Laminar two-phase boundary layers in natural convection film boiling

Zusammenfassung Es wird die zweiphasige Grenzschichtströmung behandelt, die beim laminaren Filmsieden an einer vertikalen Platte und an einem langen horizontalen Zylinder unter den Bedingungen natürlicher Konvektion entsteht. In den Rechnungen wird der gesamte Bereich derjenigen Dampfüberhitzung miterfasst, in welchem rein konvektiver Wärmetransport möglich ist unter Einschluss des Impulstransports an der Phasengrenzfläche. Neue Heliummessungen und bekannte Versuchsergebnisse stimmen in befriedigender Weise mit den Rechnungen überein, wenn man berücksichtigt, dass ein vereinfachtes Strömungsmodell mit einer glatten Phasengrenzfläche vorausgesetzt ist.     

Laminar natural convection boundary layers on horizontal circular discs

Summary A first-order boundary-layer analysis, including variable properties effects, is given for laminar natural convection on a horizontal circular disc. It is shown that the result for the Nusselt number is very close to that for a square plate.

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Zusammenfassung Eine Grenzschichtsanalyse erster Ordnung, einschliesslich der Wirkung variabler Stoff-Eigenschaften, wird für laminare natürliche Konvektion auf einer horizontalen kreisrunden Platte angegeben. Es wird gezeigt, dass das Ergebnis hinsichtlich der Nusselt-Zahl demjenigen für eine viereckige Platte sehr ähnlich ist.

     

Laminar natural convection boundary layers on horizontal surfaces possessing a circular cut-out

Summary A first-order boundary-layer analysis, including variable properties effects, is given for laminar natural convection on a horizontal surface possessing a circular cut-out. The latter provides the leading edge of the boundary layer and it is shown that, initially, boundary layer development occurs as if the leading edge were straight. Far from the cut-out, boundary layer development is shown to become completely axially symmetric.     

Free convection boundary layers near corners and sharp trailing edges

The problems of how a free convection boundary layer adjusts to a small but sudden change in plate angle, and how it leaves a body with a wedge-shaped trailing edge are discussed. The angle is taken to beO(G _r ^–3/28) so that the flow near the discontinuity can be described by a double deck structure. For plates with very small angle changes, the lower deck equations can be linearised and the resulting equations have been solved using Fourier transforms. In the general case, the lower deck equations have to be solved numerically and results are given for both concave and convex corners and for a range of wedge angles. It is found that for sufficiently large angles there is a region of reversed flow in each case, in the first and last cases in a region centred on the discontinuity, while for convex corners it is downstream of the corner.

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Zusammenfassung Es wird das Problem gelöst wie sich bei freier Konvektion die Grenzschicht an eine kleine aber plötzliche Änderung des Plattenwinkels anpaßt, und wie sie eine keilförmige Hinterkante verläßt. Der Winkel wird von der GrößenordnungO(G _r ^–3/28) gewählt, damit die Strömung nahe zur Diskontinuität durch eine Zweischichtstruktur beschrieben werden kann. Für Platten mit sehr kleinen Winkeländerungen kann man die Gleichungen der unteren Schicht linearisieren und durch eine Fourier-Transformation lösen. Für den allgemeinen Fall müssen die Gleichungen der unteren Schicht numerisch gelöst werden, und wir geben Resultate für konkave und konvexe Ecken sowie für einen Reihe von Keilwinkeln an. Für hinreichend große Winkel konnten wir in allen Fällen eine Umkehr-Strömung feststellen, im ersten und letzten Fall in einem Gebiet mit der Diskontinuität als Zentrum, und für konvexe Ecken stromabwärts von der Ecke.

     

Laminar flow of viscoelastic liquids

The effect of normal stresses on the behavior of the fluidity of viscoelastic liquids is considered. Expressions are proposed for calculating the velocity fields and skin friction coefficients in circular pipes.Institute of Thermophysics, Siberian Division, Academy of Sciences of the USSR, Novosibirsk. Translated from Mekhanika Polimerov, No. 6, pp. 1123–1126, November–December, 1970.     

An application of a boundary element method to natural convection

The direct boundary element method is applied to the numerical modelling of thermal fluid flow in a transient state. The Navier-Stokes equations are considered under the Boussinesq approximation and the viscous thermal flow equations are expressed in terms of stream function, vorticity, and temperature in two dimensions. Boundary integral equations are derived using logarithmic potential and time-dependent heat potential as fundamental solutions. Boundary unknowns are discretized by linear boundary elements and flow domains are divided into a series of triangular cells. Charged points are translated upstream in the numerical evaluation of convective terms. Unknown stream function, vorticity, and temperature are staggered in the computational scheme.

