A REVIEW OF DISPERSION MODELLING AND PARTICLE TRAJECTORIES IX ~ ' T ~ I V I O S P H E R I C FLOWS

Jai Singh Sachdev

A thesis submitted in conformity with the requirements for the degree of Masters of Applied Science

Graduate Depart ment of Aerospace Engineering University of Toronto

Copyright @ 2000 by Jai Singh Sachdev

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Abstract

A Review of Dispersion Modelling and Particle Trajectories in Atmospheric Flows

Jai Singh Sachdev

h s t e r s of Applied Science

Graduate Department of Aerospace Engineering

Cniversity of Toronto

'2000

Ar niospheric and turbulence rnodelling, boundary layer parameterizat ion. and current dispclr-

sion rnodelling techniques are reviewed. Dispersion rnodels for atmospheric How conditioiis

iricliide the &Diffusion model. Gaussian Plume model. and Lagrangian parricle tracking

techniques (Gaussian Puff model), and their applications in CALPCFF (a corrimercial soft-

ware package) are described. Studies of Gaussian Plume and Puff models in stable. neutraliy

stable. and unstable boundary layers show that the puff rnethod reproduces acceptable pluriic

niodel results in simple Born cases. The influence drag and gravity forces have on the niotion

of pollutant particles are investigated by means of derivation and analysis of the equatiori-

s of translational motion. Incorporation of the drag and gravity forces into the Gaussian

Puff dispersion niodel shows that the effect these forces have on the growth of a pollutarit

plume depends on the magnitude and frequency of the turbulent Aow-field fluctuations. The

Gaussian Puff model is flexible in that it can be applied in comples Row conditions: Iione\--

er. the assumption of the Gaussian distribution of pollutants does not account for particle

deposit ion.

Acknowledgement s

1 would like to thank Prof. Gottlieb for providing me with the opportunity to study with

him at the Masters level. for seeing the ability in me to accomplish this research. and for Iiis

guidalice. friendship, and support.

To mu parents and brothers who have been a source of never ending support and inspiratioii.

And finally. thanks to the rnany friends at CTIAS for their friendship and encouragenierit

over the past two years.

Contents

1 Introduction l.

1 . I Background and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Conservlition Equations for .-\ tmospheric Flows . . . . . . . . . . . . . . . . . 4

- 1.3 Turbulence klodelling for Atmospheric Flow . . . . . . . . . . . . . . . . . . 1

1 .4 Similarity Theory and Pararneterization . . . . . . . . . . . . . . . . . . . . 1'1

2 Particle Dispersion Modeis for Atmospheric Flows 18

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction LS

2.2 K-Diffusion Mode[ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Gaussian Plume Mode1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Lagrangian Particle Dispersion Models . . . . . . . . . . . . . . . . . . . . . '23

. . . . . . . . . . . . . 2.3 CXLPUFF: -4 Modern Dispersion Modelling Package 25

3 Theory for Particle Trajectories 29

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

. . . . . . . . . . . . . . 3.2 Derivation of the Basic Equation of Particle Motion 30

iv

3.3 Scale Analysis and Derivation of Simplified Models . . . . . . . . . . . . . . 34

3.4 Corrections to the Stokes Drag Law . . . . . . . . . . . . . . . . . . . . . . . 3;

3.5 Addition of Turbulent Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Numerical Simulation of Part ide Traject ories 41

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 . 2 Numerical Modelling of Particle Trajec tories . . . . . . . . . . . . . . . . . . 42

4.3 The Kraichnan Turbulence Mode1 . . . . . . . . . . . . . . . . . . . . . . . . 4-4

4.4 Physicai Characteristics of the Particles . . . . . . . . . . . . . . . . . . . . . 44

4 . S Particle Trajectories in a Steady Flow . . . . . . . . . . . . . . . . . . . . . . 47

4.6 Particle Trajectories in an Lnsteady Flow . . . . . . . . . . . . . . . . . . . -19

5 Numerical Cornparison of Atmosplieric Dispersion Models 60

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GO

5 . 2 htmospheric Flow Mode1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3 Cornparison of the Gaussian Plume and Gaussian Puff Dispersion .\ lodels . . 66

5.3.1 Gaussian Plume Isopleths . . . . . . . . . . . . . . . . . . . . . . . . 6s

5.3.2 Gaussian Puff Mode1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.4 Particle Trajectories in an Atmospheric Flow . . . . . . . . . . . . . . . . . . 73

- - -- 3.3 Distribution of Particles in an Atmospheric Flow . . . . . . . . . . . . . . . . r i

6 Concluding Discussion 91

A Derivation of the Space and Time Averaged Conservation Equations 98

-1.1 Space and Time Averaged Conservation Equations . . . . . . . . . . . . . . 98

h.2 Turbulent Kinetic Energy Equation . . . . . . . . . . . . . . . . . . . . . . . 100

B Derivation of the Gaussian Plume Mode1 103

C Derivation of the Random-Walk Mode1 107

D Derivation and Analysis of the Equation of Particle Motion 111

D.l Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1

D.2 Derivat ion of the Equation of Particle Motion . . . . . . . . . . . . . . . . . 1 1 1

D.3 Scale Analpis and Simplified Nociels . . . . . . . . . . . . . . . . . . . . . . 129

E Numerical Modelling of the Basset History Term 135

List of Tables

. . . . . . . . . . . . . . . . . . . . . . 1.1 Key to Pasquill stability categories [l] 9

1.2 Pasquill-Gifford horizontal dispersion parameters [l] . . . . . . . . . . . . . . 10

1.3 Pasquill-Gifford vertical dispersion paramet ers [Il . . . . . . . . . . . . . . . . 10

4.1 Particle Trajectory Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Particle trajectory data for particles in air . . . . . . . . . . . . . . . . . . . . 47

3.1 Data for Gaussian Ptrff Models . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2 Input data for Gaussian Plume llodels . . . . . . . . . . . . . . . . . . . . . 67

vii

List of Figures

3.1 Comparison of experimental and computed drag coefficients. . . . . . . . . .

. . . . . . . . . . . . . 3 . Cunningham correction coefficient for srnall particles.

-1.1 Normalized (a) x- and (b) z-direction velocities computed by the Level 1 and

Level 2 equations of particle motion versus normaliaed time for R = 10.0 p i

. . . . . . . . . . . . . . . . . . . . . . . . ancl 6 = 100 in a steady flow-field.

4.2 Relative ciifference of the x-direct ion velocity versus nornializetl t inie for var-

. . . . . . . . . . . . . . . . . . . . ious radii with = 100 in a steady Row.

4.3 (a) S-direction (m/s) and (b) 2-direction velocity (pm/s) versus tirrie (ps )

. . . . . . . . . . . . . . with R = 0.1 pm and f i = 100 in an unsteady flow.

4.4 (a) I-direction (m/s) and (b) 2-direction velocity (pm/s) versus time ( p s )

. . . . . . . . . . . . . . with R = 1.0 pm and = 100 in an unsteady flow.

4.5 (a) X-direction (m/s) and (b) 2-direction velocity (mm/s) versus time (ms)

. . . . . . . . . . . . . . with R = 10.0 pm and = 100 in an unsteady flow.

4.6 (a) S-direction (rn/s) and (b) 2-direction velocity (m/s) versus time (s) nith

. . . . . . . . . . . . . . . . R = 100.0 pm and f i = 100 in an unsteady flow.

4.7 (a) S-direction (mis) and (b) 2-direction velocity (pm/s) versus time ( p s )

. . . . . . . . . . . . . . nrith R = 0.1 pm and P = 1000 in an unsteady flow.

4.8 (a) ?<-direction (rn/s) and (b) 2-direction velocity (pm/s) versus time (ps)

. . . . . . . . . . . . . . with R = 1.0 pm and p = 1000 in an unsteady flow.

-. - Vlll

4.9 (a) .Y-direction (m/s) and (b) 2-direction velocity (mm/s) versus time (rns)

. . . . . . . . . . . . . with R = 10.0 pm and j3 = 1000 in an unsteady 0ow.

4.10 (a) '<-direction (m/s) and (b) 2-direction velocity (m/s) versus time (s) nit h

R = 100.0 pm and ,5 = 1000 in an unsteady flow . . . . . . . . . . . . . . .

Typical vertical velocity profiles in unstable, neutrally stable and stable How

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . conditions.

Ground-level isopleths of y u / Q from a source H = 10 m above the groiitid

in unstable atmospheric conditions (class B stability) from (a) the Gaussian

Plume hlodel. (b) the Gaussian Puff Model with 251 puffs. and ( c ) the Gaus-

sian Piiff mode1 with 501 puffs at t = 3600 s. . . . . . . . . . . . - . . . . . .

Ground-level isopleths of yu /Q from a source H = 10 rn above the grouricl

in neutrally stable atmospheric conditions (class D stability) from (a) the

Gaussian Phme Model, (b) the Gaussian Puff Mode1 with 251 puffs. and (c)

. . . . . . . . . . . . . the Gaussian Puff model with 501 puffs at t = 3600 S.

Ground-level isopleths of y u / Q lrorn a source H = IO rn above the grouncl in

stable atmospheric conditions (class F stability) from (a) the Gaussian Plunie

Model, (b) the Gaussian Puff Mode1 with 251 puffs. and (c) the Gaussian Puff

mode1 mith 501 puffs at t = 3600 S. . . . . . . . . . . . . . . . . . . . . . . .

Ground-level concentrations found by the Gaussian Puff model for a release

of 101 particles/puffs under unstable conditions: (a) gravity and drag forces

are neglected. (b) gravity and drag forces are included with = 2000 ancl

R = 1.0 Pm, and (c) gravity and drag forces are included rvith ,5 = 3000 and

R = 10.0 pm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ground-level concentrations found by the Gaussian Puff model for a release

of 101 particles/puffs under stable conditions: (a) gravity and drag forces

are neglected, (b) gravity and drag forces are included mith 3 = 4000 and

R = 1.0 pm? and (c) gravity and drag forces are included with f i = 3000 and

R = 10.0pm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * . ~ . ~ .

5 . Distribution of particles along the dom-wind distance a t t = 100 s for (a)

. . . . . . . . . . . . . . . . . . . . . unstable and (b) stable flow conditions.

5.8 Distribution of particles along the dom-wind distance a t t = 400 s for (a)

unstable and (b) stable flow conditions. . . . . . . . . . . . . . . . . . . . . .

3.9 Distribution of particles along the down-mind distance at t = 700 s for (a)

unstable and (b) stable 9on conditions. . . . . . . . . . . . . . . . . . . . . .

5.10 Distribution of particles along the down-wind distance at t = 1000 s for (a)

unstable and (b) stable flow conditions. . . . . . . . . . . . . . . . . . . . . .

5-11 Distribution of particles along the cross-wind distance a t t = 100 s for (a)

unstable and (b) stable flow conditions. . . . . . . . . . . . . . . . . . . . . .

3-12 Distribution of particles along the cross-wind distance a t t = 400 s for (a)

unstable and (b) stable Row conditions. . . . . . . . . . . . . . . . . . . . . .

3.13 Distribution of particles along the cross-wind distance at t = 700 s for (a)

unstable and (b) stable flow conditions. . . . . . . . . . . . . . . . . . . . . .

5-14 Distribution of particles along the cross-wind distance at t = 1000 s for (a)

unstable and (b) stable Row conditions. . . . . . . . . . . . . . . . . . . . . .

5.15 Distribution of particles along the vertical distance at t = 100 s for (a) tinstable

and (b) stable flow conditions. . . . . . . . . . . . . . . . . . . . . . . . . . .

5-16 Distribution of particles along the vertical distance at t = -LOO s for (a) unstable

and (b) stable flow conditions. . . . . . . . . . . . . . . . . . . . . . . . . . .

5.17' Distribution of particles dong the vertical distance at t = 700 s for (a) unstable

. . . . . . . . . . . . . . . . . . . . . . . . . . and (b) stable Bow conditions.

5.18 Distribution of particles along the vertical distance at t = 1000 s for (a)

unstable and (b) stable flow conditions. . . . . . . . . . . . . . . . . . . .

Chapter 1

Introduction

1.1 Background and Objectives

Dispersion modelling in atmospheric flows is an intense area of research nithiri tbc geo-

ptiysical scientific cornmunit. Governments plan to use such models to aid in regulating.

monitoring, and tracking pollutant releases from controlled burns of debris in famer's fields.

forest fires. accidental fires. and volcanos. This thesis is intended as a revieiv of the ciir-

rent dispersion rnodelling techniques. and an in depth look into particle trajectories iri an

atmospheric flom.

In 1991. a European initiative known as the "Harrnonization within Xtmospheric Dispersion

'\[odelling for Regulatory Purpose" was Iaunched to coordinate CO-operation. standarclizn-

tion. and management of atmospheric dispersion models for regulatory purposes.' This

society provides a forum for technical discussion and contains an excellent database of es-

isting dispersion modelling software.* Development of validation models and data is one of

the main areas of focus of this group. The Environmental Protection Agency (EPA).%

the United States of America, also provides an estensive library of atmospheric dispersion

models. One of the prominent dispersion models featured by the EPA is CALPCFF [2]

which is considered to be one of the most extensive and complete models in use t o d .

Smoke plumes and other pollutant releases are generally comprised of gases and solid parti-

cles. Dispersion of the pollutant occurs due to a number of physical processes. The release

velocity and temperature of the pollutant wili greatly influence the vertical rise of the plurne.

The velocity of the flow-field will have the dominant effect on the mean-direction of the pol-

lutant release and the turbulent components of the flow-field velocity are the rimin cause

of pollutant dispersion. Gravity could have an effect on the vertical distribution and the

down-wind ground-level concentration of the pollutant, depending on the particle size and

the magnitude of the vertical turbulent velocities in the Bow-field. Drag effects can be ini-

portant for larger particles in an unsteady flow when the frequency of the turbiilent Bon-field

is greater than the relaxation frequency of the particle. or when large changes occur in the

mean-flow field direction.

Atmospheric dispersion models require a meteorological flow model to provide atniospheric

data. including the velocity fields, density profiles. turbulent velocities. temperature profiles.

and so on. -4 diffusion equation (conservation of the pollutant concentration) can be useci

in conjunction with the flow-field equations to model the dispersion of a gaçeous pollutant.

This approach could be effective. since no simplification of the flow-field is requirecl: howver.

it would tend to be computationally expensive since these equations must be solved on ii

3-dimensional grid. A simplified version of this equation is the K-diffusion equatioti i i i

which a diffusion coefficient replaces the turbulent Bus terrns. Slesoscale How domains [3]

tend to be on the order of ten kilometers in length and width. and a few hundred nietres

high. Therefore. coarse grids could ease the computational effort, but this would severely

compromise the accuracy and resolut ion of the desired down-wind pollutant concent rat ions.

The Gaussian plume model was derived from the K-diffusion rnodel to simplify the compu-

tational effort required in predicting down-wind plume concentrations [3, 11. The derivat ion

of this mode1 requires that the mean Bow-field is constant and that the emisçion of pollutant

is continuous. It is assumed that the concentration distribution about the centre-line of the

plume is Gaussian. The Gaussian plume model can provide quick results for predicting the

path of a plume from a continuous source? but the assumptions and simplifications made in

the derivation limit its use to non-cornplex simulations.

Lagrangiaa particle dispersion rnodels simulate the pollutants as single particles. Plumes

can be represented by modelling thousands of particles. Fewer particles can be used if

each particle represents the centre of a pu& The Gaussian Puff model imposes a Gaussian

distribution of the pollutant concentration about the particle. These models allow for greater

Aexibility of the flow-field but dramatically increase the computational effort required.

The Lagrangian particle techniques require that the trajectories of single particles are fol-

lowed. Most of these techniques will set the particle's velocity to be equivalent to the local

flow-field velocity. However, multiple physical effects could be influeatial on the trajectory

of the particle. The importance of the unsteady drag and gravity forces are dependent on -

the size and density of the particies. L he temperature àiEerence between the parricie anci

the atmosphere and the initial particle monientum (velocity and spin) are also imporraiit

physical effects that should be considered.

Yarious atmospheric dispersion models are presented in Chapter 2. including the Ii-Diffusion

moclel. the Gaussian Plume 'vlodel. and Lagrangian Particle Dispersion Uodels. A clescrip-

cion of the CALPC'FF [2] plume modelling techniques are also included.

A detailed derivation and analysis of the equations of particle motion for a spherical particle

in an unsteady flow-field is conducted in Chapter 3. The particle's motion is only impedetl by

drag and gravity. Temperature difference and torque effects are neglected. The reader shoulcl

consult Lamb [JI? Landau Si Lifshitz [3], Happe1 & Brenner [611 and Rudinger [TI for more

information. The analysis of the equations of motion d l lead to possible simplifications.

depending on certain physical characteristics of the particle. These levels of equations arc

numerically studied in C hapter 4 to determine what simplifications are valid mhen t racking

particles in an atmospheric flow.

The Gaussian Plume and Puff models are numerically compared in Chapter 5. These models

are applied with unstable. neutrally stable' and stable flow conditions to determine mhether

or not the puR model and plume model reproduce similar results. The effects of including

gravity in the Gaussian Puff model (as opposed to assuming that the velocities of the puffs

are equiwlent to the flow velocity) are investigated as well. The construction of the Bow-field

model is also described.

Sections (1.2): (1.3), and (1.4) of this chapter are devoted to introducing the reader to the

basics of meteorological modelling, atmospheric turbulence modelling, and boundary laver

parameterization. For more detailed information on these topics, the reader is referred to

Pielke [3], Businger [8], Hanna [9], Lamb [IO], Turner [l], Wilcox [ll], Hinze [12], Lumley

[U], Lumley Sr Panofsky [U], and Haltiner & Martin [15].

1.2 Conservation Equations for At mospheric Flows

In general, the conservation equations governing the motion of a mesoscale rneteorological

system must include the effects of viscosity, cornpressibility. rotation! and chernical corn-

position. Water could appear in any of the three States: vapour. liquid droplets. or solid

particles. Pollutant materials are usually considered in the gaseous state only. but could

appear as liquid or even solid form. Chernical reactions could also take place. Derivation

of the conservation equations can be found in many references, including Landau % LX-

shitz [SI' Chandrasekhar [16], and Hinze [El. The conservation equations typically used in

atmospheric modelling, Pielke [3] or Businger [8], are given by

BP - + v (pu) = O, dt

These are the the conservation of n i a s (continuity), consemation of momenturn. energy

conservation (potential temperature), conservation of water (in solid - 1. liquid - 2 . or gaseous

- 3 state), and the conservation of poilutant species (x, represents the concentration of t tie

gaseous species m). Note that the viscosity of the air is typically ignored and chernical

equilibrium has been assurned. The ideal gas law for dry air is giwn by

ivhere R. is the universal gas constant, pd is the rnolecular weight of dry air. and & is the

gas constant of dry air. When vapour is included: the molecular weight of atmospheric air

is denoted by

in which q3 is the mass ratio of water vapour Mu over dry air Md (specific humidity) and

is the molecular weight of water. Therefore, the ideal gas law can be written as

Patm P d

when including water vapour. The virtual temperature of the çystem can then be defined as

such that the influence of the water vapour on the ideal gas law is included in the temperature

term instead of the gas constant and ji = pd/pHqo. Note that molecular weights are typically

given by p d = 29.98 and = 18.02. The ideal gas law can now be written as

The potential temperature is derived from the well-known entropy relation

For constant heat conditions (ds = 0). the above equation c m be rewritten and iiitegratcd

as

The potential temperature is defined as the equivalent temperature at a given reference state.

