Magnetohydrodynamic free convection in a strong cross-field

Résumé On étudie le courant de convection MHD, montant le long d'une plaque verticale chauffée en présence d'un puissant champ magnétique transversal. La température et la rapidité sont données par un problème particulier de perturbation, et le courant est tout à fait différent, même avec une conductivité fluide minime du courant non magnétisé correspondant. La première partie de ce travail a trait aux effets de la dissipation ohmique; dans la seconde, on néglige la dissipation pour étudier de plus près le courant vertical à température linéaire, y compris l'effet de succion ou d'injection uniforme.     

On the free convection from a horizontal plate

Zusammenfassung Die freie Konvektion an einer beheizten horizontalen Platte wird nach den üblichen Methoden der Grenzschichttheorie untersucht. Weist die wärmeübertragende Plattenseite nach oben, so ist das Problem auf dieser Basis unlösbar, wohl aber findet man eine widerspruchsfreie Lösung, wenn diese Plattenseite nach unten weist. In diesem ist die Nusselt Zahl, die den Wärmeübergang kennzeichnet, proportional der 1/5-ten Potenz der Rayleigh Zahl, welche die Beheizung charakterisiert. Die Übereinstimmung mit dem Experiment ist befriedigend.     

Nonstationary problem of free convection with radiative heat transfer

Nonstationary problem of free convection of viscous incompressible fluid in a three-dimensional domain with allowance for radiative heat transfer is studied in the framework of the diffusion P₁-approximation of the equation of radiative transfer. The solvability of the problem is proven, and sufficient conditions for the uniqueness are presented.     

An asymptotic method for free convection past a vertical plate

Zusammenfassung Das Problem der freien Strömung an einer ungleichmässig erhitzten Wand wird auf die Berechnung einer Reihe von Funktionen zurückgeführt, von denen jede einer linearen Differentialgleichung mit konstanten Koeffizienten genügt.

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Written under the sponsorship of the National Science Foundation, Contract NSF-G3050, New York University. Reproduction in whole or in part permitted for any purpose of the United States Government.     

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.     

Approximate solution of the transient free convection laminar boundary layer equations

Zusammenfassung Es wird eine angenäherte Lösung der Gleichungen der laminaren Grenzschicht in freier Konvektionsströmung an einer halbunendlichen senkrechten Platte angegeben. Obwohl bereits Ähnlichkeitsvariablen für die transiente Strömung dieses Typus angegeben wurden, ist bis heute kein numerisches Ergebnis bekannt. Daher wird im vorliegenden Aufsatz das System der nichtlinearen, gekoppelten partiellen Differentialgleichungen durch Approximation gelöst. Durch ein Integrationsverfahren kann das System der Differentialgleichungen mit drei unabhängigen Variablen (zwei Ortskoordinaten und Zeit) auf ein solches mit zwei unabhängigen Variablen zurückgeführt werden. Das so verbleibende Gleichungssystem wird auf zwei verschiedene Arten behandelt. Die eine wird als eine Methode der Integralrelationen bezeichnet, die andere als Linienmethode (ein Verfahren der Differenzenrechnung). Nach beiden Methoden werden für verschiedene Prandtl-Zahlen Geschwindigkeits- und Temperaturprofile berechnet und mit experimentellen Ergebnissen verglichen.

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Nomenclature a constant - a _i approximating function, see equation (24) - b constant - b _i approximating function, see equation (24) - d _i approximating functions, see equation (33) - e _i approximating functions, see equation (34) - f _i weighting functions - g _i weighting functions - g acceleration due to gravity - G Grashof number - h heat transfer coefficient - k thermal conductivity - L characteristic length - L(u) differential operator onu - Nu Nusselt number - P Prandtl number - q heat flux - t time - T temperature - T ambient temperature - u ₁ dimensionless characteristic velocity,Gr ^–1/2(U₁ L/) - U velocity in theX direction - U ₁ dimensional characteristic velocity,U ₁=U/f(Y/GD) - V velocity in theY direction - W _k weighting functions, see equation (5) - X distance along plate - x dimensionless distance along plate,X/L - x increment onx for method of lines - Y distance from plate - y dimensionless distance from plate,Y/L - thermal diffusivity - coefficient of thermal expansion - dimensional boundary layer thickness - dimensionless boundary layer thickness, /L - a function of - dimensionless length,Y/ - temperature excess, (T-T) - _w wall temperature difference, (T _w–T) - _k approximating functions, see equation (4) - kinematic viscosity - dimensionless time,Gr ^–1/2( _t /L ²) - approximating functions, see equation (4) Work performed under the auspices of the U.S. Atomic Energy Commission.     

Combined free and forced laminar convection in internally finned square ducts

Analysis is presented for the heat transfer performance of square ducts with internal fins from each wall in the case of combined free and forced convection by fully developed laminar flow. Numerical results are obtained for the Nusselt number and the pressure drop parameter for various values of finlengths and heat source parameter. For various values of Rayleigh numbers, the Nusselt number increases with the increase in finlength and decreases with the increase in heat source parameter.

