Statistical Properties of a Meandering Plume in Turbulent Boundary Layer

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Statistical Properties of a Meandering Plume in
Turbulent Boundary Layer

• Alex Skvortsov Ralph Gailis
• CSIRO Complex Systems Science Annual Workshop
• August 8-10, 2006

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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

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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

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The Fluctuating Plume Model
Transverse plane distance x downstream
Transverse coords xT (y, z), so that x (x, xT)
Instantaneous plume centroid xT,c (yc, zc)
Moving ref. frame fixed to plume centroid with
coords xT,r (yr, zr)
It follows that xT xT,r xT,c
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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
• PDF of relative concentration
• relies on moments of relative concentration, Cr ,
  ir
• Take moments of PDFs many statistics can be
  derived analytically

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Summary of Previous Work - The Coanda Water
Channel

• 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)

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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

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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

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2D LIF Dataset

• All data has now been collected
• Canopy Array
• No obstacles _at_ 3 different heights
• Regular Cubic arrays
• Random height arrays
• Random placement arrays
• Data processing is in progress

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Array 004 Random obstacle heights (1H, 2H 3H)
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Statistics Images (example - Array 004)
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Centroid Statistics Array 001 (left) and Array
018 (right)

• 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

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Centroid StatisticsHorizontal Meander Histograms
Array 018
Array 001

• Very self-similar - good fit for Gaussian

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Centroid StatisticsVertical Meander Histograms
Array 001
Array 018

• Lognormal fit (i.e. its Log fits Gaussian)

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Concentration Statistics Horizontal Relative
Concentration Profiles
Array 018
Array 001

• Horizontal linescan across mean centroid position
• Clear Gaussian fit up to 3s

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Theory Plume Meander Fluctuations

• Analytical self-similar solution for power-law
  velocity profile and eddy viscosity

• Large Deviations Theory ltConcentrationgtParticle
  PDF

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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)
• 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

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QQ-Plots to Validate Meander PDF
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Work with CSIRO Atmospheric Research -(Dr
M.Borgas)
Relative Dispersion Lagrangian framework pair
correlation concentration covariance
Cross section integral averaged over separation
vector angles
Analysis from COANDA 2D LIF (smooth wall)
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Relationship with separation PDF /internal
fluctuations
p from a modified Richardsons Diffusion Model
Sample fits using Richardsons Diffusion Model
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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

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ADV LDV Comparison
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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

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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