Setting Ski as the potential temperature, 8. and pz = LOO0 mbar as the reference pressure.

the equation for the potential temperature is

1000 mb " d C p

o=T\.( p (in mb) ) .

Often in atmospheric modelling the continuity equation is approxîmated by

which is known as the deep continuity equation. The density, pa. is a synoptic scale of tlie

density mhich varies mith height. Other approximations are used to simplify the equations.

such as the hydrostatic approximationt but these d l not be necessary for the discussion of

atmospheric dispersion models.

To obtain the highest resolution of the solution of the conservation equations. the dependent

variables are decomposed into mean and fluctuating values and the equations are averaged

over the volume of the grid-spacing. The conserved mriables are written as

The grid-volume averaging is obtained by integrating the conservation equations. This inte-

gration is performed by

Therefore. 3 represents the average of 4 over the finite tirne and space intervals given by At.

Ar. Ayt and Ar. The variable bl' is the deviation of qi frorn this average value. Applying

equations (1.1) and (1.2) to the consemation equation (1.1)-(1.5): 1 ) (1.8). and (1.9)

gives the following set of averaged conservation equat ions:

a p- + pu. Vu + puIf . Vuu = -Vjj + +g - 2 f l x (pu) + p ~ 2 ü . ( 1.14) dt

A complete derivation of these equations have been included in Appendis A. I\lternatively.

if equation (1.10) is used for continuity, then equation (1.13) reduces to

v (pou) z v (pu) = 0. (1.21)

This assumption alloms for the other conservation equations to be written as

- du p 7& + V(puu) + V(pul'u") = -vp + j5g - 2f2 x (pu) + p ~ ' a .

apn 1 I - + =V(püp,) + IV(=) = Sqn n = 1.2? 3, (1.24) at P P

a m 1 1 - - - + -V(jXzrn) + -V(pY1xL) = Sx , m = l , 2 ? .... JI. (1.23) at P

where the advection terms have been rewritten in the flux fonn. The sub-grid scale cor-

relation terms are also knonm as the turbulent fluxes. The task is now to find equations

CO approximate these flues. Parameterization is usually conducted based on esperimental

data and the equation of turbulent kinetic energy. as seen in the nest section.

1.3 Turbulence Modelling for At mospheric Flows

Tilt. task uE curluieuce riiucleiliug ib tu fiid apprwij'iiiiiàtiùns fur the i inknmn süb-gid cûi-

relations in terms of known flow properties such that a sufficient number of equations esist

to solve for the unknowns of the system [Il] . One of the main problems with atniospheric

dispersion modelling, and atmospheric modelling itself. is t hat the flow propert ies at the

sub-grid scale are not as well defined as what is required. This leads to the parameterizatiori

of the turbulent fluxes using experimental data and simplified physical concepts.

The turbulent kinetic equation can be derived from the momentum equation. This derivation

has been included in Appendix A. and the results are

where e = ?u"' and ë = f;;" are the turbulent kinetic energv and the average turbtileiit

kinetic energy respectively. The variable r" represents the fluctuation in the Esner funct ion.

used to scale the pressure gradient term in the momenturn equation given by

siich that the E-mer function is written as

CPTv .=.(g) =-. 19

The fourth term on the left-hand side of the turbulent energy equation (1.26) represents the

addition of kinetic turbulent energy to the system due to the esistence of an average velocity

shear and sub-grid scale velocity fluxes. This is generally referred to a s the shear production

of the turbulent kinetic energy. The final term in equation (1.26) represents the production

of kinetic turbulent energy due to buoyancy. The ratio of these two production terms gives

the flux Richardson number

The horizontal shear productions to the turbulent kinetic energy are neglected and it is

assumed that l&/dzl zz l&/a~l » Im/azl. The flux Richardson number provides in-

formation on the relative contribution of the buoyant and vertical shear of the averaged

horizontal flow velocity production of turbulent kinetic energy. The sub-grid Aus terrns. -- ~LW". wtturt, and w"u" are often approximated as

where 16 and Km are known as exchange coefficients. In general these coefficients iire iiot

constant in time or space. Substitution of the terms defined by equation (1.28) into equation

where Ri is the gradient Richardson number. The sign of Ri is determined by that of the

vertical temperature gradient. Therefore, a positive gradient Richardson number corresponds

to a stably stratified layer: Ri = O corresponds to a neutral stratification: and a negative

gradient Richardson number indicates an unstablv stratified 1-r. The unstable stratifiecl

Iayer is broken into two regions: when (Ri( 1 the shear production of sub-grid scale kinetic

energy is more important: and when lRil > 1 the buoyant production of sub-grid scale kinetic

eriergy is dominant.

Pasquill [Il categorized the intensity of the turbulence near the ground depending on the

wind-speed (at a height of 10 m). time of day, and solar radiation. These stability classes. A'

being niost unstable and *Go being the most stable, are used to determine the horizontal and

vertical dispersion parameters as defined by Pasquill and Gifford [II. S t rongly. rnoderately.

and slightly unstable conditions are represented by A'. 'B'. and *CT respectively. Xeutrally

stable conditions are denoted by 'D'. Slightly, moderately. and strongly stable sistems are

represented by 'Et. 'F', and 'G' respectively. Table (1.1) shows the stability classes for given

times, solar radiation. and surface wind speed.

Strong insolation refers to a sunny midday in summer, slight insolation corresponds to similar

conditions in winter. Xght is defined as the time interval one hour before dusk and one hour

after d a m . The neutral stability condition, 'Dy, should be used for overcast conditions

during day or night. This stability class shouId also be used during the first and last hour

of night period as defined previously.

Table 1.1: Key to Pasquill stability categories [l].

InsoIat ion Night

Wind speed Strong Moderate Slight Thinly overcast < 318 cloud

(at 10 m) m/s > 4/8 low cloud

< 2 A A - B B

2 - 3 A - B B C

3 - 5 B B - C D

5 - 6 C C - D D

> 6 C D D

The PasquiIl-Gifford dispersion paranieters offer a turbulence closure model bucd or1 the

stability of the surrounding atmosphere. These parameters are typically used in the Gaiissiari

Plume Mode1 to determine the horizontal and vertical dispersion of the plume as a function

of the downwind distance from the source.

The horizontal and verticai fluctuations of the wind are required to estimate the horizontid

and vertical dispersion of the flow. The Pasquill-Gifford technique assumes that the hori-

zontal and vertical concentration distributions are Gaussian. The plume midt h and height

are written as the standard deviations of the concentration distributions in the cross-wind

and vertical directions and are considered to be functions of only the stability class and the

downwind distance. The horizontal Pasquill-Gifford dispersion parameter is given by 1

where T is a function of the domrvind distance, x? and the Pasquill stability class as given

in table (1.2).

The vertical Pasquill-Gifford dispersion parameter is a function of the dotvn-wind distance

and two parameters based on the don=-wind distance and the stability class:

where x is in km. The values of a and b are given in table (1.3) for a given downwind distance

and stability class.

Mellor k Yamada [l'il compiled a second-moment turbulent closure mode1 for mesoscale

geophysical systems based on the prognostic equation for the turbulent kinetic energy. Their

Table 1.2: Pasquill-Gifford horizontal dispersion parameters [Il.

S tability Equations for T

Table 1.3: Pasquill-G ifford vertical dispersion parameters [l].

stabiiity distance a b O: at

(km) .=ma,

stability distance a b 0, ; ~ t

(km) -Crnil.r

E > 40.0 47.618 0.29592

20.0 - 40.0 35.420 0.36625 141.9

10.0 - 20.0 26.960 0 . 4 7 1 109.3

4.0 - 10.0 24.703 0.50527 79.1

2.0 - 4.0 22.534 0.57154 49.8

1.0 - 2.0 21.628 0.63077 33.5

0.3-1.0 21.625 0.15660 21.6

0.1 - 0.3 23.331 0.81956 S.:

< 0.1 24.260 0.83660 3.5

F > 60.0 34.219 0.21716

30.0 - 60.0 27.074 0.2'7436 83.3

15.0 - 30.0 22.651 0.32651 68.8

7.0 - 15.0 17.836 0.41500 54.9

3.0 - 7.0 16.187 0.46490 40.0

2.0 - 3.0 14.823 0.54503 2'7.0

1.0 - 2.0 13.953 0.63227 21.6

0.7 - 1.0 13.953 0.68465 14.0

0.2 - 0.7 14.457 0.7840'7 10.9

< 0.2 15.209 0.81558 4.1

mode1 incorporates various degrees of approximation. The level 2.5 mode1 is discussed and

applied in this study as done previously by Uliasz [18]. The equation for the turbulent kinetic

energv used by Mellor Sr kamada is given by

where P,, Pb, and r represent the shear production, buoq-ant production. and dissipation of

t tir bulent kinet ic energy:

The parameter 1 represents the turbulent length scale. The exchange coefficients (erlcly

diffusivities) are non-dimensionalized by

The non-dimensionalized eschange coefficients are funct ions of numerous empirical constants

(including dl from above) and two parameters G, and Go. These are given by

{ A B2, Ci: a,) = (0.92: O.T.ll 16.6. 10.1, 0.08. 23/2/16.6}. ( 1-42)

The turbulent length scale, 1. is found from

as used by Mellor Sr Yamada [17]. The parameter n is the von Karman constant and q-, is

the surface roughness parameter. A table of these values for various types of ground-cover

c m be found in Pielke's book on mesoscale meteorological modelling [3].

The horizontal eddy diffusivity K H is introduced in order to smooth the numerical solution

of the equations, not for physical reasons [18]. The parameter is found from

Cliasz applied this turbulent ciosure mode1 to a particle dispersion model using the Randoni-

Walk model of particle dispersion (181. The equations he used to determine the variarices of

nind velocity components are given by

Equations (1 As)-(1.47) can be integrated to find the required horizontal and vertical dis-

persion parameters required by the Gaussian Plume model by integrating the variances by

where Ru = Ru (A t/TL,) represents the Lagrangian auto-correlation for the time step At

normalized by the Lagrangian integral time-scale. TL,. The Lagrangian auto-correlation

function relates the time-scale of the grid to the tirne scale of the sub-grid turbulent motions

and mil1 be derived in the Random-Walk Dispersion model section.

1.4 Similarity Theory and Paramet erizat ion

Similarity theory allows for simpliiied forms of the sub-grid scale B u e s to be produced.

Parameterkation of the planetary boundary layer in conjunction with similarity theory can

be used to represent typical velocity and temperature profiles for various stabilitp conditions.

4Iuch of the parameterization is the result of dimensional analysis and matching of ernpirical

data.

-4s shown by the equation (1.28), the sub-grid fluxes can be approximated by an eschange

coefficient and the gradient of the rnean value of the flow characteristic ar hand. In turii.

the? can be represented by sub-grid scale Ruses as shown by

where arc tan(^/^) = p . The velocity scale u. = J(-m)o is the friction velocity. siich t hat

r = ,ou: represents the surface shear stress. The friction velocity can be related to the surface

roughness by

i, = u y g .

The temperature scale 8. is known as the Aux temperature. The mean horizontal flw

velocity can be represented by = d-. such that equation (1.49) and (1.50) c m be

written as

From dimensional analysis. the exchange coefficients can be written as I\m = t i x . and

= KU.. Substituting these relations into equations (1.53) and (1.51) gives

mhich can lx integrated (from z, to z) as

These provide logarithmic velocity and potential temperature profiles. It should be noted

that this development has assumed that there is no change in the mean Nind direction. The

profiles given by equations (1.56) and (1.57) represent a neutral stratification of the atmo-

spheric boundary layer. Using the velocity and potential temperature relations determined

above. the flux Richardson number defined by equation (1.29) can be rewritten as

Set ting

nhere mm = I under neutral stratification of the boundary layer (and thus equation ( 1.54)

is realized). This parameter is known as the non-dimensional wind-shear. Slultiplying the

Hus Richardson number by dm gives

The parameter L. known as the Obukhov Length. is given by

h similar relation to equation (1.39) is set for the potential temperature equation (1 .5 ) .

given by

where 00 = 1 for neutral stability conditions. The ratio of the height over the Obukhov

length. (:IL). and the parameters dm and 4s provide another method of charecterizing

the stability of the atmospheric boundary layer. When i / L < O and p,. do < 1 then the

atmosphere is unstably stratified ( w l V > 0' 8. < 0). A stable stratification is represented by

these parameters when z / L > O and dm! Qe > 1 (w"BU < 0: 8. > O). Finalle the atmosphere

is considered to be neutrally stable when dm, dB = 1 and z / L = O (L = x since u-"Bu = 0.

8. = O) as s h o w by equation (1 .54 .

Rearranging equat ion (1.59) as

and integrat ing provides

i

where - > 2 and L L

is known as the correction to the logarithmic wind profile that results from non-neutral

stratifications. -4 similar derivation for the potential temperature results in

:IL (1 - &l) ivhere On = 1

d (i) .

The region of the atmosphere directly above the surface is known as the planetary boundary

layer. The top of this layer, zi, is defined as the lowest Ievel in the atmosphere that the

dependent variables are unaffected by the ground surface through the turbulent transfer oF

air. The mind-profiles above this height have reached the free stream velocity. The planetary

boundary l q e r is usually separated into t hree discrete sublayers: the viscous su blaycr. the

surface sublayer. and the transition sublayer.

The viscous sublayer is the Iayer closest to the surface. ranging from ,- = O to :,. Tlie

potential temperature at the top of the layer can be related to that of the ground by [3]

The nest layer is the surface 1-r which estends from , to h,? where h, usually varies from

10 m to 100 m. Comrnonly used equations for the profile correction factors (1.65) and (1.67)

are take from reference (31 to be

and equations (1.59) and (1.62) are giwn by

Lurnley Sr Panofski [14] report that for a stable stratification. the correction to the velocity

profile is given by Q,(r/L) = 1 + z/ (0 ,18f 0.4) such that the velocity profile can be written

The transition sublayer extends from the top of the surface layer. h,, to the height of the

planetary boundary layer, zi. The height of the planetary boundary layer usually ranges

Uctwcn 100 in to scvcral kilomctcrs. Within this !u?er, the mezn 1.vind p e r d ! ! chan.... --OL-

direction with height. such that the wind profiles described by equations (1.39) and (1.62)

are not valid in this sublayer. The mind profiles within the transition layer can be written

ils p. 191

whcre f = 2Rsin(d) is the coriolis parameter (where d is the latitude and R = 27/24 hoiirs

is the rate of rotation of the earth). The geostrophic wind components. cg and u,. are giwn

The height of the boundary laver? :,? is usually obtained by radiosonde or other observation

platforms. Prognostic equations have been developed to dictate The height of the surface

Iayer can typically be estimated by using h, = 0 . 0 4 ~ ~ and the parameter z, can be founcl

froni 2, = Ju.L/ / for a stable boundary layer.

The Lagrangian time scales required by the Lagrangian auto-correlations discussed in regards

to equation (1.48) can be found through the parameterization and similarity theory applied

above. They can be erpressed by the peak wavelengths in spectra of the corresponding wind

components by

For an unstable boundary layer, the peak wavelengths can be given by [3. 91

Note that horizontally homogeneous turbulence has been assumed for these relations. Cnder

stable conditions, the wavelengths are given by

where the mean horizontal motion is in the x-direction. These relations were derived froni

esperimental data in the hlinnesota area. Finally. for neutral stratification. the ~vavelengtli . . 1s mi- rnn US

C C '

for al1 three velocity components. ?lote that al1 of these relations. for each stability çlass.

have beer? derived under the assurnption of a non-complex terrain.

Chapter 2

Part icle Dispersion Models for

At mospheric Flows

Introduction

Dispersion is defined as the spreading of a material released into a flow. The spatial spreading

of these materials in time is the result of many factors. including the initial velocity of the

release. the physical properties of the material released (density radius. temperature). and

meteorological variables (mean wind-flow, temperature gradient. turbulence). The niain

source of particle dispersion is due to the turbulence of the Born. ÇVind-field turbulence can

be mechanically generated or produced by buoyant colurnns of air. Mechanical turbulence

is csused by wind flowing past obstacles (buildings. vegetation) or by wind shear. Buoyant

generation of turbulence is produced by colurnns of air that have been heated at the surface

and caused to rise.

An atmospheric dispersion model must be coupled with an atmospheric wind-field model.

which provides meteorological data to the dispersion model. Typicall- this should include

the wind velocity, temperature profiles. density profiles, pressure data, terrain data. and

turbulent flux data. Knowledge of the location, type of release (point. area. or volume

sources). and data about the material released (initial velocity, chernical content, etc.) must

also be knom.

CHAPTER 2. PARTICLE DISPERSION MODELS FOR ATMOSPHERIC FLOWS 19

There are two main ways that the atmospheric model equations, (1.13), (1. 14) , (1.15). (1.16).

and (1.17) can be approached to predict the dispersion of pollutant materials. The? can

be integrated simultaneously or the species equation (1.17) can be solved separately. The

first approach would be very computationally expensive and is rarely used. The second

approximate approach assumes that there is no feed-back between the dispersion model and

atmospheric model equations. This is accurate unless the effect of the pollutant species 011

the radiative RLY (incorporated in the Ss term) becomes important.

There are three main methods of evaluating the transport and dispersion of the pollutant

species for the second approach outlined above. The first and second both simplify the

species equation (1.17) into a Gaussian Plume 4Iodel and a K-Diffusion Model. The Gaussian

Plume Mode1 assumes that the horizontal and vertical correlations are given by a Gaussia~i

distribution. The K-Diffusion Mode1 uses the assumption that the flux of the pollutant is

proportional to the rnean gradient and an exchange coefficient. The mean wind veloçitics

ancl turbulent closure data is interpolated directly from the meteorological model. The final

met hod replaces equation (1.17) tvit h a stochastic model when the meteorological niodel

predictions are used to determine the statistical properties of the pollutant dispersion. The

paths of particles are followed iri this rnethod. known as Lagrangian Particie Dispersion

IIodels. This model also tends to be computationally espensive as it tracks the trajectories

of a very large number of individual particles. However. it is a very flexible method wliich

c m be employed for very comples flows and environments, producing accurate results.

2.2 K-Diffusion Model

The K-Diffusion mode1 will be discussed before the Gaussian Plume model, because the

Gaussian Plume model can be considered a mathematical simplification of the K-Diffusion

model. Let the density be spatially constant, and let the turbulent Aux in equation (1.25)

be approximated by (similar to equation (1.35))

where hgx is the difision coefficient. Then the pollutant conservation equation can be written

CHAPTER 2. PARTICLE DISPERSION MODELS FOR ATMOSPHERIC FLOWS 20

This equation must be solved in conjunction with the other equations of motion and this

is computationally expensive. The accuracy and resolution of the solution also depends on

the density of the grid-points. Often thiç equation is simplified by ignoring the convectire

terms. This simplification leads ro the Gaussian Plume bIodel. The system of equations c m

be reduced if the wind-profile is known or assurned. In this case the system of equations is

rediiced to the continuity equation and the K-Diffusion equation.

