Conservation Element Solution Research Articles

A central-upwind scheme for two-phase shallow granular flow model

In this article, a central-upwind scheme (CUP) is extended to solve two-phase shallow granular flow (TPSF) model. The flow in this case is considered as incompressible and it is regarded to be a shallow layer of granular and liquid material. The model consists of continuity and momentum equations for both solid and liquid phases. Our main intrigue is the numerical approximation of the above-mentioned solid-liquid model, the complexity of which raises numerical challenges. The proposed method is non-oscillating upwind biased that does not require a Riemann solver at each step. A few test problems are approximated numerically to check the performance of suggested scheme. For validation the outcomes are compared with the conservation element solution element (CE/SE) and kinetic flux vector splitting (KFVS) schemes.

Analyses of non-Fourier heat conduction in 1-D spherical biological tissue based on dual-phase-lag bio-heat model using the conservation element/solution element (CE/SE) method: A numerical study

Among the various numerical methods in solving the hyperbolic equation systems, methods that are simple, have higher accuracy, and avoid creating unwanted numerical oscillations along with having shorter runtime are preferred. In the present study, two numerical methods, space-time conservation element and solution element (CESE) as well as control volume (CV) method have been utilized to estimate temperature distribution in the spherical living tissue during laser irradiation. The results of the CESE method in simulating the non-Fourier thermal wave of the dual-phase-lag (DPL) model in spherical coordinates have been validated against the available semi-analytical results as well as the results obtained using the CV method. The good correlation between the results proved that the CESE method can be applied accurately in solving the DPL model thermal waves in spherical coordinates. In addition, the advantages of the CESE method over the CV method have been investigated. No numerical oscillations are detected in the temperature distribution obtained by the CESE method during and after exposure. Furthermore, in the same circumstances, in terms of radial grid size, a larger time step can be used in CESE method which reduces runtime. Examining the four non-Fourier models, hyperbolic, wavelike, diffusive, and over-diffusive, shows that the thermal wave occurs only in two of these models, hyperbolic and wavelike, when the phase lag of the heat flux is greater than the phase lag of the temperature gradient. After stopping laser irradiation, the maximum temperature is diffused from the surface of the tissue to its inner layers. The results proved that the speed of thermal wave progression is not only constant in hyperbolic and DPL models but also decreases in different tissues with the passage of time. Regardless of the type of tissue, the wave motion towards the center of the tissue in the hyperbolic model is greater than the DPL model. In addition, studying the thermal wave propagation speed in different tissues indicated that this parameter has a direct relationship with the thermal diffusivity of the tissues. The diffusion property caused by the temperature gradient phase lag in the DPL model is the reason why the wave expansion is greater than the hyperbolic model and why a wider range of the tissue is affected by the laser irradiation.

A predictive model of organic acids separation by chromatography with strong anionic resins in sulfate form

Organic acids commonly have quite symmetrical chromatography profiles at low pH (< 1.5) with strong anionic resins, but a significant tailing can be observed with succinic and citric acids. Classical adsorption models, like the Langmuir model, fail to predict this behavior, which can have a major influence on mean retention times and profile shapes, therefore on chromatography performances.A new retention model was developed to better predict organic acid separation with strong anionic resin. This model combines a refined Langmuir adsorption model and an ion-exchange model. Organic acid adsorption is assumed to be due to hydrogen bonding with sulfate and hydrogen sulfate counter-anions on the resin. The adsorption capacity depends mostly on molecular size: up to sixteen formic acid molecules could be adsorbed per counter-anions, meanwhile only two succinic acid or one citric acid molecules could be adsorbed.This adsorption model was then embedded in a generic and accurate modeling approach (continuous column with mass balance equations solved by the conservation element/solution element (CE/SE) method). All parameters of this column model were identified by fitting the simulation to experimental results (equilibrium curves and pulse tests). Then, the column model was validated with original experimental results from a binary mixture pulse test (formic and succinic acids). Results show that simulations are much more predictive for multi-component pulse tests, both in terms of profile shape and retention time, which cannot be captured without considering ion-exchange.

