Roland Hunt

Continuous transformation computation of boundary layer equations between similarity regimes

Abstract Consideration is given to the computation of boundary layer flows displaying evolution between similarity regimes. A continuous transformation is introduced which reflects the associated evolution. When applied in conjunction with recent developments involving extrapolation on crude nets an efficient, accurate and straightforward algorithm ensues.

The rate of spreading in spin coating

In this paper we reconsider the fundamental problem of the centrifugally driven spreading of a thin drop of Newtonian fluid on a uniform solid substrate rotating with constant angular speed when surface-tension and moving-contact-line effects are significant. We discuss analytical solutions to a number of problems in the case of no surface tension and in the asymptotic limit of weak surface tension, as well as numerical solutions in the case of weak but finite surface tension, and compare their predictions for the evolution of the radius of the drop (prior to the onset of instability) with the experimental results of Fraysse & Homsy (1994) and Spaid & Homsy (1997). In particular, we provide a detailed analytical description of the no-surface-tension and weak-surface-tension asymptotic solutions. We demonstrate that, while the asymptotic solutions do indeed capture many of the qualitative features of the experimental results, quantitative agreement for the evolution of the radius of the drop prior to the onset of instability is possible only when weak but finite surface-tension effects are included. Furthermore, we also show that both a fixed- and a specific variable-contact-angle condition (or ‘Tanner law’) are capable of reproducing the experimental results well.

Magnetohydrodynamic free convection flow about a semi-infinite plate at whose surface the heat flux is uniform

SummaryThe magnetohydrodynamic free convection flow of an electrically conducting fluid past a vertical, semi-infinite plate in a strong cross field has been examined when a uniform heat flux is present at the plate. Formulation in terms of a characteristic length has enabled a full numerical solution to be obtained providing details of skin friction and heat transfer at all stations along the plate for a range of Prandtl numbers appropriate to liquid metal coolants. Series solutions are also presented which give good estimates of the skin friction and heat transfer coefficients.ZusammenfassungDie magneto-hydrodynamische Strömung mit freier Konvektion in einer elektrisch leitenden Flüssigkeit entlang einer vertikalen halbunendlichen Platte mit einem gleichförmigen Wärmefluß ist in einem starken Querfeld untersucht worden. Die Einführung einer charakteristischen Länge hat eine vollständige numerische Lösung ermöglicht, die die Reibungskraft und den Wärmeübergang entlang der Platte ergibt, für einen Bereich von Prandtlzahlen geeignet für Flüssigmetallkühler, Reihenlösungen, die gute Schätzungen von Reibungskraft und Wärmeübergangs-Koeffizienten ermöglichen.

Fourth-order method for solving the Navier-Stokes equations in a constricting channel

SUMMARY A fourth-order numerical method for solving the Navier‐Stokes equations in streamfunction=vorticity formulation on a two-dimensional non-uniform orthogonal grid has been tested on the fluid flow in a constricted symmetric channel. The family of grids is generated algebraically using a conformal transformation followed by a non-uniform stretching of the mesh cells in which the shape of the channel boundary can vary from a smooth constriction to one which one possesses a very sharp but smooth corner. The generality of the grids allows the use of long channels upstream and downstream as well as having a refined grid near the sharp corner. Derivatives in the governing equations are replaced by fourth-order central differences and the vorticity is eliminated, either before or after the discretization, to form a wide difference molecule for the streamfunction. Extra boundary conditions, necessary for wide-molecule methods, are supplied by a procedure proposed by Henshaw et al. The ensuing set of non-linear equations is solved using Newton iteration. Results have been obtained for Reynolds numbers up to 250 for three constrictions, the first being smooth, the second having a moderately sharp corner and the third with a very sharp corner. Estimates of the error incurred show that the results are very accurate and substantially better than those of the corresponding second-order method. The observed order of the method has been shown to be close to four, demonstrating that the method is genuinely fourth-order. # 1977 John Wiley & Sons, Ltd.

