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1 – 10 of 58Jean Batina, Serge Blancher and Tarik Kouskou
Mathematical and numerical models are developed to study the melting of a Phase Change Material (PCM) inside a 2D cavity. The bottom of the cell is heated at constant and uniform…
Abstract
Purpose
Mathematical and numerical models are developed to study the melting of a Phase Change Material (PCM) inside a 2D cavity. The bottom of the cell is heated at constant and uniform temperature or heat flux, assuming that the rest of the cavity is completely adiabatic. The paper used suitable numerical methods to follow the interface temporal evolution with a good accuracy. The purpose of this paper is to show how the evolution of the latent energy absorbed to melt the PCM depends on the temperature imposed on the lower wall of the cavity.
Design/methodology/approach
The problem is written with non-homogeneous boundary conditions. Momentum and energy equations are numerically solved in space by a spectral collocation method especially oriented to this situation. A Crank-Nicolson scheme permits the resolution in time.
Findings
The results clearly show the evolution of multicellular regime during the process of fusion and the kinetics of phase change depends on the boundary condition imposed on the bottom cell wall. Thus the charge and discharge processes in energy storage cells can be controlled by varying the temperature in the cell PCM. Substantial modifications of the thermal convective heat and mass transfer are highlighted during the transient regime. This model is particularly suitable to follow with a good accuracy the evolution of the solid/liquid interface in the process of storage/release energy.
Research limitations/implications
The time-dependent physical properties that induce non-linear coupled unsteady terms in Navier-Stokes and energy equations are not taken into account in the present model. The present model is actually extended to these coupled situations. This problem requires smoother geometries. One can try to palliate this disadvantage by constructing smoother approximations of non-smooth geometries. The augmentation of polynomials developments orders increases strongly the computing time. When the external heat flux or temperature imposed at the PCM is much greater than the temperature of the PCM fusion, one must choose carefully some data to assume the algorithms convergence.
Practical implications
Among the areas where this work can be used, are: buildings where the PCM are used in insulation and passive cooling; thermal energy storage, the PCM stores energy by changing phase, solid to liquid (fusion); cooling and transport of foodstuffs or pharmaceutical or medical sensitive products, the PCM is used in the food industry, pharmaceutical and medical, to minimize temperature variations of food, drug or sensitive materials; and the textile industry, PCM materials in the textile industry are used in microcapsules placed inside textile fibres. The PCM intervene to regulate heat transfer between the body and the outside.
Originality/value
The paper's originality is reflected in the precision of its results, due to the use of a high-accuracy numerical approximation based on collocation spectral methods, and the choice of Chebyshev polynomials basis in both axial and radial directions.
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J. Batina, M. Batchi, S. Blancher, R. Creff and C. Amrouche
The purpose of this paper is to analyse the convective heat transfer of an unsteady pulsed, laminar, incompressible flow in axisymmetric tubes with periodic sections. The flow is…
Abstract
Purpose
The purpose of this paper is to analyse the convective heat transfer of an unsteady pulsed, laminar, incompressible flow in axisymmetric tubes with periodic sections. The flow is supposed to be developing dynamically and thermally from the duct inlet. The wall is heated at constant and uniform temperature.
Design/methodology/approach
The problem is written with classical homogeneous boundary conditions. We use a shift operator to impose non‐homogeneous boundary conditions. Consequently, this method introduces source terms in the Galerkin formulation. The momentum equations and the energy equation which govern this problem are numerically solved in space by a spectral Galerkin method especially oriented to this situation. A Crank‐Nicolson scheme permits the resolution in time.
Findings
From the temperature field, the heat transfer phenomenon is presented, discussed and compared to those obtained in straight cylindrical pipes. This study showed the existence of zones of dead fluid that locally have a negative influence on heat transfer. Substantial modifications of the thermal convective heat transfer are highlighted at the entry and the minimum duct sections.
Practical implications
Pulsated flows in axisymmetric geometries can be applied to medical industries, mechanical engineering and technological processes.
Originality/value
One of the original features of this study is the choice of Chebyshev polynomials basis in both axial and radial directions for spectral methods, combined with the use of a shift operator to satisfy non‐homogeneous boundary conditions.
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Sapna Pandit, Manoj Kumar, R.N. Mohapatra and Ali Saleh Alshomrani
This paper aims to find the numerical solution of planar and non-planar Burgers’ equation and analysis of the shock behave.
Abstract
Purpose
This paper aims to find the numerical solution of planar and non-planar Burgers’ equation and analysis of the shock behave.
Design/methodology/approach
First, the authors discritize the time-dependent term using Crank–Nicholson finite difference approximation and use quasilinearization to linearize the nonlinear term then apply Scale-2 Haar wavelets for space integration. After applying this scheme on partial differential, the equation transforms into a system of algebraic equation. Then, the system of equation is solved using Gauss elimination method.
