Time Conservation Element Research Articles

Performance evaluation of the conservation element and solution element method in SMB process simulation

The computational performance of the space–time conservation element and solution element method (CESE) is evaluated in solving simulated moving bed (SMB) chromatographic models, compared with a 2nd-order upwind discretization method (UDS2) and 4th-order orthogonal collocation finite element method (OCFE4) of ASPEN chromatography (ASPEN Tech, USA). A novel finite volume discretization is proposed for the CE/SE method to treat the exit boundary condition in the adsorption column close to plug-flow operation. A single column model and a linear/equilibrium SMB model which have the analytic solutions are used for the computational performance assessment of the numerical methods. For the two examples, CESE shows the superiority over UDS2 and OCFE4 in terms of computational efficiency and accuracy. Other five practical SMB examples which have no analytic solution are used to examine computational features of the three numerical methods. Using UDS2, it shows a fast calculation but the sufficient number of mesh points is required to enhance accuracy. The accuracy of CESE is comparable with OCFE4. Since CESE enforces both local and global flux conservation in space and time and is combined with the finite volume discretization of the exit boundary condition, its numerical solution of SMB adsorption models is accurate and computationally efficient.

Two-dimensional Dual-Phase-Lag thermal behavior in single-/multi-layer structures using CESE method

The Space–Time Conservation Element and Solution Element (CESE) method is used to simulate the propagation of Dual-Phase-Lag (DPL) laser-pulsed thermal waves through single-layer and multi-layer 2D structures. Numerical solutions are presented for the temperature distributions induced within a single-layer structure by four non-Fourier propagation modes, namely hyperbolic, wavelike, diffusive and over-diffusive. In addition, the transmission–reflection phenomena induced as the thermal wave propagates through the interface between two layers with dissimilar properties are systematically examined and discussed.

An improved CE/SE scheme for multi-material elastic–plastic flows and its applications

In this paper, a new definition of SE and CE, which is based on the hexahedron mesh and simpler than Chang’s original CE/SE method (the space–time Conservation Element and Solution Element method), is proposed and an improved CE/SE scheme is constructed. Furthermore, the improved CE/SE scheme is extended in order to solve the elastic–plastic flow problems. The hybrid particle level set method is used for tracing the interfaces of materials. Proper boundary conditions are presented in interface tracking. Two high-velocity impact problems are simulated numerically and the computational results are carefully compared with the experimental data, as well as the results from other literature and LS-DYNA software. The comparisons show that the computational scheme developed currently is clear in physical concept, easy to be implemented and high accurate and efficient for the problems considered.

Validation and verification of Courant number insensitive CE/SE method for transient viscous flow simulations

In this paper, we report an extension of the space–time conservation element–solution element (CE/SE) framework-based viscous flow solver. With the accuracy of solution obtained through the use of a CE/SE-based solver closely related to the CFL number disparity across the mesh, a new formulation to make the solution insensitive to CFL number disparity is herein presented. The capability of the developed solver is then validated through simulation of 2D problems such as driven cavity, external flow over a flat plate, laminar flow over a square cylinder, etc. Investigations are also conducted to verify the sensitivity of results to grid spacing and mesh structure.

A stable numerical method for solving variable coefficient advection–diffusion models

In a recent paper [E. Defez, R. Company, E. Ponsoda, L. Jódar, Aplicación del método CE-SE a la ecuación de advección-difusión con coeficientes variables, Congreso de Métodos Numéricos en Ingenierá (SEMNI), Granada, Spain, 2005] a modified space–time conservation element and solution element scheme for solving the advection–diffusion equation with time-dependent coefficients, is proposed. This equation appears in many physical and technological models like gas flow in industrial tubes, conduction of heat in solids or the evaluation of the heating through radiation of microwaves when the properties of the media change with time. This method improves the well-established methods, like finite differences or finite elements: the integral form of the problem exploits the physical properties of conservation of flow, unlike the differential form. Also, this explicit scheme evaluates the variable and its derivative simultaneously in each knot of the partitioned domain. The modification proposed in [E. Defez, R. Company, E. Ponsoda, L. Jódar, Aplicación del método CE-SE a la ecuación de advección-difusión con coeficientes variables, Congreso de Métodos Numéricos en Ingenierá (SEMNI), Granada, Spain, 2005] with regard the original method [S.C. Chang, The method of space–time conservation element and solution element. A new approach for solving the Navier–Stokes and Euler equations, J. Comput. Phys. 119 (1995) 295–324] consists of keeping the variable character of the coefficients in the solution element, without considering the linear approximation. In this paper the stability of the proposed method is studied and a CFL condition is obtained.

