Abstract

Most numerical studies on solute mixing rely on mesh-based methods, and complicated schemes have been developed to enhance numerical stability and reduce artificial diffusion. This paper systematically studies the depth-averaged random walk scheme, which is a meshfree method with the merits of being highly robust and free of numerical diffusion. First, the model is used to solve instantaneous release problems in uniform flows. Extensive parametric studies are carried out to investigate the influences of the number of particles and the size of time steps. The predictions are shown to be independent of time steps but are sensitive to the particle numbers. Second, the model is applied to the solute transport problem along an estuary subject to extensive wetting and drying during tidal oscillations. Finally, the model is used to investigate the wind-induced chaotic mixing in a shallow basin. The effect of diffusion on the chaotic mixing is investigated. This study proposes a generic sampling method to interpret the output of the random walk method and highlights the importance of accurately taking diffusion into account in analysing the transport phenomena. The sampling technique also offers a guideline for estimating the total number of particles needed in the application.

INTRODUCTION

Pollutant transport in waters has been extensively studied since it is relevant to a wide range of environmental problems and may cause significant economic losses (Rajar 1997). Among the natural waters, many can be regarded to be shallow, such as estuaries, lakes, reservoirs and nearshore waters. The horizontal scale of these flows is much larger than their depth. Therefore, it can be assumed that the solute is well-mixed vertically over the water column, allowing a depth-integrated approach to model the solute transport process. This paper focuses on the solute transport in shallow waters.

With the development of computing techniques, many modern numerical methods have been established to solve such problems (Lin & Falconer 1997; Gupta et al. 2004; Begnudelli & Sanders 2006; Benkhaldoun et al. 2007). Previously, most research has been based on the Eulerian approach to solve the advection-diffusion equation using finite-difference, finite-volume or finite-element method (Mingham et al. 2001; Liang et al. 2006; Yang et al. 2018). However, these grid-based methods have proven to be problematic in addressing steep concentration gradients and tend to produce numerical diffusion (Wu & Liang 2018). By contrast, Lagrangian methods use a group of discrete particles to represent the pollutant cloud, and these particles are tracked independently in flows. The Lagrangian approach can usually yield a good estimation regardless of the concentration gradients (Zhang & Chen 2007; Wu et al. 2018), and the computation can be easily parallelised. Nowadays, thanks to the rapid progress in electronic technology, the high computational cost of the Lagrangian approach has become increasingly affordable. Its merits include inherently conserving mass and being free from artificial diffusion.

The computational cost of the Lagrangian approach depends on two factors: the number of particles to represent the pollutant cloud and the size of the computational time step. The choice of these two factors is crucial to the efficiency of random walk modelling. Traditionally, it is necessary to manually tune the number of particles and the size of time steps. Such a trial-and-error procedure is time-consuming and relies on the modeller's experience. This paper first uses an ideal case study to investigate the sensitivity of numerical solutions to the time step and particle number in depth-averaged random walk modelling. Then, the model is used to solve a solute oscillation problem along a hypothetical tidal estuary. The results of a TVD–MacCormack model are used as a reference. Finally, the paper describes the application of the random walk scheme to simulate the wind-driven chaotic mixing in a shallow circular lake. The results are qualitatively compared with those in previous research, and the importance of diffusion is highlighted.

DEPTH-AVERAGED ADVECTION-DIFFUSION EQUATION

The governing equation of the conservative solute transport is the advection-diffusion equation (Gresho & Sani 1998). Advection is the transport of a substance by a fluid flow as the result of the fluid's bulk motion. In this process, it is assumed that particles of the solute exactly follow the flow. Diffusion is the process whereby solute transports from higher concentration to lower concentration due to the small-scale random movements. It is assumed that the diffusive substances do not affect the motion of flows. In other words, the flow field is independent of the existence of the diffusive materials. With these assumptions, the conservation of solute mass is presented as:  
formula
(1)
where t is time; s is the depth-averaged concentration of the solute; h is the water depth; u and v represent the velocities along x and y-axis, respectively; qs is the sources term of the governing equation, representing the increase (qs>0) or decrease (qs<0) in the total amount of the solute; Dxx, Dxy, Dyx and Dyy represent the dispersion-diffusion tensor of depth-averaged mixing in Cartesian coordinates. The relationship between the streamwise-transverse system and Cartesian coordinate can be expressed as:  
formula
(2)
 
formula
(3)
 
formula
(4)
where  
formula
(5)
Ds and Dt are the streamwise and transverse diffusion coefficients, respectively; ɛs and ɛt are two dimensionless constants. θ=arctan(v/u) is the angle between the direction of the x-axis and the direction of the local flow. The shear velocity u is expressed as Equation (6) using Chézy coefficient.  
formula
(6)