2.3 Gaussian Plume Mode1

111 redit- the eschange coefficients K,, are never constant in time or space. However. if

they are assumed to be constant. then the diffusion process represents Fickian diffusion.

In addition. if the spatial variation of the density and the advection of the resolvable How

velocities are ignored. the K-Diffusion model is written in the following simplified forni ( also

ignoring the source term) :

The remainder of the conservation equations are neglected. t,hus the Gaussian Plume .\Iode1

is a major simplification of the fundamental physics of an atmospheric fiow. Steady-state

rneteorological conditions are assumed. and hence it assumes that the plume has a straight

centre-line? pointing in the wind direction. Therefore. it cannot represent recirculation of

the pollutant since complex wind conditions are not allowed. Continuous emissions are

assumed and mass is conserved; there are no material losses due to chernical reaction or

h m deposition. The solution of this equation for a release at the surface (2 = O)' through

a Fourier transform analysis as shown in Appendk B, is given by

where E is the mass of the release. This is known as the Puff Equation. The coordinates

correspond to the centre of the puff of the pollutant moving at a uniform mean horizontal

Aow given by r2 = ü2 + v*. The puff dispersion parameters are given bp

oz = , / 2 ~ ~ = t ; oY = J'LK,,~; and oz =

which represent the standard deviation of the Gaussian function (2.3). To represent a contin-

uous point source, steady-state conditions are aççumed and the coordinates are transformed

CHAPTER 2. PARTICLE DISPERSION MODELS FOR ATMOSPHERIC FLOWS 21

such that the mean Bow is in the x-direction. The mass input per x-direction dispersion

parameter, Ela,, is replaced by a pollutant emission rate (g/s) per Bow velocity*! QIü. Thus

for a release at the surface, the Gaussian Plume Mode1 is given by

- X ( X , Y, 4 = exp

It is assumed t hat the concentration distribution is Gaussian. bot h horizontally and vert i-

cally. Thuc the Pasqiiill-Gifford dispersion pararnet~rs rl~fined in section (1.3) are iised for

ug and oz. For a release a t ; = H from the surface. the Gaussian Plume equation is given

b y

The variable H is known as the effective height of the centreline of the pollutant plume [Il.

This paranieter is a function of the actual emission height. h. and the rise height of the

plume, AH. These parameters are related by the equation

The plume rise height occurs due to two physical effects: the mornentum of the release and

the buoyancy of the pollutant. The mornentum of the outflow of gas can be found directly

from its mass and release velocity The buoyancy of the gas is dependent on its density

and temperature relative to the density and temperature of the atmosphere surrounding the

release point. I t is noted by Turner [II that buoyancy generally has a greater effect than the

momentum of the release if the temperature of the gas is 10 to 15 K higher than the ambient

atmospheric temperature. The buoyancy B u , FB, must be found to calculate the buoyancy

incluced rise height of the plume. It is given by Turner (1) as

where I..; is th release velocity (or esit velocity) of the plume, d is the diameter of the area of

the plume release, and the temperatures Tp and Ta represent the plume temperature and the

ambient air temperature respectively. The plume rise equations for a buoyancy induced rise

are mostly based on empirical data for al1 stability conditions [l]. For unstable and neutrally

stable conditions, the buoyancy induced rise height is given by

2 ~ . 4 2 5 ~ i ' ~ / u , for Fs < 53, AHg =

3 8 . 7 1 ~ ~ ' ~ l t ~ h for FB 2 53,

CHAPTER 2 . PARTICLE DISPERSION MODELS FOR ATMOSPHERIC FLOWS 37 --

where u h is the wind-speed at the release height. For stable conditions, the buoyancy induced

rise height is calculated by

{ FE?)'^ ( F ; ' ~ ) } A H s = min 2.6 - ,4.0 - 'Uh s s3/5

where s is a stability parameter. This stability parameter is dependent on vertical gradient

of the potential temperature. and is determined by

-4 momentum induced plume rise for unstable and neutral conditions is given by

For stable conditions the momentum induced plume rise is determined from

The final rise height of the plume' for any stability condition. is gicen by the greater of the

risc height due to the mornentum and buoyancy:

Turner [l] also notes that the effect of the momentum of the release dissipates over a much

smaller time period than the buoyancy induced plume rise. In the case of buoyancy. the

plume may rise gradually over a long period of tirne (a few minutes). For any stability

conditions, the gradua1 buoyancy induced rise height should be calculated from

AHBgr = 1.60 (2.15) 'u h

While it is greater in value, the value of the plume rise height calculated by equation (2.15)

should be used in place of the buoyancy induced rise height equations determined before.

This value should be used until the distance to the final rise height. XI. is reached. At that

point the final rise height as calculated by equation (2.14) should be used. The distance to

the final rise can be found for unstable-neutral conditions by

and for stable conditions by

CHAPTER 2. PARTICLE DISPERSION MODEM FOR ATMOSPHERIC FLOWS

2.4 Lagrangian Particle Dispersion Models

Lagrangian particle dispersion models (also known as the conditioned particle technique or

Random-Walk model) follows the motion of single pollutant particles or a volume of pollutant

as a single particle in a flow: Uliasz [18], Walklate [20], Pielke [3], Legg Sc Raupach [XI. and

Hinch [22]. The position of these particles is defined by the following relation

Y ( t + At) = Y ( t ) + (V(t) + ~ " ( t ) ) nt.

where and V" represent the mean and turbulent flow veiocity components (resolvable anci

sub-grid scales). The mean flow cornponents are usually obtained directly h m the niete-

orological mode1 governed by the conservation equations (neglecting the pollutant species

equation). The sub-grid scale velocity components are derived from the solution of the

Langevin Stochastic Differential Equation. Legg Si Raupach [XI. as shown in Appenclis C.

and are given by

aruw V"(t) = R,,(ilt)Vtt(t - At) + [l - x ( ~ t ) ] l l ~ o , { + [l - R,(ilt)]T~,- . ('2.19) a:

rvhere Ru are Lagrangian auto-correlations for the lag time At. { are (quasi) random nunibers

with zero mean and unit standard deviation. Equations (1.45)-(1.47) can be iised for the

standard deviations of the velocity components ou, note that r,, = ow. The other tinie-

averaged turbulence velocities ru, = 6 and r,, = fi can also be found from Sfellor

Si Yamada's turbulence closure model [17]. The Lagrangian auto-correlations are shown in

Appendix C to be exponential functions of the h g time At and the Lagrangian integral time

scale Tt,

The time scales can be determined through the method s h o w in section (1.4).

The concentration of particles within an arbitrary sampling volume can be calculated from

where mp is the mass of the particles and Np is the number of particles within the sampling

volume (Ax,A~,Az,) under consideration. A very large number of particle trajectories m u t

be tracked to obtain a continuous measure of the particle concentration in the sampling

volumes. However, this method does allow for a dispersion study with a more comples

meteorological flow- field.

In this model, the mean motion of the particle is assumed to be equivalent to the mean

velocity of the atmosphere. However, this leaves out the effects of other particle physics:

such as drag, buoyanc. heat transfer, rotation, non-rigidity, and part icle shapes. Walklate

[?O] applied an equation of motion for a rigid, spherical particle into the Ranclorri-Milk

modei. This equation incorporated the effects of drag and gravit- anci is given by

- V( t ) = ü(t - At) + ( ~ ( t - At) - ü( t - l t ) ) exp(-3At) + VT, (1 - e x p ( 4 & ) ) . (2.22)

This equation is derived from the solution of the equation of particle motion given by equation

(3.3-4). derived in the nest chapter. The derivation of equation (2.22) has been includecl iri

Appendix C.

To reduce the computational effort required by the model described above for single particles.

equation (2.15) can be used to follow the motion of puffs of pollutant as clefincd by ecluiitioii

(2.3). t hereby using the Randorn-Walk and Gaussian Puff models toget her. Thr Rantloni-

LValk model is used to track the centre of each puff released by a source and the Gaussian

Puff model provides the concentration of the pollutant around the centre of the puff witli

a Gaussian distribution of the pollutant m a s . The motion of the puffs is generally foiind

using the mean wind velocities only (no particle trajectory models or turbulent velocities are

used), thus this is a highly simplified version of the Lagrangian particle dispersion niodel.

The concentration of pollutant is found by summing the contribution of the N-puffs modelled:

The coordinates (.Yn: 1;: 2,) represent the centre of the nth-puff. The standard deviations

of the Gaussian distribution O,, O,,, and 0, can be found using the rnethod described in

section (1.3). The exponential function defined by equation (2.20) is used as the Lagrangian

auto-correlation function. Integrating this equation gives [18]

oy(t + At) = c~,(t) + ov(t)At for t 5 2Tt,.

~ i ( t + At) = oi ( t ) + 2TL&(t) At for t > 2TL,.

hgain a large number of puffs must be followed to reproduce a continuous source.

CHAPTER 2. PARTICLE DISPERSION MODELS FOR ATMOSPHERIC FLOWS

2.5 CALPUFF: A Modern Dispersion Modelling

Package

CALPUFF [2] is a non-steady-state pollutant dispersion mode1 that allows for comples ter-

rain effects, over water transport, coastal interaction effects, building down-was h. wet and

dry removal, multi-species, and chemical transformation of the pollutant species. hllnine.

Dabberdt, and Simmons [231 report that it represents the state-of-the-art practice of atrno-

spheric dispersion niodelling. CALPUFF can simulate the effects of time and space-varying

meteorological conditions and meteorological fields produced by rneteorological niodels (such

as R-WS or hIhl5) or from single weather stations. It is capable of representing point sources.

line sources. area sources? and volume sources of constant or varying emission rates.

The b a i s of the dispersion met hod used by CALPUFF is the Gaussian Puff methocl outlined

in the previous section and mat hernatically stated by equation (2.23). Scire. St rimaitis. and

Yamartino (21 rewrite this equation in a form that determines the concentration of the

pollutant at a given receptor. ivith coordinates (I,, y,). while taking into account a mised-

laver height . h

where He is the effective height of the centre of the puff above the ground. da and (1, are the

distances between the centre of the puff and the receptor in the along-mind and cross-wind

directions. and the sumrnation over n in the vertical term accounts for possible reflections

off the mixing height and the ground. For a horizontally symmetric puff under honiogeneous

conditions, oz = op, equation (2.26) can be reduced to

in which s is the distance traveled by the puff and R represents the distance from the puff to

the receptor. Equations (2.23) and (2.27) represent "snapshot" descriptions of the pollutant

puff at time t. A time averaged concentration, X, can be derived by integrating equation

(2.27) over an incremental distance of travel, ds, and time-step dt:

where s, is the value of s at the beginning of the time-step. Let the initial and final positions

of the centre of the puff be denoted by (x,, y,) and (x2, y2). For simplification of the analysis.

assume that the mass is conserved ( E ( s ) = E(s , ) ) , such that there are no losses due to

pollutant removal or chemical transformations. Scire et al. [2] include these processes nithin

their analysis. Two assumptions are made based on the fact that the incremental distance

traveled by the puff. ds, is very small: the trajectory segment can be approximated bu a

straight line, and that the change in the vertical term, g (s ) , is negligible. Thus the distance

betrveen the centre of the puff and the receptor is given by

where d z = r- - rl dg = r ~ ? - -1 and p is the dirnensioniess radial distance to the receptor.

The value of p is zero at the beginning of the trajectory and one at the end of the trajectory.

Appiying t hese assumptions to equation (2.19) and transforming to pcoordinates giws

- R ~ ( ~ ) / ~ O , ' ) dp.

The solution of equation (7.30) can be presented in terms of error functions and esponelitials:

CALPUFF h a the ability to mode1 the puffs as slugs instead. The slug mode1 consists of

Gaussian packets of pollutant stretched in the along-wind direction. The length of the slug

can be given as t, = IL& where u is the velocity of the flow. and At, is the time of

emission of the pollutant pufF. The slug concentration of a slug is given by 121:

where u' = u +O, is the scalar nrind speed, F is the "causality function", and g is the vertical

contribution as given by equation (2.26). The causality function accounts for the edge effects

CHAPTER 2. PARTICLE DISPERSION MODELS FOR ATMOSPHERIC FLOWS '21

near the endpoints of the slug. The cross-slug and along-slug distances are specified by the

parameters d, and da respectively. The along-slug length is given by &, = dai i da-. where

rial and da? represent the distances from the centroid of the slug to the near and far ends of

the slug. The dispersion coefficients o,l and O,* refer to values a t the same ends.

In the case that the emission rate and meteorological conditions are constant with time. a

generalized integrated slug model can be obtained, as per the method described above for the

puif model. for a slug whose near (or p u n g j encipoint corresponcis with the source iocation.

Scire et al. [2] obtain for the integrated slug mode1

- 1 l & I F = ,erf - ( ~ 2 ) + <- - uAt, {[ceerf(<,) - cberf ((a)] + , - [exp({,') - esp(<:)]}. (2.39)

where

The parameters ce. 6. and p2 represent

the beginning of the tirne step. and the

the situation a t the end of the time step At,. at

steady-state conditions at the source. Yumerical

integration must be used for slugs that do not meet the imposed requirements outlined above

as there is no analytical solution available.

Scire et al. [2] presented a comparison of the two versions of the integrated puff model and

the slug mode1 relative to results produced from a Gaussian Plume model. One of the piiff

models, termed the local puff rnodel, utilizes local flow properties (such as the dispersion

parameters) evaluated at the mid-point of each time step. The other integrated puff niodel

uses receptor-specific properties. The second of the two models is the one used by the

C ALPCFF dispersion model.

Cnder steady-st ate condit ions, the puff and slug models should tend towards the results of

the Gaussian plume model. The local puff mode1 requires greater than 500 puffs per hour

to accurately repeat the plume results. This requires over 6500 times the amount of CPL!

time that is used by the plume model. The CALPUFF integrated puff model and the slug

model both accurately reproduce the plume steady-state results with only marginally more

computation time and only 1 puff per hour.

The CALPUFF puff and slug model results for a non-steady emission of 1 g/s over one full

hour produce similar results. The slug model, however, takes into the account the effects

at the edges of the puff more accurately and out-performs the puff model in computational

efficiency.

Chapter 3

Theory for Particle Trajectories

3.1 Introduction

The dispersion rnodels described earlier fail to fully describe the motion of single particles in

a flow. For example. the Random-Walk model usuaily assumes that the particles have tlic

same velocity as the flow. It can be estended. as has been shown. to model sorrie effccts of

particle motion in a flow. The main effects that should be considered are gravity arid drag.

Rotation, heat transfer, rigidity, and the shape of the particle could also have a considernble

effect on its motion. In section (3.'2), the basic equation of motion for a single particle under

the influence of drag and gravity is derived. This equation sirnplified to various degrees

for particles of higher density through a scale analysis and physical reasoning as shown in

section (3.3). In total. t hree additional equations of motion are determined. The derivat ions

summarized in sections (3.2) and (3.3) are fu11y presented in hppendk D. .A fifth equation

of particle motion can also included. which assumes that the effects of drag, gravity. and

buoyancy can al1 be neglected. In this case, the particle's velocity is simply equivalent to

the local fluid velocity. The basic equation of motion and the subsequent simplified versions

are only valid under Bon, conditions in which the slip and shear Reynolds numbers are mucli

less than one and the diameter of the particles are greater than the mean free path of the

Bow. Correction factors are introduced in section (3.4) that can be used to allow for srnalier

particles or higher Reynolds numbers. The addition of turbulent terms are presented in

section (3.5)

CHAPTER 3. THEORY FOR PARTICLE TRAJECTORIES

3.2 Derivation of the Basic Equation of Particle

Mot ion

The following developrnent of the momentum equation for a spherical particle is sirnilar

to that described in Mâuey 9r Riley 1241 and Landau 9L Lifshitz [SI. It is assurned chat

the particle is rigid, non-rotating, and has a temperature equivalent to the fluid. The flw

field of an incompressible, undisturbed Auid fforv u(x, t ) obeys the conservation of mass

and momentum equations. An obstacle in the flon will locally perturb the ffow field to a

disturbed field v(x. t). For a rigid. spherical particle with radius R located at Y( t ) . the tlow

field must obey the following equations of motion

v = V + Sl x [X - Y(t )] on the sphere.

v = u as lx - Y(t)l + m.

Equations (3.1) and (3.2) represent the conservation of mass and momentum. where the

incompressible stress tensor cm is given by

The boundary condit ion (3.3) represents a no-slip condition on the spliere. w hicli indicates

that the fluid velocity on the sphere is equivalent to the particle velocity V(t) and the

contribution from the particle's angular velocity n(t). From this point on. the particle i d 1

be considered to be non-rotating, which sets the angular velocity to zero. An additional

torque equation would be required to include rot ating particle effects. Two more equations

are required to include heat transfer effects between the fluid and the particle: a temperature

(energv) conservation law for the fluid and a heat transfer equation for the particle. The

boundary condition (3.4) specifies that the disturbed fluid velocity and the undisturbed fluid

velocity are equivalent far away from the particle. Then the equation of translational motion

for a particle is given by the sum of the forces acting on the particle

where the fiuid stress tensor must be evaluated over the surface of the sphere. Employing a

change of coordinates to a particle-centered Frarne, r = x -Y ( t ) and w(r. t ) = v(x. t ) - V( t )

CHAPTER 3. THEOIIY FOR PARTICLE TRAJECTORIES 31

will ease the evaluation of the fluid stress tensor. The conservation Iaws still hold for the

slip velocity. The boundary conditions speci& that the slip velocity is equal to zero on the

surface of the sphere and the far-field wlocity is given by w = u - V. Within this new

coordinate system. the slip velocity can be separated into an undisturbed Aow-field Uo and

a disturbed flow-field U, given by w = Uo + U where Uo = u - V and U = v - u. The

disturbed velocity must tend to zero far from the particle and is equivalent to zero on the

surface of the particle. Substitution of the slip velocity components into the coiiservation

laws for the slip velocity will result in separate continuity equations for the slip velocity

cornponents and two momentum equations which are coupled through the convective ternis

of the disturbed velocity momentum equation. The surface integral in equation (3.6) can

rlow be written in terms of the undisturbed and disturbed flow-field contributions

where the undisturbed and disturbed force terms are given by

The contribution to the fluid force from the undisturbed Row can be found quite generally

without any further assumptions. From Gauss's theorem. equation (3.5) can be written as

a volume integral and expanded as

assuming that the sphere is sufficiently srnall such that the viscous stress tensor acts uniform

over the surface of the particle. The first term on the right hand side is the force due to the

buoyancy of the particle (when added to the gravity term in equation (3.7)). .As expected.

a particle with the same mass as the fluid it has displaced results in a particle t hat is

neutrally buoyant. The second term is a reaction force due to the fluid acceleration. The

fluid acceleration tetm incorporates the effects of the fluid stress-gradients on the particle.

The forces due to the disturbed flow created by the particle now need to be determined.

