Conservation Element And Solution Element Method Research Articles

Numerical Study of the Effects of Injection Conditions on Rotating Detonation Engine Propulsive Performance

A three-dimensional upwind conservation element and solution element method (CESE) in cylindrical coordinates is first developed to effectively solve the unsteady reactive Euler equations governing a hydrogen–air rotating detonation engine (RDE) with coaxial structures. The effects of the annular width on the structure of the detonation front and the relationship between the thrust and mass flow rate are then investigated. Additionally, RDEs with various injection conditions are systematically analyzed regarding flow patterns and propulsion performance. The results reveal a positive correlation between the specific impulse and the area ratio of the injection slot to the head-end wall. Nevertheless, the specific impulse shows minimal dependence on the injector slot’s location when the area ratio is constant. Ultimately, it is concluded that the area ratio between the injector and the head-end wall is critical in determining the loss of specific impulse.

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.

Numerical modeling of laboratory-scale asteroid impact based on elastoplastic flow model and CESE method

Asteroid impacts are destructive and low-probability threats to the Earth. The numerical simulation is considered an applicable analysis tool in asteroid deflection programs. As a novel shock-capturing strategy, the space–time conservation element and solution element (CESE) method can reliably predict shock waves and mechanical behaviors under high pressure and large strain conditions. In this paper, based on an elastoplastic flow model and an updated CESE scheme, the laboratory-scale iron asteroid impacts are modeled numerically, and the multi-material boundary treatment and the interface tracing strategy are introduced. Under hypervelocity impacts of the projectile to the iron asteroid target, the construction and realization of morphologies of impact craters and the implantation of projectile material into the target are numerically calculated. Numerical results show that the crater diameter and depth increase with increasing impact velocity and with increasing temperature, which softens the target. Computational results are compared with experimental observations available in the open literature, and good agreement is found. Therefore, the CESE method is successfully extended for capturing the key features of laboratory-scale hypervelocity asteroid impacts.

Investigation of the confinement effects on the blast wave propagated from gas mixture detonation utilizing the CESE method with finite rate chemistry model

ABSTRACT In this paper, we investigate the effect of confinement on the explosion characteristic of a certain volume of a gas mixture in an environment. Instead of using the equivalent weight of TNT, we use a computational fluid dynamics method with a reduced chemical kinetic mechanism to simulate the gas mixture detonation. In this method, reacting flow equations including realistic finite-rate chemistry models are solved by employing the space-time conservation element and solution element (CESE). To evaluate the accuracy of the kinetic mechanism and the validity of the simulation of the volumetric explosions, including the blast wave propagation and its interaction with the walls, the explosion of a certain volume of stoichiometric propane/oxygen mixture in a semi-confined volume is investigated and the numerical results are compared with an experimental data. The excellent agreement between the numerical results and the experimental data indicates that it is possible to predict the behavior of the blast wave generated by the detonation of a gas mixture in a relatively complex environment accurately by using appropriate chemical kinetics. Therefore, we have used this method to investigate the effects of confinement on the blast wave propagation caused by gas detonation in more detail.

Space–Time Conservation Element and Solution Element Method and Its Applications

This paper reviews the development of the space–time conservation element and solution element (CESE) method and summarizes its applications in various research areas. The CESE method is a special finite-volume-type method that provides an alternative approach to numerical solutions of fluid-dynamic equations and conservation laws in various physical systems. Based on a unified treatment of time and space, this method solves the integral form of the governing equations by discretization of the space–time domain. Recent progress in CESE schemes mainly includes the construction of a family of upwind CESE schemes, the extended definitions of conservation elements and solution elements for arbitrary meshes, and a new approach to developing high-order CESE schemes. Selected applications of the CESE method (including high-speed aerodynamics, multifluid flows, detonations, and aeroacoustics) are presented. Features of the CESE method are described.

Modeling of the Hydrocracking Reactor by the CESE Method

In this article, the improved space-time conservation element and solution element (CESE) method are used to simulate the dynamic treatment of the hydrocracking reactor. The dynamic model consists of four lumps: vacuum gas oil (VGO), middle distillate, naphtha, and gas which is dissolved by this method. The offered method can solve the partial differential equations caused by the reactions inside the hydrocracking reactor. In this study, both temperature and mole fraction variables are solved explicitly and simultaneously. In the CESE method, to obtain a suitable answer to the dynamic model, a CFL insensitive scheme was used which, for the CESE method to be stable, the CFL number should be less than 1. In this work, obtained results from the CESE method, in good agreement with the data industry. Outcomes illustrate that AAD% of the yield forecast for the middle distillate, naphtha, and gas are 3.33%, 2.56 %, and 6.47%, respectively. This method unlike other contractual numerical methods treats with space and time coordinates Similar.

