Space-time Conservation Element Research Articles

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.

Application of CE/SE method to gas-particle two-phase detonations under an Eulerian-Lagrangian framework

This study aims to extend the original Eulerian space-time conservation element and solution element (CE/SE) method to the Eulerian-Lagrangian framework to solve the gas-particle two-phase detonation problems. The gas-aluminum particle two-phase detonations are numerically investigated by the developed Eulerian-Lagrangian code, in which the gas-phase compressible Euler equations are solved by our in-house CE/SE scheme based on quadrilateral meshes. Additionally, the particle-phase Lagrangian equations, together with the stiff source terms of interphase interactions and chemical reactions, are explicitly integrated via the operator-splitting technique. A dynamic data structure is introduced to store particle information to overcome the tremendous communication costs when applying message passing interface parallel to the Eulerian-Lagrangian framework. The code is shown to be of better parallel efficiency in moderate-scale computations than that uses static arrays. Comparisons with previous one-dimensional and two-dimensional simulation results and experimental observations are conducted to demonstrate the accuracy and reliability of the developed Eulerian-Lagrangian CE/SE code in gas-particle two-phase detonation simulations. Moreover, the code is also applied to simulate polydisperse gas-particle detonations which is close to a realistic scenario, and significant differences in detonation characteristics are found when compared with the monodisperse counterparts. The great demands of using the Eulerian-Lagrangian method to obtain more physics-consistent gas-particle detonation results are addressed, which the traditional Eulerian-Eulerian simulations fail to observe.

Multiple structure in oxy-hydrogen detonation propagation in rectangular tubes with an improved CE/SE scheme

An improved space-time conservation element and solution element (CE/SE) scheme is used to simulate propagation process of 3-D oxy-hydrogen detonation in rectangular tubes. Multiple detonation structure based on elementary cells of the diagonal, rectangular and spinning mode in square tubes are obtained by numerical cases with different aspect ratios of cross-section. By means of visualizing its detailed structure, the formation mechanism of 3-D multiple cellular patterns is preliminarily discussed. As the cross-sectional aspect ratio increases, new multiple cellular patterns are observed.

Numerical study on the jet formation of simple-geometry heavy gas inhomogeneities

The jet formation of gas inhomogeneities under the impact of a planar shock wave is an interesting phenomenon that is closely related to shock convergence. In this study, a series of heavy gas inhomogeneities with very simple geometries (square, rectangle, circle, and triangle) are numerically reproduced to trace the source of the jet formation. Special attention is given to the wave patterns that lead to the formation of specific jet forms. The shock-accelerated multicomponent flow is simulated by solving inviscid compressible Euler equations. An up-wind characteristic space-time conservation element and solution element scheme is adopted, and a five-equation model is used to treat the gas interface. The jet types that emerge in the experimental images are explained based on the numerical results, and a typical shock pattern that ensures the jet formation is uncovered. It is found that, physically, the jet is initiated by the impact of the internal Mach stem, and the jet growth is nourished by the high speed gas flow induced by this Mach stem. The width of the jet is determined by the height of the internal Mach stem. Geometrically, a focal wedge enveloped by slip lines emerges in the gas inhomogeneity, in which the gas is accelerated. It is found that the existence of the focal wedge and the coordinates of the wedge tip can be used as qualitative criteria to illustrate the mechanism of the jet formation. These criteria provide a more intuitive basis for understanding the direction, scale, and process of jet formation.

Acoustic characteristics of pulse detonation engine with ellipsoidal reflector**Project supported by the National Natural Science Foundation of China (Grant Nos. 11372141 and 11472138) and the National Defense Pre-Research Foundation of China (Grant No. 61426040201162604002).

Acoustic characteristics of a pulse detonation engine (PDE) with and without an ellipsoidal reflector are numerically and experimentally investigated. A two-dimensional (2D) non-splitting unstructured triangular mesh Euler solver based on the space-time conservation element and solution element (CE/SE) method is employed to simulate the flow field of a PDE. The numerical results clearly demonstrate the external flow field of the PDE. The effect of an ellipsoidal reflector on the flow field characteristic near the PDE exit is investigated. The formation process of reflected shock wave and reflected jet shock are reported in detail. An acoustic measurement system is established for the PDE acoustic testing. The experimental results show that the ellipsoidal reflector changes the sound waveform and directivity of PDE sound. The reflected shock wave and reflected jet shock result in two more positive pressure peaks in the sound waveform. The ellipsoidal reflector changes the directivity of PDE sound from 20° to 0°. It is found that the peak sound pressure level (PSPL) and overall sound pressure level (OASPL) each obtain an increment when the PDE is installed with a reflector. The maximum relative increase ratio of PSPL and OASPL are obtained at the focus point F2, whose values are 6.1% and 6.84% respectively. The results of the duration of the PDE sound indicate that the reflecting and focusing wave generated by the reflector result in the increment of A duration and B duration before and near focus point F2. Results show that the ellipsoidal reflector has a great influence on the acoustic characteristic of PDE sound. The research is helpful for understanding the influence of an ellipsoidal reflector on the formation and propagation process of PDE sound.

