THE RELATIONSHIP BETWEEN SKEWNESS AND ... - Springer Link

Boundary-Layer Meteorology (2005) 115: 341–358 DOI 10.1007/s10546-004-5642-7

Ó Springer 2005

THE RELATIONSHIP BETWEEN SKEWNESS AND KURTOSIS OF A DIFFUSING SCALAR T. P. SCHOPFLOCHER* and P. J. SULLIVAN Department of Applied Mathematics, The University of Western Ontario, Canada

(Received in final form 4 October 2004)

Abstract. It has been demonstrated that in turbulent dispersion, there exists a quadratic relationship between the skewness (S) and kurtosis (K) statistics obtained from continuous, elevated sources of scalar contaminant released into both convective and stable atmospheric boundary layers. Specifically, one observes that K ¼ AS2 þ B;

where A and B are empirically fitted constants that depend on the flow. For two reasons, this is potentially useful information in regard to modelling the probability density function (PDF) of a diffusing scalar. First, since many PDFs have a signature relationship between their skewness and kurtosis, candidate models can immediately be either accepted or rejected depending upon whether they conform to the quadratic curve that is observed experimentally. Second, if one intends to model the PDF by inverting a limited number of moments, the task is reduced when there is a functional relationship between the standardized third and fourth moments. The aforementioned relationship has been corroborated by others who have examined data over a wide range of experimental configurations. However, from one flow to another, there appears to be a non-negligible variability in the two fitting constants of the quadratic curve. In this paper we put forth a framework to help explain this phenomenon, and we also attempt to predict how these parameters vary in space and/or time. Our point is illustrated with well-resolved data from a wind-tunnel, grid-turbulence, plume experiment. Keywords: Concentration, Kurtosis, Probability density, Skewness, Turbulent diffusion.

1. Introduction When a scalar contaminant with concentration C is released into a flow, its evolution is governed by the convective diffusion equation @C ð1Þ þ u rC ¼ jr2 C; @t where j is the coefficient of molecular diffusivity, and u is the velocity field, itself governed by the Navier–Stokes equations. Due to the instability of these non-linear governing equations, measurable quantities in a turbulent system must be given a statistical treatment. Scalars such as heat or

E-mail: tschopfl@uwo.ca

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T. P. SCHOPFLOCHER AND P. J. SULLIVAN

contaminant concentration are no exception. Normally, one represents a scalar field by the probability density function (henceforth denoted as a density or a PDF), defined as pðh; x; tÞdh ¼ probfh Cðx; tÞ < h þ dhg: ð2Þ While we often suppress the notation for convenience, C ¼ Cðx; tÞ is generally a function of both position vector x and time t. The evolution of pðh; x; tÞ is extremely complicated (Chatwin, 1990). For this reason, it is often advantageous to consider the evolution of its central moments. These are defined as Z h0 ðh CÞr pðh; x; tÞdh; r 2; ð3Þ lr ðx; tÞ ¼ 0

where Cðx; tÞ ¼

Z

h0

hpðh; x; tÞdh;

ð4Þ

0

and h0 is the largest source concentration value. (Note that, by definition, C is intrinsically a non-negative quantity.) In this way, a less complicated but still intractable differential equation can be derived for the evolution of the moments (Chatwin and Sullivan, 1990.) This is a reasonable approach, since with a knowledge of all the moments, the PDF can be fully reconstructed. Even in the absence of a parametric model, and with an incomplete set of measured moments, one can still obtain an accurate representation of the density (using an orthogonal polynomial expansion, for example). In practice, the first three or four sample moments are enough to capture the bulk characteristics of the PDF (Derksen and Sullivan, 1990). If one wishes to retrieve the more subtle features of the probability distribution, higher-order moments are required. It is still possible to model the more detailed features of the PDF with r 4. However, more assumptions must be made about the general shape of the curve. For example, by invoking the theory of extreme-value analysis, the high-concentration PDF tails were modelled by both Mole et al. (1995) and Schopflocher (2001) by means of a generalized Pareto distribution. In light of the aforementioned, one would like a parametric model for the PDF (a mixture-model, for physical reasons to be outlined later) with enough flexibility to capture the various shapes that are observed in different experimental situations. Also, one would like the important tail area to be represented accurately. If one makes the correct assumptions about the functional form of the PDF, this approach makes the task of statistical inference particularly simple. The method of moments can be applied with minimal numerical difficulties in order to yield robust point estimates (Schopflocher, 1999). The goal then is to determine the parameters of the

