Time-dependent Boundary Research Articles

A new analytical method for transient heat conduction in composite disks applied to thermal plug and abandonment of oil wells

A new hybrid method for transient heat conduction problems is developed and applied to simulate a Plug and Abandonment (P&A) operation of oil wells. An oil well is approximated by concentric disks in a one-dimensional configuration, which allows for use of polar coordinates. The application of the Separation of Variables Method (SVM) is used as the analytical framework for the solution of the conductive heat transfer arising from a volumetric heat source located in the center of the disks. The SVM is able to solve only time-independent boundary conditions. However, using the Duhamel's theorem, the solution determined with the SVM can be used to achieve the solution when both time-dependent boundary conditions and internal heat generation are prescribed. Lastly, for verification purposes, a commercial software that solves the transient temperature field by means of numerical procedures, providing reliability to the analytical method proposed in this work.

Convective thermal losses of long-term underground hot water storage

A case of underground long-term hot water storage is investigated numerically. The study is based on the unsteady two-dimensional Navier-Stokes equations in Boussinesq approximation applied to a closed cavern with time-dependent temperature boundary conditions on the walls. The problem formulated in a vorticity-stream function statement is solved by finite difference method (FDM) for high values of the Rayleigh number and for the Prandtl number of water. Streamlines, velocity and temperature fields are presented graphically for given moments of time. The evolution of the thermocline thickness in the mid-section of the cavern is slow and illustrates that the hot water zone occupies more than the half of the cavern even after 6 months period. The Nusselt number on the walls shows that the convective thermal losses are small and after certain period of time tend to decrease due to the diminished temperature difference at the walls. The influence of the fluid convection on the thermal losses is evaluated quantitatively, showing high seasonal thermal efficiency of the insulated hot water storage.

Topography optimisation using a reduced-dimensional model for transient conjugate heat transfer between fluid channels and solid plates with volumetric heat source

Consideration of transient effects is important for industrial applications of heat transfer structure optimisation studies; however, the huge computational cost associated with transient problems is a pressing concern. This paper proposes an extension of a previous reduced-dimensional model to transient conjugate heat transfer between a fluid flow and solid-heated plates in a plate heat exchanger. The extended reduced-dimensional model introduces the temperature field of the plate governed by the heat conduction equation, which is coupled to the temperature field of the fluid, governed by the convection-diffusion equation, through the heat flux balance equation at the contact surface. The model is based on assumptions of fully developed flow and constant temperature profile, reducing the three-dimensional problem to a planar problem and significantly reducing computational costs. The accuracy of the model for the simulation of transient heat transfer is verified by comparison with a three-dimensional model. In this paper, the topography of the heat exchanger plate is optimised for both steady-state and transient conditions by applying the reduced-dimensional model. The effectiveness of the optimised design was demonstrated by the cross-check of both the reduced-dimensional and full three-dimensional models. Furthermore, this work considers the effect of time-independent boundary conditions and time-dependent boundary conditions on transient optimisation. The transient and steady-state optimised designs are analysed and compared for both conditions, and the necessity of transient optimisation is discussed.

Analytical Solution of the Linearized Boussinesq Equation Considering Time-Dependent Downslope Boundary, Variable Recharge and Bedrock Seepage

Analytical Solution of the Linearized Boussinesq Equation Considering Time-Dependent Downslope Boundary, Variable Recharge and Bedrock Seepage

Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s Equation

In this work, we consider Cauchy-type problems for Laplace’s equation with a dynamical boundary condition on a part of the domain boundary. We construct a discrete-in-time, meshless method for solving two inverse problems for recovering the space–time-dependent source and boundary functions in dynamical and Dirichlet boundary conditions. The approach is based on Green’s second identity and the forward-in-time discretization of the non-stationary problem. We derive a global connection that relates the source of the dynamical boundary condition and Dirichlet and Neumann boundary conditions in an integral equation. First, we perform time semi-discretization for the dynamical boundary condition into the integral equation. Then, on each time layer, we use Trefftz-type test functions to find the unknown source and Dirichlet boundary functions. The accuracy of the developed method for determining dynamical and Dirichlet boundary conditions for given over-determined data is first-order in time. We illustrate its efficiency for a high level of noise, namely, when the deviation of the input data is above 10% on some part of the over-specified boundary data. The proposed method achieves optimal accuracy for the identified boundary functions for a moderate number of iterations.

