Petrov-Galerkin methods for natural convection in directional solidification of binary alloys

A Petrov-Galerkin finite element method is presented for calculation of the steady, axisymmetric thermosolutal convection and interface morphology in a model for vertical Bridgman crystal growth of nondilute binary alloys. The Petrov-Galerkin method is based on the formulation for biquadratic elements developed by Heinrich and Zienkiewicz and is introduced into the calculation of the velocity, temperature and concentration fields. The algebraic system is solved simultaneously for the field variables and interface shape by Newton's method. The results of the Petrov-Galerkin method are compared critically with those of Galerkin's method using the same finite element grids. Significant improvements in accuracy are found with the Petrov-Galerkin method only when the mesh is refined and when the formulation of the residual equations is modified to account for the mixed boundary conditions that arise at the solidification interface. Calculations for alloys with stable and unstable solute gradients show the occurrence of classical flow transitions and morphological instabilities in the solidification system.     

Solidification of hypereutectic and hypoeutectic binary alloys with buoyancy and surface tension driven natural convection

Convection patterns and evolution of macrosegregation during solidification of hypereutectic and hypoeutectic NH₄CL-H₂O binary systems in rectangular side chilled ingots have been numerically investigated. Under the conditions of pure thermal/solutal convection, without a free surface, solidification of NH₄CL 70%-H₂O hypereutectic alloy is characterized by the formation of channels and A-segregates, while solidification of NH₄CL 10%-H₂O hypoeutectic alloy is characterized by the formation of circulation cells in the narrow melt and V-segregates. Surface tension effects during solidification of NH₄CL 70%-H₂O results in establishing a counterclockwise rotating cell at the cavity top, changing the number and orientation of developed channels, and creating an area of negative segregation at the cavity top. During solidification of NH₄CL 10%-H₂O, surface tension effects increase the intensity of flow and results in a higher degree of macrosegregation. Received on 9 June 1998     

Experimental study of air natural convection on metallic foam-sintered plate

The natural convection on metallic foam-sintered plate at different inclination angles was experimentally studied. Seven copper foam samples with different pore densities (10–40 pore per inch), porosities (0.90–0.95), and aspect ratios (the ratio of foam thickness to sample length, 0.1–0.5) were measured at inclination angles of 0° (vertical orientation), 15°, 30°, 45°, 60°, 75°, 90° (horizontal orientation). The heat conduction and natural convection inside the foam both contributed to the total heat transfer. Although, the form and viscous drag, which are influenced by permeability and viscous friction in the thermal boundary layer respectively, tend to suppress the natural convection, the heat transfer was finally enhanced by the foam sintered surface due to large surface area extension. Optimum inclination range 60–75° corresponding to maximum average Nu number was found in the heat flux range of 600–1800 W/m². The sintered foam surface with lower porosity and pore density was recommended for heat transfer enhancement. Particularly, the sample with porosity 0.9, pore density of 10 PPI, aspect ratio of 0.5 offered the highest average Nu number among the studied samples. An empirical correlation for modified Nusselt number at isoflux boundary condition considering the foam morphology parameter and inclination angle was proposed within deviation ±15% between the correlation and the experimental data.     

On the onset of turbulence in natural convection on inclined plates

The problem of determination of the turbulence onset in natural convection on heated inclined plates in an air environment has been experimentally revisited. The transition has been detected by using hot wire velocity measurements. The onset of turbulence has been considered to take place where velocity fluctuations (measured through turbulence intensity) start to grow. Experiments have shown that the onset depends not only on the Grashof number defined in terms of the temperature difference between the heated plate and the surrounding air. A correlation between dimensionless Grashof and Reynolds numbers has been obtained, fitting quite well the experimental data.     

Approximate system of equations for describing nonstationary natural concentration convection in binary gas mixtures

Analysis of the Navier-Stokes equations at small values of the hydrostatic compressibility parameter leads to the formulation of an approximate system of equations for describing nonstationary natural convection in isothermal binary mixtures of gases with arbitrary ratio of the densities. A system of equations for nonstationary concentration convection is also obtained in the Boussinesq approximation, and it is shown that its use in the solution of the considered class of problems may lead to not only quantitative errors but also to a qualitative distortion of the results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 57–61, September–October, 1980.     