SKEWNESS AND KURTOSIS OF A DIFFUSING SCALAR

343

model by the use of only three or four measured moments. The general approach is outlined by Lewis and Chatwin (1996) and Schopflocher and Sullivan (2002). While trying to model the PDF with only a few moments is an attractive idea, it is a difficult task in practice. This is largely due to the limited information available about the mathematical properties of the PDF, its moments, and any other statistic related to C. For that reason, any relationship between higher moments (empirical or otherwise) proves to be highly beneficial. Mole and Clarke (1995) presented arguments for a quadratic relationship between the standardized third and fourth moments for a diffusing scalar. Specifically, they claimed that the functional form is approximately K ¼ AS2 þ B; where skewness S and kurtosis K are defined as l3 S ¼ 3=2 l2 and K¼

l4 ; l22

ð5Þ ð6Þ

ð7Þ

and A and B are order-one constants. In that paper, Mole and Clark also considered relationships between other statistics including higher-order moments and dosage. Here, however, we focus exclusively on the relationship in (5). Using data from a continuous, elevated source in both convective and stratified conditions in the atmospheric boundary layer, Mole and Clark determined the fitted constants in (5) to be A ¼ 1:31 and B ¼ 0:77. Since the original work of Mole and Clark, data obtained from a wide range of experimental configurations have been shown to collapse onto a parabola of the same form. Lewis et al. (1997) showed a similar trend from field trials over four different locations and stability conditions for continuous releases. They found that (5) provided an adequate fit with A ¼ 4=3 and B ¼ 3. Also, Schopflocher (1999) examined data from a grid-turbulence, laboratory plume experiment. Again, there was a convincing collapse, this time with A ¼ 1:20 and B ¼ 2:74. The relationship between skewness and kurtosis has proven to be remarkably robust. For example, consider the experiments by Hall et al. (1991). These consisted of releasing a finite amount of dense gas (the Richardson number was varied over different experimental releases) into a wind-tunnel boundary layer and measuring concentration downstream. In order to compile statistics, between 50 and 100 releases were conducted for each Richardson number under consideration. (Note that some of these

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experiments included solid and crenellated fences in the flow path.) Analysis of these concentration records reveals behaviour that conforms to (5) (Heagy and Sullivan, 1996). Chatwin and Robinson (1997) present a table of values for the constants A and B over a range of release densities, both with and without the fences. In Table II of their paper, a relatively narrow range of values (1:0 A 1:5 and 1:5 B 2:5) was shown to encompass the various experimental setups. The fact that there exists such a pattern in experimental field data as well as laboratory data on clouds is of potentially great value. The expression given in (5) is ideal when modelling environmental flows. As previously stated, the equations governing the evolution of both the PDF and its moments are intractable due to non-closure. Furthermore, the validation of approximate solutions to these equations is constantly plagued by the difficulty of making adequate field measurements (or even controlled laboratory experiments in the case of repeated contaminant cloud releases). So, it is very important that data from isolated fixed-point measurements (the most common experimental set-up) can be used to corroborate (5). In Section 2 we provide the theory underlying the work herein. This is essentially an overview of the ab model due to Chatwin and Sullivan (1990). In Section 3 we derive conditions under which one should expect the relationship in (5) to apply. Section 4 provides a comparison of our results for experimental data from a grid-turbulence plume. Finally, in Section 5, concluding remarks and a brief discussion of results are presented. 2. The ab Model for Moments The fact that turbulent convective motions and molecular diffusivity act on much different time scales led Chatwin and Sullivan (1990) to derive a methodology for prescribing the moments lr ðx; tÞ. Since then, there have been some modifications to their original theory (Sawford and Sullivan, 1995); however, the ideas outlined therein laid the groundwork for a statistical model of turbulent diffusion based on physical arguments. Their reasoning begins by considering a hypothetical flow where there is no molecular diffusivity (i.e. j ¼ 0 in Equation (1)). In this case, there is no agent to bring about the mixing of a scalar with the host fluid. However, since the process of convection is still at play, one has a situation in which a mass of contaminant is distended into ever-thinning sheets and strands. This (strictly theoretical) flow would be characterized by a PDF of the form ð8Þ pðh; x; tÞ ¼ pðx; tÞdðh h0 Þ þ ð1 pðx; tÞÞdðhÞ; where p is the probability of being in marked fluid located at position x at time t, and dðÞ is the Dirac delta function.