Numerical simulation of CO2 chemical absorption in a gas-liquid bubble column using the space-time CESE method

Abstract In this study, numerical simulation of the removal process of CO2 using aqueous solutions of piperazine (PZ) in a bubble column has been done. The main focus of this work is to investigate the applicability of the space-time conservation element and solution element (CESE) method for the numerical solution of the coupling of hydrodynamics, mass transfer and chemical reactions. The reactive absorption process has been investigated by means of experiments and numerical simulation. A two-dimensional computational fluid dynamics (CFD) model in the Eulerian framework has been used to describe the gas-liquid system. To obtain the numerical solution, the CESE method based on a uniform rectangular mesh has been implemented by developing a computer code. Experimental study and CFD simulations have been carried out at different conditions of CO2 partial pressure (20 and 36 kPa) and solution concentration (0.1, 0.3 and 0.5 M). The simulation results were in reasonable agreement of 5.226 % and 4.575 % with the experimental values for gas holdup and mass transfer flux, respectively. The comparisons prove that the CESE method can be considered as a proper numerical method for the simulation of complex multiphase flows due to its simplicity, accuracy and low computational costs.

Fluid structure interaction with a finite rate chemistry model for simulation of gaseous detonation metal-forming

Numerical simulation of metal-forming by gaseous detonation in a stoichiometric mixture of H2-O2 is performed in this paper, utilizing the Space-Time Conservation Element and Solution Element (CESE) method. We employ the CESE method to solve the reacting flow equations, including realistic finite-rate chemistry model. The detonation mechanism is considered by a detailed mechanism of the seven species and sixteen reactions for H2-O2 mixture. The fluid-structure interface treatments smoothly handle between detonation wave and work piece by using the immersed boundary method (IBM). To accurately predict the behavior of the work piece, the fluid (CESE) solver was coupled with the LS-DYNA® FEM structural solver.The fluid solver applies the fluid pressure on the structural elements as external boundary conditions and feeds back the displacements and velocity of the interface from the structural solver as its new boundary, while the chemistry solver preparing all source terms in the conservation equation for fluid solver.Our purpose is to evaluate the accuracy of CESE-IBM FSI method to handle the gaseous detonation metal-forming, which is a very complex FSI problem with chemistry. After a description of CESE-IBM FSI method, finite-rate chemistry model and experimental test set-up, a comparison is performed through the detonation characteristics, midpoint deflection of the work piece, and effect of initial pressure and temperature of the gas mixture. A good agreement is obtained between numerical results and empirical data.

An upwind CESE scheme for 2D and 3D MHD numerical simulation in general curvilinear coordinates

Shen et al. [1,2] proposed an upwind space-time conservation element and solution element (CESE) scheme for 1D and 2D hydrodynamics (HD) in rectangular coordinates, which combined the advantages of CESE and upwind scheme, namely, guaranteed strictly the space–time conservation law as well as captured discontinuities very efficiently. All kinds of upwind schemes can be combined very flexibly for different problems to achieve the perfect combination of CESE and finite volume method (FVM). However, in many physical applications, we need to consider geometries that are more sophisticated. Hence, the main objective of this paper is to extend the upwind CESE scheme to multidimensional magneto-hydrodynamics (MHD) in general curvilinear coordinates by transforming the MHD equations from the physical domain (general curvilinear coordinates) to the computational domain (rectangular coordinates) and the new equations in the computational domain can be still written in the conservation form. For the 3D case, the derivations of some formulas are much more abstract and complex in a 4D Euclidean hyperspace, and some technical problems need to be solved in the debugging process. Unlike in HD, keeping the magnetic field divergence-free for MHD problems is also a challenge especially in general curvilinear coordinates. These are the main obstacles we have overcome in this study. The test results of benchmarks demonstrate that we have successfully extended the upwind CESE scheme to general curvilinear coordinates for both 2D and 3D MHD problems.

