Reactive Flow Equations Research Articles

Response of a swirl flame to inertial waves

Acoustic waves passing through a swirler generate inertial waves in rotating flow. In the present study, the response of a premixed flame to an inertial wave is scrutinized, with emphasis on the fundamental fluid-dynamic and flame-kinematic interaction mechanism. The analysis relies on linearized reactive flow equations, with a two-part solution strategy implemented in a finite element framework: Firstly, the steady state, low-Mach number, Navier–Stokes equations with Arrhenius type one-step reaction mechanism are solved by Newton’s method. The flame impulse response is then computed by transient solution of the analytically linearized reactive flow equations in the time domain, with mean flow quantities provided by the steady-state solution. The corresponding flame transfer function is retrieved by fitting a finite impulse response model. This approach is validated against experiments for a perfectly premixed, lean, methane-air Bunsen flame, and then applied to a laminar swirling flame. This academic case serves to investigate in a generic manner the impact of an inertial wave on the flame response. The structure of the inertial wave is characterized by modal decomposition. It is shown that axial and radial velocity fluctuations related to the eigenmodes of the inertial wave dominate the flame front modulations. The dispersive nature of the eigenmodes plays an important role in the flame response.

Effective Behavior Near Clogging in Upscaled Equations for Non-isothermal Reactive Porous Media Flow

For a non-isothermal reactive flow process, effective properties such as permeability and heat conductivity change as the underlying pore structure evolves. We investigate changes of the effective properties for a two-dimensional periodic porous medium as the grain geometry changes. We consider specific grain shapes and study the evolution by solving the cell problems numerically for an upscaled model derived in Bringedal et al. (Transp Porous Media 114(2):371–393, 2016. doi: 10.1007/s11242-015-0530-9 ). In particular, we focus on the limit behavior near clogging. The effective heat conductivities are compared to common porosity-weighted volume averaging approximations, and we find that geometric averages perform better than arithmetic and harmonic for isotropic media, while the optimal choice for anisotropic media depends on the degree and direction of the anisotropy. An approximate analytical expression is found to perform well for the isotropic effective heat conductivity. The permeability is compared to some commonly used approaches focusing on the limiting behavior near clogging, where a fitted power law is found to behave reasonably well. The resulting macroscale equations are tested on a case where the geochemical reactions cause pore clogging and a corresponding change in the flow and transport behavior at Darcy scale. As pores clog the flow paths shift away, while heat conduction increases in regions with lower porosity.

Analysis of the approximate slow invariant manifold method for reactive flow equations

The Approximate Slow Invariant Manifold method of Singh, Powers and Paolucci is a useful method for addressing model reduction in systems of reactive flow equations. It exploits separations of time scales between slow and fast species, and it generalizes the Intrinsic Low-Dimensional Manifold method, which was developed for model reduction in the context of reaction kinetics, to systems with diffusive and active transport. In this article, we present a mathematical analysis of the Approximate Slow Invariant Manifold method in the context of systems of reaction–diffusion equations with slow and fast reaction kinetics. Beginning with systems of two species (one slow and one fast), and then treating general systems with multiple slow and fast species, we explicitly determine the accuracy of the Approximate Slow Invariant Manifold method. We find that it is correct up to and including the terms of first order in the small parameter that measures the separation of the kinetics time scales, and that it captures many of the terms at second order, as well. Our analysis includes precise statements of the errors at second order, and we find that these are proportional to the slow components of the reaction–diffusion equation, as well as to the curvature of the critical manifold. We illustrate the results analytically on two prototypical examples, the Michaelis–Menten–Henri model with diffusion of the slow species and the Davis–Skodje model in which both the slow and fast species diffuse.

Electric Manipulation of Laminar Nonpremixed Counterflow Propane Flames

The effect of the electric field on laminar nonpremixed counterflow propane flames was analyzed computationally. The computations were conducted using ANSYS fluent platform associated with a detailed kinetic mechanism. The mechanism was supplemented with a set of three reactions accounting for the consumption/production of three chemi-ions. It was established that the position of the flame could be only controlled through altering the intensity of the applied electric field. The effect of the applied electric field was included within the reactive flow equations via introducing two distinct terms: a body force term that accounts for the electric field effects on the momentum of the reactive mixture, and an extra diffusion term that accounts for the mobility charged species, namely ambipolar diffusion. This study clearly shows that electric force provides a potential for controlling the location of propane flames without affecting their structure.