345

Clearly, this is a two-state process, since the sample space contains only two members – the uniform release concentration h0 (that of the strands), and 0 (the concentration outside the strands). Furthermore, the model contains only two parameters, p and h0 , so the system is characterized completely by its first two moments ð9Þ C ¼ ph0 ; l2 ¼ ph20 ð1 pÞ:

ð10Þ

In a real flow where j > 0, the aforementioned thinning of sheets and strands will eventually be balanced by the ‘‘thickening’’ caused by molecular diffusion. So, while we still expect the strand-like structures to exist and be of a higher concentration than the background, there will be an ambient concentration level that is certainly non-zero. This feature of the concentration field has been corroborated by high-resolution laboratory experiments (Dahm et al., 1991; Corriveau and Baines, 1994; Buch and Dahm, 1997). The formulae for the first two moments in (9) and (10) were modified to include empirical constants accounting for the effects of molecular diffusion. First, h0 in (10) was replaced by a representative sheet-strand concentration. Then, another parameter was incorporated to account for the increased background concentration. These modifications again result in a two-state process (Sullivan and Ye, 1996) where now, the delta functions from (8) have each moved closer together (away from the extremes they formerly represented). As such, one can write the second moment in terms of the first as l2 ¼ b2 CðaC0 CÞ;

ð11Þ

where b is a factor accounting for the increasing ambient levels due to molecular diffusion, and C0 is the maximum mean concentration at the time of consideration in a sudden release. For a steady release, C0 is the maximum mean concentration along the cross-section at the distance under consideration. So, C0 varies with time in the former case, and with distance downstream in the latter. Similarly, a and b are also functions of time alone in the case of a cloud, and are functions of downstream distance alone in the case of a steady release. This simplified version, as outlined by Clarke and Mole (1995), is an appropriate reference base for the discussion that follows. In order to capture the characteristics of a real flow, one can view turbulent dispersion as a regenerative process whereby a non-uniform, moving, scalar source is continually transporting central core fluid to the periphery. This means that C will take on a (continuous) spectrum of values for both the high-concentration strand values as well as the low-concentration background values. To incorporate this crucial property, a new parameter must be added for each moment that is being modelled (Sawford and Sullivan, 1995). Such a moment prescription can be written explicitly in terms of the mean C,

346


reference concentration C0 , as well as parameters a, b, and now ki . For the second, third and fourth moments, we have l2 ¼ b2 Cðr2 CÞ;

ð12Þ

l3 ¼ b3 Cðr3 3r2 C þ 2C2 Þ;

ð13Þ

l4 ¼ b4 Cðr4 4r3 C þ 6r2 C2 3C3 Þ; 2

ð14Þ 3

where r2 ¼ aC0 , r3 ¼ ðak3 C0 Þ and r4 ¼ ðak4 C0 Þ . From Equations (12)–(14), we derive an expression for the skewness and kurtosis as S¼

K¼

Cðr3 3r2 C þ 2C2 Þ C3=2 ðr2 CÞ3=2

ð15Þ

;

Cðr4 4r3 C þ 6r2 C2 3C3 Þ C2 ðr2 CÞ2

:

ð16Þ

Note that, while a is redundant now that the ki are introduced, it is convenient to keep this parameter for reference to the original formalism described above. Again, C0 , a, b and ki depend on time alone in the case of a cloud, and are functions of downstream distance alone in the case of a steady release. For clarity of presentation, and because it is the nature of the data to be examined here, we will, for the most part, discuss the steady-state situation henceforth. Nevertheless, all results generalize to the time-dependant case. The a b description has enjoyed a considerable amount of validation over a wide range of flows, including a very convincing comparison to a secondmoment profile from the atmospheric boundary layer. For further details on these analyses, the reader is referred to studies by Chatwin and Sullivan (1990) and Sawford and Sullivan (1995) – both referred to later in this paper. Detailed fits to Equations (12)–(14) for the Sawford and Tivendale (1992) data (discussed here) are shown in Sullivan (2004).

3. The Relationship between Skewness and Kurtosis It was noted by Mole and Clark (1995) that when all of the ki in (12)–(14) are unity (i.e. the two-state system that gives rise to the variance in (11)), one obtains the relationship K ¼ S2 þ 1:

ð17Þ

It can be shown (Wilkins, 1944) that this is the lower bound of the general relationship (valid for all densities whose third and fourth moments are defined)


K S2 þ 1:

