Engineering topics

Mcfadden, G. B.

Publications and source records attributed to Mcfadden, G. B..

At least 37 records · Page 2

NASA NTRS1990

The effect of an electric field on the morphological stability of the crystal-melt interface of a binary alloy. III - Weakly nonlinear theory

The effect of a constant electric current on the crystal-melt interface morphology during directional solidification at constant velocity of a binary alloy is considered. A linear temperature field is assumed, and thermoelectric effects and Joule heating are neglected; electromigration and differing electrical conductivities of crystal and melt are taken into account. A two-dimensional weakly nonlinear analysis is carried out to third order in the interface amplitude, resulting in a cubic amplitude equation that describes whether the bifurcation from the planar state is supercritical or subcritical. For wavelengths corresponding to the most dangerous mode of linear theory, the demarcation between supercritical and subcritical behavior is calculated as a function of processing conditions and material parameters. The bifurcation behavior is a sensitive function of the magnitude and direction of the electric current and of the electrical conductivity ratio.

Wheeler, A. A.↗

NASA NTRS1990

A numerical and analytical study of nonlinear bifurcations associated with the morphological stability of two-dimensional single crystals

The nonlinear stability of a two-dimensional single crystal of pure material in an undercooled melt is studied both analytically and numerically. The quasi-steady state approximation is used for the thermal fields, and the effects of different solid and liquid thermal conductivities, isotropic interfacial growth kinetics, and isotropic surface tension are included. The bifurcation analysis is performed by calculating the instantaneous value of the fundamental component of the local normal growth speed for an interface perturbed by a single Fourier shape component. Numerically, the fundamental component of the interfacial growth speed is found by Fourier analysis of the solution to an integrodifferential equation obeyed at the interface. Analytically, an expansion technique is used to derive a solvability condition defining each of these bifurcation points. The analytical and numerical results are in very close agreement. Almost all of the bifurcations are subcritical, and the results are presented by giving values of the Landau coefficient as a function of the different dimensionless parameters used in the model.

Brush, L. N.↗

NASA NTRS1990

Instability during directional solidification - Gravitational effects

After an elementary introduction to solidification, recent research that attempts to understand the interaction of fluid flow with a crystal-melt interface under gravitational influence using linear stability analysis is reviewed. Recent numerical calculations of fluid flow and interface morphology during alloy solidification are briefly discussed.

Coriell, S. R.↗

NASA NTRS1989

Elimination of spurious eigenvalues in the Chebyshev tau spectral method

Spectral methods have been used to great advantage in hydrodynamic stability calculations; the concepts are described in Orszag's seminal application of the Chebyshev tau method to the Orr-Sommerfeld equation for plane Poiseuille flow in 1971. Orszag discusses both the Chebyshev Galerkin and the Chebyshev tau methods, but presents results for the tau method, which is easier to implement than the Galerkin method. The tau method has the disadvantage that two unstable eigenvalues are produced that are artifacts of the discretization. An extremely simple modification to the Chebyshev tau method is presented which eliminates the spurious eigenvalues. First a simplified model of the Orr-Sommerfeld equation discussed by Gottlieb and Orszag was studied. Then the Chebyshev tau method is considered, which has two spurious eigenvalues, and then a modification which eliminates them is described. Finally, results for the Orr-Sommerfeld equation are considered where the modified tau method also eliminates the spurious eigenvalues. The simplicity of the modification makes it a convenient alternative to other approaches to the problem.

Mcfadden, G. B.↗

NASA NTRS1989

Buoyancy effects on morphological instability during directional solidification

The onset of morphological instability during the directional solidification of a single-phase binary alloy at constant velocity vertically upwards is treated by a linear stability analysis. The case in which a heavier solute is rejected at the solidifying interface is considered, and the effect of natural convection on the critical concentration for the onset of instability is studied. For tin containing lead, a small destabilization of the system at low growth velocities, and a large increase in the wavelength of the instability at the onset are found. Calculations show that the destabilization is enhanced as the variation of density with solute concentration is reduced, and in the limit of neutrally-dense solute, there is a long wavelength instability for which the critical solute concentration is several orders of magnitude lower than that predicted by the Mullins and Sekerka (1964) analysis in the absence of convection. For the neutrally-dense solute, a simplified analysis indicates the roles played by the interface deformation and thermal convection in promoting the instability. In particular, the destabilization is very sensitive to the ratio of crystal and melt thermal conductivities.

Coriell, S. R.↗

NASA NTRS1988

Effect of surface tension anisotropy on cellular morphologies

A three-dimensional weakly nonlinear analysis for conditions near the onset of instability at the crystal-melt interface was carried out to second order, taking into account the effects of latent heat generation and surface-tension anisotropy of the crystal-melt interface; particular consideration was given to the growth of a cubic crystal in the 001-, 011-, and 111-line directions. Numerical calculations by McFadden et al. (1987), performed for an aluminum-chromium alloy with the assumption of a linear temperature field and an isotropic surface tension, showed that only hexagonal nodes (and not hexagonal cells) occurred near the onset of instability. The results of the present analysis indicate that the nonlinear temperature field (which occurs when thermal conductivities of the crystal and the melt are different and/or the latent heat effects are not negligible) can modify this result and, for certain alloys and processing conditions, can cause the occurrence of hexagonal cells near the onset of instability.