Refraction of detonation wave at interface between condensed explosives and high sound-speed material

The denotation of condensed explosives is very vulnerable to be influenced by the character of its confinement material. Confinements of different materials on the condensed explosives can remarkably change the shock locus and propagation speed of the detonation wave. Especially, when the confinement material has a higher sound-speed than the CJ velocity of explosives, some highly complicated refraction phenomena of detonation waves would take place near the explosives-material interface. This paper aims at analyzing the refraction phenomena of detonation waves in condensed explosives in theoretical and numerical ways. Firstly, an improved shock polar theory based on ZND model of detonation is built to give the styles of the refraction in detonation waves in order to provide a leading-order prediction of the confinement interaction. The improved shock polar is established at the leading shock wave of explosives detonation, and the refraction interaction is determined by the polar curve of the leading shock waves within the unreacted explosives and the polar curve of the refraction shock waves within the confinement material. Secondly, a second-order cell-centered Lagrangian hydrodynamics method, based on the characteristics theory for two-dimensional hyperbolic partial differential equations, is developed to solve the chemically reactive flow equations by the three-term ignition-growth chemistry reaction law. The main character of this method is that the finite volume discretization is adopted and an instantaneous evolution solver from an approximate Galerkin evolution operator is applied to compute the velocity and pressure of a grid vertex in order to update the grid coordinates and evaluate the numerical flux across the cell interface. A representative experiment about the propagation of a slipping detonation wave is numerically simulated. From the theoretical and numerical results about the refraction of detonation waves while the PBX9502 explosives interacting with beryllium interface, there exist four kinds of refraction styles of the detonation wave at high sound-speed material interface: the regular refraction with reflecting shock wave, the irregular refraction with bound precursor wave, the irregular refraction with twin Mach reflection, and the irregular refraction with -wave structure. In the first style, the front of the leading shock wave is straight, the flows in the detonation reactive zone and beryllium are both supersonic, and a reflecting shock wave appears behind the leading shock wave and a refracting shock wave appears within beryllium. In the second style, the front of the leading shock wave is also straight, the flow in the detonation reactive zone is supersonic but the one in beryllium is subsonic, so a reflecting shock wave appears behind the leading shock wave and a refracting shock wave appears within beryllium too; moreover, the refracting shock wave is almost perpendicular to the material interface, that is a bound precursor wave. In the third style, the front of the leading shock wave becomes forward curve, and the flows in the detonation reactive zone and beryllium are both subsonic, i.e., a Mach item is produced at some distance above the material interface where there are two Mach reflection structures on the top and the bottom of the Mach item respectively. Obviously, the bottom Mach reflection is a free precursor wave from the refracting shock wave within beryllium. In the fourth style, the forward curve range of the front of the leading shock wave becomes very broad, and accordingly, the range of subsonic flows in the detonation reactive zone becomes very wide. This makes the top Mach reflection disappear but the bottom one still exist, so the whole structure of the reflection wave seems to be like the Greek alphabet ; meanwhile, the flow within beryllium may be all in a subsonic state.

Analysis on refraction of detonation wave at the explosive-metal interface

This paper analyzes theoretically and numerically the refraction phenomenon of detonation wave at the explosive-metal interface, motivated by the problem that there exist large discrepancies between the experimental results and the classical shock polar theory. After pointing out the major defects of the classical shock polar theory based on CJ model of detonation, an improved shock polar theory based on ZND model of detonation is presented to give the styles of the refraction of detonation wave and the pressure values at the interaction point between the refracted shock wave and the incident shock wave, to show the pressure values at free-surface of copper remarkably lower than the ones at the shock interaction point due to the attenuation effects from the chemical reaction expansion and the following Taylor rarefaction. A second-order cell-centered Lagrangian hydrodynamics method with high resolution based on the subcharacteristics theory is develped to solve the reactive flow equations of detonation in condensed explosive, and then to numerically simulate a representative refraction experiment about T2 explosive interacting with copper. The simulated pressure values at the interaction point agree well with the ones from the improved shock polar theory, and the simulated pressure values at free-surface of copper agree well with the experimental values, meanwhile, the refraction styles predicted by the improved shock polar theory are confirmed by the numerically simulated flowfield images. From the theoretical and numerical results, there exist three kinds of refraction styles of detonation waves at explosive-metal interface:i) the regular refraction with reflecting shock wave, ii) the irregular refraction with Mach reflection, and iii) the regular refraction without any reflecting wave; in particular, the regular refraction with no reflecting wave is a kind of refraction style unable to be predicted by the classical shock polar theory, meanwhile, the pressure values at the free-surface and the interaction point inside the shocked metal both monotonically decrease with the increase of the incident angle.

