Vol 57, No 7 (2016)

Some numerical results for the two- and three-dimensional de Vahl Davis benchmark are presented. This benchmark describes thermal convection in a square (cubic) cavity with vertical heated walls in a wide range of Rayleigh numbers (10⁴ to 10¹⁴), which covers both laminar and highly turbulent f lows. Turbulent f lows are usually described using a turbulence model with parameters that depend on the Rayleigh number and require adjustment. An alternative is Direct Numerical Simulation (DNS) methods, but they demand extremely large computational grids. Recently, there has been an increasing interest in DNS methods with an incomplete resolution, which, in some cases, are able to provide acceptable results without resolving Kolmogorov scales. On the basis of this approach, the so-called parameter-free computational techniques have been developed. These methods cover a wide range of Rayleigh numbers and allow computing various integral properties of heat transport on relatively coarse computational grids. In this paper, a new numerical method based on the CABARET scheme is proposed for solving the Navier–Stokes equations in the Boussinesq approximation. This technique does not involve a turbulence model or any tuning parameters and has a second-order approximation scheme in time and space on uniform and nonuniform grids with a minimal computational stencil. Testing the technique on the de Vahl Davis benchmark and a sequence of refined grids shows that the method yields integral heat f luxes with a high degree of accuracy for both laminar and highly turbulent f lows. For Rayleigh numbers up to 10¹⁴, a several percent accuracy is achieved on an extremely coarse grid consisting of 20 × 20 cells refined toward the boundary. No definite or comprehensive explanation of this computational phenomenon has been given. Cautious optimism is expressed regarding the perspectives of using the new method for thermal convection computations at low Prandtl numbers typical of liquid metals.

We consider the problem of gravitational instability (Rayleigh–Taylor instability) of a horizontal thin gas layer between two liquid half-spaces (or thick layers), where the light liquid overlies the heavy one. This study is motivated by the phenomenon of boiling at the surface of direct contact between two immiscible liquids, where the rate of the “break-away” of the vapor layer growing at the contact interface due to development of the Rayleigh–Taylor instability on the upper liquid–gas interface is of interest. The problem is solved analytically under the assumptions of inviscid liquids and viscous weightless vapor. These assumptions correspond well to the processes in real systems, e.g., they are relevant for the case of interfacial boiling in the system water-n-heptane. In order to verify the results, the limiting cases of infinitely thin and infinitely thick gas layers were considered, for which the results can be obviously deduced from the classical problem of the Rayleigh–Taylor instability. These limiting cases are completely identical to the well-studied cases of gravity waves at the liquidliquid and liquid–gas interfaces. When the horizontal extent of the system is long enough, the wavenumber of perturbations is not limited from below, and the system is always unstable. The wavelength of the most dangerous perturbations and the rate of their exponential growth are derived as a function of the layer thickness. The dependence of the exponential growth rate on the gas layer thickness is cubic.

The effect of the mesh geometry on the accuracy of solutions obtained by the finite-element method for problems of linear fracture mechanics is investigated. The guidelines have been formulated for constructing an optimum mesh for several routine problems involving elements with linear and quadratic approximation of displacements. The accuracy of finite-element solutions is estimated based on the degree of the difference between the calculated stress-intensity factor (SIF) and its value obtained analytically. In problems of hydrofracturing of oil-bearing formation, the pump-in pressure of injected water produces a distributed load on crack flanks as opposed to standard fracture mechanics problems that have analytical solutions, where a load is applied to the external boundaries of the computational region and the cracks themselves are kept free from stresses. Some model pressure profiles, as well as pressure profiles taken from real hydrodynamic computations, have been considered. Computer models of cracks with allowance for the pre-stressed state, fracture toughness, and elastic properties of materials are developed in the MSC.Marc 2012 finite-element analysis software. The Irwin force criterion is used as a criterion of brittle fracture and the SIFs are computed using the Cherepanov–Rice invariant J-integral. The process of crack propagation in a linearly elastic isotropic body is described in terms of the elastic energy release rate G and modeled using the VCCT (Virtual Crack Closure Technique) approach. It has been found that the solution accuracy is sensitive to the mesh configuration. Several parameters that are decisive in constructing effective finite-element meshes, namely, the minimum element size, the distance between mesh nodes in the vicinity of a crack tip, and the ratio of the height of an element to its length, have been established. It has been shown that a mesh that consists of only small elements does not improve the accuracy of the solution.