DEPTH-AVERAGED RANDOM-WALK METHOD

Equation reformulation

In this paper, the depth-averaged advection-diffusion equation is recast into a new form that follows a standard advection-diffusion equation to design the particle-tracking algorithm. A new concentration variable, S=sh, is introduced. Then, the new form of the equation can be written as:  
formula
(7)
 
formula
(8)
 
formula
(9)

The source term qs in the previous equation is neglected in the new form, as all the following case studies do not include any source or sink. According to the stochastic process theory, S can considered as a probability density function. U and V in Equations (8) and (9) represent the modified advective velocity components. The more detailed explanations of this scheme can be found in Wu & Liang (2018). This depth-averaged random walk model is then implemented by performing the advection and diffusion transport processes in each time step, as explained in the following sections.

Advective transport

In the advective transport process, the particles exactly follow the flow. However, most of the flow fields are solved by traditional grid-based methods. The advective velocities in Equations (8) and (9) must be evaluated at the position of each particle. In each grid cell that contains a particle, the velocity components u and v are interpolated to the second-order accuracy. The new particle position after the advective transport process can be expressed as Equations (10) and (11) using the second-order accurate iterative scheme:  
formula
(10)
 
formula
(11)
where and are the flow velocity used in calculating the particles' advective displacement in each time step. To increase the order of accuracy, they are taken to be the averaged velocity within each time step of movement.  
formula
(12)
 
formula
(13)

Diffusive transport

After the advection process, the particles undergo diffusive transport in a time step. The random streamwise and transverse velocity components are calculated as:  
formula
(14)
 
formula
(15)
Here, the random numbers rs and rt are independent and follow a normal distribution with a mean of zero and a standard deviation of unity. The subscripts s and t represent the streamwise and transverse directions, respectively. The superscript d represents the diffusion-related velocity components. In the Cartesian coordinate system, the diffusion-related velocity components are expressed as:  
formula
(16)
 
formula
(17)
Finally, particles' new coordinates at the end of one time step can be evaluated as:  
formula
(18)
 
formula
(19)

Boundary condition

Particles follow a ‘random walk’ trajectory with the flow inside the computational domain. The boundary treatment is designed to prevent them from crossing solid boundaries. In this study, the fully reflective boundary condition is applied to particles that penetrate the solid boundaries after implementing Equations (18) and (19). When the time step is not too large, such a treatment then reflects the particles back into the computational domain.

SOLUTE TRANSPORT IN UNIFORM FLOWS

The instantaneous release problem in uniform flows is tested first using the random walk model. When the uniform flow only follows the x-axis (v = 0, Dxy = 0, Dxx=Ds, Dyy=Dt), the analytical solution of this ideal test case can be expressed as:  
formula
(20)

In this case study, the total amount of solute material M = 233.06 kg is released suddenly at the initial location (x0, y0). As shown in Figure 1, the water depth is set to be h = 1 m, and the Chézy coefficient is 40 m1/2/s over the whole study area. The streamwise dispersion and transverse diffusion coefficients are typical values of 13.0 and 1.2, respectively, in straight open channel flows (Falconer 1991). Two flow conditions are considered in this section. The first one is a uniform flow with u = 1 m/s along the x-axis (θ = 0). The solute material is initially located at (x0, y0) = (0, 400 m). The second one is a uniform flow aligned diagonal direction (θ = 45°). The velocity is set to be u = v = 1/m/s. For computations in both scenarios, the particle numbers P is 2.33 × 106 and the time step is 1 s.

Figure 1

The instantaneous release problem in a uniform flow: (a) x-direction flow and (b) diagonal-direction flow.

Figure 1

The instantaneous release problem in a uniform flow: (a) x-direction flow and (b) diagonal-direction flow.

The development of concentration contours for the flows along the x-axis direction and diagonal direction are presented in Figure 1(a) and 1(b), respectively. It is notable that the major axis of the elliptical patches is along the flow direction. The cloud of the solute experiences rapid elongation along its path. The reason is that streamwise dispersion is over ten times larger than the transverse diffusion. The results for the same test case obtained by a grid-based method, TVD–MacCormack, can be found in Liang et al. (2010).