Findings
Present method is the extension of the method (Jiwari, 2012). The numerical solutions using Scale-2 Haar wavelets prove that the proposed method is reliable for planar and non-planar nonlinear Burgers’ equation and yields results better than other methods and compatible with the exact solutions.
Originality/value
The numerical results for non-planar Burgers’ equation are very sparse. In the present paper, the authors identify where the shock wave and discontinuity occur in planar and non-planar Burgers’' equation.
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J. FRÖHLICH and R. PEYRET
The low Mach number approximation of the Navier—Stokes equations is of similar nature to the equations for incompressible flow. A major difference, however, is the appearance of a…
Abstract
The low Mach number approximation of the Navier—Stokes equations is of similar nature to the equations for incompressible flow. A major difference, however, is the appearance of a space‐ and time‐varying density that introduces a supplementary non‐linearity. In order to solve these equations with spectral space discretization, an iterative solution method has been constructed and successfully applied in former work to two‐dimensional natural convection and isobaric combustion with one direction of periodicity. For the extension to other geometries efficiency is an important point, and it is therefore desirable to devise a direct method which would have, in the best case, the same stability properties as the iterative method. The present paper discusses in a systematic way different approaches to this aim. It turns out that direct methods avoiding the diffusive time step limit are possible, indeed. Although we focus for discussion and numerical investigation on natural convection flows, the results carry over for other problems such as variable viscosity flows, isobaric combustion, or non‐homogeneous flows.
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R. RAMAKRISHNAN, ALLAN R. WIETING and EARL A. THORNTON
An adaptive mesh refinement procedure that uses nodeless variables and quadratic interpolation functions is presented for analysing transient thermal problems. A temperature based…
Abstract
An adaptive mesh refinement procedure that uses nodeless variables and quadratic interpolation functions is presented for analysing transient thermal problems. A temperature based finite element scheme with Crank‐Nicolson time marching is used to obtain the thermal solution. The strategies used for mesh adaptation, computing refinement indicators, and time marching are described. Examples in one and two dimensions are presented and comparisons are made with exact solutions. The effectiveness of this procedure for transient thermal analysis is reflected in good solution accuracy, reduction in number of elements used, and computational efficiency.
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Gopal Priyadarshi and B.V. Rathish Kumar
In the past few years, Haar wavelet-based numerical methods have been applied successfully to solve linear and nonlinear partial differential equations. This study aims to propose…
Abstract
Purpose
In the past few years, Haar wavelet-based numerical methods have been applied successfully to solve linear and nonlinear partial differential equations. This study aims to propose a wavelet collocation method based on Haar wavelets to identify a parameter in parabolic partial differential equations (PDEs). As Haar wavelet is defined in a very simple way, implementation of the Haar wavelet method becomes easier than the other numerical methods such as finite element method and spectral method. The computational time taken by this method is very less because Haar matrices and Haar integral matrices are stored once and used for each iteration. In the case of Haar wavelet method, Dirichlet boundary conditions are incorporated automatically. Apart from this property, Haar wavelets are compactly supported orthonormal functions. These properties lead to a huge reduction in the computational cost of the method.
Design/methodology/approach
The aim of this paper is to reconstruct the source control parameter arises in quasilinear parabolic partial differential equation using Haar wavelet-based numerical method. Haar wavelets possess various properties, for example, compact support, orthonormality and closed form expression. The main difficulty with the Haar wavelet is its discontinuity. Therefore, this paper cannot directly use the Haar wavelet to solve partial differential equations. To handle this difficulty, this paper represents the highest-order derivative in terms of Haar wavelet series and using successive integration this study obtains the required term appearing in the problem. Taylor series expansion is used to obtain the second-order partial derivatives at collocation points.
Findings
An efficient and accurate numerical method based on Haar wavelet has been proposed for parameter identification in quasilinear parabolic partial differential equations. Numerical results are obtained from the proposed method and compared with the existing results obtained from various finite difference methods including Saulyev method. It is shown that the proposed method is superior than the conventional finite difference methods including Saulyev method in terms of accuracy and CPU time. Convergence analysis is presented to show the accuracy of the proposed method. An efficient algorithm is proposed to find the wavelet coefficients at target time.
Originality/value
The outcome of the paper would have a valuable role in the scientific community for several reasons. In the current scenario, the parabolic inverse problem has emerged as very important problem because of its application in many diverse fields such as tomography, chemical diffusion, thermoelectricity and control theory. In this paper, higher-order derivative is represented in terms of Haar wavelet series. In other words, we represent the solution in multiscale framework. This would enable us to understand the solution at various resolution levels. In the case of Haar wavelet, this paper can achieve a very good accuracy at very less resolution levels, which ultimately leads to huge reduction in the computational cost.