Application of CESE method to simulate non-Fourier heat conduction in finite medium with pulse surface heating

This study employs the space–time conservation element and solution element (CESE) method to simulate the temperature and heat flux distributions in a finite medium subject to various non-Fourier heat conduction models. The simulations consider three specific cases, namely a single phase lag (SPL) thermal wave model with a pulsed temperature condition, a SPL model with a surface heat flux input, and a dual phase lag (DPL) thermal wave model with an initial deposition of thermal energy. In every case, the thermal waves are simulated with respect to time as the thermal wave propagates through the medium with a constant velocity. In general, the simulation results are found to be in good agreement with the exact analytical solutions. Furthermore, it is shown that the CESE method yields low numerical dissipation and dispersion errors and accurately models the propagation of the wave form even in its discontinuous portions. Significantly, compared to traditional numerical schemes, the CESE method provides the ability to model the behavior of the SPL thermal wave following its reflection from the boundary surface. Further, a numerical analysis is performed to establish the CESE time step and mesh size parameters required to ensure stable solutions of the SPL and DPL thermal wave models, respectively.

Time-domain computation of muffler frequency response: Comparison of different numerical schemes

A comparative study of the performance of different schemes used to solve one-dimensional gas flow equations when applied to the computation of the frequency response of exhaust mufflers is presented. Simple but representative systems with well-known acoustic behaviour were considered. Apart from the classical Lax–Wendroff and MacCormack schemes, the total variation diminishing schemes, flux corrected transport techniques or the innovative space–time conservation element and solution element method were considered. The results provide guidelines for a proper choice of the numerical scheme, taking into account the mesh spacing.

A HIGH-RESOLUTION SPACE–TIME CONSERVATIVE METHOD FOR NON-LINEAR HYPERBOLIC CONSERVATION LAWS

We present a second-order scheme for the numerical solution of hyperbolic systems which treats space and time in a unified manner. The flow variables and their slopes are the basic unknowns in the scheme. The scheme utilizes the advantages of the space–time conservation element and solution element (CE/SE) method of Chang [1995] as well as central schemes of Nessyahu and Tadmor [1990]. However, unlike the CE/SE method the present scheme is Jacobian-free and hence like the central schemes can also be applied to any hyperbolic system. In Chang's method, a finite difference approach is being used for the slope calculation in case of non-linear hyperbolic equations. We propose to propagate the slopes by a scheme even in the case of non-linear systems. By introducing a suitable limiter for the slopes of flow variables, we can apply the same scheme to linear and non-linear problems with discontinuities. The scheme is simple, efficient and has a good resolution especially at contact discontinuities. We derive the scheme for one- and two-space dimensions. In two-space dimensions we use structured triangular meshes. The second-order accuracy of the scheme has been verified by numerical experiments. Several numerical tests presented in this article validate the accuracy and robustness of the present scheme.

Modeling the Resistive MHD by the Cese Method

In this paper, the Space–Time Conservation Element and Solution Element (CESE) method is applied to 2.5-dimensional resistive magnetohydrodynamics (MHD) equations in Cartesian coordinates, with the purpose of modeling the magnetic reconnection study. To show the validity and capacity of its application to MHD reconnection problem, spontaneous fast reconnection and magnetic reconnection in multiple heliospheric current sheets are studied, which show good consistency with those obtained formerly by other authors. In order to assess the ∇ ⋅ B = 0 constraint numerically, the contours and evolution of ∇ ⋅ B are analyzed. The numerical results tell us that the CESE numerical scheme not only has good numerical resolution but also can keep the divergence-free condition for magnetic fields in the reconnection problems during the evolutionary process without any special treatment.

