Direct numerical simulations of transverse and spanwise magnetic field effects on turbulent flow in a 2:1 aspect ratio rectangular duct

Magnetic fields are used extensively to direct liquid metal flows in material processing. Continuous casting of steel uses different configurations of magnetic fields to optimize turbulent flows in rectangular cross-sections to minimize defects in the solidified steel product. Realizing the importance of a magnetic field on turbulent flows in rectangular cross-sections, the present work is aimed at understanding the effect of a magnetic field on the turbulent metal flow at a nominal bulk Reynolds number of ∼5300 (based upon full duct height) (Re_τ = 170, based upon half duct height) and Hartmann numbers (based upon half duct height) of 0, 6.0 and 8.25 in a 2:1 aspect ratio rectangular duct. Direct numerical simulations in a non-MHD 2:1 aspect ratio duct followed by simulations with transverse and span-wise magnetic fields have been performed with 224 × 120 × 512 cells (∼13.7 million cells). The fractional step method with second order space and time discretization schemes has been used to solve the coupled Navier-Stokes-MHD equations. Instantaneous and time-averaged natures of the flow have been examined through distribution of velocities, various turbulence parameters and budget terms. Spanwise (horizontal) magnetic field reorganizes and suppresses secondary flows more strongly. Turbulence suppression and velocity flattening effects are stronger with transverse (vertical) magnetic field.     

A model for semiconductor quantum dot molecule based on the current spin density functional theory

Based on the current spin density functional theory, a theoretical model of three vertically aligned semiconductor quantum dots is proposed and numerically studied. This quantum dot molecule (QDM) model is treated with realistic hard-wall confinement potential and external magnetic field in three-dimensional setting. Using the effective-mass approximation with band nonparabolicity, the many-body Hamiltonian results in a cubic eigenvalue problem from a finite difference discretization. A self-consistent algorithm for solving the Schrödinger-Poisson system by using the Jacobi-Davidson method and GMRES is given to illustrate the Kohn-Sham orbitals and energies of six electrons in the molecule with some magnetic fields. It is shown that the six electrons residing in the central dot at zero magnetic field can be changed to such that each dot contains two electrons with some feasible magnetic field. The Förster-Dexter resonant energy transfer may therefore be generated by two individual QDMs. This may motivate a new paradigm of Fermionic qubits for quantum computing in solid-state systems.     

A conservative discontinuous Galerkin scheme for the 2D incompressible Navier–Stokes equations

In this paper we consider a conservative discretization of the two-dimensional incompressible Navier–Stokes equations. We propose an extension of Arakawa’s classical finite difference scheme for fluid flow in the vorticity–stream function formulation to a high order discontinuous Galerkin approximation. In addition, we show numerical simulations that demonstrate the accuracy of the scheme and verify the conservation properties, which are essential for long time integration. Furthermore, we discuss the massively parallel implementation on graphic processing units.     

Finite element analysis of viscous,incompressible fluid flow: Part 1 : Basic methodology

A numerical procedure is developed for the analysis of general two-dimensional flows of viscous, incompressible fluids using the finite element method. The partial differential equations describing the continuum motion of the fluid are discretized by using an integral energy balance approach in conjunction with the finite element approximation. The nonlinear algebraic equations resulting from the discretization process are solved using a Picard iteration technique.A number of computational procedures are developed that allow significant reductions to be made in the computational effort required for the analysis of many flow problems. These techniques include a coarse-to-fine-mesh rezone procedure for the detailed study of regions of particular interest in a flow field and a special finite element to model far-field regions in external flow problems.     

