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Στην εργασία γίνεται μελέτη του προβλήματος μεταφοράς και διάχυσης σε μια και δυο διαστάσεις με αναλυτικό και αριθμητικό τρόπο για ρύπους βιομηχανικής μονάδας. / We study in the paper the advection/ diffusion problem in one and two dimension. We use analytical and arithmetical solutions to solve the air pollution problem.

This thesis develops two families of numerical methods, based upon rational approximations having distinct real poles, for solving first- and second-order parabolic/ hyperbolic partial differential equations. These methods are thirdand fourth-order accurate in space and time, and do not require the use of complex arithmetic. In these methods first- and second-order spatial derivatives are approximated by finite-difference approximations which produce systems of ordinary differential equations expressible in vector-matrix forms. Solutions of these systems satisfy recurrence relations which lead to the development of parallel algorithms suitable for computer architectures consisting of three or four processors. Finally, the methods are tested on advection, advection-diffusion and wave equations with constant coefficients.

Efficient mixing can be achieved in flows where turbulence is absent, if the trajectories of passively advected particles in the flow are chaotic. The chaotic nature of particle trajectories results in exponential stretching of material lines in the flow. Thus the interface along which diffusion occurs is stretched exponentially leading to efficient mixing. It has been demonstrated recently that regions in flow fields that exhibit poor mixing and non-chaotic particle trajectories can have an important bearing on the overall dynamics and transport of the entire domain. The space-time trajectories of physical stirrers or elliptic points in two dimensional flows can be classified according to braid groups. One can predict a lower bound on the topological entropy (i.e. exponential rate of stretching of material lines) of flows (h_f) by applying the Thurston-Nielsen classification theorems to these braids. This gives a reduced order model for the dynamics of transport of the entire flow field using just a few points. Recent work has shown that this methodology can be used to estimate a lower bound on h_f using the braids formed by Almost Cyclic Sets (ACS) in certain periodic Stokes' flows. These ACS are closely related to Almost Invariant Sets (AIS) which are identified using a probabilistic set oriented method that makes use of the descritised Perron-Frobenius operator of the flow map. This work extends this approach to flows at non-zero Reynolds numbers, which take into account the effects of inertia. The role of Finite Time Coherent Structures (FTCS) in the dynamics of flow fields is also investigated. Unlike ACS, the FTCS approach is more general as it can be applied to aperiodic flow fields. Further, the relationship between mixing efficiency and the topological entropy of flow fields at non-zero Reynolds numbers is also studied. / Ph. D.

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We are interested in examining the long-time decay rate of a passive scalar in two-dimensional flows. The focus is on the effect of boundary conditions for kinematically prescribed velocity fields with random or periodic time dependence. Scalar evolution is followed numerically in a periodic geometry for families of flows that have either a slip or a no-slip boundary condition on a square or plane layer subdomain D. The boundary conditions on the passive scalar are imposed on the boundary C of the domain D by restricting to a subclass invariant under certain symmetry transformations. The scalar field obeys constant (Dirichlet) or no-flux (Neumann) conditions exactly for a flow with the slip boundary condition and approximately in the no-slip case. At late times the decay of a passive scalar, for example temperature, is exponential in time with a decay rate gamma(kappa), where kappa is the molecular diffusivity. Scaling laws of the form gamma(kappa) ~ C*kappa^alpha for small kappa are obtained numerically for a variety of boundary conditions on flow and scalar, and supporting theoretical arguments are presented. In particular when the scalar field satisfies a Neumann condition on all boundaries, alpha ~ 0 for a slip flow condition; for a no-slip condition we confirm results in the literature that alpha ~ 1/2 for a plane layer, but find alpha ~ 2/3 in a square subdomain D where the decay is controlled by stagnant flow in the corners. For cases where there is a Dirichlet boundary condition on one or more sides of the subdomain D, the exponent measuring the decay of the scalar field is alpha ~ 1/2 for a slip flow condition and alpha ~ 3/4 for a no-slip condition. The scaling law exponents alpha for chaotic time-periodic flows are compared with those for similarly constructed random flows. Motivated by the theory of passive scalar field, in Part II of this work we extend the investigation of the evolution of passive scalar for the flows addressed specifically in Part I. Based on an ensemble averaging over random velocity fields, the theoretical results obtained confirm the scaling laws computed numerically for a single, long realisation of random flows. In analogy with Lebedev and Turitsyn (2004) and Salman and Haynes (2007) our results show very good agreement between such an ensemble theory and applications. In part III of our study, we expand upon the work set out in the previous parts of this thesis in terms of the polar-co-ordinate system. We analyse the structures of flows driven near to a corner with a link to Moffatt corner eddies. A long-time exponential decay rate gamma(kappa)=C*kappa^alpha has been obtained confirming our numerical and theoretical results predicted in Part I and Part II in this work. The exponent alpha is determined in a structure of Moffatt corner eddies.

Surface mining of oil sands in northern Alberta requires stripping of saline-sodic shale overburden, which is typically placed in large upland overburden dumps. Due to the chemical nature of this shale, engineered soil covers must be constructed over the shale to support the growth of forest vegetation. A research site on South Bison Hill (SBH), a shale overburden dump at the Syncrude Canada Ltd. Mildred Lake Mine, has been used by researchers over the past decade to study the performance of a reclamation cover. This study was undertaken to improve the understanding of salt and moisture dynamics in the cover-shale system. In particular, the objective of this study was to develop an estimate of the net percolation rate through the cover soil and into the shale overburden. Stable isotope (ä2H and ä18O) measurements obtained from the pore water of soil samples were used to develop stable isotope profiles at various sampling locations along the slope and plateau of the SBH. Simulated profiles were then generated using 2D, finite element numerical modelling software and compared to the measured profiles. Model parameters were obtained from testing and the work of previous researchers. The model results revealed that the net percolation is greatest (32-50 mm/yr) for the plateau and mid-slope bench sample locations. Net percolation rates for sample locations on the slope were lower at 0-12 mm/yr. The results from the stable isotope modelling were utilized in a SO42- transport model to ascertain if calculated net percolation rates could explain measured salinity profiles. This modelling exercise revealed that calculated SO42- profiles are highly dependent on the assumed SO42- production rates in the shale, which is primarily attributed to pyrite oxidation. The model results showed the isotope-based net percolation rates could explain the measured SO42-profiles for a reasonable range SO42- production rates. The SO42- production rates calculated in the model were greatest for the plateau and mid-slope bench locations and lesser for the sloped locations. The model also showed that the mass of SO42- removed by interflow was minimal compared to the mass generated by pyrite oxidation and that net percolation is the dominant flushing mechanism at net percolation rates of 8 mm/yr or more.