Positivity-preserving CE/SE schemes for solving the compressible Euler and Navier–Stokes equations on hybrid unstructured meshes

We construct positivity-preserving space–time conservation element and solution element (CE/SE) schemes for solving the compressible Euler and Navier–Stokes equations on hybrid unstructured meshes consisting of triangular and rectangular elements. The schemes use an a posteriori limiter to prevent negative densities and pressures based on the premise of preserving optimal accuracy. The limiter enforces a constraint for spatial derivatives and does not change the conservative property of CE/SE schemes. Several numerical examples suggest that the proposed schemes preserve accuracy for smooth flows and strictly preserve positivity of densities and pressures for the problems involving near vacuum and very strong discontinuities.

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.

The space–time CESE scheme for shallow water equations incorporating variable bottom topography and horizontal temperature gradients

The effects of bottom topography and horizontal temperature gradients on the shallow water flows are theoretically investigated. The considered systems of partial differential equations (PDEs) are non-strictly hyperbolic and non-conservative due to the presence of non-conservative differential terms on the right hand side. The solutions of these model equations are very challenging for a numerical scheme. Thus, our primary goal is to introduce an improved numerical scheme which can handle the non-conservative differential terms efficiently and accurately. In this paper, the space–time conservation element and solution element (CESE) method is extended to approximate these model equations. The proposed scheme has capability to overcome all difficulties posed by this nonlinear system of PDEs. The performance of the scheme is analyzed by considering several case studies of practical interest and the results of suggested scheme are compared with those of central NT scheme. The accuracy of the scheme is verified qualitatively and quantitatively.

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.

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.

An upwind space-time conservation element and solution element scheme for solving dusty gas flow model

An upwind space-time conservation element and solution element (CE/SE) scheme is extended to numerically approximate the dusty gas flow model. Unlike central CE/SE schemes, the current method uses the upwind procedure to derive the numerical fluxes through the inner boundary of conservation elements. These upwind fluxes are utilized to calculate the gradients of flow variables. For comparison and validation, the central upwind scheme is also applied to solve the same dusty gas flow model. The suggested upwind CE/SE scheme resolves the contact discontinuities more effectively and preserves the positivity of flow variables in low density flows. Several case studies are considered and the results of upwind CE/SE are compared with the solutions of central upwind scheme. The numerical results show better performance of the upwind CE/SE method as compared to the central upwind scheme.

A space–time CESE scheme for shallow water magnetohydrodynamics equations with variable bottom topography

The dynamics of thin layer incompressible and electrically conducting fluids is described by shallow water magnetohydrodynamics equations (SWMHD), for which the evolution is nearly two dimensional and the magnetic equilibrium lies in the third direction. In this article, a space–time conservation element and solution element (CESE) scheme is extended for numerical investigation of one- and two-dimension shallow water magnetohydrodynamics (SWMHD) with variable bottom topography. The presence of non-conservative terms and the inclusion of variable bottom topography on the right hand side of SWMHD equations poses some major challenges for any numerical scheme. The proposed CESE scheme differs from the previous techniques because of global and local flux conservation in a space–time domain without restoring to interpolation and extrapolation. To check the validity of the scheme, a number of test problems are considered. The results are also compared with the already existing results in the literature by central-upwind scheme.

Maximum-principle-satisfying space-time conservation element and solution element scheme applied to compressible multifluids

A maximum-principle-satisfying space-time conservation element and solution element (CE/SE) scheme is constructed to solve a reduced five-equation model coupled with the stiffened equation of state for compressible multifluids. We first derive a sufficient condition for CE/SE schemes to satisfy maximum-principle when solving a general conservation law. And then we introduce a slope limiter to ensure the sufficient condition which is applicative for both central and upwind CE/SE schemes. Finally, we implement the upwind maximum-principle-satisfying CE/SE scheme to solve the volume-fraction-based five-equation model for compressible multifluids. Several numerical examples are carried out to carefully examine the accuracy, efficiency, conservativeness and maximum-principle-satisfying property of the proposed approach.