WAMR: An adaptive wavelet method for the simulation of compressible reacting flow. Part I. Accuracy and efficiency of algorithm

The Wavelet Adaptive Multiresolution Representation (WAMR) method provides a robust method for controlling spatial grid adaptation — fine grid spacing in regions where a solution varies greatly (i.e., near steep gradients, or near-singularities) and a much coarser grid where the solution varies slowly. Subsequently, a wide range of spatial scales, often demanded in challenging continuum physics problems, can be efficiently captured. Furthermore, the wavelet transform provides a direct measure of local error at each collocation point, effectively producing automatically verified solutions.The method is applied to the solution of unsteady, compressible, reactive flow equations, and includes detailed diffusive transport and chemical kinetics models. Accuracy and performance of the method are examined on several test problems. The sparse grids produced by the WAMR method exhibit an impressive compression of the solution, reducing the number of collocation points used by factors of many orders of magnitude when compared to uniform grids of equivalent resolution.

CFD Study of NOx Emissions in a Model Commercial Aircraft Engine Combustor

Air worthiness requirements of the aircraft engine emission bring new challenges to the combustor research and design. With the motivation to design high performance and clean combustor, computational fluid dynamics (CFD) is utilized as the powerful design approach. In this paper, Reynolds averaged Navier-Stokes (RANS) equations of reactive two-phase flow in an experimental low emission combustor is performed. The numerical approach uses an implicit compressible gas solver together with a Lagrangian liquid-phase tracking method and the extended coherent flamelet model for turbulence-combustion interaction. The NOx formation is modeled by the concept of post-processing, which resolves the NOx transport equation with the assumption of frozen temperature distribution. Both turbulence-combustion interaction model and NOx formation model are firstly evaluated by the comparison of experimental data published in open literature of a lean direct injection (LDI) combustor. The test rig studied in this paper is called low emission stirred swirl (LESS) combustor, which is a two-stage model combustor, fueled with liquid kerosene (RP-3) and designed by Beihang University (BUAA). The main stage of LESS combustor employs the principle of lean prevaporized and premixed (LPP) concept to reduce pollutant, and the pilot stage depends on a diffusion flame for flame stabilization. Detailed numerical results including species distribution, turbulence performance and burning performance are qualitatively and quantitatively evaluated. Numerical prediction of NOx emission shows a good agreement with test data at both idle condition and full power condition of LESS combustor. Preliminary results of the flame structure are shown in this paper. The flame stabilization mechanism and NOx reduction effort are also discussed with in-depth analysis.

Revisiting Shock Initiation Modeling of Homogeneous Explosives

Shock initiation of homogeneous explosives has been a subject of research since the 1960s, with neat and sensitized nitromethane as the main materials for experiments. A shock initiation model of homogeneous explosives was established in the early 1960s. It involves a thermal explosion event at the shock entrance boundary, which develops into a superdetonation that overtakes the initial shock. In recent years, Sheffield and his group, using accurate experimental tools, were able to observe details of buildup of the superdetonation. There are many papers on modeling shock initiation of heterogeneous explosives, but there are only a few papers on modeling shock initiation of homogeneous explosives. In this article, bulk reaction reactive flow equations are used to model homogeneous shock initiation in an attempt to reproduce experimental data of Sheffield and his group. It was possible to reproduce the main features of the shock initiation process, including thermal explosion, superdetonation, input shock overtake, overdriven detonation after overtake, and the beginning of decay toward Chapman-Jouget (CJ) detonation. The time to overtake (TTO) as function of input pressure was also calculated and compared to the experimental TTO.