Shape sensitivity analysis with design-dependent loadings—equivalence between continuum and discrete derivatives

The purpose of this paper is twofold: (1) showing equivalence between continuum and discrete formulations in sensitivity analysis when a linear velocity field is used and (2) presenting shape sensitivity formulations for design-dependent loadings. The equations for structural analysis are often composed of the stiffness part and the applied loading part. The shape sensitivity formulations for the stiffness part were well-developed in the literature, but not for the loading part, especially for body forces and surface tractions. The applied loads are often assumed to be conservative or design-independent. In shape design problems, however, the applied loads are often functions of design variables. In this paper, shape sensitivity formulations are presented when the body forces and surface tractions depend on shape design variables. Especially, the continuum–discrete (C–D) and discrete–discrete (D–D) approaches are compared in detail. It is shown that the two methods are theoretically and numerically equivalent when the same discretization, numerical integration, and linear design velocity fields are used. The accuracy of sensitivity calculation is demonstrated using a cantilevered beam under uniform pressure and an arch dam crown cantilever under gravity and hydrostatic loading at the upstream face of the structure. It is shown that the sensitivity results are consistent with finite difference results, but different from the analytical sensitivity due to discretization and approximation errors of numerical analysis.     

An efficient coupling of FORM and Karhunen–Loève series expansion

The topic of this paper is the solution of reliability problems where failure is influenced by the spatial random fluctuations of loads and material properties. Homogeneous random fields are used to model this kind of uncertainty. The first step of the investigation is the random field discretization, which transforms a random field into a finite set of random variables. The second step is the reliability analysis, which is performed using the FORM in this paper. A parametric analysis of the reliability index is usually performed with respect to the random field discretization accuracy. This approach requires several independent reliability analyses. A new and efficient approach is proposed in this paper. The Karhunen–Loève series expansion is combined with the FEM for the discretization of the random fields. An efficient solution of the reliability problem is proposed to predict the reliability index as the discretization accuracy increases.     

A posteriori error analysis of an augmented fully-mixed formulation for the stationary Boussinesq model

In this paper we undertake an a posteriori error analysis along with its adaptive computation of a new augmented fully-mixed finite element method that we have recently proposed to numerically simulate heat driven flows in the Boussinesq approximation setting. Our approach incorporates as additional unknowns a modified pseudostress tensor field and an auxiliary vector field in the fluid and heat equations, respectively, which possibilitates the elimination of the pressure. This unknown, however, can be easily recovered by a postprocessing formula. In turn, redundant Galerkin terms are included into the weak formulation to ensure well-posedness. In this way, the resulting variational formulation is a four-field augmented scheme, whose Galerkin discretization allows a Raviart–Thomas approximation for the auxiliary unknowns and a Lagrange approximation for the velocity and the temperature. In the present work, we propose a reliable and efficient, fully-local and computable, residual-based a posteriori error estimator in two and three dimensions for the aforementioned method. Standard arguments based on duality techniques, stable Helmholtz decompositions, and well-known results from previous works, are the main underlying tools used in our methodology. Several numerical experiments illustrate the properties of the estimator and further validate the expected behavior of the associated adaptive algorithm.     

Streamwise computation of duct flows

A computational method is presented to calculate momentum and energy transport in two-dimensional viscous compressible duct flows. The flow in the duct is partitioned into finite streams. The difference equations are then obtained by applying momentum and energy conservation principles directly to the individual streams. The method is applicable to laminar and turbulent flows.     

High-Resolution Finite Volume Methods on Unstructured Grids for Turbulence and Aeroacoustics

In this paper we focus on the application of a higher-order finite volume method for the resolution of Computational Aeroacoustics problems. In particular, we present the application of a finite volume method based in Moving Least Squares approximations in the context of a hybrid approach for low Mach number flows. In this case, the acoustic and aerodynamic fields can be computed separately. We focus on two kinds of computations: turbulent flow and aeroacoustics in complex geometries. Both fields require very accurate methods to capture the fine features of the flow, small scales in the case of turbulent flows and very low-amplitude acoustic waves in the case of aeroacoustics. On the other hand, the use of unstructured grids is interesting for real engineering applications, but unfortunately, the accuracy and efficiency of the numerical methods developed for unstructured grids is far to reach the performance of those methods developed for structured grids. In this context, we propose the FV-MLS method as a tool for accurate CAA computations on unstructured grids.     

