Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue8.2000:

Adaptive hierarchical grid model of water-borne pollutant dispersion

A.G.L. Borthwick

a,*

, R.D. Marchant

a,b

, G.J.M. Copeland

b

a

Department of Engineering Science, Oxford University, Parks Road, Oxford OX1 3PJ, UK b

Department of Civil Engineering, Strathclyde University, 107 Rottenrow, Glasgow G4 0NG, UK

Abstract

Water pollution by industrial and agricultural waste is an increasingly major public health issue. It is therefore important for water engineers and managers to be able to predict accurately the local behaviour of water-borne pollutants. This paper describes the novel and ecient coupling of dynamically adaptive hierarchical grids with standard solvers of the advection±di€usion equation. Adaptive quadtree grids are able to focus on regions of interest such as pollutant fronts, while retaining economy in the total number of grid elements through selective grid re®nement. Advection is treated using Lagrangian particle tracking. Di€usion is solved separately using two grid-based methods; one is by explicit ®nite di€erences, the other a di€usion-velocity approach. Results are given in two dimensions for pure di€usion of an initially Gaussian plume, advection±di€usion of the Gaussian plume in the rotating ¯ow ®eld of a forced vortex, and the transport of species in a rectangular channel with side wall boundary layers. Close agreement is achieved with analytical solutions of the advection±di€usion equation and simulations from a Lagrangian random walk model. An application to Sepetiba Bay, Brazil is included to demonstrate the method with complex ¯ows and topography. Ó _{2000 Elsevier}

Science Ltd. All rights reserved.

1. Introduction

Water pollution has long been recognised as of ex-treme importance as a public health issue. For example, the Victorian water supply and sewerage system in London was constructed as a direct consequence of the cholera epidemic in 1854 centred at the Broad Street Pump. In the late 20th century, water pollution from industrial and agricultural e‚uent a€ects all types of water source, ranging from groundwater aquifers to oceans. Examples include the dumping of sludge and other toxic wastes at sea, oil spills from ships, the dis-charge of low-grade radioactive waste from nuclear power stations into coastal waters, runo€ of agricultural chemicals (fertilisers and pesticides) into watercourses, municipal and industrial e‚uent discharge into rivers from outfalls, and algal blooms in shallow lakes and reservoirs.

Accurate, ecient modelling of water-borne pollution is essential for an understanding of the dispersion of industrial and agricultural materials that enter overland or groundwater ¯ows. In general, water-borne pollution commences with a highly concentrated and possibly sudden discharge of toxic material. Inert pollutants are

advected by the ¯ow velocities and di€used by turbulent mixing and, to a lesser extent, molecular di€usion. The termdispersion is commonly used to describe pollutant spreading. For typical watercourses, the ¯ow is fully turbulent and mixing due to eddies is several orders of magnitude larger than mixing by molecular di€usion. It is vital that any model of pollution transport should be able to simulate the localised features such as occur at pollutant fronts or in oscillating ¯ows. It is to this end that the work presented here is directed.

Mathematical modelling of water-borne pollutant transport is usually based on the governing ¯ow equa-tions (mass and momentum conservation) coupled with a species transport equation which linearly combines advection, di€usion and source/sink terms. Various nu-merical solution techniques have been devised for dis-cretising and solving the species transport equation. One important approach is Lagrangian particle tracking whereby the continuous mass of pollutant is discretised into a prescribed set of masslets, each of which is ad-vected with the ¯ow velocity and then di€used by further particle motions (in keeping as far as possible with the mathematical solution of the pure di€usion equation). For example, Jozsa [16] applied Gaussian random walks to each particle, whereas Addison [1] employed a frac-tional Brownian motion approach in conjunction with white noise. Lagrangian techniques have the advantage www.elsevier.com/locate/advwatres

*

Corresponding author.

that the advection process is correctly modelled, but the drawback that a very large number of masslets is re-quired to deal with the di€usion step accurately. It should also be noted that it is necessary to couple the masslet motions to the ¯ow solver, which is invariably undertaken in an Eulerian frame of reference. An in-terpolated velocity ®eld has to be used which has a smoothing e€ect on the solution. Another approach is to solve the advection±di€usion equation entirely on a ®xed Eulerian grid. In this case, the di€usion calculation is relatively straightforward, but great care has to be taken with the ®rst derivative advection terms to avoid instability or numerical dissipation. For example, Ek-ebjñrg and Justesen [9] used a cartesian grid-based ex-plicit third-order accurate ®nite-di€erence method. Borthwick and Kaar [3] solved the curvilinear species equation on boundary-®tted grids with the donor cell method for advection.

