Additional Boundary Conditions Research Articles

A Method for Obtaining Analytical Solutions to Boundary Value Problems by Defining Additional Boundary Conditions and Additional Sought-For Functions

A method of obtaining analytical solutions to thermal conduction problems is considered. It is based on a separation of thermal conduction into two stages of its time evolution using additional boundary conditions and additional sought-for functions in the heat balance integral method. Solving the partial differential equations is thus reduced to the integration of two ordinary differential equations for some additional sought-for functions. The first stage is characterized by fast convergence of an analytical solution to the exact one. For the second stage, an exact analytical solution is obtained. The additional boundary conditions for both stages are such that their fulfillment by the sought-for solution is equivalent to the fulfillment of the original equation at the boundary points and at the temperature perturbation front. It is shown that if the equation is valid at the boundary points, it is also valid inside the domain.

Simulation and design of folded perovskite x-ray detectors

A variety of medical, industrial, and scientific applications requires highly sensitive and cost-effective x-ray detectors for photon energies ranging from keV to MeV. Adapting the thickness of polycrystalline or single crystal conversion layers especially to high-energy applications increases the complexity of fabrication and potentially decreases the performance of conventional direct conversion x-ray detectors. To tackle the challenges with respect to the active layer thickness and to combine the superior performance of single crystal materials with the low-cost nature of polycrystalline conversion layers, we investigate thin film x-ray detector technologies based on a folded device architecture. Analytical models simulating the sensitivity and the detective quantum efficiency (DQE) are used to evaluate the performance of folded detectors based on polycrystalline organic-inorganic perovskite semiconductors in various layout configurations and for different photon energies. Simulations of folded perovskite devices show high sensitivities. The DQE analysis introduces additional noise related boundary conditions for the folding length. A comparison with conventional detectors based on state of the art conversion materials at different photon energies demonstrates the potential of the folded detector layout as simulated sensitivities are comparable to single crystal detectors.

Investigation of Free Surface Turbulence Damping in RANS Simulations for Complex Free Surface Flows

The modelling of complex free surface flows over weirs and in the vicinity of bridge piers is presented in a numerical model emulating open channel flow based on the Reynolds Averaged Navier-Stokes (RANS) equations. The importance of handling the turbulence at the free surface in the case of different flow regimes using an immiscible two-phase RANS Computational Fluid Dynamics (CFD) model is demonstrated. The free surface restricts the length scales of turbulence and this is generally not accounted for in standard two-equation turbulence modelling approaches. With the two-phase flow approach, large-velocity gradients across the free surface due to the large difference in the density of the fluids can lead to over-production of turbulence. In this paper, turbulence at the free surface is restricted with an additional boundary condition for the turbulent dissipation. The resulting difference in the free surface features and the consequences for the solution of the flow problem is discussed for different flow conditions. The numerical results for the free surface and stream-wise velocity gradients are compared to experimental data to show that turbulence damping at the free surface provides a better representation of the flow features in all the flow regimes and especially in cases with rapidly varying flow conditions.

The Global-in-Time Existence of a Classical Solution for Some Free Boundary Problem

We consider the problem with free (unknown) boundary for the one-dimensional diffusion–convection equation. The unknown boundary is found from the additional condition on the free boundary. A dilation of the variables reduces the problem to an initial-boundary value problem for a strictly parabolic equation with unknown coefficients in the known domain. These coefficients are found from an additional boundary condition, which makes it possible to construct a nonlinear operator whose fixed points define the solution to the initial problem.