347

ð18Þ

While the two state-model is an idealized scenario, it should be accurate as a limiting case. As discussed above, molecular diffusion is a slow process when compared with turbulent convection. Therefore, while difficult to achieve experimentally, at very small times after release (in the case of a cloud) or very small distances downstream (in the case of a steady source) the effects of j will be negligible and (17) will be a good approximation. If one views the real-flow scenario (modelled by (12)–(14)) as a perturbation to the hypothetical one (modelled by a double delta function PDF), then it is reasonable to expect that one would observe an approximate quadratic collapse as described in (5) (Mole and Clarke, 1995). In this situation, it is likely that a graph of skewness against kurtosis will lie on a parabola slightly above the lower bound described in (17). It has been shown that the PDF for C is represented in a very natural way as a mixture distribution (Chatwin and Sullivan, 1989). One writes pðh; !Þ ¼ pfðh; !Þ þ ð1 pÞgðh; !Þ; ð19Þ where f is the PDF describing the concentration distribution of the strands, g is the PDF describing the concentration distribution of the background, p is the probability of being in the high concentration material, and ! is a parameter vector. Lewis et al. (1997) put forth the conditions under which the model in (19) would give rise to the relationship in (5). In particular, they considered the conditions under which kurtosis goes like the square of skewness when the PDFs f and g from (19) are a generalized Pareto distribution and an exponential distribution, respectively. It is interesting that Ye (1995) obtained a reasonably good fit to some laboratory plume data by convolving a Gaussian with the (two-state) PDF associated with the moments in (12)–(14) when the ki ¼ 1. While the ab model (with ki ¼ 1) cannot capture the required concentration range, it is still useful for describing bulk characteristics of the diffusion process. Specifically, the means and relative magnitudes of the components for the mixture in (19) should be consistent. A further comparison was made by examining the individual component statistics of a double beta PDF (Schopflocher and Sullivan, 2002). In that paper, it was shown that the two-state process does a good job modelling the mixing ratio p as well as the means of the individual beta densities. If we choose to model the PDF for C via (19), then there are some immediate consequences regarding the skewness versus kurtosis graph. Consider (2) and (3). With a simple change of variable, these can be rewritten as Z 1 ðh CÞr dP ð20Þ lr ¼ 0

348


and mr ¼

Z

1

hr dP;

ð21Þ

0

where the distribution function (DF) P ¼ probfC hg. In this formulation, where the variate is a function of the DF, h ¼ hðPÞ is called the quantile function. In general, for quantile function h and parameter vector ! ¼ ðt1 ; . . . ; tk Þ, if one can write ð22Þ hðP; t1 ; . . . ; tk Þ ¼ t1 þ t2 hðP; 0; 1; t3 ; . . . ; tk Þ; then we call t1 and t2 location and scale parameters, respectively. The remaining k2 are called shape parameters. Denoting ð23Þ h01 ¼ hðP; 0; 1; t3 ; . . . ; tk Þ; we write the rth central moment as r Z 1 Z 1 r lr ¼ t2 h01 h01 dP dP: 0

Finally, we express the skewness and kurtosis as 3 R 1 R1 h h dP dP 01 01 0 0 S¼ 2 3=2 ; R1 R1 0 h01 0 h01 dP dP R 1

ð24Þ

0

4 h dP dP 01 0 0 K¼ 2 2 : R1 R1 0 h01 0 h01 dP dP h01

ð25Þ

R1

ð26Þ

The important thing to notice in Equations (25) and (26) is that there is no dependence on the location or scale parameters. So, we have ð27Þ S ¼ Sðt3 ; . . . ; tk Þ; and K ¼ Kðt3 ; . . . ; tk Þ:

ð28Þ

If K and S depend on a single shape parameter, then a plot of K versus S results in a single curve. If there are two or more shape parameters, with the exception of special circumstances, the relationship of K and S will be represented by a two-dimensional region on the plot. In the event that one observes a single curve in the skewness versus kurtosis plane, one can model the PDF of C with a one (shape) parameter density. One notes the success of Lewis and Chatwin (1996) in using a 3-parameter model PDF. From the ab model described in Section 2, we know that for a steady flow, the parameters in (15) and (16) are constant along a cross-section of the


349

flow. We will demonstrate that for a given downstream location, both S and K depend primarily on one parameter. Furthermore, this parameter varies in a systematic way as one moves downstream (away from the source). To begin, re-write (5) in terms of the parametric representations of skewness and kurtosis in (15) and (16). Expanding both sides in powers of C and equating the corresponding coefficients, one arrives at the following conditions required in order for (5) to hold true, k34 ; k43

ð29Þ

B ¼ 4A 3;

ð30Þ

L1 ¼ 6k34 k23 þ 4k34 3k43 þ 4k63 þ k34 k43 ¼ 0;

ð31Þ

L2 ¼ 4k34 k23 3k34 þ 3k43 4k63 ¼ 0:

ð32Þ

A¼

It was mentioned above that the case where the ki ¼ 1 (for which A ¼ B ¼ 1, and (31) and (32) are satisfied exactly) corresponds to a double delta function PDF. Using (15) and (16) as well as (29)–(32), (5) can be expressed as K ¼ AS2 þ B þ

L1 þ L2 C k43 ð1 CÞ3

ð33Þ

;

where C ¼ C=aC0 . By direct observation of the ab prescription for skewness and kurtosis given in (15) and (16), it is clear that when C 1, and K and S are large, then K ðk34 =k43 ÞS2 since aki are order unity. It is instructive to express the ki values as a perturbation of the two-state model as ð34Þ k3 ¼ 1 þ 3 ; k4 ¼ 1 þ 4 :