Numerical methods of higher order of accuracy for incompressible flows

The work deals with numerical solution of the Navier–Stokes equations for incompressible fluid using finite volume and finite difference methods. The first method is based on artificial compressibility where continuity equation is changed by adding pressure time derivative. The second method is based on solving momentum equations and the Poisson equation for pressure instead of continuity equation. The numerical solution using both methods is compared for backward facing step flows. The equations are discretized on orthogonal grids with second, fourth and sixth orders of accuracy as well as third order accurate upwind approximation for convective terms. Not only laminar but also turbulent regimes using two-equation turbulence models are presented.     

Application of mesh-adaptation for pollutant transport by water flow

An adaptive finite volume method is proposed for the numerical solution of pollutant transport by water flows. The shallow water equations with eddy viscosity, bottom friction forces and wind shear stresses are used for modelling the water flow whereas, a transport-diffusion equation is used for modelling the advection and dispersion of pollutant concentration. The adaptive finite volume method uses simple centred-type discretization for the source terms, can handle complex topography using unstructured grids and satisfies the conservation property. The adaptation criteria are based on monitoring the pollutant concentration in the computational domain during its dispersion process. The emphasis in this paper is on the application of the proposed method for numerical simulation of pollution dispersion in the Strait of Gibraltar. Results are presented using different tidal conditions and wind-induced flow fields in the Strait.     

Visualization of vorticity and vortices in wall-bounded turbulent flows

This study was initiated by the scientifically interesting prospect of applying advanced visualization techniques to gain further insight into various spatio-temporal characteristics of turbulent flows. The ability to study complex kinematical and dynamical features of turbulence provides means of extracting the underlying physics of turbulent fluid motion. The objective is to analyze the use of a vorticity field line approach to study numerically generated incompressible turbulent flows. In order to study the vorticity field, we present a field line animation technique which uses a specialized particle advection and seeding strategy. Efficient analysis is achieved by decoupling the rendering stage from the preceding stages of the visualization method. This allows interactive exploration of multiple fields simultaneously, which sets the stage for a more complete analysis of the flow field. Multifield visualizations are obtained using a flexible volume rendering framework which is presented in this paper. Vorticity field lines have been employed as indicators to provide a means to identify "ejection" and "sweep" regions; two particularly important spatio-temporal events in wall-bounded turbulent flows. Their relation to the rate of turbulent kinetic energy production and viscous dissipation, respectively, have been identified.     

Streamwise computation of three-dimensional parabolic flows

A new approach to calculate three-dimensional parabolic flows is presented. The flow field is computed by calculating velocity along a set of streamlines. The dependent variables commonly used in the computation of three-dimensional flows are the three velocity components. In contrast, the dependent variables in the present approach are the streamwise velocity and the two coordinates, in the cross-stream plane, of the chosen streamlines. The streamwise velocity is calculated from the finite difference equations obtained by applying Euler's momentum theorem to streamtubes constructed around the chosen streamlines; the streamline coordinates are calculated from the conservation of mass. Results of the calculations, based on the present approach, are compared with the experimental data for flow through rectangular ducts; the agreement is satisfactory.     

Application of mesh-adaptation for pollutant transport by water flow

An adaptive finite volume method is proposed for the numerical solution of pollutant transport by water flows. The shallow water equations with eddy viscosity, bottom friction forces and wind shear stresses are used for modelling the water flow whereas, a transport-diffusion equation is used for modelling the advection and dispersion of pollutant concentration. The adaptive finite volume method uses simple centred-type discretization for the source terms, can handle complex topography using unstructured grids and satisfies the conservation property. The adaptation criteria are based on monitoring the pollutant concentration in the computational domain during its dispersion process. The emphasis in this paper is on the application of the proposed method for numerical simulation of pollution dispersion in the Strait of Gibraltar. Results are presented using different tidal conditions and wind-induced flow fields in the Strait.     