Simple iteration is found to converge to the quasi steady-state flow. Boundary solutions for two-dimensional examples at a Reynolds number 100 and Grashoff number 10⁷ are obtained.     

Free convection boundary layers on a vertical surface in a heat-generating porous medium

The natural convection boundary-layer flow on a solid verticalsurface with heat generated within the boundary layer at a rateproportional to (T – T)^p (p 1) is considered. The surfaceis held at the ambient temperature T except near the leadingedge where it is held at a temperature above ambient. The behaviourof the flow as it develops from the leading edge is examinedand is seen to become independent of the initial heat input;however, it does depend strongly on the exponent p. For 1 p 2, the local heating eventually dominates at large distancesand there is a convective flow driven by this mechanism. Forp 4, the local heating does not have a significant effect,the fluid temperature remains relatively small throughout andthe heat transfer dies out through a wall jet flow. For 2 <p < 4, the local heating has a significant effect at relativelysmall distances, with a thermal runaway developing at a finitedistance along the surface.     

Laminar boundary layers along an infinite porous plate in the presence of a transverse magnetic field

Exact solutions of the Navier-Stokes equations are derived by a Laplace-transform technique for two-dimensional, incompressible flow of an electrically conducting fluid past on infinite porous plate. It is assumed that the flow is independent of the distance parallel to the plate and that the velocity component normal to the plate is constant. A general formula is derived for the velocity distribution in terms of the given external velocity. The skin friction is obtained and some special cases are considered.     

An optimal asymptotic-numerical method for convection dominated systems having exponential boundary layers

The convection dominated diffusion problems are studied. Higher order accurate numerical methods are presented for problems in one and two dimensions. The underlying technique utilizes a superposition of given problem into two independent problems. The first one is the reduced problem that refers to the outer or smooth solution. Stretching transformation is used to obtain the second problem for inner layer solution. The method considered for outer or degenerate problems are based on higher order Runge–Kutta methods and upwind finite differences. However, inner problem is solved analytically or asymptotically. The schemes presented are proved to be consistent and stable. Possible extensions to delay differential equations and to nonlinear problems are outlined. Numerical results for several test examples are illustrated and a comparative analysis is presented. It is observed that the method presented is highly accurate and easy to implement. Moreover, the numerical results obtained are not only comparable with the exact solution but also in agreement with the theoretical estimates.     

A finite time singularity of the boundary layer equations of natural convection

Summary The unsteady natural convection flow about a horizontal circular cylinder at whose surface the heat flux is constant is considered. A numerical solution of the equations valid in the vicinity of the uppermost generator of the cylinder displays singular behaviour after a finite time. It is demonstrated that the breakdown of the solution is consistent with a triple layer structure characterised by an inertia dominated central region.

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Zusammenfassung Wir untersuchen die nicht-stationäre natürliche Konvektion entlang eines horizontalen Zylinders mit konstantem Oberflächenwärmefluß. Eine in der Umgebung des obersten Generators des Zylinders gültige numerische Lösung führt nach endlicher Zeit zu einem singulären Verhalten. Es wird gezeigt, daß das Zusammenbrechen der Lösung konsistent ist mit einer Dreischichtstruktur in der der mittlere Bereich von der Trägheit dominiert wird.

     

Investigation of a powerful analytical method into natural convection boundary layer flow

The natural convection boundary layer flow modeled by a system of nonlinear differential equations is considered. By means of similarity transformation, the non-linear partial differential equations are reduced to a system of two coupled ordinary differential equations. The series solutions of coupled system of equations are constructed for velocity and temperature using homotopy analysis method (HAM). Convergence of the obtained series solution is discussed. Finally some figures are illustrated to show the accuracy of the applied method and assessment of various prandtl numbers on the temperature and the velocity is undertaken.     

Sharpened bounds for corner singularities and boundary layers in a simple convection–diffusion problem

A singularly perturbed convection–diffusion problem posed on the unit square is considered. Its solution may have exponential and parabolic boundary layers, and corner singularities may also be present. Sharpened pointwise bounds on the solution and its derivatives are derived. The bounds improve bounds near an outflow corner of the problem that were derived in an earlier paper of the authors. Application is made to an error analysis of a finite element method for the problem.     