Efficient Simulation of Chromatographic Processes Using the Conservation Element/Solution Element Method

Chromatographic separation processes need efficient simulation methods, especially for nonlinear adsorption isotherms such as the Langmuir isotherms which imply the formation of concentration shocks. The focus of this paper is on the space–time conservation element/solution element (CE/SE) method. This is an explicit method for the solution of systems of partial differential equations. Numerical stability of this method is guaranteed when the Courant–Friedrichs–Lewy condition is satisfied. To investigate the accuracy and efficiency of this method, it is compared with the classical cell model, which corresponds to a first-order finite volume discretization using a method of lines approach (MOL). The evaluation is done for different models, including the ideal equilibrium model and a mass transfer model for different adsorption isotherms—including linear and nonlinear Langmuir isotherms—and for different chromatographic processes from single-column operation to more sophisticated simulated moving bed (SMB) processes for the separation of binary and ternary mixtures. The results clearly show that CE/SE outperforms MOL in terms of computational times for all considered cases, ranging from 11-fold for the case with linear isotherm to 350-fold for the most complicated case with ternary center-cut eight-zone SMB with Langmuir isotherms, and it could be successfully applied for the optimization and control studies of such processes.

Accelerating the discontinuous Galerkin cell-vertex scheme solver on GPU-powered systems

The discontinuous Galerkin cell-vertex scheme (DG-CVS) is a high-order space-time Riemann-solver-free numerical solver for general hyperbolic conservation laws. It fuses the discontinuous Galerkin (DG) method and the conservation element/solution element (CE/SE) method to take advantage of the best features of both methods. In DG-CVS, the time derivatives of the solution are treated as independent unknowns together with spatial derivatives of the solution, which is amendable to a GPU's parallel execution style. In a GPU environment, this type of scientific application poses challenges, such as high thread divergence, low kernel occupancy, and hardware-unfriendly memory access patterns. This paper presents various optimisations that address those issues. Our proposed optimisations include thread remapping, register pressure reduction as well as software-managed cache memory utilisation. DG-CVS is accelerated by up to 54% on AMD HD7970 GPU, when compared to CPU only execution.

Large Eddy Simulation of Planar Free Jet Flow Using the Space-Time Conservation Element/Solution Element Scheme

In order to investigate the characteristics of planar free jet flows, the Monotonically Integrated Large Eddy Simulation (MILES) method with the high-resolution space-time conservation element/solution element (CE/SE) scheme was used to simulate the two-dimensional free jet flows. For free jet flows within the Reynolds number range 35300~2200, the flow structures, averaged and instant velocity, turbulence intensity, velocity profiles of different cross-sections were examined respectively. The simulation results agree well with the experiment data. A region of constant height exists right after the jet outlet, in which the average centerline velocity and turbulence intensity maintains same with the jet outlet, and the instant fluctuation amplitudes are also small. For turbulent flows, the length of the undisturbed region is equal to the length of region of constant height. While near the laminar region, the length of undisturbed region is slightly larger than the length of zone of constant height. The vortex structures start to generate after the undisturbed region, develop coherently in the region of flow establishment, and break down gradually in the region of fully established flow region. As the Reynolds number increases, the length of zone of constant height, the length of undisturbed region and the length of potential core region all decrease correspondingly.

Numerical simulation of oxide nanoparticle growth characteristics under the gas detonation chemical reaction by space-time conservation element–solution element method

Under harsh conditions (such as high temperature, high pressure, and millisecond lifetime chemical reaction), a long-standing challenge remains to accurately predict the growth characteristics of nanosize spherical particles and to determine the rapid chemical reaction flow field characteristics. The growth characteristics of similar spherical oxide nanoparticles are further studied by successfully introducing the space-time conservation element–solution element (CE/SE) algorithm with the monodisperse Kruis model. This approach overcomes the nanosize particle rapid growth limit set and successfully captures the characteristics of the rapid gaseous chemical reaction process. The results show that this approach quantitatively captures the characteristics of the rapid chemical reaction, nanosize particle growth and size distribution. To reveal the growth mechanism for numerous types of oxide nanoparticles, it is very important to choose a rational numerical method and particle physics model.

