Distance-neighbour Function Research Articles

Relative dispersion of a scalar plume in a turbulent boundary layer

The relative dispersion of a scalar plume is examined experimentally. A passive fluorescent tracer is continuously released from a flush-bed mounted source into the turbulent boundary layer of a laboratory-generated open channel flow. A two-dimensional particle image velocimetry–laser-induced florescence (PIV–LIF) technique is applied to measure the instantaneous horizontal velocity and concentration fields. Measured results are used to investigate the relationship between the boundary-layer turbulence and the evolution of the distance-neighbour function, namely the probability density distribution of the separation distance between two marked fluid particles within a cloud of particles. Special attention is paid to the hypothesis that a diffusion equation can describe the evolution of the distance-neighbour function. The diffusion coefficient in such an equation, termed the ‘relative diffusivity’, is directly calculated based on the concentration distribution. The results indicate that the relative diffusivity statistically depends on particle separation lengths instead of the overall size of the plume. Measurements at all stages of the dispersing plume collapse onto a single curve and follow a 4/3 power law in the inertial subrange. The Richardson–Obukhov constant is estimated from the presented dataset. The relationship between the one-dimensional (1D) representation of the distance-neighbour function and its three-dimensional (3D) representation is discussed. An extended model for relative diffusivity beyond the inertial subrange is proposed based on the structure of the turbulent velocity field, and it agrees well with measurements. The experimental evidence implies that, while the diffusion of the distance-neighbour function is completely determined by the underlying turbulence, the overall growth rate of the plume is affected by both the turbulent flow and its actual concentration distribution.

An experimental study of turbulent relative dispersion models

We report measurements of the spreading rate of pairs of tracer particles in an intensely turbulent laboratory water flow. We compare our measurements of this turbulent relative dispersion with the longstanding work of Richardson and Batchelor, and find excellent agreement with Batchelor's predictions. The distance neighbour function, the probability density function of the relative dispersion, is measured and compared with existing models. We also investigate the recently proposed exit time analysis of relative dispersion.

Similarity Scaling Of Surface-Released Smoke Plumes

Concentration fluctuation data from surface-layer released smokeplumes have been investigated with the purpose of finding suitable scaling parametersfor the corresponding two-particle, relative diffusion process.Dispersion properties have been measured at downwind ranges between 0.1 and 1 km from a continuous, neutrally buoyant ground level source. A combinationof SF6 and chemical smoke (aerosols) was used as tracer. Instantaneous crosswind concentration profiles of high temporal (up to 55 Hz) and spatialresolution (down to 0.375 m) were obtained from aerosol-backscatter Lidar detectionin combination with simultaneous gas chromatograph (SF6) reference measurements. The database includes detailed crosswind concentration fluctuation measurements. Each experiment, typically of 1/2-hour duration, containsplume mean and variance concentration profiles, intermittency profiles andexceedence and duration statistics. The diffusion experiments were accompanied by detailed in-situ micrometeorological mean and turbulence measurements. In this paper, a new distance-neighbour function for surface-released smoke plumes is proposed, accompanied by experimental evidence in its support. The new distance-neighbour function is found to scale with the surface-layer friction velocity,and not with the inertial subrange dissipation rate, over the range of distance-neighbour separations considered.

An experimental investigation of the relative diffusion of particle pairs in three-dimensional turbulent flow

The particle tracking (PT) technique is used to study turbulent diffusion of particle pairs in a three-dimensional turbulent flow generated by two oscillating grids. The experimental data show a range where the Richardson–Obukhov law 〈r2〉 = Cεt3 is satisfied, and the Richardson–Obukhov constant is found to be C = 0.5. A number of models predict much larger values. Furthermore, the distance–neighbour function is studied in detail in order to determine its general shape. The results are compared with the predictions of three models: Richardson (1926), Batchelor (1952) and Kraichnan (1966a). These three models predict different behaviours of the distance–neighbour function, and of the three, only Richardson's model is found to be consistent with the measurements. We have corrected a minor error in Kraichnan's (1996a) Lagrangian history direct interaction calculations with the result that we had to increase his theoretical value from C = 2.42 to C = 5.5.

