Comments on ''Magnetospheric free-boundary problem in axial symmetry''.
Magnetospheric free boundary representation synthesis problem, discussing relaxation solution, dipole moment and stream direction
PLASMA PHYSICS↗
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Solution of free-boundary problems using finite-element/Newton methods and locally refined grids - Application to analysis of solidification microstructure
A new method is presented for the solution of free-boundary problems using Lagrangian finite element approximations defined on locally refined grids. The formulation allows for direct transition from coarse to fine grids without introducing non-conforming basis functions. The calculation of elemental stiffness matrices and residual vectors are unaffected by changes in the refinement level, which are accounted for in the loading of elemental data to the global stiffness matrix and residual vector. This technique for local mesh refinement is combined with recently developed mapping methods and Newton's method to form an efficient algorithm for the solution of free-boundary problems, as demonstrated here by sample calculations of cellular interfacial microstructure during directional solidification of a binary alloy.
Tsiveriotis, K.↗
Magnetospheric free-boundary problem in axial symmetry.
Approximate solution of magnetospheric free boundary for steady state interaction between field-free streaming plasma and magnetic dipole
FREE BOUNDARY↗
Boundary perturbation method for free boundary problem in convectively cooled continuous casting
Novel mathematical techniques are here used to treat general boundary conditions in convectively cooled continuous ingot casting problems, applying a constant convective heat transfer coefficient at the ingot sides (so that the temperature distribution along the sides is unknown). The problem is first inverted to treat the physical coordinates as dependent variables of temperature and heat flow, mapping the ingot into a region that is nearly rectangular. A boundary perturbation method is then used to obtain an analytical solution in this near-rectangular region. Solidification interface shapes depend on two dimensionless parameters: one is a function of the casting velocity, while the other is dependent on the convective heat transfer coefficient at the cooled wall.
Siegel, R.↗
Some free boundary value problems in elasticity with electroelastic applications
Free boundary value problems in elasticity, applying plane linear theory to electroelasticity
Sloss, J. M.↗
Some free boundary value problems in elasticity applications
Electroelastic applications of free boundary value problems in elasticity theory
Slose, J. M.↗
Finite-element methods for steady solidification problems
Four Galerkin finite-element methods are tested for solving the free-boundary problem that describes steady solidification. The formulations differ in the solution method used to account for the unknown shape of the melt/solid interface, in the interphase condition (either balance of heat flux or equilibrium of temperature) distinguished for locating the interface, and in the technique used for solving the systems of algebraic equations that result from the finite-element approximations. Methods that use the melting point isotherm to locate the melt/solid interface are found more accurate and efficient than formulations based on the interfacial energy balance. Solution by a Galerkin-Newton algorithm of the free-boundary problem transformed to a fixed domain is most efficient when the field problem in each phase is made nonlinear by including radiation from the melt and solid to the surroundings.
Ettouney, H. M.↗
Finite element analysis in fluids; Proceedings of the Seventh International Conference on Finite Element Methods in Flow Problems, University of Alabama, Huntsville, Apr. 3-7, 1989
Recent advances in computational fluid dynamics are examined in reviews and reports, with an emphasis on finite-element methods. Sections are devoted to adaptive meshes, atmospheric dynamics, combustion, compressible flows, control-volume finite elements, crystal growth, domain decomposition, EM-field problems, FDM/FEM, and fluid-structure interactions. Consideration is given to free-boundary problems with heat transfer, free surface flow, geophysical flow problems, heat and mass transfer, high-speed flow, incompressible flow, inverse design methods, MHD problems, the mathematics of finite elements, and mesh generation. Also discussed are mixed finite elements, multigrid methods, non-Newtonian fluids, numerical dissipation, parallel vector processing, reservoir simulation, seepage, shallow-water problems, spectral methods, supercomputer architectures, three-dimensional problems, and turbulent flows.