ð35Þ

Retaining only linear terms in i , L1 ¼ L2 ¼ 43 34 ¼ D

ð36Þ

and (33) becomes K ¼ ð1 þ DÞS2 þ

1þD 4þ

1 ð1 CÞ2

!! ;

ð37Þ

or K ¼ A0 S2 þ B0

ð38Þ

where, for C 1 A0 ¼ 1 þ D;

ð39Þ

B0 ¼ 1 þ 5D:

ð40Þ

350


At a fixed distance downstream the only term in (37) that varies over the cross-section is C. This dependence is expected to be slight and certainly negligible as C 1. For example, if we assume values C ¼ C0 , a ¼ 1:5, and D ¼ 0:1, then the last bracketed term on the right-hand side of (37) is 3.2. This is to be compared with the value of 1.5 that the term would assume if C ¼ 0. Thus anticipating i to be small, the ab prescription of moments suggests that, at any fixed distance downstream for a continuous emission, (38) will provide a very good approximation to virtually all data on that cross-section. Like A and B, the values of A0 and B0 depend on the ki and change with downstream distance. It is conceivable that these parameters become asymptotically constant. If this is the case, all values of K and S2 should fall on the line in (38) with constant values of A0 and B0 (neglecting the minor contribution from C in (37)). Analogous statements apply to clouds. To avoid any confusion in notation, let us clarify the distinction between (5) and (38). We will continue to refer to (5) as the general relationship between skewness and kurtosis. However, for the purposes of this paper, constants A and B are obtained through a straightforward regression, and constants A0 and B0 are derived from measured values of k3 and k4 as described above.

4. Analysis and Comparison to Experiment It is remarkable that a quadratic relationship between skewness and kurtosis has been observed in both field and laboratory experiments (the latter including contaminant clouds). In order to assess the position put forward in Section 2, we take advantage of the well-controlled, steady, laboratory experiments of Sawford and Tivendale (1992) where many sampling positions in the flow have already been used to validate the ab moment prescription. Here, one is assured that sufficient record length is available to approximate the ensemble-averaged third and fourth moments. Also, reasonable temporal and spatial resolution has been achieved in these measurements. The experiments of Sawford and Tivendale (1992) were conducted in a suction wind tunnel with mean wind speed U ¼ 5 m s1 . Allowing this air to pass through a grid with mesh spacing M ¼ 0:0254 m, they obtained a fully turbulent flow with Reynolds number R ¼ UM=m 8500, where m is kinematic viscosity. The contaminant source consisted of a heated wire placed downstream from the grid and span-wise across the tunnel, thus producing a line source of heat. Measurements were taken by fast-response, cold-wire anemometers placed transversely across the plume at approximately logarithmically spaced distances downstream. All experiments comprise 16 measurements, each taken at a different cross-stream location, across a


351

fixed downstream distance x (between 2 and 2600 mm). Since the mean flow field is accurately described by a normal distribution across the plume, it is natural to measure the distance from the centreline in terms of the (in general, non-integer) number of associated standard deviations r. So, a particular run might be conducted at, say, 100 mm and consist of 16 measurements ranging from about 3:5r to 3:5r. For more information regarding the experimental set-up, the reader is referred to the original work by Sawford and Tivendale (1992) as well as the analysis by Sawford and Sullivan (1995). At all downstream locations there are stations that are presumably too far from the centreline to give reasonable results. Once the probe is more than about 2:5r off the centreline, the signal-to-noise ratio becomes unacceptable (Schopflocher and Sullivan, 2002). Since the third and fourth moments tend to diverge at these locations, they are omitted from the discussion. Another source of error arises if measuring stations are too close to the source. When the probe is less than about 10 mm downstream it is difficult to obtain adequate resolution across the plume (Sullivan, 2004). Analysis of these near-source stations is not discussed. In total, we analyze 13 downstream positions and 10 cross-stream positions. This means that we should calculate 130 pairs of S and K. However, all experiments were repeated so that we actually have multiple measurements for all downstream stations (between 2 and 5 in each case). This raises the number of pairs to 42 10 ¼ 420. For each experiment, we regress K against S2 in the ordinary least squares sense, using only the 10 cross-stream stations of a given run. However, when we consider the trend downstream, we will average the point estimates of A and B for experiments with a common downstream location x. Note that the variance associated with these mean values was typically very small. All 42 graphs of kurtosis against skewness-squared were convincingly linear. Shown in Figures 1a–1d are different plots from representative downstream locations (15, 150, 1600, 2600 mm). To explore further the prediction of (5) from the ab prediction description at a given downstream distance, we make use of the fact that the values of mean concentration in these experiments were accurately Gaussian over each of the measurement cross-sections. Values of K and S2 were computed from (15) and (16) using a Gaussian function for C. These are shown on Figure 2 for some representative downstream distances. The linear relationship between S2 and K appears to describe the points for each of the downstream locations shown on Figure 2. Therefore, there is little evidence of dependence on C in (37) for values of S and K approximately equal to unity. Note that even if S ¼ 0, such a distribution would likely be positioned at an off-axis location where the value of C is low. For such an example, see Figure 1a in Sawford and Sullivan (1995) and Figure 7 in Chatwan and Sullivan (1990).

