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.     

Special case of a traveling wave in one model of a multicomponent medium

The Kuropatenko model for a multicomponent medium whose components are polytropic gases is considered. It is assumed that, as x → ±∞, the multicomponent medium is in a homogeneous state with constant gas-dynamic parameters — velocity, pressure, and temperature. For the traveling wave flows, conditions similar to the Hugoniot conditions are obtained and used to uniquely determine the flow parameters for x → −∞ from the flow parameters x → +∞ and traveling wave velocity. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 39–47, July–August, 2009.     

New phenomenological model of multicomponent gas diffusion

A new phenomenological model is proposed for diffusion of multicomponent gas mixtures. Obvious general structural relations between the mass diffusion fluxes and the gradients of gas mixture components are obtained within the framework of the model in the case of the multicomponent diffusion coefficients expressed in terms of the “intrinsic” diffusion coefficients of individualmixture components. The molecular mean free path model can be used to estimate the latter.     

A simple semiempirical model for the effective viscosity of multicomponent suspensions

We propose a methodology to approximate the viscosity of multicomponent suspensions. The procedure consists of successive applications of expressions for the viscosity of binary mixtures, originally written as the product of monomodal stiffening functions. First, the viscosity of a binary mixture made of the two smallest components is calculated. This allows to extract a volume fraction that will be used, together with the volume fraction of the third component, to feed the next iteration of the procedure to calculate the viscosity of a trimodal mixture and so on. The application of this approach to arbitrary mixtures requires the detailed knowledge of the geometry of the system in the form of size ratios and compositions. When this information is unknown, an approximation of the model can still be used as a fitting tool. With that purpose, the final expression for the viscosity is written in terms of an effective volume fraction that is further approximated by the use of a (1,2) Padé approximant. This approximation allows to incorporate the crowding effects due to different species in a volume fraction-dependent crowding factor that can be used as a fitting parameter to match experimental or simulation data. We have applied the model to mixtures of particles with different sizes and tested its accuracy comparing with experimental results obtaining very good agreement.     

Diffusion in multicomponent gaseous mixtures in a model of local thermodynamic equilibrium

Diffusion of chemical elements into an ionized multicomponent gaseous mixture is considered in a model of local thermodynamic equilibrium. A linear dependence of the mass flows of chemical elements and the heat flow on the temperature gradient, mass fractions of the chemical elements, and the electric field is obtained. An example is given of a calculation of the effective diffusion coefficients for a hydrogen-helium mixture.     

A pseudopotential-based multiple-relaxation-time lattice Boltzmann model for multicomponent/multiphase flows

In this paper,a pseudopotential-based multiplerelaxation-time lattice Boltzmann model is proposed for multicomponent/multiphase flow systems.Unlike previous models in the literature,the present model not only enables the study of multicomponent flows with different molecular weights,different viscosities and different Schmidt numbers,but also ensures that the distribution function of each component evolves on the same square lattice without invoking additional interpolations.Furthermore,the Chapman-Enskog analysis shows that the present model results in the correct hydrodynamic equations,and satisfies the indifferentiability principle.The numerical validation exercises further demonstrate that the favorable performance of the present model.     

Development of a computer program for the prediction of flow in a rotating cavity

A computer program has been developed to predict laminar source-sink flow in a rotating cylindrical cavity. Although the program is based on a standard finite difference technique for recirculating flow, it incorporates two novel features. Step changes in grid size are employed to obtain sufficient resolution in the boundary layers and special treatment is given to the solution of the pressure correction equations, in the ‘SIMPLE’ algorithm, in order to improve the convergence properties of the method. Results are presented both for the flow in an infinite rotating cylindrical annulus and a finite rotating cylindrical cavity, with the inner cylindrical surface acting as a uniform source and the outer cylinder as a sink. These show good agreement with existing analytical solutions and illustrate some of the problems associated with the computation of rapidly rotating flows.     

