Peter K. Jimack

Gravity-driven flow of continuous thin liquid films on non-porous substrates with topography

A range of two- and three-dimensional problems is explored featuring the gravity-driven flow of a continuous thin liquid film over a non-porous inclined flat surface containing well-defined topography. These are analysed principally within the framework of the lubrication approximation, where accurate numerical solution of the governing nonlinear equations is achieved using an efficient multigrid solver. Results for flow over one-dimensional steep-sided topographies are shown to be in very good agreement with previously reported data. The accuracy of the lubrication approximation in the context of such topographies is assessed and quantified by comparison with finite element solutions of the full Navier–Stokes equations, and results support the consensus that lubrication theory provides an accurate description of these flows even when its inherent assumptions are not strictly satisfied. The Navier–Stokes solutions also illustrate the effect of inertia on the capillary ridge/trough and the two-dimensional flow structures caused by steep topography. Solutions obtained for flow over localized topography are shown to be in excellent agreement with the recent experimental results of Decre & Baret (2003) for the motion of thin water films over finite trenches. The spread of the ‘bow wave’, as measured by the positions of spanwise local extrema in free-surface height, is shown to be well-represented both upstream and downstream of the topography by an inverse hyperbolic cosine function. An explanation, in terms of local flow rate, is given for the presence of the ‘downstream surge’ following square trenches, and its evolution as trench aspect ratio is increased is discussed. Unlike the upstream capillary ridge, this feature cannot be completely suppressed by increasing the normal component of gravity. The linearity of free-surface response to topographies is explored by superposition of the free surfaces corresponding to two ‘equal-but-opposite’ topographies. Results confirm the findings of Decre & Baret (2003) that, under the conditions considered, the responses behave in a near-linear fashion.

A fully implicit, fully adaptive time and space discretisation method for phase-field simulation of binary alloy solidification

A fully implicit numerical method based upon adaptively refined meshes for the simulation of binary alloy solidification in 2D is presented. In addition we combine a second-order fully implicit time discretisation scheme with variable step size control to obtain an adaptive time and space discretisation method. The superiority of this method, compared to widely used fully explicit methods, with respect to CPU time and accuracy, is shown. Due to the high nonlinearity of the governing equations a robust and fast solver for systems of nonlinear algebraic equations is needed to solve the intermediate approximations per time step. We use a nonlinear multigrid solver which shows almost h-independent convergence behaviour.

Flow of evaporating, gravity-driven thin liquid films over topography

The effect of topography on the free surface and solvent concentration profiles of an evaporating thin film of liquid flowing down an inclined plane is considered. The liquid is assumed to be composed of a resin dissolved in a volatile solvent with the associated solvent concentration equation derived on the basis of the well-mixed approximation. The dynamics of the film is formulated as a lubrication approximation and the effect of a composition-dependent viscosity is included in the model. The resulting time-dependent, nonlinear, coupled set of governing equations is solved using a full approximation storage multigrid method. The approach is first validated against a closed-form analytical solution for the case of a gravity-driven, evaporating thin film flowing down a flat substrate. Analysis of the results for a range of topography shapes reveal that although a full-width, spanwise topography such as a step-up or a step-down does not affect the composition of the film, the same is no longer true for th...

On spatial adaptivity and interpolation when using the method of lines

The solution of time-dependent partial differential equations with discrete time static remeshing is considered within a method of lines framework. Numerical examples in one and two space dimensions are used to show that spatial interpolation error may have an important impact on the efficiency of integration. Analysis of a simple problem and of the time integration method is used to confirm the experimental results and a computational test for monitoring the impact of this error is derived and tested.

A new parallel domain decomposition method for the adaptive finite element solution of elliptic partial differential equations

We present a new domain decomposition algorithm for the parallel finite element solution of elliptic partial differential equations. As with most parallel domain decomposition methods each processor is assigned one or more subdomains and an iteration is devised which allows the processors to solve their own subproblem(s) concurrently. The novel feature of this algorithm however is that each of these subproblems is defined over the entire domain—although the vast majority of the degrees of freedom for each subproblem are associated with a single subdomain (owned by the corresponding processor). This ensures that a global mechanism is contained within each of the subproblems tackled and so no separate coarse grid solve is required in order to achieve rapid convergence of the overall iteration. Furthermore, by following the paradigm introduced in 15 , it is demonstrated that this domain decomposition solver may be coupled easily with a conventional mesh refinement code, thus allowing the accuracy, reliability and efficiency of mesh adaptivity to be utilized in a well load‐balanced manner. Finally, numerical evidence is presented which suggests that this technique has significant potential, both in terms of the rapid convergence properties and the efficiency of the parallel implementation. Copyright