352


Kurtosis

(a)

60

60

50

50

40

40

30

30

20

20

10

10

Data Fit

0

0 0

5

10

15

20

25

30

35

40

45

50

Skewness Squared

Kurtosis

(b)

90

90

80

80

70

70

60

60

50

50

40

40

30

30 20

20 Data Fit

10

10 0

0 0

10

20

30

40

50

60

70

Skewness Squared

(c)

80 70

Kurtosis

60 50 40 30 20 Data Fit

10 0 0

5

10

15

20

25

30

35

40

45

50

Skewness Squared

Kurtosis

(d)

30

30

25

25

20

20

15

15

10

10

5

5 Data Fit

0 0

2

4

6

8

10

12

14

16

0 18

Skewness Squared

Figure 1. (a) K plotted against S2 for the 15-mm station. Also shown is the linear fit with A ¼ 1:16; B ¼ 1:85; (b) K plotted against S2 for the 150-mm station. Also shown is the linear fit with A ¼ 1:24, B ¼ 1:94; (c) K plotted against S2 for the 1600-mm station. Also shown is the linear fit with A ¼ 1:56, B ¼ 1:78; (d) K plotted against S2 for the 2600-mm station. Also shown is the linear fit with A ¼ 1:40, B ¼ 2:26.

353


The a–b analysis suggests that the constants A and B in (5), as well as constants A0 and B0 in (38), are not independent. Combining (39) and (40), we obtain B0 ¼ 5A0 4:

ð41Þ

In order to test this relationship, the quantity 5A4 is calculated and compared with parameter B (see Table I). The comparison is reasonable and consistent with the procedures used in arriving at the average values for A and B (also shown in Table I). Another significant finding is that the fit constants do indeed vary in a systematic way as we move downstream. This result is anticipated from the ab model since the values for k3 , k4 and D (shown in Table I) appear to satisfy the criteria in (37) for each cross-section. In Figure 3, four of the 13 regressions K ¼ AS2 þ B (x = 10, 200, 700, 2600 mm) are plotted together to illustrate the systematic movement away from the lower bound K ¼ S2 þ 1 (also on Figure 3). From the a–b analysis, we expect that parameters A and B (as well as A0 and B0 ) will depend on k3 and k4 . The directly fitted values listed in Table I (and shown graphically in Figure 4) indicate that A and B will generally increase as one proceeds downstream. In Figure 4 the slope parameter A is observed to start at 1.14, rise rapidly, and then level off at 1.43 as we move far from the source. The intercept parameter B appears to have less variation (1:82 B 2:22) and far more irregularity. The magnitude and trend of the directly fitted values of A and B appear to be similar to those of A0 and B0

14 12

Kurtosis

10 8 6 4

15 mm 150 mm 1600 mm 2600 mm

2 0 0

1

2

3

4

5

6

7

8

9

Skewness Squared

Figure 2. K plotted against S2 for ab parameters and a Gaussian mean profile.

10

354


TABLE I Parameter estimates for the 13 downstream locations of the CSIRO experiments. x (mm)

k3

k4

10 15 20 20 50 70 100 150 200 300 700 1600 2600

1.043 1.052 1.060 1.068 1.073 1.079 1.080 1.077 1.084 1.092 1.078 1.057 1.056

1.087 1.102 1.115 1.124 1.142 1.154 1.157 1.154 1.170 1.181 1.166 1.136 1.129

A

B

1.139 1.156 1.157 1.171 1.174 1.199 1.189 1.221 1.249 1.254 1.287 1.435 1.434

2.019 1.869 1.841 1.838 1.876 1.819 1.979 1.973 2.069 2.184 1.978 2.103 2.222

A0

B0

D

5A)4

1.088 1.097 1.106 1.100 1.133 1.145 1.151 1.156 1.174 1.177 1.185 1.179 1.163

1.440 1.485 1.530 1.500 1.665 1.725 1.755 1.780 1.870 1.885 1.925 1.895 1.815

0.088 0.097 0.106 0.100 0.133 0.145 0.151 0.156 0.174 0.177 0.185 0.179 0.163

1.695 1.781 1.786 1.855 1.872 1.995 1.947 2.103 2.245 2.269 2.435 3.176 3.168

100 Lower Bound

90 80

10mm

Kurtosis

70 200mm

60 50

700mm

40 2600mm

30 20 10 0 1

2

3

4

5

6

7

8

9

10

11

12

13

14

Skewness Squared

Figure 3. Fitted K ¼ AS2 þ B lines for 10, 200, 700 and 2600 mm downstream locations. Also shown is the theoretical lower bound, K ¼ S2 þ 1.

