Numerical solution of the eXtended Pom-Pom model for viscoelastic free surface flows

In this paper we present a finite difference method for solving two-dimensional viscoelastic unsteady free surface flows governed by the single equation version of the eXtended Pom-Pom (XPP) model. The momentum equations are solved by a projection method which uncouples the velocity and pressure fields. We are interested in low Reynolds number flows and, to enhance the stability of the numerical method, an implicit technique for computing the pressure condition on the free surface is employed. This strategy is invoked to solve the governing equations within a Marker-and-Cell type approach while simultaneously calculating the correct normal stress condition on the free surface. The numerical code is validated by performing mesh refinement on a two-dimensional channel flow. Numerical results include an investigation of the influence of the parameters of the XPP equation on the extrudate swelling ratio and the simulation of the Barus effect for XPP fluids

A numerical method for junctions in networks of shallow-water channels

There is growing interest in developing mathematical models and appropriate numerical methods for problems involving networks formed by, essentially, one-dimensional (1D) domains joined by junctions. Examples include hyperbolic equations in networks of gas tubes, water channels and vessel networks for blood and lymph in the human circulatory system. A key point in designing numerical methods for such applications is the treatment of junctions, i.e. points at which two or more 1D domains converge and where the flow exhibits multidimensional behaviour. This paper focuses on the design of methods for networks of water channels. Our methods adopt the finite volume approach to make full use of the two-dimensional shallow water equations on the true physical domain, locally at junctions, while solving the usual one-dimensional shallow water equations away from the junctions. In addition to mass conservation, our methods enforce conservation of momentum at junctions; the latter seems to be the missing element in methods currently available. Apart from simplicity and robustness, the salient feature of the proposed methods is their ability to successfully deal with transcritical and supercritical flows at junctions, a property not enjoyed by existing published methodologies. Systematic assessment of the proposed methods for a variety of flow configurations is carried out. The methods are directly applicable to other systems, provided the multidimensional versions of the 1D equations are available

Transition to branching flows in optimal planar convection

We study steady flows that are optimal for heat transfer in a two-dimensional periodic domain. The flows maximize heat transfer under the constraints of incompressibility and a given energy budget (i.e. mean viscous power dissipation). Using an unconstrained optimization approach, we compute optima starting from 30--50 random initializations across several decades of Pe, the energy budget parameter. At Pe between 10$^{4.5}$ and 10$^{4.75}$, convective rolls with U-shaped branching near the walls emerge. They exceed the heat transfer of the simple convective roll optimum at Pe between 10$^{5}$ and 10$^{5.25}$. At larger Pe, multiple layers of branching occur in the optima, and become increasingly elongated and asymmetrical. Compared to the simple convective roll, the branching flows have lower maximum speeds and thinner boundary layers, but nearly the same maximum power density.Comment: 18 pages, 10 figure

Hydraulisen mallin kehitys ja sen soveltaminen pieneen kaupunkiuomaan

Understanding flow in urban streams is important, for example to prevent damages caused by flooding or to be able to estimate pollution transport in a stream. In this thesis a one-dimensional hydraulic model was developed to simulate gradually varied subcritical steady or unsteady open channel flow in a small urban stream or a branched stream network. The model was implemented using Fortran 95 programming language and it was connected to PostgreSQL/PostGIS spatial database using a middleware program implemented in Perl. The database stores the input data and the simulation results of the model, whereas the middleware program enables the interaction between these components and provides a user interface for the system. It was observed that simulating flow in a small urban stream is challenging for many reasons. In small streams estimation of flow resistance is difficult, since the flow varies on a large scale and the variations are usually rapid, the methods commonly used to determine resistance are not suitable for small streams, and local energy losses have a significant impact. Lateral inflow to the stream might also be so significant that is should be taken account in modeling, but quantifying its amount is difficult. Furthermore, low flows are rather common in small streams from the computational point of view, and these may cause numerical difficulties in simulation. The developed model was applied to a small Ridalinpuro stream in Nummela, Southern Finland, to verify that it works. To confirm that the model works also in a branched channel network and in other situations it is intended to, more tests are needed with appropriate data. However, in its present state the model provides a tool that can be used to simulate flow in single reaches of open channel, and which can easily be modified and extended to suit further needs.Veden liikkeiden ymmärtäminen kaupunkiuomissa on tärkeää esimerkiksi tulva-vahinkojen ehkäisemiseksi ja veden mukana kulkeutuvien saasteiden leviämisen arvioimiseksi. Tässä diplomityössä kehitettiin yksiulotteinen hydraulinen malli simuloimaan tasaisesti muuttuvaa stationaarista tai epästationaarista verkasvirtausta pienessä kaupunkiuomassa tai haarautuvassa uomaverkostossa. Malli toteutettiin Fortran 95-ohjelmointikielellä ja se yhdistettiin PostgreSQL/PostGIS-tietokantaan käyttäen Perl-kielellä ohjelmoitua väliohjelmistoa. Tietokantaa käytetään mallin tarvitsemien lähtötietojen ja simulointitulosten tallennuskohteena, kun taas väli-ohjelmiston tarkoituksena on yhdistää malli tietokantaan ja toimia käyttöliittymänä kehitetylle malli-tietokanta-systeemille. Pienten kaupunkiuomien virtausmallinnuksen havaittiin olevan hankalaa monista eri syistä johtuen. Ensinnäkin virtausvastusten arviointi on vaikeaa, sillä virtaaman muutokset ovat yleensä suuria ja nopeita, tavallisesti käytetyt virtausvastusten arviointimenetelmät eivät sovellu pieniin uomiin ja koska paikalliset energiahäviöt voivat olla suuria. Uomaan sivuilta tuleva vesimäärä saattaa myös olla niin suuri, että se tulisi ottaa huomioon mallinnuksessa, mutta sen arvioiminen on hankalaa. Lisäksi pienet virtaamat ovat laskennallisesta näkökulmasta yleisiä pienissä uomissa, ja nämä saattavat aiheuttaa numeerisia hankaluuksia mallinnuksessa. Kehitetyn mallin verifioimiseksi sitä käytettiin virtaaman mallintamiseen Nummelan Ridalinpurossa. Jotta voitaisiin varmistua, että malli toimii myös haarautuvassa uomaverkostossa ja muissa suunnitelluissa tilanteissa, lisää testejä tulisi suorittaa sopivaa dataa käyttäen. Nykyisellään malli tarjoaa työkalun, jota voidaan käyttää virtaaman mallintamiseen yksittäisessä avouomassa ja jota voidaan helposti muokata ja laajentaa vastaamaan tulevia tarpeita

