P. C. Chatwin

On the longitudinal dispersion of passive contaminant in oscillatory flows in tubes

The paper examines how a passive contaminant disperses along the axis of a tube in which the flow is driven by a longitudinal pressure gradient varying harmonically with time. This problem is of intrinsic interest and is relevant to some important practical problems. Two examples are dispersion in estuaries and in the blood stream. By means both of statistical arguments and an analysis like that used by Taylor (1953) in the case of a steady pressure gradient it is shown that eventually the mean distribution of concentration satisfies a diffusion equation (and is therefore a Gaussian function of distance along the axis) with an effective longitudinal diffusion coefficient K ( t ) which is a harmonic function of time with a period equal to one half of that of the imposed pressure gradient. Contrary to the supposition made in most previous work on this problem it is shown by examining some special cases that the harmonic terms in K ( t ) may have a noticeable effect on the dispersion of the contaminant and in particular on the rate at which it is spreading axially. The size of the effect depends on both the frequency and the Schmidt number and is particularly large at low frequencies. The paper concludes with an analysis of a model of dispersion in estuaries which has been used frequently and it is concluded that here too oscillatory effects may often be noticeable.

The approach to normality of the concentration distribution of a solute in a solvent flowing along a straight pipe

Taylor (1953, 1954 a ) showed that, when a cloud of solute is injected into a pipe through which a solvent is flowing, it spreads out, so that the distribution of concentration C is eventually a Gaussian function of distance along the pipe axis. This paper is concerned with the approach to this final form. An asymptotic series is derived for the distribution of concentration based on the assumption that the diffusion of solute obeys Ficks law. The first term is the Gaussian function, and succeeding terms describe the asymmetries and other deviations from normality observed in practice. The theory is applied to Poiseuille flow in a pipe of radius a and it is concluded that three terms of the series describe C satisfactorily if Dt/a 2 > 0·2 (where D is the coefficient of molecular diffusion), and that the initial distribution of C has little effect on the approach to normality in most cases of practical importance. The predictions of the theory are compared with numerical work by Sayre (1968) for a simple model of turbulent open channel flow and show excellent agreement. The final section of the paper presents a second series derived from the first which involves only quantities which can be determined directly by integration from the observed values of C without knowledge of the velocity distribution or diffusivity. The latter series can be derived independently of the rest of the paper provided the cumulants of C tend to zero fast enough as t → ∞, and it is suggested, therefore, that the latter series may be valid in flows for which Ficks law does not hold.

The effect of aspect ratio on longitudinal diffusivity in rectangular channels

In a recent paper Doshi, Daiya & Gill (1978) showed that the value of Taylors longitudinal diffusivity D for laminar flow in a channel of rectangular cross-section of breadth u and height b is about 8 D 0 , for large values of the aspect ratio a / b , where Do is the value of the longitudinal diffusivity obtained by ignoring all variation across the channel. This superficially surprising result is confirmed by an independent method, and is shown to be caused by the boundary layers on the side walls of the channel. The primary purpose of the paper, however, is to consider the value of D in turbulent flow in a flat-bottomed channel of large aspect ratio, for which arguments based on physics are adduced in support of the formula D ≈[1 + B][1 - λ( b / u )], where B and λ are positive constants independent of b . It is shown that this result is consistent with laboratory experiments by Fischer (1966). The paper concludes with a discussion of the practical effects of aspect ratio on longitudinal dispersion in channels whose cross-section is approximately rectangular.

The relative diffusion of a cloud of passive contaminant in incompressible turbulent flow

A problem of major practical interest is the variation with x and t of the statistical properties of Γ( x , t ), the distribution of concentration of a contaminant in a cloud containing a finite quantity Q of contaminant, released in a specified way at t = 0 over a volume of order L 3 0 . Of particular relevance is the case of relative diffusion (when x is measured throughout each realization relative to the centre of mass of the cloud), when important properties are L ( t ), the linear dimension of the cloud, C ( x , t ), the ensemble mean concentration,

The initial development of longitudinal dispersion in straight tubes

\overline{c^2}({\bf x}, t)

Measurements of concentration fluctuations in relative turbulent diffusion

, the variance of the concentration, and p ( y , t ), the distance-neighbour function. Much fundamental work has led to a knowledge of the way L varies with t , but not of the way the other properties vary. Hitherto therefore, prediction of such variation has normally used unjustifiable empirical concepts such as eddy diffusivities, but this is ultimately unsatisfactory, practically as well as theoretically. Hence the exact equations have been used to obtain a quite new description of the structure of a dispersing cloud, which it is hoped will serve as a basis for future practical work. When κ = 0 (where κ is the molecular diffusivity) the magnitude of p ( y , t ) is of order Q / L 3 for most y , but of order Q / L 3 0 when | y | is very small. By a variety of arguments it is shown that these facts can be explained (for many, if not all, flows) only if the distributions of C and

Some turbulent diffusion invariants

\overline{c^2}

The dispersion of contaminant released from instantaneous sources in laminar flow near stagnation points

, as well as that of p , have a core-bulk structure. In the bulk of the cloud C and

On the interpretation of some longitudinal dispersion experiments

\overline{c^2}

The initial dispersion of contaminant in Poiseuille flow and the smoothing of the snout

have magnitudes of order Q / L 3 and Q 2 / L 3 0 L 3 respectively, but there is a core region of thickness decreasing to zero surrounding the centre of mass within which they have much greater magnitudes. In one case, examined in some detail, the magnitudes in the core are of order Q / L 3 0 and Q 2 / L 6 0 . It is then shown that the core and bulk exist even in the real case when κ ≠ 0. In the real case the core thickness no longer tends to zero but to a constant of order λ c , the conduction cut-off length. As a consequence almost entirely of molecular diffusion acting in the core region, the magnitudes of C and