence of the flow. In time-dependent flows ip is not conserved along the trajectories, hence trajectories are no longer restricted to the streamlines. The structure of the trajectories in the phase space can be visualized on a Poincare section that contains the intersection points of the trajectories with a plane corresponding to a specified fixed phase of the flow, tq. On this stroboscopic section the advection dynamics can be defined by the stroboscopic Lagrangian map... 

Since the time-dependence of the velocity field is restricted to a finite region the complex chaotic orbits are also limited to this region. Advection in such flows is a chaotic scattering process (Ott and Tel, 1993 Ott, 1993) in which fluid elements approach the mixing zone along the inflow streamlines, they follow chaotic trajectories inside... 

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