Spectral Boundary Element Method Research Articles

Droplet motion in a microconfined shear flow via a three-dimensional spectral boundary element method

A 3D spectral boundary element method is employed to compute the dynamics of a single droplet in a microconfined shear flow. Comparisons have been made for the motion of an initially spherical droplet near a single wall and that between two parallel plates. Investigations are conducted for the influences of the capillary number, viscosity ratio, and initial location of the droplet on the droplet deformation, orientation, velocities, as well as the transition between the initial rapid deformation and the subsequent relaxation stage. Computational results for the deformation and velocities are compared with analytical predictions. It is found that the analytical predictions are limited for small deformations, large droplet-wall distances, and near equiviscous droplets.

Migration and deformation of bubbles rising in a wall-bounded shear flow at finite Reynolds number

The quasi-steady migration and deformation of bubbles rising in a wall-bounded linear shear flow are investigated experimentally in the low-but-finite-Reynolds-number regime. A travelling optical device that follows the bubble is used for this purpose. This apparatus allows us to determine accurately the bubble radius, contour and rising speed, together with the distance between the bubble and the wall. Thereby the transverse component of the hydrodynamic force is obtained for Reynolds numbers Re (based on the bubble diameter and slip velocity of the bubble in the undisturbed shear flow) less than 5. The results indicate that in the range 0.5 Re J. Fluid Mech ., vol. 246, 1993, pp. 249–265) for rigid spheres, whereas it becomes larger than the theoretical prediction for Re > 1.5. In the regime in which bubble deformation is significant, the shape of the bubble and the deformation-induced transverse force are determined both experimentally and computationally, using a spectral boundary element method. Both estimates are found to be in good agreement with each other, while the theory of Magnaudet, Takagi & Legendre ( J. Fluid Mech ., vol. 476, 2003, pp. 115–157) is found to predict accurately the deformation but fails to predict quantitatively the deformation-induced transverse force.

Interfacial dynamics in Stokes flow via a three-dimensional fully-implicit interfacial spectral boundary element algorithm

Since the pioneering work of Youngren and Acrivos [G.K. Youngren, A. Acrivos, On the shape of a gas bubble in a viscous extensional flow, J. Fluid Mech. 76 (1976) 433–442] 30 years ago, interfacial dynamics in Stokes flow has been implemented through explicit time integration of boundary integral schemes which require that the time step is sufficiently small to ensure numerical stability. To avoid this difficulty, we have developed an efficient, fully-implicit time-integration algorithm based on a mathematically rigorous combination of implicit formulas with a Jacobian-free three-dimensional Newton method. The resulting algorithm preserves the stability of the employed implicit formula and thus it has strong stability properties, e.g. it is not affected by the Courant condition or by physical stiffness such as that associated with the critical conditions of interfacial deformation. In our work, the numerical solution of our implicit algorithm is achieved through our spectral boundary element method. Our numerical results for free-suspended droplets are in excellent agreement with experimental findings, analytical predictions and earlier computational results at both subcritical and supercritical conditions, and establish the properties of our fully-implicit spectral boundary element algorithm.

Interception of two spheroidal particles in shear flow

A numerical method is developed for simulating the motion of particles with arbitrary shape in an effectively infinite or bounded viscous flow. The particle translational and angular velocities are computed directly by solving an integral equation of the second kind arising from the double-layer representation for Stokes flow, subject to a specified force and torque. The deflated integral equation is solved by the method successive substitutions using a spectral boundary-element method. In the case of force- and torque-free particles, the contribution of the particles to the effective viscosity of the suspension is expressed in terms of the distribution density of the double-layer potential over the particle surfaces. The method is applied to investigate the interception of two spheroidal particles in simple shear flow, with emphasis on the net particle displacement and shift in the phase of rotation after separation, and on the transient signature of the interception on the effective viscosity of the suspension. The net particle displacement normal to the streamlines of the shear flow is found to have both positive and negative values depending on the relative particle configuration before interception. For moderate particle separations, the phase of rotation is shifted only slightly with respect to the Jeffery orbit executed by a particle in isolation.

Stochastic BEM with spectral approach in elastostatic and elastodynamic problems with geometrical uncertainty

The present paper proposes a method for stochastic problems that have uncertainty in the boundary geometry. The method is developed by applying the spectral stochastic approach to the boundary element method and is called the spectral stochastic boundary element method (SSBEM). In the SSBEM, the uncertainty in the boundary geometry is represented by the Karhunen–Loève expansion. It is shown that, by utilizing material derivative, variation of boundary element matrices associated with the geometrical fluctuation of the boundary can be approximated by the Taylor expansion. The solution is represented by a stochastic process expressed in the form of polynomial chaos expansion. The stochastic equation is then projected on a homogeneous chaos space. This procedure reduces a stochastic equation to an ordinary linear matrix equation that can be solved by conventional schemes. The SSBEM can estimate not only mean values and variances of the solutions but also their probability density functions. In order to examine the performance, the SSBEM is applied to two-dimensional elastostatic and elastodynamic problems with geometrical boundary uncertainty. Computation results of the SSBEM exhibit good agreement with those obtained by Monte Carlo simulation. The efficiency of the SSBEM is verified by comparison of their computation times.

