Nonstationary Boundary Conditions Research Articles

Asymptotic boundary condition for numerical modeling of wave packets in a supersonic boundary layer

Within the framework of linear stability theory, a simple non-stationary boundary condition is developed to simulate the far-field asymptotic behavior of linear wave packets propagating in a compressible boundary layer. This condition allows us to skip the linear stage of instability evolution in direct numerical simulations (DNS) of laminar-turbulent transition. It is also suggested to perform numerical simulations of the linear stage without recalculating the Jacobi matrix in the Newton iteration procedure. Robustness and efficiency of the both approaches are evaluated by computations of the first-mode wave packet propagating in the boundary layer on a flat plate at freestream Mach number 2. DNS of the nonlinear stage show that the wave packet quickly breaks down into a turbulent spot, if its hump amplitude is umax′≥0.1U∞. This confirms an amplitude nature of the transition onset criterion. The spectral analysis indicates that the breakdown mechanism differs from the oblique, fundamental or subharmonic resonance and requires additional studies.

Dynamic characteristics and energy efficiency evaluation of a novel solar seasonal thermal storage - heating system

This paper proposes solar seasonal thermal energy storage system compounded with long-term and short-term energy storage tanks for a single-family dwelling, which using assisted water source heat pump to further improve the systems stability. With the objective of improving the solar energy utilization efficiency and reducing the primary energy consumption, the dynamic simulation model of the “heat collection-heat storage-heat supply” system under the non-stationary boundary conditions throughout the year is established. The energy transfer and conversion law of each subsystem and the dynamic operation characteristics are analyzed through simulation method. The system cooperative control strategy is proposed and parameter optimization is carried out. Results indicate that the annual heat collection efficiency of the water source heat pump composite system has been improved by 2.2 % compared with the electric auxiliary heating system. Meanwhile, the heat storage efficiency and the solar energy guarantee rate increased by 5.8 % and 8.3 %, respectively. Moreover, the optimal water source heat pump composite system not only improves energy efficiency and achieves the cost savings compared with the electric auxiliary heating system, but also reduces carbon dioxide emission by 10.3 %, which have great advantages for energy saving and emission reduction.

THERMODIFFUSION SATURATION OF THE SURFACE LAYERS OF α-TITANIUM BY OXYGEN, NITROGEN, CARBON

The purpose of the study is to analytically assess the depth of the gas-saturated zone in the case of a single-component diffusion saturation of alpha-titanium with nitrogen, oxygen and carbon from a rarefied controlled gas environment. Results. Based on the analysis of the literature data, the work schematically shows the interaction of alpha titanium with the elements of implementation and presents the processes with the corresponding parameters that characterize. It is shown that the surface impurity concentration is equal to the equilibrium concentration and is established instantly and does not depend on time. Consequently, with the proposed generalized nonstationary boundary condition in the absence of diffusion of impurities into the volume of the metal, the time dependence of its surface concentration is given, determined by the intensity of surface processes. The dependence of the relative change in the microhardness in the diffusion zone of titanium due to dissolved nitrogen (without taking into account the contribution of nitride inclusions) is presented. Analytically calculated concentration profiles of nitrogen generally correlate well with the distribution of the corresponding relative changes in microhardness in the surface layer. Analytical calculations of the concentration profiles of oxygen, nitrogen and carbon in titanium at a saturation temperature of 700 °C are presented. Practical value. The results obtained will make it possible to preliminarily estimate the size of the fortified near-surface layer depending on the parameters of chemical-thermal treatment and select the optimal parameters of thermal diffusion treatment to ensure the formation of reinforced layers on products made of alpha-titanium based on elements of interstitial in order to increase the functional properties.

A New Formulation of Maxwell’s Equations

In this paper, new forms of Maxwell’s equations in vector and scalar variants are presented. The new forms are based on the use of Gauss’s theorem for magnetic induction and electrical induction. The equations are formulated in both differential and integral forms. In particular, the new forms of the equations relate to the non-stationary expressions and their integral identities. The indicated methodology enables a thorough analysis of non-stationary boundary conditions on the behavior of electromagnetic fields in multiple continuous regions. It can be used both for qualitative analysis and in numerical methods (control volume method) and optimization. The last Section introduces an application to equations of magnetic fluid in both differential and integral forms.

