Stephen George Wallis

Development, calibration and evaluation of two mathematical models for pollutant transport in a small river

The present research has two main objectives (1) to build two models for concentration prediction in a stream subject to a pollutant release and (2) to investigate options for estimating the parameters of the models. The models rely on the fundamental advection-dispersion equation and were developed, calibrated and evaluated using tracer data from experiments conducted in the Murray Burn in Edinburgh, UK. During the evaluation by comparison against field data both models were able to predict the main features of the observations at the first three monitoring sites, but results at the final site were less good. These types of models rely very much on the ability to make good estimates of velocity and dispersion coefficients along the stream. Although these parameters could be estimated using tracer experiments, it would be easier if they could be estimated from other hydraulic data such as the water flow rate and the channel characteristics. For the Murray Burn such models for parameter estimation were developed in the form of non-linear relationships with flow rate. This approach could be used to calculate the parameters for other similar streams, if the coefficients in the equations were similar. Alternatively, further work would be needed to explore how these coefficients vary between streams.

ACCURATE NUMERICAL SIMULATION OF ADVECTION USING LARGE TIME STEPS

We describe a technique for achieving accurate numerical simulations of advective transport at large Courant numbers using large time steps. The scheme is called ULTIMATE DISCUS and it implements Leonards universal flux limiter and QUICKEST algorithms within a semi-Lagrangian treatment of advection. This enables the scheme to achieve monotonic solutions, mass conservation and, most importantly, high accuracy without any limit on the time step (or Courant number). The results of numerical experiments of advection over a fixed distance show that the accuracy of the method increases with increasing spatial resolution and generally increases (but in a non-trivial manner) with increasing Courant number. Accuracy is exact at all integer values of Courant number ; for Courant numbers increasing between zero and one, accuracy improves rapidly and monotonically ; for other integer-integer ranges of Courant number there is a minimum of accuracy close to the mid-range value. This behaviour is explained in terms of the known accuracy of the QUICKSET algorithm as a function of Courant number and the reducing number of interpolative steps required in the simulations as the Courant number increases.

A conservative, semi-Lagrangian fate and transport model for fluvial systems-I. Theoretical development

Abstract In this paper a new conservative, semi-Lagrangian fate and transport model for fluvial systems is presented. Following an outline of the current trends in water quality modelling, the main features of Eulerian and semi-Lagrangian modelling approaches are discussed. It is argued that an inherently conservative semi-Lagrangian approach offers a way of meeting the future needs of the industry. The semi-Lagrangian philosophy for advection is then outlined and further developed with the authors explaining how dispersive transport, simple first-order decay terms and inflow boundary conditions can be incorporated within the algorithm. The method (termed DISCUS) possesses high accuracy but, in contrast to other semi-Lagrangian fluvial transport codes, it is centred around a finite-volume based mass balance, which guarantees mass conservation and makes it amenable to flux limiting techniques for suppressing non-physical grid-scale oscillations. The method’s semi-Lagrangian character allows it to work at very large time steps, which makes it efficient for long-term and multiple scenario simulations. The method’s robustness and inherent stability make it particularly useful for highly non-uniform streams and variable loading conditions and also for complex interacting chemical species.

Assessment of heavy metals emission from traffic on road surfaces

This study aims to analyse RDS heavy metal concentrations on road deposited sediment (RDS) using Riccarton Campus of Heriot-Watt University, Edinburgh, Scotland as a study site. RDS samples were collected at two transverse positions from different sites over a 4 month period in order to describe the influence of traffic on heavy metal emissions. The heavy metal concentrations of the RDS were determined by strong nitric acid digestion and atomic absorption spectrometry. The mean concentrations for Zn, Cu, Cd, Cr, Ni, Pb and Fe were found to be 213, 57, 1, 16, 15, 118, and 13497 mg kg-1 from samples near to the curb and 211, 79, 2, 15, 9, 35, and 14276 mg kg−1 from samples 1 m from the curb respectively. Furthermore for both positions the highest concentrations for all metals were associated with the finer fraction (<63 µm) and stronger correlations between the metals were found further from the curb than near the curb, indicating that metals accumulating on the road surface further from the curb may likely be from the same source (traffic), while the sources of metals near the curb are more diverse.