The influence of time steps

The advantage of the random walk model includes high accuracy and small numerical diffusion. However, the disadvantage that is associated with this Lagrangian approach is its high computational cost. As seen in the scheme description, the amount of the computation depends on two factors: the size of the computational time step and the number of particles to present the pollutant cloud. Therefore, the choices of these two factors are crucial to the efficiency of the random walk model. The following part of this section is to discuss the influence of the two parameters on the random walk model when applied to the instantaneous release problem in uniform flows.

A great deal of research on time steps has been undertaken for the Euler scheme. The size of is restricted by the Courant–Friedrichs–Lewy condition. Usually, the smaller the time step is, the more stable the simulation will be. However, the behaviour of the random walk model is very different. Figure 2 presents the longitudinal concentration profile at 600 s after the solute release. Regardless of the time step of 600 s or 0.01 s, the same concentration profile is predicted by the model. Variations of the peak concentration are not affected by the size of time steps as well, as seen in Figure 3.

Figure 2

Concentration distributions at 600 s with changes in time steps (x-direction flow).

Figure 2

Concentration distributions at 600 s with changes in time steps (x-direction flow).

Figure 3

Development of peak concentrations with changes in time steps (x-direction flow).

Figure 3

Development of peak concentrations with changes in time steps (x-direction flow).

This independency of time steps can be proved through the time marching procedure of the calculation. After n time steps, the position of the particles at t is expressed as:  
formula
(21)
 
formula
(22)
If the time step changes to be , n/m times of iterations are needed to reach the same time level. Then, the new position of the particles at t time is expressed as:  
formula
(23)
 
formula
(24)
In the present model, the random numbers ri follow a normal distribution with a mean of zero and a standard deviation of unity, as shown in Equation (25). According to the properties of a normal distribution (Bryc 2012), and have the same expectation and deviation:  
formula
(25)
 
formula
(26)
 
formula
(27)

Therefore, the results will obey the distribution of no matter what size of the time step is used. It can be concluded that the change of the time step does not affect the accuracy of the computation for this case of uniform flow with constant water depth.

The influence of particle numbers

Figure 4 shows a qualitative illustration of the solute transport process predicted by the random walk model using different numbers of particles. In general, the contour turns out to be more reasonable with the particle number increases. Larger particle numbers significantly improve the visualisation of the elliptical cloud for solute distribution.

Figure 4

Evolution of the solute cloud in x-direction uniform flows (P0 = 233).

Figure 4

Evolution of the solute cloud in x-direction uniform flows (P0 = 233).

To get a quantitative analysis, the predicted results are compared with analytical solutions. Figure 5 shows variations of peak concentration with different particle numbers used in the simulation. The size of sampling bins is set to be a circle with a radius of 1 m. It is apparent that small particle numbers tend to produce numerical oscillations. On the contrary, by introducing a larger number of particles in the model, the prediction further approaches the analytical solution. For example, when 2.33 × 107 particles are introduced in the model, the peak concentration is identical to the analytical solution. The relative errors with different particle numbers and different sampling bin sizes are compared in Figure 6(a). In the legend, r represents the radius of the bin. It is clear that, with the increase of particle numbers involved in the simulation, the error decreases significantly. The error for insufficiently small particle numbers can jump beyond 18%, which is unacceptable in most simulations. When the number of particles in each sampling bin approaches 200 to 300, the relative error is reduced to less than 5%. This means that the particle should be guaranteed to be higher than hundreds per bin to give a reasonable estimation of the solute concentration. In addition, it is worth noting that the error for both x-direction and diagonal flows is always very similar, as shown in Figure 6(b). This means the random-walk model is not affected by the flow direction, unlike most of the mesh-based models.

Figure 5

Variation of peak concentration for x-direction flows.

Figure 5

Variation of peak concentration for x-direction flows.

Figure 6

Relative errors of peak concentration: (a) with the changes in particle numbers in each sampling bin and (b) with changes in time.

Figure 6

Relative errors of peak concentration: (a) with the changes in particle numbers in each sampling bin and (b) with changes in time.

