Two-dimensional Nonstationary Problem Research Articles

Development of the wave motion induced by near-bottom periodic disturbances in a two-layer shear current

The behavior of wave motion arising in an ideal incompressible homogeneous fluid under switching-on periodic bottom disturbances is studied in the linear approximation for the two-dimensional non-stationary problem. In the undisturbed state, the velocities of two-layer fluid flow are linear functions of the vertical coordinate in each of the layers with different gradients and coincide on the boundary of the layers. The upper boundary of fluid can be either free or bounded by the rigid cover. The dispersion dependences and the group velocities of the appearing wave modes are determined. The vertical displacements of the free surface and the interface between the layers are calculated. A comparison with the solution for a single-layer fluid is carried out.

Splitting schemes for non-stationary problems with a rational approximation for fractional powers of the operator

In this paper we study the numerical approximation of the solution of a Cauchy problem for a first-order-in-time differential equation involving a fractional power of a self-adjoint positive operator. One popular approach for the approximation of fractional powers of such operators is based on rational approximations. The purpose of this work is to construct special approximations in time so that the solution at a new time level is produced by solving a set of standard problems involving the self-adjoint positive operator rather than its fractional power. Stable splitting schemes with weight parameters are proposed for the additive representation of the rational approximation of the fractional power of the operator. Finally, numerical results for a two-dimensional non-stationary problem with a fractional power of the Laplace operator are also presented.

Modelling of the processes of impact of a projectile with elements of individual defence

A two-dimensional and three-dimensional non-stationary problem of the interaction of a homogeneous impactor and a heterogeneous structure made of steel and ceramics and placed in a Kevlar pocket is considered. The model of the human body is a plate of gelatine with cylindrical inserts-imitators of human bones. The results of numerical simulation using different approaches for describing heterogeneous media are compared. On the basis of direct numerical simulation, it is shown that the gradient armor plate (steel + B4C) has the best weight and size parameters.

Оценки сходимости итерационного алгоритма численного решения двумерных нестационарных задач магнитной гидродинамики

Оценки сходимости итерационного алгоритма численного решения двумерных нестационарных задач магнитной гидродинамики

Investigation of the influence of gravitational forces on the process of displacement of viscoplastic fluids

The object of research is a numerical simulation of the process of two-dimensional two-phase filtration of viscoplastic oil and water, taking into account the gravitational forces, some properties of liquids, as well as relative phase permeabilities and capillary forces.As is known, the problems of multiphase filtration have specific features. Therefore, there is a need to develop difference schemes in adaptive grids that reduce the artificial viscosity and oscillation of the numerical solution. They also make it possible to obtain acceptable results with a small number of nodes in the computational grid.To take into account the singularities of the solution, a difference-iteration method is used in moving grids. Based on the computational experiment, the influence of the initial pressure gradient and gravity on the displacement process is investigated.Economical difference schemes that combine the advantages of explicit and implicit schemes are constructed and make it possible to reduce the two-dimensional problem to a chain of one-dimensional problems. A difference-iterative method is also proposed in moving grids for solving two-dimensional (axisymmetric) non-stationary filtration problems of anomalous liquids, by means of which an iterative process is constructed to find the distribution of water saturation.The carried out calculations to determine the influence of gravity on the displacement process have shown that at z=0, even at low productive-bed thicknesses, gravitational forces influence the displacement process. And over time this influence increases: if at the time t=0.08 on the circuit the difference of water saturation was 0.0077; at t=0.24–0.0122, then at t=1.04 it becomes equal to 0.0292.It is shown that when modeling the process without taking gravity into account it is expedient to simplify the geometry of the filtration region, i. e., to consider a plane-radial flow in view of the considerable simplicity of the calculations.The developed algorithms can be used for hydro-gas dynamic calculations related to the development and operation of oil fields containing anomalous oil.