Nozzle external geometry as a boundary condition for the azimuthal mode selection in an impinging underexpanded jet

The role of the external boundary conditions of the nozzle surface on the azimuthal mode selection of impinging supersonic jets is demonstrated for the first time. Jets emanating from thin- and infinite-lipped nozzles at a nozzle pressure ratio of $3.4$ and plate spacing of $5.0D$, where $D$ is the nozzle exit diameter, are investigated using high resolution particle image velocimetry (PIV) and acoustic measurements. Proper orthogonal decomposition is applied to the PIV fields and a difference in dominant instability mode is found. To investigate possible explanations for the change in instability mode, additional nozzle external boundary conditions are investigated, including the addition of acoustic dampening foam. A difference in acoustic feedback path is suggested to be the cause for the change in dominant azimuthal modes between the flows. This is due to the thin-lip case containing a feedback path that is concluded to be closed exclusively by a reflection from the nozzle base surface, rather than directly to the nozzle lip. The ability of the flow to form a feedback path that maximises the impingement tone gain is discussed with consideration of the numerous acoustic feedback paths possible for the given nozzle external boundary conditions.

Method of additional unknown functions in heat conductivity problems with variable physical properties of the medium

Finding analytical solutions to the problems of thermal conductivity with variable physical properties of the medium by classical analytical methods is very complicated mathematically. The known expressions repre-senting complex infinite series including two types of Bessel functions and gamma-functions are, in fact, numerical as they require a numerical solution to complex transcendental equations with eigenvalues of the boundary problem. Such solutions can hardly be used in engineering applications, especially in cases when a solution to a certain problem is only an intermediate stage in other problems (such as thermoelasticity and control problems, inverse problems, etc.) which can be solved effectively only by finding analytical solutions to the initial problems. Therefore, an urgent problem now is to develop new methods of obtaining analytical solutions to the abovementioned problems, at least approximate ones. The study employed methods of additional boundary conditions and additional unknown functions in the integral method of heat balance. High-precision approximate analytical solutions to the transient heat conduction problem with nonhomogeneous physical properties of the medium for an infinite plate under symmetric boundary conditions of the first type have been obtained. The initial problem for partial differential equations is reduced to two problems in which ordinary differential equations are integrated. Additional boundary conditions are defined in such a way that their fulfillment in accordance with the new method is equivalent to the result of solving the initial partial differential equation at the boundary points and at the temperature perturbation front (for the first stage of the process). By combining methods with finite and infinite heat propagation rate we have been able to obtain high-precision analytical solutions for the whole time range of the unsteady process including its small and ultra small values. The solutions look like simple algebraical polynomials not including special functions (Bessel, Legendre, gamma-functions and others). Since it is not necessary to directly integrate the initial equations by the space variable and to reduce them to ordinary differential equations with additional unknown functions, the considered method can be used for solving complex boundary problems in which differential equations do not allow distinguishing between the variables (into nonlinear, with linear boundary conditions and heat sources, etc.).

Analytical solution to the problem of heat transfer using boundary conditions of the third kind

Non-stationary heat transmission within solid bodies is described using parabolic and hyperbolic equations. Currently, numerical methods for studying the processes of heat and mass transfer in the flows of liquids and gases have disseminated. Modern programs allow the automatic construction of computational grids, solutions to the systems of equations and offer a wide range of tools for analysis. Approximate analytical solutions have significant advantages compared to numerical ones. In particular, the solutions obtained in an analytical form allow performing parametric analysis of the system under study, configuration and programming of measurement devices, etc. Based on the joint use of additional desired function and additional boundary conditions in the integral method of heat balance, a method of mathematical modeling for the heat transfer process in a plate under symmetric boundary conditions of the third kind has been developed. Using the heat flux density as a new desired function, the method for solving heat conduction problems with boundary conditions of the third kind has been proposed. Finding a solution to the partial differential equation with respect to the temperature function presents integrating an ordinary differential equation with respect to the heat flux density on the surface of the studied zone. It has been shown that isotherms appear on the surface of the plate with a certain initial velocity which depends on the heat transfer intensity. The calculation results have been compared to the exact solution. The presented method can be used in determining the heat flux density of buildings and heating devices, finding heat losses during convective heat transfer and designing heat transfer equipment. The results can be applied to increase the validity and reliability of the calculation of actual losses and balance of thermal energy. The method reliability, validity and a high degree of approximation with about 3% inaccuracy have been demonstrated. The accuracy of the solution depends on the number of approximations performed and is determined by the degree of the approximating polynomial.

