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Extreme Values of Contaminant Concentration
in theAtmospheric Boundary-Layer
Nils Mole (University of Sheffield)Thom Schopflocher
Paul Sullivan (University of Western Ontario)
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Background
Typical diffusing plume
The texture of the contaminant field consists of concentration in
very thin sparsely distributed sheets.
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Background…
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Background…
Fixed point measurements showing large peaks of concentration
-40-20
020406080
100120140160
0 10 20 30 40 50 60
Time (s)
Conc
entra
tion
Fluc
tuat
ions
(arb
itrar
y un
it)
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High Concentration Tails
max asgp
gfp 1
dtxprobdtxp ,,;
max
o
nn dpm
As the sum of two functions,
Such that
Moments of p(θ),
For sufficiently large n, max
o
nn dgm
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Generalized Pareto Density function
.max
o
nn dgm
nkkkandg
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
ak
ag 2max0
ka
ak
nanank
mm
n
n
1111
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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.
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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.
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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.
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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).
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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.
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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. ,1 211
1
nn
n
nnux
U
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Examples of the possible GPD shapes.
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Variation of centreline η with downstream distance X.