On the use of solute water fronts to measure nitrate adsorption in a Red Ferrosol

Nitrogen leaching in soils depends on the mobility of nitrate with, and relative to, water. This mobility is affected by anion adsorption at soil surfaces. Our work examines nitrate (NO3−) adsorption during non‐steady, unsaturated absorption of NO3− solution by columns of dry Red Ferrosol (Inceptic Eutrudox), approximating solute transport in unsaturated field soils. The experiments, in which no water flows past the end of the column, show that the application of conventional water and reactive solute flow equations is valid. The profiles of water and NO3−, measured at different horizontal infiltration times, coalesce on to unique curves when plotted as a function of the Boltzmann variable, λ. Anion adsorption delays the NO3− advance relative to that of water. Pragmatically, we use Freundlich and Langmuir equations as convenient mathematical descriptions of how the NO3− is partitioned between the soluble and adsorbed concentrations and conclude that the latter is appropriate for our dataset. We advocate the use of the Langmuir equation (non‐linear form) to obtain the fitting parameters to avoid the need to use ratios of small values, which results in large errors. Absorption of a solution into unsaturated columns of soil provides a powerful method for exploring adsorption: the experiment implicitly tests the water flow equation, the salt flow equation and solute equilibrium between the water‐soluble and adsorbed phases.

Thermal Behavior of a Flow in a Pulsed Detonation Thermal Spraying Device

In the present work, a numerical investigation has been performed to study the thermal behavior of a flow through a pulsed detonation thermal spray device that employs a hydrogen-air mixture. In this device, a detonation wave travels inside a detonation tube and eventually exits the tube and impinges against a substrate. Two types of substrates have been investigated, namely, a sphere and a flat-plate. The present model is based on solving the two dimensional, axisymetric Euler reactive flow equations with temperature-dependent specific heats. A global single step finite rate reaction has been considered. The present results demonstrate the effects of two main parameters on the thermal performance of the device under consideration. These parameters are: the size and shape of the substrate, and the stand off distance.

Two-scale expansion with drift approach to the Taylor dispersion for reactive transport through porous media

In this work we study reactive flows through porous media. We suppose dominant Péclet's number, dominant Damköhler's number and general linear reactions at the pore boundaries. Our goal is to obtain the dispersion tensor and the upscaled model. We introduce the multiple scale expansions with drift for the problem and use this technique to upscale the reactive flow equations. Our result is illustrated with numerical simulations for the dispersion tensor.

A study of detonation propagation and diffraction with compliant confinement

Previous computational studies of diffracting detonations with the ignition-and-growth (IG) model demonstrated that, contrary to experimental observations, the computed solution did not exhibit dead zones. For a rigidly confined explosive it was found that while diffraction past a sharp corner did lead to a temporary separation of the lead shock from the reaction zone, the detonation re-established itself in due course and no pockets of unreacted material remained. The present investigation continues to focus on the potential for detonation failure within the IG model, but now for a compliant confinement of the explosive. The aim of the present paper is two-fold. First, in order to compute solutions of the governing equations for multi-material reactive flow, a numerical method is developed and discussed. The method is a Godunov-type, fractional-step scheme which incorporates an energy correction to suppress numerical oscillations that occur near material interfaces for standard conservative schemes. The accuracy of the solution method is then tested using a two-dimensional rate-stick problem for both strong and weak confinements. The second aim of the paper is to extend the previous computational study of the IG model by considering two related problems. In the first problem, the corner-turning configuration is re-examined, and it is shown that in the matter of detonation failure, the absence of rigid confinement does not affect the outcome in a material way; sustained dead zones continue to elude the model. In the second problem, detonations propagating down a compliantly confined pencil-shaped configuration are computed for a variety of cone angles of the tapered section. It is found, in accord with experimental observation, that if the cone angle is small enough, the detonation fails prior to reaching the cone tip. For both the corner-turning and the pencil-shaped configurations, mechanisms underlying the behaviour of the computed solutions are identified.