A method of obtaining analytical solutions to boundary value problems based on defining additional boundary conditions and additional desired functions

A method of obtaining analytical solutions to boundary value problems based on defining additional boundary conditions and additional desired functions

Evaluation of OpenFOAM against CODASC wind tunnel database and impact of heating on the flow in an idealised street canyon

This paper focuses on examination of the CFD simulated impact of irregularly heated surface on the flow recirculation and pollutant distributions in a tree-free idealised street canyon. For that purpose, the buoyantBoussinesqPimpleFoam solver of the CFD model OpenFOAM was tested against the CODASC wind tunnel database (http://www.windforschung.de/CODASC.htm). The wind tunnel experiments were run only under neutral atmospheric stratification conditions, so in order to be examined the impact of the heating on the flow, OpenFOAM was set for two general cases. In the first one, the model was run with initial conditions which correspond to the physical conditions in the wind tunnel. The simulated data is then compared to that measured in the tunnel. For the second case, an additional boundary condition for the air temperature near the surface of the street canyon was set. An intercomparison of the modelled data from both cases is then made.

Additional waves and additional boundary conditions in local quartic metamaterials

Additional electromagnetic waves and additional boundary conditions (ABCs) in non-local materials attract a lot of attention. Here we report the possibility of additional propagating and evanescent waves in local anisotropic and bi-anisotropic linear materials if non-reciprocity is present. We also show that additional waves cannot propagate in local reciprocal bi-anisotropic materials. We investigate the possible options for ABCs and describe how to complement the conventional 4 Maxwell’s boundary conditions in the situations when there are more than 4 waves that need to be matched at the boundary of local and linear quartic metamaterials. We show that these ABCs must depend on the properties of the interface and require the introduction of the additional effective material parameters describing this interface, such as surface conductivities.

Heat Exchange in a Cylindrical Channel with Stabilized Laminar Fluid Flow

Based on the determination of the temperature perturbation front and additional boundary conditions, an approximate analytical solution is obtained for the stationary heat exchange problem when fluid flows in a cylindrical channel with a constant parabolic velocity profile (the Gretz–Nusselt problem), which allows us to investigate the temperature distribution in the fluid in a wide range of distances from the pipe inlet, including small and very small distances. Based on the data of numerical calculations of temperature change at a certain value of the spatial variable using the solution obtained by solving the inverse heat conduction problem, the Peclet number was found (in the case where it is unknown in the solution obtained), from which we can determine the velocity profile and the flow rate of the liquid. Graphs of the distribution of isotherms and the their velocities in space over time are plotted.

Semi-analytic solution of Eringen’s two-phase local/nonlocal model for Euler-Bernoulli beam with axial force

Eringen’s two-phase local/nonlocal model is applied to an Euler-Bernoulli nanobeam considering the bending-induced axial force, where the contribution of the axial force to bending moment is calculated on the deformed state. Basic equations for the corresponding one-dimensional beam problem are obtained by degenerating from the three-dimensional nonlocal elastic equations. Semi-analytic solutions are then presented for a clamped-clamped beam subject to a concentrated force and a uniformly distributed load, respectively. Except for the traditional essential boundary conditions and those required to be satisfied by transferring an integral equation to its equivalent differential form, additional boundary conditions are needed and should be chosen with great caution, since numerical results reveal that non-unique solutions might exist for a nonlinear problem if inappropriate boundary conditions are used. The validity of the solutions is examined by plotting both sides of the original integro-differential governing equation of deflection and studying the error between both sides. Besides, an increase in the internal characteristic length would cause an increase in the deflection and axial force of the beam.