On the onset of natural convection of Bingham liquid in rectangular enclosures

The problem of initiation of thermal convection in rectangular cavities filled with viscoplastic material is considered. A two-sided asymptotic expansion is combined with numerical modelling in order to obtain an expression for the critical yield number, which is valid for the entire range of the cavity's aspect ratio.     

The onset of natural convection in a horizontal air layer heated from below

In this work, the beginning of the instability (onset of convection) of an air layer of infinite width and depth heated from underneath with a constant heat flux is studied. In the theoretical part, the instability is studied using the quasi-static assumption. The functional relationship of Rayleigh number vs. horizontal wave number of the disturbance is obtained in a digital computer using Green's Fonctions for the case of Prandtl number equal to one. Furthermore in order to make a comparison with similar investigations, limiting cases of infinite and very small Prandtl numbers are also taken into consideration.-In the experimental part, Rayleigh numbers corresponding to the onset of manifest convection based on visual observations are investigated by optical methods under various heat fluxes. The average value measured is found to be 145.     

A boundary element formulation for natural convection problems

This paper presents a boundary element formulation employing a penalty function technique for two-dimensional steady thermal convection problems. By regarding the convective and buoyancy force terms in Navier-Stokes equations as body forces, the standard elastostatics analysis can be extended to solve the Navier-Stokes equations. In a similar manner, the standard potential analysis is extended to solve the energy transport equation. Finally, some numerical results are included, for typical natural convection problems, in order to demonstrate the efficiency of the present method.     

A fixed domain model for microsegregation during solidification of binary alloys

A fixed-domain numerical model for microsegregation during alloy solidification is developed. The phenomena of solute partitioning at the moving solid/liquid interface and subsequent redistribution by diffusion in the solid and liquid phases have been formulated using volumetric terms. A solute balance equation valid for the whole domain comprising the solid and liquid phases has been obtained in terms of the liquid concentration. The effects of microstructure coarsening on microsegregation has been described and included in the present model. Numerical experiments and comparisons have been carried out between the present fixed-domain model, previous deforming-domain models, and the exact analytical solutions available in the literature. Good agreement has been observed between the predictions of the present fixed-domain model and the exact analytical solutions. Further extensions of the present model for the analysis of two-dimensional microsegregation have been also reported.     

A higher-order analysis of natural convection in a corner

A study of natural convection flow in a right angled corner formed by a semi-infinite vertical plate, which is maintained at the ambient temperature, and a semi-infinite horizontal plate, which is prescribed with a uniform heat flux, is carried out for moderately large values of the Grashof number by the method of matched asymptotic expansions. Higher-order corrections are found for the velocity and temperature fields as well as for the heat transfer and skin friction coefficients. The interaction between the two boundary-layers, which form on the vertical and horizontal plates, takes place through an isothermal outer flow. Eigenvalues and their corresponding eigenfunctions which are associated with the inner expansions have been sought. We are able to continue the solution up to the contribution played by the first eigenvalue and to uniquely find the first eigensolution. Numerical results have been obtained for a wide range of values of the Prandtl number, , but the results are only presented for =0.72 (air) and 6.7 (water). It is found that higher-order corrections to the classical boundary-layer theory are quite significant even for Grashof numbers of order 10⁹.Eine Untersuchung über freie Konvektionsströmung in einer rechtwinkligen Ecke, die aus einer horizontalen und vertikalen, halbunendlichen Platte gebildet wird, wurde für ziemlich hohe Werte der Grashofzahl mit der asymptotisch, erwartungstreuen Schätzung durchgeführt. Bei der vertikalen Platte wird die Umgebungstemperatur konstant gehalten und die Horizontale Platte steht unter gleichmäßigem Wärmefluß. Für die Geschwindigkeits-und Temperaturfelder sind gleich gute Verbesserungen höherer Ordnung gefunden worden, wie für die Wärmeübertragung und den Oberflächenreibungskoeffizient. Die gegenseitige Beeinflussung der beiden Grenzschichten, die sich auf der horizontalen und vertikalen Platte bilden, entstand durch eine isotherme Außenströmung. Eigenwerte und ihre darauf bezogenen Eigenfunktionen, die mit der Expansion verbunden sind, wurden ermittelt. Wir sind in der Lage, die Lösung durch eine Approximation hoher Ordnung zu ersetzen und dadurch eine einzige Eigenlösung zu finden. Die numerischen Ergebnisse sind für weite Bereiche der Prandtlzahl, erhalten worden. Dargestellt wurden jedoch nur die für =0,72 (Luft) und 6,7 (Wasser). Es wurde herausgefunden, daß die Verbesserungen höherer Ordnung für die klassische Grenzschichttheorie sehr bedeutend sind, besonders für die Grashofzahlen der Ordnung 10⁹.     