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Jiawei Wang, Feng Chen, Jinghui Shao, Weichen Zhang and Xikui Ma
This paper aims to present a novel hybrid time integration approach for efficient numerical simulations of multiscale problems involving interactions of electromagnetic fields…
Abstract
Purpose
This paper aims to present a novel hybrid time integration approach for efficient numerical simulations of multiscale problems involving interactions of electromagnetic fields with fine structures.
Design/methodology/approach
The entire computational domain is discretized with a coarse grid and a locally refined subgrid containing the tiny objects. On the coarse grid, the time integration of Maxwell’s equations is realized by the conventional finite-difference technique, while on the subgrid, the unconditionally stable Krylov-subspace-exponential method is adopted to breakthrough the Courant–Friedrichs–Lewy stability condition.
Findings
It is shown that in contrast with the conventional finite-difference time-domain method, the proposed approach significantly reduces the memory costs and computation time while providing comparative results.
Originality/value
An efficient hybrid time integration approach for numerical simulations of multiscale electromagnetic problems is presented.
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Caroline Nore, Houda Zaidi, Frederic Bouillault, Alain Bossavit and Jean-Luc Guermond
The purpose of this paper is to present a new formulation for taking into account the convective term due to an imposed velocity field in the induction equation in a code based on…
Abstract
Purpose
The purpose of this paper is to present a new formulation for taking into account the convective term due to an imposed velocity field in the induction equation in a code based on Whitney elements called DOLMEN. Different Whitney forms are used to approximate the dependent variables. The authors study the kinematic dynamo action in a von Kármán configuration and obtain results in good agreement with those provided by another well validated code called SFEMaNS. DOLMEN is developed to investigate the dynamo action in non-axisymmetric domains like the impeller driven flow of the von Kármán Sodium (VKS) experiment. The authors show that a 3D magnetic field dominated by an axisymmetric vertical dipole can grow in a kinematic dynamo configuration using an analytical velocity field.
Design/methodology/approach
Different Whitney forms are used to approximate the dependent variables. The vector potential is discretized using first-order edge elements of the first family. The velocity is approximated by using the first-order Raviart-Thomas elements. The time stepping is done by using the Crank-Nicolson scheme.
Findings
The authors study the kinematic dynamo action in a von Kármán configuration and obtain results in good agreement with those provided by another well validated code called SFEMaNS. The authors show that a 3D magnetic field dominated by an axisymmetric vertical dipole can grow in a kinematic dynamo configuration using an analytical velocity field.
Originality/value
The findings offer a basis to a scenario for the VKS dynamo.
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Yuri N. Skiba and Denis M. Filatov
The purpose of this paper is to suggest a new approach to the numerical simulation of shallow‐water flows both in plane domains and on the sphere.
Abstract
Purpose
The purpose of this paper is to suggest a new approach to the numerical simulation of shallow‐water flows both in plane domains and on the sphere.
Design/methodology/approach
The approach involves the technique of splitting of the model operator by geometric coordinates and by physical processes. Specially chosen temporal and spatial approximations result in one‐dimensional finite difference schemes that conserve the mass and the total energy. Therefore, the mass and the total energy of the whole two‐dimensional split scheme are kept constant too.
Findings
Explicit expressions for the schemes of arbitrary approximation orders in space are given. The schemes are shown to be mass‐ and energy‐conserving, and hence absolutely stable because the square root of the total energy is the norm of the solution. The schemes of the first four approximation orders are then tested by simulating nonlinear solitary waves generated by a model topography. In the analysis, the primary attention is given to the study of the time‐space structure of the numerical solutions.
Originality/value
The approach can be used for the numerical simulation of shallow‐water flows in domains of both Cartesian and spherical geometries, providing the solution adequate from the physical and mathematical standpoints in the sense of keeping its mass and total energy constant even when fully discrete shallow‐water models are applied.
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Large‐eddy simulation (LES) of a turbulent channel flow is performed using different subfilter‐scale (SFS) models and test filter functions. The SFS models used are the dynamic…
Abstract
Large‐eddy simulation (LES) of a turbulent channel flow is performed using different subfilter‐scale (SFS) models and test filter functions. The SFS models used are the dynamic Smagorinsky model (DSM) and the dynamic mixed model (DMM). The DMM is a linear combination between the scale‐similarity model and the DSM. The test filter functions investigated are the sharp cut‐off (in spectral space) and smooth filter that is commutative up to fourth‐order. The filters are applied either in the homogeneous directions or in all three spatial directions. The governing equations are discretized using a fourth‐order energy‐conserving finite‐difference scheme. The influence from the test filter function and the SFS model on the LES results are investigated and the effect of two‐dimensional versus three‐dimensional test filtering are investigated. The study shows that the combination of SFS model and filter function highly influences the computational results; even the effect on the zeroth‐order moment is large.
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