On vibrations of FG nanobeams

In the current study, the vibration behavior of axially functionally graded nanobeams is investigated in the framework of Eringen's two-phase local-nonlocal model which considers both local and nonlocal phases in modelling nano-scale structures. Material properties are varied through the length using power-law functions. By employing the Hamilton's principle and a conversion between integral-differential, higher order equation of motion with two additional higher order non-classical boundary conditions are achieved. A modified differential quadrature method is used to solve the problem and the accuracy of the current methodology is approved by comparing the results with previous studies for simplified models. In order to indicate the influence of different parameters such as nonlocal term, phase coefficients, material varying function, etc., a comprehensive parametric study is presented and nanobeam with different boundary conditions are discussed. For different boundary conditions, variation of frequency terms is presented especially for cantilever axially functionally graded nanobeams in which differential form of Eringen's nonlocal theory is unable to model accurately. It is shown that the two-phase nonlocal model has the capability to model the axially functionally graded nanobeams with different boundary conditions and material varying terms.

The offshore boundary condition in surf zone modeling

Numerical models predicting surfzone waves and shoreline runup in field situations are often initialized with shoreward propagating (sea-swell, and infragravity) waves at an offshore boundary in 10–30 m water depth. We develop an offshore boundary condition, based on Fourier analysis of observations with co-located current and pressure sensors, that accounts for reflection and includes nonlinear phase-coupling. The performance of additional boundary conditions derived with limited or no infragravity observations are explored with the wave resolving, nonlinear model SWASH 1D. In some cases errors in the reduced boundary conditions (applied in 11 m depth) propagate shoreward, whereas in other cases errors are localized near the offshore boundary. Boundary conditions that can be implemented without infragravity observations (e.g. bound waves) do not accurately simulate infragravity waves across the surfzone, and could corrupt predictions of morphologic change. However, the bulk properties of infragravity waves in the inner surfzone and runup are predicted to be largely independent of ig offshore boundary conditions, and dominated by ig generation and dissipation.

Ray-tracing semiclassical (RTS) low frequency acoustic modeling validated for local and extended reaction boundaries

The recently introduced acoustic ray-tracing semiclassical (RTS) method is validated for a set of practically relevant boundary conditions. RTS is a frequency domain phased geometrical acoustics method which directly reproduces the acoustic Green's function. As previously demonstrated for a rectangular room and weakly absorbing boundaries with a real and frequency-independent impedance, RTS is capable of modeling also the lowest modes of such a room, which makes it a useful method for low-frequency sound field modeling in enclosures. In practice, rooms are bounded with diverse types of materials, resulting in a frequency-dependent, phase-modifying absorption/reflection. In a realistic setting, we test the RTS method with two additional boundary conditions: a local-reaction boundary simulating a resonating absorber and an extended-reaction boundary representing a porous layer backed by a rigid boundary described within the Delany-Bazley-Miki model, as well as a combination thereof. The RTS-modeled spatially dependent pressure response, frequency-band decay curves and reverberation times are compared to those obtained by the finite element method. The RTS results show systematic agreement for all boundary conditions and disclose some interesting particularities of the method.

Comparison of Hydrodynamic Models for the Electromagnetic Nonlocal Response of Nanoparticles

AbstractSince deep nanoscale systems are increasingly studied, accurate macroscopic theories dealing with quantum mechanical effects are in high demand. Concerning the electromagnetic response of nanoplasmonic systems, several hydrodynamic models have been proposed, each introducing an additional boundary condition (ABC) to describe the behavior of the plasmonic electron cloud. Four hydrodynamic approaches with four different boundary conditions are compared: the hard wall hydrodynamic model with the Sauter ABC, the curl‐free hydrodynamic model with the Pekar ABC, the shear forces hydrodynamic model with the specular reflection ABC, and the quantum hydrodynamic model with the corresponding ABC. This is done by investigating near‐ and far‐field features of a metallic nanosphere. It is shown that different hydrodynamic models may result in an entirely different prediction of the nanoparticle's response. These models are validated by using other local and nonlocal response models and experimental results.