Crystallographically based model for transformation-induced plasticity in multiphase carbon steels

The microstructure of multiphase steels assisted by transformation-induced plasticity consists of grains of retained austenite embedded in a ferrite-based matrix. Upon mechanical loading, retained austenite may transform into martensite, as a result of which plastic deformations are induced in the surrounding phases, i.e., the ferrite-based matrix and the untransformed austenite. In the present work, a crystallographically based model is developed to describe the elastoplastic transformation process in the austenitic region. The model is formulated within a large-deformation framework where the transformation kinematics is connected to the crystallographic theory of martensitic transformations. The effective elastic stiffness accounts for anisotropy arising from crystallographic orientations as well as for dilation effects due to the transformation. The transformation model is coupled to a single-crystal plasticity model for a face-centered cubic lattice to quantify the plastic deformations in the untransformed austenite. The driving forces for transformation and plasticity are derived from thermodynamical principles and include lower-length-scale contributions from surface and defect energies associated to, respectively, habit planes and dislocations. In order to demonstrate the essential features of the model, simulations are carried out for austenitic single crystals subjected to basic loading modes. To describe the elastoplastic response of the ferritic matrix in a multiphase steel, a crystal plasticity model for a body-centered cubic lattice is adopted. This model includes the effect of nonglide stresses in order to reproduce the asymmetry of slips in the twinning and antitwinning directions that characterizes the behavior of this type of lattices. The models for austenite and ferrite are combined to simulate the microstructural behavior of a multiphase steel. The results of the simulations show the relevance of including plastic deformations in the austenite in order to predict a more realistic evolution of the transformation process. This work is part of the research program of the Netherlands Institute for Metals Research (NIMR) and the Stichting voor Fundamenteel Onderzoek der Materie (FOM, financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)). The research was carried out under project number 02EMM20 of the FOM/NIMR program “Evolution of the Microstructure of Materials” (P-33).     

Comparison of multicomponent fuel droplet vaporization experiments in forced convection with the Sirignano model

The vaporization of multicomponent fuel droplets was studied experimentally in a heated flow and the results were compared to the model proposed by Abramzon and Sirignano. The droplet was suspended on a permanent holder which was set up in a thermal wind-tunnel. This wind-tunnel was fitted with a video recording system and an infra-red camera. The period during which the droplet was suspended on the holder before the opening of the hot air flow damper was recorded. This first sequence corresponds to the droplet vaporization in natural convection, whose initial experiment conditions, especially diameter, temperature, composition of the droplet, are well known. Then the damper was turn on, and the sequence of forced convection begun. The initial diameter of the droplet was recorded by the video system. The other initial conditions of this second sequence cannot be determined experimentally. The distribution of temperature in the droplet and the surface temperature, the mass fraction distribution in the droplet and the surface mass fraction were unknown. These unknown parameters were determined by coupling our experiment with a model using “the film concept” in natural convection. Experimental results were compared with the calculations and found satisfactory, in natural convection as well as in forced convection initiated by this method. The method was tested in the case of a fuel mixture droplets (heptane–decane) for different initial concentrations and variable durations of the sequence in natural convection.     

Interaction of a wave with an obstacle in a multicomponent two-phase medium

The author's model [1] of a multicomponent liquid medium with nonlinear limiting compression diagrams and constant coefficient of viscosity is improved by the introduction of a coefficient of viscosity that varies during the deformation. The new model is used to obtain a numerical solution to the problem of the propagation of a plane wave produced by a shock load and the interaction of the wave with a fixed obstacle. Such a problem was solved earlier [2] in the case of a viscous medium for linear diagrams of static and dynamic compression and constant coefficient of viscosity. It is shown that the nonlinearity of the diagram of static compression leads with increasing pressure first to an increase in the reflection coefficient and then to a decrease of it. If the load has a sufficient duration, the initial section of the obstacle is subject to a succession of several waves, the number of which increases with increasing duration and amplitude of the load. The calculation was made for glycerine with air bubbles. It is assumed that at pressures up to 400·10⁵ N/m² glycerine is a linearly elastic medium In this case, the dynamic compression diagram of the two-component glycerine—gas-bubble medium is linear.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 183–187, May–June, 1981.I thank Yu. A. Sozonenko for discussion and valuable comments.     

On the use of a two-layer model for large-eddy simulations of supersonic boundary layers with separation

The two-layer modeling approach has become one of the most promising and successful methodology for simulating turbulent boundary layers in the past ten years. In the present study, a mixed wall model for large-eddy simulations (LES) of high-speed flows is proposed which combine two approaches; the thin-Boundary Layer Equations (TBLE) model of Kawai and Larsson (1994) and the analytical wall-layer model of Duprat et al. (2011) for streamwise pressure gradients. The new hybrid model has been efficiently implemented into a three-dimensional compressible LES solver and validated against DNS of a spatially-evolving supersonic boundary layer (BL) under moderate and strong pressure gradients, before being employed for the prediction of nozzle flow separations at different flow conditions, ranging from weakly to highly over-expanded regimes. A good agreement is obtained in terms of mean and fluctuating quantities compared to the DNS results. Particularly, the current wall-modeled LES results are found to perfectly match the DNS data of supersonic BL with/out pressure gradient. It is also shown that the model can account for the effect of the large-scale turbulent motions of the outer layer, indicating a good interaction between the inner and the outer part of the wall layer. In terms of simulations costs and improvements of computing power, the obtained results highlight the capability of the current wall-modeling LES strategy in saving a considerable amount of computational time compared to the wall-resolved LES counterpart, allowing to push further the simulations limits. Furthermore, the application of these computationally low-costly LES simulations to nozzle flow separation allows to clearly identify the origin of the shock unsteadiness, and the existence of broadband and energetically-significant low-frequency oscillations (LFO) in the vicinity of the separation region.     