A two-dimensional nonstationary problem of elastic diffusion for an orthotropic one-component layer

A 2 D nonstationary problem of an orthotropic one-component elastic layer, accounting for diffusion, is considered. A locally balanced model of elastic diffusion is used, which includes a coupled set of equations of elastic body motion and a mass-transfer equation. On the layer boundaries normal displacement, tangential stress and diffusive flow are assumed. At an initial time, the layer is in an undisturbed state. The solution is sought in an integral form, as a double convolution in time and along the space coordinate of Green functions and the right-hand sides of the boundary equations. To construct a solution of the initial problem, Laplace time transform, exponential Fourier transform along the space coordinate in the direction of the layer surface, reduction to zero boundary conditions, and expansion into incomplete Fourier series along a coordinate directed through the depth of the layer are successively used. Thus, the initial problem is reduced to a set of linear algebraic equations, from which forms of the sought functions are found. It is shown that the necessary condition of the applicability of the present algorithm is orthotropy of the medium in question. Inversion of Laplace transform of the sought functions is reduced to computing the originals of the rational functions, which is done analytically, using deductions. It is found that all Green functions are even (odd) relative to Fourier transform parameters, which makes it possible, when the right-hand sides of the boundary conditions are even (odd), to reduce the inversion of exponential Fourier transform to the inversion of sinus-, cosine-transform. In that case, quadrature formulas of medium rectangles are used to find the originals of Fourier transformants. An example is considered, where surface perturbation is modeled with Heaviside time function and a Gaussian along the space coordinate. A homogenized medium of interlaced layers of aluminum and copper is used as a model of an orthotropic medium. The computational results are presented both analytically and in the form of 3D diagrams of the diffusant concentration increment and displacement vector components.

Heat transfer in heated industrial premises with using radiant heating system

The results of mathematical modeling of heat transfer processes in a closed air volume surrounded by enclosing constructions, heated by supplying energy to the upper contour of gas infrared radiators are represented. Regimes of turbulent natural conjugate convection in the region bounded by solid walls are investigated. Two-dimensional nonstationary problem is solved in the framework of the Navier -Stokes equations for gas and thermal conductivity for solid walls. Nonstationary processes of heat propagation in course of time and essential heterogeneity of temperature fields and heat fluxes are established.

Numerical simulation of the flow of a gas-liquid medium in a circular channel at cooling a high-temperature metal cylinder with a variable cross-section

The numerical simulation results of the flow of a gas-liquid medium over the high-temperature metal cylinder surface in a circular channel are presented. The flow over the solid metal cylinder with a variable cross-section by a gas-liquid medium is considered. Based on the mathematical model taking into account the cooling-media-flow axial symmetry relative to the cylinder longitudinal axis, the results of the numerical simulation of the process of solving the two-dimensional nonstationary problem of the coupled heat exchange of the gas-liquid medium flow and metal cylinder are obtained. The control volume approach is used for solving the differential equation system. The flow field parameters are calculated using the algorithm SIMPLE. For the iterative solution of linear algebraic equations, Gauss-Seidel method with under-relaxation is used. The numerical calculation results are obtained for the flow hydrodynamic parameters of the gas-liquid medium at cooling the high-temperature metal cylinder by taking into account vaporization. The values of temperatures, flow velocities, and vapour distribution in the computational domain are determined. The variation intensity of the metal cylinder surface temperature depending on the gas-liquid flow velocity is analyzed.

Determination of the optimal working conditions of cable lines in an underground cable channel

Mathematical modeling of heat transfer processes inside a cable channel taking into account natural convection and radiation between the cable surface and tube and the loss in shields was carried out. An underground cable channel consisting of eight lines with a voltage of 6 kV and six lines with a voltage of 35 kV was investigated. The conductor section is 150 mm2. Cables of APvVng-LS brand were placed in a triangle inside polyethylene tubes. The patterns of temperature fields obtained under variance of different various factors were determined, the permissible currents were calculated, and the optimal circuits of connection of cable lines depending on the selected criterion were proposed. The two-dimensional nonstationary problem of complicated heat and mass transfer and heat conductivity inside the underground cable channel was solved by the finite elements method using the ANSYS software package. The range of permissible operating currents depending on the location of cable lines and priority of their connection was determined. The effect of the temperature of the environment on the permissible current was considered. The optimal ranges of thermal permissible currents taking into account the shield loss were calculated. The optimal connection circuits of the cable lines depending on the problem being solved were obtained. An estimation of transferred power for every considered connection method is presented. Conclusions for practical application of the method of connection of the cable line were drawn. The fields of application of this mathematical model were proposed.

