Development of a one-dimensional contaminant model for streams and rivers

The Contaminant Model for Streams (CMS) was developed for use in studies where both data and resources for model application are limited. CMS can be quickly and easily applied, yet it is still versatile enough to be used for a variety of conditions ranging from short term spill modeling to multi-year simulations of contaminant fate in stream water and bottom sediments. The model can be applied for both organic and inorganic contaminants. Suspended solids can be transported or a steady-state concentration may be input. Steady-state, uniform hydraulic conditions are assumed within the modeled reach, which greatly reduces model complexity. A model application may consist of one or more reaches connected in series or in a branched network. Possible sediment model configurations include: 1) water column only, 2) water column and mixed sediment layer, and 3) water column, mixed sediment layer, and deep sediment layer

Development of a one-dimensional contaminant model for streams and rivers

The Contaminant Model for Streams (CMS) was developed for use in studies where both data and resources for model application are limited. CMS can be quickly and easily applied, yet it is still versatile enough to be used for a variety of conditions ranging from short term spill modeling to multi-year simulations of contaminant fate in stream water and bottom sediments. The model can be applied for both organic and inorganic contaminants. Suspended solids can be transported or a steady-state concentration may be input. Steady-state, uniform hydraulic conditions are assumed within the modeled reach, which greatly reduces model complexity. A model application may consist of one or more reaches connected in series or in a branched network. Possible sediment model configurations include: 1) water column only, 2) water column and mixed sediment layer, and 3) water column, mixed sediment layer, and deep sediment layer

The ADI-FDTD Method for High Accuracy Electrophysics Applications

The Finite-Difference Time-Domain (FDTD) is a dependable method to simulate a wide range of problems from acoustics, to electromagnetics, and to photonics, amongst others. The execution time of an FDTD simulation is inversely proportional to the time-step size. Since the FDTD method is explicit, its time-step size is limited by the well-known Courant-Friedrich-Levy (CFL) stability limit. The CFL stability limit can render the simulation inefficient for very fine structures. The Alternating Direction Implicit FDTD (ADI-FDTD) method has been introduced as an unconditionally stable implicit method. Numerous works have shown that the ADI-FDTD method is stable even when the CFL stability limit is exceeded. Therefore, the ADI-FDTD method can be considered an efficient method for special classes of problems with very fine structures or high gradient fields. Whenever the ADI-FDTD method is used to simulate open-region radiation or scattering problems, the implementation of a mesh-truncation scheme or absorbing boundary condition becomes an integral part of the simulation. These truncation techniques represent, in essence, differential operators that are discretized using a distinct differencing scheme which can potentially affect the stability of the scheme used for the interior region. In this work, we show that the ADI-FDTD method can be rendered unstable when higher-order mesh truncation techniques such as Higdon's Absorbing Boundary Condition (ABC) or Complementary Derivatives Method (COM) are used. When having large field gradients within a limited volume, a non-uniform grid can reduce the computational domain and, therefore, it decreases the computational cost of the FDTD method. However, for high-accuracy problems, different grid sizes increase the truncation error at the boundary of domains having different grid sizes. To address this problem, we introduce the Complementary Derivatives Method (CDM), a second-order accurate interpolation scheme. The CDM theory is discussed and applied to numerical examples employing the FDTD and ADI-FDTD methods

Locally Implicit Time Integration for Linear Maxwell\u27s Equations

This thesis is concerned with the full discretization of Maxwell\u27s equations in cases where the spatial discretization has to be carried out with a locally refined grid. In such situations locally implicit time integrators are an appealing choice for the time discretization since they overcome the grid-induced stiffness of these problems. We analyze such a locally implicit time integrator in the case where the space discretization stems from a central fluxes discontinuous Galerkin method. In fact, we prove its stability under a CFL condition which solely depends on the coarse part of the spatial grid and give a rigorous error analysis showing that the integrator is second order convergent. Moreover, we extend this time integrator so that it can be applied to an upwind fluxes discontinuous Galerkin space discretization. We show that this novel integrator preserves the second order temporal convergence and that it inherits the improved properties of an upwind fluxes discretization (better stability, higher spatial convergence rate) compared to the central fluxes case