THE USE OF THE METHOD OF SEPARATION OF VARIABLES (FOURIER) FOR SOLVING BOUNDARY PROBLEMS OF UNSTATIONARY HEAT CONDUCTIVITY IN A MULTILAYER MEDIUM

Anticorrosive coating of pipes, precipitation of hardness salts on the walls of pipes in hot water supply systems, heating, refrigeration systems, in coils for charging and discharging heat accumulators, coils of heat exchangers and a multilayer structure of metal-plastic pipes for the heating system and for the floor heating systems is an additional thermal resistance for the transfer of heat from the coolant to the environment. This resistance must be taken into account when calculating heat transfer from the pipe surface, both to increase the efficiency of heat exchangers and to calculate unproductive heat losses to the environment from main pipelines and risers of engineering systems. During the design process, the specialists are faced with the question of how to reduce or increase the heat transfer of 1 m of a steel pipe. To increase, you need to change the infrared radiation up. This is done with paint. Red color enhances heat dissipation. It is better if the paint is matte. Another approach is to install ribs. They are mounted externally. This will increase the heat transfer area. And for a "warm floor" system or a coil immersed in a solid medium (for example, a ground collector of a heat pump or a coil for charging / discharging a heat accumulator with a solid heat storage material), it is necessary to correctly calculate the heat transfer of 1 m of the pipe in order to ensure heating or cooling of this pipe to the required parameters. When do you need to decrease the parameter? The need arises when optimizing a section of the pipeline located outside the residential area, in an unheated volume. Then experts recommend insulating the site - isolating it from the external environment. According to M. Jacob's data, the formulas for heat transfer by thermal conductivity through flat walls can often be applied with sufficient accuracy for cylindrical walls, if the heat transfer surface is taken along the average thickness. Moreover, a large number of practical problems of calculating temperature fields in multilayer objects can be calculated as one-dimensional. Previously, an analytical solution to the homogeneous problem of non-stationary heat conduction in multilayer objects with stationary boundary conditions of the third kind was proposed. This article discusses the issues of unsteady heat conduction in multilayer objects. A solution to the homogeneous boundary problem with nonstationary boundary conditions of the third kind is proposed. The solution is based on the Fourier variable separation method by the eigenfunctions of the problem and the Duhamel integral. The proposed solution formula has an explicit form and, due to the recurrent form of writing the basic relations, can be useful in numerous calculations.

Nonstationary 1D Dynamics Problems for Heteromodular Elasticity with Piecewise-Linear Approximation of Boundary Conditions

The paper provides the investigation of a heteromodular elastic medium under dynamic loading. The heteromodularity (when the stress - strain relation depends on the deformation direction) is a distinctive feature of many natural and structural materials: rocks, porous and cohesive bulk media, fibrous and granular composites, some metal alloys, etc. The fact that the listed materials show the heteromodular property at the stage of elastic deformation should be especially taken into account when solving problems of their shock dynamics. To describe the heteromodular behavior of an elastic medium in terms of small strains we use the physically nonlinear model of V.P. Myasnikov. The accepted assumption about the one-dimensional straining reduces the nonlinear relationship of stresses and small strains to piecewise linear equations. In the case of dynamic shock deformation, the initial nonlinearity of the model is concentrated in the equations which define the velocity of the shock wave abruptly transforming the heteromodular medium from a stretched to a compressed state. In this paper we investigate the processes of generation, motion, and possible interactions of plane one-dimensional deformation waves (including shock ones) in a heteromodular elastic half-space. The points of the half-space boundary undergo one-dimensional motions according to a given non-linear law corresponding to the “stretching-compression” mode. We suggest replacing the nonstationary boundary condition of the problem by its piecewise linear approximation and constructing a connected sequence of analytical solutions with a linear boundary condition at each local time interval. The proposed approach is the basis of the numerical solving algorithm for a boundary value problem with a given nonlinear condition. It is shown that the general solution behind the shock wave consists of several local layers, which number is related to the quantity of nodes in the piecewise linear decomposition of the boundary condition. In these layers, the compression deformation is defined by the relevant part of the boundary condition and simultaneously “stores” information on the preliminary tension, which should be considered an important feature of the heteromodular medium dynamics.

Study of convective structures and phase transition induced in a water layer by non-stationary boundary conditions

The problem of reconstructing interference and Hilbert structures from a numerical model of the evolution of the thermal field of convective flows in a water layer bounded by flat heat-exchanging surfaces under unsteady boundary conditions in the monotonic cooling mode and taking into account density inversion at a temperature of +4 °C was solved. The simulation of the thermal fields of convective flows in the form of the dynamic structure of isotherms was carried out taking into account the nonlinear dependence of thermal conductivity and water density on temperature. The field of the phase function, its Hilbert image and the interference field, which are compared with the results of the interference and Hilbert visualization of the fields of phase optical density obtained in the experiment, were reconstructed from the isotherms field supplemented by calculating of the velocity field and of the temperature gradients field. The presented films illustrate the qualitative adequacy of the coevolution of numerical models and real processes.