(also shown in Table I) found using the approximation given in (39) and (40) and calculated with measured values of k3 and k4 . The scatter plots and associated regression lines for A versus A0 and B versus B0 are shown in Figures 5a and 5b. The directly fitted values A and B

355

SKEWNESS AND KURTOSIS OF A DIFFUSING SCALAR 1.6

2.3 2.2 2.1

1.4 2 1.9

1.3

1.8

Value of Parameter B

Value of Parameter A

1.5

1.2 1.7 A B

1.1

1

1.6 1.5

10

15

20

30

50

70 100 150 200 300 Distance Downstream (in mm)

700

1600 2600

Figure 4. Fit parameters A and B as functions of downstream distance.

appear to be consistently larger than the calculated values A0 and B0 . This discrepancy can, in part, be accounted for by the different numerical procedures used to arrive at the original ki and the estimates for A and B offered here. There is a reasonable amount of variation in the experimental values of k3 and k4 (especially at the 1600 and 2600 mm downstream stations) shown on Figure 3 in Sawford and Sullivan (1995) that were used in the calculations of A0 and B0 . This variation is reflected in the scatter evident in the comparisons of A versus A0 and B versus B0 (where the variability in the latter comparison is magnified relative to that of the former). 5. Discussion There is a remarkably small amount of variability in the directly fitted values of A and B from (5) over a wide range of experimentally challenging situations such as the atmospheric boundary layer, laboratory-produced dense clouds, and the well-controlled experimental plume discussed in this paper. In the plume study, a systematic variation of the values of A and B was shown to occur with downstream distance, with essentially no variation across the plume cross-section. These variations conformed to the ab prescription provided in Section 2. We have shown herein that a quadratic relationship between skewness and kurtosis is consistent with the ab model. That is, the underlying physical

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(a)

1.5 1.45 1.4 1.35

A

1.3 1.25 1.2 1.15 1.1 1.08

1.1

1.12

1.14

1.16

1.18

1.2

1.8

1.9

2

A'

(b) 2.25 2.2 2.15 2.1 2.05 B 2 1.95 1.9 1.85 1.8 1.4

1.5

1.6

1.7 B'

Figure 5 (a). Fitted parameter A versus approximation A0 with a trend-line. (b) Fitted parameter B versus approximation B0 with a trend-line.

structure in turbulent flows is one in which the concentration field is comprised of relatively high concentration sheets and strands, as well as a lowlevel, ambient background. That the relationship given in (5) implies the validity of the ab prescription of moments is a valuable observation. It would be extremely difficult to validate such a claim with either continuous or sudden releases in the atmospheric boundary layer (or even in the laboratory, in the case of clouds). So, access to isolated, fixed-point measurements has been essential to this study. The lower-ordered moments in (12)–(14) depend on the mean


357

concentration Cðx; tÞ, a and b (which are functions of downstream distance), as well as additional functions ki for each higher moment (these are functions of time in the case of a cloud). Remarkably, as few as the first four of these moments can be inverted to yield a reliable estimate of the PDF of concentration C. The equations that describe the evolution of these functions are relatively straightforward and invite the use of simple closure schemes for their solution (see Labropulu and Sullivan, 1995; Mole and Clarke, 1995; Sullivan, 2004). A satisfactory solution scheme for the mean concentration for both clouds and continuous elevated sources in the atmospheric boundary layer has been given in Sullivan and Yip (1991). The important event of the sudden release of contaminant into the atmospheric boundary layer, as may result from and accidental spill, has been investigated in Labropulu and Sullivan (1995); there, the functions aðtÞ and bðtÞ were used to calculate the second central moment using (12). In principle, this scheme can be employed to calculate k3 and k4 , which would enable an approximation of the PDF via a four-moment inversion scheme. Acknowledgements This research received financial support from the National Science and Engineering Research Council of Canada. The authors wish to express their gratitude to Brian Sawford and Charles Tivendale at the Commonwealth Scientific and Industrial Research Organisation in Aspendale, Australia for generously making available their data and for providing valuable comments. Thanks are also extended to Philip Chatwin and Nils Mole at the University of Sheffield, and David Lewis at the University of Liverpool. References Buch, K. A. and Dahm, W. J.: 1997, ‘Experimental Study of the Fine-Scale Structure of Conserved Scalar Mixing in Turbulent Shear Flows. Part 2. Sc 1’, J. Fluid Mech. 364, 1– 29. Chatwin, P. C.: 1990, ‘Hazards due to Dispersing Gases’, Environmetrics 1, 143–162. Chatwin, P. C. and Robinson, C.: 1997, ‘The Moments of the PDF of Concentration for Gas Clouds in the Presence of fences’, Il Nuovo Cimento 20(3), 361–383. Chatwin, P. C. and Sullivan, P. J.: 1989, ‘The Intermittency factor of Scalars in Turbulence’, Phys. Fluids A 4, 761–763. Chatwin, P. C. and Sullivan, P. J.: 1990, ‘A Simple and Unifying Physical Interpretation of Scalar Fluctuation Measurements from Many Turbulent Shear Flows’, J Fluid Mech 212, 533–556. Clarke, L. and Mole, N.: 1995, ‘Modelling the Evolution of Moments of Contaminant Concentration in Turbulent Flows’, Environmetrics 6(6), 607–617.