Simulation of turbulent dispersion using a simple random model of the flow field

A method is devised to simulate the movement and spreading of a patch of contaminant in two-dimensional turbulent flow. The turbulent motion is exponentially divided into components of differing wave number, adjacent components being made to have correlation times differing by a factor of two. The turbulent motion is then reconstructed by replacing each component with a sinusoidal advection field having a randomly directed wave number. Contaminant particles are advected by each of the reconstructed components, the smallest scale components being applied first. A computer simulation was performed, using a Kolmogorov k - 5 3 turbulent energy spectrum. Batchelor's σ ∝ t 3 2 law for the spreading of a contaminant patch was reproduced, approximately, as was Richardson's non-Gaussian asymptotic form of the distance-neighbour function.

Some turbulent diffusion invariants

In view of the importance of concentration fluctuations in practical and theoretical problems of turbulent diffusion, there are presented here some invariant properties of the distribution of fluctuations associated with a cloud of contaminant containing a finite quantity Q of material. These properties are invariant provided only that Q is conserved, no assumption whatsoever being made about the random turbulent velocity field. Consequences of the results for (i) steady plumes, (ii) the representation of the distribution of concentration by series of (generalized) Hermite polynomials, and (iii) the relationship of the ensemble mean concentration with the distance–neighbour function, are discussed using experimental evidence.

Measurements of concentration fluctuations in relative turbulent diffusion

In 1965 Sullivan made many measurements of concentration in dye plumes in the surface layer of Lake Huron with the primary purpose of estimating the distance–neighbour function (Sullivan 1965, 1971). This paper presents the results of a recent analysis of the concentration fluctuations in these experiments for, despite their great practical and theoretical importance, there are very few published reports of such measurements from natural environments. One reason for this apparent neglect has undoubtedly been the anticipated high noise level, and the present results confirm this expectation. The experimental analysis uses the framework of relative diffusion since this has great advantages compared with that of absolute diffusion. Despite the noise, the results are consistent, to the degree of spatial resolution attained, with the self-similar structure anticipated for relative, but not absolute, diffusion. Further interesting features of the results are that changes in the form of the statistical properties across the plume indicate an unexpectedly strong influence of the central regions, and that certain statistical properties have much less noisy profiles than that of the mean square fluctuations. The influence of molecular diffusion is shown to be strong. Interpretation of the results is based partly on the extension of the theory recently developed by Chatwin & Sullivan (1979a) for a cloud, although the limited spatial resolution attained did not allow direct critical examination of this work.

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 L30. 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, $\overline{c^2}({\bf x}, t)$, 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/L3 for most y, but of order Q/L30 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 $\overline{c^2}$, as well as that of p, have a core-bulk structure. In the bulk of the cloud C and $\overline{c^2}$ have magnitudes of order Q/L3 and Q2/L30L3 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/L30 and Q2/L60.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 λc, the conduction cut-off length. As a consequence almost entirely of molecular diffusion acting in the core region, the magnitudes of C and $\overline{c^2}$ in both the core and the bulk decay to zero in a way which depends on the details of the fine-scale structure of the velocity field. Several examples of the decay are discussed.

The approach to normality of the distance-neighbour function when used to describe relative turbulent dispersion

Batchelor [1] suggested that the Distance-Neighbour Function when used to describe the relative turbulent dispersion of a cloud of marked fluid, whose diameter is well within the length scale range of the universal inertial sub-range of turbulence, would become of Gaussian form. This paper determines that a necessary condition for the Gaussian form to apply is that the non-dimensional time of (ɛ/v)1/2t ∼ 300, following the release of the marked fluid with an initial diameter <(v3/t)1/4, be attained.