A seale analysis shows that in the Iow slip Reynolds number Re, = RFV0/v « 1 and the

low shear Reynolds number b i t Re,(R2/u)(Uo/L) < 1, that the advective terms in the

CHAPTER 3. THEORY FOR PARTICLE TRAJECTORIES 32

disturbed velocity momentum equation may be omitted [24]. The resulting momentum

equation for the disturbed velocity is given by

which corresponds to an unsteady Stokes flow problem. The undisturbed flow-field is as-

sumed to be uniform, which requires that V x Uo = O and VnUo = O where n > 1. In

spherical coordinates. the polar âuis is parallel to Uo and therefore al1 quantities are func-

tioris of r and the polar angle 0 only [SI. The drag force (or the force on the sphere due to

the rnoving fluid) is parallel to the velocity Uo. The force over the entire sphere. projected

in the direction of Uo is given by

ac;, w here & = 2p-,

dr

Therefore. it is required to find the disturbed velocity and pressure profiles about the sphere.

Taking the curl of the disturbed momentum equation (3 .1 1) results in

d -(V x U) = V V ~ V x U). (3.13) dt

Since the curl of a gradient is zero, V x Vp = O. Therefore. the curl of the disturbed

fluid velocity must satisfy the heat conduction equation. Solution of equation (3.15) will

provide the required velocity profile which can be used to determine the pressure profile by

integrating equation (3.11). The velocity profile c m be determined by assuming the velocity

is of the form

U = V x (V x ( fUo)) = exp(- iwt) V x (V x (fu,)) (3.16)

such that the velocity Uo is given by Uo = exp(-iwt)uo which indicates that it oscillates

116th a frequency (J? and f is a function of r only, f = f (r). I f t e r a long and involved

derivation, which is presented in Appendk D, the velocity and pressure profiles are given by

t lie following equations

CHAPTER 3. THEORY FOR PARTICLE TRAJECTORIES

The polar and axial velocity components of equation (3.17) are used to determine the fiuid

stress tensor components found from equations (3.13) and (3.14). Subsequently. these resiilts

and the pressure profile are substituted into equations (3.12) to determine the unsteady force

on the particle. Applying the undisturbed flow-field velocity relation. Uo = u - V. the tinal

form of the unsteady force on the particle becomes

d / d r ( ~ ( t ) - u[Y( t ) , t ) ] - i a 2 ~ ' u ~ u ( l J -67ri2p 1; ( ) dr. (3.19)

(av(t - r ) ) f

The three terms on the right-hand side of the above equation are known as the force diic

to added nias. the Stokes drag force, and the Basset h is toq force. These terrns have bern

derived using the low slip and shear Reynolds numbers approsirnation (Re,. Re, « 1). and

are not valid in higher flow regimes.

The added mass term represents the virtual mass added to the particle since the acceleration

of the particle requires the surrounding fluid to accelerate. The volume of the addecl mass

is equal to half of the volume of the particle.

The Basset history force depends on the past particle motiont weighted by the kernel ( t -7) f . where ( t - r) represents the elapsed time since the past acceleration. It acts as an augmented

viscous drag and depends on the viscosity of the fluid. the acceleration of the particle. and

the acceleration of the fluid.

The terrns involving the Laplacian of the flow-field V2u are k n o m as the Faxen terms.

Siniply stated. the Faxen terms account for the non-uniformitp (or curvature) of the flow-

field. The Faxen terms are generally smdl when compared with the other terms and so are

usually neglected, as will be s h o m in the next section.

Substituting the results for the undisturbed (3.10) and disturbed (3.19) flow-field force con-

tributions into equation (3.7) gives the final form of the equation of translational particle

CHAPTER 3. THEORY FOR PARTICLE TEUJECTORIES 34

motion (or the momentum equation of the particle) for a rigid, spherical, non-rotating, and

constant temperature part ide

T V( t ) - u[Y ( t ) , t ) ] - + 2 ~ k h u ( t ) ) - 6 7 d P 1; ( dr. (3.20)

( i i ~ ( t - T ) ) +

The position of the particle is d Y ,,

by integrating the velocity over time.

3.3 Scale Analysis and Derivation of Simplified Models

Following Mei. Adrian. and Hanratty (251, a scale analysis of the particle equation of motion

provides information on which of the terms are the most dominant. and will allow for sini-

plifications of the equation of motion for certain conditions. A steady state enalysis of the

equation of particle motion will give the terminal velocity? VT, of the particle as

where VT, is the terminal settling velocity given by

i -

where 3 = p J p f is the non-dimensional density factor. Relating the particle velocity in

terms of the terminai settling velocity and a fluctuating velocim v(t). by V( t ) = VTs +v(t)

and substituting into equation (3.20) gives the equation of particle motion in terms of the

fluctuating velocity v as

CHAPTER 3. THEORY FOR PARTICLE TRAJECTORIES

Equation (3.23) cm be non-dirnensionalized by introducing the following parameters:

The non-dimensional equation of part icle mot ion is given by

where the non-dimensional numbers are

Equations (3.26), (3.27). and (3.25) represent the Stokes number. a non-dimensional tirrie-

scale. and a non-dimensional length-scnle. The pararneter c ~ o represents the typical frequency

of the fluctuations in the flow.

The Stokes nurnber is a product of sio and the diffusive scale. R2/u . it represents the ratio

of the frequency of the turbulent fluctuations in the flow to the frequency of the viscous

damping.

The non-dimensional time-scale, 3: is cornrnonly referred to as the particle relaxation tinie

[il. The factor is due to the added mass contribution to the equation of motion.

The dimensionless length scale is the ratio of the size of the sphere to the typical length of

the fluctuations in the flow. It is required that the sphere size must be small relative to the

length scale such that f! = Rko z O if the Faven terms are to be neglected. If this condition

is met, the non-dimensional equation of particle motion (3.25) can be simplified to

The factor f i is proportional to 2-2t so that the particle acceleration term on the left-hand

side of the equation of particle motion is of the order of P. In the lom Reynolds number

CHAPTER 3. THEORY FOR PARTICLE TRAJECTORIES 36

approximation, the Stokes number is required to be much less than one, E < 1. It can

be seen from equation (3.29) that the particle motion is dorninated by the drag terms of

0(1), which also includes the buoyancy effects. The Basset history forces are of O(;). and

the Buid acceleration and added mass are both O(?). Therefore, for light particles. such

that = $ 5 O(l) , al1 of the terms in equation (3.29) must be included in the solutiori

of the particle velocity. However, for the case of heaw particles. such that the density

ratio is of order then the particle acceleration term is of O(;). Therefore. for h e a y

particles. equation (3.29) can be solved for O(;)? which means that the added mass and Buid

acceleration terms can be neglected and the equation of particle motion simplifies to

which contains only the buoyancy effects. Stokes drag, and the Basset history integrah.

For even heavier particles. with a density ratio of O(?') or greater. al1 of the terrns i i i

equation (3.30) may be neglected other than the particle acceleration term and the Stokes

drag/buoyancy terms. which gives

Note that in the last t ~ w mode1 equations. (3.30) and (3.31). the factor - in the particle

acceleration term rnay also be neglected without any further loss in accuracy. In terrns of

the dimensional values. and the particle velocity V(t), equations (3.29). (3.30): and (3.3 1)

c m be written as

respectively. The factors E and ,b' are given by

CHAPTER 3. THEORY FOR f ARTICLE TRAJECTOWES 37

The simplest equation of particle motion assumes that the drag and buoyancy forces on

the particle are negligible and hence the particle velocity is simply equivalent to the fluid

velocity. that is

3.4 Corrections to the Stokes Drag Law

The Stokes Drag Law is valid only for flows with a Reynolds much less than one. Re,. Re, < 1. and for particles with a diameter much greater than the mean free path of the flow. d > A.

Correction factors must be used for cases not meeting these conditions. For the first case.

a nonlinear drag law must be used in place of the Stokes Drag Law and a slip correction is

used to address the Row continuum effects.

The Stokes Drag Law gives the drag coefficient as

where Re,

Force on a

FD =

To extend

- - 2 p f R lu - Vl and lu - Vl = d(u - V) (u - V). Lsing this relation. the drag P

particle is given by

the drag law to higher Reynolds numbers, the following equations can be used:

The first equation is known as the Oseen Drag Correction [26] and is limited to flows with

Re, < 40. The next equation [27, 71 can be used for flows satisffing Re, < 1000. Figure

(3.1) displays a plot of the Stokes drag, Oseen Drag Correction, equation (3.41), and es-

perimentdly observed drag values [27] for Reynolds numbers ranging from 0.1 to 70.000.

Note that the unsteady drag force, equation (Ul), derived in section (3.2) is only valid

CHAPTER 3. THEORY FOR PARTICLE TRAJECTORIES

Oseen Correction

Experimental Data

1 . L . . . . . . 1 . . . * . . . . l . 1 . r a . . . . . 1 .

1 O-' 1 oO 10' 1 O' 1 0' 1 O' Reynolds Number

Figure 3.1: Cornparison of experimental and computed drag coefficients.

for Re, « 1. Therefore. these drag corrections can only be used in conjunction ivith t lie

equation of motion given by equation (3 .34 , and should be written as

d V 1 m - = -pfACDlu - Vl(u - V) - (m, - mf)g .

dt 2

Note that al1 of the drag latvs stated above can be summarized by CD = f (Re,). mhere

for the Stokes Drag Law f (Rep) is equal to 1. Substituting this relation and mp = p,V, for

the mass of the sphere. equation (3.42) can be written as

Note the factor ;+ = !i is essentially the same a s the parameter 3 given by equation

(3.36). The difference of the factor added to the non-dimensional density ratio in (3.36) is

due to the fact that the unsteady force contribution has been neglected in the derivation of

equation (3.43).

Aerosol and srnoke paxticles can be very small. Smoke particles are generally considered to

be only 0.01 - 1 pm in diameter [7]. If the particle size is s m d or comparable to the mean

CHAPTER 3. THEORY FOR PARTICLE TRAJECTORIES 39

free path of the Buid, A, then the assurnption of continuum flow is invalid (note that the

mean free path of atmospheric air is approximately 0.1 pm). The drag laws discussed above

will then over-predict the drag force on the pasticle. A correction factor based on the particle

size and the mean free path of the fluid must be introduced to account for this effect. This

correction factor is known as the Cunningham correction [Ti, 7]or the slip factor as given by

nhere l in is the Knudsen number defined by Kn = X/3R. Figure (3.2) presents the irii-

portance of the Cunningham correction factor for very small particle sizes. This correct iuii

factor should be introduced into the drag force by

Parücle Radius (m)

Figure 3.2: Cunningham correction coefficient for srnall particles.

C HAPTER 3. THEORY FOR PARTICLE TEWJECTORIES

3.5 Addition of Turbulent Terms

Turbulent veiocities are incorporated into the equations of particle motion by writing the fliiid - velocity as u = u + ut', where ü and u" represent the mean and turbulent Rom velocities.

Substituting this relation into the various equations of motion given by equation (3.3'2).

(3.33). and (3.34) gives

The turbulent velocities must be obtained through a turbulent closure model. Ttie randoni-

walk mode1 combined with Mellor Sr khmada's second moment closure model [L;] coulci

adequately provide these values. Kraichnan [28] produced another rnodel that woiild provide

the turbulent velocities based upon a turbulent energy spectrum. This mode1 is discussetl

further in the nest chapter.

Chapter 4

Numerical Simulation of Part ide

Trajectories

Introduction

The basic equation of particle motion derived in section (3.2) includes steady and uristeady

force effects due to gravity and drag. In section (3.3). physica! reasoning allowed for sinipli-

fications of this equation based on the magnitude of the density ratio P = p p / p f . Equations

(3.32). (3.33), and (3.34) are the resulting simplified equations from the scale analysis in

dimensional form. For the remainder of this chapter. these equations will be terrned the

Level 3. 1. and 1 equations respectively for decreasing cornplexity. For convenience. thesc

equat ions are repeated here:

1 dV Level 3 --

$8 dt

1 dV Level 1 --

,L3 dt + V = u f VTs.

Xumerical modelling of the equations of motion are discussed in section (4.2). These equa-

tions ni11 be studied for steady and unsteady flows. The unsteady Bow-field is realized by

superirnposing turbulent values onto the steady Bow-field. The Kraichnan turbulence mode1

is used to create the turbulent velocity components. -4 summary of this rnethod is presented

in section (4.3). Determination of the physical characteristics of particles of various sizes aiid

densities is conducted in section (4.4) to gain knowledge about the motion of the particles

before they are numerically solved. In the following section, the Level 1 and 2 equations of

particle motion are numerically cornpared in a steady flow to determine the magnitude of the

effect that the Basset history terni has on the drag of the particle. Al1 three equations art3

stiidied in unsteady flow conditions to find the effect of the added-mass and Eiuid-acceleratioii

terms of the Level 3 equation of particle motion, to provide insight into which Level of ecpa-

tion should be conçidered for particle dispersion modelling in atmospheric flows. This stutly

is conducted in section (4.6). Note. that the range of pollutant particle sizes. R. lalls within

IO-' rn to 1 0 - ~ rn [il. Particle to air density ratios. P o are typically found to be around 1000.

4.2 Numerical Modelling of Particle Trajectories

The nunierical solution of the equations of particle motion is rat her straight-forward. The

Level 1 equation of particle motion has been solved using 4IacCormrrckk predictor-corrector

nierhod. This method c m be summarized by

predictor 4 = #(") dm (")

+h- dt

corrector dnf ') = @(") +

The time-step, h. is chosen such that it is much Iess than the particle relaxation time.

h « 113. The algorithm for solving the Level 1 equation of motion is then given by

The level 2 equation is more difficult to solve numerically due to the Basset history term.

As given by Reeks & McKee [29], the Basset history terms for the particle and fluid velocity

contributions can be numerically modelled by the following equations

where the time derivative of the Buid velocity following the moving sphere is given by

Derivation of equations (4.5) and (4.6) are included in append~u E. The- represent only a

first order approximation to the integral term. However. the cornputational complesity iintl

nieniory intensi ty drast ically increases. especially when solving using the predictor-correct or

method discussed earlier. This is due to the fact the history term depends on al1 of th(.

values of the required variable from the beginning of tlie simulation. The tinie derivat ive ut'

the Buid velocity following the moving sphere in the fluid history term requires knoaleclgc

of the flow velocity surrounding the particle's position over time as c m be seen wtien i t is

represented numerically by

Given the numerical schemes introduced for the Basset history terms. the Level Xquatiori

( m i l ) (m) ( m i l ) (ni+l)

of particle motion can be solved erplicitly by

au + Lc; -

The Level 3 equation of' particle motion becomes even more complicated due to tlie added

U i - ui + C; ut i l - %-1 - - h

. (4.7)

mass and fluid acceleration terms. This equation can be written as

% (m+i) ~ A X ]

4 2 D ~ i + -6 -1 (n) Dt (n)

where the fluid acceleration as observed at the centre of the particle is given b>*

C HAPTER 4. NUMERICAL SIMULATION OF PARTICLE T RAJECTORIES

4.3 The Kraichnan Turbulence Mode1

The turbulent Bow-field model varies the flow velocity around a specified mean flow as derivecl

by Iiraichnan [28]. This mode1 simulates a normal. incompressible. s ta t ionan isotropie

velocity field, realized in the form

k n where u,(kn) = x - kn

lknl? ~ b ( k n ) = Cn x -

lknl'

The vectors <, and G. and the time constants, un, are chosen from a Gaussian distribution

with a mean of zero and a standard deviation of one and w, respectively. The wave-numbers.

kn, are also picked from a Gaussian distribution with mean k, and standard cleviation of

/col?. It is required that

energy spectrum. E ( k ) ,

b y

the vector kn is selected such that the desired shape of the turbulent

is realized iri the limit that .V -t S. The energy spectrum is giveri

The purpose of this turbulence model is to aid in studying the unsteady flow effects on the

particle motion. as similarly used by Kraichnan (281, Reeks Sr 'iIcKee [29]. h x e y [30]. and

Wang Si 'uIauey [31].

4.4 Physical Characteristics of the Particles

The physical characteristics of particles of various sizes and densities are determined to

gain insight into the motion of the particles, the equations that can be used to simulate

t heir trajectories. and the terminal velocities that they should attain. Information on the

transient response time to disturbances in the flow are gained from the particle relaxation

frequency. 3.

For small particles (of the same order of magnitude as the mean free path of the Bon- or

slightly greater) and for flow with a slip Reynolds number that does not meet the required

condition Re, < 1, the modifications to the Stokes drag law hom section ( 3 . 4 must be

included. The steady-state (or terminal) velocities for the particle in a steady flow can then

be given by

2R where Re,, = - J(UZ0. - V T ~ ) ~ + (,uy, - vTY)2 + (uZCs - I+: )~ .

V

The addition of the correction factors cause the terminal velocity relation to beconie non-

linear. The x and y-direction terminal velocities are simply given by \+= = u,, = u.

I = uDcs = O' where the steady flow-field is given as u = ( u . O , O). In the vertical direction.

the hr-field velocity u:, is also given as zero. Hence, the steady-state Reynolds nuniber

ZR Re,, = - u b+:.

Therefore. the steady-state particle velocities in the z-direction can be given by

c c cc - c c \ , I'T, = -

f (Re,,) '' = 2/3 1 + h ~ e , , 1 + 6 u

Re-arranging this equation for the vertical terminal velocity gives

This equation can be solved using numerical root finding methods to proride the s t e a d -

state vertical velocity to compare with those obtained by the numerical simulation in the

next section.

The vertical terminal velocity and other physical characteristics for particles of rarious radii

and densities are presented in table (4.1). As known frorn equation (3.38)' the Stokes nurnber.

c. increases witith the size of the particle. The significance of this trend will be discussed

in the next section. The Cunningham Correction factor, Cc, decreases as the particle size

increases and is important For particles with a radius less than 100 pm. These tn70 parameters

are dependent only on the size of the particle; however, the particle re la~at ion frequency

3, the terminal settling velocity VTs? and the vertical terminal velocity C+, al1 require a

specified particle to air density ratio as well. The particle relaxation frequency is largest for

the srnallest and lightest paxticle. This corresponds to the smallest particle relaxation (or

response) time, which indicates that smaller particles require a very short time to obtain their

Table 4.1: Particle Trajectory Data

terminal velocities. As expected. the terminal settling velocity and vertical terminal velocity

is smallest for the lightest particle. The terminal velocities increase as the size and density of

the particle increases and the particle relaxation frequency decreases. Therefore. the particle

response time for the heaviest particle is quite large which indicates that the particle will

take a long time to obtain its terminal velocities. The drag force is more important for large

particles than for small particles.

As outlined in section (3.3), the magnitude of the particle to air density ratio' as compared

to the inverse of the Stokes number e, dictates the level of simplification of the basic equation

of particle motion. The Stokes number is proportional to the square of the particle radius.

Therefore. the inverse of the Stokes number decreases as the radius increases. The Level

3 equation of particle motion must be used to calculate the velocity for a particle with a

density ratio less t han the inverse of the Stokes number. Simplifyîng the equation of motion

to the Level 3 equation is valid if the density ratio is greater than the inverse of the Stokes

number. The Level 1 equation can only be used if the density ratio exceeds the square of the

inverse of the Stokes number. Xote that for a highly viscous flow. the inverse of the Stokes

number is very large. For these flows, simplifications in the equation of particle motion are

less likely to be justifiable. The Stokes oumber, inverse of the Stokes nurnber? and the square

of the inverse of the Stokes number are presented in table (4.2) for various particle radii in

atmospheric air. This table shows that for a particle with a radius of 1.0 Pm. a density ratio

greater than 5400 is required to justi- computing the velocity using the Level 2 equation

of particle motion, and a density ratio greater than 29-million is required to use the Level L

equation.

