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Partial characteristic decomposition for multi-species Euler equations
Abstract
High-order finite difference schemes employing characteristic decomposition are widely used for the simulation of compressible gas flows with multiple species. A challenge for the computational efficiency of such schemes is the quadratically increasing dimensionality of the convective flux eigensystem as the number of species increases. Considering the sparsity of the multi-species eigensystem, a remedy is proposed to split the eigensystem into two parts. One is the gas mixture part, which is subjected to the established characteristic decomposition schemes for single-fluid Euler equations. The other part corresponds to the species partial mass equations, which can be solved directly in physical space as the decoupled sub-eigensystem for the species part is composed of two diagonal identity matrices. This property relies on the fact that species are advected with the same convective velocity. In this way, only the gas mixture part requires a characteristic decomposition, resulting in a much higher efficiency for the convective-flux calculation. To cure the inconsistency due to splitting, a consistent update of species mass fractions is proposed. Non-reactive and reactive test cases demonstrate that the proposed scheme reduces the computational cost without deteriorating high-order accuracy.
... where is the number of reactions; is a constant ℎ 0 standing for enthalpy related to chemical bonds for each species; is the reaction rate; is the stoichiometry coefficient for species in reaction . Different from many other works [5][6][7] , is arranged in ℎ 0 the source terms but not in the total energy E, which facilitates the utilization of Patankar modification for the energy equations and helps preserve the pressure positivity, as shown in our previous work 8 . Denoting as the formula for each , = 1,2,⋯, species, the general form for a reaction is written as ...
... This case was investigated by many researchers, to name a few, Oran 44 . The Δ = 0.003 cm results are consistent with those obtained by Wang et al. 7 with the same chemical mechanism. It is shown that the proposed PMPRK scheme can resolve the detonation wave even on coarse grids and numerical experiments are advanced without negative concentrations. ...
... (2) on an adaptive Cartesian grid [25], flow solver in the present study can conveniently employ high-order low-dissipation shock-capturing schemes to reconstruct the inviscid convective flux terms, as well as high-order central difference schemes to compute the viscous diffusive fluxes. In this study, the 5th-order WENO-LLF [28] finite difference scheme for multi-species reactive flows, based on characteristic decomposition [29] and upwind flux splitting, and a simple 4th-order central difference scheme are used, respectively. To balance the overall accuracy in time and the computational cost, the temporal integration utilizes the strong stability preserving (SSP) 2nd-order explicit P e e r R e v i e w O n l y ...
An immersed boundary method (IBM) has been developed to handle the solid body embedded flowfield simulation for compressible reactive flows, paving the way of application for a wide range of fluid‐solid interaction problems. Previously, the Brinkman penalization method (BPM), originated from porous media flows, has been successfully used for incompressible Navier‐Stokes equations by adding penalization terms to momentum equations. However, it is non‐trivial to solve the compressible form due to the penalized continuity equation that usually poses severe numerical stiffness. In order to circumvent this issue, an extending procedure for relevant variables from the fluid to solid domain is considered, by analyzing the ordinary differential equations remained after operator splitting. Density can be then determined with the help of an equation of state. Meanwhile, efforts of enforcing the Neumann boundary condition, e.g., the adiabatic wall condition, on the fluid‐solid interface can be minimized by extending temperature across the interface directly. One more advantage of the extending step lies in that it can quickly reach a steady state when performed within a narrow band around the interface. Implemented into an adaptive Cartesian grid based ow solver for compressible Navier‐Stokes equations with chemical reaction source terms, the present variable‐extended IBM is validated by numerical examples ranging from single‐species non‐reactive to multi‐species deto‐native flows in one‐ and two‐dimensional domains. Numerical results show 1) the successful specification of slip or non‐slip, adiabatic or isothermal wall condition on the fluid‐solid interface and 2) loss of total energy in the original BPM being avoided and the numerical accuracy being improved especially for energy‐sensitive reactive flows.
... Therefore we solve the multi-component flow with a gas mixture consisting of 21% O 2 and 79% N 2 , which results in a similar γ. Following the initial condition suggested in [71], we design the following shock tube as ...
