Space-time Conservation Element Research Articles

Interior ballistic two-phase flow model and its calculation for a mixed charge structure

The mixed charge for guns generally consists of two or more kinds of propellants. Medium and large caliber guns usually adopt the typical combination of tubular and granular propellants, not only meeting interior ballistic indexes but also improving launch safety. However, numerical investigations, charge design, and optimizations of such charge structures rely heavily on classical interior ballistic theory. To gain insight into some crucial phenomena in this complex interior ballistic process, a two-phase flow dynamic model of a 125 mm smoothbore gun with mixed charge (granular propellants, tubular propellants, and a combustible cartridge case) is developed. Unlike the previous form of solid governing equations, the conservation equations, surface temperature and combustion law functions are simultaneously established. The space-time conservation element and solution element (CE/SE) method is used to solve the mathematical model, simplifying the calculations of partial derivatives in source terms. Results show that the model can describe the combustion and flow, and the predicted projectile velocity agrees well with the measured one. In addition, the effects of the mass ratio of mixed propellants on interior ballistic performance are discussed in detail. As the tubular propellant mass decreases, ignition consistency deteriorates, particle accumulation intensifies, and the maximum pressure exceeds the permitted one, all of which may affect launch safety. The work in this paper provides a powerful tool for observing and predicting the interior ballistic process of the mixed charge.

On the irregular jet formation of shock-accelerated spherical heavy gas bubbles

The behaviors of shock-accelerated heavy gas bubbles are numerically investigated, focusing on clarifying the forming mechanisms of the bubble jets in different types. The present study categorizes heavy bubble jets into two types, regular jets, and irregular jets. The present shock-accelerated multi-component flows are simulated by solving inviscid compressible Euler equations. An upwind characteristic space-time conservation element solution element scheme is adopted, and a five-equation model is used to treat the gas interface. Bubbles of R22, SF6, and Kr in ambient N2 and air are examined, and the incident shock Mach numbers are 1.1 and 1.23. The numerical results demonstrate that the bubble jet formation and its shape are very sensitive to the test gas species and incident shock strength. It is found that the tiny upstream jet formed in the single-shocked SF6/air scenario results from a very small Mach stem impingement onto the bubble upstream interface, the type II shock-shock interaction features the flow mechanism. While the large upstream jet formed in the re-shock SF6/air scenario is a combined result of the re-shock convergence and later vortex stretching. For the complex Kr/air scenario, the upstream jet results from the vorticity-induced inward jet stretching, and the downstream hollow jet results from the slip line guided tip extension. The measurements of bubble volumes, gas mixings, and material line lengths suggest that, although the jet formation greatly changes the bubble morphology, it makes a minor contribution to the bubble overall integral properties.

Numerical simulation of the influence of confined multi-point ignited H2–O2 mixture on the propagation of shock waves towards a deformable plate

The study of shock wave propagation in a detonation chamber is of great importance as a part of the plate forming process. Investigations related to the effects of premixed gas detonation on the deflection of a plate require in-depth examination. An Eulerian-Lagrangian numerical simulation is conducted using the space-time conservation element and solution element method of LS-DYNA software to study the effect of confined multi-point ignited gaseous mixture on the dynamic response of thin plates clamped at the end of a combustion chamber. The FSI couples a Lagrangian finite element solver with a Eulerian fluid solver in a 2D space with detailed chemistry of H2–O2 mixture. The solution contains the detonation wave propagation through the combustion chamber and its interaction with the plate. The influence of variation in the multi-point ignition locations and combustion chamber dimensions on the pressure history and plate deflection is studied. To verify the model, a comparison with the experimental study is carried out using an adjustable model representative of the real experiment. The verified model is used to link the evolution of plate shape with the arrival time and intensity of shock waves within the chamber. It is found that a longer distance between the ignition point and the plate intensifies the ultimate deflection of the plate. In addition, a fairly large combustion area employed in a direction rather than transverse to the plate surface is unable to influence the ultimate deformation of the plate.