A computational algorithm for constructing a two-dimensional heat wave generated by a non-stationary boundary condition

The paper discusses solutions of the nonlinear heat equation, which have the form of a heat wave propagating on a zero background with a finite velocity. Such solutions are not typical for parabolic equations, and their existence is associated with the degeneration of the problem at the wave (zero) front. We propose a numerical algorithm for constructing a two-dimensional heat wave, symmetrical with respect to the origin, with a non-zero boundary condition defined on the moving boundary. The main difficulty of the new task is that at each time point a heat wave front (a domain boundary) is unknown. The solution is carried out in two stages. At first, we change the roles of unknown function and radial polar coordinate. For a new unknown function at each time point, we obtain a boundary value problem for the Poisson equation in a known region. The step-by-step solving of this problem by the method of boundary elements at a given time interval allows us to determine the law of the zero front moving. At second, we approximate the found zero front by an analytical function and construct a generalized self-similar solution. The developed algorithm is implemented and tested on a task set.

К РЕШЕНИЮ ЗАДАЧ НЕСТАЦИОНАРНОЙ ДИФФУЗИИ В СЛОИСТЫХ СРЕДАХ

The issues of nonstationary diffusion in layered structures are considered. When designing the devices for implementing mass transfer processes, it is necessary to take into account the properties of the substance and the nature of the processes. Design time reduces significantly and the efficiency of the devices is higher if a good physical model is built and a mathematical analysis with kinetics of the processes is applied. The difficulties of theoretical analysis and calculation of mass transfer are determined by the complexity of the transfer mechanism to and from the phase boundary. Therefore, simplified models of mass transfer processes are used in which the mass transfer mechanism is characterized by a combination of molecular and convective mass transfer. Many important practical problems involve the calculation of nonstationary diffusion (Fick's second law of diffusion) for a certain volume of substance (substances). For qualitative evaluation of processes, in the case of symmetry, volumetric issues can be considered as one-dimensional tasks, i.e. dependent on one coordinate. The general solution of the non-stationary diffusion equation for layered environments is proposed. The case of non-stationary boundary conditions of the third kind on the external surface and boundary conditions of the fourth kind conjugation for contiguous layers has been considered. The solution is obtained by separating the Fourier variables by the eigenfunctions of the problem using the Duhamel integral. The proposed solution is explicit and due to the recurrent form of the basic relations can be useful in numerical calculations

Application of modified Navier-Stokes equations to determine the unsteady force effects of a heterogeneous liquid

The article is focused on calculating the force effects of a heterogeneous liquid on pipe walls. The solution is based on the concentration of solid particles. The base fluid is assumed to be incompressible. The solution will apply Euler-Lagrange's solution principle. Two tasks will be solved; with a rigid and a flexible tube wall. The solution will be carried out with non-stationary boundary conditions that were determined experimentally. Interaction of a heterogeneous fluid with a flexible wall assumes its deformation. The force effects will be solved by two methods; FSI simulation using ANSYS FEA solvers and CFD solvers ANSYS Fluent and using Navier-Stokes equations by direct integration through liquid volume. In this case, the unsteady term of the Navier-Stokes equations will be modified so that the Gauss Ostrogradsky theorem can be used to calculate the force. At the end, the force effects on the rigid and compliant wall will be compared with the unsteady turbulent flow of the heterogeneous liquid.

Optical diagnostics of convective structures induced by non-stationary boundary conditions in a vertical water layer

Optical diagnostics of convective structures induced by non-stationary boundary conditions in a vertical water layer

Evaluation of the Thermal Resistance of Structure with Reflective Insulation: Measurement under Non-Stationary Boundary Conditions

Some manufacturers of reflective insulation products claim that their relatively thin products have the same thermal insulation properties as common insulation with much times higher thickness. This paper deals with experimental measurements of the thermal insulation properties of real structures with reflective insulation. Several experimental sequences were conducted to determine the thermal resistance of the structure. The evaluation was conducted according to the procedures specified in the international standard (ISO 9869 - 1994). Effect of products (based on the reflection of thermal radiation) on the thermal insulation properties of the structure will be determined by evaluation. This effect will be compared with the effects of common insulation (with the intention of reduce of the heat transfer by the conduction).