Interference features in scanning gate conductance maps of quantum point contacts with disorder

We consider quantum point contacts (QPCs) defined within disordered two-dimensional electron gases as studied by scanning gate microscopy. We evaluate the conductance maps in the Landauer approach and wave function picture of electron transport for samples with both low and high electron mobility at finite temperatures. We discuss the spatial distribution of the impurities in the context of the branched electron flow. We reproduce the surprising temperature stability of the experimental interference fringes far from the QPC. Next, we discuss -- previously undescribed -- funnel-shaped features that accompany splitting of the branches visible in previous experiments. Finally, we study elliptical interference fringes formed by an interplay of scattering by the point-like impurities and by the scanning probe. We discuss the details of the elliptical features as functions of the tip voltage and the temperature, showing that the first interference fringe is very robust against the thermal widening of the Fermi level. We present a simple analytical model that allows for extraction of the impurity positions and the electron gas depletion radius induced by the negatively charged tip of the atomic force microscope, and apply this model on experimental scanning gate images showing such elliptical fringes

Considering the settling of dispersed water in the water barrier when calculating the explosion-proof distance at the methane explosion in a mine

Gas dynamics equations are used to simulate the interaction of shock waves with water or rock–dust barriers. The model is enhanced with the presence of dispersed water in the flow and its settling on the walls of the working. An approach to the implementation of the method for solving the problem of the propagation of shock waves in a branched network of mine workings, considering the interaction of shock waves with water barriers has been developed. The approach is based on the use of the numerical method of S.K. Godunov. Examples of solving the problem of the propagation of shock waves from a methane explosion in simulated networks of coal mine workings with water barriers placed in them are give

Dispersive to non-dispersive transition and phase velocity transient for linear waves in plane wake and channel flows

In this study we analyze the phase and group velocity of three-dimensional linear traveling waves in two sheared flows, the plane channel and the wake flows. This was carried out by varying the wave number over a large interval of values at a given Reynolds number inside the ranges 20-100, 1000-8000, for the wake and channel flow, respectively. Evidence is given about the possible presence of both dispersive and non-dispersive effects which are associated with the long and short ranges of wavelength. We solved the Orr-Sommerfeld and Squire eigenvalue problem and observed the least stable mode. It is evident that, at low wave numbers, the least stable eigenmodes in the left branch of the spectrum beave in a dispersive manner. By contrast, if the wavenumber is above a specific threshold, a sharp dispersive to non-dispersive transition can be observed. Beyond this transition, the dominant mode belongs to the right branch of the spectrum. The transient behavior of the phase velocity of small three-dimensional traveling waves was also considered. Having chosen the initial conditions, we then show that the shape of the transient highly depends on the transition wavelength threshold value. We show that the phase velocty can oscillate with a frequency which is equal to the frequency width of the eigenvalue spectrum. Furthermore, evidence of intermediate self-similarity is given for the perturbation field.Comment: 19 pages, 11 figures. Text and discussion improved with respect to the first version. Accepted for publication on Physical Review