Development of a parallel computational fluid dynamics algorithm on a hypercube computer

One of the main factors limiting the widespread use of computational fluid dynamics codes for engineering design is their very large requirements both in terms of computer memory and CPU time. Distributed memory parallel computers offer both the potential for a dramatic improvement in cost/performance over conventional supercomputers and the scalability to large numbers of processors that is required if performance beyond that of current supercomputers is to be achieved. As part of an evaluation to explore the potential of such machines for computational fluid mechanics applications, a concurrent algorithm for the solution of the Navier-Stokes equations has been developed and demonstrated on a hypercube parallel computer. The algorithm is based on a domain decomposition of a well-established serial pressure correction algorithm. The algorithm is demonstrated on both a 32-node scalar and eight-node vector Intel iPSC/2 for complicated two-dimensional laminar and turbulent flow problems with different grid sizes and numbers of processors. Speed-ups relative to a single processor of 12.9 with 16 processors and 20.2 with 32 processors are achieved on a scalar iPSC/2, demonstrating the parallel efficiency of the algorithm. Measured performance on a 32-node scalar iPSC/2 exceeds one-sixth that of a Cray X-MP running the original serial algorithm. The performance of the algorithm on an eight-node vector iPSC/2 exceeds that of the larger scalar hypercube and is about one-fifth that of the Cray X-MP. With cost/performance more than 10 times better than the Cray, these results dramatically show the cost effectiveness of vector hypercubes for this class of fluid mechanics algorithm.     

Asymptotic solution of the equations for a three-dimensional multicomponent laminar boundary layer with large injection

The asymptotic method of outer and inner expansions is used to analyze the flow of a multicomponent gas in a three-dimensional boundary layer on a smooth blunt body with large injection. Asymptotic expressions are derived for the friction coefficients, the heat and diffusion fluxes of the components on the surface of the body, and the velocity, temperature, and concentration profiles of the components across the layer of injected gases. It is shown that with large injection the limiting (bottom) streamlines on the surface of the body coincide in the first approximation with the vectorial lines of the pressure gradient.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 47–56, March–April, 1975.The author is indebted to G. A. Tirskii for a discussion of the work.     

A computer model of the two-roll mill for generalised Newtonian flow

We investigate the flow of a generalised Newtonian liquid between two contra-rotating cylinders of equal size — the so called two-roll mill problem. A finite element method is used to obtain solutions for the case of a Carreau model and for a power law model. Consideration is given to the influence of shearthinning on the flow pattern around the cylinders. Results are presented for different speeds of rotation of the cylinders and for various values of fluid parameters. A comparison is made with the analytical solution of Jeffrey.     

Determination of thermal-hardening stress in steels by use of thermoplasticity theory

The problems of quenching-stress analysis are critically reviewed and an adequate simple theory is discussed. The theory accounts for both plasticity and volumetric changes due to phase transformation accompanying the thermal-hardening of a group of simple steels that are characterized by C-shaped time-temperature-transformation diagrams. The volume dilatation in the absence of stress is assumed to be a linear function of the separate specific volumes and weight fractions of the constituents (pearlite, austenite and martensite). With use of the classical relationships of a formal theory of transformation kinetics, the amounts of pearlite and martensite are expressed in terms of the temperature and the temperature-history. The specific forms of such functions are given. In order to account for the influence of phase transformation on plastic properties, the non-isothermal plastic flow-rule is generalized, and a thermal-hardening parameter is introduced which is identified with the amount of pearlite. Variational principles and bounding inequalities associated with the fundamental rate-problem are considered. As an example, the problem for a rapidly, uniformly-cooled half-space is solved. The variations of the residual stress and the final amount of martensite with distance from the outer surface are given, for several values of the rate-of-cooling. The results suggest that the residual stress vanishes on the plane containing approximately 30–35 per cent of martensite.