Extension of a Fluctuating Plume Model to Urban Areas
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Transcript Extension of a Fluctuating Plume Model to Urban Areas
Statistical Properties of a Meandering
Plume in Turbulent Boundary
Layer
Alex Skvortsov & Ralph Gailis
CSIRO Complex Systems Science Annual Workshop
August 8-10, 2006
Outline
Very Brief Overview of the Fluctuating Plume Model
Summary of Previous Work
New Results
2D LIF/Theory
Meander Fluctuation Theory
Future Work & Integration Plans
Motivation
A general model for dispersion, including
concentration fluctuations:
shear boundary layers and canopies
relatively fast calculations
as many analytical components as possible
integrate with/complement other models of
dispersion and wind flow
Use the fluctuating plume paradigm
The Fluctuating Plume Model
Transverse plane
distance x
downstream
3
Transverse coords
xT = (y, z), so that
x = (x, xT)
Moving ref. frame
fixed to plume
centroid with coords
xT,r = (yr, zr)
It follows that
xT = xT,r + xT,c
Stirring and mixing
2
Concentration
Instantaneous
plume centroid
xT,c = (yc, zc)
rc(t)
Instantaneous
Centroid
In-plume fluctuation scale = ir
r
1
< r >
<>
0
Inner plume scale = r
Outer plume scale = a
-1
-10
-5
0
r
5
10
Basic Concept of the Model
The total concentration PDF is the average of the
conditional PDF of instantaneous concentration c
over the fluctuations in centroid motion xT,c
f ( ; ( x, xT )) f r ( ; ( x, xT xT ,c )) f c ( x, xT .c ) dxT ,c
PDF of relative concentration
relies on moments of relative concentration, Cr , ir
Take moments of PDFs – many statistics can be
derived analytically
Summary of Previous Work - The Coanda
Water Channel
Flow conditioning elements
(square mesh array and sawtooth fence)
Two-component fibre-optic
LDV system
Three-axis traverse system
designed to provide
maximum flexibility for data
collection
Coanda meteorological water channel (10 m long, with cross-section 1.5 m wide
and 1.0 m high) with model canopy (flat plate array) installed on the channel floor
Square bar array and saw-tooth fence at channel inlet used in acceleration of
development of deep boundary layer over the model canopy
Dispersing dye dispersion measured using Laser Induced Fluorescence (LIF)
Obstacle Arrays
Originally analysed ‘No Obstacles’
‘Canopy’ and ‘Urban Arrays’ config.
These cases can be seen to be
baseline and “extreme” obstacle
arrays
Analysis of 2D LIF is now well
underway
Array004
flow
Regular placement of obstacles with random
heights of 1H, 2H, or 3H.
New Experiment - 2D Data Collection
Fluctuating position of plume centroid against fixed
1D LIF beam may cause data inconsistencies in
relative frame
More reliable way to collect data is to employ 2D
transverse scan
2DLIF Scan
Urban Array
2D LIF Dataset
All data has now been collected
Canopy Array
No obstacles @ 3 different heights
Regular Cubic arrays
Random height arrays
Random placement arrays
Data processing is in progress
Array 004 – Random obstacle heights (1H, 2H &
3H)
Statistics Images (example - Array 004)
Centroid Statistics:
Array 001 (left) and Array 018 (right)
Centroid Position - mm, Different Distance from Source
18
16
14
Z Axis - mm
12
10
8
6
4
2
0
-3
-2
-1
0
1
2
Y Axis - mm
3
4
5
6
7
Array 001 – regular array of 1H obstacles
Array 018 – random placement of 1H obstacles
Spread of centroid position in each cross-section does not
seem to be self-similar and depends on particular obstacle
configuration, source and measurement points
Centroid Statistics:
Horizontal Meander Histograms
Horizontal Centroid Position - Statistics
2
10
Horizontal Centroid Position - Statistics
2
10
adm2479
adm2480
adm2481
adm2482
adm2491
adm2492
adm2310
adm2312
adm2315
adm2323
adm2325
adm2334
1
10
1
% of Observations
% of Observations
10
0
10
-1
-1
10
10
-2
-2
10
0
10
0
5
10
15
Y Axis - mm
Array 001
20
25
30
10
0
5
10
15
Y Axis - mm
Array 018
Very self-similar - good fit for Gaussian
20
25
30
Centroid Statistics:
Vertical Meander Histograms
Vertical Centroid Position - LOGNORMAL FIT
2
10
1
10
Vertical Centroid Position - LOGNORMAL FIT
2
adm2310
adm2312
adm2315
adm2323
adm2325