Shock characteristics evolution of detonation waves forward impacting on the solid wall

The forward reflection of detonation waves on the solid wall will lead to a high pressure rise. The research systematically introduced the theoretical, numerical, and experimental exploration on the shock propagation characteristics of detonation waves forward impacting on a solid wall in the present work. The one-dimensional shock theory was carried out to solve the pressure rise ratio in this process. The exact solution and its variation law of a positive increase with filling pressure were expressed. One-dimensional simulations based on the space-time conservation element and solution element method were utilized to reveal the pressure decrease and velocity increase laws for the reflected shock wave. The blockage, oscillation, and attenuation phenomena of detonation waves and reflected shock waves under the effect of the tube–wall reflection were demonstrated in two-dimensional works. Experimental results from the detonation tube pressure test system showed a larger amplitude and duration of the reflected shock wave than the detonation wave. Pressure evolution and the formation of pressure plateaus were consistent with the simulation results. In addition, the time required for the pressure plateaus to decay to 0.5 times the Chapman-Jouget (C–J) detonation pressure is relatively constant under different filling conditions.

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.

Compressional wave propagation in saturated porous media and its numerical analysis using a space-time conservation element and solution element method.

Compressional waves in saturated porous media are relevant to many fields from oil exploration to diagnostic of human cancellous bone and can be used to interpret physical behaviors of materials. In this work, based on Biot's theory in the low frequency range, a key finding is that there exists a critical frequency of Biot's theory in the low frequency range, which determines the coincidence of the properties of Biot waves of the first and second kinds. Furthermore, we have investigated the dispersion and attenuation of the coalescence of the first and second compressional waves in the low frequency range. The coalescence of the first and second waves is strongly attenuated with a moderate phase velocity and shows the in-phase feature. In addition, acoustic wave propagation has been calculated numerically using the space-time conservation element and solution element (CESE) method. The CESE-simulated results are compared to the experimental data and to those of the classical transfer function approach. We show that the CESEscheme preserves the local and global flux conservations in the solution procedure of Biot's theory. It is found that the CESEmethod provides more accurate predictions of high dispersion and strong attenuation of compressional waves in the low frequency range and is well suitable for predicting compressional wave fields in saturated porous media.

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.

Numerical analysis of heat flow in wall of detonation tube during pulse detonation cycle

The thermal ablation of the detonation tube is a key factor in pulse detonation engine (PDE) research. In this study, a symmetric two-phase detonation model and a cylinder heat conduction model are used for studying the features of transient heat transfer. The effects of the tube material and size on the heat transfer are analyzed using this model. To capture the shock wave simply and accurately, the space–time conservation element and solution element method and the finite-difference method are used for simulating the interior flow and the heat-transfer process of the detonation tube, respectively. The numerical results are consistent with the experimental results. The temperature of the PDE wall increases by 1.5 K per four detonation cycles at a frequency of 20 Hz. Then, the effects of the materials of the tube wall and the diameter and length of the tube are investigated. The inner flow greatly affects the inner-wall temperature, particularly the influence is obvious when the tube materials change. Increasing the diameter can improve the heat dissipation, and reducing the length smooths the temperature gradient along the axis. The results of this study provide guidance for the design of PDEs.

Numerical simulation of transient heat conduction in a multilayer material by the conservation elements/solution elements (CE/SE) method

ABSTRACT The space-time Conservation Element Solution Element (CE/SE) method is used to solve the 1D unsteady equation of thermal diffusion in a multilayer material, with application to thermal batteries. A conservative form of the equation is derived and solved using the CE/SE method. At the interface between the layers, local thermodynamics properties are discontinuous. Their spatial derivatives are calculated using central differences scheme. First, a material with two layers is simulated. The results show a very good agreement with analytical and COMSOL Multiphysicső calculations, and the Fourier Number Insensitivity property is validated, even for values beyond the 0.5 stability criterium. In a second test, a material stack representative of a thermal battery is considered. Obtained results are also very close to the COMSOL simulations.