The flow of wet steam in a one-dimensional nozzle

The formation of water droplets in a low-pressure steam turbine, seriously degrades the efficiency of the generator. A model has been developed which includes the nucleation and subsequent growth of the droplets as the extra equations to the usual Euler equations for dry steam. A feature of this work is that all the equations are cast in Eulerian form compared to much of the previous work which considered the droplets in Lagrangian form. The ensuing equations are solved using a second-order upwind TVD scheme which can cope with the steep gradients which occur in the solution. The results for a 1-D nozzle are presented and compared with experimental results. Copyright

LOW PRANDTL NUMBER MAGNETOHYDRODYNAMIC NATURAL CONVECTION IN A STRONG CROSS FIELD

The low Prandtl number flow of a conducting fluid about a semi-infinite vertical plate in the presence of a strong cross magnetic field is investigated numerically. The range of Prandtl numbers examined extends down to values appropriate to liquid-metal reactor coolants. A numerical scheme is employed that takes advantage of the established limiting similarity states at the leading edge and downstream.

THE TWO-DIMENSIONAL LAMINAR VERTICAL JET WITH POSITIVE OR ADVERSE BUOYANCY

A comprehensive appraisal of the title problem is presented in terms of a characterizing nondimensional coordinate f that incorporates the initial momentum of the jet and the conserved heat flow. The integrated forms of the governing equations provide monitors on the consistency of regular and singular perturbation series solutions in 4 about limiting similarity states associated with the flow. In the case of a positively buoyant source of momentum the formulation provides the basis for a complete numerical integration over the semi-infinite region downstream of the jet initiation. Accordingly, details of velocity and temperature along the axis of the jet are obtained, and undetermined coefficients in the asymptotic downstream perturbation solution may be estimated. A numerical integration of the negatively buoyant case leads to a breakdown characterized by unbounded growth of the boundary-layer width. An inviscid structure for this breakdown within the framework of the governing boundary-layer equations ...

The numerical solution of elliptic free boundary problems using multigrid techniques

Abstract A multigrid algorithm is developed for the numerical solution of elliptic free boundary problems. The domain of the problem is mapped onto a rectangle and the governing equations discretized using finite differences. The resulting algebraic system is solved iteratively using a multigrid V cycle. For a convergent relaxation procedure it is necessary to use line iteration perpendicular to the free boundary simultaneously altering the values of the dependent variable and the position of the boundary, which is conveniently done using a single Newton iteration. Three problems are considered, a Poisson type problem, a steady state heat transfer problem, and one from electrochemical machining. The first two problems rapidly converge in a few multigrid cycles, the third converges less rapidly though adequately. Since full multigrid (FMG) is used, the results on the three finest grids could be combined to give accurate results of sixth order.

The numerical solution of parabolic free boundary problems arising from thin film flows

Abstract A numerical solution of high accuracy is obtained for the large Reynolds number, thin film flow over a horizontal flat plate, cylinders and spheres resulting from a vertical jet of liquid falling on the surface. A coordinate transformation is used which simultaneously maps the film thickness onto the unit interval and removes the singularity at the leading edge. The resulting equations are parabolic and these are solved using the Keller box method which is modified to accommodate the outer, free boundary. Using an extrapolation technique, results of sixth order accuracy are obtained with an estimated accuracy of six significant figures. For the flat plate, solutions for both 2-dimensional and axisymmetric jets are obtained and, for cylinders and spheres, different Froude numbers and volume flows are considered. The method can be used to solve any parabolic problem with a free boundary.

Vertical mixed convection flow about a horizontal line source of heating or cooling

Abstract The vertical mixed convection flow of a uniform stream, about a horizontal line source generating favourable buoyancy effects, may be characterized by an evolution between a weak and strong plume. This appraisal of the developing flow field provides the basis for an efficient formulation of the problem. Comprehensive solutions within this framework are obtained for a wide range of Prandtl numbers. In contrast the vertical mixed convection flow about a horizontal line source resulting in adverse buoyancy forces may be expected ultimately to display stagnation. Numerical solutions of the boundary layer equations governing this adverse case reveal that the anticipated stagnation is accompanied by a singular behaviour characterized by unbounded growth of the shear layer.