CFD Analysis of Transient Conjugated Forced Convection in Minipipes with Time-Dependent Periodical Convection Boundary Conditions

CFD Analysis of Transient Conjugated Forced Convection in Minipipes with Time-Dependent Periodical Convection Boundary Conditions

Perfect plasticity versus damage: an unstable interaction between irreversibility and Γ-convergence through variational evolutions

This paper addresses the question of the interplay between relaxation and irreversibility through quasi-static evolutions in damage mechanics, by inquiring the following question: can the quasistatic evolution of an elastic material undergoing a rate-independent process of plastic deformation be derived as the limit model of a sequence of quasi-static brittle damage evolutions? This question is motivated by the static analysis performed in [J.-F. Babadjian et al. Commun. Pure Appl. Math. 74 (2021) 1803–1854], where the authors have shown how the brittle damage model introduced by Francfort and Marigo (see [G.A. Francfort and J.-J. Marigo, Eur. J. Mech. A Solids 12 (1993) 149–189, G.A. Francfort and J.-J. Marigo, J. Mech. Phys. Solids 46 (1998) 1319–1342]) can lead to a model of Hencky perfect plasticity. Problems of damage mechanics being rather described through evolution processes, it is natural to extend this analysis to quasi-static evolutions, where the inertia is neglected.We consider the case where the medium is subjected to time-dependent boundary conditions, in the one-dimensional setting. The idea is to combine the scaling law considered in [J.-F. Babadjian, et al. Commun. Pure Appl. Math. 74 (2021) 1803–1854] with the quasi-static brittle damage evolution introduced in [G.A. Francfort and A. Garroni, Arch. Rational Mech. Anal. 182 (2006) 125–152] by Francfort and Garroni, and try to understand how the irreversibility of the damage process will be expressed in the limit evolution. Surprisingly, the interplay between relaxation and irreversibility is not stable through time evolutions. Indeed, depending on the choice of the prescribed Dirichlet boundary condition, the effective quasi-static damage evolution obtained may not be of perfect plasticity type.

Semi-analytical models for two-dimensional multispecies transport of sequentially degradation products influenced by rate-limited sorption subject to arbitrary time-dependent inlet boundary conditions

Most of the semi-analytical and analytical models employed to depict multidimensional, multispecies transport of sequentially degrading reaction products are built upon solving a set of coupled advection-dispersion equations (ADEs). Within these equations, sorption is considered as being equilibrium-controlled. However, it has been demonstrated that more realistic predictions of the transport of contaminants in the groundwater could be obtained by the use of a rate-limited sorption process instead of an equilibrium sorption assumption. This study is thus designed to devise semi-analytical models for the two-dimensional multispecies transport of a chemical mixture comprised of a parent compound and its degradation-daughter products which is influenced by rate-limited sorption subject to arbitrary time-dependent inlet boundary conditions. Three integral transforms are applied to generate a set of linear algebraic equations (AEs) resulting from the reductions of the ADEs. The contaminant concentration of each species is calculated by solving these AEs and then retransforming the solutions back to the original time-space domain. The simulation results obtained with our newly developed semi-analytical model are nearly identical to those generated using a numerical model based on the Laplace transform finite difference (LTFD) method. This confirms the validity of the new semi-analytical model. An investigation of the effect of rate-limited sorption on the migration of the contaminant plume is carried out using various time-dependent inlet boundary conditions. The sorption rate values vary from low to high, specifically, 0.05, 0.5, 5, and 50 years−1. The results show that the predicted concentrations of all contaminants within in the decay chain decrease as the sorption rate constant increases, for both constant and exponentially time-dependent boundary sources. However, under pulse loading boundary conditions, the concentrations of the later degradation products tend to increase with the sorption rate constant. The semi-analytical models created in this study, allow for different inlet boundary conditions, so can be utilized to simulate the transport of sequentially degrading reaction products. This greater accuracy makes them useful for many applications.

Analysis of Multilayer Cylindrical Thermal Conduction With a Time-Varying Convective Boundary Condition

Abstract Heat transfer in a multilayer body plays a key role in design and optimization of several engineering systems. While the analysis of simple multilayer problems is quite straightforward, realistic scenarios such as time-dependent boundary conditions result in significant complications in analysis. This work presents thermal analysis of a one-dimensional heat-generating multilayer cylinder with time-varying convective heat transfer at the boundary. Such a scenario may occur in applications such as nuclear reactors, jet impingement cooling, turbine blade heat transfer, as well as casting and related manufacturing processes. Analysis is presented for both annular and solid cylinders. A derivation for the temperature distribution is carried out, using a shifting function to split the time-dependent boundary condition into two parts, followed by appropriate mathematical substitution. For particular special cases, the analytical results derived here are shown to reduce exactly to results from past work. Good agreement of the theoretical results with numerical simulations is also demonstrated. Thermal response to various realistic time-dependent boundary conditions is analyzed. This work contributes towards the design of realistic multilayer problems and may facilitate the optimization of engineering systems where multilayer thermal conduction plays a key role.