A Riemann-Solver Free Spacetime Discontinuous Galerkin Method for General Conservation Laws

This paper summarizes a Riemann-solver-free spacetime discontinuous Galerkin method developed for general conservation laws. The method integrates the best features of the spacetime Conservation Element/Solution Element (CE/SE) method and the discontinuous Galerkin (DG) method. The core idea is to construct a staggered spacetime mesh through alternate cell-centered CEs and vertex-centered CEs within each time step. Inside each SE, the solution is approximated using high-order spacetime DG basis polynomials. The spacetime flux conservation is enforced inside each CE using the DG concept. The unknowns are stored at both vertices and cell centroids of the spatial mesh. However, the solutions at vertices and cell centroids are updated at different time levels within each time step in an alternate fashion. Thanks to the staggered spacetime formulation, there are no left and right states for the solution at the spacetime interface. Instead, the solution available to evaluate the flux is continuous across the interface. Therefore, no (approximate) Riemann solvers are needed to provide a unique numerical flux. The current method can be used to solve arbitrary conservation laws including the compressible Euler equations, shallow water equations and magnetohydrodynamics (MHD) equations without the need of any form of Riemann solvers. A set of benchmark problems of various conservation laws are presented to demonstrate the accuracy of the method.

A New Space-Time CE/SE Numerical Tracking of Contaminant Transport in Fractured Stratified Geologic Profiles

To date, efficient numerical simulation of contaminant transport in geologic porous media is challenged by parametric jumps resulting from stratification and the use of ideal initial/boundary conditions. Thus, to resolve some contaminant hydrology problems, this work presents the development of the Space-Time Conservation Element/Solution Element (CE/SE) scheme for advection-dispersion-reaction a-d-r transport in geologic media. The CE/SE method derives from the native form of Gauss conservation law. Therefore, it is able to effectively handle non-trivial discontinuities that may exist within the problem domain. In freshwater aquifer, stratification and other parametric jumps are examples of such discontinuity. To simulate the Nigerian experience of nitrate pollution of freshwater aquifers; the a-d-r contaminant transport model is herein solved under a time periodic nitrate fertilizer loading condition on farmlands. Results show that this approach is able to recover the well-known field pattern of nitrate profiles under farmlands. Cyclic loading impacts more on the dispersivity of an aquifer. Hence, dispersion coefficient modulates the response of aquifers to loading frequency. However, aquifers with conductivity less than 10-6 m/day are almost insensitive to periodic loads. The CE/SE method is able to sense slight (i.e. order of 10-3) variation in hydrological parameters. Also, CE/SE computes contaminant concentration and its flux simultaneously. Thus, it facilitates a better understanding of some reported phenomena such as contaminant accumulation and localized reverse transport at the interface between fracture and matrix in geologic medium. Clearly, CE/SE is an efficient and admissible tool into the family of numerical methods available for tracking contaminant transport in porous media.

EXTRAPOLATION OF THE SOLAR CORONAL MAGNETIC FIELD FROMSDO/HMI MAGNETOGRAM BY A CESE-MHD-NLFFF CODE

Due to the absence of direct measurement, the magnetic field in the solar corona is usually extrapolated from the photosphere in numerical way. At the moment, the nonlinear force-free field (NLFFF) model dominates the physical models for field extrapolation in the low corona. Recently we have developed a new NLFFF model with MHD relaxation to reconstruct the coronal magnetic field. This method is based on CESE--MHD model with the conservation-element/solution-element (CESE) spacetime scheme. In this paper, we report the application of the CESE--MHD--NLFFF code to \SDO/HMI data with magnetograms sampled for two active regions (ARs), NOAA AR 11158 and 11283, both of which were very non-potential, producing X-class flares and eruptions. The raw magnetograms are preprocessed to remove the force and then inputted into the extrapolation code. Qualitative comparison of the results with the \SDO/AIA images shows that our code can reconstruct magnetic field lines resembling the EUV-observed coronal loops. Most important structures of the active regions are reproduced excellently, like the highly-sheared field lines that suspend filaments in AR 11158 and twisted flux rope which corresponds to a sigmoid in AR 11283. Quantitative assess of the results shows that the force-free constraint is fulfilled very well in the strong-field regions but apparently not that well in the weak-field regions because of data noise and numerical errors in the small currents.

