Showing papers on "Boundary value problem published in 1997"

Long-scale evolution of thin liquid films

Abstract: Macroscopic thin liquid films are entities that are important in biophysics, physics, and engineering, as well as in natural settings. They can be composed of common liquids such as water or oil, rheologically complex materials such as polymers solutions or melts, or complex mixtures of phases or components. When the films are subjected to the action of various mechanical, thermal, or structural factors, they display interesting dynamic phenomena such as wave propagation, wave steepening, and development of chaotic responses. Such films can display rupture phenomena creating holes, spreading of fronts, and the development of fingers. In this review a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations. As a result of this long-wave theory, a mathematical system is obtained that does not have the mathematical complexity of the original free-boundary problem but does preserve many of the important features of its physics. The basics of the long-wave theory are explained. If, in addition, the Reynolds number of the flow is not too large, the analogy with Reynolds's theory of lubrication can be drawn. A general nonlinear evolution equation or equations are then derived and various particular cases are considered. Each case contains a discussion of the linear stability properties of the base-state solutions and of the nonlinear spatiotemporal evolution of the interface (and other scalar variables, such as temperature or solute concentration). The cases reducing to a single highly nonlinear evolution equation are first examined. These include: (a) films with constant interfacial shear stress and constant surface tension, (b) films with constant surface tension and gravity only, (c) films with van der Waals (long-range molecular) forces and constant surface tension only, (d) films with thermocapillarity, surface tension, and body force only, (e) films with temperature-dependent physical properties, (f) evaporating/condensing films, (g) films on a thick substrate, (h) films on a horizontal cylinder, and (i) films on a rotating disc. The dynamics of the films with a spatial dependence of the base-state solution are then studied. These include the examples of nonuniform temperature or heat flux at liquid-solid boundaries. Problems which reduce to a set of nonlinear evolution equations are considered next. Those include (a) the dynamics of free liquid films, (b) bounded films with interfacial viscosity, and (c) dynamics of soluble and insoluble surfactants in bounded and free films. The spreading of drops on a solid surface and moving contact lines, including effects of heat and mass transport and van der Waals attractions, are then addressed. Several related topics such as falling films and sheets and Hele-Shaw flows are also briefly discussed. The results discussed give motivation for the development of careful experiments which can be used to test the theories and exhibit new phenomena.

A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers

Abstract: The numerical implementation of an ocean model based on the incompressible Navier Stokes equations which is designed for studies of the ocean circulation on horizontal scales less than the depth of the ocean right up to global scale is described. A "pressure correction" method is used which is solved as a Poisson equation for the pressure field with Neumann boundary conditions in a geometry as complicated as that of the ocean basins. A major objective of the study is to make this inversion, and hence nonhydrostatic ocean modeling, efficient on parallel computers. The pressure field is separated into surface, hydrostatic, and nonhydrostatic components. First, as in hydrostatic models, a two-dimensional problem is inverted for the surface pressure which is then made use of in the three-dimensional inversion for the nonhydrostatic pressure. Preconditioned conjugate-gradient iteration is used to invert symmetric elliptic operators in both two and three dimensions. Physically motivated preconditioners are designed which are efficient at reducing computation and minimizing communication between processors. Our method exploits the fact that as the horizontal scale of the motion becomes very much larger than the vertical scale, the motion becomes more and more hydrostatic and the three- dimensional Poisson operator becomes increasingly anisotropic and dominated by the vertical axis. Accordingly, a preconditioner is used which, in the hydrostatic limit, is an exact integral of the Poisson operator and so leads to a single algorithm that seamlessly moves from nonhydrostatic to hydrostatic limits. Thus in the hydrostatic limit the model is "fast," competitive with the fastest ocean climate models in use today based on the hydrostatic primitive equations. But as the resolution is increased, the model dynamics asymptote smoothly to the Navier Stokes equations and so can be used to address small- scale processes. A "finite-volume" approach is employed to discretize the model in space in which property fluxes are defined normal to faces that delineate the volumes. The method makes possible a novel treatment of the boundary in which cells abutting the bottom or coast may take on irregular shapes and be "shaved" to fit the boundary. The algorithm can conveniently exploit massively parallel computers and suggests a domain decomposition which allocates vertical columns of ocean to each processing unit. The resulting model, which can handle arbitrarily complex geometry, is efficient and scalable and has been mapped on to massively parallel multiprocessors such as the Connection Machine (CM5) using data-parallel FORTRAN and the Massachusetts Institute of Technology data-flow machine MONSOON using the implicitly parallel language Id. Details of the numerical implementation of a model which has been designed for the study of dynamical processes in the ocean from the convective, through the geostrophic eddy, up to global scale are set out. The "kernel" algorithm solves the incompressible Navier Stokes equations on the sphere, in a geometry as complicated as that of the ocean basins with ir- regular coastlines and islands. (Here we use the term "Navier Stokes" to signify that the full nonhydrostatic equations are being employed; it does not imply a particular constitutive relation. The relevant equations for modeling the full complex- ity of the ocean include, as here, active tracers such as tem- perature and salt.) It builds on ideas developed in the compu- tational fluid community. The numerical challenge is to ensure that the evolving velocity field remains nondivergent. Most