On the stability of viscous free–surface flow supported by a rotating cylinder

Using an adaptive finite–element (FE) scheme developed recently by the authors, we shed new light on the long–standing fundamental problem of the unsteady free–surface Stokes flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field. For supportable loads, we observe that the steadystate is more readily attained for near–maximal fluid loads on the cylinder than for significantly sub–maximal loads. For the latter, we investigate large–time dynamics by means of a finite–difference approximation to the thin–film equations, which is also used to validate the adaptive FE simulations (applied to the full Stokes equations) for these significantly sub–maximal loads. Conversely, by comparing results of the two methods, we assess the validity of the thin–film approximation as either the load is increased or the rotation rate of the cylinder is decreased. Results are presented on the independent effects of gravity, surface tension and initial film thickness on the decay to steady–state. Finally, new numerical simulations of load shedding are presented.

A comparison of some dynamic load-balancing algorithms for a parallel adaptive flow solver

In this paper we contrast the performance of a number of different parallel dynamic load-balancing algorithms when used in conjunction with a particular parallel, adaptive, time-dependent, 3D flow solver. An overview of this solver is given along with a description of the dynamic load-balancing problem that results from its use. Two recently published parallel dynamic load-balancing software tools are then briefly described and a number of recursive parallel dynamic load-balancing techniques are also outlined. The effectiveness of each of these algorithms is then assessed when they are coupled with the parallel adaptive solver and used to tackle a model 3D flow problem.

Finite Element Simulation of Three-Dimensional Free-Surface Flow Problems

An adaptive finite element algorithm is described for the stable solution of three-dimensional free-surface-flow problems based primarily on the use of node movement. The algorithm also includes a discrete remeshing procedure which enhances its accuracy and robustness. The spatial discretisation allows an isoparametric piecewise-quadratic approximation of the domain geometry for accurate resolution of the curved free surface. The technique is illustrated through an implementation for surface-tension-dominated viscous flows modelled in terms of the Stokes equations with suitable boundary conditions on the deforming free surface. Two three-dimensional test problems are used to demonstrate the performance of the method: a liquid bridge problem and the formation of a fluid droplet.

A domain decomposition preconditioner for a parallel finite element solver on distributed unstructured grids

Abstract A number of practical issues associated with the parallel distributed memory solution of elliptic partial differential equations (p.d.e.s) using unstructured meshes in two dimensions are considered. The first part of the paper describes a parallel mesh generation algorithm which is designed both for efficiency and to produce a well-partitioned, distributed mesh, suitable for the efficient parallel solution of an elliptic p.d.e. The second part of the paper concentrates on parallel domain decomposition preconditioning for the linear algebra problems which arise when solving such a p.d.e. on the unstructured meshes that are generated. It is demonstrated that by allowing the mesh generator and the p.d.e. solver to share a certain coarse grid structure it is possible to obtain efficient parallel solutions to a number of large problems. Although the work is presented here in a finite element context, the issues of mesh generation and domain decomposition are not of course strictly dependent upon this particular discretization strategy.

A Weakly Overlapping Domain Decomposition Preconditioner for the Finite Element Solution of Elliptic Partial Differential Equations

We present a new two-level additive Schwarz domain decomposition preconditioner which is appropriate for use in the parallel finite element solution of elliptic partial differential equations (PDEs). As with most parallel domain decomposition methods each processor may be assigned one or more subdomains, and the preconditioner is such that the processors are able to solve their own subproblem(s) concurrently. The novel feature of the technique proposed here is that it requires just a single layer of overlap in the elements which make up each subdomain at each level of refinement, and it is shown that this amount of overlap is sufficient to yield an optimal preconditioner. Some numerical experiments---posed in both two and three space dimensions---are included to confirm that the condition number when using the new preconditioner is indeed independent of the level of mesh refinement on the test problems considered.