Mathematical modelling of complex dynamics

Soft materials have a wide range of applications, which include the production of masks for nano–lithography, the separation of membranes with nano–pores, and the preparation of nano–size structures for electronic devices. Self–organization in soft matter is a primary mechanism for the formation of structure. Block copolymers are long chain molecules composed of several different polymer blocks covalently bonded into a single macromolecule, which belong to an important class of soft materials which can self–assemble into different nano–structures due to their natural ability to microphase separate. Experimental and theoretical studies of block copolymers are quite challenging and, without computer simulations, it is difficult and problematic to analyse modern experiments. The Cell Dynamics Simulation (CDS) technique is a fast and accurate computational technique, which has been used to investigate block copolymers. The stability has been analysed by making use of different discrete Laplacian operators using well–chosen time steps in CDS. This analysis offers stability conditions for phase–field, based on the Cahn–Hilliard Cook (CHC) equations of which CDS is the finite difference approximation. To overcome grid related artefacts (discretization errors) in the computational grid, the study has been done for employing an isotropic Laplacian operator in the CDS framework. Several 2D and 3D discrete Laplacians have been quantitatively compared for their isotropy. The novel 2D 9–point BV(D2Q9) isotropic stencil operators have been derived from the B.A.C. van Vlimmeren method and their isotropy measure has been determined optimally better than other exiting 2D 9–point discrete Laplacian operators. Overall, the stencils in 9–point family Laplacians in 2D and the 19–point stencil operators in 3D have been found to be optimal in terms of isotropy and time step stability. Considerable implementation of Laplacians with good isotropy has played an important role in achieving a proper structure factor in modelling methods of block copolymers. The novel models have been developed by implementing CDS via more stable implicit methods, including backward Euler, Crank–Nicolson (CN) and Alternating Direction Implicit (ADI) methods. The CN scheme were implemented for both one order and two order parameter systems in CDS and successful results were obtained compared to forward Euler method. Due to the implementation of implicit methods, the CDS has achieved second–order accuracy both in time and space and it has become stronger, robust and more stable technique for simulation of the phase–separation phenomena in soft materials

Analysis of Heat Partitioning During Sliding Contact At High Speed and Pressure

This research develops a mathematical formulation and an analytical solution to frictional heat partitioning in a high speed sliding system. Frictional heating at the interface of sliding materials impacts temperature and the wear mechanisms. The heat partition fraction for a sliding system is an important parameter in calculating the distribution of frictional heat flux between the contacting surfaces. The solution presented in this dissertation considers the characteristics of the slipper\u27s frictional heat partition values along with the experimental loading data. With a physics based, rather than a phenomenological approach, this solution improves the estimate for the slipper\u27s heat partition function. Moreover, this analytical solution is practical in calculating the average surface temperature and estimating the total melt wear volume. The heat partition function compares favorably with existing experimental and analytical data. Using the Strang\u27s Splitting and ADI methods, a numerical method for surface temperature and corresponding wear percentage under dynamic bounce conditions was extensively developed

Finite Difference Computing with PDEs: A Modern Software Approach

finite difference methods; programming; python; verification; numerical methods; differential equation