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Zusammenfassung Es wird eine Analyse für den Wärmeaustausch von quadratischen Rohren mit inneren Rippen an jeder Wand im Falle einer Kombination von freier und erzwungener Konvektion bei voll entwickelter laminarer Strömung gegeben. Numerische Resultate für die Nusselt-Zahl und den Druckabfall-Koeffizienten für verschiedene Rippenbreiten und Parameter der Wärmequelle werden erhalten. Für einige Werte der Rayleighzahl wächst die Nusselt-Zahl mit der Rippenbreite und fällt mit wachsendem Parameter der Wärmequelle.

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Nomenclature A cross sectional area of the duct - B _2k Bernoulli numbers - c _p specific heat at constant pressure - D _h hydraulic diameter of finless duct - E _n complex constants (20) - F heat source parameter,Q/c _p - F _n () defined by Equation (14) - G(, , , ) Green's function (15, 16) - g gravitational acceleration - H() Heaviside function - h() defined by Equation (22) - i imaginary unit,i ²=–1 - ImW imaginary part ofW - K(,t) kernel of the integral equation, defined by (25) - k thermal conductivity - L pressure drop parameter, –D _h ² (p/x+ _w )/ - l fin length of each fin, Figure (1) - N _u Nusselt number, Equation (32) - p pressure - Q heat generation rate - R() defined by Equation (26) - R _A Rayleigh number, _w gc _p D _h ⁴ /k - ReW real part ofW - T dimensionless temperature, (t–t _w )/(c _p D _h ² /k) - T _mx dimensionless mixed mean temperature, Equation (33) - t fluid temperature - t ₀ reference temperature atx=0 - u local axial velocity - mean axial velocity - V u/ - W complex function defined by Equation (6) - w suffix denoting wall conditions - W ₀ defined by Equation (9) - W ₁ W–W ₀, Equation (18) - x axial coordinate along the length of the duct - y, z cross-sectional coordinates - constant temperature gradient, t/x - coefficient of thermal expansion of the fluid - fluid density - _n - dynamic viscosity - () Dirac delta function - ² Laplacian operator, ²/y ²/²/z ² - , y/D _h ,z/D _h      

Effect of double stratification on MHD free convection in a micropolar fluid

This paper analyzes the flow and heat and mass transfer characteristics of the free convection on a vertical plate with variable wall temperature and concentration in a doubly stratified micropolar fluid. A uniform magnetic field is applied normal to the plate. The governing non-linear partial differential equations are transformed into a system of coupled non-linear ordinary differential equations using similarity transformations and then solved numerically using the Keller-box method. The numerical results are compared and found to be in good agreement with previously published results as special cases of the present investigation. The non-dimensional velocity, microrotation, temperature and concentration are presented graphically for various values of magnetic parameter, coupling number, thermal and solutal stratification parameters. In addition, the Nusselt number, the Sherwood number, the skin-friction coefficient, and the wall couple stress are shown in a tabular form.     

Parameter-dependent systems on the half-line and free convection boundary-layers in porous media

In this paper we use a recently elaborated abstract method, for parameter-dependent ODE systems over the interval (0, ∞), to obtain existence results for the problem of self-similar solutions in boundary-layer free convection in porous media. Using a generalization of the method to exponentially decaying solutions, we are able to recover some known results, and to obtain a new branch of solutions in the case of the so-called backward boundary-layer. The arguments involve the derivation of suitable a priori estimates for the solutions of the problem.     

Numerical prediction of recirculation flows with free convection encountered in gas-agitated reactors

Flows in a gas-agitated reactor have been predicted by a finite difference procedure. The free-convection phenomena in the gas-liquid mixtures have been accounted for by the calculation of a void fraction determined from the gas flow rate. Computations have been performed for two different situations: first, with the allowance of slip between gas and liquid phases, and second, without any slip. Reasonable agreement has been achieved between the measurements.     

Combined forced and free convection heat transfer from a horizontal flat plate

The problem of combined forced and free convection heat transfer in boundary-layer flow past a heated or cooled, horizontal flat plate is studied in the limits of large and small Prandtl number using the technique of matched asymptotic expansions. It is found that the effects of natural convection are confined to small changes in the basic forced convection velocity and temperature fields providedGr/Re ^5/2 Pr ^2/3 1 forPr1, orGr/Re ^5/2 Pr ^1/2 1 forPr1. The governing similarity equations at the first two orders in these small parameters are solved and the results compared with the numerical solutions of Sparrow and Minkowycz (1962).