An experimental and theoretical study of plastic shear localization mechanisms observed under dynamic deformation using the shear–compression scheme on a Hopkinson–Kolsky bar has been carried out using specimens of AMg6 alloy. The mechanisms of plastic shear instability are associated with collective effects in the microshear ensemble in spatially localized areas. The lateral surface of the specimens was photographed in the real-time mode using a CEDIP Silver 450M high-speed infrared camera. The temperature distribution obtained at different times allowed us to trace the evolution of the localization of the plastic strain. Based on the equations that describe the effect of nonequilibrium transitions on the mechanisms of structural relaxation and plastic flow, numerical simulation of plastic shear localization has been performed. A numerical experiment relevant to the specimen-loading scheme was carried out using a system of constitutive equations that reflect the part of the structural relaxation mechanisms caused by the collective behavior of microshears with the autowave modes of the evolution of the localized plastic flow. Upon completion of the experiment, the specimens were subjected to microstructure analysis using a New View-5010 optical microscope–interferometer. After the dynamic deformation, the constancy of the Hurst exponent, which reflects the relationship between the behavior of defects and roughness induced by the defects on the surfaces of the specimens is observed in a wider range of spatial scales. These investigations revealed the distinctive features in the localization of the deformation followed by destruction to the script of the adiabatic shear. These features may be caused by the collective multiscale behavior of defects, which leads to a sharp decrease in the stress-relaxation time and, consequently, a localized plastic flow and generation of fracture nuclei in the form of adiabatic shear. Infrared scanning of the localization zone of the plastic strain in situ and the subsequent study of the defect structure corroborated the hypothesis about the decisive role of non-equilibrium transitions in defect ensembles during the evolution of a localized plastic flow.

Methods for solving shallow-water equations that describe flows in rotating annular channels are considered and the results of numerical calculations are analyzed for the possible generation of global large-scale flows, narrow jets, and numerous small-scale vortices in laboratory experiments. External effects in fluids are induced using a mass source–sink and the MHD-method of interaction of radial electric current with the magnetic field generated by the field of permanent magnets. A central–upwind scheme modified to suit the specific aspects of geophysical hydrodynamics. Initially, this method was used to solve shallow-water equations only in hydraulic problems, such as for flows in dam breaks, channels, rivers, and lakes. Geophysical hydrodynamics (in addition to free surface and topography) requires a rotation of the system as a whole, which is accompanied by the appearance of a complex system of vortices, jets, and turbulence (these should be taken into account in the formulation of the problem). Accordingly, the basic features of the central–upwind method should be changed. The modifications should ensure that the scheme is well-balanced and choose interpolation methods for desired variables. The main result of this modification is the control over numerical viscosity affecting the fluid motion variety. The active dynamics of a large number of vortices transformed into jets or generating large-scale streams is the general result of modifications suitable for geophysical hydrodynamics. Because there are technical difficulties in the creation of an appropriate laboratory setup for modeling of geophysical flows with the help of numerous source–sinks, it will be appropriate to use numerical experiments for studying the motions generated by this method. Unlike this method, the MHD-method can be rather easily used in laboratory conditions to generate a large variety of flows and vortex currents in the channel by a relatively small number of permanent magnets. Specifically, this method made it possible to obtain large-scale circular flows over the entire channel area, jets, and systems of interacting vortices. For the purpose of experiments, the distributions of source–sinks and systems of permanent magnets over the bottom of annular channels are determined.

The structure of a convective flow of molten magnesium in a metallothermic titanium reduction apparatus has been studied numerically for various retort heating and cooling configurations. The mathematical model is based on the thermogravitational convection equations for a singlephase fluid in the Boussinesq approximation. A nonuniform computational grid with a total of 5 million grid points was used. The LES (Large Eddy Simulation) technique was applied to take into account the turbulence. The problem was considered in a three-dimensional nonstationary formulation, which allowed one to construct the instantaneous and average characteristics of the process and to analyze the velocity and temperature pulsation fields. It has been found that steady axisymmetric flows occur at moderate Grashof numbers (Gr ~ 10⁷–10⁸), while unsteady turbulent flows take place at Grashof numbers corresponding to the actual titanium reduction process (Gr ~ 10¹²). The influence of the degree of nonhomogeneity of the heat release due to the titanium reduction reaction that runs mainly on the magnesium surface has been studied. The computations have been performed for two configurations of the system for maintaining the apparatus heating regime: with furnace heaters operating at full power and with switched-off furnace heaters. Fundamental differences in convective flow structure for these two heating methods have been revealed. It has been established that the most intense velocity and temperature pulsations arise in the region adjacent to the interface between the cooled and heated parts of the retort side surface.