An attractive option is to use Lagrangian particle tracking or the method of characteristics for advection (e.g., [6]), and then to solve di€usion on an Eulerian grid. Holly and Usseglio-Polatera [15] employed ®nite di€erences for di€usion and a high-order method of characteristics for advection. Karpik and Crockett [18] combined ®nite volumes on a curvilinear grid for di€u-sion with method of characteristics for advection. Bru-neau et al. [4] have reviewed use of the method of characteristics as opposed to ®nite di€erences for con-vection dominated problems. For mixed Lagrangian± Eulerian schemes, it is necessary to use interpolations when exchanging species information between the grid nodes and the masslets. High grid resolution may be extremely important when simulating di€usion. Pollu-tant fronts are often sharp and the transportive motions are tied into the local ¯ow ®eld, which may vary greatly in scale. It is also possible for advecting pollutants to exhibit chaotic trajectories because of time-dependent ¯ow ®elds. An example where the local ¯ow ®eld is ex-tremely important is at the end of an outfall where ef-¯uent is discharged into tidal receiving waters.

Anastasiou and Chan [2] used non-adapted unstruc-tured triangular grids generated using the advancing front technique in demonstrating the ability of a ®nite-volume scheme to model the linear advection of steep fronts. Gaspar et al. [12] used hierarchical, but non-adaptive, multi-grid methods for di€usion, with particle tracking for advection. The foregoing references show that unstructured and hierarchical grids are well suited to local mesh re®nement.

This paper describes an adaptive hierarchical grid-based numerical scheme for predicting localised water-borne pollutant transport in two-dimensions. The grid generator utilises a novel approach whereby grid regu-larisation is undertaken eciently in a single sweep. This permits dynamic coupling between the adaptive quad-tree grid and the advection±di€usion solver. In the

present work, the prescribed initial pollutant mass is discretised into a set of masslets and the ¯ow domain rescaled within the unit square. The solution process is as follows: (i) move the masslets in a Lagrangian manner according to the ¯ow velocity components, (ii) create a re®ned hierarchical grid about the masslets, (iii) calcu-late the concentration at the centre of each cell, (iv) solve the di€usion equation for each cell either by explicit ®-nite di€erences or using a di€usion-velocity approach and (v) repeat the calculations from (i). Results are presented for standard benchmark tests (for which an-alytical and alternative numerical solutions are avail-able), and the model is found to be accurate and computationally ecient. An application to a real lo-cation (Sepetiba Bay, Brazil) is also presented, which demonstrates the ability of the model to represent complex topography and an evolving plume from a continuous discharge.

2. Hierarchical grids

In order to achieve local grid densities which relate properly to evolving pollutant and ¯ow features, it is necessary to use either unstructured or hierarchical adaptive grids. Hierarchical grids involve recursive spatial decomposition and represent a sensible com-promise between control of local mesh re®nement, speed of generation and array storage requirements. Quadtrees have been applied to two-dimensional domains in image processing (e.g., [20]) and increasingly in computational ¯uid dynamics (e.g., [8,12,13,17,22]). Here, a square quadtree grid generator has been selected because of its robustness and relative ease of generation in comparison with other unstructured grid generation techniques.

The quadtree grid generation method is summarised below:

(i) create a set of seeding points according to the dis-cretised pollutant distribution and ¯ow boundaries; (ii) rescale the ¯ow domain to ®t within the unit square (i.e., grid level 0);

(iii) input the subdivision level of the ®nest grid,m; (iv) divide the square into four panels (i.e., grid re-®nement level 1);

level, the local reference number is determined from which quadrant the panel lies in as follows: north-westˆ11; north-eastˆ21; south-westˆ12; and south-eastˆ22. Trailing zeros are then added to the cell identi®cation number to ensure commonality of length. In order to facilitate neighbour ®nding, pointers to the cell parent and its children, if any, are stored with the cell identi®cation number. For any cell, its level is equal to the number of pairs of non-zero digits in the identi-®cation number. The cell-centre co-ordinates are calcu-lated from

xˆ0:5‡X

l

mˆ1

1

2m‡1…2imÿ3†; im2O …1a†

and

yˆ0:5ÿX

l

mˆ1

1

2m‡1…2imÿ3†; im2E; …1b†

wherelis the cell level, andimare successive odd (O) or

even (E) digits of the cell identi®cation number. In the above system, the co-ordinates start from the south-west corner of the unit square root panel.