Physical Time and Human Time

This paper is a comment on both Bunamano and Rovelli (Bridging the neuroscience and physics of time arXiv:2110.01976. (2022)) and Gruber et al. (in Front. Psychol. Hypothesis Theory, 2022) and which discuss the relation between physical time and human time. I claim here, contrary to many views discussed there, that there is no foundational conflict between the way physics views the passage of time and the way the mind/brain perceives it. The problem rather resides in a number of misconceptions leading either to the representation of spacetime as a timeless Block Universe, or at least that physically relevant universe models cannot have preferred spatial sections. The physical expanding universe can be claimed to be an Evolving Block Universe with a time-dependent future boundary, representing the dynamic nature of the way spacetime develops as matter curves spacetime and spacetime tells matter how to move. This context establishes a global direction of time that determines the various local arrows of time. Furthermore time passes when quantum wave function collapse takes place to an eigenstate; during this process, information is lost. The mind/brain acts as an imperfect clock, which coarse-grains the physical passage of time along a world line to determine the experienced passage of time, because neural processes take time to occur. This happens in a contextual way, so experienced time is not linearly related to physical time in general. Finally I point out that the Universe is never infinitely old: its future endpoint always lies infinitely faraway in the future.

Temperature Response of Double-Layer Steel Truss Bridge Girders

Double-layer steel truss continuous girders are prone to significant temperature stress, deviation, torsion, and warping, thus causing adverse temperature structural responses, and also affecting the safety and durability of bridge structures. This paper presents an investigation on time-dependent characteristics in the temperature field and temperature response of double-layer steel truss continuous bridge girders, fully considering the shielding effect subjected to different solar radiation angles during the high-temperature season. The time-dependent thermal boundary conditions and support conditions provided for the steel truss bridge structure were determined. Subsequently, a thermal analysis model for the entire structure of double-layer steel truss continuous girders was established to attain the temperature distribution law. The research results show that significant differences occur in the position and temperature difference of temperature gradients exhibited in the vertical, horizontal, and longitudinal directions in the double-layer steel truss bridge structure. The temperature distribution pattern within the chord section is mainly influenced by the environmental temperature and solar radiation intensity, along with the heat exchange between different panels. Thereafter, a validated temperature gradient formula for the component section has been proposed. The time-dependent laws in structural displacement, stress, and rotation angle under daily temperature cycling conditions have been revealed, thereby providing a theoretical basis for the life cycle construction and safety maintenance of double-layer steel truss structure bridges.

Numerical investigation on tornado-like flows and immersed bodies using vortex models

A numerical investigation to evaluate tornado flows with immersed bodies is presented in this work, where vortex profile models are adopted to generate the flow field based on time-dependent boundary conditions. In order to reproduce flow conditions obtained from field data and experimental analyses, a parametric study is performed considering the main model parameters influencing the tornado flow characteristics. A Taylor–Galerkin finite element model is used for flow simulation, where eight-node hexahedral elements with one-point integration are adopted. Vortex flow fields are generated numerically using the Modified Rankine Combined Vortex Method (MRCVM) and a fully three-dimensional approach is utilized for tornado-like flows considering the Vastistas vortex model. Turbulence is analyzed using Large Eddy Simulation (LES) with Smagorinsky's sub-grid scale modeling. The vortex formulations are verified considering a circular cylinder submitted to vortex flows, where aerodynamic force coefficients are obtained. A tornado flow field generated experimentally is reproduced utilizing the vortex models proposed here and a cubic building subject to different tornado flow conditions is also analyzed. Results demonstrate that velocity profile models are able to reproduce tornado flow fields and aerodynamic forces on immersed bodies satisfactorily since the boundary conditions and vortex model parameters are calibrated properly.