Use of Lagrangian-Eulerian Computing Method for Systems with Phase Change at the Solution of Stefan Boundary Value Problem

The problem of a planar solidification of a material with an additional nonstationary radiant of heat on a semi-infinite plane has been solved. For a solution the condition of Stefan was used. Results have been compared with an analytical solution in case of the absence of an additional radiant of heat, as well as with a solution obtained by perturbations method. A more complicated two-dimensional nonstationary problem of a solidification of a liquid with interface free-boundary has been also solved. The purpose of this problem solution is to predict position of a material phase boundary, as well as the temperature distribution in a layer of PCM (Phase-Change Material) with boundary conditions of Dirichlet.

Two-dimensional nonstationary problem of elastic diffusion for an isotropic one-component layer

A locally equilibrium model of mechanodiffusion which comprises a coupled system of motion equations for an elastic body and a mass transfer equation is used to solve the two-dimensional nonstationary problem of elastic diffusion for an isotropic one-component layer. The solution is constructed using Fourier series, Laplace time transforms, and Fourier transforms for the spatial coordinate. The Laplace transform originals are found analytically, and the Fourier transforms are inverted by quadrature formulas.

Numerical Investigations of Mixed Convection of Incompressible Viscous Fluid in LNG Storage with a Various Locations of Input and Output Mass

The article shows the results of mathematical simulation of mixed convection in the low-temperature storage of liquefied natural gas with a regenerative cooling. The regimes of mixed convection in a closed area with the different arrangement of the input and output sections of the masses are investigated. Two-dimensional nonstationary problem in the model of the Navier-Stokes in dimensionless variables “vorticity - stream function - temperature” was examined. Are obtained distributions of the hydrodynamic parameters and temperatures, characteristic basic laws governing the processes being investigated. Detailed circulating currents and carried out analysis of the mechanism of vortices formation and the temperature distribution in the solution for mixed convection mode with low Reynolds and Grashof numbers (Gr = 10 6 , 100 <Re <1000). Is established the significant influence of the geometrical arrangement of the input and output mass sections and input stream velocity on the structure of liquid flow and temperature in the low temperature LNG storage tanks.

Mathematical Simulation of Convective Heat Transfer in the Low-Temperature Storage of Liquefied Natural Gas

The article shows the results of mathematical modeling of convective heat transfer in the low-temperature storage of liquefied natural gas. Regime of natural convection in an enclosure with different intensity of the heat flux at the external borders are investigated. Was examined two-dimensional nonstationary problem within the model of Navier-Stokes in dimensionless variables vorticity - stream function - temperature. Distributions of hydrodynamic parameters and temperatures that characterize the basic regularities of the processes are obtained. Circulating flows are determined and carried out the analysis of vortices formation mechanism and the temperature distribution in solution at conditions of natural convection when the Grashof number (Gr = 10 6 ). A significant influence of heat transfer rate on solutions boundary on flow structure and temperature field in LNG storage tanks.

Space–time adaptive solution of inverse problems with the discrete adjoint method

This paper develops a framework for the construction and analysis of discrete adjoint sensitivities in the context of time dependent, adaptive grid, adaptive step models. Discrete adjoints are attractive in practice since they can be generated with low effort using automatic differentiation. However, this approach brings several important challenges. The space–time adjoint of the forward numerical scheme may be inconsistent with the continuous adjoint equations. A reduction in accuracy of the discrete adjoint sensitivities may appear due to the inter-grid transfer operators. Moreover, the optimization algorithm may need to accommodate state and gradient vectors whose dimensions change between iterations. This work shows that several of these potential issues can be avoided through a multi-level optimization strategy using discontinuous Galerkin (DG) hp-adaptive discretizations paired with Runge–Kutta (RK) time integration. We extend the concept of dual (adjoint) consistency to space–time RK-DG discretizations, which are then shown to be well suited for the adaptive solution of time-dependent inverse problems. Furthermore, we prove that DG mesh transfer operators on general meshes are also dual consistent. This allows the simultaneous derivation of the discrete adjoint for both the numerical solver and the mesh transfer logic with an automatic code generation mechanism such as algorithmic differentiation (AD), potentially speeding up development of large-scale simulation codes. The theoretical analysis is supported by numerical results reported for a two-dimensional non-stationary inverse problem.