A high-order CESE scheme with a new divergence-free method for MHD numerical simulation

In this paper, we give a high-order space–time conservation element and solution element (CESE) method with a most compact stencil for magneto-hydrodynamics (MHD) equations. This is the first study to extend the second-order CESE scheme to a high order for MHD equations. In the CESE method, the conservative variables and their spatial derivatives are regarded as the independent marching quantities, making the CESE method significantly different from the finite difference method (FDM) and finite volume method (FVM). To utilize the characteristics of the CESE method to the maximum extent possible, our proposed method based on the least-squares method fundamentally keeps the magnetic field divergence-free. The results of some test examples indicate that this new method is very efficient.

Modeling and numerical simulation of afterburning of thermobaric explosives in a closed chamber

An Eulerian-Lagrangian simulation of the thermobaric explosion and the propagating detonation waves with aluminum particle combustion in a closed chamber is carried out and validated very favorably with the pressure histories measured experimentally. A novel numerical framework based on the space-time conservation-element and solution-element method (CE/SE) was employed to simulate the complex detonating flow with detailed chemistry. A discrete particle technique was adopted with either a mono-dispersed or poly-dispersed particle distribution. The results between the experiment and simulation on pressure histories and corresponding impulse forces are compared and showed good agreement. In addition, the effects of metal combustion on afterburning were discussed in detail in terms of the reaction mechanisms and particle dynamics.

An Eulerian model for nonlinear waves in elastic rods, solved numerically by the CESE method

In this paper, we use a comprehensive perturbation theory to develop an Eulerian model for nonlinear longitudinal waves in tapered elastic rods. The equations of motion, when cast in their Eulerian form, are ideal for fixed-grid numerical techniques such as finite-difference methods, finite-volume methods, and modern high-resolution shock-capturing schemes. The leading-order equations in our perturbation formalism are verified using a control-volume analysis, then linearized to recover a classical model for longitudinal waves in ultrasonic horns. We assess the mathematical structure of our leading-order model, deduce the eigenvalues and eigenvectors of the Jacobian matrix, and verify its hyperbolicity. Solutions are obtained using the space-time conservation element and solution element (CESE) method, a novel numerical technique for first-order hyperbolic systems. In our simulations, the CESE method is shown to effectively capture the evolution of traveling waves such as shocks and rarefactions, as well as salient features of nonlinearity.

An improved CESE method and its application to steady-state coronal structure simulation

This paper presents an improved space-time conservation element and solution element (CESE) method by applying a non-staggered space-time mesh system and simply improving the calculation of flow variables and applies it to magnetohydrodynamics (MHD) equations. The improved CESE method can improve the solution quality even with a large disparity in the Courant number (CFL) when using a fixed global marching time. Moreover, for a small CFL (say < 0.1), the method can significantly reduce the numerical dissipation and retain the solution quality, which are verified by two benchmark problems. And meanwhile, comparison with the original CESE scheme shows better resolution of the improved scheme results. Finally, we demonstrate its validation through the application of this method in three-dimensional coronal dynamical structure with dipole magnetic fields and measured solar surface magnetic fields as the initial input.

Numerical Analysis of Hypervelocity Impact Problems with Large Deformations, High Strain Rates and Spall Fractures

In the study, a software program named “SUPER CE/SE” is developed for the simulation of hypervelocity impact problems with large deformations, high strain rates and spall fractures. In the software program, an Eulerian method consisting of an improved CE/SE (Space-time Conservation Element and Solution Element Method) scheme is used. A void growth model which takes the Bauschinger Effect (BE) into account and a newly proposed front tracking method are adopted in the simulation. The formation and propagation of a crack is described by a newly developed automatic crack growth algorithm. Numerical simulation of spall fracture in a plate when impacted by a spherical projectile at a velocity of 6.0 km/s is carried out. The numerical results are in qualitative agreement with the corresponding experimental data. It turns out that the BE has obvious influence on the length of the crack and better agreement with the experiment is obtained when the BE is considered. It is also validated that the newly proposed front tracking method is feasible and reliable for representing the cracks in the problems with large deformation and high strain rates. According to those research results, it is proved that the software program SUPER CE/SE is robust and effective in the simulation of hypervelocity impact problems.

