Journal of Applied Mathematics and Physics (ZAMP) Vol. 40, March 1989
0044-2275/89/020297q)3 $ 1.50 + 0.20/0 9 1989 Birkh~iuser Verlag, Basel
Near-source dispersion of contaminant from an elevated line-source By P. J. Sullivan and H. Yip, Dept. of Applied Mathematics, The University of Western Ontario, London, Ontario, Canada N6A 5B9 This note is an extension of Sullivan and Yip [1] and [2] where a flexible solution scheme, based on the asymptotic, small-time, solution to the convective-diffusion equation for an instantaneous point-source of contaminant concentration was used, through superposition, to find solutions to non-uniform continuous and instantaneous sources. It was recognized, and evident upon comparison with experimental results of Raupach and Legg [3], that some form of time dependence should be incorporated in the eddy-diffusivity to affect a better representation of the data at the closest measuring station from the source. Formerly the solution-scheme had but one empirical constant and here it is shown that the time-dependent aspect of the near-source diffusivity can be adequately included at the expense of one additional empirical constant. The statistics of the motion of fluid particles in a turbulent shear flow are nonstationary. The variance growth-rate was shown in Dewey and Sullivan [4] to depend on the integral over time of the Lagrangian autocorrelation function of the particle velocity and that this integral could be decomposed into three distinct components. These components are in essence made up of the change in location of the mean displacement, the correlation over the turbulent fluctuating velocities and the correlation between the turbulent fluctuating velocities and the variation in the mean velocity field sampled by the particles. Of relevance here, in using an eddy-diffusivity, is the time-dependent integral of the correlation over the fluctuating velocity field. It is to be noted that, following a small time interval from release, the dominant mechanism for the variance growth-rate of a contaminant cloud is spreading due to turbulent fluctuations convecting contaminant into regions with different values of mean velocity (i.e., the third term in the Dewey and Sullivan [4] decomposition, see also Sullivan and Yip [1]). In modelling turbulent diffusion with an eddy-diffusivity one anticipates dominant effects to arise from the presence of impermeable boundaries and that the interaction between crossstream mixing and mean velocity gradients does not depend critically on the detailed structure of the turbulent fluctuating velocities. The eddy-diffusivity arising from the streamwise (x) component of the Lagrangian fluctuating velocity field u(t; h) as given in Dewey and Sullivan [4] can be expressed as,
K(t; h) = (u(h)2) l/2(u(z(t))2)1/2 /'.ttoR(z; h) dr,
(D
where,
R(t; h) -
u(0; h)u(t; h)
(u(h) 2),/2(u(~(0)2) ~/2
(2)
when u(h) 2 and u(;~(t)) 2 are the mean-square Eulerian values at release height h at t = O
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P . J . Sullivan and H. Yip
ZAMP
and at the location of the mean cross-stream displacement s at time interval t following release respectively. On the expectation that the integral of (2) appearing in (1) converges to the value T(h) at t -* oo and further that,
T(h) oc h/(u(h) 2) 1/2,
(3)
then from (1),
K(t; h) oc (u(~(t)) 2) t/2h.
(4)
Within the constant-stress layer (see Csanady [5] p. 115) u(z) 2 oc u ~,,
(s)
where u, is the constant friction velocity such that the t --* oo asymptotic diffusivity K(h) becomes,
K(h) = ku,h,
(6)
and the Reynold's analogy with k approximately equal to von Karmans constant is recovered. An estimate of the Lagrangian integral time scale for the vertical component of the fluctuating velocities in the neutral atmospheric boundary layer of
V - a(h/u,),
(7)
where b represents the ratio of two constants given in Eq. 3.12 of Hunt and Weber [6]. This estimate suggests the correlation to be short-lived and comparable with the time h/u, required for fluid particles to reach the boundary. The form of the autocorrelation for a Markov process, R(~) = e-~/T,
6.0 - -
(8)
x/h = 2.5
Figure la
6.0 - -
x/h = 7.5
Figure 1b
3.0-
3.0--
k
o
I
I
0.5
1.0 C{xlh,z/h)/8 *
I 1.5
I 2.0
o
I 0.5
I
I
I
1.0
1.5
2.0
C(xlh,zlh)le*
Figure 1 Figures la and lb show a comparison between the measured concentration values of Raupach and Legg [3] ( 9 ) and the concentration values from the solution-scheme ( ) when a time-dependent diffusivity was used at x/h = 2.5 and x/h = 7.5, respectively. The dashed lines represent the concentration values from the solution-scheme without a time-dependent diffusivity. At x/h = 2.5 and 7.5, tiT = 1.26 and 5.47 respectively.
Vol. 40, 1989
299
Near-source dispersion of contaminant from an elevated line-source
is found to be a reasonable representation of experimental d a t a except for very small values of z (see Csanady [5] p. 54, Taylor [7] and measurements of Sullivan [8]). F o r the variable eddy-diffusivity it is useful to use (8), in (1) and take into account (6) to retrieve
K(z, t) = •(z)( 1 - e -t/r),
(9)
where release height h is contained in T and ~c(z) in the constant stress region is
~c(z) = U .Zok[C2(zl/zo) 2 + (Z /Zo) 2] 1/2.
(10)
The parameters Co, zo, and z~ are described in Sullivan and Yip [2]. It is expected that the time required for the cloud to sample an appreciable variation in ~(z) is larger than T. The relative change in K experienced by a cloud released at z = h over the interval T,
l dK
i.e. K-d-z-zAz, can be estimated using the cloud dimension Az ~- , ~ ( ( h ) T ( h )
and (6) and
(7) with b = 0.19 to be
1 . d K . Az ~ x/(.19)k - 0 . 2 7 . K dz
(11)
In Figure 1 the concentration as a function of vertical distance z/h for an elevated, continuous, line-source as measured by Raupach and Legg [3] are shown to be in rather good agreement with the solution-scheme when (9) is used with (10) (the values for the parameters in (10) are identical to ones used in Sullivan and Yip [2]), and b = 0.19 used in (7). This value of b is within the range given by Hunt and Weber [6] and leads to agreement that is within the experimental accuracy given by Raupach and Legg [3].
Acknowledgments
The authors received financial support from the Natural Sciences and Engineering Research Council of Canada. References
P. J. Sullivan and H. Yip, ZAMP 36, 596 (1985). P. J. Sullivan and H. Yip, ZAMP 38, 409 (1987). M. R. Raupach and B. J. Legg, J. Fluid Mech. 136, 111 (1983). R. Dewey and P. J. Sullivan, ZAMP 30, 601 (1979). G. T. Csanady, Turbulent diffusion in the environment, D. Reidel Publ. Co., Dordrecht 1973. [6] J. C. R. Hunt and A. H. Weber, Quart. J. R. Met. Soc. 105, 423 (1979). [7] G. I. Taylor, Proc. London Math. Soc. a20, 196 (1922). [8] P. J. Sullivan, J. Fluid Mech. 49, 551 (1971).
[1] [2] [3] [4] [5]
Abstract
The use of a time-dependent-diffusivity in the solution of the convective-diffusion equation is explored. The results are shown to compare favourably with near-source experimental data. (Received: June 27, 1988; revised: October 24, I988)