Abrupt Changes In Boundary Conditions Research Articles

Transient thermal boundary value problems in the half-space with mixed convective boundary conditions

Numerous problems in thermal engineering give rise to problems with abrupt changes in boundary conditions particularly as regards structural and construction applications. There is a continual need for improved analytical and numerical techniques for the study and analysis of such canonical problems. The present work considers transient heat propagation in a two-dimensional half-space with mixed boundary conditions of the Dirichlet and the Robin (convective) type. The exact temperature field is obtained in double integral form by applying the Wiener–Hopf technique, and this integral is then efficiently evaluated numerically with the help of numerical algorithms developed for the present problem and by exploiting the properties of the integrand. The results are then compared to the numerical solutions using finite element methods, and excellent agreement is shown. Some potential application areas of the theory are also discussed, specifically in the regime where edge effects can influence thermal convection processes.

Formation of the turbulent boundary layer at air blowing through a wall with an abrupt change in boundary conditions

Development of an incompressible turbulent boundary layer with air blowing through a finely perforated flat surface, consisting of a permeable region and impermeable region behind, was studied experimentally. The mass flow rate of injected air Q per an area unit was varied from 0 to 0.2 (kg/s)/m2. Detailed data about the internal structure of the boundary layer in the flow region, characterized by an abrupt change in the flow conditions at the boundary of permeable and impermeable regions, were obtained. A consistent decrease in the local values of skin friction coefficient along a permeable sample and with an increase in the values of Q, reaching 90% at maximal Q, is shown. The role of the flow region behind the zone with an abrupt change in the boundary conditions, essential from the viewpoint of skin friction reduction, is revealed.

Numerical analysis of edge singularities in three‐dimensional elasticity

A numerical method is described for the computation of eigenpairs which characterize the exact solution of linear elastostatic problems in three-dimensions in the vicinity of edge singularities. These may be caused by re-entrant corners, abrupt changes in boundary conditions or material properties. Such singularities are of great interest from the point of view of failure initiation: The eigenpairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes. For this reason, failure theories directly or indirectly involve the eigenpairs and their amplitudes. This paper addresses the problem of determining the edge eigenpairs numerically on the basis of the modified Steklov formulation (presented in Reference 1 in a 2-D framework) in conjunction with the p-version of the finite element method. Numerical results are presented for several cases including isotropic as well as anisotropic multi-material interfaces. © 1997 John Wiley & Sons, Ltd.

Numerical analysis on singular solutions of the Poisson equation in two-dimensions

A numerical method for extracting the coefficients of the asymptotic series solution of the Poisson equation in two dimensions in the vicinity of singular points is presented. This method is an extension of that presented in (Szabo and Yosibash 1996) to non-homogeneous boundary value problems, and is general in the sense that it is applicable to almost any type of point singularity. Numerical experiments for crack-tip singularities, re-entrant corner singularities, abrupt change in boundary conditions, and singularities associated with a multi-material inclusion are presented to substantiate the proposed techniques. Constant as well as varying non-homogeneous “right-hand-side” functions are studied.

Computing edge singularities in elastic anisotropic three-dimensional domains

Computation of eigen-pairs characterizing the linear elastostatic solution in three-dimensional anisotropic domains in the vicinity of edge singularities is addressed. The singularities may be caused by re-entrant corners, abrupt changes in boundary conditions or material properties. Edge singularities in three-dimensional domains are of great interest from the point of view of failure initiation: The eigen-pairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes. For this reason, failure theories directly or indirectly involve the eigen-pairs and their amplitudes. Herein we address the problem of determining the edge eigen-pairs numerically on the basis of the modified Steklov formulation in conjunction with the p-version of the finite element method. The method is very accurate, efficient and robust, and provides complex eigen-pairs if they exist. Several practical problems are studied, and examples are presented for cases including multi-material inclusion problems, cracks in dissimilar materials, and multi-material interfaces at free and clamped edges.

Numerical analysis of singularities in two‐dimensions part 1: Computation of eigenpairs

AbstractA numerical method is described for the computation of eigenpairs which characterize the exact solution of linear second‐order elliptic partial differential equations in two dimensions in the vicinity of singular points. The singularities may be caused by re‐entrant corners and abrupt changes in boundary conditions or material properties.Such singularities are of great interest from the point of view of failure initiation: The eigenpairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes. For this reason, failure theories directly or indirectly involve the eigenpairs and their amplitudes.This paper addresses the problem of determining the eigenpairs numerically on the basis of the Steklov formulation. Numerical results are presented for several cases. Importantly, the method is applicable to three‐dimensional cases.

An efficient finite element method for treating singularities in Laplace's equation

We present a new finite element method for solving partial differential equations with singularities caused by abrupt changes in boundary conditions or sudden changes in boundary shape. Terms from the local solution supplement the ordinary basis functions in the finite element solution. All singular contributions reduce to boundary integrals after a double application of the divergence theorem to the Galerkin integrals, and the essential boundary conditions are weakly enforced using Lagrange multipliers. The proposed method eliminates the need for high-order integration, improves the overall accuracy, and yields very accurate estimates for the singular coefftcients. It also accelerates the convergence with regular mesh refinement and converges rapidly with the number of singular functions. Although here we solve the Laplace equation in two dimensions, the method is applicable to a more general class of problems.

Liquid flow on a vertical plane with abrupt change in boundary conditions

Liquid flow on a vertical plane with abrupt change in boundary conditions

Basal Stress Concentrations Due to Abrupt Changes in Boundary Conditions: A Cause for High Till Concentration at the Bottom of a Glacier

The existence of cold patches at the base of a glacier suggests that the sliding law will depend on these patches, which will essentially affect the viscosity constant. In a poly thermal glacier, such as a glacier which is cold in its lower part and temperate in its upper part, basal boundary conditions change from no-slip to viscous sliding. It is anticipated that the viscosity constant of this sliding law will depend on the distance from the transition line between cold and temperate ice.The mixed boundary conditions, namely no-slip where the ice is cold and viscous sliding where it is temperate, induce large stresses and velocity changes close to the transition line. In fact, it is shown that, for a Newtonian fluid and all investigated discontinuities of boundary data, square-root singularities of the stresses will develop at the transition line. Asymptotic expressions for the basal stresses are derived. The explicit forms of these asymptotic expansions depend on the form of the spatial dependence of the sliding law and, furthermore, on the numerical values of the viscosity coefficient. It is, moreover, argued that the stress concentrations are sufficiently pronounced to account for the removal of basal rock especially in regions of high cleavage concentrations, the details again depending upon the sliding coefficients.No mathematical details of the problem solved are presented as attention is focused on the physical processes.

Laminar Wake Flame Heights

One important measure of material fire hazard is the flame height a given polymer produces upon burning in a specified ambience. Six systems are considered here—two fuel geometries: wall-mounted and free standing; and three flow fields: forced, free and mixed-mode. In each case, the extent of the combusting gas downstream of a pyrolyzing slab is obtained as a function of the fuel’s thermochemical properties. The flow is modeled as a steady, laminar, two-dimensional, nonradiative boundary layer. The combustion is described by a single Shvab-Zeldovich energy-species equation assuming unit Lewis number and a fast one-step overall gas phase reaction. Numerical methods are employed due to the abrupt change in boundary conditions at the end of the pyrolyzing slab. However, an approximate similarity solution is found for forced flow which yields explicit flame heights. Based on these results, explicit functional fits to numerical flame heights are obtained for free and mixed-mode flows. Comparisons between theory and experiment indicate quantitative agreement.