Table 4.3: Particle trajectory data for particles in air.

4.5 Particle Trajectories in a Steady Flow

The Level 1 and 2 equations of particle motion are solved nurnerically and solution's cont-

pared for the case of uniform steady flom. in order to determine the magnitude of the effect

that the Basset history term has on the drag of the particle. The steady flow-field assunied

a constant non-zero flow in the x-direction only, given by u = (u. O? 0).

Figure (4.1) contains plots of the (a) x-direction velocity and the (b) z-direction velocity

versus time b r a particle with a radius of 10.0 pm and a density ratio of 100. The velocities

are non-dirnensionalized by the theoretically deterrnined terminal velocities and the time is

non-dimensionalized by the particle relaxation frequency . The plots display the particle

velocities as computed by the Level 1 and Level 2 equations of particle motion. Since the

total drag force is smaller, the velocities detennined from the Level 1 equation reach the

terminal velocity well before the velocity computed by the Level 2 equation. Xote that the

terminal velocities agree very well mith the theoreticdy determined d u e s .

The relative difference between the particle velocity detemined from the Level 1 (VL ' ) and

Level 2 ( V L 2 ) equations of motion can be computed by (vL1 - v ~ ' ) / v ~ ' . These values are

plotted in figure (4.2) versus time normalized by the particle relaxation frequency for the four

radii values in tables (4.1) and (4.2) and a constant density ratio of 100. As shown in figure

(4.2). the case with the largest radius produces the smallest relative difference. However.

according to the phpical reasoning from above, none of these particle cases can justifiably

use the Level 1 equation of particle motion, and only in the case when the radius is 100 pni

can the Level 2 equation be used to determine the particle's velocity. Figure (4.2) indicates

that the Basset history effect on the particle motion has a large effect ori the short-terni

niotion of the particle but becomes negligible in the long-term. It should be noted that the

dimensional time values for the four particle radii are one second for the largest particle.

R = 100.0 pm and much less for the others.

Figure 4.1: Xormalized (a) x- and (b) z-direction velocities computed by the Level 1 and

Level 2 equations of particle motion versus normalized time for R = 10.0 pm and f i = 100

in a steady flow-field.

Figure 4.2: Relative difference of the x-direction velocity versus norrnalized time for viirioiis

radii with 8 = 100 in a steady Row.

4.6 Particle Trajectories in an Unsteady Flow

The Basset history effect is a valid force only for an unsteady flow. The above study indicates

the relative magnitude of the effect, but the results are not physically realistic. Note that

only the Basset history effect on the particle velocity is determined since the acceleration of

the fluid velocity is zero. Figures (4.3) to (-4.10) compare the x- and 2-direction velocities

computed by the Levels 1, 2! and 3 equations of particle motion for particle radii of 0.1 /cm.

1.0 Pm. 10.0 prnt 100.0 pm with a constant density ratio of P = 100 released in an unsteady

Bow-field. These Bow conditions were achieved by super-imposing a turbulent flowfield. de-

termined by the Kraichnan turbulence model. ont0 the steady Bow-field from section (4.5).

The turbulent flow values were dampened to produce fluctuations in the floa of a magnitucle

of one-tenth the terminal velocities of the flow to ensure that the turbulent fluctuations did

not dominate the force effects that were being studied (the drag and buoyancy forces). 1,Vith

the density ratio specified above, the use of the Level 2 equation of particle motion is only

valid for the particle with the radius of 100.0 pm as shown by table (4.2). The Level 1

equation should not be used with this density ratio.

The tirne-step was chosen such that it vas less than the particle relaxation time 110. For

the small particles, this would require the tirne-step to be of the order of 10-8 S. The particle

velocity and the Buid velocity surrounding the particle in its trajectory was required for the

computation of the Basset history terms over the entire trajectory. This required extensive

computational effort and memory usage which restricted the total trajectory time simulated.

The drag terms impede the acceleration of the particle to the velocity of the Aow. The total

cirag for the Levei i equaciori is sriiaiier tiiari producd Liy the Lçvel 2 or Levrl 3 rqiiirtiüiis

and therefore, the transient response time should be less when the velocity is calculatecl

using the Level 1 equation. Figures (4.3) to (4.6) show that the difference between the

velocities calculated by the Levels 2 and 3 equations of particle motion are negligible for

this density ratio. This indicates that the contribution to the drag on the particle motion

from the added m a s and Buid acceleration terms are very srnall. For the case of the sniallest

particle. R = 0.1 Pm, t here is no difference in the veloci ties calculated by the Levels 1. 2 . and

3 equations of motion. This particle has a very large particle response frequency. 3. wliich

corresponds to a srnall response time to fluctuations in the Bow. Therefore. particles of this

size will attain their terminal velocity in a very short tirne (< 1.0 ps). The difference in th.

transient time between the Level 1 and the Level 2 and 3 equations of motion becornes larger

as the particle size increases. As indicated previouslg, it has been justified that the Lewl 2

quat ion of particle motion can only be used to calculate the velocity of the particle when

the particle radius is R = 100.0 p. The transient response time of the particle in tiiis case

is quite large, and depending on the magnitude and frequency of the turbulent fluctuations.

the drag terms could have an impact on dispersion models applied in atmospheric flows.

Figures (4.7) to (4.10) present the results for a similar study as discussed above. escept for

a constant density ratio of 1000. Note that for this weight, from the data presented in table

(4.2): use of the Level2 equation to determine the velocities is valid for the particles with radii

R = 10.0 pm and 100.0 Pm. However, the difference between the veloci- t ime plots for the

three equations of particle motion proved to be insignificant for the flow conditions studied.

Use of the Level 1 equation is marginally justifiable for the largest particle. Therefore.

depending on the magnitude and frequency of the turbulent fluctuations, the Basset history.

fluid acceleration, and added-mass terms cm be neglected in pollutant dispersion mode1

applications for heavy particles. The transient response time is quite long for the larger

particles: therefore, the steady drag term should be considered in atmospheric dispersion

models.

This study has shown that for large particles (R 2 100.0 pm) and heavy particles ( f i 2 1000)

that the steady drag terms (Stokes drag) has a large enough effect on the transient response

time of the particle that it might have a significant influence on the trajectory of a pollutarit

plume in an atmospheric flow. The unsteady drag terms found in the Level2 and 3 equations

of particle motion are important factors for simulations concerning extremely light-weight

particles ( P < 100) - especially if the light particles have large radii. These terms are also

more influential in highly viscous Buids or flows with short time-scales or simulation times.

1 1 .S time (ps)

Y

>x

0.4

0.2

Figure 4.3: (a) S-direction (m/s) and (b) Z-direction velocity (pm/s) versus tirne (ps) with

R = 0.1 pm and j = 100 in an unsteady Born.

O* I 1 I 1

0.5 1 1.5 2 2.5 time (ps)

-

r

-

time (ps)

tinte (ps)

Figure 4.4: (a) X-direction (mis) and (b) 2-direction velocity (pm/s) venus time (ps) with

R = 1.0 pm and p = 100 in an unsteady flow.

1 t 1 1 1

1 5 10 15 20 25 time (ms)

time (ms)

Figure 4.5: (a) S-direction (m/s) and (b) Z-direction velocity (mm/s) versus time (ms) with

R = 10.0 prn and = 100 in an unsteady flow.

I 1 I 1

Level 1

time (s)

Level 1

O 0.05 0.1 0.15 0.2 0.25 time (s)

Figure 4.6: (a) '<-direction (m/s) and (b) 2-direction velocity (m/s) versus time (s) with

R = 100.0 prn and P = 100 in an unsteady flonr.

time (jcs)

10 15 time (ps)

Figure 4.7: (a) .Y-direction (m/s) and (b) 2-direction velocity (prn/s) versus time ( p s ) Rith

R = 0.1 pm and f i = 1000 in an unsteady flow.

time (ps)

Figure 4.8: (a) .Y-direction (m/s) and (b) 2-direction velocity (pm/s) versus time (ps) with

R = 1.0 pm and ,ij = 1000 in an unsteady flow.

t I 1 I 1 O 5 10 15 20 25

time (rns)

time (ms)

Figure 4.9: (a) X-direction (rn/s) and (b) 2-direction velocity (mm/s) versus time (ms) wit.h

R = 10.0 pm and = 1000 in an unsteady fiow.

l 1 1 l 1 0.05 0.1 0.1 5 0.2 0.25

tirne (s)

Level 1, 2

I I 1 1

0.1 O. t5 time (s)

Figure 4.10: (a) X-direction (m/s) and (b) Z-direction velocity (m/s) versus time (s) n i th

R = 100.0 prn and ,ii = 1000 in an unsteady flow.

Chapter 5

Numerical Cornparison of

Atmospheric Dispersion Models

5.1 Introduction

The Gaussian Plume model. section (2.3), is a widely used and accepted model for prc-

viding quick estimates of the pollutant concentration dom-wind frorn a specified source.

The dispersion parameters required by the plume model and presented in tables (1.1). ( 1.2).

and (1.3) were determined from experimental data from the dispersion of a plume over a

Rat grass-land [II. This should indicate that the assumption of a Gaussian distribution is

valici and it is used as reference for validation by some [2]. -4 recent study by Hangan 13-1

compared dispersion results from mind tunnel tests, computational simulations (from a coni-

mercial software package), and a Gaussian Plume model for non-uniform flow conditions

influenced by building wake entrainment and topographical effects. Though the Gaiissian

Plume niodel can provide useful preliminary predictions. Hangan showed it is ineffective at

accurately modelling the concentrations in a pollutant plume, especially mhen topographical

or other disturbances are present. The computational simulations generally over-predicted

the experimental data. Pielke [3] Iisted a number of limitations that the Gaussian Plume

mode1 is based on, including: major simplification of the conservation equations governing

the atmospheric flow, its inability to represent pollutant dispersai in cornplex flows (recir-

culation, changes in mean-wind direction), and that the dispersion of pollutants is assumed

to be governed by a Gaussian distribution about the plume centre-line. The Gaussian Puff

dispersion model presents a flexible alternative to the plume mode1 which addresses sonic

of the concerns noted above. A meteorological flow-field model is required to provide at-

mospheric data to the puff model. Therefore, the puff model's ability to simulate pollutant

dispersion in complex flows is limited by the flow-field model. A Gaussian distribution of

pollutant rnaterial is still assumed.

Sirice the Gaussian Plume rnodel was derived for a release over a flat-terrain. the Gaussiari

Puff model must be able to reproduce the plume model's results to be a valid alternative. Tu

hcilitate this study an atmospheric Bow rnodel rvas constructed to simulate stable. neutrally

stable. and unstable meteorological conditions. This flow model will be described in dctail iri

section (3.2) . Ground-level concentrations produced by the Gaussian Plume and Puff niodels

Froni a point source are compared in section (5.3). This gives an indication of how well the

puff model reproduces the plume model in each of the flow conditions listed preriously.

The velocity of the particles representing the centre of the puffs are typically assurnecl to t ~ e

equal to the local flow velocity. As shown by the numerical results in section (4.5). the tran-

sient t h e it takes a particle to achieve its terminal velocity in a steady flow is insignificant

relative to the simulation time of an atmospheric dispersion model. Honever. a meteoro-

logical flow-field consists of a non-constant velocity profile due to the Row-field interaction

wit h the ground. other objects within the flow domain. and temperature differences. \*eloc-

ity gradients. temperature gradients, and other factors (surface roughness. etc.) are main

producers of atmospheric turbulence as discussed in section (1.3). Therefore. neglect ing the

effects of drag and gravity on the particles may not be physically accurate. In section (5.4).

the application of the Level 1 equation of particle motion from chapter 4 is applied in con-

junction with the Gaussian Puff model in unstable and stable flow conditions. The results

are compared to those from the conventional method of determining the velocity of the puffs.

The Gaussian Plume and Puff models are similar in that they both represent the pollutant

concentration by a Gaussian distribution. The validity of this assumption is investigated by

plotting the distribution of a number of particles a t different times in the flow and fitting

the appropriate Gaussian profile to the data. The dom-wînd, cross-wind. and verticai

distributions are determined for the simulation of 1000 particles a t various times. This

investigation is present ed in section (5.5).

CHAPTER 5. NUMERICAL COMPARISON OF ATMOSPHERIC DISPERSION MODELS 6'2

5.2 Atmospheric Flow Mode1

The construction of a simple, yet realistic, flow-field mode1 was required to accurately rep-

resent an atmospheric flow for comparing the various dispersion models. The Similarity

Theory presented in section (1.4) produced the desired vertical velocity and temperaturc

profiles. Spe~ifically~ equations (1 .il) and (1.72) were used to represent the atmospheric

901:: For ~?nçtab!e, neutrally stable, and stable atmorph~rir ronditions nwr ii Rat. grassu ter-

rain. Note that these flow conditions correspond to Pasquill stability classes B. D. and F

respectively. Turbulent flow values were found from the velocity and temperature profiles

using the method described in section (1.4). The variance of the velocity components n e r e

foiincl from equations (1.45)-(1.47).

The niean flow (down-wind) direction tvas set as the x-direction. such t hat the flow velocity is

given by u = ( ~ ( 2 ) . 0, O). Therefore. the s-direction velocity and the temperature gradients

cari be writ ten as

These equations can be integrated analytically (from z = 2, to I) to pro1

and temperature profiles, given by

;ide the

where the integration functions I, and le are

( j . 2 )

velocity

1

where z = (1 - 15:)', q, = (1 - 1 5 3 ~ ;

The Obukhov length, LI was defined by equation (1.61). The value of velocity scale is found

From ~qiiation (1.52) where the surface roughness. a. is set to 0.03 m for a grassv terrain !3!.

The temperature scale. 8. t hen defined the stability conditions of the atmosphere. Stable.

neutrally stable' and unstable atniospheric stratifications are governed by positive. zero. and

nega t ive values of B. respect ively.

Turbulent fluctuations can be derived from the mean flow-field. given by the sirnilarit? re la

tions above, by the second-moment turbulent closure model discussed in section (1.3) [LI.

The turbulent closure mode1 produces the standard velocity deviations required to produce

turbulent velocity components by the random-walk model presented in section (2.4) [XI. The turbulent fiow is a result of shear production due to the velocity gradient and buoyant

production due to the temperature gradient.

The random-walk model turbulent velocity components are derived from the Langevin

S tochastic Different ial equation [21] as shown by equation (2.19) and rewrit ten here as

h, 1"

u l ( t ) = &, ( A t ) ~ y ( t - At) + [1 - et (At)] l'bu, ci + [l - &, (A t ) ] TLu, -. (5 .3) d z

The Lagrangian auto-correlations are given by equation (2.20) for the time-step At. Srel-

lor 8~ k'amada's level 2.5 turbulent closure model is based on the pro-gnostic equation for

the turbulent kinetic energy given by equation (1.32). For convenience. equation ( 1 . X ) is

repeat ed here

üliasz [lS] uses an implicit two-cycle time-splitting finite-difference method to solve t his

three-dimensional equation. This integration technique for the time-step 6t = t("+ '1 - t(") is

given by

The finite difference operators -1,: A,, and Ar for this scheme are given by

aiid the vertical force term. F:, represents the contribution to the kinetic turbulent cncrgy

from the shear production. buoyant production. and the dissipation term

Note that the turbulent kinetic energy in the dissipative term has been linearized by

wliich provides the dissipative contribution in the 2-direction finite difference operator. .\, . The boundary conditions for the turbulent kinetic energy are given by

The shear production, buoyant production. and dissipation of turbulent kinetic energy were

defined by equations (1.33), (1.34): and (1.35). The exchange coefficients (and their non-

dimensional counter-parts) are calculated at each time-step outlined by the time-marching

method given above. The turbulent length scale, defined by equation (1.43) should a h be

re-evaluated at each tirne-step. The turbulent length scaie factor, L',, c m be determined

using simple numerical integration techniques, such as Simpson's rule.

The hIellor & kamada turbulence mode1 can be simplified greatly when using the similariry

relations to provide the meteorological wind-field. According to equations (Z.3) and (5.4)

the x-direction velocity and temperature profiles are only a function of the height 2 froni the

ground. This allows the irnplicit tirne-marching scheme outlined by equations (Z.7) to (5.1 1)

to be summarized as a single step

Non the exchange coefficients and turbulent length scales only need to be cleterniinecl at

time-step (n). The difference operator. .IL simplifies to

Though this great ly simplifies the cornputational effort involved in simulating ii tiirbiilmt

meteorological wind-field. the flow-field has become limited in its flesibility. Even thoiigh

the flow-field is three-dimensional in nature. it is represented by a quasi-one-diniensional

problem (since the tliree-dimensional turbulent kinetic energy field has been reduced to a

single vertical profile). The mean-direction of the Bow is unchangeable and any disturbances

other thari those from the turbulence produced by the flow conditions are impossible to

incorporate. This mode1 does perform well for its intended purpose and should be liniitecl

to short simulation times (one to two hours).

The boundary conditions required to find the turbulent kinetic energy profile are given by

equation (3.17) only. .in initial condition mhich satisfies the boundary conditions is set as

where e. is the amplitude of the turbulent kinetic energy which Nil1 control the magnitude

of the turbulent fluctuations and H is the height of the vertical domain.

Once the turbulent kinetic energy profile has been deterrnined, the vaxiances in the wind

velocities, au: ou, and a, can be determined from equations (1.43), (1.46), and (1.47). which

are in tum used to determine the turbulent velocity components from equation (2.19). The

turbulent Bow-field is then super-imposed onto the velocity profiles found from equation (5.3).

Figure (5.1) presents the vertical velocity profiles in the x-direction for unstable. neut rally

stable. and stable atmospheric conditions corresponding to Pasquill stability classes B. D.

and F.

Figure 5.1: Typical vertical velocity profiles in unstable. neutrally stable and stable Hotv

condit ions.

5.3 Cornparison of the Gaussian Plume and Gaussian

Puff Dispersion Models

Three test cases were designed to compare the Gaussian Plume and Puff Models in an

unstable. neutraily stable, and stable atmospheric conditions. The pollutant release is set

at a height of 10 m, from a source of 0.10 m in diameter with a release velocity of 1.00 ni/s

in the vertical direction. ;\ pollutant plume is reproduced by the Gaussian Puff rnodel by

releasing puffs of a certain m a s in time intervals such that the total emission rate of the puff

rnodel corresponds to the emission rate of the plume model. Two pu£€ cases are set to try

to reproduce a pollutant plume with an emission rate of 0.010 kg/s over a simulation tirne

of one hour. These cases are surnméu-ized in table (5.1) and were set according to i.. similar

simulation conducted by Scire et al. [2].

Table 5.1: Data for Gaussian Puff hlodels.

Case 1 Case 2

Number of puffs 250 500

Puff rnass, E (kg) 0.144 0.0'72

Release interval. At (s) 14.4 7.2

The input data required for the Gaussian Plume model, such that the atmospheric conditions

correspond to the three flow-field cases. are given in table (5.1).

