Some observations on the simple exponential function as a Lagrangian velocity correlation function in turbulent diffusion

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SOME OBSERVATIONS ON THE SIMPLE EXPONENTIAL FUNCTION AS A LAGRAN~IAN VELOCITY CORRELATION FUNCTION IN TURBULENT DIFFUSION J. NEUMANN Department of Atmospheric Sciences, The Hebrew University, Jerusalem, Israel

Abstract - In view of the repeated statements that a function of the form t-” (t = travel time, L = Lagrangian integral time scale of turbulence in the velocity component of concern) provida a good approximation to the Lagrangian velocity correlation functions R(t) in turbulcot flow, a check is made how welt does the expression for crosswind spread ensuing from the caption R(t) = c-“‘ represent the data embodied in the rccently published updated Hoska-Briggs-Gifford-Pasquiil graphs for spread. The comparisons are made with the graphs for horizontal crosswind spread u1 for Pasquili’s stability categories A to F and for the vertical spread a, for category D (neutral). The agrcemcrttis cx&cttt. Since the above graphs include the effect of dry deposition and since the long-travel distauce ends of the graphs show close agreement with Taylor’s prediction for spread at large travel times (strictly, for a homogeneous and stationary turbulence), based on a theory that does not account for the deposition process, it is pointed out that the assumption of a constant fractional depletion by deposition along any horizottta! crosswind line renders Taylor’s theory valid also for the case where dry deposition is in operation.

6, being the horizontal crosswind spread of the particles. It follows from the right side of (1) that

1. INTRODl.JCIlON It has been pointed out over the years by various writers on turbulent diffusion* that the simple exponential function of the form emriL,where t = travel time from source and L = Lagrangian integral time scale of turbulence in the velocity component of concern, appears to approximate the Lagrangian correlation functions of turbulent velocity components rather wdf. If then R(t) = eetfr+is the correlation function of, say, the horizontal crosswind component v of turbulent velocity, then following Taylor’s (1921) wellknown result for small particles dispersing from a continuous and steady point source in a fluid, in conditions of a homogeneous and stationary turbulence, +22 ’ (t-r’)R(t’)dt’ i 0

’ (t-f’)e-“‘Ldt’,

f: 27

s

(1)

0

* For a recent set of experimental data, see Fig. I in Hart and Frenzen (1976); for a com~rativejy recent text-book statement, see Csanady (1973. p. 65). t Taylor was the first, in his analysis of “diffusion by continuous movements”, to use the function e-I’& as an approximation in a Lagrangian type of analysis. Equation (2) can be found in his 1921 paper, see his Equation (23). The writer is indebted to Dr. F. A. Gifford, Atmospheric Turbuience and Diffusion Laboratory, NOAA, Oak Ridge, Tenn., for pointing out this fact in a letter. Equation (1) is the form in which Kampi de Firiet (1939) expressed Taylor’s result.

u> = Z;?L(t -41

-e-‘/L)).

(2)

Equation (2) should give us, within the goodnesssfapproximation to R(t), the variation of 6, with trave1 time.? The Lagrangian integral time scale L of the turbulence is defined by m L=

R(t)dt. 5

0

It can be considered a measure of the time during which the particles persist, in the mean, in a motion in a given direction. Despite the relatively frequent statements reiterating the goodness-of-approximation to the Lagrangian velocity correlation functions by the simple exponential function e-*‘L, several investigators proposed to represent uf as pure power functions of travel distance or time, see a brief review in PasqJill (1974, pp. 182-185). However, if the Lagrangian velocity correlation functions are well represented, indeed, by e-I”&, then pure power functions cannot give good predictions for uj in the medium range of travel distances (meaning travel times t not greatly different from t). The expression (1 -e-r’L) in (2) is equivalent to an infinite series whose terms are integral powers oft from 1 to co, and no pure power function, i.e. no single power, can well approximate this over a wide range oft (except, of course, at r CCL when terms quadratic or

1965

J.