Grid regularisation and neighbour ®nding are un-dertaken together in strict order, starting from the smallest cells (i.e., highest level). For any object cell, its identi®cation number is manipulated to give the identi-®cation numbers of eight hypothetical (same size) ad-jacent and corner neighbour cells. Full details of the cell number manipulations are given by Greaves and Borthwick [14], and so only a brief summary follows. Cell number manipulation initially involves removing trailing zeros, and splitting the remaining non-zero in-tegers into odd and even strings as before. From the quadtree structure, one of the east or west (and north or south) neighbours must have the same parent as the object cell. This neighbour will hereafter be referred to as the sister cell, and by de®nition, each cell will have one east or west lateral, one north or south lateral, and one diagonal sister neighbour. The other neighbours are more distantly related in the quadtree, entailing a search upwards in the tree via parents of parents, etc., and then downwards via an aunt or great aunt, etc., until the more distant cousin neighbour is located. In practice, hypothetical sister neighbour numbers are obtained by converting the ®nal digit inOorEfrom 1 to 2 (or vice versa), recombining the new and oldOandEdigits, and adding back the trailing zeros. It should be noted that the initial conversion is equivalent to a re¯ection in the central axis of the parent cell. Each cousin neighbour is located by a direct re¯ection in one of the central axes of its nearest common ancestor (i.e., the (lowest-level) an-cestor shared by the cell and its neighbour). Due to the quadtree structure, both the object and neighbouring cousin cells border this axis, while occupying separate quadrants of the nearest common ancestor. In

imple-mentation, cousins are identi®ed as follows. Retain the O and E digits of the nearest common ancestor (e.g., grandparent in the case of ®rst cousins). Swap the next integer (1 for 2, or vice versa) in either, or both, of theO orEdigits. Set the remaining digits to zero. Recombine to form the cousin cell identi®cation number. Sister and cousin cell identi®cations are illustrated in the example application given below.

In practice, the quadtree is searched for the existence of each hypothetical neighbour cell because a same size neighbour cannot exist if an undivided lower-level cell is encountered at the end of the tree path. If the tree search fails for a particular hypothetical neighbour cell, then the following strategy is adopted. The tree is descended from the nearest common ancestor using pointers to children until a child cell identi®cation number is found, which is the same as that of an existing cell. This de-termines the identi®cation number of the neighbour cell. If it is more than four times the size of the object cell, the neighbour cell is further subdivided recursively and the new cells numbered accordingly. Hence, any new neighbour cells created are always one level lower than the object cell, and are therefore considered later with cells at the next (lower) level. This has the major (and novel) advantage that regularisation spreads out from regions of high cell density, and is therefore completed in a single sweep.

The following example illustrates the relationship between the quadtree grids and the integer tree. Fig. 1 depicts the non-regularised level 4 quadtree grid that would be created about a pair of seeding points located at (0.3, 0.3) and (0.35, 0.35) within the unit square. The associated tree structure is shown in Fig. 2. Cell infor-mation is held in the following arrays:

To illustrate the neighbour ®nding techniques, con-sider cell 10 in Fig. 3, which has identi®cation number 12211221. The odd and even digits are Oˆ …1;2;1;2† and Eˆ …2;1;2;1†. After conversion, we have

Onewˆ …1;2;1;1† and Enewˆ …2;1;2;2†. Combining

Onew with E, gives 12211211 corresponding to the

identi®cation number of cell 9, which is the west sister of cell 10. Similarly,Ocombined withEnewgives 12211222,

which refers to the south sister cell 12. Now return to cell 10 with identi®cation number 12211221. Its parent and grandparent are cells 8 and 5 with identi®cation numbers 12211200 and 12210000, respectively. First, cousins are obtained by integer manipulation of the ®nal two digits of the parent cell to give 12211100 (i.e., north

cell 6), 12212200 (i.e., east cell 13) and 12212100 (i.e., north-west cell 7). Second, cousins are obtained using the grandparent cell number and again altering the last pair of digits to obtain 1211 (cell 4), 1222 (cell 15) and 1212 (cell 14).