Calculation of unsteady natural concentration convection in binary mixtures of gases with arbitrary ratio of their densities

The approximate system of equations formulated by Nikulin, Potekhin, and Strelets [1] can, in contrast to the system in the Boussinesq approximation [2], be used to describe natural concentration convection in the presence of significant changes in the composition in gas mixtures with arbitrary ratio of the molecular weights of the components. In the present paper the possibilities opened up by the use of this system of equations are illustrated by the example of a numerical investigation of unsteady free convection of an isothermal binary gas mixture in a closed rectangular region in a wide range of variation of the Archimedes number and the ratio of the molecular weights of the components of the mixture.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 27–31, July–August, 1982.     

The onset of penetrative double-diffusive convection

The onset of double-diffusive convection in a horizontal fluid layer is studied. The density is assumed to depend quadratically on the temperature and linearly on the solute concentration. Under the Boussinesq approximation, the linear stability of the conduction state is investigated with respect to the oscillatory and steady convection modes. For steady onset, the critical thermal Rayleigh number is found to be a double-valued function of the solutal Rayleigh number as long as the relative maximum of the density profile exists within the fluid layer. Driving mechanisms of the steady convections are discussed.     

A two-phase,two-component model for natural convection in a porous medium

A numerical study of natural convection in a two-phase, two-component flow in a porous medium heated from below is presented. Interphase mass and energy transfer, latent heat and bouyancy effects are major physical features. This study extends earlier studies of natural convection based on single-phase, saturated porous medium models. The appearance of two-phase heat pipe zones in the flow has a marked effect on the fluid and heat flows as well as on the performance of the numerical methods. The numerical techniques for handling phase change, Jacobian construction and time step selection are discussed.     

Experimental techniques in natural convection

Experimental techniques in natural convection heat transfer employed in the author's laboratory are introduced. The techniques are mostly related to visualization of flow, temperature field, and heat flux distribution in fluids. Three topics are presented, the first being natural convection in a horizontal rectangular liquid layer driven by surface tension and buoyancy. The patterns of flow were visualized by suspending fine aluminum flakes in the liquid. At the same time, the distribution of the temperature gradient in the liquid was visualized by an optical method making use of the refraction of light. The second topic is the onset of oscillatory convection in the Czochralski growth melt. In this case a forced flow due to rotation of the crystal and the vessel is superimposed on the buoyancy convection, resulting in an oscillatory flow under certain circumstances. The flow pattern and the temperature distribution in the liquid were visualized simultaneously by suspending in the liquid a microencapsulated temperature-sensitive liquid crystal. Periodical oscillation of the flow and the temperature was clearly recognized. The third topic is the rollover of double liquid layers that were stratified stably due to a density difference. A small-scale experiment was carried out to clarify the basic mechanism of rollover. The tracer method was used to visualize boundary layer flow along the vertical side wall and the shadowgraph technique to visualize the density distribution in the liquid layers. The article emphasizes the importance of visual observation in the investigation of natural convection phenomena.     