Accuracy,robustness and efficiency comparison in iterative computation of convection diffusion equation with boundary layers

Nine‐point fourth‐order compact finite difference scheme, central difference scheme, and upwind difference scheme are compared for solving the two‐dimensional convection diffusion equations with boundary layers. The domain is discretized with a stretched nonuniform grid. A grid transformation technique maps the nonuniform grid to a uniform one, on which the difference schemes are applied. A multigrid method and a multilevel preconditioning technique are used to solve the resulting sparse linear systems. We compare the accuracy of the computed solutions from different discretization schemes, and demonstrate the relative efficiency of each scheme. Comparisons of maximum absolute errors, iteration counts, CPU timings, and memory cost are made with respect to the two solution strategies. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 379–394, 2000     

A note on the similarity equations arising in free convection boundary layers with blowing and suction

The similarity equations for the free convection boundary-layer flow on a vertical plate with prescribed wall temperature and transpiration velocity are considered. The range of existence of solution is discussed first. For blowing this is seen to be independent of the transpiration parameter , depending only on the Prandtl number. For suction this range of existence of solutions is seen to depend on as well. Asymptotic solutions for strong suction and strong blowing are derived and compared with numerical solutions of the similarity equations.     

Mixed convection heat transfer in the boundary layers on an exponentially stretching surface with magnetic field

An analysis has been carried out to describe mixed convection heat transfer in the boundary layers on an exponentially stretching continuous surface with an exponential temperature distribution in the presence of magnetic field, viscous dissipation and internal heat generation/absorption. Approximate analytical similarity solutions of the highly non-linear momentum and energy equations are obtained. The present results are found to be in excellent agreement with previously published work on various special cases of the problem. Numerical results for temperature distribution and the local Nusselt number have been obtained for different values of the governing parameters. The numerical solutions are obtained by considering an exponential dependent stretching velocity and prescribed boundary temperature on the flow directional coordinate. The effects of various physical parameters like Prandtl number, Hartman number, Grashof number on dimensionless heat transfer characteristics are discussed in detail. In particular, it has been found that increase in Prandtl number decreases the skin-friction coefficient at the stretching surface, while increase in the strength of the magnetic field leads to increase in the local Nusselt number.     

Finite difference method of solution of unsteady magnetic hydrodynamic boundary layer equations in a natural convection

The paper presents an investigation of the effect of a transverse magnetic field on an unsteady natural convection along a heated vertical plate. Finite difference method is employed to solve the non-linear equations governing the motion. The difference scheme satisfies consistency and stability conditions. The stability condition depends on time step which has to be satisfied in every time step. Proper care has been taken for the non-linear terms. The solutions have been deduced for different values of the magnetic field intensity. The results are satisfactory. In long run our results coincides with the steady state solutions already existing in the literature.     

Laminar free convection due to a point source of Buoyancy

Summary Similarity solutions are found for the laminar flow arising from a point source of buoyancy. Approximate analytic solutions in closed form are derived for the non-dimensional velocity and density distributions as a function of the Prandtl number. These approximations are shown to compare favourably with numerical solutions given by Fujii and to include the exact solutions of Yih for Prandtl numbers of 1 and 2.

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Résumé Il existe des solutions de similitude pour l'écoulement laminaire provenant d'une source ponctuelle de poussée. On obtient des solutions analytiques approximatives closes, sous forme de fonction du nombre de Prandtl, pour les distributions non dimensionnelles de vitesse et de densité. On démontre que ces approximations se comparent avantageusement aux solutions numériques présentées par Fujii, en plus de fournir les solutions exactes de Yih pour les nombres de Prandtl 1 et 2.

     

Laminar free convection due to a point source of Buoyancy

Similarity solutions are found for the laminar flow arising from a point source of buoyancy. Approximate analytic solutions in closed form are derived for the non-dimensional velocity and density distributions as a function of the Prandtl number. These approximations are shown to compare favourably with numerical solutions given by Fujii and to include the exact solutions of Yih for Prandtl numbers of 1 and 2.     

Soft boundary layers in polymeric films

Application of a polymer on a substrate, e. g. a metal sheet, generates a boundary layer in the polymer. In this boundary layer the properties of the polymer differ from the bulk properties. In the present contribution a mechanical theory based on a structural parameter is presented that allows for the existence of such interphases and, therefore, leads to effective material properties depending on the thickness of the layer. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)