SOLUTE OSCILLATION ALONG A ONE-DIMENSIONAL TIDAL ESTUARY

This section considers the situation where the solute is transported forward and backward in a hypothetical tidal estuary. As shown in Figure 7, the boundary condition for the left end is regarded as a sinusoidal tide, while the right side of this estuary is considered as a vertical wall. The total length of the estuary is 13,800 m, and the bed level changes from −5 m at the end of the seaward side to 0 m at the right end. In this case, the tidal flow has an average water level of 0 m and amplitude of 2 m, rising from the average water level at the beginning of the simulation. Before the pollutant transport simulation, the predictions of the flow field were obtained by using TVD–MacCormack scheme to solve the SWEs (shallow water equations). Detailed information can be found in Liang et al. (2010). The discretised velocity field is then reconstructed over the whole domain using linear interpolation.

Figure 7

One-dimensional hypothetical tidal estuary.

Figure 7

One-dimensional hypothetical tidal estuary.

The initial concentration is set to be 100 units at the cell located at x = 10 km, while the concentration at the other positions is set to zero. An illustration of the particles' position and their distribution at different times is given by Figure 8. It is well presented that particles oscillate with tidal flows and gradually disperse in the longitudinal direction. The instantaneously released solute moves with the rising tide to the closed wall under the influence of tidal currents, while the receding tide carries the particles back to the open boundary. At the same time, the spread of the solute grows because of the effect of dispersion.

Figure 8

Distribution of solute particles along the one-dimensional hypothetical estuary.

Figure 8

Distribution of solute particles along the one-dimensional hypothetical estuary.

The aforementioned sampling algorithm is required to convert the scatter of particles, as illustrated in Figure 8, to the concentration profile, as illustrated in Figure 9. This sampling algorithm is crucial in the interpretation of the results of the random walk method and in the evaluation of the concentrations. Traditionally, this concentration profile, i.e., histogram, uses sampling bins of fixed length. However, this fixed length may not suit all the positions. In this work, the length of the bin is chosen automatically by including 200 particles within each bin. This dynamic determination of the bin size is found to be able to avoid spurious fluctuations and achieve good comparisons with analytical and previous results. A series of smooth concentration profiles along the estuary at both high and low water levels are shown in Figure 9. When the solute cloud moves close to the landward position, the tidal flow generally decelerates because of the wall boundary. It is notable that the peak concentration at 27 hours is even higher than that at 9 hours, as the water depth is smaller near the right-side wall which amplifies the value of the concentration to conserve mass of the solute.

Figure 9

Distribution of the concentration along the one-dimensional hypothetical estuary.

Figure 9

Distribution of the concentration along the one-dimensional hypothetical estuary.

The numerical solution of the TVD–MacCormack model is used as a reference for this case study. For such a grid-based method, solutions are sensitive to the grid size. The finer the grid is, the more accurate the simulation will be. The numerical diffusion for the Eulerian model is most obvious at the early stage of the simulation because of the relatively sharp concentration gradient. Take Figure 10(a) as an example, which presents the concentration profile at 3 hours. Several different cell sizes, from 3 m to 270 m, are used in the mesh-based simulations. When the mesh size is increased to 270 m, the concentration distribution is far too flattened because of large numerical diffusion. The peak concentration for the grid size of 3 m is nearly three times higher than that for the grid size of 90 m. However, very small grid size leads to very large computational cost. On the contrary, the random walk model does not involve numerical diffusion. A narrower distribution and a higher peak concentration are predicted by the present model, and it is similar to the prediction of the mesh-based method with very small grid size.

Figure 10

Distribution of the concentration along the estuary with the changes in grid size: (a) 3 hours; (b) 9 hours; (c) 27 hours; (d) 33 hours; (e) 51 hours.

Figure 10

Distribution of the concentration along the estuary with the changes in grid size: (a) 3 hours; (b) 9 hours; (c) 27 hours; (d) 33 hours; (e) 51 hours.

WIND-DRIVEN CHAOTIC MIXING IN A SHALLOW CIRCULAR LAKE

In this section, the random walk model is applied to simulate the particle motion due to wind-driven mixing in a shallow lake. As shown in Figure 11, the property of the circular lake is suggested by Kranenburg (1992). The velocity field is described by a stream function in the polar coordinate system:  
formula
(28)
Figure 11

Aerial view of the model lake with a periodic sequence of storm events.

Figure 11

Aerial view of the model lake with a periodic sequence of storm events.

The von Kármán constant κ is set to be 0.4; the mean depth H is 0.5 m and the radius of the lake R0 is 120 m; z0 is a roughness height of the bed, set to 2.8 mm. The water depth h is described by a function of radial distance r from the basin centre as follows:  
formula
(29)

These physical parameters are set to be the same as those in Kranenburg (1992) for comparison purposes.