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.

A space time conservation element and solution element method for solving two-species chemotaxis model

This work is related to the numerical investigation of two species chemotaxis models in both one and two dimensions. This system in its simpler form is a set of non-linear partial differential equations. First equation represents the dynamics of cell densities and the others are responsible for chemoattractant concentration. The system is known to produce delta-type singularities in finite time. Therefore, a less order accurate numerical schemes are not capable of handling such a complicated spiky solution. A Space Time Conservation Element and Solution Element (CE/SE) method is proposed for solving such systems. This method has distinct attributes which includes treatment of space and time variables in unified fashion. Furthermore, numerical diffusion is reduced on the basis that both conserved quantities and derivatives are anonymous. Due to this feature, diffusion in numerical schemes is inherently reduced. The Nessyahu–Tadmor (NT) central scheme is also implemented for validation and comparison. Several one and two dimensional case studies are carried out. Numerical results obtained through proposed numerical method are analyzed in comparison to Nessyahu–Tadmor (NT) central scheme. Moreover, effects of the different densities and concentration functions are explored to see generic applicability of the proposed method for current system of equations. It is observed that the CE/SE and NT central method has the capability to capture the delta type singularities in the solution profile. However, CE/SE resolves the solution peaks better as compare to NT central scheme.

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.

Space-time CE/SE method for solving repulsive chemotaxis model

This work is concerned with the numerical investigation of nonlinear mathematical systems that describe repulsive chemotaxis phenomena in biology. A space-time conservation element (CE) and solution element (SE) method-based splitting procedure is proposed in one space dimension. In chemotaxis, the orientation of the cells changes. They move to the location which are chemically more favorable or move away from repellents or toxin. Mathematically, these are the systems of nonlinear coupled partial differential equations. The analytical solutions for this set of equations is not possible. A traditional less order accurate solution may fail to describe the underlying phenomena. Therefore, the state of the art numerical procedures is the need for such systems to get physically reliable solution in acceptable computational cost. Unlike tradition numerical procedures, the CE/SE-method has distinct attributes like it treats time and space in a unified fashion. Both conserved quantities and corresponding derivatives are considered to be unknowns due to which inherited numerical diffusion is reduced. Several benchmark numerical test problems are simulated for validation of the scheme. The numerical solutions of case studies are obtained for one space dimension. Moreover, one more central numerical scheme introduced by Nessyahu–Tadmor (NT) is also adapted on staggard grids which is considered to be a black box solver for such models for comparison. It was observed that both schemes perform well for the current mathematical model. However, the CE/SE scheme is more capable of capturing the peaks produced in the solution profile. Further, convergence study is also carried out from both schemes which reveal that the proposed method is fast as compared to NT central scheme.

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.

Numerical analysis of drop coalescence in cavity flow using CESE method

On the basis of the space-time conservation element and solution element method and by applying the hybrid particle level-set function, the drop coalescence in a dual-layer vertical channel has been numerically studied. Interactions between the flow field and the drop migration are observed. Main eddies and secondary vortexes induced by the drop coalescence are examined. Deformation and migration of the drop are mainly controlled through interfacial tension and the shear stress gradient. Drop coalescence is related to a viscous dissipation of energy.