A New Hybrid Numerical Scheme for Two-Dimensional Ideal MHD Equations

We present a new hybrid numerical scheme for two-dimensional (2D) ideal magnetohydrodynamic (MHD) equations. A simple conservation element and solution element (CESE) method is used to calculate the flow variables, and the unknown first-order spatial derivatives involved in the CESE method are computed with a finite volume scheme that uses the solution of the derivative Riemann problem with limited reconstruction to evaluate the numerical flux at cell interface position. To show the validation and capacity of its application to 2D MHD problems, we study several benchmark problems. Numerical results verify that the hybrid scheme not only performs well, but also can retain the solution quality even if the Courant number ranges from close to 1 to less than 0.01.

The space–time CESE method for solving special relativistic hydrodynamic equations

The special relativistic hydrodynamic equations are more complicated than the classical ones due to the nonlinear and implicit relations that exist between conservative and primitive variables. In this article, a space–time conservation element and solution element (CESE) method is proposed for solving these equations in one and two space dimensions. The CESE method has capability to capture sharp propagating wavefront of the relativistic fluids without excessive numerical diffusion or spurious oscillations. In contrast to the existing upwind finite volume schemes, the Riemann solver and reconstruction procedure are not the building blocks of the suggested method. The method differs from previous techniques because of global and local flux conservation in a space–time domain without resorting to interpolation or extrapolation. The scheme is efficient, robust, and gives results comparable to those obtained with more sophisticated algorithms, even in highly relativistic two-dimensional test problems.

Analysis of Momentum Transfer in a Lid-Driven Cavity Containing a Brinkman–Forchheimer Medium

The CE/SE (the space-time conservation element and solution element method) scheme with the second-order accuracy has been proposed. And the pretreatment method has been introduced to convert the parabolic equations to the hyperbolic equations, which are accurately solved by the CE/SE method. The lid-driven rectangular cavity containing a porous Brinkman–Forchheimer medium is studied in this numerical investigation. The Brinkman–Forchheimer equation is used such that both the inertial and viscous effects are incorporated. The governing equations are solved by the improved CE/SE approach. The characteristics of the flow are analyzed with emphasis on the influence of the Darcy number and the cavity depth. It is found that the porous medium effect decreases both the strength and the number of eddies, especially for deep cavities.

Solving the Euler and Navier–Stokes equations by the AMR–CESE method

The application of the AMR–CESE method for solving the Euler and Navier–Stokes equations is presented. The method is a combination of the space–time conservation element and solution element (CESE) method and the adaptive mesh refinement (AMR) technique. Its implementation is based on modification of the original CESE method and utilization of the framework of a parallel-AMR package PARAMESH to manage the block-AMR grid system. Furthermore, a variable time step algorithm is introduced to realize adaptivity of the method in both space and time. A test suite of standard problems for Euler and Navier–Stokes flows are calculated and the results show high resolution, high efficiency and versatility of the method for both shock capture and resolving boundary layer.

Numerical Model of Natural River Flow by Unstructure Space-Time SE-CE Method

A new horizontal two-dimensional mathematic scheme namely space-time Conservation Element and Solution Element (CE-SE) is introduced in this paper. The CE-SE method has some features which space and time are unified and treated on the same footing, both local and global flux are enforced conservation. The proposed model is applied to solve the 2-D shallow water equations in the triangular mesh, and tested by using the field data in Yangzi River. It shows that the hydraulics characters in natural rivers which are with complex boundaries and topography can be well simulated by using this new method.

Velocity-Stress Equations for Waves in Solids of Hexagonal Symmetry Solved by the Space-Time CESE Method

This paper reports an extension of the space-time conservation element and solution element (CESE) method to simulate stress waves in elastic solids of hexagonal symmetry. The governing equations include the equation of motion and the constitutive equation of elasticity. With velocity and stress components as the unknowns, the governing equations are a set of 9, first-order, hyperbolic partial differential equations. To assess numerical accuracy of the results, the characteristic form of the equations is derived. Moreover, without using the assumed plane wave solution, the one-dimensional equations are shown to be equivalent to the Christoffel equations. The CESE method is employed to solve an integral form of the governing equations. Space-time flux conservation over conservation elements (CEs) is imposed. The integration is aided by the prescribed discretization of the unknowns in each solution element (SE), which in general does not coincide with a CE. To demonstrate this approach, numerical results in the present paper include one-dimensional expansion waves in a suddenly stopped rod, two-dimensional wave expansion from a point in a plane, and waves interacting with interfaces separating hexagonal solids with different orientations. All results show salient features of wave propagation in hexagonal solids and the results compared well with the available analytical solutions.