Graham Wilks

COMBINED FORCED AND FREE CONVECTION FLOW ON VERTICAL SURFACES

NOMENCLATURE h dimensionless stream function ; Gr,, Grashof number, g/?A7’x3!v2 ; Gr:. modified Grashof number, gbqx4/kv2 ; 9. acceleration due to gravity ; h. local heat transfer coefficient. q/AT. k, thermal conductivity; Nu,, local Nusselt number, hxik; Prandtl number, V/K; heat flux constant ; Reynolds number, Uxiv; temperature; ambient temperature; local plate temperature : free stream velocity; streamwise velocity : transverse velocity: streamwise coordinate; transverse coordinate : T, TO: coefficient of thermal expansion ; dimensionless temperature : pseudo-similarity variable ; coefficient of thermal conductivity; kinematic viscosity ; dimensionless streamwise coordinate: stream function. criticisms made by Merkin [2] with respect to attempted series solutions about the basic free and forced convection flows as outlined by Szewczyk [3]. Essentially the error in [3] could be attributed to the choice ofa parameter expansion as opposed to the correct choice, namely a coordinate expansion. In this light it is seen that account must be taken of difficulties, which arise in the form of eigen-solutions, in the perturbed free convection solution. These difficulties are associated with the asymptotic nature of this solution at a distance far removed from the leading edge and indeed reflect the solutions’ unawareness of the precise location of the leading edge (see Stewartson [4]). ‘In [2]‘ a dimensional analysis of the governing equations yields the choice of nondimensional coordinate as

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.

Magnetohydrodynamic free convection about a semi-infinite vertical plate in a strong cross field

SummaryThe magnetohydrodynamic free convection flow of an electrically conducting fluid past a semi-infinite plate in a strong cross-field has been examined. 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. An estimate of an indeterminacy in asymptotic solutions allows favourable comparison to be made between series solutions estimates of these quantities and their exact numerical values.ZusammenfassungDie magneto-hydrodynamische Strömung mit freier Konvektion in einer elektrisch leitenden Flüssigkeit entlang einer halbunendlichen Platte 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. Eine Abschätzung von einer Unbestimmtheit in den asymptotischen Lösungen erlaubt einen Vergleich zwischen den durch Reihelösungen erhaltenen Werten dieser Grössen und ihren exakten numerischen Werten.

The flow of a uniform stream over a semi-infinite vertical flat plate with uniform surface heat flux

Abstract The effect of buoyancy forces on the boundary-layer flow over a semi-infinite vertical flat plate is investigated. The buoyancy forces are favourable, resulting from a uniform flux of heat from the surface of the plate, and their interaction with the boundary-layer flow associated with a uniform stream is examined. Two series solutions are obtained, one valid near the leading edge and the other downstream. An accurate numerical method is used to describe the flow in the region where the series are not valid. Comparison of results leads to some confidence in the merit of the series solutions for Prandtl number of 0 (1).

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.

Heat transfer in the flow of a cold, two-dimensional vertical liquid jet against a hot, horizontal plate

Abstract A cold, thin film of liquid impinging on an isothermal hot, horizontal surface is modelled as a two-dimensional jet of prescribed uniform velocity, film thickness and temperature. An approximate solution for the velocity and temperature distributions in the flow along the horizontal surface is developed, which exploits the Watson hydrodynamic similarity solution for thin film flow. A numerical solution of high accuracy has also been obtained. Comparisons indicate that the approximate solution may provide a valuable basis for assessing flow and heat transfer in more complex settings modelled by jet impingement such as cylinder inundation flows.

Mixed-convection laminar film condensation on a semi-infinite vertical plate

A comprehensive study of the problem of laminar film condensation with both a gravitational type body force and a moving vapour concurrent and parallel to the surface has been presented here. It demonstrates where both the body force and vapour velocity are significant through a comprehensive numerical solution obtained by a modified Keller box method. Important parameters governing condensation and heat transfer of pure vapour are determined. A perturbation analysis is applied in the leading edge and downstream regimes. The thin film approximations for the both regimes are obtained and compared with exact numerical solutions.

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.

An accurate numerical method for systems of differentio-integral equations associated with multiphase flow

An accurate numerical method which is applicable to systems of differentio-integral equations with quite general boundary conditions has been developed. The method is a useful extension of the Keller box scheme designed to facilitate the solution of differential systems involving integral operators which naturally arise in multiphase flows. A combination of merging and reduction procedures is introduced to handle the multilayer and integral operator features of such problems. The development of the method is demonstrated in the context of laminar film condensation in the presence of both external forcing and body forces.

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 ...