Nonlinear two-dimensional transient thermoelastic analysis of functionally graded composite plates subjected to localised cooling loads

In this study, for the first time, localized thermal cooling is considered for the transient thermoelastic response of an axisymmetric plate, addressing the underexplored area of localized low-temperature thermoelasticity. Functionally graded composite plates are analyzed which consist of a mixture of stainless steel (SUS304) and low-carbon steel (AISI1020). A two-dimensional transient heat conduction equation is employed to capture localized cooling effects, featuring various adaptive time-dependent thermal boundary conditions which mirror both loading and unloading scenarios, effectively simulating real-world phenomena. Novel parameters are introduced to facilitate a comprehensive understanding of various aspects of thermal load handling and their impact on thermoelastic responses. The heat conduction equations are solved using the Generalized Differential Quadrature Method (GDQM) and the Crank-Nicolson scheme. The nonlinear governing equations, incorporating geometrical nonlinearity through the von Kármán assumption within the First Shear Deformation Theory (FSDT) framework, are solved using the GDQM and the Picard's technique. The model is validated using Finite Element Method (FEM) through Abaqus. A comprehensive analysis is provided that considers influence of the ratio of thermally affected and unaffected plate surface, thermal load magnitude, rapidity of thermal loading and unloading, duration of the cooling load, geometrical nonlinearity, and temperature dependence of material. The study shows that the maximum von Mises stress within the structure remains consistent regardless of the duration of the cooling load, as long as the rate at which the cooling load is applied stays the same. Additionally, the findings reveal that even minor localized thermal loads can produce substantial stress, especially at the intersection of thermally affected and non-affected zones on the exposed surface in some situations.

From Transience to Recurrence for Cox–Ingersoll–Ross Model When b < 0

We consider the Cox–Ingersoll–Ross (CIR) model in time-dependent domains, that is, the CIR process in time-dependent domains reflected at the time-dependent boundary. This is a very meaningful question, as the CIR model is commonly used to describe interest rate models, and interest rates are often artificially set within a time-dependent domain by policy makers. We consider the most fundamental question of recurrence versus transience for normally reflected CIR process with time-dependent domains, and we examine some precise conditions for recurrence versus transience in terms of the growth rates of the boundary. The drift terms and the diffusion terms of the CIR processes in time-dependent domains are carefully provided. In the transience case, we also investigate the last passage time, while in the case of recurrence, we also consider the positive recurrence of the CIR processes in time-dependent domains.

Numerical analysis for electromagnetic scattering with nonlinear boundary conditions

This work studies time-dependent electromagnetic scattering from obstacles whose interaction with the wave is fully determined by a nonlinear boundary condition. In particular, the boundary condition studied in this work enforces a power law type relation between the electric and magnetic fields along the boundary. Based on time-dependent jump relations of classical boundary operators, we derive a nonlinear system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. These fields can subsequently be computed at arbitrary points in the exterior domain by evaluating a time-dependent representation formula. Fully discrete schemes are obtained by discretizing the nonlinear system of boundary integral equations with Runge–Kutta based convolution quadrature in time and Raviart–Thomas boundary elements in space. Error bounds with explicitly stated convergence rates are proven, under the assumption of sufficient regularity of the exact solution. The error analysis is conducted through novel techniques based on time-discrete transmission problems and the use of a new discrete partial integration inequality. Numerical experiments illustrate the use of the proposed method and provide empirical convergence rates.

Avoiding order reduction phenomenon for general linear methods when integrating linear problems with time dependent boundary values

When applied to stiff problems, the effective order of convergence of general linear methods is governed by their stage order, which is less than or equal to the classical order of the method. This produces an order reduction phenomenon, present in all general linear methods except those with high stage order, in a manner similar to that observed in other time integrators with internal stages.In this paper, we investigate the order reduction which arises when general linear methods are used as time integrators when using the method of lines for solving numerically initial boundary value problems with time dependent boundary values.We propose a technique, based on making an appropriate choice of the boundary values for the internal stages, with which it is possible to recover one unit of order, as we prove in this work. As expected, this implies a considerable improvement for the general linear methods suffering order reduction. Moreover, numerical experiments show that the improvement is not only in these cases, but that, even when the order reduction is not expected, the size of the errors is drastically reduced by using the technique proposed in this paper.