ADDITIONAL SOUGHT-FOR FUNCTIONS AND LOCAL COORDINATE SYSTEMS APPLIED IN THE HEAT CONDUCTIVITY PROBLEMS FOR MULTILAYERED BUILDING STRUCTURES

The solution problems of the additional the sought-for function and additional boundary conditions based when using local coordinate systems, an approximate analytical solution of the heat conduction problem for a double-layer plate is obtained for symmetric boundary conditions of the fi rst kind. The use of the additional sought-for function in the integral method of heat balance makes it possible to reduce the solution of the partial diff erential equation to the integration of an ordinary diff erential equation.

Modelling of Existing Heating Plant Replacement with a Waste to Energy Plant and a Peak-Load Natural Gas Boiler

The paper assesses the possibility of replacement of a heating plant with a waste-to-energy (WtE) plant as base heat source, together with a natural gas boiler for covering power peaks and as thermal backup. The decision of the Commission of the EU Member States of July 2017 shows a tightening of the emission limits for NOx, SO2 and other substances for thermal sources with a power over 50 MWt to enter into force in 2021. In the Czech Republic, this restriction may affect a number of heating plants, which are currently in operation and they can become economically unsustainable. In addition to the requirements for greening of obsolete coal-fired heat sources, decreasing cost of electricity or the inappropriate dimensioning of due to current or future heat demand in the district heating system (DHS) can be other factors that affect the economy of these sources negatively. This fact, together with the planned ban on landfilling of untreated municipal waste from 2024 in the Czech Republic, represents the potential for the implementation of WtE projects in localities with DHS. Using the techno and economic mathematical models, the methodology of designing the proper processing capacity of a WtE plant and the gas boiler heat output is discussed. Achieving the best economy is prerequisite of the calculations. Additional boundary conditions are considered, such as the coverage of heat demand or the operating parameters of both heat sources. Furthermore, the economically optimal mode of operation of the two sources is presented on the basis of the operational data from real heating plants. The typical profile of demand for heat during the year and fluctuations at different time levels are taken into account. The paper also summarizes the relationship between key parameters such as the heat price and the available waste treatment cost.

Solution of an inverse transient heat conduction problem in a part of a complex domain

The purpose of this work is to formulate two simple methods which can be used to solve nonlinear inverse heat conduction problems in a part of a complex-shaped component in the on-line mode. The proposed methods can be useful if temperature measurements are carried out only in a selected part of an outsize element or if the whole element cannot be analysed due to the inverse heat conduction problem being ill-conditioned. It is assumed that the conductive heat transfer occurs through the surfaces separating the domain from the rest of the component. It is shown that the simplifying assumption of thermal insulation on these surfaces, which is often presented in literature, can cause significant errors. Compared to works published previously, if two additional unknown boundary conditions are introduced on the separating surfaces, the inverse problem conditioning deteriorates substantially. Despite this, stable solutions are achieved for “noisy measured data”.The presented methods can be used to optimize the power unit start-up and shutdown operation. They may also enable a reduction in the heat loss arising during the process and extend the power unit life. The methods presented herein can be applied in monitoring systems working both in conventional and nuclear power plants.

Method of Decreasing the Order of a Partial Differential Equation by Reducing to Two Ordinary Differential Equations

Using additional unknown functions and additional boundary conditions in the integral method of heat balance, we obtain approximate analytic solutions to the non-stationary thermal conductivity problem for an infinite solid cylinder that allow to estimate the temperature state practically in the whole time range of the non-stationary process. The thermal conducting process is divided into two stages with respect to time. The initial problem for the partial differential equation is represented in the form of two problems, in which the integration is performed over ordinary differential equations with respect to corresponding additional unknown functions. This method allows to simplify substantially the solving process of the initial problem by reducing it to the sequential solution of two problems, in each of them additional boundary conditions are used.