An improved CE/SE scheme for incompressible multiphase flows and its applications

A new space-time conservation element-solution element (CE/SE) scheme, which is based on the hexahedron mesh, is proposed by constructing a new structure of solution elements (SEs) and conservation elements (CEs) and extended to two-dimensional incompressible multiphase flows. This scheme is applied to several incompressible flow problems, such as lid-driven cavity flow, dam break flow, motions of falling droplets, flow instability and water-kerosene interaction. The numerical results and their comparison with lecture results and experimental data are presented. The comparisons demonstrate that the computational scheme developed currently is clear in physical concept, easy to be implemented and highly accurate and efficient for the above-mentioned problems. The highlight is that the pretreatment method is proposed, transforming the elliptic-parabolic equations to the hyperbolic equations, which can be directly simulated by the improved CE/SE scheme. Thus, the improved CE/SE scheme can be applied to incompressible multiphase flows widely.

Applications of the CE/SE Scheme to Incompressible Viscous Flows in Two-Sided Lid-Driven Square Cavities

The spacetime conservation element-solution element (CE/SE) method is extended to two-dimensional incompressible viscous flow in a two-sided lid-driven square cavity. Based on the SIMPLE method concept, the preconditioned dual-time scheme is introduced for unsteady computations. The CE/SE-based code is validated by simulating one-sided lid-driven cavity flows. The two-sided lid-driven square cavity problem involves several interesting characteristics being successfully predicted, including development of a pair of off-corner vortices and a free shear layer in the case of parallel wall motion, as well as the appearance of corner vortices for lower Reynolds numbers in the case of anti-parallel motion of the walls. It is found that both the Reynolds number and the direction of the moving walls affect the fluid flow in the cavity.

Dynamic Simulation of a Pilot Scale Vacuum Gas Oil Hydrocracking Unit by the Space‐Time CE/SE Method

AbstractThis work introduces a modified space‐time conservation element/solution element (CE/SE) method for the simulation of the dynamic behavior of a pilot‐scale hydrocracking reactor. With this approach, a four‐lump dynamic model including vacuum gas oil (VGO), middle distillate, naphtha and gas is solved. The proposed method is capable of handling the stiffness of the partial differential equations resulting from the hydrocracking reactions. To have a better judgment, the model is also solved by the finite difference method (FDM), and the results from both approaches are compared. Initially, the absolute average deviation of the cold dynamic simulation using the CE/SE approach is 8.98 %, which is better than that obtained using the FDM. Then, the stability analysis proves that for achieving an appropriate response from the dynamic model, the Courant number, which is a function of the time step size, mesh size and volume flow rate through the catalytic bed, should be less than 1. Finally, it is found that, following a careful selection of these parameters, the CE/SE solutions to the hydrocracking model can produce higher accuracy than the FDM results.

Development of a Self-Consistent Truly Multiphysics Algorithm Based Upon the Courant-Insensitive Space–Time Conservation-Element Solution-Element Method

This paper reports on the theoretical aspects and current development status of a self-consistent truly multiphysics algorithm. The algorithm is based upon the Courant-insensitive space-time conservation-element solution-element methodology. Previous attempts for electromagnetic solutions have applicability only in constant material domains with PEC boundary conditions. This paper reports on the extension of this algorithm for the solution of the generalized Maxwell equations, including linear-dispersive materials. The numerical solution is shown to be extremely accurate on highly nonuniform meshes and reduces to the classical Yee FDTD error properties in the uniform Cartesian grid limit. Validation problems and comparison with the ubiquitous baseline FDTD algorithm will be presented in 1-D (2-D space-time). Results show that the second-order CESE method has an accuracy equivalent to fourth-sixth order FDTD for equal grids with highly discontinuous coefficients (e.g., permittivity).

A Compact High Order Space-Time Method for Conservation Laws

AbstractThis paper presents a novel high-order space-time method for hyperbolic conservation laws. Two important concepts, the staggered space-time mesh of the space-time conservation element/solution element (CE/SE) method and the local discontinuous basis functions of the space-time discontinuous Galerkin (DG) finite element method, are the two key ingredients of the new scheme. The staggered space-time mesh is constructed using the cell-vertex structure of the underlying spatial mesh. The universal definitions of CEs and SEs are independent of the underlying spatial mesh and thus suitable for arbitrarily unstructured meshes. The solution within each physical time step is updated alternately at the cell level and the vertex level. For this solution updating strategy and the DG ingredient, the new scheme here is termed as the discontinuous Galerkin cell-vertex scheme (DG-CVS). The high order of accuracy is achieved by employing high-order Taylor polynomials as the basis functions inside each SE. The present DG-CVS exhibits many advantageous features such as Riemann-solver-free, high-order accuracy, point-implicitness, compactness, and ease of handling boundary conditions. Several numerical tests including the scalar advection equations and compressible Euler equations will demonstrate the performance of the new method.