A Computationally Efficient Method for the Thermo-Structural Analysis of a Turbocharger Turbine Wheel Under Engine Transient Operation

Abstract The accurate evaluation of thermally-induced stresses is of particular importance for life-cycle analysis of modern turbochargers. Conjugated heat transfer (CHT), steady-state calculations represent the state-of-the-art, whose temperature distribution is usually imported into a structural solver as a thermal load for carrying out thermo-structural analyses. In the case of transient analyses, however, issues arise. Indeed, due to the different timescales of convective and conductive heat-transfer mechanisms, unsteady CHT simulations are burdened by high computational costs. Moreover, in the specific case of turbocharged engines, the timescales of engine load transients and turbocharger response may differ by a factor of up to 105. The present study proposes an approach for carrying out transient thermo-structural analyses of the solid domain only, characterized by a physical time in the order of 101 s, without the need for unsteady fluid flow calculations, which would imply the use of a solver time-step in the order of 10−5 s. In the proposed methodology, the effects of the fluid flow in the heat-transfer process are modeled by imposing adequate boundary conditions to wet surfaces of the solid domain with minimum computational effort. Indeed, by performing two steady-state CHT calculations and one steady-state computational fluid dynamics (CFD) calculation, time-dependent boundary conditions can be derived in terms of convective heat transfer coefficient and near-wall fluid temperature. The case study used for model development is a high-performance turbocharger turbine for automotive applications, responding to engine load transients of tenths of seconds. Initially, steady-state CFD and CHT models of the turbine stage are validated against experimental data. Then, transient analyses on the turbine wheel are carried out, and their results are imported as thermal loads into an unsteady structural solver, allowing for accurate transient thermo-structural analyses at sustainable computational cost.

Weakly nonlinear instability analysis of triple diffusive convection under internal heat generator and modulated boundaries

A nonlinear instability analysis of triple diffusive convection under the time-dependent heat and mass transfer boundary conditions in the presence of internal heat source is evaluated in this study. On various physical parameters, the momentary behavior of both Sherwood and Nusselt number profiles is examined. In the geometry, we have considered two parallel infinite horizontal plates acting gravity vertically downward z-direction. By using the weakly nonlinear analysis, the Ginzburg–Landau equation is generated for the rate of heat and mass transport. Here, we have considered the temperature and concentration of two solutes. The temperature and first concentration of the solute at the lower plate are higher than the upper plate, while the second concentration of the solute at the upper plate is higher compared to that of the lower plate. According to the different modulation, we have considered four cases based on the phase angle of the modulations. The convective heat and mass transports are measured as a function of the Nusselt number (Nu) and Sherwood number (Sh1 and Sh2) for both the concentration. From the results, it is found that the first Lewis number increases all the considered profiles, while Ri increases the Nusselt number profile only. The principal discovery elucidated by this article resides in the observation that the internal heat source, subject to modulated boundaries, maintains the convective instability if different solutes are used from both ends.

Boundary corrections on multi-dimensional PDEs

Two new boundary correction techniques are proposed in order to mitigate the order reduction phenomenon associated with the numerical solution of initial boundary value problems for parabolic partial differential equations in arbitrary spatial dimensions with time-dependent Dirichlet boundary conditions. The new techniques are based on the idea of discretizing the PDE problem at the boundary points as similar as possible to that of the interior points of the domain. These new techniques are considered for the time integration with W-methods based on approximate matrix factorization. By suitably modifying the internal stages of the methods on the boundary points, it is illustrated by numerical testing with time-dependent boundary conditions that the new boundary correction techniques are able to keep the same accuracy and order of convergence that the method reaches in the case of homogeneous boundary conditions.

Towards resilient operation of photovoltaic-thermal collector with incorporated organic phase change material: Numerical and experimental investigation

The resilient operation of the novel free-standing photovoltaic-thermal (PVT) collector with an integrated phase change material (PCM) in variable weather conditions was examined by developing the computational fluid dynamics (CFD) 3D model to perform numerical analysis of heat transfer. The developed transient multi-parameter numerical model was directly compared with experimental measurements for the case of variable cloudiness, unstable ambient temperature, and fluctuating wind. Due to volatile weather conditions, the numerical analysis included variable time-dependent boundary conditions implemented using user-defined functions (UDF). The investigation considered a novel multi-parameter numerical approach, together with novel experimental data for unconventional organic PCM. Specifically, the thermal characteristics of unconventional organic PCM, i.e., pork fat, were experimentally determined with the differential scanning calorimetry (DSC) method and embedded in the numerical model. It was found that melting of the crystal phase of pork fat PCM starts at 8.3 °C and ends at 45.2 °C with a latent heat of melting being 45.4 Jg−1. The numerical model was successfully validated with experimental data. The mean absolute percentage deviation of the operating temperature numerical forecast compared with the experiment was from about 2.9% to 4.9%, depending on the considered domain. The findings of this investigation can be used to improve the existing designs of PVT-PCM collectors. Furthermore, a validated numerical model can be used to investigate the impact of various operating parameters on the system performance to enhance the resilience aspect in the early development phase of novel designs.