On pressure and velocity boundary conditions for the lattice Boltzmann BGK model

Abstract: Pressure ~density! and velocity boundary conditions are studied for 2-D and 3-D lattice Boltzmann BGK models ~LBGK! and a new method to specify these conditions is proposed. These conditions are constructed in consistency with the wall boundary condition, based on the idea of bounceback of the non-equilibrium distribution. When these conditions are used together with the incompressible LBGK model @J. Stat. Phys. 81 ,3 5 ~1995!# the simulation results recover the analytical solution of the plane Poiseuille flow driven by a pressure ~density! difference. The half-way wall bounceback boundary condition is also used with the pressure ~density! inlet/outlet conditions proposed in this paper and in Phys. Fluids 8, 2527 ~1996! to study 2-D Poiseuille flow and 3-D square duct flow. The numerical results are approximately second-order accurate. The magnitude of the error of the half-way wall bounceback boundary condition is comparable with that of other published boundary conditions and it has better stability behavior. © 1997 American Institute of Physics. @S1070-6631~97!03406-5#

A general boundary condition for liquid flow at solid surfaces

Abstract: Modelling fluid flows past a surface is a general problem in science and engineering, and requires some assumption about the nature of the fluid motion (the boundary condition) at the solid interface. One of the simplest boundary conditions is the no-slip condition1,2, which dictates that a liquid element adjacent to the surface assumes the velocity of the surface. Although this condition has been remarkably successful in reproducing the characteristics of many types of flow, there exist situations in which it leads to singular or unrealistic behaviour—for example, the spreading of a liquid on a solid substrate3,4,5,6,7,8, corner flow9,10 and the extrusion of polymer melts from a capillary tube11,12,13. Numerous boundary conditions that allow for finite slip at the solid interface have been used to rectify these difficulties4,5,11,13,14. But these phenomenological models fail to provide a universal picture of the momentum transport that occurs at liquid/solid interfaces. Here we present results from molecular dynamics simulations of newtonian liquids under shear which indicate that there exists a general nonlinear relationship between the amount of slip and the local shear rate at a solid surface. The boundary condition is controlled by the extent to which the liquid ‘feels’ corrugations in the surface energy of the solid (owing in the present case to the atomic close-packing). Our generalized boundary condition allows us to relate the degree of slip to the underlying static properties and dynamic interactions of the walls and the fluid.

Parabolized stability equations

Abstract: Parabolized stability equations (PSE) have opened new avenues to the analysis of the streamwise growth of linear and nonlinear disturbances in slowly varying shear flows such as boundary layers, jets, and far wakes. Growth mechanisms include both algebraic transient growth and exponential growth through primary and higher instabilities. In contrast to the eigensolutions of traditional linear stability equations, PSE solutions incorporate inhomogeneous initial and boundary conditions as do numerical solutions of the Navier-Stokes equations, but they can be obtained at modest computational expense. PSE codes have developed into a convenient tool to analyze basic mechanisms in boundary-layer flows. The most important area of application, however, is the use of the PSE approach for transition analysis in aerodynamic design. Together with the adjoint linear problem, PSE methods promise improved design capabilities for laminar flow control systems.

Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice boltzmann bgk model

Abstract: In this paper we analytically solve the velocity of the lattice Boltzmann BGK equation (LBGK) for several simple flows. The analysis provides a framework to theoretically analyze various boundary conditions. In particular, the analysis is used to derive the slip velocities generated by various schemes for the nonslip boundary condition. We find that the slip velocity is zero as long as Σαfαeα=0 at boundaries, no matter what combination of distributions is chosen. The schemes proposed by Nobleet al. and by Inamuroet al. yield the correct zeroslip velocity, while some other schemes, such as the bounce-back scheme and the equilibrium distribution scheme, would inevitably generate a nonzero slip velocity. The bounce-back scheme with the wall located halfway between a flow node and a bounce-back node is also studied for the simple flows considered and is shown to produce results of second-order accuracy. The momentum exchange at boundaries seems to be highly related to the slip velocity at boundaries. To be specific, the slip velocity is zero only when the momentum dissipated by boundaries is equal to the stress provided by fluids.

Turbulence Modeling Validation, Testing, and Development

Abstract: The primary objective of this work is to provide accurate numerical solutions for selected flow fields and to compare and evaluate the performance of selected turbulence models with experimental results. Four popular turbulence models have been tested and validated against experimental data often turbulent flows. The models are: (1) the two-equation k-epsilon model of Wilcox, (2) the two-equation k-epsilon model of Launder and Sharma, (3) the two-equation k-omega/k-epsilon SST model of Menter, and (4) the one-equation model of Spalart and Allmaras. The flows investigated are five free shear flows consisting of a mixing layer, a round jet, a plane jet, a plane wake, and a compressible mixing layer; and five boundary layer flows consisting of an incompressible flat plate, a Mach 5 adiabatic flat plate, a separated boundary layer, an axisymmetric shock-wave/boundary layer interaction, and an RAE 2822 transonic airfoil. The experimental data for these flows are well established and have been extensively used in model developments. The results are shown in the following four sections: Part A describes the equations of motion and boundary conditions; Part B describes the model equations, constants, parameters, boundary conditions, and numerical implementation; and Parts C and D describe the experimental data and the performance of the models in the free-shear flows and the boundary layer flows, respectively.

OPPDIF: A Fortran program for computing opposed-flow diffusion flames

Abstract: OPPDIF is a Fortran program that computes the diffusion flame between two opposing nozzles. A similarity transformation reduces the two-dimensional axisymmetric flow field to a one-dimensional problem. Assuming that the radial component of velocity is linear in radius, the dependent variables become functions of the axial direction only. OPPDIF solves for the temperature, species mass fractions, axial and radial velocity components, and radial pressure gradient, which is an eigenvalue in the problem. The TWOPNT software solves the two-point boundary value problem for the steady-state form of the discretized equations. The CHEMKIN package evaluates chemical reaction rates and thermodynamic and transport properties.

Simulation of rayleigh-benard convection using a lattice boltzmann method

Abstract: Rayleigh-B\'enard convection is numerically simulated in two and three dimensions using a recently developed two-component lattice Boltzmann equation (LBE) method. The density field of the second component, which evolves according to the advection-diffusion equation of a passive scalar, is used to simulate the temperature field. A body force proportional to the temperature is applied, and the system satisfies the Boussinesq equation except for a slight compressibility. A no-slip, isothermal boundary condition is imposed in the vertical direction, and periodic boundary conditions are used in horizontal directions. The critical Rayleigh number for the onset of the Rayleigh-B\'enard convection agrees with the theoretical prediction. As the Rayleigh number is increased higher, the steady two-dimensional convection rolls become unstable. The wavy instability and aperiodic motion observed, as well as the Nusselt number as a function of the Rayleigh number, are in good agreement with experimental observations and theoretical predictions. The LBE model is found to be efficient, accurate, and numerically stable for the simulation of fluid flows with heat and mass transfer.