Some data on the distance-neighbour function for relative diffusion

Repeated observations of dye plumes on Lake Huron are interpreted according to the theoretical proposals of Richardson (1926) and Batchelor (1952) about the characteristics of a dispersing cloud of marked fluid within a field of homogeneous turbulence. The results show the average of several instantaneous concentration distributions about their centre of gravity to be approximately Gaussian and the distance-neighbour function to be of approximately Gaussian form. The data are consistent with the theoretical description given by Batchelor, namely, \[ q(y,t) = (2\pi\overline{y^2})^{-\frac{1}{2}}\exp (-y^2/2\overline{y^2}),\quad (\overline{y^2} = (\frac{2}{3}\alpha t)^3), \] where q(y, t) is the distance-neighbour function and α is the constant of the ‘4/3-power law’. The average value of α is estimated to be 0·12 cm2/3 sec−1. The rate of turbulent energy dissipation in the near-surface currents of Lake Huron is estimated as ε ∼ 2·1 × 10−3 cm2sec−3.

Analytical theory of turbulent diffusion

Recently Kraichnan (1959) has propounded a theory of homogeneous turbulence, based on a novel perturbation method, that leads to closed equations for the velocity covariance. In the present paper, this method is applied to the theory of turbulent diffusion and closed equations are derived for the probability distributions of the positions of marked fluid elements released in a turbulent flow.Two topics are discussed in detail. The first is the probability distribution, at timet, of the displacement of an element from its initial position. In homogeneous flows, this distribution is found to resemble that for classical diffusion but with a variable coefficient of diffusion which is proportional to$v^2_0 t$for$t \ll l|v_0$and which approaches a constant value [eDot ]lv0fortt[Gt ]l/v0(l= macroscale,v0= r.m.s. turbulent velocity).The second topic treated is the joint probability distribution of the displacements of two fluid elements. Particular attention is focused upon the probability distribution of relative displacement, i.e. Richardson's distance-neighbour function. This is found to be Gaussian for separationsrwhich are large ([Gt ]l). For smaller separations (r[Lt ]l), its behaviour at high Reynolds numbers is found to be quite well expressed in terms of a variable diffusion coefficientK(r,t), as suggested by Richardson (1926). For all but extremely short times,K(r,t) is found to depend only onrand on the form of the inertial range spectrumE(k). On assuming$E(r) \propto v^2_0 l(kl)^{- \frac {3}{2}}$as results from Kraichnan's approximation (1959), one finds$E(r) \propto v_0 l(r|l)^{ \frac {3}{2}}$. On the basis of similarity arguments of the Kolmogorov type, which give$E(r) \propto v^2_0 l(kl)^{- \frac {5}{2}}$, one finds$E(r) \propto v_0 l(r|l)^{ \frac {4}{3}}$as, in fact, Richardson originally proposed. The dispersionr2is proportional to$l^2(v_0 t|l)^4$on Kraichnan's theory; while$\langle r^2 \rangle \propto l^2 (v_0 t|l)^3$on the similarity theory. This illustrates that the behaviour of$\langle r^2 \rangle$is very sensitive to the spectrum.

Transforms for the eddy-diffusion of clusters

Fresh interest in the ‘distance-neighbour function’ has been aroused partly by the recent observations of Stommel, who found that the diffusivity for neighbours varied as the 4/3 power of their separation on the sea, the same law that L. F. Richardson had noticed in 1926 in the atmosphere; and partly also by a rather similar law which has been deduced by Weizäcker and by Batchelor from the theory of isotropic turbulence, started by Kolmogoroff in 1941. Some doubt has hitherto peristed as to the circumstances in which a given distance-neighbour function could determine the distribution of diffusing particles in space. A recent objection by Ichiye is shown to be ill-founded. The obscurities almost all disappear when the dis­tribution of the diffusing particles is analyzed into an odd and an even function of the co-ordinate, and when the distance-neighbour functions of the odd and even parts are formed separately. Definite transformations are then available forward and backward. The manner of diffusion of the distance-neighbour function of the odd part is not fully known. The inquiry concerning it has been reduced to instructions to future observers of fluids in large volumes. For the even part an observational test is proposed by equation (74 a ). Wind tunnels are not considered.