adm2334
Gauss
10
adm2479
adm2480
adm2481
adm2482
adm2491
adm2492
Gauss
1
PDF - % of Observations
PDF - % of Observations
10
0
10
0
10
-1
10
-1
10
-2
10
-2
0
5
10
15
Z Axis - mm
Array 001
20
25
30
10
0
5
10
15
Z Axis - mm
Array 018
Lognormal fit (i.e. its Log fits Gaussian)
20
25
30
Concentration Statistics:
Horizontal Relative Concentration Profiles
Concentration - Horizontal Profile, Different Distance from Source
0
10
Concentration - Horizontal Profile, Different Distance from Source
0
10
-1
10
-1
10
-2
10
-2
C/Cmax
C/Cmax
10
-3
10
-3
10
-4
10
-4
10
-5
10
-5
10
-6
10
-20
-6
10
-20
-15
-10
-5
0
5
Y/Sigma
-15
-10
-5
0
Y/Sigma
5
10
15
20
Array 001
Horizontal linescan across mean centroid position
Clear Gaussian fit up to 3
Array 018
10
15
Theory: Plume Meander Fluctuations
Analytical self-similar solution for power-law velocity profile
and eddy viscosity
Large Deviations Theory: <Concentration>=Particle PDF
Theory: Relative Fluctuation Intensities
The relative PDF is given by the formula below, and is
dependent on the relative fluctuation intensity (k = i-1/2)
kk
f r ( ; ( x, xT ,r ))
Cr ( k ) Cr
k 1
k
exp
Cr
Up to now we have used a ‘bulk’ fluctuation intensity,
assuming it remains constant over a constant y-z plane
Gives analytical results, but is an over simplification:
3
3
x = 186.8 mm
x = 261.5 mm
2
2
2 Cr
1
ir (1 ir 0 )
Cr 0
ir
ir
2
1
1
(a)
(b)
0
0
-3
-2
-1
0
yr /ry
1
2
3
-3
-2
-1
0
yr /ry
1
2
3
QQ-Plots to Validate Meander PDF
Work with CSIRO Atmospheric Research (Dr M.Borgas)
Relative Dispersion: Lagrangian framework
pair correlation ≡ concentration covariance
1
2
C x, y, z C x, y r cos , z r sin ddzdy
0 0
Cross section integral
averaged over separation
vector angles
Analysis from COANDA 2D LIF (smooth wall)
7.5x10-3
5.0x10
x/h=2
x/h=4
x/h=6
x/h=10
x/h=16
x/h=26
-3
C2(r)
C2 r , x
2
h=37.5 mm
pixel = 1.45 mm
2.5x10-3
0.0x100
0
10
20
r (pixels)
30
Relationship with: separation PDF /internal fluctuations
1
3
1
p d t 1 2 / 3
C2 r C2 0
4 0
0
p exp
2/ 3
r t
3/ 2
9 / 2
pr t
3/ 2
1
2 / 3 3 / 2
d
p from a modified
Richardson’s
Diffusion Model
tx U
can be predicted for atmospheric
flows
7.5x10-3
x/h=2
x/h=4
x/h=6
x/h=10
x/h=16
x/h=26
Model Fit
Richardson
Sample fits using
Richardson’s Diffusion
Model
-3
C2(r)
5.0x10
h=37.5 mm
pixel = 1.45 mm
2.5x10-3
0.0x100
0
10
20
30
r (pixels)
Internal plume fluctuations directed related to C2(0)
iR
Plume meander has a weak effect in internal
fluctuations, so the cross section integral is a
robust estimate.
Velocity Field Characterisation
In UrbanArray water channel experiments we used an LDV
(Laser Doppler Velocimeter) to characterise flow field
The LDV only measures two components of the flow at a
time, so we considered the use of a Sontek acoustic
Doppler velocimeter (ADV)
The sampling volume of the Sontek is much larger than for
the LDV, so we expect some trade-off in the detail we see
for small-scale structure of the flow
ADV & LDV Comparison
5.0
4.5
A2UV
B2UV
C2UV
K2UV
UVudm0524
ASontek
Bsontek
Csontek
Ksontek
Array001 unit cell -1,6
4.0
A2
3.5
B2
F2
height (z/H)
D2
E2
flow
3.0
C2
G2
I2
J2
K2
L2
H2
2.5
2.0
1.5
1.0
Approach
Flow
0.5
0.0
-0.05
0.00
0.05
0.10
0.15
0.20
U (m/s)
0.25
0.30
0.35
0.40
Initial Analysis Results
We expect that this data will provide a better
understanding of how the upstream obstacles
influence the flow field around the source
Also plan to use this detailed measured data as
flow input during testing of our initial prototype
urban fluctuating plume model
Preliminary analysis of this dataset shows
variations in the height and arrangement of
obstacles in both the near and far field has a
surprisingly small impact on the flow statistics
Future Work
Complete analysis of 2D dispersion data
Analytical Model for plume internal fluctuations
Analysis of Velocity data to aid theoretical
development of model and as a data input for
testing prototype concentration model
Interface with a Lagrangian particle model
a “higher level” layer, interpreting stochastic model
output
gives higher order concentration moments or the full
concentration PDF
can then simulate concentration time series