Solving the MHD equations by the space–time conservation element and solution element method

We apply the space–time conservation element and solution element (CESE) method to solve the ideal MHD equations with special emphasis on satisfying the divergence free constraint of magnetic field, i.e., ∇ · B = 0. In the setting of the CESE method, four approaches are employed: (i) the original CESE method without any additional treatment, (ii) a simple corrector procedure to update the spatial derivatives of magnetic field B after each time marching step to enforce ∇ · B = 0 at all mesh nodes, (iii) a constraint-transport method by using a special staggered mesh to calculate magnetic field B, and (iv) the projection method by solving a Poisson solver after each time marching step. To demonstrate the capabilities of these methods, two benchmark MHD flows are calculated: (i) a rotated one-dimensional MHD shock tube problem and (ii) a MHD vortex problem. The results show no differences between different approaches and all results compare favorably with previously reported data.

Application of space–time CE/SE method to shallow water magnetohydrodynamic equations

In this article we apply a space–time conservation element and solution element (CE/SE) method for the approximate solution of shallow water magnetohydrodynamic (SMHD) equations in one and two space dimensions. These equations model the dynamics of nearly incompressible conducting fluids for which the evolution is nearly two-dimensional with magnetic equilibrium in the third direction. In this article we are using a variant of the CE/SE method developed by Zhang et al. [A space–time conservation element and solution element method for solving the two-dimensional unsteady Euler equations using quadrilateral and hexahedral meshes, J. Comput. Phys. 175 (2002) 168–199]. This method uses structured and unstructured quadrilateral and hexahedral meshes in two and three space dimensions, respectively. In this method, a single conservation element at each grid point is employed for solving conservation laws no matter in one, two, and three space dimensions. The present scheme use the conservation element to calculate flow variables only, while the gradients of flow variables are calculated by central differencing reconstruction procedure. We give both one- and two-dimensional test computations. A qualitative comparison reveals an excellent agreement with previous published results of wave propagation method and evolution Galerkin schemes. The one- and two-dimensional computations reported in this paper demonstrate the remarkable versatility of the present CE/SE scheme.

Analysis of numerical methods to solve one-dimensional fluid-dynamic governing equations under impulsive flow in tapered ducts

A comparative study of different schemes to solve one-dimensional (1-D) gas flow equations in tapered pipes was conducted in this paper. The total variation diminishing schemes, flux corrected transport techniques or the innovative space–time conservation element and solution element method have been contrasted with experimental data. The experimental results were obtained with an impulsive flow test rig, where isolated pressure pulses can be generated in order to study the flow-dynamic behaviour of different components, including tapered pipes. The results of the different schemes were compared to the experimental data and a fairly good agreement was achieved when high-resolution schemes were used but with a considerable increment of computational effort. A mass conservation study through the tapered ducts with impulsive flow was also carried out, which demonstrated that it is not the resolution level of the scheme but the governing equations arrangement that has decisive importance in mass conservation.

Space–Time Method for Detonation Problems with Finite-rate Chemical Kinetics

The space–time conservation element and solution element (CE/SE) method originally developed for non-reacting flows is extended to accommodate finite-rate chemical kinetics for multi-component systems. The model directly treats the complete conservation equations of mass, momentum, energy, and species concentrations. A subtime-step integration technique is established to handle the stiff chemical source terms in the formulation. In addition, a local grid refinement algorithm within the framework of the CE/SE method is incorporated to enhance the flow resolution in areas of interest. The capability and accuracy of the resultant scheme are validated against several detonation problems, including shock-induced detonation with detailed chemical kinetics and multi-dimensional detonation initiation and propagation.

Extension of CE/SE method to 2D viscous flows

The space–time conservation element–solution element (CE/SE) method is extended to two-dimensional viscous flow problems. The formulation is presented and discussed. The extended CE/SE method is applied to several 2D viscous flow problems, such as boundary layer flow, entrance of channel flow, backward facing step flow and cavity flow. The numerical results and their comparison with analytical result and experimental data are presented. The capability of this method for predicting viscous flow feature and heat transfer is demonstrated. Also the capability of the method to solve both high-speed flows and low-Mach number flows is demonstrated.