Chung, T. J.↗
Analysis of shape of porous cooled medium for an imposed surface heat flux and temperature
The surface of a porous cooled medium is to be maintained at a specified design temperature while being subjected to uniform heating by an external source. An analytical method is given for determining the shape of the medium surface that will satisfy these boundary conditions. The analysis accounts for temperature dependent variations of fluid density and viscosity and for temperature dependent matrix thermal conductivity. The energy equation is combined with Darcy's law in such a way that a potential can be defined that satisfies Laplace's equation. All of the heat-transfer and flow quantities are expressed in terms of this potential. The determination of the shape of the porous cooled region is thereby reduced to a free-boundary problem such as in inviscid free jet theory. Two illustrative examples are carried out: a porous leading edge with coolant supplied through a slot and a porous cooled duct with a rectangular outer boundary.
Siegel, R.↗
Natural convection in steady solidification - Finite element analysis of a two-phase Rayleigh-Benard problem
Galerkin finite-element approximations and Newton's method for solving free boundary problems are combined with computer-implemented techniques from nonlinear perturbation analysis to study solidification problems with natural convection in the melt. The Newton method gives rapid convergence to steady state velocity, temperature and pressure fields and melt-solid interface shapes, and forms the basis for algebraic methods for detecting multiple steady flows and assessing their stability. The power of this combination is demonstrated for a two-phase Rayleigh-Benard problem composed of melt and solid in a veritical cylinder with the thermal boundary conditions arranged so that a static melt with a flat melt-solid interface is always a solution. Multiple cellular flows bifurcating from the static state are detected and followed as Rayleigh number is varied. Changing the boundary conditions to approach those appropriate for the vertical Bridgman solidification system causes imperfections that eliminate the static state. The flow structure in the Bridgman system is related to those for the Rayleigh-Benard system by a continuous evolution of the boundary conditions.
Chang, C. J.↗
Thermocapillary flows and their stability
Steady convective flows for geometries of varying complexity are described. This in general involves the solution of nonlinear free boundary problems. The stability of these flows were determined to more complicated (perhaps time dependent) modes of fluid motion. These include shear flow and free surface instability modes, with possible coupling between them. The effect of surface chemical species is described, including impurities which act as surfactants and insoluble oxide films, on the convective patterns and their stability.
Davis, S.↗
The fully nonlinear development of Goertler vortices in growing boundary layers
The fully nonlinear development of small wavelength Goertler vortices in a growing boundary layer is investigated using a combination of asymptotic and numerical methods. The starting point for the analysis is the weakly nonlinear theory of Hall (1982b) who discussed the initial development of small amplitude vortices in a neighborhood of the location where they first become linearly unstable. That development is unusual in the context of nonlinear stability theory in that it is not described by the Stuart-Watson approach. In fact the development is governed by a pair of coupled nonlinear partial differential evolution equations for the vortex flow and the mean flow correction. Here the further development of this interaction is considered for vortices so large that the mean flow correction driven by them is as large as the basic state. Surprisingly it is found that such a nonlinear interaction can still be described by asymptotic means. It is shown that the vortices spread out across the boundary layer and effectively drive the boundary layer. In fact the system obtained by writing down the equations for the fundamental component of the vortex generate a differential equation for the basic state. Thus the mean flow adjusts so as to make these large amplitude vortices locally neutral. Moreover in the region where the vortices exist the mean flow has a square-root profile and the vortex velocity field can be written down in closed form. The upper and lower boundaries of the region of vortex activity are determined by a free-boundary problem involving the boundary layer equations. In general it is found that this region ultimately includes almost all of the original boundary layer and much of the free-stream. In this situation the mean flow has essentially no relationship to the flow which exists in the absence of the vortices.
Hall, Philip↗
Fully implicit shock tracking
A tracking method for solving hyperbolic free boundary problems arising in fluid dynamics is presented. The principal feature of the new approach is that the implicit form is used to simultaneously treat both interior points and boundary conditions. In particular, the location of the free boundary (shock) surface is treated implicitly and coupled with all other unknowns. The method is presented in the context of unsteady one-dimensional flow in a variable area duct with an internal shock wave. The fully implicit method and other strategies for advancing the shock are compared for computing a steady solution via a time asymptotic approach. Issues regarding extension of the method to multiple dimensions are also discussed.