The application of CE/SE scheme in the simulation of 2-dimension expansion of ablation plasma jet in open air

The 2-dimensional unsteady expansion process of plasma jet ablated in capillary because of the discharging of pulse power is simulated with CE/SE (conservation element / solution element) scheme in this paper. The high resolution computational space-time CE/SE scheme is simplified and imp roved to solve the axisymmetric two dimensional Euler equations of the unsteady expansion process of ablation plasma jet in open air. The variation and distribution of pressure and velocity of ablation plasma jet in flow field are illustrated and analysed . The shock wave and discontinuity of the jet are clearly depicted according to results of simulation, and simulating results are well agreement with experiment.

Hypersonic viscous flow over large roughness elements

Viscous flow over discrete or distributed surface roughness has great implications for hypersonic flight due to aerothermodynamic considerations related to laminar–turbulent transition. Current prediction capability is greatly hampered by the limited knowledge base for such flows. To help fill that gap, numerical computations are used to investigate the intricate flow physics involved. An unstructured mesh, compressible Navier–Stokes code based on the space–time conservation element, solution element (CESE) method is used to perform time-accurate Navier–Stokes calculations for two roughness shapes investigated in wind tunnel experiments at NASA Langley Research Center. It was found through 2D parametric study that at subcritical Reynolds numbers, spontaneous absolute instability accompanying by sustained vortex shedding downstream of the roughness is likely to take place at subsonic free-stream conditions. On the other hand, convective instability may be the dominant mechanism for supersonic boundary layers. Three-dimensional calculations for both a rectangular and a cylindrical roughness element at post-shock Mach numbers of 4.1 and 6.5 also confirm that no self-sustained vortex generation from the top face of the roughness is observed, despite the presence of flow unsteadiness for the smaller post-shock Mach number case.

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.

The Direct Aero-Acoustics Simulation of Flow around a Square Cylinder Using the CE/SE Scheme

The space-time conservation element solution element (CE/SE) method is applied in the classical around square cylinder flow problem for aeroacoustics simulation. The unsteady Navier-Stokes equation and Euler equation are solved to directly numerical simulate the aero-acoustics field aroused in unsteady turbulent flow. The computation domain is decomposed into sound-near field and sound-far field, they are computed and analyzed. Large eddy simulation (LES) is used to compute the acoustic source while the Ffowcs Williams-Hawkings (FW-H) acoustic analogy is applied to this problem also. Comparing the two methods, qualitative agreement is demonstrated using the frequency-spectrum of a monitor point located at (50D,0). The CE/SE method is able to extract more details about the non-linear relation between the flow and sound.

Computer-aided model analysis for ionic strength-dependent effective charge of protein in ion-exchange chromatography

A packed-bed chromatographic model developed in this study includes adsorption isotherms considering steric mass action (SMA) as well as non-linearity in liquid/solid phases, adsorption kinetics and mass transport. For solving the nonlinear and nonequilibrium adsorption model described by a partial differential algebraic equation (PDAE) system, a fast and accurate numerical method (i.e., conservation element/solution element (CE/SE) method), is proposed. Sensitivity and elasticity of the model parameters (e.g., steric/shape factors, adsorption heat coefficient, effective protein charge, equilibrium constant, mass transfer coefficient, axial dispersion coefficient and bed voidage) are analyzed for a BSA–salt system in a low protein concentration range. Within a low concentration range of bovine serum albumin (BSA) where linear adsorption isotherms are shown, the adsorption heat coefficient, shape and steric factors have little effect on adsorption isotherms and the retention time. However, the effective protein charge which is related to pH and ionic strength affects the retention time significantly even if the change is slight. The mass transfer coefficient (i.e., adsorption rate coefficient) has an effect on the peak height of the elution curve at the same retention time. Sensitivity of the retention time to the bed voidage is relatively large, since the bed voidage changes the fluid velocity of the mobile phase. Comparing the proposed model with experimental elution curves obtained at different salt concentrations, it is proposed that the effective protein charge could depend upon the salt concentration (or ionic strength). The reason for this dependence may be a steric hindrance of protein binding sites combined with a salt shielding effect neutralizing the surface charges of the protein.