Propagation and stability of waves of electrical activity in the cerebral cortex

Abstract: Nonlinear equations are introduced to model the behavior of the waves of cortical electrical activity that are responsible for signals observed in electroencephalography. These equations incorporate nonlinearities, axonal and dendritic lags, excitatory and inhibitory neuronal populations, and the two-dimensional nature of the cortex, while rendering nonlinear features far more tractable than previous formulations, both analytically and numerically. The model equations are first used to calculate steady-state levels of cortical activity for various levels of stimulation. Dispersion equations for linear waves are then derived analytically and an analytic expression is found for the linear stability boundary beyond which a seizure will occur. The effects of boundary conditions in determining global eigenmodes are also studied in various geometries and the corresponding eigenfrequencies are found. Numerical results confirm the analytic ones, which are also found to reproduce existing results in the relevant limits, thereby elucidating the limits of validity of previous approximations.

An Earth-like numerical dynamo model

Abstract: The mechanism by which the Earth and other planets maintain their magnetic fields against ohmic decay is among the longest standing problems in planetary science. Although it is widely acknowledged that these fields are maintained by dynamo action, the mechanism by which the dynamo operates is in large part not understood. Numerical simulations of the dynamo process in the Earth's core1,2,3,4 have produced magnetic fields that resemble the Earth's field, but it is unclear whether these models accurately represent the extremely low values of viscosity believed to be appropriate to the core. Here we describe the results of a numerical investigation of the dynamo process that adopts an alternative approach5 to this problem in which, through the judicious choice of boundary conditions, the effects of viscosity are rendered unimportant. We thereby obtain a solution that at leading order operates in an Earth-like dynamical regime. The morphology and evolution of the magnetic field and the fluid flow at the core–mantle boundary are similar to those of the Earth, and the field within the core is qualitatively similar to that proposed on theoretical grounds6.

Artificial Neural Networks for Solving Ordinary and Partial Differential Equations

Abstract: We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the boundary (or initial) conditions and contains no adjustable parameters. The second part is constructed so as not to affect the boundary conditions. This part involves a feedforward neural network, containing adjustable parameters (the weights). Hence by construction the boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ODE's, to systems of coupled ODE's and also to PDE's. In this article we illustrate the method by solving a variety of model problems and present comparisons with finite elements for several cases of partial differential equations.

Numerical schemes for conservation laws

Abstract: Initial Value Problems for Scalar Conservation Laws in 1-D. Initial Value Problems for Scalar Conservation Laws in 2-D. Initial Value Problems for Systems in 1-D. Initial Value Problems for Systems of Conservation Laws in 2-D. Initial Boundary Value Problems for Conservation Laws. Convection-Dominated Problems. List of Figures. References. Index.

Numerical Simulation in Fluid Dynamics: A Practical Introduction

Abstract: Notation 1. Numerical simulation - a key technology of the future 2. The mathematical description of flows 3. The numerical treatment of the Navier-Stokes equations 4. Visualization techniques 5. Example applications 6. Free boundary value problems 7. Example applications for free boundary value problems 8. Parallelization 9. Energy transport 10. Turbulence 11. Extension to Three dimensions 12. Concluding remarks Appendix A Appendix B Bibliography Index.

The boundary node method for potential problems

Abstract: The Element-Free Galerkin (EFG) method allows one to use a nodal data structure (usually with an underlying cell structure) within the domain of a body of arbitrary shape. The usual EFG combines Moving Least-Squares (MLS) interpolants with a variational principle (weak form) and has been used to solve two-dimensional (2-D) boundary value problems in mechanics such as in potential theory, elasticity and fracture. This paper proposes a combination of MLS interpolants with Boundary Integral Equations (BIE) in order to retain both the meshless attribute of the former and the dimensionality advantage of the latter! This new method, called the Boundary Node Method (BNM), only requires a nodal data structure on the bounding surface of a body whose dimension is one less than that of the domain itself. An underlying cell structure is again used for numerical integration. In principle, the BNM, for 3-D problems, should be extremely powerful since one would only need to put nodes (points) on the surface of a solid model for an object. Numerical results are presented in this paper for the solution of Laplace's equation in 2-D. Dirichlet, Neumann and mixed problems have been solved, some on bodies with piecewise straight and others with curved boundaries. Results from these numerical examples are extremely encouraging. © 1997 by John Wiley & Sons, Ltd.