Kranenburg (1992) found that the particle motion becomes chaotic when surface wind stress periodically changes its direction. Therefore, a sequence of periodic storm events is designed for this case study. During the first and second halves of a period, the direction of the storm wind alternates between the north-east and north-west directions. It is assumed that the wind stress suddenly changes its direction every half period, while its intensity is constant. Another assumption is that the velocity field instantaneously adapts to wind conditions. The resulting flow field is governed by a dimensionless storm duration parameter μ as follows.  
formula
(30)
where ts is the storm duration, i.e., half of the period T. For all the cases considered, Poincaré sections are used to illustrate the mixing properties, which is the superposition of particle trajectories at the end of each cycle.

For the purpose of comparison with previous studies, the diffusive process is first ignored so as to test the behaviour of the advection part of the random walk model. These particles are tracked for 500 periods. For small μ, as shown in Figure 12(a), the superposed particles with the same phase form elliptical patterns that occupy the left and right halves of the lake. The area of the chaotic region increases with the increase of μ, while the sizes of the noticeable elliptical patterns decrease. When μ is beyond 0.70, the regular patterns are no longer obvious, and only chaotic distribution is noticed (for μ = 0.84). These findings are consistent with previous results of Kranenburg (1992).

Figure 12

Poincaré sections for wind periodically blowing from the north-east and north-west with different μ (without diffusive process).

Figure 12

Poincaré sections for wind periodically blowing from the north-east and north-west with different μ (without diffusive process).

Then, both advection and diffusion processes are considered as tracer particles are also spread by turbulence diffusion and shear dispersion in reality. Figure 13 depicts the impact of streamwise dispersion (the transverse diffusion is set to 0 in this case study) on this chaotic mixing phenomenon. For equal to 0.001, it is still clear that the circular lake is divided into two halves and patterns of the elliptic movement are still obvious. With the increase of the streamwise dispersion coefficient, this pattern becomes more and more blurred until the KAM (Kolmogorov–Arnold–Moser) (Kranenburg 1992) curves no longer exist ( > 0.1). It is worth mentioning that the growth of the streamwise dispersion will not change the mean position of points, but it will remove the large local concentration gradients by increasing the random motion.

Figure 13

Poincaré sections for both advection and diffusion processes with different (μ = 0.14).

Figure 13

Poincaré sections for both advection and diffusion processes with different (μ = 0.14).

For μ = 0.28 and 0.42, the entire calculation domain is chaotic when the dispersion is as small as 0.1, as shown in Figures 14 and 15. This means that the increase in the duration of the periodic storm will emphasise the influence of the diffusion coefficient, making it easier to achieve a chaotic state. Figure 16 shows the advection and diffusion behaviour of a line of 10,000 particles after 32 periods of the storm event. The line is initially positioned along the x-axis. The whorl and tendril structures coexist when μ is 0.28. With the increase of the dispersion parameter, these fine structures both disappear and particles are spead anywhere in the lake, although it can still be seen that the particle distributions in the left and right halves are symmetrical.

Figure 14

Poincaré sections for both advection and diffusion processes (μ = 0.28).

Figure 14

Poincaré sections for both advection and diffusion processes (μ = 0.28).

Figure 15

Poincaré sections for both advection and diffusion processes (μ = 0.42).

Figure 15

Poincaré sections for both advection and diffusion processes (μ = 0.42).

Figure 16

Snapshots of the particle distribution in Kranenburg's model lake (t = 32 T; μ = 0.28).

Figure 16

Snapshots of the particle distribution in Kranenburg's model lake (t = 32 T; μ = 0.28).

CONCLUSIONS

The traditional random walk model has been extended to solve the depth-integrated advection-diffusion equation. First, this model is verified by solving an instantaneous release problem in a uniform flow. The analytical solution is used as a reference. The results reveal several advantages of this model, including high accuracy, stability and simplicity. Extensive parametric studies have been carried out to investigate the sensitivity of the predictions to the computational parameters. It has been found that simulations are independent of the size of the time step. The particle number significantly influences the performance of the random walk model. Too few particles degrade the visual inspection and make it impossible to quantitatively examine the solute distribution. In uniform-flow applications, a relatively large time step will reduce the computation expense without compromising accuracy. Particle numbers should be chosen to ensure at least 200–300 particles in each sampling bin. Subsequently, investigations are carried out regarding the oscillation of a pollutant cloud in a tidal estuary. The sampling method is optimised to convert the particle distributions into concentration profiles. The random walk simulations display high accuracy, which can only be achieved by the mesh-based simulations with extremely fine resolutions. The mesh-based methods are shown to be highly sensitive to grid resolution. Finally, the model is used to simulate the chaotic mixing process in a circular lake. The results for the case of pure advection are consistent with the findings reported by previous researchers. Because of the presence of turbulent diffusion and bottom friction in any real-world lake, the advection is always accompanied by transverse diffusion and longitudinal dispersion. This study shows that diffusion plays an important role in the mixing. The well-structured chaotic mixing pattern produced by pure advection cannot be observed in reality.