After regularisation, the quadtree grid and integer tree are as indicated in Figs. 3 and 4 with the cell in-formation arrays becoming:

Cell number

Identi®cation number

Parent pointer

Child pointers

0 00000000 ± 1 2 3 16

1 11000000 0 ± ± ± ±

2 21000000 0 ± ± ± ±

3 12000000 0 4 5 14 15

4 12110000 3 ± ± ± ±

5 12210000 3 6 7 8 13

6 12211100 5 ± ± ± ±

7 12212100 5 ± ± ± ±

8 12211200 5 9 10 11 12

9 12211211 8 ± ± ± ±

10 12211221 8 ± ± ± ±

11 12211212 8 ± ± ± ±

12 12211222 8 ± ± ± ±

13 12212200 5 ± ± ± ±

14 12120000 3 ± ± ± ±

15 12220000 3 ± ± ± ±

16 22000000 0 ± ± ± ±

Cell number

Identi®cation number

Parent pointer

Child pointers

0 00000000 ± 1 2 3 16

1 11000000 0 29 30 31 32

2 21000000 0 37 38 39 40

3 12000000 0 4 5 14 15

4 12110000 3 17 18 19 20

Fig. 3. Regularised quadtree grid.

Fig. 4. Tree structure for regularised grid. Fig. 2. Tree structure for non-regularised grid.

Fig. 5 shows the quadtree grid generated about 3143 boundary seeding points, which represent a river bifur-cation. The grid has a maximum level of 9 and is com-posed of 9220 panels. The grid is identical to that produced by Yiu et al. [23].

3. Governing equations

In vector form, the Fickian species transport equation governing the spatial and temporal distribution of pol-lutant concentration c at any point in a continuous medium is

o_c

o_t‡urcÿ r…Drc† ˆs …2†

in whichtis time,u the ¯ow velocity vector,Dthe dif-fusivity tensor, s a source term, and r is the spatial

gradient operator. In two dimensions, assumingDto be the molecular di€usivity, which is constant in space and time,

where uand vare the velocity components in the hori-zontal cartesian-coordinate directions,xandy. The two-dimensional species equation is subject to the following boundary conditions. At solid walls, there is no mass ¯ux and so the concentration gradient is zero from FickÕs law. Thus,

open boundaries, it is assumed that pollutant advects across the boundary at the ¯ow velocity.

Free surface ¯ows which transport water-borne pol-lutants are almost invariably turbulent. It is usual to de®ne the instantaneous concentration and instanta-neous velocity vector as

cˆc‡c0 and uˆu‡u0; …5†

where the overbar and prime denote temporal mean and ¯uctuating components. Time-averaging the species transport equation, we obtain

o_c

o_t‡urcÿ r Drc

ÿu0_c0_{ˆs: …6†}

Taking the turbulent concentration ¯ux to be analogous to Reynolds stresses in the momentum equations, we may assume

ÿu0_c0_{ˆerc; …7†}

where e is a turbulent di€usivity coecient. Turbulent ¯uxes are much larger than molecular ¯uxes, and so

o_c

o_t‡urcÿ r erc

ˆs: …8†

For nearly horizontal ¯ows in shallow water, where there are no ¯uxes through the bed or free surface, the depth-averaged form of Eq. (8) is

o_C

o_t ‡UrCÿ 1

qHr…qHerC† ˆS …9†

Uˆ 1

Similarly the depth-averaged source term is

Sˆ 1

H

Z f

ÿd

sdz: …12†

Writing out Eq. (9) assuming isotropic turbulent di€u-sivity, we have

It should be noted that a more complicated discretisa-tion scheme than presented in this paper would be needed to deal with the additional second order cross derivative terms encountered for anisotropice, which is particularly important in groundwater ¯ow. For uni-forme, Eq. (13) reduces to

For uniform ¯uid density and negligible variation in total depth, further simpli®cation gives

o_C

which is exactly equivalent to Eq. (3), but with depth-averaged instead of instantaneous two-dimensional quantities. For depth-averaged pollutant transport, the solid wall boundary conditions are

o_C

o_x ˆ0 at xˆxB and o_C

o_y ˆ0 aty ˆyB: …16†

4. Numerical solver

In a similar manner to Chorin_Õs [5] operator splitting method, the advection and di€usion processes are treated separately. The advection step is undertaken using Lagrangian particle tracking. The new particle positions are then used as seeding points, about which a regularised quadtree grid is generated. Cell-centre con-centrations are determined from the numbers of mass-lets in each cell times their individual masses divided by the mass of water in the cell. Two methods are imple-mented to model the di€usion step: a predictor±correc-tor ®nite-di€erence scheme and a di€usion-velocity approach. In the former, the concentration values are updated, and a new distribution of masslets created Fig. 5. Example quadtree grid for river geometry.

according to the nodal concentrations, ready for the next advection step. In the latter scheme, concentration gradients are computed for each cell and interpolation used to compute the di€usion-velocity components. The masslets are then moved again in a Lagrangian sense by the di€usion velocities to new positions, ready for the next advection step.