Table 5.2: Input data for Gaussian Plume Models

Unstable Neutra1 Stable

Pasquill stability class

Source diameter, d (m)

Release height. h (m)

Release velocity, V, (m/s)

Release temperat ure. T, (K)

Atmospheric temperature, Ta (K)

Flow velocity at h? uh (m/s)

Temperature gradient at h, 9 (K/m)

Effective plume height, H (m)

Note that the release temperature is assumed to be equivalent to the synoptic atmospheric

temperature since neither the puff model or the flow-field model incorporates temperature

effects of the pollutant release. Therefore, the effective plume height for the Gaussian Puff

model, is dependent o d y on the rnomentum of the release as given by equation (2.14).

CHAPTER 5. NUMERICAL COMPARISON OF ATMOSPHEMC

5.3.1 Gaussian Plume Isopleths

Ground-level concentration isopleths are plotted for a Gaussian

DISPERSION MODELS 68

Plume model. Concentration

isopleths consist of lines corresponding to values of p h / Q as a function of the down-wind (x)

and cross-wind (y) distance. Equation (2.6) gives the pollutant concentration as a funct ion

of x o y: 2, and H for the Gaussian plume rnodel. This equation c m be re-written in ternis

of a functiori of the concentration in the down-wind distance and the vertical plane. r i r d ari

esponential governing the plume growth in the cross-wind ais.

The centre-plane concentration, ~ ( x , O. r: H), is given by

Equation ( 5 . 2 2 ) can be solved for the cross-wind distance y as

Frorn this equation. the cross-wind distance can be found for given concentration and clown-

wind distance values, determining the concentration isopleths. Figures ( 5 . h ) . (5.3a). and

(%la) display the ground-Ievel isopleths for the three stability cases. Sote that the meaii

wind-field is in the s-direction and that the concentrations are mirrored in the s-asis. Tlie

cross-wind dispersion of the pollutant material is much greater for the unstable Bow-field

conditions. As expected, the least amount of dispersion occurs in the stable case.

5.3.2 Gaussian Puff Mode1

The concentration of pollutant material resulting from N puffs is given by equation (2.23)

The trajectory of the centre of each puff (&,&,Zk) is modelled using equation (3.37). which

neglects al1 external forces. The particles are released in in temls defined in table (5 .1 ) .

Figures (3.2b), (5.3b), and (5.4b) present the ground-level concentration isoplet hs for t Lie

'LX puff case for unstable, neutral, and stable flow conditions. The results for the 301 puff

case are presented by figures (5 .2~) . (5.3c), and ( 5 . 4 ~ ) for the same Bow conditions.

In general. the Gaussian PUR method is capable of reproducing the results of the Gaussian

Plume model. For unstable ffow conditions, it is noted that the simulation was not long

enough to obtain the .'steady-state" assumption made by the plume model. It aas notetl

by Pielke 131 t hat mesoscale rneteorological applications should fa11 wit hin the t imc range

of one to twelve hours. The one hour that al1 three cases tvere run for mas enoiigh tinie

For the stable and neutrally stable cases to form a fdly developed plume. Therefore. tlic

puff niodel under-predicts the plume model in the down-wind dispersion for unstable flw

conditions. The lateral dispersion of pollutant in the early section of the pliirrie froni rhc

puff model over-predicts the results from the plume model. for ail three ff ow conditions. Tlic

puff triodels in the stable and neutrally stable cases produced similar lateral dispersion at

the furthest down-wind section of the plume that the plume model produced. honever. the

dom-wind dispersion was not as great. For al1 three Bow conditions. the two Gaiissian Ptiff

cases produced very similar results. which indicates that over 251 puffs do not need to be

modelled to accurately represent the pollutant plume.

Figure 3.2: Ground-level isopleths of p / Q from a source H = 10 m above the ground in

unstable atrnospheric conditions (class B stability) from (a) the Gaussian Plume Model. (b)

the Gaussian Puff Model with 231 puffs. and ( c ) the Gaussian Puff mode1 with 501 puffs a:

t = 3600 S.

Figure 3.3: Ground-level isopleths of p / Q from a source H = 10 m above the ground

in neutrally stable atmospheric conditions (class D stability) from (a) the Gaussian Plume

Model, (b) the Gaussian Puff Model with 251 puffs, and (c) the Gaussian Puff mode1 with

501 puffs at t = 3600 S.

Figure 3.4: Ground-level isopleths of XU/& from a source H = 10 m above the gound in

stable atmospheric conditions (class F stability) fIom (a) the Gaussian Plume Model. (b)

the Gaussian Puff Model with 251 pufi , and ( c ) the Gaussian Puff mode1 rrith JO1 puffs at

t = 3600 S.

CHAPTER 5. NUMERICAL COMPARISON OF ATMOSPHERIC DISPERSION MODELS 73

5.4 Particle Trajectories in an Atmospheric Flow

Trajectories of single particles were computed in an atmospheric flow usirig the equatioiis of

particle motion that neglect and include gravity and drag effects. The purpose of this study is

ro determine the impact that drag and gravity have on the distribution of pollutant particles

(or puff centres) in atmospheric dispersion modelling. Drag and gravity effects require t h

racii uf ~ i i e puliutaiit puffs have à prrdaterrniiied density and radiüs. Thc rcsülting p!ÿmcs

determined by the puff model that included the gravity and drag forces used two sets of

particle data: R = 1.0 pm with ,ij = 2000 and R = 10.0 pm with @ = 3000. The gravit? ancl

drag forces will be much greater on the second set. The magnitude and the frequency of the

turbulent fluctuations in the atmospheric Born model could be of the same order or greater

chan the magnitude of the drag and force effects on the trajectories of the puffs. If that is

the case. then the drag and gravity forces will have no or little effect on the motion of t h

puffs. The two particle data cases are set to determine how large and heavy the particles

Iiüve to be to have a n impact on the trajectories of the pollutant particles.

For each case. one particle was released every second for a simulation time of 100 sccontls

(for a total of 100 particles) in unstable and stable atmospheric conditions. Sote tliat the

Cunningham Correction. equation (3.44). and the non-linear drag law. equation (3.41). ncre

inclucled in the drag term of the particle momentum equation. Figures (5.5,). (5.5b). and

(3 .5~) display the ground-level pollutant concentrations as given by the Gaussian Puff model

for unstable condit ions. The results for stable conditions are included in figures (L6a). ( X b ) .

and (5 .6~) . Plots (5.5a) and (5.6a) present the results for the puff model that neglecteci the

effects of gravity and drag. Figures (J.5b, c) and (5.6b, c) include the effects of the drag and

buoyancy forces for the two particle data sets described previously.

The results of this study show that there was not much difference between the case where

gravity and drag were included for the first set of particle data (the lighter and smaller

particles) compared to when t hey were neglected. Counter-intuitive results occur. such

that the plume from the puff model that included drag and gravity effects actually traveled

further than the case nrithout drag and gravity. This is due to the turbulence of the Boa.

The magnitude of the turbulent fluctuations were greater than the effect of gravity and so

dominated the vertical motion of the particles. The drag has little effect on the particles

since the particle relaxation time (transient response time) was small compared to the time

scale of the turbulent fluctuations and the magnitude of the turbulent components aere

small enough such tbat the particle mas able to attain the "terminal velocity!' rather quickly

However, the results for the heavier and bigger particles as shown by figures ( 5 . 5 ~ ) and ( 5 . 6 ~ )

prove that these forces can be important. The motion of the particles is dominated bu the

drag force as the plume has only travelled a fraction of the distance as the previous cases. -4

higher rate of particle deposition also occurs due to the higher settling velocity of the heiivier

part,icle. This is evident due to the higher ground-level pollutant concentration in the centre

of the plumes.

For light (6 5 '1000) and small particles (R 5 1.0 pm) the turbulent fluctuations of the

flow doniinate the motion of the particles. and therefore the drag and gravity forces can be

neglected. However. for larger and heavier particles. these forces become important factors

in the simulation of the particle trajectories. Most dispersion modelling applications (for

example: controlled burns, forest fires. and volcanos), the pollutant particles are of various

sizes and weights. Improvements must be made with the Gaussian Puff dispersion niodel to

account for a distribution of particle characteristics to mode1 the effects of drag and gravity

forces.

puii . z ' ~ ~ . ~ ~ ~ ((3 p m *md 0-1 = 8 pm oooz = @ q q p papnpn? a n saxo3 Pelp puz  p e f i

(q) -pa?salSau am saxo3 9e1p pue QpvB (e) :suoypnoJ alqvls Japnn sgnd/sap!ued 101

CHAPTER 5 . NUMERICAL COMPARISON OF ATMOSPHERIC DISPERSION MODELS 77

5.5 Distribution of Particles in an Atmospheric Flow

This numerical experiment was designed to look a t the actual distribution of the particles

at different times in the simulation. This will provide some insight into the validity of

the assumption of the Gaussian distribution of pollutants used by the Gaussian Puff and

Plume models. The trajectories of 1000 particles were calculated in unstable and stable

f l u ~ a . Gravit? àiid Jrag fûrces on the particles rerc ncglccted such that thc turbulcn:

Huct iiat ions in the flow are the only factors effecting the dispersion of the part icles. A siniilar

s t udy incorporat ing the drag and gravity forces requires major computat ional resoiirces.

Figures (5.7) to (5.10) provide plots for the distribution of the particles in the s-direction for

simulation times of 100. 400, 700. and LOO0 seconds for unstable and stable Row conditions.

Note that 'i is the concentration (number) of particles contained a t each distance x. Sirnilar

plots have been produced for the cross-wind direction g (plots (5.11) to (5.14)) aricl tlir

vertical direction 2 (plots (5.13) to (5.18)).

Each of thcsc plots display solid dots for the concentration values at their respective distariccs

and a Gaussian distribution function fit to the data. For the down-wind and cross-~vind

distance plots. the Gaussian distribution fit is based on the function

The distribution function for the vertical direction. is taken frorn equation (5 .25) for the puH

concentration function as

The first plots a t t = 100 s for the down-wind direction distributions. show a good agree-

ment between the actual particle locations and the Gaussian distribution. However. as time

increases the distribution of' the particles do not agree with the Gaussian profile. This occurs

due to the deposition of particles that corne into contact with the ground. The use of the

Gaussian distribution function does not allow for the loss of pollutant particles to deposition.

The plots of the lateral dispersion of the particles indicate that this process is fairly ne11

represented by a Gaussian distribution throughout the entire simulation. The height of the

fitted Gaussian is somewhat exaggerated in the unstable cases, but the standard deviation

from the mean value is correct. Note, that the standard deviation from the mean is greater

for the unstable flow-field than for the stable Bow-field, as erpected.

The distribution of the pollutant particles in the vertical direction hold a Gaussian form

given by equation (5.27) about the release height for the first 400 seconds of the simulation.

For the unstable flow-field, this trend continues up until 700 seconds. but from then on the

rate of deposition of the particles is much more rapid than what equation (5 .27) dictates.

The same result occurs for the stable Hm-tield. starting at a rnuch earlier time (at

seconds). It is evident from these plots that the rate of deposition is much greater for the

stable fiow-field than for the unstable case. This happens since the turbulent production

due to buoyancy is much greater for the unstable case as a result of its inverted vertical

terriperature profile.

down-wind distance (m)

down-wind distance (m)

Figure 5.7: Distribution of particles dong the down-wind distance at t = 100 s for (a)

unstable and (b) stable flow conditions.

l I I 1

Figure 5.8: Distribution of particles along the dom-wind distance at t = 400 s for (a)

unstable and (b) stable flow conditions.

a

4000

1

0.9

0.8

O.?

0.6 X m

0.5 \

Z

0.4

0.3

1 1

500 1 O00 1 500 2000 2500 down-wind distance (m)

-

down-wind distance (m)

4

a

a

a

Da

8 a

0.2 -

0.t - -

0 - 1 I I 1

O 500 1000 1500 2000 2500 3000 3500

I 1 I I 1 i

-

-

-

-

-

-

-

-

- -

-

-

a -,

: d

a 4)

a

-

O*

O a

m , 0 - 4

4

a

O

a O

Figure 5.9: Distribution of particles dong the dom-wind distance at t = 700 s for (a)

1

0.9 -

0.8 -

0.7 -

0.6 - . e 0 x m *a

m. 0 . o . .

0.4 - 0 m . m

0.3 -

0.2 - -

4

L

O 500 1000 1500 2000 2500 3500

unstable and (b) stable tlow conditions.

4000 down-wind distance (m)

9 9

0.1 - *a - i. - 0

O O I

O 500 1000 1500 2000 2500 down-wind distance (rn)

\ m -9 -- -

O 1 1 I 1 1

O 500 1000 1500 2000 2500 3000 3500 4000 down-wind distance (m)

Figure 5.10: Distribution of particles along the down-mind distance at t = 1000 s for (a)

unstable and (b) stable flow conditions.

O 1 1 1 1 i I

1 -300 -200 -100 O 100 200 300 400 cross-wind distance (m)

cross-wind distance (m)

Figure 3.11: Distribution of particles along the cross-nrind distance at t = 100 s for (a)

unstable and (b) stable flow conditions.

cross-wind distance (m)

Figure 5.12: Distribution of particles dong the cross-wind distance at t = 400 s for (a)

1 l

0.9 - 0.8 -

0.7 -

0.6 - x m E 0.5 - z - Z

0.4 - -

0.3 - -

0.2 - -

0.1 - ..

O I 1

-250 -200 -150 -100 -50 O 50 100 150 200

unstable and (b) stable flow conditions.

250 cross-wind distance (m)

cross-wind distance (m)

cross-wind distance (m)

Figure 5.13: Distribution of particles along the cross-nrind distance at t = 700 s for (a)

unstable and (b) stable flow conditions.

cross-wind distance (m)

-vkl -200 -150 -100 -50 O 50 100 150 200 250 cross-wind distance (m)

Figure 3.14: Distribution of particles almg the cross-wind distance at t = 1000 s for (a)

unstable and (b) stable flow conditions.

vertical distance (m)

Figure 5.13: Distribution of particles dong the vertical distance at t = 100 s for (a) unstable

verticai distance (m)

and (b) stable flow conditions.

30 vertical distance (m)

Figure 5.16: Distribution of particles dong the vertical distance at t = 400 s for (a) unstable

1

-

-

-

-

0.3 -

0.2 - - 0.1 -

I 1

O 10 20 30 40 50

and (b) stable flow conditions.

60 vertical distance (m)

vertical distance (m)

Figure 5.17: Distribution of particles dong the vertical distance at t = 700 s for (a) unstable

and (b) stable flow conditions.

30 vertical distance (m)

0.2 - 0 0 - O . .a 0

0.1 - mm* - 0

O 1 1

O 1

10 20 30 40 50 60 vertical distance (m)

Figure 5.18: Distribution of particles along the vertical distance at t = 1000 s for (a) unstable

and (b) stable Bow conditions.

Chapter 6

Concluding Discussion

Tlic intention of this thesis nas to provide a review of the current dispersion riiotirllirig

tecliriicpes. and to take an in depth look into particie trajectories in an atniospheric Hoiv.

In particular. an investigation into the effect of drag and gravity on the dispersion of tlit:

pollutant material was conducted. This was accomplished by deriving and studying the

equations of motion for a pollutant particle in an unsteady Bon. and by applying drag a~ id

gravity terms to the Gaussian Puff approach to dispersion modelling in armosphcric floivs.

The three main methods of evaluating the transport and dispersion of the pollutant species

were discussed in Chapter 2. Simplification of the species conservation equation (1.17) leatl

to the I<-Diffusion Model and the Gaussian Plume Model. The &Diffusion Mode1 uses

the assurnption that the flux of the pollutant is proportional to the mean gradient and an

eschange coefficient. The mean Rrind velocities and turbulent closure data is interpolated

directly from the meteorological model. The Gaussian Plume Model assumes that the hor-

izontal and vertical specie correlations are given by a Gaussian distribution. S teadj-state

meteorological conditions are assumed, and hence it assumes that the plume has a straight

centre-line. pointing in the wind direction. Therefore. it cannot represent recirculation of

the pollutant since cornplex wind conditions are not allowed. Only one stability class can be

considered at a time. Continuous emissions are assumed and mass is conserved: t.here are no

material losses due to chemicd reaction or from deposition. This model is derived through

major simplifications and hence it is not always physically accurate. It does. however. pro-

vide a quick and simple prediction of the plume trajectory and dom-nind concentration

values.

The final method replaces the species conservation equation with a Lagrangian part icle track-

ing approach, in which the trajectories of single pollutant particles are followed. Dispersion

of the pollutant particles is modelled for a stochastic process, for which the statistical prop-

erties are deterrnined from the meteorological Aow model. Lagrangian particle dispersiori

modelling tends to be computationally expensive as the trajectories of a very large nuniber

of individual particles is required to represent a pollutant plume. However. this merhod is

ver? flexible and it can produce accurate results for complex Bows and environrnents. The

Gaussian Puff model regards the particles being followed as the centre of pollutant puffs. A

Gaussian distribution of the pollutant material is assumed around the centre of each piiff.

The velocities of the particles (or piiff centres) in the Lagrangian particle tracking approach

are typically modelled to be equal to the Bow-field velocity. However, in some situations.

the drag and gravity forces on the particles could have important effects. To fully st udy the

effect of drag ancl gavity on the dispersion of pollutant rnaterial in an atmospheric flom. a

general equation of translational particle motion was derived under unsteady flow conditions

for a rigid, spherical part icle. The result ing equation included Stokes drag, buoyancy force.

a fluid-acceleration term which accounts for the pressure of the flow-field. and higher-ordcr

driig terms (the Basset history and added-mass terms). I t was determined [rom the riiinieriral

simulations that the higher-order drag and fluid-acceleration terms have important effects

for extremely light particles ( f i < 100) and highly viscous flows. These effects ivould Iiave a

inore influential role for scenarios with short simulation times and length scales. Note thnt

the Basset History force has a large short term effect, but becomes negligible in the long

term. The Stokes drag and buoyancy force do increase the transient response time of the

particle significantly enough that these forces could influence the trajectory of a pollutant

plume modelled by Lagrangian particle dispersion techniques. depending on the Frequency

and magnitude of the turbulent fluctuations in the atmospheric flow.

Xumerical simulations were designed to study the Gaussian Puff model in more depth. The

required standard deviation parameters used by the Gaussian Plume model. despite al1 of

the assumptions and simplifications required to derive it, were determined from experimental

results over a flat-plain. Therefore, the Gaussian Puff model should reproduce the Gaussian

Plume model in such a scenario under stable, neutrally stable and unstable atmospheric

conditions. It was found that the lateral dispersion found from the puff model produced

similar results as the plume model, however, the dom-wind dispersion fkom the puff model

was not as great as from the plume model. In general, the G a w i a n Puff model reproduced

the Gaussian Plume model fairly well, without using a large number. The computatiori

time for the puff model far exceeded the relatively instantaneous plume model. Therefore.

the plume model is still a viable option when a quick plume trajectory and down-wirid

concentration prediction is required. This model can also be used in Rat-plain releases in

flow condit ions t hat can be considered steady-state (mean-wind direction and st abilitx class

mil1 not change for the time the prediction is required). The Gaussian Puff model shoiild

be used in situations that include comples flow conditions or the presence of buildings aiid

varied geography (mountains, hills. forests? bodies of water).