In order to simulate cryogenic H2−O2 jets under subcritical condition, a numerical model is constructed to solve compressible reactive multi-component flows which involve complex multi-physics processes such as moving material interfaces, shock waves, phase transition and combustion. The liquid and reactive gaseous mixture are described by a homogeneous mixture model with diffusion transport for heat, momentum and species. A hybrid thermodynamic closure strategy is proposed to construct an equation of state (EOS) for the mixture. The phase transition process is modeled by a recent fast relaxation method which gradually reaches the thermo-chemical equilibrium without iterative process. A simplified transport model is also implemented to ensure the accurate behavior in the limit of pure fluids and maintain computational efficiency. Last, a 12-step chemistry model is included to account for hydrogen combustion. Then the developed numerical model is solved with the finite volume method where a low dissipation AUSM (advection upstream splitting method) Riemann solver is extended for multi-component flows. A homogeneous reconstruction strategy compatible with the homogeneous mixture model is adopted to prevent numerical oscillations across material interfaces. Having included these elements, the model is validated on a number of canonical configurations, first for multiphase flows, and second for reactive flows. These tests allow recovery of the expected behavior in both the multiphase and reactive limits, and the model capability is further demonstrated on a 2D burning cryogenic H2−O2 jet, in a configuration reminiscent of rocket engine ignition.
... To achieve high-order FD schemes for Eqs. (1) and (2), Jacobian system including the left and right eigen-vectors as well as Roe-averages of pressure derivatives to calculate the speed of sound c at the cell face needs to be considered; see details in [19]. It should be noted that, in this study, the adjacent two sets of cell-centered pressure derivatives such as ∂p ∂ρ , ∂p ∂e and ∂p ∂yi are directly obtained by outputs from the thermo-solver, i.e. ...
In this study, a mass-fraction based fully conservative multi-component two-phase flow solver is considered using characteristic-wise finite-difference (CWFD) discretization with the 5th-order WENO scheme, in order to reduce numerical interface smearing and oscillations. Real-fluid thermodynamic properties are accounted for by a vapor-liquid equilibrium (VLE) model according to the local total density, internal energy and composition of the homogeneous mixture, with each phase being separatedly described by its Peng-Robinson equation of state (PR-EoS). A multicomponent Roe averaging for the cell-face eigen-system in CWFD methods has been developed with pressure derivatives resulted from the VLE model. Several 1D testing examples, e.g. interface advection, shock-tube problems and double-expansion cavitation, have been examined to demonstrate the low-oscillation, low-dissipation and robust performance of the present solver, in comparison with finite-volume schemes. A 2D transcritical injection process has also been simulated. It has been shown that high-order numerical schemes, such as the current CWFD method may be the way to reduce the smearing in diffuse interface modelling.
Les interactions entre les jets propulsifs et le sillage d’un lanceur spatial génèrent des contraintes mécaniques et thermiques pouvant compromettre la réalisation des objectifs de vol. Pour étudier ces écoulements instationnaires, la mise en place de méthodes numériques précises et abordables repose sur la recherche d’un compromis entre le niveau de résolution des échelles turbulentes et la complexité de la modélisation physique des jets. Pour contribuer à cette recherche, cette thèse est consacrée au développement et à l’évaluation d’une approche numérique permettant de réaliser des simulations ZDES bi-espèces sur des configurations représentatives des lanceurs de nouvelle génération. Les différents outils composant cette approche, dont un schéma hybride permettant une adaptation locale de la dissipation numérique et les versions bi-espèces des trois modes de la ZDES, sont évalués avec succès sur des cas d’études intégrant graduellement les phénomènes fluides pilotant les interactions de jet pour les arrière-corps de lanceurs spatiaux. L’approche est ensuite employée pour réaliser le 1er calcul RANS/LES de la littérature sur une configuration multi-tuyère de référence. Les apports d’une telle approche par rapport aux modélisations RANS pour l’évaluation quantitative des caractéristiques instantanées, statistiques et spectrales du champ de pression s’exerçant sur les parois sont exposés. Les méthodes numériques développées durant la thèse constituent ainsi des nouveaux outils permettant d’accompagner la conception des lanceurs spatiaux mais également d’autres véhicules soumis à des écoulements compressibles, turbulents et multi-espèces.
Essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes were designed for solving hyperbolic and convection–diffusion equations with possibly discontinuous solutions or solutions with sharp gradient regions. The main idea of ENO and WENO schemes is actually an approximation procedure, aimed at achieving arbitrarily high-order accuracy in smooth regions and resolving shocks or other discontinuities sharply and in an essentially non-oscillatory fashion. Both finite volume and finite difference schemes have been designed using the ENO or WENO procedure, and these schemes are very popular in applications, most noticeably in computational fluid dynamics but also in other areas of computational physics and engineering. Since the main idea of the ENO and WENO schemes is an approximation procedure not directly related to partial differential equations (PDEs), ENO and WENO schemes also have non-PDE applications. In this paper we will survey the basic ideas behind ENO and WENO schemes, discuss their properties, and present examples of their applications to different types of PDEs as well as to non-PDE problems.