4.1. Advection

Initially, the total mass of pollutant, M, is divided equally amongst a prescribed integer number of mass-lets,N. These are distributed in proportion to the initial concentration pro®le, and a grid generated with the masslets as seeding points. Cells in regions with non-zero concentrations but which contain no masslets are each assigned a single masslet. For each cell, the mass of each masslet, mc, is then revised so that mcˆMc=Nc

where Mc is the cellÕs pollutant mass and Nc is the

number of masslets in the cell. Once the number of masslets per cell has been determined, their starting positions within each cell are speci®ed by random dis-tribution. The masslets serve two purposes: one to control mesh density; the other to enable advection to be calculated by Lagrangian particle tracking.

The advection step is accomplished by moving masslets with the ¯ow velocity components such that

dx

dt ˆU and

dy

dt ˆV: …17†

Using the predictor±corrector scheme, this becomes

xn‡1_k ˆxn

time tˆnD_t. The corresponding ¯ow velocity compo-nents are (Uk;Vk), and are either interpolated from an

underlying grid in an engineering application or pre-scribed analytically for the validation tests. After the masslets have been advected, their new positions are used as seeding point locations for the construction of a new grid. By numerical experimentation, it was found that sensible results were achieved by setting the panel subdivision parameter pˆ10, for a total of 10,000 masslets. This represents a compromise between res-olution and computational eciency, and worked sat-isfactorily within the range of test cases considered herein. Cell-centre concentration values are then calcu-lated (ready for the di€usion step) by summing the mass contributions from each masslet in the cell. The indi-vidual masslet positions are stored at this point in order to minimise numerical di€usion on their subsequent re-placement.

4.2. Di€usion: ®nite-di€erence scheme

The regularised quadtree grid contains 12 cell con-®gurations. For each con®guration, the Laplacian (dif-fusion) operator was discretised using general Taylor expansions [10]. Using the predictor±corrector scheme to discretise the temporal derivative, an explicit al-gebraic formula was derived for each cell con®guration, such that

o_C o_t

n

c

ˆ 2e 15h2 6C

n n

ÿ

‡6Cne‡3C n

s‡C

n

se‡6C

n

wÿ22C

n c

;

…30†

and

o_C o_t

n

c

ˆ e 9h2 6C

n w

ÿ

‡6Cnn‡6C n

e‡3C

n

s‡2C

n

seÿ20C

n s

:

…31†

On updating the concentrations, a new distribution of masslets was calculated. For each cell, the cell-centre concentration was assumed to be the mean value over the cell, from which the new mass of pollutant in the cell

is computed asMc. Hence, the number of new masslets

in the cell, Nc, is the nearest integer value of…McN=M†.

There is an integer round-o€ error, which a€ects mass conservation. Here, this e€ect was counteracted by summing the total number of new masslets created per cell, and reassigning each masslet with a new mass

mcˆMc=Nc. As in all classical explicit methods, to

en-sure stability of the scheme a time-step criterion must be adhered to, and in this case is determined by the di€u-sion coecient and the grid spacing (rather than an advection-based criterion). The one employed here is given by Fletcher [10], and relates to the smallest (i.e., highest level) cells. For the scheme to remain stable across the entire grid, it was necessary to ensure that Fig. 6. Cell con®gurations.

eD_t=h2_{60:5. However, in order to reduce numerical}

di€usion caused by reseeding particles in cells of di-mension h at each time step D_t, it is required that

h2_{=Dte. This is inconsistent with the stability}

re-quirement, and so another strategy has been used to reduce numerical di€usion. This strategy ensures that particles are reseeded as close as possible to their pre-vious locations. It should be noted that random posi-tioning is used if the new number of masslets is greater than the old. It was found that the stability requirement for a small time step could be relaxed somewhat from that of an Euler integration by using the predictor± corrector scheme. This e€ectively reduced numerical

di€usion further by allowing a larger time step to be used.