Numerical Investigation of Transonic Resonance with a Convergent-Divergent Nozzle

At pressure ratios lower than the design value, convergent ‐divergent nozzles often undergo a e ow resonance accompaniedbytheemissionofacoustictones.Thephenomenon,drivenbytheunsteadyshockwithinthedivergent section of the nozzle, has been studied experimentally previously. The space ‐time conservation element solution element method is employed to numerically investigate the phenomenon. The computations are performed for a given nozzle geometry forseveral different pressure ratios. Sustained limit-cycleoscillations are encountered in all cases. The oscillation frequencies and their variation with pressure ratio, including a stage jump, agree well with the experimental results. The unsteady e ow data cone rm that stage 1 of the resonance (fundamental) involves a one-quarter standing wave, whereas stage 2 (third harmonic ) involves a three-quarter standing wave within the divergent section of the nozzle. Details of the shock motion, the e ow, and the near acoustic e eld are documented for one case each of stages 1 and 2. I. Introduction T HIS paper is concerned with an aeroacoustic resonance often encountered with convergent‐ divergent nozzles when run near transonic conditions. The resonance is usually accompanied by the emission of intense acoustic tones. Whereas a casual observer may easily confuse it with the well-known screech tone, it has been shown to be different in character as well as origin. The frequency of the tone increases with increasing plenum pressure. The frequency variation may involve a staging behavior, that is, an abrupt jump in frequency. Whereas odd harmonic stages take place at lower pressures, the fundamental takes place over a wide rangeof higher pressures. Depending on nozzle geometry, the fundamental has been found to persist to pressure ratios as high as 5. The phenomenon has been identie ed and studied experimentally by Zaman etal. 1 For a discussion of background, technological relevance, and pertinent past work from the literature, see Ref. 1. A discussion of numerical works from the literature apparently capturing the same phenomenon, and the implication of those results, will be deferred to the discussion in Sec. V. During the course of the experimental study a numerical study was initiated to complement the investigation. The space‐ time conservation-element and solution-element (CE/SE) method, to be elaborated shortly, was employed because of its past success with e ows involving shocks and acoustic waves. 2 The calculations were e rst performed for a nozzle geometry and pressure ratio that corresponded to an experimental test condition. The result was encouraging in that the e ow not only exhibited a quasi periodicity, but the frequency was also in agreement with the experimental result. Subsequent calculations at two other pressures captured the right trend in the frequency variation as well as the stage jump. These promising results prompted a further study. Thiswas deemedwell-justie ed

Comparison of numerical methods for the simulation of dispersed phase systems

This contribution deals with a new numerical method for an accurate and efficient simulation of particulate processes. As an example for dispersed phase systems a detailed model for crystallization processes is considered. After the model derivation, which is based solely on physical principles, different techniques for the numerical simulation of the mathematical model are discussed. State-of-the-art finite volume schemes based on the ‘method of lines’ approach are compared to the recently published ‘Space–time conservation element and solution element method’. The presented simulation results show a strong dependence on the chosen numerical method. Guidelines for a proper selection of numerical methods for the treatment of population balance based models are given.

Application of the Space–Time Conservation Element and Solution Element Method to One-Dimensional Convection–Diffusion Problems

In the space–time conservation element and solution element (CE/SE) method, the independent marching variables used comprise not only the mesh values of the physical dependent variables but also, in contrast to a typical numerical method, the mesh values of the spatial derivatives of these physical variables. The use of the extra marching variables results from the need to construct the two-level, explicit and nondissipative schemes which are at the core of the CE/SE development. It also results from the need to minimize the stencil while maintaining accuracy. In this paper, using the 1D a–μ scheme as an example, the effect of this added complication on consistency, accuracy, and operation count is assessed. As part of this effort, an equivalent yet more efficient form of the a–μ scheme in which the independent marching variables are the local fluxes tied to each mesh point is introduced. Also, the intriguing relations that exist among the a–μ. Leapfrog, and DuFort–Frankel schemes are further explored. In addition, the redundancy of the Leapfrog, DuFort–Frankel, and Lax schemes and the remedy for this redundancy are discussed. This paper is concluded with the construction and evaluation of a CE/SE solver for the inviscid Burgers equation.