The influence of the viscous boundary layer on the oblique detonation wave structure has been simulated and analyzed by solving the two-dimensional Navier–Stokes equations containing the hydrogen/air elementary reaction model. It is found that the effect of the viscous boundary layer can be neglected in the smooth transition initiation structure, but not in the abrupt transition initiation structure. The interaction of the lateral shock wave and the viscous boundary layer results in the formation of a high-temperature recirculation zone near the wall of the initiation zone, a novel structure that does not appear in inviscid formulations. Within this structure, the chemical reaction is triggered to occur in advance, eventually leading to the equilibrium initiation position relocating upstream compared with the inviscid case. Nevertheless, the viscous boundary layer has a limited impact on the main flow region downstream of the oblique detonation. The shock wave structures and pressure distributions at various Mach numbers are obtained computationally and analyzed in detail.
The structure of an oblique detonation wave (ODW) induced by a wedge is investigated via numerical simulations and Rankine–Hugoniot analysis. The two-dimensional Euler equations coupled with a two-step chemical reaction model are solved. In the numerical results, four configurations of the Chapman–Jouguet (CJ) ODW reflection (overall Mach reflection, Mach reflection, regular reflection, and non-reflection) are observed to take place sequentially as the inflow Mach number increases. According to the numerical and analytical results, the change of the CJ ODW reflection configuration results from the interaction among the ODW, the CJ ODW, and the centered expansion wave.
Abstract A new technique for a finite-difference weighted essentially nonoscillatory scheme (WENO) on curvilinear grids to preserve freestream is introduced. This technique first divides the standard finite-difference WENO into two parts: (1) a consistent central difference part and (2) a numerical dissipation part. For the consistent central difference part, the conservative metric technique is directly adopted. For the numerical dissipation part, it is proposed that the metric term should be frozen for constructing the upwinding flux. This treatment only affects the numerical dissipation part, and the order of accuracy is maintained. With this technique, the freestream is perfectly preserved, and the flow fields are better resolved on wavy and random grids.
Detailed chemical kinetic reaction mechanisms have been developed to describe the pyrolysis and oxidation of nine n-alkanes larger than n-heptane, including n-octane (n-C8H18), n-nonane (n-C9H20), n-decane (n-C10H22), n-undecane (n-C11H24), n-dodecane (n-C12H26), n-tridecane (n-C13H28), n-tetradecane (n-C14H30), n-pentadecane (n-C15H32), and n-hexadecane (n-C16H34). These mechanisms include both high temperature and low temperature reaction pathways. The mechanisms are based on our previous mechanisms for the primary reference fuels n-heptane and iso-octane, using the reaction class mechanism construction first developed for n-heptane. Individual reaction class rules are as simple as possible in order to focus on the parallelism between all of the n-alkane fuels included in the mechanisms, and these mechanisms will be refined further in the future to incorporate greater levels of accuracy and predictive capability. These mechanisms are validated through extensive comparisons between computed and experimental data from a wide variety of different sources. In addition, numerical experiments are carried out to examine features of n-alkane combustion in which the detailed mechanisms can be used to compare reactivities of different n-alkane fuels. The mechanisms for all of these n-alkanes are presented as a single detailed mechanism, which can be edited to produce efficient mechanisms for any of the n-alkanes included, and the entire mechanism, with supporting thermochemical and transport data, together with an explanatory glossary explaining notations and structural details, will be available for download from our web page.
This report describes and documents the subroutine CHEMEQ2, used to integrate stiff ordinary differential equations arising from reaction kinetics. This is a second generation improvement of CHEMEQ using a new quasi-steady-state predictor-corrector method that is A-stable for linear equations and second-order accurate. A single integration method can now be used for all species, regardless of the timescales of the individual equations. Start-up costs and memory requirements are low, so CHEMEQ2 is ideal for multi-dimensional reacting-flow simulations which require the solution of a process-split, initial-value problem in every computational cell for every global timestep. The algorithm, its implementation, the FORTRAN code, the internal variables and the argument lists are presented, along with several test problem results.
We present the basic ideas and recent developments in the construction, analysis, and implementation of ENO (essentially non-oscillatory) and WENO (weighted essentially non-oscillatory) schemes and their applications to computational fluid dynamics. ENO and WENO schemes are high-order accurate finite difference or finite volume schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in computational fluid dynamics and other applications, especially in problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics.
VODE is a new initial value ODE solver for stiff and nonstiff systems. It uses variable-coefficient Adams-Moulton and Backward Differentiation Formula (BDF) methods in Nordsieck form, as taken from the older solvers EPISODE and EPISODEB, treating the Jacobian as full or banded. Unlike the older codes, VODE has a highly flexible user interface that is nearly identical to that of the ODEPACK solver LSODE.