4.3. Di€usion: di€usion-velocity method

This method is analogous to that proposed by Ogami and Akamatsu [19] for viscous di€usion in momentum transport. In two dimensions, the species equation, without the source term, may be rewritten as

o_C o_t ‡

o

o_x U

ÿe

C

o_C o_x

C

‡ o

o_y V

ÿe

C

o_C o_y

C

ˆ0:

This indicates that the pollutant concentration disperses with a velocityUÿ …e=C†…o_C=o_{x†in thex-direction and} a velocityV ÿ …e=C†…o_C=o_{y†in they-direction. SinceU} and V are the advection velocities, Ogami and Aka-matsu therefore de®ned the di€usion velocities as

Udˆ ÿ

The (predictor±corrector) di€usion step for the kth masslet is therefore

where D_{t is the time step. The di€usion velocities}

are proportional to the negative of the concentration gradient in accordance with FickÕs law of di€usion, and inversely proportional to the concentration. In order to evaluate the di€usion velocities, values of concentration and concentration gradients are estimated at the masslet positions using Shepard-type scattered data interp-olation [11]. The interpinterp-olation formula is

f…x;y† ˆ

where f…x;y† is the interpolated function of interest, Kis the total number of nodal values of f used in the interpolation, and w is an inverse distance weighting function speci®ed by

The interpolations were implemented by ®rst estimating values of pollutant concentration at the corners of each grid cell using the cell-centre values. Weighting func-tions were applied according to the six possible vertex topologies illustrated in Fig. 7. Concentration gradients were then calculated along the cell edges using the in-terpolated corner concentrations. Concentration gradi-ents were obtained within the cell by linear interpolation of the cell edge gradients, and the Shepard-type scat-tered data interpolation used to estimate the concen-trations at each masslet position. Di€usion velocities were then calculated from (33).

4.4. Di€usion: random walks

For comparison purposes, di€usion was also simu-lated using a gridless random walk technique, analogous to Brownian motion. It is assumed that the masslets_Õ displacements and velocities have the same standard deviation as the Gaussian solution of pure di€usion. In this case, the random walk di€usion velocities are

Udˆb

where b is a mean zero random number with unit standard deviation andD_{t is the time step.}

4.5. Initial and boundary conditions

The initial concentration pro®le is speci®ed. For ex-ample, a Gaussian distribution is applied for the still water pure di€usion test, and for the rotating ¯ow case. An initial regular grid of level 5 is employed to position pseudo-masslets according to the speci®ed pro®le. The pseudo-masslets are then used as seeding points for a revised quadtree grid, which replaces the regular grid. The pseudo-masslets are discarded, and a revised initial distribution of masslets created about the quadtree grid. In order to simulate a continuous discharge of strength J units sÿ1 _{(noting that the source SˆJ=h}2_H

pollution units mÿ3_sÿ1_{), masslets are injected at a rate of}

nevery time stepD_{t. The initial mass of each masslet,m}i c,

is therefore JD_{t=n units. These masslets are released}

uniformly within an area described by a radiusri_.

At solid boundaries, masslets are re¯ected whether advecting or subject to a di€usion velocity or random walk. For the ®nite-di€erence di€usion step, a von Neumann condition was applied whereby the concen-tration gradient normal to the boundary is set to zero.

Advecting masslets are directly transported across open boundaries by the ¯ow ®eld. In the random walk and di€usion-velocity schemes, the di€using masslets also crossed the open boundaries by means of the walk or di€usion-velocity component themselves (i.e., anal-ogous to advection). Reintroduction of masslets was undertaken at periodic open boundaries, according to the local velocities. In the ®nite-di€erence scheme, the boundary concentration values were determined using linear extrapolation (with a cut-o€ value of zero con-centration to prevent negative concon-centrations being in-troduced at the boundary).

5. Results

Three cases and one application are used to validate the model and indicate its potential for simulating water-borne pollutant transport.

5.1. Pure di€usion

An initial Gaussian pro®le is speci®ed from the ana-lytic solution of the pure di€usion equation [7] as

C…x;y;t† ˆ M0 4p_etexp

ÿ x

2_‡y2

… †

4et

; …38†

whereM0ˆ4pis the initial pollutant mass at initial time

tˆt0ˆ2:5 s. The di€usion coecient was set to eˆ0:001 m2 _sÿ1_{, corresponding to a maximum initial}

concentration of 400 units mÿ2_{, and the source term S}

was omitted. The starting grid was of level 5 and con-tained 10,235 masslets (after integer rounding) distrib-uted according to the concentration pro®le. The time step was D_{tˆ0:05 s. Fig. 8 shows the concentration}

pro®le (depicted using MATLAB as a surface plot) and the adapted quadtree grids (with two-dimensional shading) at timestˆ2:55, 3.5 and 4.5 s, respectively, for the ®nite-di€erence di€usion scheme. Corresponding concentration pro®les obtained using the di€usion-ve-locity and random walk di€usion schemes are presented in Figs. 9 and 10. In general, the concentration distri-butions are in reasonably close agreement; the ®nite-di€erence scheme gives the smoothest representation, whereas the random walk results are much rougher as would be expected. Figs. 11 and 12 show the concen-tration pro®le diagonally across the computational do-main at times tˆ2:75 and 4.0 s. The ®nite-di€erence scheme gives results that are almost identical to the analytical solution. This demonstrates that the scheme replicates the correct level of di€usion without con-tamination by numerical di€usion. The di€usion-vel-ocity model is also in reasonable agreement, but the random walk model is the least accurate. The

®nite-di€erence scheme results are reasonably symmetric, in-dicating that there is little or no in¯uence from the cartesian quadtree grid.