The effects of drag and buoyancy forces on the pollutant plume development as modelled by

the Gaussian Puff mode1 was investigated by cornputing the trajectories of the puff centres

witti equations tliat included and neglected these forces. Depending on the magnitude niid

the frequency of the turbulent fluctuations in the flow-field. it was found that these forctls

coiild have a profound effect on the dispersion of pollutant material. Therefore. al1 Gaussiari

Plume models stiould include these forces when calculating the trajectories of the puffs.

Neglection of tliese forces are valid if it is known that the weight and the size of the pollutant

particles are extremely small (R < 1.0 pm and p < 2000). such that its particle resporisc

tinie (113) is insignificant.

The assuniption of the Gaussian distribution of pollutants \.as also studied in Chapter 5 .

Drag and gravity forces were neglected such that atmospheric turbulence mas the only driving

factor in the dispersion of the particles. It was found that the Gaussian distributions matched

the results well escept that they do not account for deposition of pollutant material.

The basics of meteorological modelling, atrnospheric turbulence modelling, and boundary

1-er pararneterization were introduced in Chapter 1. The meteorological model constriicted

for the purpose of studying the dispersion models in Chapter 5 was based on the boundary

layer pararneterization. For a more detailed investigation into dispersion modelling in atmo-

spheric flows. a more cornplete meteorological flow model must be used. such as RALIS [33].

hfeteorological modelling, and the application of dispersion models to atmospheric flows. is

a major area of interest because of the desire to predict the weather and path that pollutant

releases will travel. This could aid government management of controlled burns in farmer's

fields and forests, and in the combat of forest fires. An immense amount of data is required

to initiate a mesoscale rneteorological flow-field model. Wind-profiles, temperature profiles.

humidity data, atmospheric density and pressure profiles are some of the important data

that must be known. Information on the topology and about the pollutant source must be

known as well. Determination of the initial values and boundary conditions is one of the

major difficulties that arise when conducting atmospheric simulations. Weather information

can be determined from satellite images and local radiosonde data. This data could be too

sparse. and more data is required for more accurate meteorological and pollutant dispersion

predictions. hircraft mounted instruments such as Aventech's XIMSIS-10 (Aircraft Integrat-

ed Neteorological Ueasurement Systern)' can be used to provide more detailed data [34]. h

niisture of slant-vertical and horizontal Right trajectories are flown to obtain real-tinie three

tlirnensional temperature. humidity and wind velocity profiles.

The Gaussian Puff model. and Lagrangian tracking models in general. have proved to be

good models due to their flexibility in dealing with flows containing major disturbances.

how the- interact with the meteorological Elow model. and how force effects on the particlc

can be easily included. This study h a . also s h o w that the drag and gavity forces on the

particle can have an important effect on the growth of a pollutant plume and should be

considered. However. the direct simulation of a large number of particles or puff centres

is coniputationally expensive. especially when the effects of gravity and drag forces arc

included. Improvement on the direct simulation approach has been the focus of study for a

group of scientists undertaking the National Science Foundation's computational challenge

on the simulation of the motion of particles in flowing liquids.?

The study conducted on the motion of particles was limited to purely translational motion

of rigid. spherical particles. Rotational motion and heat transfer effects could have major

influences on the particle trajectories. The buoyancy of a particle will be altered tintil the

system decays to thermal equilibrium. Therefore, the temperature of a pollutant release

dictates whether or not the resulting plume will initially rise or faIl. The initial rotation of

the pollutant particles and the effect of the torque due to external and fluid forces couic1

have an influence on trajectories of the particles (or puff centres).

-- - -

'Aven~edi Research Ine., 2700 Steeles ivenue West, Suite 202, Concord, ON L4K 3C8. 2http://www.aem.umn.edu/Solid-Liquid_F10ws/

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Appendix A

Derivation of the Space and Time

Averaged Conservation Equat ions

A. 1 Space and Time Averaged Conservation

Equat ions

This sectiou of the appendiv presents the derivations for the averaged conservation consena-

tion equations. given by equations (1.13)-(1.20). The derivation begins with the equations

of motion given by equations (1.1)-(1.5). (1.9), (1.7) and (1 3). -411 variables are espressed

in terms of a mean and fluctuating value, as demonstrated by

Denote an integration over Axl Ay, Az, and At by

w here the following hold

For each of the conservation equations, put each of the dependent variables in terms of the

values given by equation (-4.1) and integrate as per equation (A.2). Start with the continuity

APPENDIX A. DERIVATION OF THE AVERAGED CONSERVATION EQUATIONS

equation

%J - + v . ( p u ) = O, dt

d -(p+ 4') + v - ( ( p + 4')(ü + u")) = 0. &

equation (note, put = a)

where the second pressure term is usually neglected (assurning y « 1). giving

Potential temperature

APPENDIX A. DERIVATION OF THE AVERAGED CONSERVATION EQUATIONS

Similarly the nrater and species concentration conservation

The potential temperature, \+tual temperature. and ideal gas law are given by

Eqiiatio~is (-4.3)-(A.9) are equivalent to equations (1.13)-( 1.20).

A. 2 Turbulent Kinetic Energy Equation

The foilowing is a derivation of the turbulent kinetic energy equation (1 26). The mean value

of a variable $ can be written in terms of its synoptic scale atmospheric condition ou iirid

the a mesoscale deviation d'! given by

i t is assumed that the hydrostatic equation holds on the synoptic scale. Thus. the vertical

pressure gradient term on the synoptic scale is given by

The Esnor function is defined to scale the pressure gradient in terms of the virtual ccmper-

ature. pressure. and the synoptic scale pressure tenn. It is defined by

Gsing this value, the pressure gradient term c m be written as

APPENDIX A. DERIVATION OF THE AVERAGED CONSERVATION EQUATIONS

and the hydrostatic equation can be written as

lap, 9, ara --= - = -9. (A. 11) at ad The momentum equation rewritten to include the Exnor function (and ignoring the coriolis

and viscous terms) is given by

In terms of the mean and fluctuating terms defined by equation (A.1) and using the syriopric

scale and mesoscale deviation terms for the mean vales of the potential temperature aritl the

Esnor function as defined in equation (A.10), the momentum equation expands to

d -(a + u") + (ü + u") . V(ü + u") = -(Bo + 8' + B")V(iro + T' + 7") + g. dt

Csing tlie hydrostatic equation for the vertical synoptic scale pressure gradient we fiiid t hat

Substituting this relation back into the momentum equation gives

d -(a + u") + (ü + u") . V(ü + u") = dt

Xeglect mesoscale and turbulent variations in the potential temperature except for tlie ternis

incorporating the gravity effects (Boussinesq approximation): (Ot/8)g and (B"/B)g. The

moment um equat ion now simplifies to

a - (ü + ut') 4- (ü + u") . V(ü + u") = d t

Integrate equation (A. 13) as given by (-4.2). This results in

a -(ü+U") + (Üfu") -V(Ü+u") = at

APPENDIX A. DERIVATION OF THE AVERAGED CONSERVATION EQUATIONS

'low subtract equation (A. l-l) from equation (-4.13)

.\ lultiply equation (A. 15) by u" and define the turbulent kinetic energy as e = ~ u " d l - = - u'"

Now integrate this equation as shown by equation (A.2)

Equation (-4. 16) is the turbulent kinetic energy equation as given by equation ( 1.26).

Appendix B

Derivation of the Gaussian Plume

This appendix presents the derivation of the Gaussian Plume Dispersion mode1 for a pol-

lutant release from a ground source. This is accomplished by a Fourier analysis of the

I<-diffusion equation given by equation (2.2). re-written here as

31 a2 ., a2 k - = Ks, ,, + Ki; - + KZ\ 7 d t au' d t -

wit h boundary conditions

initial conditions : ~ ( x : y, z. 0) = 0, x # O? y # 0. 2 # O.

b o u n d a ~ conditions : ~ ( f m. t ) + O: V ~ ( k m . t ) + 0.

continuity : /a [ O E [OO y (x. y. 2. t ) dxdydz = E.

The forward and reverse three-dimensional Fourier transform of the

The derivative of equation (B.5) with respect to time is

species is giren by

dxdydz. (8.3)

dk,dk,dk:. (B-6)

Substitution of equation (B. 1) gives

Evaluation of the integral term in equation (B.7) for the x-component can be found as follows

The inner-integal can be solved by integrating by parts

The first

equation

bu parts

term in this equation can be neglected since the boundary condition spccifietl by

(B.3) indicates that this term is approximated zero. Integrating the second rerrri

and enforcing the same boundary conditions gives

Substitution of this result back into equation (B.8) gives

( i k J 2 y exp(-ik *x) dxdydr.

Repeating this procedure for the y- and :-componets of the species concentration results

in

- exp(-i k . x) dxdydz = (ik,)' x exp(-ik + X) d rdydz (B.10)

/f_ /P_ /or 2 esp(-i k a-x) dzdydz = (ikE)* x esp(-ik --x) dxdydz. (B.11)

Substitution of (B.9), (B.10), and (B.11) back into equation (B.7) gives

APPENDIX B. DERIVATION OF THE GAUSSIAN PLUME ~ ~ O D E L

Gising the defintion (B.5) for F, the above equation reduces to

Which can be integrated as

The initial condition defined by (B.?) the definition of F given by (B.5). and the contiriuity

equation are used to define the constant B as

F(k.0) = B = ( x , O ) e x p ( i k - x ) drdyd- ( 2 ~ p

Therefore. equation (B.12) is given by

E F(k. t ) = -

('ln)3/? esp(- h',, kZt) exp(-h,, k i t ) exp(-Iï,. - k ! t ) . -

To determine y (x. t ) . substitute equation (B. 13) into equation (B.6) and solve:

(B. 14)

The individial integrations are carried out as follows:

00

exp(- K,, k:t) exp(ik,x) dk, - - Lm e x p ( - 4 , k l t ) cos(k,x) + i sin(k,r) 1 dk,

C10

+ i /__ exp(-& k:t) sin(k,r) dk,.

Yote that the integration over infinite limits of an odd function is zero. Continuing then

gives

exp(-&, kl t) exp(ik,x) dk, - ~X~(-K~J~$!) cos(k,r) dk,

- - exp j - - ,/kr Kxz L , -ik,t, I

Repeatirig this procedure for the y- and z-components gives

Substitute equations (B. 15). (B.15). and (B.17) into equation (B.14)

This is the Gaussian Puff equation for a ground-level pollutant release. The paranieters O,.

0,. and oz represent the standard deviations of the Gaussian distribution in the x. y. and :

directions. and are given by

Appendix C

Derivat ion of the Random- Walk

Mode1

The b a i s of the Random-Walk mode1 is that the particle position is given by

where V and V" are the velocity components representing the effects due to fluid-particle

interaction (mean velocity) and turbulence (fluctuating velocity). The mean velocity coni-

ponents are produced from the solution of equation of motion given bu eqiiation (3.34). This

equation is repeated here for convenience

The solution of this equation is given by

APPENDIX C. DERIVATION OF THE RANDOM-WALK MODEL

At time step ( n ) , equation (C.3) is discretized as

-(O) solving for V

At time step (n+l). equation (C.3) is discretized as

Substituting for ? O ) gives

Therefore. equation (C.3) can be advanced in time using

~ ( n + l ) = u<") + (v'"' - dn)) esp(-3At) + vTs (1 - e s p ( - . M t ) ) (Cs-!)

The turbulent velocity components are produced from the solution of' the Langevin Differ-

ential equation [21] given by

where TL, is the Lagrangian tirne constant defined in section (1.3), the coefficient X are

coefficients to be described later, and ( ( t ) is a random-number chosen with a Gaussian

probability density function with a zero mean and a covariance of <(r)<(t) = d(t - 7) . For

sirnplicity in solving this equation, rewrite it as

where F represents the Iast term in equation (C.5) and cu = l /TL,. This equation c m bc

solved simply using the Laplace Transform rnethod, where L{VU) = v and L i t } = f

Saking the reverse transform of this equation and using the convolution integral for the

middle term on the left-hand side, equation (C.5) is solved by

V"(t) = V"(0) exp [ dz - ( ) (C.6)

Eqiiation (C.6) represents a random process with mean, fluctuation about the rnean. rari-

ance. and covariance given by

v1l1jo)vrtt(t) = ~ " t ( û ) ~ f i l ( i ) esp (5) Tt,

respectively. Yote that {(r)<(t) = S(T - s). If the turbulence is stationary. characteristics

do not change with time. then ri, = Vtll(0)V1t'(O) = V"'(t)VU'(t). Therefore. the value of X

can be set frorn the variance equation LE

Equntion (C.6) represents a Markov process. -4 hIarkov process is defined by Legg & Raupacli

[XI as a stochastic process whose behaviour only depends on values after some initial time

and not prior to that time. A Markov sequence can be constructed from the llarkov process

where Cn is a random number chosen from a Gaussian distribution and standard deviation

of 1. The coefficients a, b, and c are to be selected based on the velocity functions and

t ime-scales defined above t O give the sequence the required variance O, and int egral-t ime

scale TL,. Coefficents a and c can be selected from direct cornparison of equations (C.6) and

(C.7)

TO provide the correct variance for the b1,larkov sequence, the factor b is found from conipar-

ison of the central term of equation (C.7) and the variance of equation (C.6)

Substituting these values for the coefficients back into the Markov sequence defined by e-

quation (C.7) gives the equation for the turbulent velocity components to be used by the

Rauduiii-CValk u i Lagraugiau clisyersiuu i i idri

Appendix D

Derivation and Analysis of the

Equation of Particle Mot ion

Introduction

The equations studied in sections (3.2) and (3.3) of chapter 3 are derived in full in this

appendk. The following derivation does not include the terrns dependent on the Laplacian

of the flow-field. V2Uo. knom as the Faxen terms. Sirnply stated, the Fasen term accoiints

for the non-uniformity of the flow-field. It is assumed at the start of the derivation that the

undisturbed Aow-field is uniform? and hence these terms do not evolve from the derivation.

This assumption was done for the sake of simplicity and brevity of the derivation.

D.2 Derivation of the Equation of Particle Motion

The following development of the moment um equation for a sphencal part ide is siniilar

to that described in Mauey & Riley [24] and Landau 9r Lifshitz [SI. The flow field of an

incompressible, undist urbed fluid flow is given by u(r t ) , and O beys the follonring equat ions

of mot ion

APPENDIX D. DERIVATION AND ANALYSIS OF THE EQUATION OF PARTICLE MOTION^^^

Du P ~ E = PI (g + U - VU) = pfg + V p + ,Lv2u.

An obstacle in the flow will alter the flow field to a disturbed field denoted by v(x. t ) . For a

rigid. spherical particle with radius R located at Y ( t ) , the florv field must obey the following

equations of motion

v = V + x [X - Y(t ) ] on the sphere.

v = u as l x - Y ( t ) l + ra.

Equation (D.5) represents a no-slip condition on the sphere. which indicates that the fluicl

velocity on the sphere is equivalent to the particle velocity V(t) and the coritri but ion frorii

the particle's angular velocity R(t). From this point on- the particle will be considered to bc

non-rotating, d i ich sets the angular velocity to zero. An additional torque equation \voulti

be required to include rotating particle effects. Similarly, an additional heat equation rvould

be required to mode1 temperature difference effects. A difference in temperature betweeii

the particle and the fluid would also require that the energy consenation equation for t h

fluid must be considered as local compressibility effects could occur. The boundary condition

specified by equation (D.6) specifies that the disturbed fluid velocity and the disturbed Huicl

velocity are equiwlent far away from the particle. Equation ( D . 4 can be written in the torr11

where a, is the incompressible fluid stress tensor defined by

Then the equation of motion for a particle is given by the sum of the forces acting on the

part icle dV

m , ~ = r n , g + i o , - n d s (D-9)

where the Buid stress tensor must be evaluated over the surface of the sphere. Employing a

change of coordinates to a particle-centered frame Rn11 ease the evaluation of the Buid stress

tensor. üsing the following change of variables (note t = t )

APPENDIX D. DERIVATION AND ANALYSIS OF THE EQUATION OF PARTICLE hd0~10~113

The continuity equation becomes

The momentum equation in the new coordinate system is given by

where V = V(t) such that V V = O and V'V = O

(D. 13)

where the incompressible stress tensor is now given by

The no-slip bolindary and far-field boundary conditions become

w = V + n x r-V=O for Ir1 = R.

w = u - V as Ir1 + m.

It is convenient to separate the flow-field into two cornponents. the undisturbed Hon-field

and the disturbed flow-field. The undisturbed Aow field. Uo, corresponds to the steadp Han-

forces due to the Ruid motion, whereas the disturbed Bow, U. corresponds to the uiisteacly

Forces acting on the particle. These flow fields relate to the particle centered field by

w=U0+U, U o = u - V ? U = V - U . (D. 17)

Substituting the relation for the undisturbed flow into the original equations of motion for

an undisturbed flow, equations (D. 1) - (D.2), determines the cont inuity and moment om

equations for this flow-field

APPENDIX D. DERIVATION AND ANALYSIS OF THE EQUATION OF PARTICLE M 0 ~ 1 0 ~ 1 1 - t

where the undist urbed incompressible stress tensor is

Substituting relation (D.17) into the continuity equation (D.12), results in

duCl equiiti~zi (D.18) shûws that c~nt inüi ty must hûld separatel-,* fsr îhc distürbcd and ündis-

tiirbed cases

Espansion of the momentum equation (D. 13) gives

Subtracting the undisturbed flow momentum equation (D.19) From the above equation results

in the disturbed Rom momentum equation

where the disturbed flow incompressible stress tensor is given by

The no-slip and far-field boundary conditions are given by

w = U o + U = O for Ir1 = R1

U = -Uo = -(u - V) for Ir1 = R.

w = U o + U = u - V as Irl-tw,

U = - U o + u - V = O as Ir1 + W .

The integral term in the equation for particle motion (D.9) c m now be rvritten in terms of

the undisturbed and disturbed flow-field contributions

APPENDIX D. DERIVATION AND ANALYSE OF THE EQUATION OF PARTICLE MOTIONL~J

where the force due to the undisturbed flow, FI^^, and the force on the sphere due to the

disturbed Bow, FI^, are given by

The contribution to the Buid force from the undisturbed flow can be found quite generally.

mit hout any furt her assumptions. From Gauss's theorem, equation (D.27) can be written as

a volume integral and expanded as

assuming that the sphere is sufficiently small such that the viscous stress tensor acts iiniform

over the surface of the particle. Substitution of equation (D. 19) gives

The equation of particle motion can be summarized as

The Brst two terrns on the ri&-hand side of the equation above are the buoyancy force.

and the force due to fluid acceleration. The fluid acceleration term incorporates the effects

of the fluid stress-gradients on the particle.

The forces due to the disturbance flow created by the particle now need to be determinecl.