1966

NEUMANN

higher in t can be neglected ; and when t a L, the exponential function may be equated to zero). published The recently updated Hosker-Briggs-Gifford-Pasquill (HBGP) graphs (Hosker, 1974, Fig. 1, involving recent contributions by B&s), which like the earlier Gifford-Pasquill (Gifford, 1960) graphs are presented on double logarithmic charts and which extend, again as the earlier Gifford-Pasquill graphs, over travel distances from 100 m to 100 km, indicate an appreciable curvature, a fact that is not compatible with a pure power-function representation across the whole of the aforementioned range of travel distances. Actually, earlier, a marked curvature was exhibited by the results of the “Green Glow” experiments (Fuquay et cl., 1964, Fig. 5). Still earlier, in 1959, a critique of the use of power functions for c, was put forward by Barad and Haugen (1959, p. 18). A particularly strong case against the use of power functions is found in a recent paper by Gifford (1976, pp. 74-75). In view of all this, the relative simplicity of (2) raises the question why should we strive at the use of power functions? One can retort, perhaps, that (2) requires estimates of 7 and L, quantities which are seldom available, a situation that renders the use of (2) impractical. But since the adoption of the form u/ = UX~, where a and p are constants, also requires adoption of values for these constants and since the relationships of these to the correct physical variables are not known, one could fit curves of the form of (2) to the best available diffusion data, such as the HBGP curves, and obtain in this fashion estimates of the physical parameters figuring in (2) pertinent to each of the Pasquill stability categories. In Table 1 we list values of ‘;r and L for each of the HBGP graphs for specified values of U, the mean wind. The purpose of this note is then to draw attention to the excellent degree of approximation offered by (2) when applied to the HBGP crosswind spread diagrams. An additional purpose is to consider the effect of dry deposition on the form of af, a factor whose importance has gained appreciation over the past decade or so.

homogeneity being an assumption in Taylor’s theory. Most diffusion experiments are reported in terms of travel distance rather than travel time, a statement that is true in respect of the HBGP graphs. The transformation from time to distance and vice-versa can be achieved, of course, through the relationship x = St, x being the travel distance from source, if the magnitude, say the r.m.s., of the fluctuations of turbulent velocity are small relative to the mean wind. Such a relationship is generally found to hold in the atmosphere and is embodied in the oft-made statement “the intensity of turbulence in the atmosphere is small”. The applicability of (3) to atmospheric diffusion was shown by Hay and Pasquill (Pasquill, 1974, pp. 184-185). As to (4), a reference to the long traveldistance ends of the HBGP curves indicates that it is well satisfied by them. The latter fact is especially noteworthy as Taylor’s (1921) results were derived at a time when the process of dry deposition was not appreciated and, consequently, not accounted for. In contrast, Hosker’s work does take the process into consideration. In Section 3 below we will point to a simple assumption which goes some way to explain why (4) should hold near the earth’s surface without altering the form of (4). A particularly simple check on the validity of predictions of (2) is to take the case t = L. Dropping the subscript y and denoting the value of u pertinent to t = L by u,,, we have from (2), 0: =

2e-‘vfL2 L 0.74pL2

Now, 7 and L can be estimated from the HBGP graphs by applying (3) and (4) and the relation x = it. Let (x,,u,) and (x2,u2) be travel distances and the associated u values at points where, respectively, (3) and (4) are satisfied. Similarly, let x0 be the travel distance where u = uc. At that point, x,, = UL. Then, from (3) and (4) x,=tiL=-.

x;u; 2x, u;

(6)

From (6) (5) and (3), *; = 0.74s. x: 2. SOME RELATIONSHH’S CHECKING EQUATION

FOR (2)

Equation (2) requires a knowledge of 2 and L. These can be estimated by reference to the “asymptotic” forms of (2) oiz. c> = v2t2 for t cc L (3) and c$=2u2Lt

for

t>>L,

(4)

provided that (3) and (4) hold for the atmosphere where turbulence near the surface is not homogeneous,

Let us apply (7) to the HBGP curves. Six curves are involved for u*, each corresponding to cne of the stability categories A-F introduced by Pasquill. Additionally, we can apply (7) to the a, curve for vertical spread for stability D (neutral). In Table 1 we list values of (x1. a,) based on the short travel-distance ends and those of (x2, 17~)based on the long travel-distance ends of the curves. The values of x,, and u,, in the Table were computed from (6) and (7). Further, the Table lists the experimental values of u0 read off the curves at the points x,, calculated as described above. It is clear from the table that the agreement is excellent. Finally, the table lists values of 2 and L for each of

F D

E

B c D

A

Stability category 22 14 12 8.3 6 4 5

2, 6800 4800 3200 2250 1600 1200 400

pm’,

(70)

2400 1700 700 750 :z!