Mcfadden, G. B.↗

NASA NTRS1988

The effect of an electric field on the morphological stability of the crystal-melt interface of a binary alloy

A fully time-dependent linear stability analysis of the morphological stability of a planar interface during directional solidification of a binary alloy at constant velocity in the presence of an electric field, is performed. The electromigration of solute and the differing electrical conductivities of solid and liquid for a model in which the temperature gradient is constant are taken into account. The present results are compared with the constitutional supercooling criterion, and it is shown there may be substantial differences. A modified constitutional supercooling criterion which is valid over a large range of conditions is derived. It is also found under certain conditions that the onset of instability may be time dependent.

Wheeler, A. A.↗

NASA NTRS1988

Initial conditions implied by t exp 1/2 solidification of a sphere with capillarity and interfacial kinetics

The paper explores the initial conditions implied by t exp 1/2 growth of a spherical crystal solidifying from a pure, undercooled melt, including the effects of both capillarity and interface kinetics, and relates the findings to initial conditions that would be expected on the basis of classical nucleation theory. For crystal sizes near the nucleation radius, the calculated temperature profiles show a cold region ahead of the advancing interface that is even more undercooled than the undercooled bath. This cold region acts as a local heat sink that compensates for the reduced growth speed that would otherwise result from capillarity and kinetics, leading to precisely the same t exp 1/2 growth law that would have been obtained had both capillarity and kinetics been neglected.

Sekerka, R. F.↗

NASA NTRS1988

Numerical simulation of morphological development during Ostwald ripening

The morphological evolution of a small number of particles undergoing Ostwald ripening in two dimensions was investigated using a boundary integral technique. The numerical calculations show that, in many cases, interparticle diffusional interactions do not induce large distortions in particle morphology from a circle, due to the strong local interaction between regions of different curvature on the same particle. Particle migration due to interparticle diffusional interactions was observed, with the migration distance larger than the initial radius of a particle, which is linked to the particle spatial arrangement. As diffusion is the mechanism for the migration, it is expected that particle movement is a generic aspect of the coarsening process at high volume fractions of coarsening phase and, possibly, at low volume fractions. The calculations support unambiguously the Ostwald ripening mechanism for flat formation during liquid-phase sintering. As such, the appearance or regions of low interfacial curvature is a dynamic phenomenon dependent on interparticle diffusion.

Voorhees, P. W.↗

NASA NTRS1988

Solutal convection during directional solidification

During directional solidification of a binary alloy at constant velocity, buoyancy-driven fluid flow may occur due to the solute gradients generated by the solidification process. Numerical calculations of the solute and fluid flow fields in the melt have been carried out using finite differences in a two-dimensional, time-dependent model that assumes a planar crystal-melt interface and allows time-dependent gravitational accelerations. The container walls are rigid and perfectly insulating for solute. For constant vertical gravitational accelerations, as the solutal Rayleigh number is varied, multiple steady states and time-dependent states may occur. The bifurcation from the quiescent state may be subcritical or transcritical, depending on the aspect ratio of the container. Calculations have also been performed for a gravitational acceleration that is assumed to be uniform in magnitude with its direction rotating uniformly. Numerical results have been obtained for a Schmidt number of 10 and a gravitational acceleration of 0.0001 G. The maximum variation in the solute concentration at the crystal-melt interface is calculated for various values of the rotation rate of the gravitational acceleration.

Mcfadden, G. B.↗

NASA NTRS1987

Nonplanar interface morphologies during unidirectional solidification of a binary alloy. II - Three-dimensional computations

A finite difference method is used to obtain three-dimensional steady-state solutions for nonplanar interface morphologies in order to study the situation of equal thermal properties in the crystal and melt with negligible latent heat release. Stable steady-state solutions corresponding to two-dimensional bands and three-dimensional hexagonal nodes, as well as to rectangular interface planiforms, are found using a model of an aluminum-chromium alloy with a distribution coefficient of greater than one. Hexagonal nodes are predicted near the onset of instability, in agreement with weakly nonlinear theory.

Mcfadden, G. B.↗

NASA NTRS1987

Thermosolutal convection during directional solidification. II - Flow transitions

The influence of thermosolutal convection on solute segregation in crystals grown by vertical directional solidification of binary metallic alloys or semiconductors is studied. Finite differences are used in a two-dimensional time-dependent model which assumes a planar crystal-melt interface to obtain numerical results. It is assumed that the configuration is periodic in the horizontal direction. Consideration is given to the possibility of multiple flow states sharing the same period. The results are represented in bifurcation diagrams of the nonlinear states associated with the critical points of linear theory. Variations of the solutal Rayleigh number can lead to the occurrence of multiple steady states, time-periodic states, and quasi-periodic states. This case is compared to that of thermosolutal convection with linear vertical gradients and stress-free boundaries.

