Two-dimensional Steady-state Problems Research Articles

Numerical modelling of convection-diffusion problems with first-order chemical reaction using the dual reciprocity boundary element method

PurposeThe purpose of this paper is to describe an extension of the boundary element method (BEM) and the dual reciprocity boundary element method (DRBEM) formulations developed for one- and two-dimensional steady-state problems, to analyse transient convection–diffusion problems associated with first-order chemical reaction.Design/methodology/approachThe mathematical modelling has used a dual reciprocity approximation to transform the domain integrals arising in the transient equation into equivalent boundary integrals. The integral representation formula for the corresponding problem is obtained from the Green’s second identity, using the fundamental solution of the corresponding steady-state equation with constant coefficients. The finite difference method is used to simulate the time evolution procedure for solving the resulting system of equations. Three different radial basis functions have been successfully implemented to increase the accuracy of the solution and improving the rate of convergence.FindingsThe numerical results obtained demonstrate the excellent agreement with the analytical solutions to establish the validity of the proposed approach and to confirm its efficiency.Originality/valueFinally, the proposed BEM and DRBEM numerical solutions have not displayed any artificial diffusion, oscillatory behaviour or damping of the wave front, as appears in other different numerical methods.

On the Effect of Velocity Profile Type on the Thermocapillary Deformation of Locally Heated Thin Liquid Film

Differences in the temperature field and deformation of locally heated thin liquid film are studied theoretically for various velocity profiles: parabolic velocity profile in the case of gravity-driven film flow, and uniform one in the case of moving heat source in zero-gravity conditions. Conjugate (hydrodynamic and heat) two-dimensional steady-state problem is solved using long-wave approximation. It is shown that under equal other conditions the flow with uniform velocity profile is characterized by dramatic increase in the thermocapillary deformation of the free surface. At the same time this regime possesses higher stability for dry spot formation comparatively with the shear-driven flow regime.

Proper orthogonal decomposition methods for noise reduction in particle-based transport calculations

Proper orthogonal decomposition techniques to reduce noise in the reconstruction of the distribution function in particle-based transport calculations are explored. For two-dimensional steady-state problems, the method is based on low rank truncations of the singular value decomposition of a coarse-grained representation of the particle distribution function. For time-dependent two-dimensional problems or three-dimensional time-independent problems, the use of a generalized low-rank approximation of matrices technique is proposed. The methods are illustrated and tested with Monte Carlo particle simulation data of plasma collisional relaxation and guiding-center transport with collisions in a magnetically confined plasma in toroidal geometry. It is observed that the proposed noise reduction methods achieve high levels of smoothness in the particle distribution function by using significantly fewer particles in the computations.

Groundwater flow to a system of drainage canals

A classical model of groundwater flow to a system of drainage canals is considered within a framework of a two-dimensional steady-state problem. The solution to this problem is derived based on the Riemann-Schwarts principle of symmetry. This solution yields simple analytical relationships expressed in terms of special or elementary functions. Numerical calculations are used to analyze in detail the effect of all physical characteristics of the model on the flow pattern. In particular, it was established that the presence of water in channels has a significant effect on the flow regime. The limiting cases of the scheme are considered, and simple approximated formulas are derived in these cases for the flow and discharge components.

High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations

This paper deals with a high-order accurate discontinuous finite element method for the numerical solution of the Euler equations. The method combines two key ideas which are at the basis of the finite volume and of the finite element method, the physics of wave propagation being accounted for by means of Riemann problems and accuracy being obtained by means of high-order polynomial approximations within elements. We focus our attention on two-dimensional steady-state problems and present higher order accurate (up to fourth-order) discontinuous finite element solutions on unstructured grids of triangles. In particular we show that, in the presence of curved boundaries, a meaningful high-order accurate solution can be obtained only if a corresponding high-order approximation of the geometry is employed. We present numerical solutions of classical test cases computed with linear, quadratic, and cubic elements which illustrate the versatility of the method and the importance of the boundary condition

On the two-dimensional steady-state problem of a viscous gas in an exterior domain

We investigate plane steady flows of a viscous, isothermal, compressible fluid past an obstacle, with nonzero velocity at innity. Using the decomposition of the velocity eld onto its compressible and incompressible parts, we prove an existence and uniqueness theorem, under the assumption that the external data are \suciently small.

DIRECT SENSITIVITY COEFFICIENT METHOD FOR SOLVING TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEMS BY FINITE-ELEMENT SCHEME

Abstract A simple, but versatile, algorithm called the direct sensitivity coefficient (DSC) method has been developed to deal with multidimensional inverse heat conduction (IHC) problems. Different sensitivity coefficients have been used to assess the physical responses of the domain considered under unit loading conditions. The concept of the finite-element discretization has been applied to evaluate the global responses under total loading conditions. The method developed is capable of dealing with unknown surface heat flux, surface temperature, heal transfer coefficient, or any combination of them. In this article the formulation as well as the theory adopted is discussed. The finite-element procedures for solving a two-dimensional steady-state problem art specifically illustrated. Analysis for several selected benchmark problems is then presented, emphasizing accuracy estimations. Both linear and nonlinear problems are considered. Recommendations for further work are also included in this article.

Coupled thermal and stress-strain models for the continuous casting of steels

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Some equations of the Fuchs class in hydro- and aeromechanics

The analytic theory of linear differential equations of the Fuchs class makes it possible to consider a wide range of two-dimensional steady-state problems of the theory of jets, flow through porous media, gas dynamics, etc. Solutions of problems of finding the function that realizes the conformal mapping of the circular polygons occurring in complex flow velocity domains onto the auxiliary canonical domain reduce to such equations. For polygons with a large number of vertices, in the coefficients of these equations, in addition to the unknown affixes of the vertices, there also appear additional (so-called accessory) parameters, which are not completely determined by the position of the singular points of the equations and the exponents in them [1–3], so that the determination of the parameters is a very complicated task. It is shown that in a whole series of cases in which the polygon has angles that are multiples of π/2 and branch cuts (as is often encountered in hydro-, gas- and aeromechanics), the problem of determining the unknown parameters can be completely solved.

A heat meter of finite thickness

The two-dimensional steady-state problem is solved for a heat meter of finite thickness. Corrections to the heat meter readings are estimated.

Interacting continuous medium composed of an elastic solid and an incompressible newtonian fluid

The constitutive equations for an interacting continuum composed of an elastic solid and an incompressible Newtonian fluid are developed. The condition of incompressibility alters the equations previously obtained by Green and Steel. The concept of thermodynamic pressure is introduced and the new equations are compared with those developed by previous investigators.Using these constitutive equations, methods of solution are presented in terms of displacements or a stress function for the steady state condition. The equations developed are shown to reduce to Darcy's law for flow of fluids through a rigid porous medium and Biot's equation of fluid flow through a linear elastic solid. One and two-dimensional steady-state problems are solved as examples.

Moving step load on the surface of a half-space of granular material

The two-dimensional steady-state problem of the effect of a step pressure traveling with superseismic velocity on the surface of a half-space is treated for an elastic-plastic material. The plasticity condition selected is a function of the first and second invariants of the stress tensor, and is suitable for a granular medium where inelastic deformations are due to internal slip subject to Coulomb friction.The problem is inherently nonlinear and leads to a system of coupled nonlinear differential equations which are solved by digital computer. The character of the solutions is radically dependent on the significant nondimensional parameters, i.e. the Mach number, Poisson's ratio and a value α defining the angle of internal friction. A table giving the solutions for various combinations of the parameters is given.