In summary, this study demonstrates that the random-walk model is highly stable and free of artificial diffusion in solving the solute transport problems in aquatic environments. In particular, this study proposes a generic sampling technique to convert the scatter of discrete particles to the distribution of solute concentration. In order to obtain a good estimate of the solute concentration, this study proposes a guideline for estimating the total number of particles needed in the simulation.

ACKNOWLEDGEMENTS

We are grateful for financial support provided by the Royal Academy of Engineering UK-China Urban Flooding Research Impact Programme (No. UUFRIP\100051), the 111 Project (No. B17015) and the China Scholarship Council (No. 201708060090).

REFERENCES

REFERENCES
Benkhaldoun
F.
,
Elmahi
I.
&
Seaı
M.
2007
Well-balanced finite volume schemes for pollutant transport by shallow water equations on unstructured meshes
.
Journal of Computational Physics
226
(
1
),
180
203
.
Bryc
W.
2012
The Normal Distribution: Characterizations with Applications
.
Springer Science & Business Media
,
New York
,
USA
.
Falconer
R. A.
1991
Review of modelling flow and pollutant transport processes in hydraulic basins
. In:
Water Pollution: Modelling, Measuring and Prediction
(
Wrobel
C. A.
&
Brebbia
C. A.
eds).
Springer
,
Dordrecht
,
The Netherlands
, pp.
3
23
.
Gresho
P. M.
&
Sani
R. L.
1998
Incompressible Flow and the Finite Element Method
, Vol.
1
.
Advection-diffusion and isothermal laminar flow
.
Wiley
,
New York
,
USA
.
Gupta
I.
,
Dhage
S.
,
Chandorkar
A. A.
&
Srivastav
A.
2004
Numerical modeling for Thane creek
.
Environmental Modelling & Software
19
(
6
),
571
579
.
Kranenburg
C.
1992
Wind-driven chaotic advection in a shallow model lake
.
Journal of Hydraulic Research
30
(
1
),
29
46
.
Liang
D.
,
Falconer
R. A.
&
Lin
B.
2006
Comparison between TVD-MacCormack and ADI-type solvers of the shallow water equations
.
Advances in Water Resources
29
(
12
),
1833
1845
.
Liang
D.
,
Wang
X.
,
Falconer
R. A.
&
Bockelmann-Evans
B. N.
2010
Solving the depth-integrated solute transport equation with a TVD-MacCormack scheme
.
Environmental Modelling & Software
25
(
12
),
1619
1629
.
Lin
B.
&
Falconer
R. A.
1997
Tidal flow and transport modeling using ULTIMATE QUICKEST scheme
.
Journal of Hydraulic Engineering
123
(
4
),
303
314
.
Mingham
C. G.
,
Causon
D. M.
&
Ingram
D. M.
2001
A TVD MacCormack scheme for transcritical flow
.
Proceedings of the Institution of Civil Engineers-Water and Maritime Engineering
148
(
3
),
167
175
.
Rajar
R.
1997
The role of mathematical models, physical models and field measurements in water pollution problems
.
Transactions on Ecology and the Environment
14
,
545
555
.
Wu
X. F.
&
Liang
D.
2018
Study of pollutant transport in depth-averaged flows using random walk method
.
Journal of Hydrodynamics
2018
,
1
13
.
Wu
X. F.
,
Yang
F.
&
Liang
D. F.
2018
Study of pollutant transport in environmental flows using depth-averaged random walk method
.
EPiC Series in Engineering
3
,
2342
2350
.
Yang
F.
,
Liang
D.
&
Xiao
Y.
2018
Influence of Boussinesq coefficient on depth-averaged modelling of rapid flows
.
Journal of Hydrology
559
,
909
919
.