Proposed Inflow/Outflow Boundary Condition for Direct Computation of Aerodynamic Sound

Abstract: A simple new zonal boundary condition has been proposed. It is based upon the addition of dissipative and convective terms to the compressible Navier Stokes equations

A kind of approximate solution technique which does not depend upon small parameters — II. An application in fluid mechanics

Abstract: In this paper, the non-linear approximate technique called Homotopy Analysis Method proposed by Liao is further improved by introducing a non-zero parameter into the traditional way of constructing a homotopy. The 2D viscous laminar flow over an infinite flat-plain governed by the non-linear differential equation f′''(η) + f(η)f″(η) 2 = 0 with boundary conditions f(0) = f′(0) = 0, f′(+ ∞) = 1 is used as an example to describe its basic ideas. As a result, a family of approximations is obtained for the above-mentioned problem, which is much more general than the power series given by Blasius [Z. Math. Phys. 36, 1(1908)] and can converge even in the whole region η ϵ [0, + ∞). Moreover, the Blasius' solution is only a special case of ours. We also obtain the second-derivative of f(η) at η = 0, i.e. f″(0) = 0.33206, which is exactly the same as the numerical result given by Howarth [Proc. Roy. Soc. London A164, 547 (1938)].

Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain

Abstract: The exact boundary controllability of linear and nonlinear Korteweg-de Vries equation on bounded domains with various boundary conditions is studied. When boundary conditions bear on spatial derivatives up to order 2, the exact controllability result by Russell-Zhang is directly proved by means of Hilbert Uniqueness Method. When only the first spatial derivative at the right endpoint is assumed to be controlled, a quite different analysis shows that exact controllability holds too. From this last result we derive the exact boundary controllability for nonlinear KdV equation on bounded domains, for sufficiently small initial and final states.

Quasilinear Elliptic Equations with Degenerations and Singularities

Abstract: Presents modern methods and techniques for solving boundary value problems for nonlinear elliptic operators. Focus is upon existence and bifurcation results in appropriate Sobolev spaces. The subject of this book is the theory of nonlinear boundary value problems for elliptic operators with degeneration and singularity. It focuses on the existence and bifurcation of weak solutions in appropriate weighted Sobolev spaces. The main tools are functional analytic methods based on the variational approach and the theory of topological degree of monotone type nonlinear mapping. Topics covered include: existence results for higher order boundary value problems in a rather general setting. An extensive study of the p-Laplacian and its degenerated and singular modifications, mainly existence and bifurcation results on bounded domains as well as on the whole Euclidean space. The text requires some basic knowledge of nonlinear functional analysis and differential equations. Elementary facts on function spaces and the theory of nonlinear operators are discussed in the first chapter.

Complex coordinate stretching as a generalized absorbing boundary condition

Abstract: By complex coordinate stretching and a change of variables, it is shown simply that PML is reflectionless for all frequencies and all angles. Also, Maxwell's equations for PML media reduce to ordinary Maxwell's equations with complex coordinate systems. Many closed-form solutions for Maxwell's equations map to corresponding closed-form solutions in complex coordinate systems. Numerical simulations with the closed-form solutions show that metallic boxes lined with PML media are highly absorptive. These closed-form solutions lend a better understanding to the absorptive properties of PML media. For instance, they explain why a PML medium is absorptive when a dielectric or metallic interface extends to the edge to a simulation region where PML media reside. More importantly, the complex coordinate stretching method can be generalized to non-Cartesian coordinate systems, providing absorbing boundary conditions in these coordinate systems. © 1997 John Wiley & Sons, Inc. Microwave Opt Technol Lett 15: 363–369, 1997.

