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Relative diffusion

   

The term relative diffusion is used to describe relative motion of pairs of particles viewed as a diffusion process. Richardson  explained the importance of using relative, rather than absolute, motion in turbulence studies  [13]. His main intention was to separate the turbulent variations in the velocity field from the average velocity field.

We can look at relative diffusion in a form similar to that used for absolute diffusion
 eqnarray374
but it is more common to look at the behaviour of tex2html_wrap_inline4389, as Richardson did, or at the behaviour of tex2html_wrap_inline4343 as Kolmogorov did.

If you notice that tex2html_wrap_inline4393 is the average squared separation of a pair of particles, and assume that there exists a constant, tex2html_wrap_inline4337, such that tex2html_wrap_inline4397, or tex2html_wrap_inline4399, it is possible to show that both tex2html_wrap_inline4389 and tex2html_wrap_inline4343 can be written as powers of R which only depends on tex2html_wrap_inline4337.
eqnarray385
Both Richardson's data and Kolmogorov's derivations gives the same value for tex2html_wrap_inline4337 in three-dimensional turbulence:
eqnarray400

When we look at tex2html_wrap_inline4343 or tex2html_wrap_inline4389, we have freed ourselves of the problem of tracking every pair of particles back to a moment where they were close together. We can just look at their relative motion as a function of their separation.

Even though the formal descriptions of the quantities Richardson and Kolmogorov looked at are equivalent, their selections of what data we should look at are quite different. Richardson talks about a collection of particles, which initially are very close together, whereas Kolmogorov looks at a randomly selected collection of points throughout the whole body of fluid in question. Richardson's view is equivalent to studies of chaos , where you look at the exponential growth of an infinitely small difference in the initial conditions.

In both of the two extreme types motion; Brownian  and ballistic , tex2html_wrap_inline4393 is equal to D(t), only scaled by respectively a factor of 2 and a factor of 4.

The relative motion of two random walkers (Brownian motion) for a time, t, is equivalent to one random walker moving for twice as long a time:
eqnarray436

The relative motion of two particles in ballistic motion for a time, t, is similarly equivalent to one particle in ballistic motion for twice that period of time, but here is tex2html_wrap_inline4429, and thus:
eqnarray439

So we find that for velocity fields with absolutely no spatial correlations tex2html_wrap_inline4389 is independent of R and for velocity fields with strong spatial correlations tex2html_wrap_inline4389 is proportional to R.


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Next: Weak turbulence Up: Statistical measures of hydrodynamics Previous: Absolute diffusion

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