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Corriveau, A. and Baines, W. D.: 1994, ‘Diffusive Mixing in Turbulent Jets as Revealed by a pH Indicator’, Experiments in Fluids 16, 129–136. Dahm, W. J., Southerland, K. and Buch, K. A.: 1991, ‘Direct, High Resolution, Four Dimensional Measurements of the Fine Scale Structure of Sc 1 Molecular Mixing in Turbulent Flows’, Phys. Fluids A3, 1115–1127. Derksen, R. W. and Sullivan, P. J.: 1990, ‘Moment Approximations for Probability Density Functions’, Combust. Flame 81, 378–391. Hall, D. J., Waters, R. A., Marsland, G. W., Upton, S. L. and Emmott, M. A.: 1991, ‘Repeat Variability in Instantaneously Released Heavy Gas Clouds—Some Wind Tunnel Model Experiments’, Technical Report LR 804 (PA), National Energy Technology Centre, AEA Technology, Abington, Oxfordshire, U.K. 74 pp. Heagy, W. K. and Sullivan, P. J.: 1996, ‘The Expected Mass Fraction’, Atmos. Environ. 30, 35– 47. Labropulu, F. and Sullivan, P. J.: 1995, ‘Mean-Square Values of Concentration in a Contaminant Cloud’, Environmetrics 6(6), 619–625. Lewis, D. M. and Chatwin, P. C.: 1996, ‘A Three Parameter PDF for the Concentration of an Atmospheric Pollutant’, J. Appl. Meteorol. 36, 1064–1075. Lewis, D. M., Chatwin, P. C. and Mole, N.: 1997, ‘Investigation of the Collapse of the Skewness and Kurtosis exhibited in Atmospheric Dispersion Data’, Il Nuovo Cimento 20, 385–397. Mole, N., Anderson, C. W., Nadarajah, S. and Wright, C.: 1995, ‘A Generalized Pareto Distribution Model for High Concentrations in Short-Range Atmospheric Dispersion’, Environmetrics 6(6), 595–606. Mole, N. and Clarke, L.: 1995, ‘Relationships Between Higher Moments of Concen- tration and of Dose in Turbulent Dispersion’, Boundary-Layer Meteorol. 73, 35–52. Sawford, B. L. and Sullivan, P. J.: 1995, ‘A Simple Representation of a Developing Contaminant Concentration Field’, J. Fluid Mech. 289, 141–157. Sawford, B. L. and Tivendale, C. M.: 1992, ‘Measurements of Concentration Statistics Downstream of a Line Source in Grid Turbulence’, in Proceedings of the 11th Australian Fluid Mechanics Conference, pp. 945–948. Schopflocher, T. P.: 1999, ‘The Representation of the Scalar Concentration PDF in Turbulent Flows as a Mixture’, Ph.D. thesis, The University Of Western Ontario, 101 pp. Schopflocher, T. P.: 2001, ‘An Examination of the Right-Tail of the PDF of a Diffusing Scalar in a Turbulent Flow’, Environmetrics 12, 131–145. Schopflocher, T. P. and Sullivan, P. J.: 2002, ‘A Mixture Model for the PDF of a Diffusing Scalar in a Turbulent Flow’, Atmos. Environ. 36, 4405–4417. Sullivan, P. J.: 2004, ‘The Influence of Molecular Diffusion on the Distributed Moments of a Scalar PDF’, Environmetrics 15, 173–191. Sullivan, P. J. and Ye, H.: 1996, ‘Moment Inversion for Contaminant Concentration in Turbulent Flows’. Can. Appl. Math. Quart. 4(3), 301–310. Sullivan, P. J. and Yip, H.: 1991, ‘Contaminant Dispersion from an Elevated Point Source’, ZAMP 42, 315–318. Wilkins, E.: 1944, ‘A Note on Skewness and Kurtosis’, Ann. Math. Stat. 15, 133–135. Ye, H.: 1995, ‘A New Statistic For the Contaminant Dilution Process in Turbulent Flows’, Ph.D. thesis, The University of Western Ontario, 107 pp.