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Zusammenfassung Das Problem der kombinierten erzwungenen und freien Konvektion in der Grenzschichtströmung entlang einer geheizten oder gekühlten horizontalen ebenen Platte wurde untersucht im Grenzfall von kleinen und grossen Prandtl-Zahlen, mit Hilfe der angepassten asymptotischen Entwicklungen. Der Effekt der natürlichen Konvektion ergab sich als kleine Korrektur der Geschwindigkeits-und Temperatur-Felder, fallsGr/Re ^5/2 Pr ^2/3 1 fürPr1 oderGr/Re ^5/2 Pr ^1/2 1 fürPr1. Die Ähnlichkeitslösungen der beiden ersten Ordnungen in diesen Parametern wurden bestimmt, und die Resultate wurden mit der numerischen Lösung von Sparrow und Minkowycz (1962) verglichen.

     

Combined free and forced convection effects on the magnetohydrodynamic flow through a channel

Zusammenfassung Die freie Konvektion einer elektrisch leitenden Flüssigkeit in einer horizontalen Kanalströmung in Anwesenheit eines homogenen vertikalen Magnetfeldes wird analysiert. Für die Erwärmung der oberen Wand ergibt sich eine Neigung der Strömung zur Instabilität. Der starke Einfluss des Magnetfeldes auf die Strömung wird diskutiert.     

Neural networks analysis of free laminar convection heat transfer in a partitioned enclosure

The feasibility of using neural networks (NNs) to predict the complete thermal and flow variables throughout a complicated domain, due to free convection, is demonstrated. Attention is focused on steady, laminar, two-dimensional, natural convective flow within a partitioned cavity. The objective is to use NN (trained on a database generated by a CFD analysis of the problem of a partitioned enclosure) to predict new cases; thus saving effort and computation time. Three types of NN are evaluated, namely General Regression NNs, Polynomial NNs, and a versatile design of Backpropagation neural networks. An important aspect of the study was optimizing network architecture in order to achieve best performance. For each of the three different NN architectures evaluated, parametric studies were performed to determine network parameters that best predict the flow variables.A CFD simulation software was used to generate a database that covered the range of Rayleigh number Ra = 10⁴–5 × 10⁶. The software was used to calculate the temperature, the pressure, and the horizontal and vertical components of flow speed. The results of the CFD were used for training and testing the neural networks (NN). The robustness of the trained NNs was tested by applying them to a “production” data set (1500 patterns for Ra = 8 × 10⁴ and 1500 patterns for Ra = 3 × 10⁶), which the networks have never been “seen” before. The results of applying the technique on the “production” data set show excellent prediction when the NNs are properly designed. The success of the NN in accurately predicting free convection in partitioned enclosures should help reduce analysis-time and effort. Neural networks could potentially help solve some cases in which CFD fails to solve because of numerical instability.     

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

Summary The 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.

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Zusammenfassung Die 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.

     

Oscillatory free convection flow of a second order fluid from a vertical plate

This paper investigates the problem of free convection flow of a second order liquid in the boundary layer from a semi-infinite vertical plate in which the mean surface temperature varies as a function of the distance from the leading edge of the plate. Study of the oscillatory flow is restricted to small amplitudesε only. Several graphs have been drawn and tables have been presented to depict the effect of elasticity of the liquid and Prandtl number on the velocity and temperature distributions and Nusselt number.     

Combined effects of free and forced convection on magnetohydrodynamic flow in a rotating channel

The steady flow in a channel rotating with an angular velocity \(\vec \Omega \) and subjected to a constant transverse magnetic field is analysed. An exact solution of the governing equations is obtained. The solution in the dimensionless form contains three parameters: the Grash of number,G, the Hartmann number,M ² and the rotation parameter,K ². The effects of these parameters on the velocity and magnetic field distributions are studied. For large values ofK ² andM ², there arise thin boundary layers on the walls of the channel which may be identified as the Ekman-Hartmann layers.     

Combined free and forced convection effects on the magnetohydrodynamic flow through a porous channel

The combined effect of free and forced convection on the flow of an electrically conducting liquid between two horizontal parallel porous walls has been studied. There is a transverse magnetic field at the walls. The equations of motion and energy have been solved by a small perturbation method. The flow phenomenon has been characterized by the non-dimensional numbers like R (cross-flow Reynolds number), K (Brinkman number), G (Grashof number), M (magnetic number) and the effects of these numbers on the velocity and temperature fields, induced magnetic field, electric field and shearing stress at the walls have been studied.     

Effects of external circuit and wall conditions on magnetohydrodynamic free convection between two heated walls

Zusammenfassung Die Einwirkungen der äusseren elektrischen Schaltungsbedingungen und der elektrischen Leitfähigkeiten der Wände auf die magneto-hydrodynamische freie Konvektion einer Flüssigkeit zwischen zwei erhitzten Wänden werden studiert. Eine Annäherungsmethode wird angewendet zur Lösung des das Problem beherrschenden Systems von nichtlinearen Integro-Differentialgleichungen. Es zeigt sich, dass die Strömung vom Hartmannschen Typ ist und auch entsprechende charakteristische Eigenschaften zur Elektrizitätserzeugung hat. Der erzeugte Strom ist proportional zur Temperaturdifferenz zwischen der Flüssigkeit und den Wänden.