Interfacial instability at a heavy/light interface induced by rarefaction waves

The interaction of rarefaction waves and a heavy/light interface is investigated using numerical simulations by solving the compressible Euler equations. An upwind space–time conservation element and solution element (CE/SE) scheme with second-order accuracy in both space and time is adopted. Rarefaction waves are generated by simulating the shock-tube problem. In this work, the SF6/air interface evolution under different conditions is considered. First, the gas physical parameters before and after the rarefaction waves impact the interface are calculated using one-dimensional gas dynamics theory. Then, the interaction between the rarefaction waves and a single-mode perturbation interface is investigated, and both the interface evolution and the wave patterns are obtained. Afterwards, the amplitude growth of the interface over time is compared between cases, considering the effects of the interaction period and the strength of the rarefaction waves. During the interaction of the rarefaction waves with the interface, the Rayleigh–Taylor instability induced by the rarefaction waves is well predicted by modifying the nonlinear model proposed by Zhang & Guo (J. Fluid Mech., vol. 786, 2016, pp. 47–61), considering the variable acceleration. After the rarefaction waves leave the interface, the equivalent Richtmyer–Meshkov instability is well depicted by the nonlinear model proposed by Zhang et al. (Phys. Rev. Lett., vol. 121(17), 2018, 174502), considering the growth rate transition from Rayleigh–Taylor instability to Richtmyer–Meshkov instability. The differences in the heavy/light interface amplitude growth under the rarefaction wave condition and the shock wave condition are compared. The interface perturbation is shown to be more unstable under rarefaction waves than under a shock wave.

The space-time conservation element and solution element scheme for simulating two-phase flow in pipes

In this article, the space-time conservation element and solution element scheme is extended to simulate the unsteady compressible two-phase flow in pipes. The model is non-conservative and the governing equations consist of three equations, namely, two mass conservation equations for each phase and one mixture-momentum equation. In the third equation, the non-conservative source term appears, which describes the sum of gravitational and frictional forces. The presence of source term and two mass conservation equations in considered model offers difficulties in developing the accurate and robust numerical techniques. The suggested space-time conservation element and solution element numerical scheme resolves the volume-contact discontinuities efficiently. Furthermore, the modified central upwind scheme is also extended to solve the same two-phase flow model. The number of test problems is considered, and the results obtained by space-time conservation element and solution element scheme are compared with the solutions of modified central upwind scheme. The numerical results show better performance of the space-time conservation element and solution element method as compare to the modified central upwind scheme.

A rezoning-free CESE scheme for solving the compressible Euler equations on moving unstructured meshes

We construct a space-time conservation element and solution element (CESE) scheme for solving the compressible Euler equations on moving meshes (CESE-MM) which allow an arbitrary motion for each of the mesh points. The scheme is a direct extension of a purely Eulerian CESE scheme that was previously implemented on hybrid unstructured meshes (Shen et al. (2015) [43]). It adopts a staggered mesh in space and time such that the physical variables are continuous across the interfaces of the adjacent space-time control volumes and, therefore, a Riemann solver is not required to calculate interface fluxes or the node velocities. Moreover, the staggered mesh can significantly alleviate mesh tangles so that the time step can be kept at an acceptable level without using any rezoning operation. The discretization of the integral space-time conservation law is completely based on the physical space-time control volume, thereby satisfying the physical and geometrical conservation laws. Plenty of numerical examples are carried out to validate the accuracy and robustness of the CESE-MM scheme.

A review of Space-time conservation element solution element (CESE) schemes

The space-time conservation element and solution element (CESE) method is one of the recent advancements in CFD. This paper attempts to review various schemes along with their limitations starting with a non-dissipative a -scheme. The following a- ε scheme has an added numerical dissipation term. This numerical dissipation is capable of damping out numerical instabilities that arise only from the smooth region of the solution but fails to suppress numerical wiggles. The a-ε-α-β scheme is augmented with the ability to suppress the wiggles introduced due to discontinuities, using another added term to the solution. Hence a-ε-α-β scheme is capable of capturing both small disturbances and sharp discontinuities simultaneously. Its advantage over a-ε scheme as well as the suppression of numerical wiggle at the discontinuity will be demonstrated using Sod’s shock tube problem. All of these CESE schemes described so far become excessively diffusive at low Courant number and Mach number. Hence the construction of a Courant number insensitive scheme which has a weighted average sense. Its advantage over a-ε-α-β in terms of stability will be explained further in the paper using Sod’s shock tube problem as an example and its ability to resolve contact discontinuities without any ad-hoc parameter over CFL number ranging from one to close to 0.001