Bell, J. B.↗
Design of supercritical swept wings
Computational fluid dynamics are used to discuss problems inherent to transonic three-dimensional flow past supercritical swept wings. The formulation for a boundary value problem for the flow past the wing is provided, including consideration of weak shock waves and the use of parabolic coordinates. A swept wing code is developed which requires a mesh of 152 x 10 x 12 points and 200 time cycles. A formula for wave drag is calculated, based on the idea that the conservation form of the momentum equation becomes an entropy inequality measuring the drag, expressible in terms of a small-disturbance equation for a potential function in two dimensions. The entropy inequality has been incorporated in a two-dimensional code for the analysis of transonic flow over airfoils. A method of artificial viscosity is explored for optimum pressure distributions with design, and involves a free boundary problem considering speed over only a portion of the wing.
Garabedian, P.↗
Profiles of electrified drops and bubbles
Axisymmetric equilibrium shapes of conducting drops and bubbles, (1) pendant or sessile on one face of a circular parallel-plate capacitor or (2) free and surface-charged, are found by solving simultaneously the free boundary problem consisting of the augmented Young-Laplace equation for surface shape and the Laplace equation for electrostatic field, given the surface potential. The problem is nonlinear and the method is a finite element algorithm employing Newton iteration, a modified frontal solver, and triangular as well as quadrilateral tessellations of the domain exterior to the drop in order to facilitate refined analysis of sharply curved drop tips seen in experiments. The stability limit predicted by this computer-aided theoretical analysis agrees well with experiments.
Basaran, O. A.↗
Eruptive solar magnetic fields
The quasi-steady evolution of solar magnetic fields in response to gradual photospheric changes is considered, with particular attention given to the threshold of a sudden eruption in the solar atmosphere. The formal model of an evolving, force-free field dependent on two Cartesian coordinates is extended to a field which is not force free but in static equilibrium with plasma pressure and gravity. The basic physics is illustrated through the evolution of a loop-shaped electric current sheet enclosing a potential bipolar field with footpoints rooted in the photosphere. A free-boundary problem is posed and then solved for the equilibrium configuration of the current sheet in a hydrostatically supported isothermal atmosphere. As the footpoints move apart to spread a constant photospheric magnetic flux over a larger region, the equilibria available extend the field to increasing heights.
Low, B. C.↗
Boundary-layer model of pattern formation in solidification
A model of pattern formation in crystal growth is proposed, and its analytic properties are investigated. The principal dynamical variables in this model are the curvature of the solidification front and the thickness (or heat content) of a thermal boundary layer, both taken to be functions of position along the interface. This model is mathematically much more tractable than the realistic, fully nonlocal version of the free-boundary problem, and still recaptures many of the features that seem essential for studying dendritic behavior, for example. Preliminary numerical solutions produce snowflakelike patterns similar to those seen in nature.
Ben-Jacob, E.↗
The configuration of slow-mode shocks
A simple method for computing blunt-body flows containing slow shocks is presented. The method is based on the solution of a free-boundary problem for the shock configuration that matches a postshock potential flow to a uniform preshock flow. The method is applied to shocks arising from flow about spherical and paraboloidal obstacles. The results of a parametric study of shock formation as a function of sonic and Alfven Mach numbers showed that when either Mach number decreased, the standoff distance increased. The results should give estimates of the location and the configuration of slow shocks, provided that two as yet unverified assumptions are correct: (1) that the standoff distance and the slow-shock configuration along the stagnation streamline are not affected significantly by portions of the shock still forming far upstream of the shock nose, and (2) that fast-mode waves propagating at small angles to the magnetic field do not significantly affect the formation of the slow shock.
Wolfson, Richard↗