Influence of a periodic pulsed energy source on the regime of supersonic flow past a body

A two-dimensional nonstationary problem on the interaction of a trace developed behind a periodic pulsed energy source and a body occurring in a supersonic flow is considered. The results of numerical calculations are compared to the available experimental data.

The reliability of the measurement of the melting point by the thin-plate method

The measurement of the melting point of titanium by the thin-plate method based on a numerical solution of the two-dimensional nonstationary heat-conduction problem with complex boundary conditions, corresponding to the experimental conditions, and taking into account the temperature dependence of the thermal properties of the material of the sample, is validated.

Asymptotic behavior and time discretization analysis for the non-stationary Navier-Stokes problem

The asymptotic behavior and the Euler time discretization analysis are presented for the two-dimensional non-stationary Navier-Stokes problem. If the data ν and f(t) satisfy a uniqueness condition corresponding to the stationary Navier-Stokes problem, we then obtain the convergence of the non-stationary Navier-Stokes problem to the stationary Navier-Stokes problem and the uniform boundedness, stability and error estimates of the Euler time discretization for the non-stationary Navier-Stokes problem.

Algorithm for simulation of electromagnetic processes using combined finite and boundary elements method

This paper deals with the numerical simulation of two-dimensional non-stationary electromagnetic problems. The transient behaviour is described by: the Maxwell equations, the possibly non-linear constitutive equations and by the motion equations. For the solution of the electromagnetic problem the finite and boundary elements method (CFBE-method) is used. A new algorithm for the realization of the CFBE-method is proposed.

One approach to the analytical solution of a two-dimensional nonstationary problem of heat conduction in regions with moving boundaries on the model of a half-space

With the use of the solution of the Dirichlet nonstationary problem with discontinuous unmixed boundary conditions on the surface of an isotropic half‐space a two‐dimensional model of the problem with a moving phase boundary is considered. The problem models, for example, the processes of freezing of moist ground or the processes of formation of ice in stagnant water if a temperature lower than the freezing temperature is prescribed on the boundary surface in a circular region of finite radius. The classical one‐dimensional result follows as a particular case from solution of this problem for an infinite radius of the circle.

Unique Solvability of a Two-Dimensional Nonstationary Problem for the Convection Equations with Temperature-Dependent Viscosity

Numerous papers deal with the mathematical justication of models of viscous uid dynamics. Various aspects of the theory of the viscous incompressible uid equations, including the construction of generalized solutions, are presented in the monographs [1, 2] and also in the monograph [3], devoted to inhomogeneous uids (see also the bibliography therein). Important results in the mathematical theory of viscous incompressible ow with constant viscosity coecient were obtained in most of these papers. In problems of hydrodynamics and heat and mass transfer arising in models of many technological processes, it is important to take account of the dependence of the transport coecients on temperature. The importance of the analysis of mathematical models of these phenomena is justied by numerous applications, experimental data, and numerical investigations [4{6]. Problems in which the transport coecients depend on temperature were discussed in [7], where a result on the existence of at least one generalized solution in a model of an inhomogeneous uid was apparently announced for the rst time. It is very dicult to take account of the dependence of the transport coecients on temperature in theoretical analysis of mathematical models, since this results in an additional nonlinearity in the equations. The papers [8, 9] deal with the existence and uniqueness issues for stationary boundary value problems for the free convection equations with temperature-dependent viscosity coecients. It was proved in [10] that the three-dimensional nonstationary problem for these equations is solvable. The solution obtained there is a Hopf type solution. In the present paper, we continue the analysis of well-posedness of the nonstationary boundary value problem for equations generalizing the Oberbeck{Boussinesq equations to the case in which the viscosity coecient depends on temperature: ~t + ~