5.2. Advection±di€usion of rotating Gaussian

A Gaussian distribution of the pollutant (identical to the above case) was initially located with its peak con-centration at (0.7, 0.7) and then rotated anticlockwise in an analytic ¯ow ®eld about the centre of the unit square domain by an angular displacement of p_{=12 each time}

step. In this case, an initial 10,000 masslets are created about a quadtree grid of minimum level 4 and maximum level 6, and di€usion incorporated witheˆ0:001 m2_sÿ1_,

Fig. 9. Pure di€usion case: di€usion-velocity scheme: (a) concentration pro®le,tˆ2:55 s; (b) concentration pro®le,tˆ3:5 s; (c) concentration pro®le,tˆ4:5 s.

Fig. 10. Pure di€usion case: random walk scheme: (a) concentration pro®le,tˆ2:55 s; (b) concentration pro®le,tˆ3:5 s; (c) concentration pro®le,tˆ4:5 s.

scheme. Fig. 13 shows concentration plots as shaded grids and pro®les along the south-west to north-east diagonal across the grid for the Lagrangian advection and ®nite-di€erence di€usion scheme at times tˆ2:8 and 3.1 s. These times correspond to one-half and a full revolution, respectively, of the Gaussian. Note that

t0ˆ2:5 s. Close agreement is achieved with the analytic

solution, also presented in Fig. 13 as concentration pro®les. This case demonstrates that the model is ca-pable of reproducing accurately an advection dominated

transport process while properly accounting for di€u-sion. In the ®nite-di€erence di€usion scheme, no nega-tive concentrations resulted. Moreover, negligible arti®cial di€usion was found to occur, even though the ®nite-di€erence scheme involved regridding, interpola-tion and asymmetric grids at each time step. There was no perceptible radial shift in the centre of mass of the pollutant, which con®rms that the predictor±corrector calculations are properly tracking the advection transport.

This case was also used to examine the trade-o€ be-tween accuracy and computational eciency. After one revolution, the error between the predicted and analyt-ical solutions was computed as a normalised standard error, whereby

Eˆ

 1=…n…nÿ1††Pn

iˆ1…CiÿC† 2

q

1=nPn iˆ1Ci

;

Fig. 11. Pure di€usion concentration distributions,tˆ2:75 s.

Fig. 12. Pure di€usion concentration distributions,tˆ4:0 s.

where Ci is the ith computed concentration, C the

corresponding analytical value and n the number of cells that lie along the pro®le. The values were found to be Eˆ0:066, 0.017, 0.011, and 0.008 for 1000, 10,000, 20,000 and 50,000 masslets, respectively for

pˆ10. Since computing time increases approximately linearly with the number of masslets, a satisfactory compromise between accuracy and run time should be sought for each application. The accuracy was found not to be very sensitive to the value of p, provided that more than 1000 masslets were used. For low masslet numbers, the accuracy becomes poor as may be expected and depends in a complex way on p. It should be noted that the quadtree gridding does lead to signi®cant savings in storage. The grids in Fig. 13 consist of 778 and 823 panels, respectively, a uniform grid at the ®nest resolution would comprise 4096 panels.

5.3. Advection±di€usion of Gaussian in a shear ¯ow

Fig. 14 illustrates the transport of a Gaussian pollu-tant spill placed in a laminar ¯ow ®eld with no-slip side walls. The velocity pro®le is parabolic in thex-direction with peak velocity of 3 m sÿ1_{, and the ¯ow is sheared}

because of the lateral boundary layers. The mean aver-age velocity is 2 m sÿ1_{. The initial Gaussian is again the}

same as in the pure di€usion case, with its peak value at

(0.25, 0.5). Shaded quadtree grids and longitudinal concentration pro®les are presented at times tˆ0:45 and 1.175 s; the simulation starting at time t0ˆ2:5 s.