Three-dimensional detonation cellular structures in rectangular ducts using an improved CESE scheme**Project supported by the National Natural Science Foundation of China (Grant Nos. 10732010 and 10972010).

The three-dimensional premixed H2-O2 detonation propagation in rectangular ducts is simulated using an in-house parallel detonation code based on the second-order space–time conservation element and solution element (CE/SE) scheme. The simulation reproduces three typical cellular structures by setting appropriate cross-sectional size and initial perturbation in square tubes. As the cross-sectional size decreases, critical cellular structures transforming the rectangular or diagonal mode into the spinning mode are obtained and discussed in the perspective of phase variation as well as decreasing of triple point lines. Furthermore, multiple cellular structures are observed through examples with typical aspect ratios. Utilizing the visualization of detailed three-dimensional structures, their formation mechanism is further analyzed.

Numerical simulation of deflagration to detonation transition in granular HMX explosives under thermal ignition

To study the deflagration to detonation transition process in granular explosive under thermal ignition condition, the conductive burning is introduced into the classical model. The novel space–time conservation element and solution element method is applied to solve the mathematical governing equations. The transition process in granular HMX bed which is packed to 85 % of theoretical maximum density is simulated. The development of conductive burning, convective burning and detonation is analyzed. In the early stage, the combustion propagates slowly. It moves no more than 0.2 mm within 8.16 ms. After onset of convective burning, it takes only 20 μs to form a steady detonation with the velocity of 8165 m s−1. The time to detonation increases with the decrease in particle diameter.

Hyperbolicity of pressure–velocity equations for computational hydro acoustics

A set of model equations is proposed to simulate waves generated by unsteady, low-speed, nearly incompressible air and water flows. The equations include the continuity and momentum equations with pressure and velocity as the unknowns. Compressibility effect associated with waves motion is directly tracked by time-accurate calculation of pressure fluctuations. The corresponding density changes are modeled by using the bulk modulus of the medium. The three-dimensional equations are shown to be hyperbolic by analyzing eigenvalues and eigenvectors of the composite Jacobian matrix of the equations. Specifically, the matrix is shown to be diagonalizable and have a real spectrum. Moreover, an analytical form of the Riemann invariants of the one-dimensional equations is derived. To validate the model equations, the space–time Conservation Element and Solution Element (CESE) method and the SOLVCON code are employed to solve the two-dimensional equations. Aeolian tones generated by air and water flows passing a cylinder and over an open cavity are simulated. Numerical results compare well with previously reported data.

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.

A characteristic space–time conservation element and solution element method for conservation laws II. Multidimensional extension

The characteristic space-time conservation element and solution element (CE/SE) schemes proposed by Shen et al. (2015) 15 are straightforward extended to multidimensional schemes on 2D rectangular meshes which strictly follow the space-time conservation law. The schemes for solving both scalar conservation laws and compressible Euler equations with shock waves are developed. They are accurate and robust with CFL widely ranging from 0 to nearly 1. In Euler solvers, the frequently-used Harten, Lax and van Leer (HLL) Riemann solver, contact discontinuity restoring HLLC Riemann solver and Roe Riemann solver are employed to calculate the upwind fluxes as examples. When standard grid-aligned Riemann solvers are employed, the carbuncle phenomena are significantly suppressed when comparing with conventional upwind schemes. If rotated Riemann solvers are employed, nearly carbuncle-free results are obtained. Several well understood numerical examples are carried out to demonstrate that the 2D characteristic CE/SE schemes can simultaneously capture shocks and details of complex flow structures very well.

Combination of space–time conservation element/solution element method and continuous prediction technique for accelerated simulation of simulated moving bed chromatography

The model equation of simulated moving bed (SMB) process is solved by a space–time conservation element and solution element (CE/SE) method, which is found to be more efficient than other methods in the two case studies. Furthermore, this CE/SE method was combined with continuous prediction technique (CPM) to further accelerate determination of the cyclic steady state in SMB operation. Comparing with conventional method of lines at the same required precisions, CE/SE with CPM was proved to save the CPU time by 77% for the pseudo-linear system of glucose–fructose separation and 96% for nonlinear enantioseparation. Finally, the combination of CE/SE and CPM was applied to the optimization study to determine the optimum switching time and flow rate. To ensure the method be convergent, the four Courant numbers in four zones need to fall between 0.1 and 1. The method for fulfill this constraint is also developed.