Modelling of Solute Transport in Rivers Under Different Flow Rates: A Case Study Without Transient Storage

A methodology to derive solute transport models at any flow rate is presented. The novelty of the proposed approach lies in the assessment of uncertainty of predictions that incorporate parameterisation based on flow rate. A simple treatment of uncertainty takes into account heteroscedastic modelling errors related to tracer experiments performed over a range of flow rates, as well as the uncertainty of the observed flow rates themselves. The proposed approach is illustrated using two models for the transport of a conservative solute: a physically based, deterministic, advection-dispersion model (ADE), and a stochastic, transfer function based, active mixing volume model (AMV). For both models the uncertainty of any parameter increases with increasing flow rate (reflecting the heteroscedastic treatment of modelling errors at different observed flow rates), but in contrast the uncertainty of travel time, computed from the predicted model parameters, was found to decrease with increasing flow rate.

A conservative semi-Lagrangian algorithm for one-dimensional advection-diffusion

We describe the incorporation of diffusive transport into the original semi-Lagrangian DISCUS algorithm for pure advection. An explicit treatment of diffusion is adopted following the approach used in the QUICKEST algorithm for advection-diffusion. The semi-Lagrangian treatment of the advection term relaxes the small time step restriction normally associated with Eulerian treatments of advection, but the Eulerian treatment of the diffusion term imposes conventional limitations on the scheme. Numerical experiments of advection-diffusion, however, indicate that DISCUS has advantages over the QUICKEST scheme for advection-diffusion in three key areas: stability, accuracy and computational efficiency

Accurate simulation of transport processes in two‐dimensional shear flow

We describe a substantial improvement to an existing modelling approach for two-dimensional solute transport. The flow field is represented by a series of streamtubes in which advection is simulated using a highly accurate semi-Lagrangian numerical scheme. Transverse mixing between the streamtubes is treated with a standard numerical method for diffusion. Numerical experiments demonstrate that accurate simulations of the longitudinal dispersion caused by the interaction of a simple transverse velocity profile and transverse diffusion can be obtained over a wide range of time-steps provided that the diffusion occurring during the time-step is properly accounted for. The model is ultimately limited by the relatively poor numerical treatment of transverse diffusion, but this could be remedied by employing an enhanced numerical method for this term

Flow Dependence of the Parameters of the Transient Storage Model

Using 25 tracer experiments, parameters of the transient storage model (TSM) were evaluated for a reach of the river Brock in north-west England with the primary aim of investigating their dependence on flow rate. Since only a very few previous studies have considered this issue, these new results aid our understanding on how the TSM could be applied to a reach at flow rates beyond the range of flow rates for which observations of solute transport exist. Velocity increased with increasing flow rate in a manner consistent with current knowledge. In contrast, and unexpectedly, the dispersion coefficient reduced (weakly) with increasing flow rate and the values were rather scattered. The transient storage exchange rate increased with increasing flow rate, which corroborates some of the sparse existing knowledge of this parameter’s behaviour. The ratio of transient storage area to main channel area was essentially constant over the range of flow rates examined, which is consistent with some studies on single reaches.

A massively parallel semi-Lagrangian algorithm for solving the transport equation

The scalar transport equation underpins many models employed in science, engineering, technology and business. Application areas include, but are not restricted to, pollution transport, weather forecasting, video analysis and encoding (the optical flow equation), options and stock pricing (the Black-Scholes equation) and spatially explicit ecological models. Unfortunately finding numerical solutions to this equation which are fast and accurate is not trivial. Moreover, finding such numerical algorithms that can be implemented on high performance computer architectures efficiently is challenging. In this paper the authors describe a massively parallel algorithm for solving the advection portion of the transport equation. We present an approach here which is different to that used in most transport models and which we have tried and tested for various scenarios. The approach employs an intelligent domain decomposition based on the vector field of the system equations and thus automatically partitions the computational domain into algorithmically autonomous regions. The solution of a classic pure advection transport problem is shown to be conservative, monotonic and highly accurate at large time steps. Additionally we demonstrate that the algorithm is highly efficient for high performance computer architectures and thus offers a route towards massively parallel application.

ON OPTIONS FOR THE NUMERICAL MODELLING OF THE DIFFUSION TERM IN RIVER POLLUTION SIMULATIONS

In this paper five numerical methods for modelling the diffusion term in a one-dimensional advection-diffusion equation are compared. The motivation behind the work is to find a computationally efficient method for modelling diffusion for incorporating in a semi-Lagrangian approach for advection. Three of the schemes are traditional Eulerian implicit methods (backward, Crank-Nicholson, optimised time-weighted): the other two are based on work by Teixeira (Teixeira, 1998) who proposed a method based on a discrete view of diffusion.