Here, eˆ0:001 m2 _sÿ1_{, Sˆ0, and Dtˆ0:025 s. This}

case demonstrates the model_Õs ability to simulate the transport of a sharp front in a shear ¯ow. There is some evidence of the build up of a tail and a slightly sharper front, which compares favourably with similar analyti-cal results presented by Smith [21]. Here, the mean velocity of 2 m sÿ1_{is equivalent to Smith}

Õs bulk velocity, and corresponds to a Peclet number of 1000, as con-sidered in SmithÕs Fig. 1.

6. Application

The method was applied to a tidal embayment (Se-petiba Bay, Brazil; Latitude 23°₀0 _{South, Longitude}

44°₀0 _{West) which has a complex coastal topography}

with several islands, see Fig. 15. The overall east to west extent of the computational domain was 50 km and the north±south extent was 19 km. The coastline was rep-resented by a series ofx,ydata points which were used to seed the quadtree representation of the coastal boundaries, see Fig. 16, in which the smallest coastline cell is at level 9 (i.e., 2ÿ9 _{of the maximum domain}

di-mension which is about 100 m). These coastline seeding points together with the e‚uent masslets are used to generate the computational quadtree mesh. Note that all information was scaled to the unit square and that part of the square which lay outside of the rectangular do-main shown in these ®gures was excluded from the calculations and is not shown in the results. A linear scaling factor r was de®ned as the ratio of the unit square (1 m) to the longest dimension of the domain, in this case about 50,000 m.

The tidal ¯ow ®elds were calculated on a regular grid of cell size 250 m. An explicit ®nite-volume code was used to solve the shallow water equations (swe) with a QUICK scheme to treat the convective accel-eration terms. A simple interpolation scheme was used to deliver the current velocities to each masslet position at each time in the dispersion calculation. The veloci-ties were also scaled by the factor r; time remained unscaled. A simulated discharge of sewage e‚uent was created with a strength Jˆ109 _{bacteria s}ÿ1_{. Bacterial}

decay was represented by a T90ˆ4 hr. Decay was

in-corporated into the model calculations by reducing the concentrations in each cell by a factor of 10ÿD_t=T90

each time step. Isotropic turbulence was assumed and a value of eˆ0:25 m2 _sÿ1 _{was used to represent}

proto-type conditions. This value had to be scaled by a factor of r2 _{to represent conditions in the model unit}

square.

The adapting quadtree meshes representing the mid-®eld plume are shown in Figs. 17(a) and (b) in which Fig. 14. Advection±di€usion of a Gaussian in a shear ¯ow:

®nite-dif-ference scheme: (a) shaded quadtree grids,tˆ0:45 and 1.175 s; (b) concentration pro®les,tˆ0:45 and 1.175 s.

the smallest cell is of level 11 which is a cell size of about 25 m.

7. Conclusions

In the present work, details have been presented of an adaptive hierarchical grid generator, with local mesh enrichment and coarsening controlled by the level of pollution concentration. The quadtree stores grid

infor-mation eciently, while allowing the grid to have fractal-like re®nements. In a novel approach, grid regularisation is achieved in a single sweep starting from the ®nest cells and moving up the quadtree. The species equation is solved using Lagrangian particle tracking for advection, regridding about the new masslet positions, estimation of cell-centre concentrations and di€usion by either dif-fusion-velocity or ®nite-di€erence grid-based schemes. Good agreement is obtained with analytical solutions and numerical random walk simulations of standard Fig. 15. Sepetiba Bay, Rio de Janeiro State, Brazil. The bay opens into the Atlantic Ocean at the south-west corner of the ®gure.

advection±di€usion test cases, with the ®nite-di€erence scheme giving the most accurate results. It is clear that localised features such as sharp pollutant fronts can be captured properly by the quadtree-based model, while retaining computational eciency in terms of storage and CPU time. The method has been demonstrated to be e€ective at prototype scale and with real topography.

Acknowledgements

This work has been supported by the UK Engineer-ing and Physical Sciences Research Council through EPSRC Grant GR/L11861, and co-investigated by Professor R Eatock Taylor (Oxford University) and Dr A Folkard (Strathclyde University). The authors would like to acknowledge Dr J Jozsa of the Technical Uni-versity of Budapest, Hungary, who substantially con-tributed to the concepts described herein. The contribution of Mr. Scott Couch (Strathclyde Univer-sity) is also acknowledged who developed the swe solver used in the application to Sepetiba Bay and to Prof. Teo®lo Monteiro (Fiocruz, Rio de Janeiro), who started the project in this area.

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