Extreme Values of Contaminant Concentration

in theAtmospheric Boundary-Layer

Nils Mole (University of Sheffield)Thom Schopflocher

Paul Sullivan (University of Western Ontario)

2

Background

Typical diffusing plume

The texture of the contaminant field consists of concentration in

very thin sparsely distributed sheets.

3

Background…

4

Background…

Fixed point measurements showing large peaks of concentration

-40-20

020406080

100120140160

0 10 20 30 40 50 60

Time (s)

Co

nce

ntr

atio

n F

luct

uat

ion

s(a

rbit

rary

un

it)

5

High Concentration Tails

max asgp

gfp 1

dtxprobdtxp ,,;

max

o

nn dpm

pdf

As the sum of two functions,

Such that

Moments of p(θ),

For sufficiently large n, max

o

nn dgm

6

Generalized Pareto Density function

.max

o

nn dgm

nkkk

andg

n

o

n

1...211

!max

θ2 is the largest source concentration and a, k > 0.

for sufficiently large n,

Moment ratios are linear in 1/n, which should yield the values of a, k & θmax = a/k.

n can be found from

,11

11

k

a

k

ag

2max0 k

a

a

k

nana

nk

m

m

n

n

1111

7

Experimental data

Linear fits to the moment ratios for data from Sawford & Tivendale (1992). X is the downstream distance from the source, Z is the cross-stream distance

from the centreline, and L is the mean plume width.

8

Experimental data

The measured pdf of θ/Co (points), and GPD for fitted k and a values (curves). The right-hand panels show blow-ups of the tails. The dashed lines mark the

estimated values of θc/Co.

9

Experimental data

Variation with downwind distance X of the percentage range and area accounted for by the GPD, i.e. by θ ≥ θc. (a) Range 1 − θc/θmax, (b) area A. The squares represent centreline measurements, and the crosses measurements at

about 1L from the centreline.

10

Experimental data

Variation on the centreline of GPD parameters (estimated from the data of Sawford & Tivendale 1992) with downstream distance X. (a) θmax/Co

(squares), a/Co (triangles) and 10k (crosses), (b) θmax/Co (squares), and the approximation to θmax/Co in the no diffusion case based (curve).

11

Experimental data

Variation across the plume of GPD parameters normalized by their centrelinevalues. (a) θmax, (b) k, (c) a, (d) η. The line in (d) is that on which the

normalized value of η equals C/Co.

12

Conclusions

Have shown the evolution of the high concentration tails, including θmax, in terms of lower order moments of pdf.

Further work (not shown here) using existing models of the moments provides a five parameter representation.

Some promising schemes for the solution of the moments equation,

are currently under investigation.

,1 2111

nn

n

nnux

U

13

Examples of the possible GPD shapes.

14

Variation of centreline η with downstream distance X.