In the process, several algorithmic improvements have been made in VODE, aside from the new user interface. First, a change in stepsize and/or order that is decided upon at the end of one successful step is not implemented until the start of the next step, so that interpolations performed between steps use the more correct data. Second, a new algorithm for setting the initial stepsize has been included, which iterates briefly to estimate the required second derivative vector. Efficiency is often greatly enhanced by an added algorithm for saving and reusing the Jacobian matrix J, as it occurs in the Newton matrix, under certain conditions. As an option, this Jacobian-saving feature can be suppressed if the required extra storage is prohibitive. Finally, the modified Newton iteration is relaxed by a scalar factor in the stiff case, as a partial correction for the fact that the scalar coefficient in the Newton matrix may be out of date.
The fixed-leading-coefficient form of the BDF methods has been studied independently, and a version of VODE that incorporates it has been developed. This version does show better performance on some problems, but further tuning and testing are needed to make a final evaluation of it.
Like its predecessors, VODE demonstrates that multistep methods with fully variable stepsizes and coefficients can outperform fixed-step-interpolatory methods on problems with widely different active time scales. In one comparison test, on a one-dimensional diurnal kinetics-transport problem with a banded internal Jacobian, the run time for VODE was 36 percent lower than that of LSODE without the J-saving algorithm and 49 percent lower with it. The fixed-leading-coefficient version ran slightly faster, by another 12 percent without J-saving and 5 percent with it. All of the runs achieved about the same accuracy.
The present manuscript reports a numerical verification study based on a series of tests that allows to evaluate the numerical performance of a compressible reactive multicomponent Navier–Stokes solver. The verification procedure is applied to a density-based finite difference numerical scheme suited to compressible reactive flows representative of either combustion in high speed flows or detonation. The numerical algorithm is based on a third-order accurate Total Variation Diminishing (TVD) Runge Kutta time integration scheme. It employs a seventh-order accurate Weighted Essentially Non-Oscillatory (WENO) scheme to discretize the non-linear advective terms while an eighth-order accurate centered finite difference scheme is retained for the molecular viscous and diffusive terms. These molecular contributions are evaluated with the library EGLIB that accounts for detailed multicomponent transport including Soret and Dufour effects. The developed numerical solver thus offers an interesting combination of existing methods suited to the present purpose of studying combustion in high speed flows and/or detonations. The numerical solver is verified by considering a complete procedure that gathers eight elementary verification subsets including, among others, the classical Sod’s shock tube problem, the ignition sequence of a multi-species mixture in a shock tube, the unsteady diffusion of a smoothed concentration profile and a one-dimensional laminar premixed flame. Such verification analyses are seldom reported in the literature but constitute an important part of computational research activities. It is presently completed with the application of the verified finite difference scheme to the numerical simulation of (i) shock (reactive) mixing layer interaction and (ii) combustion ignition downstream of a highly under-expanded jet. The corresponding results shed some light onto the robustness (stability) and performance of the numerical scheme, and also provide some very valuable insights onto the complex physics that prevails in the development of chemical reactions in such situations, which are considered as representative of the discharge or accidental release of high pressure flammable mixtures into the atmosphere.
In this paper we focus our attention on obtaining well-balanced schemes for balance laws by using Marquina’s flux in combination with the finite difference and finite volume WENO schemes. We consider also the Rusanov flux splitting and the HLL approximate Riemann solver. In particular, for the presented numerical schemes we develop corresponding discretizations of the source term, based on the idea of balancing with the flux gradient. When applied to the open-channel flow and to the shallow water equations, we obtain the finite difference WENO scheme with Marquina’s flux splitting, which satisfies the approximate conservation property, and also the balanced finite volume WENO scheme with Marquina’s solver satisfying the exact conservation property. Finally, we also present an improvement of the balanced finite difference WENO scheme with the Rusanov (locally Lax–Friedrichs) flux splitting, we previously developed in [Vuković S, Sopta L. ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations. J Comput Phys 2002;179:593–621].
Detonation structures generated by wedge-induced, oblique shocks in hydrogen–oxygen–nitrogen mixtures were investigated by time-dependent numerical simulations. The simulations show a multidimensional detonation structure consisting of the following elements: (1) a nonreactive, oblique shock, (2) an induction zone, (3) a set of deflagration waves, and (4) a ‘‘reactive shock,’’ in which the shock front is closely coupled with the energy release. In a wide range of flow and mixture conditions, this structure is stable and very resilient to disturbances in the flow. The entire detonation structure is steady on the wedge when the flow behind the structure is completely supersonic. If a part of the flow behind the structure is subsonic, the entire structure may become detached from the wedge and move upstream continuously.
The present approximate and linearized Riemann solver of Euler gas dynamics equations in one dimension, with a general convex equation-of-state, is applied to a standard shock-reflection test problem. The algorithm used is computationally efficient, and can be extended to three dimensions by incorporating operator splitting. Attention is given to the details of this extension, together with a two-dimensional calculation of the flow in a tunnel with a step involving interacting waves.