Bound-Preserving Discontinuous Galerkin Methods with Modified Patankar Time Integrations for Chemical Reacting Flows

In this paper,we develop bound-preserving discontinuous Galerkin(DG)methods for chemical reactive flows.There are several difficulties in constructing suitable numerical schemes.First of all,the density and internal energy are positive,and the mass fraction of each species is between 0 and 1.Second,due to the rapid reaction rate,the system may contain stiff sources,and the strong-stability-preserving explicit Runge-Kutta method may result in limited time-step sizes.To obtain physically relevant numerical approximations,we apply the bound-preserving technique to the DG methods.Though traditional positivity-preserving techniques can successfully yield positive density,internal energy,and mass fractions,they may not enforce the upper bound 1 of the mass fractions.To solve this problem,we need to(i)make sure the numerical fluxes in the equations of the mass fractions are consistent with that in the equation of the density;(ii)choose conservative time integrations,such that the summation of the mass fractions is preserved.With the above two conditions,the positive mass fractions have summation 1,and then,they are all between 0 and 1.For time discretization,we apply the modified Runge-Kutta/multi-step Patankar methods,which are explicit for the flux while implicit for the source.Such methods can handle stiff sources with relatively large time steps,preserve the positivity of the target variables,and keep the summation of the mass fractions to be 1.Finally,it is not straightforward to combine the bound-preserving DG methods and the Patankar time integrations.The positivity-preserving technique for DG methods requires positive numerical approximations at the cell interfaces,while Patankar methods can keep the positivity of the pre-selected point values of the target variables.To match the degree of freedom,we use polynomials on rectangular meshes for problems in two space dimensions.To evolve in time,we first read the polynomials at the Gaussian points.Then,suitable slope limiters can be applied to enforce the positivity of the solutions at those points,which can be preserved by the Patankar methods,leading to positive updated numerical cell averages.In addition,we use another slope limiter to get positive solutions used for the bound-preserving technique for the flux.Numerical examples are given to demonstrate the good performance of the proposed schemes.

LARGE-EDDY SIMULATION OF TWO-PHASE REACTING FLOW IN MODEL COMBUSTOR

The gas-droplet two-phase reacting flow in a model combustor with the V-gutter flame holder is studied by an Eulerian-Lagrangian large-eddy simulation (LES) approach. The k-equation subgrid-scale model is used to simulate the subgrid eddy viscosity, and the eddy-break-up (EBU) combustion subgrid-scale model is used to determine the chemical reaction rate. A two-step turbulent combustion subgrid-scale model is employed for calculating carbon monoxide CO concentration, and the NO subgrid-scale pollutant formation model for the evaluation of the rate of NO formation. The heat flux model is applied to the prediction of radiant heat transfer. The gas phase is solved with the SIMPLE algorithm and a hybrid scheme in the staggered grid system. The liquid phase equations are solved in a Lagrangian frame in reference of the particle-source-in-cell (PSIC) algorithm. From simulation results, the exchange of mass, moment and energy between gas and particle fields for the reacting flow in the afterburner with a V-gutter flame holder can be obtained. By the comparison of experimental and simulation results, profile temperature and pollutant of the outlet are quite in agreement with experimental data. Results show that the LES approach for predicting the two-phase instantaneous reacting flow and pollutant emissions in the afterburner is feasible.

THE NONLINEAR STABILITY OF TRAVELLING WAVE SOLUTIONS FOR A REACTING FLOW MODEL WITH SOURCE TERM

In this paper, author considers a 3 x 3 system for a reacting flow model propesed by [9]. Since this model has source term, it can be considered as a relaxation approximation to 2 x 2 systems of conservation laws, which include the well-known p-system. From this viewpoint, the author establishes the global existence and the nonlinear stability of travelling wave solutions by L-2 energy method.

High-order discontinuous Galerkin method for applications to multicomponent and chemically reacting flows

This article focuses on the development of a discontinuous Galerkin (DG) method for simulations of multicomponent and chemically reacting flows. Compared to aerodynamic flow applications, in which DG methods have been successfully employed, DG simulations of chemically reacting flows introduce challenges that arise from flow unsteadiness, combustion, heat release, compressibility effects, shocks, and variations in thermodynamic properties. To address these challenges, algorithms are developed, including an entropy-bounded DG method, an entropy-residual shock indicator, and a new formulation of artificial viscosity. The performance and capabilities of the resulting DG method are demonstrated in several relevant applications, including shock/bubble interaction, turbulent combustion, and detonation. It is concluded that the developed DG method shows promising performance in application to multicomponent reacting flows. The paper concludes with a discussion of further research needs to enable the application of DG methods to more complex reacting flows.

