DIFFUSION BY CONTINUOUS MOVEMENTS

q (x,y,z) Temperature at point (x,y,z) at time t

u Velocity of a particle of fluid in direction of x

r s Heat capacity of unit volume of the fluid

< u ^> Mean value of u

C Plane perpendicular to the x-axis

H Averagerate of heat being conveyed across the plane Cc₀ Unit area of C

< u^{2 >} Mean energy of the motion

d , t Step size and time between steps

X Distance traversed by a particle in time T

p Pressure field of the flow

c Correlation coefficient in discontinuous case

Rx Correlation coefficient in continuous case

The relationship between the rate of diffusion of molecules in a liquid and the properties of the liquid has been known since the work of Einstein (1905). However, in turbulence, there is no fundamental understanding of how the diffusion constants characterizing the turbulence relates to the properties of the turbulence. The results from the Eddy Motion in the Atmosphere paper showed that turbulence motion is capable of diffusing heat and other diffusible properties through the interior of the fluid as similar to the molecular diffusion. These results motivated Taylor to ask how these diffusion constants are determined by the properties of the turbulence.

The modelconsidered here is one-dimensional transmission of heat in a turbulent fluid. The fluid is incompressible. At time t = 0, q depends only on x and increases or decreases uniformly with x. Hence the rate of change of q is constant, and lets call it b .

Let q (a,0) be the temperature at x = a and t = 0, consider a plane C perpendicular to the x-axis then the averagerate of heat being conveyed across the plane is Hc₀ = - r s b * Avg(u(x-a)). The <u(x-a)> of course depends on the nature of the turbulent motion, hence the question to ask here is what types of turbulent motion are capable of producing the observed distribution of temperature.

The turbulent motion is consider to be uniformly distributed throughout space. Therefore, < u(x-a)> equals to the mean value throughout space. To calculate < u(x-a)> , let’s look at a particle of fluid and noticed that

= < x₁² + x₂² + … + x_n² > + 2< x₁x₂ + x₁x₃ + ¼ + x_i x_i+1 +¼ >

= nd 2 + 2< x₁x₂ + x₁x₃ + ¼ + x_i x_i+1 +¼ > (3)

If there is no correlation between two x’s, < x_i x_i+1 > = 0 i.e. where the is at the i^th step doesn’t related to where it goes at (i+1)^th step, then

X_n² = nd 2

which gives the dispersion relation of random walker. From this relation we now can relate the variance of the total distance to the diffusivity constant D of the fluid by substitute T/t for n in the above expression and we found

X_n² = DT where D=d 2/t .

However, in turbulent fluid, the movement of the fluid particle from one step to the next is correlated. Let c be the correlation coefficient, then the correlation of the location of the fluid particle is defined as follow,

< x_i x_i+1 > = cd ², < x_i x_i+2 > = c² d ², ….. , < x_i x_i+n > = c^{nd 2} (4)

Here we assumed that the partial correlation of x_i with x_i+2, x_i+3,… are all zero i.e. the location of the particle at the i^th step is related only to its very next step by c, we don’t define other correlation coefficients to relate it to other future steps.

c²

x₁ x₂ x₃ x₄ …

c c c

Substitute (4) into (3) we found,

X_n² = nd 2 + 2d 2[(n- 1)c + (n- 2) c² + (n- 3) c³ +…+ c^{n- 1}] (5)

= d 2[n + ]

Replacing n by T/t , and d /t by v which is the initial uniform velocity of a fluid particle we have,

X_n² = [] (6)

In order to approximate this to continuous migration we need to decrease t . In this continuous limit X_n, v, and T are finite and tend to a define limit which required that and must also tend to a definite limit i.e. 1- c µ t .

As t decrease to zero, t / (1-c) tend to a limit A, then

X_n² ® v²[2AT –2A²(1- e^-T/A)]

Now let X denote the distance traversed by a fluid particle over T we have

< X² > = v²[2AT–2A²(1- e^-T/A)]

The behaviors of X² divided into two regimes of time,

For short time T, X² = v²T² , which is what we would expected. Since, when T is short the fluid particle has not moved far or much so the correlation coefficient c in this case is not much differ from 1. In this case we have ballistic behavior.

For long time T, X² = 2v²AT , the amount of diffusion of the fluid particles is proportional to T just as in the case of no correlation. Here A is the measurement of the rate at which the correlation coefficient between the direction of an infinitesimal path in the migration and that of an infinitesimal path at a time T falls off with increasing values of T. In this regime of time we have diffusive behavior (fig1).

However, the above theory is still discontinuous and depending on the assumption that there is no partial correlation between non-neighbor directions. Therefore the correlation coefficient c will be restricted. As we define the correlation between the direction of an infinitesimal path and the path occurs at time T to be cⁿ, we can rewrite it as follows,

[1- (1- c)]ⁿ = (1- t /A)ⁿ = (1- t /A)^T/t (7)

when t is small, (7) tends to the limit e^-T/t . Hence, we only consider the type of motion which has the correlation coefficient of the type e^-T/t .

Fig1

The above discussion gives us an idea how to treat the problems of diffusion by continuous movement. First, some statistical properties of continuously varying quantities, like pressure, will be discussed. The standard deviation of pressure from its mean value during a year is constant from year to year. Let p be the deviation from the mean pressure or the pressure field in flow. The statistical properties of barograph curve of pressure is that < p^2> is constant during successive period. Hence, from a give barograph, we then can find standard deviation of p, dp/dt, d²p/dt², … , d ⁿp/dtⁿ, … which assume to be constant in time.

Since < p^2> is constant in time, its time derivative is zero, hence we have

® (8)

which implies that there is no correlation between p and dp/dt.

Differentiate (8) with respect to t again gives,

(9)

Similarly we can treat dp/dt, d²p/dt², … , d ⁿp/dtⁿ, …in the same ways since they are constant also. Then we found the generalized correlations of p and its differential coefficients as follow,

and (10)

Let’s Rx be the correlation coefficient, p_t be the pressure at time t, and p_t+x be the pressure at time t + x , then

, since the standard deviation of p is constant. (11)

Taylor expansion of p_t+x together with (10) then (11) becomes,

(12)

Now back to our question that what types of turbulent motion are capable of producing the observed temperature distribution. With the statistical properties established above, it will enable us to answer this question and hence set down a remarkable relationship between the diffusion constant and the flow, which is known as the law of diffusion. The law which governs the average distribution of particles initially concentrated at one point, at any subsequent time.

Consider the case when turbulence in a fluid is uniformly distributed. Let u be the velocity, which parallel to x-axis, of the fluid particle that we observe. Now apply the above theory to u where Rx is the correlation coefficient between u_t and u_t+x . Suppose that < u^2> and Rx are known. By mean of (11) and (12) the value of definite integral

(13)

bring the integral inside the mean value and notice that the distance has moved by the fluid particle is

and (14)

incorporate (14) into (13) we arrive at the relation which simplifies the problem of diffusion and turbulent motion as follows,

(15)

Again we will be looking at the two regimes when T is small and T is large. The physical behaviors in these regime are different.

When T is small Rx doesn’t change much from 1 over the time interval so (15) reduced to

® (16)

in this case we have ballistic behavior. On the other hand when T is large, for large value of x , R(x ) goes off to 0 from either side (Fig.2). Define a time interval T₁ such that the velocity of the particle at the end of the interval has no correlation to the one at the beginning then

where A = as t® ¥ .

so in this case the behavior is diffusive.

diffusive.