Bquation (8)

vz 0.19 0.24 0.36 0.17 0.044 0.0064 0.0

(m’ smf) Equation (8)

Note: Most values of u2, L, x0 and u. have been rounded off to the nearest multiple of 50

68 cd “d cb “d “/ c*

-

Crosswind spread 2 3.5 5 5 3.5 2 5

Ii (ms-‘)

3550 3700 3550 4500 350

4800 5900

x0 = liL

tz 150 15

-900 -750 -350 -250 150-200 - 150 * 15

900 700 350

$1 Observed

Equation (5)

$1

‘fable 1. Computed values of L, the Lagrangian integral time scale, and of the crosswind spread uc at the travel distance x0 = tiL, compared with the value of a0 reported in the Hosker-Briggs-Gilford-Pasquig graphs (8x all stability categories x, =c 100 m, x2 = 100 km ; the values of ut and us are from the graphs; the valuea of ri are in the range stated by Hosker, 1974)

J. NEUMANN

1968

the HBGP curves using (3), (4) and the relation x = tit. The relevant equations are

3. DRY DEF’CSITION It was pointed out earlier that at the time when Taylor (1921) developed his theory, Equation (l), nothing was known about dry deposition. Thus, (1) takes no account of it though the process is of considerable importance for most pollutants at large travel distances, see relevant data and diagrams in a paper by Tadmor (1971). It is interesting to note that the HBGP curves as well as some of the other curves which show the variation of od with travel time or distant, up to large values of the latter two, indicate that the validity of (4) is not impaired in spite of the inevitably significant depletion by dry deposition. We can understand this situation if we make the assumption that deposition reduces the concentration of particles at the same rate percentagewise at all distances from thecloud axis. Suppose that the deposition process is not operating and that the variance of the horizontal crosswind positions y of the particles (y = 0 on the axis) would then be, at any given values of x and z, in terms of class intervals:

where n, is the number density of particles in the ith interval. Now, if we make the assumption that the dry deposition process does operate and that the fractional depletion rate, say, is independent of y, then +

C(l-a)ni i C(1-4&

Yf ’

(10)

which equals (9). Thus, under the assumption of the independence from axis distance of the fractional depletion rate by deposition, (4) remains valid.

4. DISCUSSION

As has already been noted, the agreement between the values read off the HBGP graphs and those computed from (5) involving (2), is excellent. Additionally, Dr. M. Epstein (Dept. of Atmospheric Sciences, The Hebrew University) has computed the

values of ud for four more travel distances for each of the curves A-F. The maximum discrepancy between the computed and observed values is not in excess of 15%; in fact, in most cases the difference is very small. Though it is not easy to read the observed values from the graphs to great “precision” and any attempt at “precision” in the case under study is meaningless, it is probably fully justified to say that the agreement is very close. One other noteworthy feature of the Table is the rather large values of L. The few values of L published in the literature are mostly for elevations close to the surface and these are of the order of a few seconds (Haugen, 1966, Table 5; Hart and Frenzen, 1976, Table 1) up to about 10s. We expect, however, the Lagrangian scale to increase with height and it is probably correct to suggest that the large values in Table 1 are connected with the fact that most of the data incorporated in the HBGP graphs represent diffusion from elevated sources.

Acknowledgement - The writer is indebted to Dr. M. Epstein for discussions on the subject of this paper.

REFERENCES Barad M. L. and Haugen D. A. (1959) A preliminirry evaluation of Sutton’s hypothesis for diffusion from a continuous point source. J. ~0~0s. Sci. 16, 12-20. Csanady G. T. (1973) Turbule~ Diction in fhe ~nvironme~. Reidel, Dordrecht. Fuquay J. J., Simpson C. L. and Hinds T. W. (1964) Estimates of groundtevel air exposures resulting from protracted emissions from “IO-meterstacks at Hanford. J. appl. Met. 3,

760-771. GitTord F. A. (1960) Atmospheric dispersion calculation using the generalized Gaussian plume model. Nucl. Saf: 1.56-59. Gifford F. A. (1976) Turbulent diffusion-typing schemes : a review. Nucf. Sa$ 17, 68-86. Hart R. L. and Frenzen P. (1976) Atmospheric dispersion over open water inferred from Lagrangian statistics. In Radiological and Environmental Research Division Annual Report, Atmospheric Physics, January-December 1976, pp. 90-97. Argonne National Lab., ANL-76-88, Pt. IV. Haugen D. A. f 1966) Some Lagrangian properties of turbulence deduced from atmospheric diffusion experiments. J. appf. Met. 5, 646-652.

Hoskcr R. P. (1974) Estimates of dry deposition and plume depletion over forests and grassland. In Physical Eehuuior o/ Radioactive Contaminants in the Atmosphere. Symposium Proc., pp. 291-308. IAEA, Vienna. Karnti de FCriet M. J. (1939) Les fonctions alCtoirs stationnaires et la theorie statistique de la turbulence homoghe. Ann. Sac. Sci. Bruxefles 59, 145-194. Pasquill F. (1974) Atmospheric LX&ion, 2nd Ed. Horwood, Chichester. Tadmor J. (i971 f Consideration ofdeposition of pollutants in the design of stack height. Atmospheric Environment. 5, 473-482. Taylor G. I. (1921) Diffusion by continuo~ movements. Proc. Land. math. Sot. 20, 196-202.