Validation of the RANS-SOM Combustion Model Using Direct Numerical Simulation of Incompressible Turbulent Reacting Flows

The second-order moment combustion model, proposed by the authors is validated using the direct numerical simulation （DNS） of incompressible turbulent reacting channel flows. The instantaneous DNS results show the near-wall strip structures of concentration and temperature fluctuations. The DNS statistical results give the budget of the terms in the correlation equations, showing that the production and dissipation terms are most important. The DNS statistical data are used to validate the closure model in RANS second-order moment （SOM） combustion model. It is found that the simulated diffusion and production terms are in agreement with the DNS data in most flow regions, except in the near-wall region, where the near-wall modification should be made, and the closure model for the dissipation term needs further improvement. The algebraic second-order moment （ASOM） combustion model is well validated by DNS.

GLOBAL EXISTENCE AND LARGE TIME BEHAVIOR OF SOLUTION TO REACTING FLOW MODEL WITH BOUNDARY EFFECT

The initial boundary value problem (IBVP) for the 3×3 hyperbolic system of reacting flow with source term proposed by R.J.LeVeque and others (see [8]) is considered.It is shown, in the present paper, that if the initial data are a suitable perturbation of a shiftcd shock profile which is suitably away from the boundary, then there exists a unique smooth solution in R2+ to the IBVP of the 3×3 hyperbolic system, which tends to another shifted shock profile of this system as t →∞.

NUMERICAL CALCULATIONS OF GASEOUS REACTING FLOWS IN A MODEL OF GAS TURBINE COMBUSTORS

This paper describes the numerical calculations of gaseous reaction flows in a model of gas turbine combustors. The profiles of hydrodynamic and thermodynamic patterns in a three-dimensional combustor model are obtained by solving the governing differential transport equations. The well-established numerical prediction algorithm SIMPLE, the modified k-ε turbulence model and k-ε-g turbulent diffusion flame model have been adopted in computations. The β function has been selected as probability density function. The effect of combustion process on flow patterns has been investigated. The calculated results have been verified by experiments. They are in remarkably good agreement.

Turbulence Characteristics of Swirling Reacting Flow in a Combustor with Staged Air Injection

This paper presents an experimental investigation of the turbulent reacting flow in a swirl combustor with staged air injection. The air injected into the combustor is composed of the primary swirling jet and the secon-dary non-swirling jet. A three dimension-laser particle dynamic analyzer (PDA) was employed to measure the in-stantaneous gas velocity. The probability density functions (PDF) for the instantaneous gas axial and tangential ve-locities at each measuring location, as well as the radial profiles of the root mean square of fluctuating gas axial and tangential velocities and the second-order moment for the fluctuating gas axial and tangential velocities are ob-tained. The measured results delineate the turbulence properties of the swirling reacting flow under the conditions of staged combustion.

Modeling and simulation of chemically reacting flows in gas-solid catalytic and non-catalytic processes

This paper gives an overview of the recent development of modeling and simulation of chemically react- ing flows in gas-solid catalytic and non-catalytic processes. General methodology has been focused on the Eulerian-Lagrangian description of particulate flows, where the particles behave as the catalysts or the reactant materials. For the strong interaction between the transport phenomena （i.e., momentum, heat and mass transfer） and the chemical reactions at the particle scale, a cross-scale modeling approach, i.e., CFD-DEM or CFD-DPM, is established for describing a wide variety of complex reacting flows in multiphase reactors, Representative processes, including fluid catalytic cracking （FCC）, catalytic conversion of syngas to methane, and coal pyrolysis to acetylene in thermal plasma, are chosen as case studies to demonstrate the unique advantages of the theoretical scheme based on the integrated particle-scale information with clear physical meanings, This type of modeling approach provides a solid basis for understanding the multiphase reacting flow problems in general.

EXISTENCE OF ENTROPY SOLUTIONS TO TWO-DIMENSIONAL STEADY EXOTHERMICALLY REACTING EULER EQUATIONS

We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function φ（T ） is assumed to have a positive lower bound. We first consider the Cauchy problem （the initial value problem）, that is, seek a supersonic downstream reacting flow when the incoming flow is supersonic, and establish the global existence of entropy solutions when the total variation of the initial data is sufficiently small. Then we analyze the problem of steady supersonic, exothermically reacting Euler flow past a Lipschitz wedge, generating an ad-ditional detonation wave attached to the wedge vertex, which can be then formulated as an initial-boundary value problem. We establish the global existence of entropy solutions containing the additional detonation wave （weak or strong, determined by the wedge angle at the wedge vertex） when the total variation of both the slope of the wedge boundary and the incoming flow is suitably small. The downstream asymptotic behavior of the global solutions is also obtained.