Numerical prediction of sound generated from flows with a low Mach number

Numerical computations of sound generated from flows with a low Mach number are presented based on Lighthill’s acoustic analogy with an assumption that sound does not alter the flow field from which it is generated. The source fluctuations of the flow field are computed by a large-eddy simulation (LES) with Dynamic Smagorinsky Model (DSM) and they are fed to the following acoustical computation as input data. An explicit/implicit finite element method with second order accuracy both in time and space is used for flow field discretization. The method is applied to the prediction of sound in three different classes of problems: far-field sound generated from flow around a bluff body, sound resulting from blade-stator interaction of turbomachinery and sound due to a turbulent boundary layer on an aerofoil. The computed frequency spectra of the sound show a fairly good agreement with the measured spectra for all the cases.     

Large eddy simulation of turbulent heat transport in the Strait of Gibraltar

We develop a numerical model for large eddy simulation of turbulent heat transport in the Strait of Gibraltar. The flow equations are the incompressible Navier–Stokes equations including Coriolis forces and density variation through the Boussinesq approximation. The turbulence effects are incorporated in the system by considering the Smagorinsky model. As a numerical solver we propose a finite element semi-Lagrangian method. The solution procedure consists of combining a non-oscillatory semi-Lagrangian scheme for time discretization with the finite element method for space discretization. Numerical results illustrate a buoyancy-driven circulations along the Strait of Gibraltar and the sea-surface temperature is flushed out and move to northeast coast. The Ocean discharge and the temperature difference are shown to control the plume structure.     

Large eddy simulation of turbulent heat transport in the Strait of Gibraltar

We develop a numerical model for large eddy simulation of turbulent heat transport in the Strait of Gibraltar. The flow equations are the incompressible Navier–Stokes equations including Coriolis forces and density variation through the Boussinesq approximation. The turbulence effects are incorporated in the system by considering the Smagorinsky model. As a numerical solver we propose a finite element semi-Lagrangian method. The solution procedure consists of combining a non-oscillatory semi-Lagrangian scheme for time discretization with the finite element method for space discretization. Numerical results illustrate a buoyancy-driven circulations along the Strait of Gibraltar and the sea-surface temperature is flushed out and move to northeast coast. The Ocean discharge and the temperature difference are shown to control the plume structure.     

Numerical solutions of the one-phase classical Stefan problem using an enthalpy green element formulation

The enthalpy method is exploited in tackling a heat transfer problem involving a change of state. The resulting governing equation is then solved with a hybrid finite element - boundary element technique known as the Green element method (GEM). Two methods of approximation are employed to handle the time derivative contained in the discrete element equation. The first involves a finite difference method, while the second utilizes a Galerkin finite element approach. The performance of both methods are assessed with a known closed form solution. The finite element based time discretization, despite its greater challenge, yields less reliable numerical results. In addition a numerical stability test of both methods based on a Fourier series analysis explain the dispersive characters of both techniques, and confirms that replication of correct results is largely attributed to their ability to handle the harmonics of small wavelengths which are usually dominant in the vicinity of a front.     

Convergence of a lowest-order finite element method for the transmission eigenvalue problem

The transmission eigenvalue problem arises in scattering theory. The main difficulty in its analysis is the fact that, depending on the chosen formulation, it leads either to a quadratic eigenvalue problem or to a non-classical mixed problem. In this paper we prove the convergence of a mixed finite element approximation. This approach, which is close to the Ciarlet–Raviart discretization of biharmonic problems, is based on Lagrange finite elements and is one of the less expensive methods in terms of the amount of degrees of freedom. The convergence analysis is based on classical abstract spectral approximation result and the theory of mixed finite element methods for solving the stream function–vorticity formulation of the Stokes problem. Numerical experiments are reported in order to assess the efficiency of the method.