Dynamic Heat Transfer through the External Wall of a Timber Structure

The article compares results of temperature and heat flux measurements in the external wall of a real timber structure with results obtained by numerical modeling using the finite element method in the ANSYS software. The measured temperature values are compared with results obtained from numerical simulation of dynamic heat transport using non-stationary boundary conditions. In the article there is evaluated a suitability of theoretical numerical calculations for a thermal field and heat flux prediction in a building structure.

Analysis of Experimental Measurements and Numerical Simulations of a Heat Field in the Light Weight Building Structure

The paper analyses results of the experimental measurements and numerical simulations of the winter and summer temperature response in the light timber structure. In the article there is evaluated the suitability of using of the theoretical numerical methods for a thermal field prediction in a building structure exposed to non-stationary boundary conditions.

Thermostressed state of a cylinder with thin near-surface layer having time-dependent thermophysical properties

We develop a method for the solution of boundary-value problems of the thermostressed state of cylindrical bodies with thin near-surface layers having time-dependent thermophysical properties. The presence of a thin near-surface layer in a long solid cylinder is taken into account in the formulated boundary-value problem by means of a nonclassical nonstationary boundary condition with variable coefficients. The replacement of this condition by the classical first-kind condition enables one to reduce the original problem to finding the solution of an integro-differential equation with variable coefficients. This equation has a Volterra-type integral operator and is solved by using spline approximations. The efficiency of this method has been partially verified on the well-known problem of heating of a thin homogeneous plate with variable Biot number by environment. We investigate the thermostressed state of a cylinder for both linear and exponential normalized surface parameters of heat transfer and heat capacity over time. For different stages of heating, we analyze at what constant values of parameters the thermal regime of the cylinder is closest to the mode with variable parameters. The obtained solutions and data on their properties can be used in solving the problems of boundary parametric identification.

Improved version of the synthetic eddy method for setting nonstationary inflow boundary conditions in calculating turbulent flows

One method for generating synthetic turbulence, i.e., the synthetic eddy method, is tested by means of the example of canonical turbulent shear flows (plane-channel and boundary-layer flows). A modification of the method, differing from the original version by the determination of the linear scale of synthetic eddy structures, is proposed. The synthetic field of turbulent fluctuations evolves more quickly to the physically realistic one when the modified method is used instead of the original one. The friction coefficient and profiles of the average velocity and Reynolds stresses also deviate less and recover faster if the modified method is used.

A Mixed Problem for Quasilinear Impulsive Hyperbolic Equations with Non Stationary Boundary and Transmission Conditions

The initial-boundary value problem for a class of linear and nonlinear equations in Hilbert space is considers. We prove the existence and uniqueness of solution of this problem. The results of this investigation are applied to solvability of initial-boundary value problems for quasilinear impulsive hyperbolic equations with non-stationary transmission and boundary conditions.

Численно-аналитическое моделирование тепловых процессов при нестационарных граничных условиях

On the basis of partnering implicit difference scheme on a temporary variable, regional-structural and variation methods the numerically-analytical modelling methodology of thermal processes in constructive elements is offered under non-stationary boundary conditions.

Modeling of Diffusion Saturation of (α + β) Titanium Alloys by Nitrogen in the Rarefied Medium

The diffusion saturation of (α + β) titanium alloy by nitrogen under rarefied medium taking into account the peculiarities of interaction both at the interface and in the metal core is modeled. The corresponding diffusion problem using the diffusion equations in the medium consisting two families of diffusion paths and non-stationary boundary condition at interface is analytically solved. The effect of the temperature and time parameters, volumetric quantity of β- phase on the kinetics of nitriding is estimated. It is established that with the increase of isothermal exposure temperature and quantity of β-phase the surface saturation by nitrogen decreases. It is caused by more intensive nitrogen penetration into bulk and formation of a deeper diffusion zone.

ИССЛЕДОВАНИЕ ГИДРАВЛИЧЕСКОГО УДАРА В ЖИДКОСТИ ПРИ КОЛЕБАНИЯХ ТРУБОПРОВОДА

A problem of modeling a hydraulic shock in a moving spatial pipeline is studied. The analysis of a non-stationary liquid flow resulting from the vibration of the pipeline is reduced to solving a mixed problem with non-stationary boundary conditions. To analyze the problem, a numerical algorithm is developed, where the method of characteristics and a first-order finite-difference scheme are implemented. The results of comparing the test calculations with an analytical solution and the results of the analyzing hydro-dynamic processes in a spatial pipeline for a preset seismic load corresponding to a typical earthquake are presented. Key words: modeling, hydraulic shock, spatial pipeline, mixed problem, nonstationary boundary conditions, seismic load.