Effects of Gas Temperature Fluctuation on the Soot Formation Reactions

The effects of gas temperature fluctuations on soot formation and oxidation reactions are investigated numerically in a reacting flow. The instantaneous variations of soot mass fraction with time are obtained under the time-averaged gas temperature of 1500-1700 K. The simulation results show that the gas temperature fluctuation has obvious influence on the instantaneous processes of soot formation and oxidation. Within the present range of gas temperature, the gas temperature fluctuation results in generally lower soot mass fraction comparing to that without gas temperature fluctuation. The increase in the fluctuation amplitude of gas temperature leads to decrease in time-averaged soot mass fraction and increase in time-averaged soot particle number density.

Finite Element Simulation of an Unimolecular Thermal Decomposition inside a Reactor

This numerical study investigates the steady state unimolecular thermal decomposition of a chemical dissolved in water inside a parallel-plate reactor containing four heated circular rods using a penalty Galerkin finite element approach. The reactant fluid enters from the left inlet and exits from the right outlet of the reactor. All solid walls of the reactor are assumed to be thermodynamically isolated. The aim of the investigation is to illustrate the effects of the energy expelled during the reaction, temperature of the heated rods and fluid inlet velocity on the thermal field and concentration of the heat sensitive chemical. The simulation is conducted for different values of inlet velocity and rods temperature taking into consideration and neglecting the reaction energy. From the results, it is concluded that the thermal field and decomposition process of the chemical are significantly influenced by fluid velocity, rods temperature and the reaction type.

A data-driven reduced-order model for stiff chemical kinetics using dynamics-informed training

A data-based reduced-order model(ROM)is developed to accelerate the time integration of stiff chemically reacting systems by effectively removing the stiffness arising from a wide spectrum of chemical time scales.Specifically,the objective of this work is to develop a ROM that acts as a non-stiff surrogate model for the time evolution of the thermochemical state vector(temperature and species mass fractions)during an otherwise highly stiff and nonlinear ignition process.The model follows an encode-forecast-decode strategy that combines a nonlinear autoencoder(AE)for dimensionality reduction(encode and decode steps)with a neural ordinary differential equation(NODE)for modeling the dynamical system in the AE-provided latent space(forecasting step).By means of detailed timescale analysis by leveraging the dynamical system Jacobians,this work shows how data-based projection operators provided by autoencoders can inherently construct the latent spaces by removing unnecessary fast timescales,even more effectively than physics-based counterparts based on an eigenvalue analysis.A key finding is that the most significant degree of stiffness reduction is achieved through an end-to-end training strategy,where both AE and neural ODE parameters are optimized simultaneously,allowing the discovered latent space to be dynamics-informed.In addition to end-to-end training,this work highlights the vital contribution of AE nonlinearity in the stiffness reduction task.For the prediction of homogeneous ignition phenomena for H2-air and C2H4-air mixtures,the proposed ROM achieves several ordersof-magnitude increase in the integration time step size when compared to(a)a baseline CVODE solver for the full-chemical system,(b)statistical technique–principal component analysis(PCA),and(c)computational singular perturbation(CSP),a vetted physics-based stiffness-reducing modeling framework.

Numerical simulation of high speed reacting shear layers using AUSM+-up scheme-based unstructured finite volume method solver

Development of an Advection Upstream Splitting Method(AUSM+-up)scheme-based Unstructured Finite Volume(UFVM)solver for the simulation of two-dimensional axisymmetric/planar high speed compressible turbulent reacting shear layers is presented.The inviscid numerical flux is evaluated using AUSM+-up upwind scheme.An eight-step hydrogen–oxygen finite rate chemistry model is used to model the development of chemical species in a supersonic reacting flow field.The chemical species terms are alone solved implicitly in this explicit flow solver by rescaling the equation in time.The turbulence modeling has been done using RNG-based k–
model.Three-stage Runge–Kutta method has been used for explicit time integration.The nonreacting twodimensional Cartesian version of the same solver has been successfully validated against experimental and numerical results reported for the wall static pressure data in sonic slot injection to supersonic stream.Detailed validation studies for reacting flow solver has been done using experimental results reported for a coaxial supersonic combustor,in which species profile at various axial locations has been compared.Present numerical solver is useful in simulating combustors of high speed air-breathing propulsion devices.