Homogeenisen seosmallin verifiointi vapaan nestepinnan ongelmaan

In this thesis, the applicability of the homogeneous mixture model of Finflo for the free surface problem is studied. The free surface problem is fundamental in marine hydrodynamics, and a special case in two phase flows. The work explores the basis of this type of modelling from mathematical and numerical viewpoint, and verifies the mixture model for the problem. The mathematical background of the problem is presented, together with the nature of it from the perspective of marine hydrodynamics. The bulk flow equations are usually averaged conditionally such that the governing equations of the multiphase model are formally the same as in the case of single phase flow. It can be shown that one additional equation suffices for the description of the segregated phases. Here, the convection equation of the void fraction is utilized. The void fraction equation is derived in conservative form based on the incompressibility constraint of the individual phases. The convection of the void fraction corresponds to the so-called Riemann problem. This is studied thoroughly by developing a two-dimensional solver for the comparison of some well-known schemes for the spatial discretisation of the convective quantity. This solver is applied to the convection of a discontinuous distribution of the void fraction. In addition, the so-called SUPERBEE limiter is implemented to the Finflo code for the extrapolation of the convective void fraction. The numerical solution of the Navier-Stokes equations for simulations of two phase flows is covered comprehensively. The code Yaffa, developed at the Aalto University, has a modern VOF model implemented, and for this reason, it is here used as a reference code. The solution algorithms, the computation of the convective quantities, the pressure correction stages as well as the treatment of the segregated phases in both of the codes are discussed in detail. The two phase flow over a submerged ground elevation is computed using the codes Finflo and Yaffa, and the forming free surface wave is compared to those found from the literature. The aim of this thesis is to get acquainted with the nature of the problem in conjunction with the specific methodology used to solve such flows. This is done in order to understand the requirements and possible modifications needed for the model when we wish to accurately predict ship flow phenomena that are not solvable using the traditional free surface tracking strategies. This way, the verification of the mixture model of Finflo is achieved.Tässä työssä tutkitaan Finflon homogeenisen seosmallin soveltuvuutta vapaan nestepinnan ongelmaan. Vapaan nestepinnan ongelma on keskeinen laivahydrodynamiikassa, ja samalla monifaasivirtauksien erikoistapaus. Työssä perehdytään tällaisen mallinnuksen perusteisiin matemaattisessa ja numeerisessa mielessä, ja verifioidaan samalla seosmallia tälle ongelmalle. Työssä esitetään ongelman matemaattinen tausta sekä sen luonne laivahydrodynamiikan kannalta. Virtausta kuvaavat yhtälöt yleensä keskiarvostetaan ehdollisesti se. käytettävän monifaasimallin perusyhtälöt ovat muodollisesti samat, kuin yksifaasisessakin tapauksessa. Voidaan osoittaa, että tässä tapauksessa erillisten faasien kuvaukseen riittää yksi lisäyhtälö, joksi työssä otetaan aukko-osuuden konvektioyhtälö. Aukko-osuusyhtälö johdetaan säilymismuodossa perustuen faasien kokoonpuristumattomuusoletukseen. Mainittu lisäyhtälö vastaa luonteeltaan konvektioyhtälön ns. Riemann-probleemaa, ja tätä käsitellään perusteellisesti. Työssä kehitetään kaksidimensioinen ratkaisija, jolla vertaillaan tunnettuja menetelmiä konvektoituvan suureen paikkadiskretoinnille soveltamalla sitä epäjatkuvan aukko-osuusjakauman konvektioprobleemalle. Lisäksi implementoidaan Finfloon ns. SUPERBEE-rajoitin konvektoituvan aukko-osuuden ekstrapolointiin. Työssä käsitellään kattavasti Navier-Stokes -yhtälöiden numeerista ratkaisua kaksifaasivirtausimulointimenetelmien kannalta. Referenssikoodiksi otetaan Aalto-yliopistossa kehitetty Yaffa, johon nykyaikainen VOF-malli on implementoitu. Muiden muassa koodien ratkaisualgoritmi, konvektoituvien suureiden laskenta, painekorjausvaihe sekä erottuneiden faasien käsittely kuvataan perusteellisesti. Finflo- ja Yaffa -koodeilla lasketaan kaksifaasivirtaus vedenalaisen kummun yli, ja syntynyttä aaltokuviota verrataan myös kirjallisuudesta löytyviin tuloksiin. Työn ajatuksena on tutustua vapaan nestepinnan ongelman luonteeseen yhdessä tällaisen yleisemmän ratkaisutavan kanssa. Tavoitteena on ymmärtää mallille asetettavia vaatimuksia sekä sitä, millaisia modifikaatioita siihen tulisi tehdä, kun esim. pyritään ennustamaan tarkasti sellaisia laivavirtauksiin liittyviä ilmiöitä, joihin perinteiset pintaa seuraavat mallit eivät pysty. Tällä tavalla saatiin Finflon seosmallin verifiointi aikaiseksi