A pulsed-wire technique for velocity and temperature measurements in natural convection flows

A pulsed-wire probe based on the use of one or two parallel wires, capable of measuring the velocity and the temperature in natural convection flows is described. These measurements are based on the analysis of the relaxation response of a pulsing wire, submitted to a very short electrical pulse. The analysis of the temperature variation on an optional second receiver wire, gives information about the velocity direction. The implementation simplicity of this probe, its good spatial precision, the lack of thermal contamination of the flow, as well as the possibility of obtaining simultaneous velocity and temperature measurements, allow the integration of the device in a multi-point measurement network, capable to deliver thermal and dynamic cartographies of unsteady convection flows.     

A more comprehensive method for yield locus construction for metallic alloys and composites

Conventional methods for constructing yield loci rely on the assumption that nonlinear strains are permanent strains, which is not always the case. A nickel-base alloy, SiC fiber-reinforced titanium, an aluminum alloy, and particlereinforced aluminum have been observed to violate this assumption. We present a method for constructing yield loci using a proof strain criterion for the permanent strain that relies on cyclic, proportional, probes of the yield surface. Two criteria are implemented: one for stress reversal and one for yielding. The method is demonstrated by the construction of initial and subsequent yield loci in the axial-shear stress plane using thin-walled tubular specimens. Results are presented for 6061-T6 aluminum as well as for 6092/SiC/17.5p-T6, which is 6092 aluminum reinforced with 17.5 volume percent silicon carbide particulate. The centers of the initial yield loci for the composite are eccentric to the origin of the stress plane most likely because of the residual stresses induced during processing. Material hardening due to multiaxial stress states can be described by tracking evolution of the subsequent yield surfaces and here hardening of the particulate composite was primarily kinematic     

Non-linear approximations for natural convection in a horizontal annulus

Two different problems are proposed as approximations of the usual system modelling natural convection under the Oberbeck-Boussinesq assumptions. The error is evaluated by means of the norm of its gradient in the Hilbert space. The average Nusselt number is also estimated.     

Onset of oscillatory binary fluid convection in three-dimensional cells

The purpose of this work is to investigate the influence of the transverse walls on the onset of convection in a horizontal rectangular cavity of infinite length filled with a binary mixture when heated from below. For the first time we take into account the effect of the third dimension without making any approximation and considering realistic boundary conditions. In previous numerical works the width of the cell was either taken to be infinity (bulk mixtures) or different approximations usually valid in the narrow cell limit were assumed (i.e., Hele–Shaw and non-ideal Hele–Shaw approximations). The results we find show that the presence of the walls has a considerable effect on the onset of convection even for intermediate transverse aspect ratio cells. They also show that the approximations generally assumed fail to reproduce the correct behaviour of the critical parameters in an important range of aspect ratio values when the primary bifurcation is oscillatory. We have compared the critical values of the Rayleigh number and the frequency with those reported in the literature (Ohlsen et al. [1]) and we find a quantitative agreement within the experimental error. PACS 47.20.Bp, 47.27.Te, 47.54.+r     

Measurements in a natural convection loop

Fluid temperature measurements are reported for a natural convection loop operating in the steady state regime. Averaged and fluctuating measurements were made for a toroidal thermosyphon oriented in the vertical plane that was heated over the lower half and cooled over the upper half. The measurements were made along the tube diameter at the entrance and exit of the heating sections for heat inputs from 1400 to 8410 w/m². The amplitude of the temperature fluctuations exceeded 10 °C near the inner walls indicating the effects from recirculating flow regions at the entrances to the heating and cooling sections.

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Messungen in einem Kreislauf mit Naturumlauf

Zusammenfassung Es wird über Messungen der Fluid-Temperatur in einem mit freier Konvektion stationär betriebenen Kreislauf berichtet. Die Messungen wurden bei leichten Fluktuationen an einem toroidalen Thermosyphon mit vertikaler beheizter Platte durchgeführt, bei dem die untere Hälfte beheizt und die obere gekühlt war. Die Temperaturenmessungen wurden über den Rohrdurchmesser am Eintritt und Austritt der beheizten Strecke für Wärmestromdichten von 1400-8410 w/m² durchgeführt. Die Amplituden der Temperaturfluktuationen überstiegen 10 K in der Nähe der inneren Wand und zeigten dabei Rezirkulationseffekte der Strömung am Eintritt des beheizten und des gekühlten Abschnitts an.