A scale analpis of equation (D.22) shows that in the low Reynolds number limit. defined by

the slip R e ~ o l d s number RÇV0/v « 1 and the shear Reynolds number (R2/v) (&IL) « 1.

that the advective terms in equation (D.22) may be omitted. Using the following parameters

APPENDIX D. DERIVATION AND ANALYSIS OF THE EQUATION OF PARTICLE MOTION^^^

to non-dimensionalize equation (D.22):

- t te- O - - -

U L R G= vü, = -VUo, vu = -vu. R2/v' bVo ' VO Ch

Nom non-dimensionalize equation (D.22) using the defined parameters

cri, au w,? - - - tr;, ur, tv* _ - - P X - -- - +-u,.vu+- ü ~ v ü , + ~ u * v ~ = - , ~ i l ~ + ~ R y u & R L R Pf IL

R ~ U SLultiply through by - to get cv,

Therefore, frorn the low Reynolds number assumption, the equation for the disturbance force

rcduces to

with boundary conditions still defined by equations (D.24) and (D.35). We are now rcquiretl

to find the surface stress produced by the unsteady Stokes flow. governed by ecpations

(D.21)' (D.'Z4)? (D.25). and (D.32). Note that the pressure term d l be abbreriated as p as

it will be assumed that it is understood that it is the disturbed pressure during the folioaing

development. Similar abbreviations will be made for the force and stress-censor terms.

The undisturbed flow-field is assumed to be uniform, which requires that V x Uo = O and

VnUo = O where n 2 1. In spherical coordinates, the polar avis is parallel to Uo and

therefore al1 quantities are functions of r and the polar angle 19 only [SI. The drag force (or

the force on the sphere due to the rnoving Buid) is parallel to the velocity Uo. The force

over the entire sphere, projected in the direction of Uo is given by

Therefore, it is required to find the disturbed velocity and pressure profiles about the sphere.

Taking the curl of the disturbed momentum equation (D.32) resuits in

d -(V x U) =vv2(v x U) a (D.36)

since the curl of a gradient is zero, V x Vp = O. Therefore, the curl of the disturbed fluid

velocity must satisfy the heat conduction equation. Assume the velocity is of the forrn

U = V x (V x ( f U o ) ) = exp(-iwt) V x (V x (f u,)) (D.37)

such that the velocity Uo is given by Uo = exp(-iwt)uo which indicates that it oscillates

with a frequency w. and f is a function of r only, f = f (r). Taking the curl of the Buid

velocity

since the divergence of the curl of a vector function is zero. VV + V x ( f u o ) = O. The

derivatire with respect to time and the Laplacian of V x U of equation (D.38) gives

substitution of equations (D.39) and (D.40) into (D.36) gives

i w e q + i w t ) ~ ' ( ~ x ( f ~ o ) ) + v e ~ ~ ( - i w t ) ~ . ' ( ~ x ( f u o ) ) = O

v 4 ( V x (fuO)) + y 2 ( V x ( fuO)) = O (D.4 1)

note that, for uniforrn motion? V x ( l u o ) can be expanded as

such that equation (DA) becomes

v4(v f x uo) + evyv f x u0) = 0.

These ttvo terms can be expanded as follows

V'(VP x u0) = v2v f x uo + VV f x Vu* + V f x v2uo =v2vf x uo

V4(V f x uo) = V4V f x lQ, + v3v f x Vuo + vZv f x v2uo +VV f x v3uo + V f X V4u0

= v4v f x uo

APPENDIX D. DERIVATION AND ANALYSE OF THE EQUATION OF PARTICLE hd0~10~118

where Vnuo = 0. Application of these relations with equation (D.42) results in

since uo is uniform and non-zero, the differential equation

niiist be satisfied. The first integration of this equation gives

iw v.' f + -v2 f = constant u

From the far-field boundary condition. the velocity U must be zero at infinity and so rriiist

its derivatives. The above equation contains 6th order derivatives in f which correspond co

-Ph order derivatives in the velocitl so the constant must be equal to zero. Therefore. ttiis

different i d equat ion simplifies to

The solution to this equation. in terms of V". is

exp (ikr ) V' f = -4

r

where k = @& = i ( i + 1 ) J F ' = k(i + 1)/6. This differential equation can be solvecl

as follows

set the constants a = .A/ik and b = .4B/ik

APPENDIX D. DERIVATION AND ANALYSIS OF THE EQUATION OF PARTICLE MOTION^^^

The velocity profile is dependent on the first derivative, second derivative, and the Laplacian

of f . Therefore, further integration is not necessary since only the deriwtives of f are

required to find the Buid velocity. The second derivative with respect to r and the Laplacian

of f are aven by

aexp(ikr) + aik r - - esp( ikr ) r3 ( ;:) ]

- - -2df -- t [a oxp(ikr) + aikr ~xp!!kr) - o ~ ? r ~ ! i k r ) r d r r -1

1 1

-4 aik V' f = - exp(ikr) = - exp(ikr).

r r

Note that the Ruid velocity profile. equation (D.37), can be expanded for uniforni motion

(suc11 that V -Uo = O. V x Uo. and VnUo for n 2 1) as follows

U = V x (V x ( f ~ o ) )

= v(v-(fuo)) -v2(fuo) where V x (V x (QA)) = VV . (dA) - V'(OA)

= ~ ( ( v f ) . U o + f ( V ~ u o ) ) - ~ ' ( f U o )

where V - (@A) = (VO) - A + d(V &)

d J - mhere V f = -n dr

- - d'f - n ( n - ~ ~ ) + ~ { f i ~ v u ~ + u ~ - ~ f i + f i ~ dr2 d~ (V x u o )

+LIo x (V x fi)) - v 2 ( f U o )

where V(A-B)=A-VB+B*VA+Ax(VxB)+B x (Vx A)

- d'f - ~ . V U ~ + U ~ . V + n - ~ v ~ ~ ) ~ - f i m ~ ~ dr2 d r

where A X ( V X B ) = A . ( V B ) ~ - A ~ V B

APPENDLX D. DERIVATION AND ANALYSIS OF THE EQUATION OF PARTICLE h d 0 ~ 1 0 ~ 1 2 0

where V x (#A) = Vd x A + @ V x A

+Vf + VU^)^

where V x r = O, Vr = i, and r x r = O

where v'( f uo) = (v' f )uo + 2V f VUo + f V ' U ~

for a uniform flow. this relation simplifies to

- - d2 f ,ô(n uo) - qn uo) - uo} - (v2/) UO. dr-

Substitution of equations (D.45). (D.46). and (D.47) into (DA) gives the velocity profile in

ternis of the unknown coefficients a and 6

1 aik - [a(r - $) exp(ikr) + h (ii(ii uo) - uo) - - rxp(ikrju0 r3 r

The boundary condition a t the surface of the sphere, r = R? requires that the disturbed

Bow-field is equivalent to the undisturbed flow-field, U = Uo = uo el^(-iwt).

aik 3 3 1 uo = -exp( ikR) -)fi(fi-u0) 1----

R - ( ikR k2R2

APPENDIX D. DERIVATION AND ANALYSE OF THE EQUATION OF PARTICLE ~ I O T I O N ~ ? ~

This rnust hold for al1 fi, therefore al1 coefficients of uo and îi(îi. uo) must vanish:

1 = - -exp(zkR) 1---- b [R ( ~ k r k2r2

O = [ g e x p ( i k R ) ( l - - akr - - k2r2 ) ] a + R 3 . 36

These conditions provide two equations for the two unknown coefficients a ancl b. Solution

of these two equations give

Therefore. substituting equations the values of a and b given by equations (D.53) and (D.S-4)

into equation (D.49) gives the final solution for the disturbed fluid velocity profile

The polar and axial velocity components are required to determine the stress tensors defiricd

by equations (D.3-L) and (D.35). The polar velocity component is found by taking the dot-

product of the velocity with the unit-vector in the polar auis. fi:

- - 1 1 3 3 ( k - ( 1 - - tkr - -) k2r2 - (1 - - tkr - -)]

The axial velocity component can be found by taking the magnitude of the cross-product

between the velocity U and the polar unit vector B :

. ~PPENDIX D. DERIVATION AND ANALYSE OF THE EQUATION OF PARTICLE ? V ~ O T I O N ~ ' ~ -

The derivative of U,. with respect to r is

Evaluation of ttiis derivative at r = R gives

The derivative of Li, with respect to 0 is

b.

Evaluation of this derivative at r = R gives

The u i a l velocity component. fi, evaluated at

The derivative of the asial velocity component. &, with respect to r is

i3L; - = 3 3 --Uosin9 - 1----

R - -esp ( i k ( r - ~ ) ) ( 2 - ikr - - - - dr 3 d [::( ikR k2R2) r2

Evaluation of this derivative at r = R gives

- 3 Ua sin O - - 2 R

(1 - i k ~ ) .

APPENDIX D. DERIVATION AND ANALYSIS OF THE EQUATION OF PARTICLE MOTION^^^

substituting (D.58) into (D.34) gives

substituting (D.59), (D.60), and (D.61) into (D.35) gives

&sin0 3Uosinû C&, sin O + - R ( 1 - i k ~ ) +

The pressure profile about the sphere can be found by substituting the velocity. U. back

into the momentum equation (D.32). The momentum can be re-arranged as

Re-ivrite the velocity profile (D.37) as

v = V x ( V x ( f U o ) )

= vo (f U,) - v'( f u,) = V V ~ ( f U o ) - ~ o ~ - f - 2 V / V U o - jv2uo = V V . (fUo) - u0v2f

where VUo and V2Uo = O. Çubstituting this result into equation (D.64) gives

iw duo where from (D.42) v4f = --v2f? and - = -iwUo

v d t

Therefore. the pressure profile is found by integrating the equation above

APPENDIX D. DERIVATION AND ANALYSIS OF THE EQUATION OF PARTICLE MOTION^^^

Note that the first term in equation (D.65) can be evaluated as follows

aik 1 = -Uo fi - (; - ik) exp(ikr)

r

- 3 R i k 1 - - ik esp ( i k ( r - R)) - uOf i ---( 2 i k r r ) 3 R 1

= Ua fi -- (- - i i ) exp ( i k ( r - R ) ) 'Zr r

The second term. involving the time derivative. can be evaluated as

3 1 1 exp ik ( r - R) ) + !-(l- - 3 - -)] 3 = ~ ~ . i i i w ~ [ ~ ( - + - ) ikr k2r2 ( 2 r" I k ~ p ~ z

iwR 3 3 R2 3 3 = & c o s D - [ ( - + - ) e x p ( i k ( ~ - R ) ) + - & - ~ ~ ) ] . 2 ikr k2r2

Substituting equations (D.66) and (D.67) back into (D.65) gives

mhere dZ = 2u/u is h o n m as the depth penetration of the disturbed motion. I t should be

noted that the magnitude of the penetration decreases as the frequency of the oscillations

APPENDIX D. DERIVATION AND ANALYSIS OF THE EQUATION OF PARTICLE MOTION^.)^

increase. Evaluating equation (D.68) on the surface of the spherical particle r = R gives

The force on the sphere due to the interaction with the fluid can now be found from equation

(D.33) from substitution of equations (D.62), (D.63), and (D.69)

ipR = ( - p. COS 8 - a (1 - i k ~ ) (cor2 8 + sin' 8) + - L; COS' 19 R d2

4 where ji$osBdS-0. d S = 4 . ~ ~ . and cos28dS=-TR' f 3

where k = I ( i + l ) /d

duo - -iwUo where - - d t

This represents the force of the sphere on the fluid. The drag force is defined as the force

of the fluid on the sphere, and therefore is the negative of equation (D.70). Note that if

the frequency w is set to zero. then the force simplifies to F = G?ipR&, which is known as

Stokes Drag. This equation can be rnanipulated to remove the explicit dependence on the

frequency of the oscillating flow. Rearranging equation (D.70) and setting it as the drag

APPENDIX D. DERIVATION AND ANALYSIS OF THE EQUATION OF PARTICLE MOTION^^^

force gives

The velocity LÏo can be represented as a Fourier integral:

Thus. the drag on the sphere can be written as

3 du, &(1+ i) du, F = +3---

R JiS d t 1 1 du,

R exp(-zwt)dw 1

The integral term cm be solved as follows

Oc 1 du, 1 dw esp(-iwt)dw = ? ( l + i ) l --

,/G dt esp(-iwt)dw

- - (1 + 2) 1.. 1.. 1 duo ( r ) exp ( - iw (t - r)) dudr

7r -oJ O fi dt

APPENDIX D. DERIVATION AND ANALYSIS OF THE EQUATION OF PARTICLE MOTION^'^^

(r) exp ( - iw(t - r ) ) d w d r 7r

( r ) exp ( i w ( r - t ) ) dwdr 1 = - - exp ( - iw( t - r ) ) d d r +

7r

lm b- /- -!- esp ( i w j i - t ) ) d d r dt O \/55

For simplification. set a = t - r and b = r - t. and note that coniples esponents can bc

espandecl as

Csing these parameters and relations gives

( i ) d& /= 1 ( esp(- iwt)dw = - - - I 1

-cm dt O J7 cos(nu) - i s i n ( n z ) ) c l d r

These integral terms can be ewluated by the following Fourier cosine and sine transforms

Substitut ing these into the above integration

APPENDE D. DERIVATION AND ANALYSE OF THE EQUATION OF PARTICLE MOTION^.)^

where (1 + i) (1 - i) = 2 and (1 + i) (1 + i) = 2. Taking the real part of t his integral

substituting back into the force equation iD.71 j gives che drag Force on ciie spiiere

Equation (D.72) does not include the Faxen terms that were used by .LIauey Sr Riley [24]

and others. These terms are dependent on the Laplacian of the flow-field such that V2Uo

does riot equal to zero. Simply stated. it accounts for the non-uniformity of the flow-field.

It was rtssurried at the start of this derivation that the undisturbed Aow-field was indeed

xxen ternis arc uniform. and hence these terms did not evolve from the derivation. The F. grn~ra l ly small when compared with the other terms and so are usually neglected. as will be

stioan in the nest section. In vector form. with the Faxen terms included. and substituting

for Uo = u - V. gives the final form of the disturbed flow-field force on the sphere

- 1 FIU - (V - u[Y(t ) , t ) ] - -a'v2u/ ) - p z - 10 Y(t )

The three terrns on the right-hand side of the above equation are known as the force due

to added m a s ' the Stokes drag force, and the Basset history force. These terms have been

derived using the low slip and shear Reynolds numbers approximation (Re, < 1) and so

they are not valid in higher Bow regimes.

The added mass term represents the virtual mass added to the particle since the acceleration

of the particle requires the surrounding fluid to accelerate. The volume of the added mass

is equal to half of the volume of the particle.

L The Basset history force depends on the past particle motion, weighted by the kernel ( t -7) i.

Where ( t -T ) represents the elapsed time since the past acceleration. It acts as an augmented

APPENDIX D. DERIVATION AND ANALYSIS OF THE EQUATION OF PARTICLE MOTION^^^

viscous drag and depends on both the viscosity of the fluid, the acceleration of the particle.

and the acceleration of the fluid.

Substituting the result for the disturbed flow-field force contribution a s given by equation

(D.73) into equation (D.31) gives the final form of the equation of particle motion (or the

momentum equation of the particle) for a rigid, spherical, non-rotating, and constant tem-

perat ure part ide

The position of the particle is found by integrating the velocity over time

D.3 Scale Analysis and Simplified Models

Following Mei et al [XI, a scale analysis of the particle equation of motion will provitle

information on which of the terms are the most dominant. and will allow for simplifications

of the equation of motion for certain conditions. -1 steady state analysis of the eqiiation of

particle motion will derive the terminal velocity. VT, of the particle.

APPENDIX D. DERIVATION AND ANALYSIS OF

where = p p / p f is the non-dimensional density factor. and VTs is the terminal settling

velocity given by

Reliiring the particle velocity in terms of the terminal settling velocity and a Ructuating

velocity. v(l). by V(t) = VT, + v(t) and substituting into equation (D.74) gives

where $(I+,) = O and V, = i d 3 .

APPENDIX D. DERIVATION AND ANALYSE OF THE EQUATION OF PARTICLE k10~10~131

now substituting for the terminal settling velocity from equation (D .ï6)

Therefore. the equation of particle motion in terms of the fluctuating velocity v. is

1 2 2 + L ~ ( U - v + -R V uIY,,, ' f i dt 10

Equation ( D . 7 can be non-dimensionalized by introducing the follorving parameters

Conducting the non-dimensionalization term-by-term:

Recombining these tenns and arranging such that the terms involving the particle velocity

are on the left hand-side, and those involving the fluid velocity are on the right hand-side

APPENDIX D. DERIVATION AND ANALYSE OF THE EQUATION OF PARTICLE N ~ O T I O N ~ ~ ~

gives

9 v Divide the equation by 5 7 ~ 0 - pR-

Finally, the non-dimensional equation of particle motion is given by

where the non-dimensional nurnbers are

APPENDIX D. DERIVATION AND ANALYSIS OF THE EQUATZON OF PARTICLE MOTION^^^

where equations (D.8U), (DM) , and (D.82) represent the Stokes number, a non-dimensional

tirne-scale, and a non-dimensional length-scale. The parameter wo represents a typical fre-

quency of the fluctuations in the flow.

The Stokes number is a product of and the diffusive scale, R2/v. It represents the ratio

of the frequency of the turbulent fluctuations in the flow to the frequency of the tiscoiis

damping.

The non-dimensional tirne-scale. 6. is cornrnonly referred to as the particle relaxation tirne

[Tl. The $ - factor is due to the added mass contribution to the equation of motion.

The dimensionless Length scale is the ratio of the size of the sphere to the typical length of

the fiiictuations in the flow. It is required that the sphere size rnust be srndl relative to clie

lengtti scale such that t = Rko zz O if the Fasen terms are to be neglected. If this conditioii

is met. the non-dimensional equation of particle motion (D.79) can be siniplified as

The factor 3 is proportional to E - ~ ? so that the particle acceleration term on the left-hard

side of the equation of particle motion is of the order of f . In the low Reynolds nuniber

approximation, the Stokes number is required to be much less than one. E' << 1. It can

be seen from equation (D.83) that the particle motion is dominated by the drag terms of

0(1)? which also includes the buoyancy effects. The Basset history forces are of O(?). and

the fluid acceleration and added mass are both O(?). Therefore. for light particles. suçli

chat 6 = $ 5 0 ( i ) , al1 of the terms in equation (D.83) must be included in the solution

of the particie velocity. However, for the case of heaw particles. such that the density

ratio is of order E- '? then the particle acceleration term is of O(<). Therefore. for hewy

particles: equation (D.83) can be solved for O(& mhich means that the added mass ancl

fluid acceleration terms can be neglected and the equation of particle motion simplifies ro

which contains only the buoyancy effects, Stokes drag, and the Basset history integrals.

For even heavier particles, with a density ratio of or greater: al1 of the term in

equation (D.84) may be neglected other than the particle acceleration term and the Stokes

APPENDIX D. DERIVATION AND ANALYSE OF THE EQUATION OF PARTICLE b 1 0 ~ 1 0 ~ 1 3 4

drag/buoyancy terms.

Yote in the last two mode1 equations, (D.84) and (D.85). the factor in the particle accel-

eration term may also be neglected without any further loss in accuracy.

Appendix E

Numerical Modelling of the Basset

History Term

The particle Basset history term is modelled as

The fluid Basset history term is solved by

where the time derivative of the Buid velocity following the rnoving sphere is given by