Displacement of fluid droplets from solid surfaces in low-Reynolds-number shear flows

The yield conditions for the displacement of fluid droplets from solid boundaries are studied through a series of numerical computations. The study includes gravitational and interfacial forces, but is restricted to two-dimensional droplets and low-Reynolds-number flow. A comprehensive study is conducted, covering a wide range of viscosity ratio λ, Bond number Bd, capillary number Ca and contact angles θA and θR. The yield conditions for drop displacement are calculated and the critical shear rates are presented as functions Ca(λ, Bd, θA, Δθ) where Δθ=θA−θR is the contact angle hysteresis. The numerical solutions are based on the spectral boundary element method, incorporating a novel implementation of Newton's method for the determination of equilibrium free surface profiles. The numerical results are compared with asymptotic theories (Dussan 1987) based on the lubrication approximation. While excellent agreement is found in the joint asymptotic limits Δθ[Lt ]θA[Lt ]1, the useful range of the lubrication models proves to be extremely limited. The critical shear rate is found to be sensitive to viscosity ratio with qualitatively different results for viscous and inviscid droplets. Gravitational forces normal to the solid boundary have a significant effect on the displacement process, reducing the critical shear rate for viscous drops and increasing the rate for inviscid droplets. The low-viscosity limit λ→0 is shown to be a singular limit in the lubrication theory, and the proper scaling for Ca at small λ is identified.

Permeability of three-dimensional models of fibrous porous media

Viscous flow through three-dimensional models of fibrous porous media is analysed. The periodic models are based on ordered networks of cylindrical fibres on regular cubic lattices. Numerical solutions are obtained using the spectral boundary element method introduced by Muldowney & Higdon (1995). Results are presented for solid volume fractions ranging from extreme dilution to near the maximum volume fraction for permeable media. Theoretical estimates are derived using slender-body theory and lubrication approximations in the appropriate asymptotic regimes. Comparisons are made with model predictions based on properties of two-dimensional media (Jackson & James 1986), and with results for disordered dispersions of prolate spheroids (Claeys & Brady 1993b).

Resistance functions for spherical particles, droplets and bubbles in cylindrical tubes

Numerical computations are performed to evaluate the resistance functions for low Reynolds number flow past spherical particles, droplets and bubbles in cylindrical domains. Spheres of arbitrary radius a and radial position b move with arbitrary velocity U within a cylinder of radius R. The undisturbed fluid may be at rest, or subject to a pressure-driven flow with maximum velocity U0. The spectral boundary element method is employed to compute the resistance force for torque-free bodies in three cases: rigid solids, fluid droplets with viscosity ratio λ = 1, and bubbles with viscosity ratio λ = 0. A lubrication theory is developed to predict the limiting resistance of bodies near contact with the cylinder walls. Compact algebraic expressions are developed which accurately represent the numerical data over the entire range of particle positions 0 < b/(R − a) < 1 for all particle sizes in the range 0 < a/R < 0.9. The resistance functions are consistent with known analytical results and are presented in a form suitable for further studies of particle migration in cylindrical vessels.

Fast fourier transform of spectral boundary elements for transient heat conduction

The spectral boundary element method for solving two‐dimensional transient heat conduction problems is developed. This method is combined with the fast Fourier transform (FFT) to convert the solution between the time and frequency domains. The fundamental solutions in the frequency domain, required for the present method, are discussed. The resulting line integrations in the frequency domain are discretized using constant boundary elements and used in a Fortran boundary element program. Three examples are used to illustrate the accuracy and effectiveness of the method in both the frequency and time domains. First, the frequency domain solution procedure is verified using the steady‐state example of a semi‐infinite half space with a heat flux applied to a patch of the surface. This spectral boundary element method is then applied to the problem of a circular hole in an infinite solid subjected to a time‐varying heat flux, and solutions in both the frequency and time domains are presented. Finally, the method is used to solve the circular hole problem with a convection boundary condition. The accurary of these results leads to the conclusion that the spectral boundary element method in conjunction with the FFT is a viable option for transient problems. In addition, this spectral approach naturally produces frequence domain information which is itself of interest.

Spectral boundary element method: a new window on boundary elements in rock mechanics : Peirce, A P; Spottiswoode, S; Napier, J A L Int J Rock Mech Min SciV29, N4, July 1992, P379–400

Spectral boundary element method: a new window on boundary elements in rock mechanics : Peirce, A P; Spottiswoode, S; Napier, J A L Int J Rock Mech Min SciV29, N4, July 1992, P379–400

The spectral boundary element method: a new window on boundary elements in rock mechanics

This paper describes a novel spectral method based on the FFT for solving boundary integral equations incorporating the effect of non-linear material behaviour. The mathematical properties of this method are developed and illustrated by means of a simple model problem. The spectral boundary element technique is shown to provide a new framework for volumetric modelling with easy access to a number of interesting features not available to spatially implemented algorithms. A fundamental set of point Fourier kernels is introduced from which a variety of approximation schemes (including standard piecewise polynomial approximations) can be constructed in the frequency domain by introducing high frequency filters. The frequency domain implementation of these approximation schemes avoids the tedious integrations associated with spatial discretizations of the integral equations and provides considerable flexibility for general-purpose user-defined approximation schemes. Techniques are described to overcome the periodicity constraint imposed by the FFT so that general non-repeating geometries can be modelled. It is shown how the same periodicity can also be exploited to model repeating geometries. Two novel iterative methods are described to solve the discretized BE equations efficiently. The first method uses the information provided by the FFT to construct an approximate inverse extremely efficiently for use in a preconditioned conjugate gradient algorithm. This method can reduce the operation count for solution of the discretized problem to O(N log N) operations. The second method is an adaptation of Jacobi iteration which can roughly double the convergence rate for linear problems and can help to inhibit undesirable simultaneous failure of neighbouring elements when modelling brittle rock fracture. Two appendices containing expressions for the boundary element kernels and their Fourier transforms are provided.