Mcfadden, G. B.↗

NASA NTRS1987

Double diffusive convection during directional solidification - Nonlinear analysis

The dynamics of doubly diffusive convective flow during the constant-velocity vertical solidification of binary melts to form single-phase solids is examined, summarizing the results of recent experimental and analytical investigations. The effects of horizontal temperature gradients are considered; the nonlinearities of the system and their numerical treatment are discussed; and numerical results are presented graphically for the solidification of a Pb melt containing 0.0015 wt pct Sn, with growth velocity 0.0002 cm/s and a temperature gradient of 200 K/cm in the liquid.

Coriell, S. R.↗

NASA NTRS1986

The effect of fluid flow due to the crystal-melt density change on the growth of a parabolic isothermal dendrite

The Ivantsov (1947) analysis of an isolated isothermal dendrite (with zero surface tension) growing into a supercooled liquid is extended to include the effects of the fluid flow due to volume contraction or expansion upon solidification. For an axisymmetric paraboloidal dendrite, an analytic solution to the Navier-Stokes equations is obtained. The magnitude of the flow is proportional to the relative density change epsilon, and the flow becomes negligible far from the surface of the dendrite. The temperature field consistent with this flow can also be found explicitly. The well-known expression that relates the dimensionless supercooling to the Peclet number in the absence of fluid flow is modified for nonzero epsilon, but the effect is of order epsilon and hence is seen to be minor for most values of epsilon and dimensionless supercooling that occur in practice.

Mcfadden, G. B.↗

NASA NTRS1986

Interaction of flows with the crystal-melt interface

The coupling between a crystal-melt interface and fluid flow is investigated. The solidification boundary conditions at the crystal-melt interface are described. The influences of morphological and double-diffusive instabilities during directional solidification on the interface are studied. The experiments by Glicksman and Mickalonis (1982) and Fang et al. (1985) which examine the relationship between the hydrodynamic state of the phase melt and phase-change interface are analyzed. The Rayleigh-Benard problem, and the effect of crystal-melt interaction on the Rayleigh number are examined. Various applications for melt-flow interactions with solid-liquid interface to engineering and welding are discussed.

Glicksman, M. E.↗

NASA NTRS1985

Double-diffusive convection with sidewalls

Stommel et al. (1956) have first described an instability, known as thermosolutal convection, thermohaline convection, or double-diffusive convection. This instability may occur in the case of a fluid in a gravitational field with two diffusing components present. The present study is concerned with the effect of sidewalls on flow in the fingering regime in the absence of applied horizontal gradients. The work was motivated by numerical results obtained on the basis of a simulation of thermosolutal convection occurring during the unidirectional solidification of a binary alloy. In this case, the unperturbed solute field in the liquid ahead of the solidifying planar interface has an exponential vertical profile because of the rejection or preferential incorporation of solute by the solid phase.

Mcfadden, G. B.↗

NASA NTRS1985

Convection During Unidirectional Solidification

The fluid flow and solute segregation which occur during directional solidification, including effects of gravity and microgravity were studied. The nature of the fluid flow, its effects on the shape of the crystal-metal interface and the resulting distribution of solutes are examined. During solidification of a binary alloy at constant velocity vertically upwards, thermosolutal convection occurs if the solute rejection or preferential incorporation at the crystal-melt interface decreases the density of the melt. Numerical calculations of the solute, temperature, and flow fields are carried out by using previously developed algorithms for Schmidt numbers Sc = 1, 10, 81. The time periodic flow, which occurs over a narrow range of parameters, is investigated.

Coriell, S. R.↗

NASA NTRS1985

Convective influence on the stability of a cylindrical solid-liquid interface

Experiments in which a long vertical, heated wire is surrounded by concentric annuli of a melt and its crystalline solid show that the convection state changes from a stable unicell surrounded by a stationary cylindrical solid-liquid interface, to a complex time-dependent flow surrounded by a rotating, helical solid-liquid interface. This transition occurs at a Grashof number of approximately 150, which is an order of magnitude less than the critical Grashof number calculated for a liquid annulus surrounded by rigid walls. A linear stability analysis has been carried out for an infinitely tall vertical annulus. When the deformable nature of the crystal-melt interface is taken into account in the boundary conditions, two new modes of instability arise. The most dangerous mode is asymmetrical and corresponds to helical waves travelling vertically upwards. The critical Grashof number and the scaling properties of the eigenstate agree with experiments. The results clearly demonstrate the coupling of convection with crystal-melt interfacial instabilities.

Fang, Q. T.↗