A free-surface boundary condition for including 3D topography in the finite-difference method

Abstract: A flexible and simple way of introducing stress-free boundary conditions for including three-dimensional (3D) topography in the finite-difference method is presented. The 3D topography is discretized in a staircase by stacking unit material cells in a staggered-grid scheme. The shear stresses are distributed on the 12 edges of the unit material cell so that only shear stresses appear on the free surface and normal stresses always remain embedded within the solid region. This configuration makes it possible to implement stress-free boundary conditions at the free surface by setting the Lame coefficients λ and μ to zero without generating any physically unjustified condition. Arbitrary 3D topographies are realized by changing the distribution of λ and μ in the computational domain. Our method uses a parsimonious staggered-grid scheme that requires only 3/4 of the memory used in the conventional staggered-grid scheme in which six stress components and three velocity components need to be stored. Numerical tests indicate that 25 grids per wavelength are required for stable calculation. The finite-difference results are compared with those of the boundary-element method for the two-dimensional (2D) semi-circular canyon model. We also present the responses of a segment of semi-circular canyon and hemispherical cavity to vertically incident plane P, SV , and SH waves and discuss the response of a Gaussian hill to an isotropic point source embedded in the hill. In the segment of semi-circular canyon, the later portions of the synthetics are characterized by phases scattered from the two vertical side walls. The hemispherical cavity and 2D semi-circular canyon both show focusing of energy at the bottom of the cavity, although the focusing effect is stronger in the former geometry. Focusing and defocusing effects due to the strong topography of the Gaussian hill produce a strong amplification of displacements at a spot located on the flank opposite to the source. Backscattering from the top of the hill is also clearly seen.

Adjoint equations in CFD: duality, boundary conditions and solution behaviour

Abstract: The first half of this paper derives the adjoint equations for inviscid and viscous compressible flow, with the emphasis being on the correct formulation of the adjoint boundary conditions and restrictions on the permissible choice of operators in the linearised functional. It is also shown that the boundary conditions for the adjoint problem can be simplified through the use of a linearised perturbation to generalised coordinates. The second half of the paper constructs the Green's functions for the quasi-lD and 2D Euler equations. These are used to show that the adjoint variables have a logarithmic singularity at the sonic line in the quasi-lD case, and a weak inverse square-root singularity at the upstream stagnation streamline in the 2D case, but are continuous at shocks in both cases.

Micromechanical models for graded composite materials

Abstract: Elastic response of selected plane-array models of graded composite microstructures is examined under both uniform and linearly varying boundary tractions and displacements, by means of detailed finite element studies of large domains containing up to several thousand inclusions. Models consisting of piecewise homogeneous layers with equivalent elastic properties estimated by Mori-Tanaka and selfconsistent methods are also analysed under similar boundary conditions. Comparisons of the overall and local fields predicted by the discrete and homogenized models are made using a C/SiC composite system with very different Young's moduli of the phases, and relatively steep composition gradients. The conclusions reached from these comparisons suggest that in those parts of the graded microstructure which have a well-defined continuous matrix and discontinuous second phase, the overall properties and local fields are predicted by Mori-Tanaka estimates. On the other hand, the response of graded materials with a skeletal microstructure in a wide transition zone between clearly defined matrix phases is better approximated by the self-consistent estimates. Certain exceptions are noted for loading by overall transverse shear stress. The results suggest that the averaging methods originally developed for statistically homogeneous aggregates may be selectively applied, with a reasonable degree of confidence, to aggregates with composition gradients, subjected to both uniform and nonuniform overall loads.

Helmholtz equation-least-squares method for reconstructing the acoustic pressure field

Abstract: A method using spherical wave expansion theory to reconstruct acoustic pressure field from a vibrating object is developed. The radiated acoustic pressures are obtained by means of an expansion of independent functions generated by the Gram–Schmidt orthonormalization with respect to the particular solutions to the Helmholtz equation on the vibrating surface under consideration. The coefficients associated with these independent functions are determined by requiring the assumed form of solution to satisfy the pressure boundary condition at the measurement points. The errors incurred in this process are minimized by the least-squares method. Once these coefficients are specified, the acoustic pressure at any point, including the source surface, is completely determined. In this paper, this method is used to reconstruct the surface acoustic pressures based on the measured acoustic pressure signals in the field. It is shown that this method can be applied to both separable and nonseparable geometries, and the...