An adaptive local grid refinement and peak/valley capture algorithm to solve nonlinear transport problems with moving sharp-fronts

Highly nonlinear advection–dispersion-reaction equations govern numerous transport phenomena. Robust, accurate, and efficient algorithms to solve these equations hold the key to the success of applying numerical models to field problems. This paper presents the development and verification of a computational algorithm to approximate the highly nonlinear transport equations of reactive chemical transport and multiphase flow. The algorithm was developed based on the Lagrangian-Eulerian decoupling method with an adaptive ZOOMing and Peak/valley Capture (LEZOOMPC) scheme. It consists of both backward and forward node tracking, rough element determination, peak/valley capturing, and adaptive local grid refinement. A second-order tracking was implemented to accurately and efficiently track all fictitious particles. Shanks’ method was introduced to deal with slowly converging case. The accuracy and efficiency of the algorithm were verified with the Burger equation for a variety of cases. The robustness of the algorithm to achieve convergent solutions was demonstrated by highly nonlinear reactive contaminant transport and multiphase flow problems.

Simulation of Combustion in a Coal-Fired Boiler with a Fixed Bed

In this study, steady-state combustion in boilers with a fixed bed has been investigated. The solution method and the discretization of the governing equations of two-dimensional turbulent reactive flow in the combustion chamber and one-dimensional coal combustion in the fixed bed are explained. Control volume and finite difference methods are used in the discretization of the equations in the combustion chamber and in the fixed bed, respectively. Kinetic calculations are employed in the solution of turbulent reactive flow equations in the combustion chamber. The combustion of coal particles in the fixed bed is ruled by the reaction-controlled mechanism. The distributions of temperature and some of the chemical species through the combustion chamber and the fixed bed are presented. Also, the variation of coal particle size with the fixed bed height is given.

Some aspects of the mathematical modelling of fireballs

The development of large-scale fireballs is a typical result of explosions associated with accidental fuel releases in the oil and process industries. The two major mechanisms that lead to fireball formation are vapour cloud explosions and boiling liquid expanding vapour explosions (BLEVEs). An ability to model fireballs is extremely important from the point of view of safety and risk analysis. The present paper provides a state-of-the-art review of the computational techniques available for the mathematical modelling of fireballs. The emphasis is placed on computational fluid dynamics (CFD) modelling, based on Reynolds-averaged Navier—Stokes (RANS) equations for reactive flow. The results of calculations and estimations of fireball hazards are compared with available experimental data. New results are presented for problems that are usually given less attention in fireball studies. These concern estimations of air entrainment rates into fireballs and potential strategies for fireball suppression.

A multigrid upwind strategy for accelerating steady‐state computations of waves propagating with curvature‐dependent speeds

AbstractA multigrid strategy using upwind finite differencing is developed for accelerating the steady state computations of waves, [14] propagating with curvature‐dependent speeds. This will allow the rapid computation of a “burn table.” In a high explosive material, a burn table will allow the elimination of solving chemical reaction ODEs by feeding in source terms to the reactive flow equations for solution of the system of ignition of the high explosive material. Standard iterative methods show a quick reduction of the residual followed by a slow final convergence to the solution at high iterations. Such systems, including a nonlinear system such as this, are excellent choices for the use of multigrid methods to speed up convergence. Numerical steady‐state solutions to the eikonal equation on several test grids are conducted. Results are presented for these cases in 2D and a cubic grid in 3D using a Runge‐Kutta time iteration for the smoothing operator until steady state is reached. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 179–192, 2002; DOI 10.1002/num.1002

Causes and consequences of flow organization during melt transport: The reaction infiltration instability in compactible media

Geochemical and field evidence suggest that melt transport in some regions of the mantle is localized into mesoscale “channels” that have widths of 0.1–100 m or larger. Nevertheless, the mechanisms for formation of such channels from a grain‐scale distribution of melt are poorly understood. The purpose of this paper is to investigate one possible mechanism for channel formation: the reaction infiltration instability (RII). We present numerical solutions of the full equations for reactive fluid flow in a viscously deformable, permeable medium. We show that dissolution in a compactible solid with a vertical solubility gradient can lead to significant flow localization such that >90% of the melt flux is channelized in <20% of the available area. In particular, the ability of the solid to compact enhances the instability by forming impermeable regions between channels. The combination of reaction, diffusion, and solid compaction leads to strong selection of preferred length scales with channel spacing smaller than the compaction length (δ ∼ 102 − 104 m). We explore the evolution of dissolution channels over parameter space and show that the behavior of the full nonlinear solutions is consistent with predictions from linear stability analysis. We also briefly consider the behavior of the instability in the presence of melting due to adiabatic decompression and demonstrate that significant localization can occur even in the presence of uniform melting and compaction. Using the linear analysis to extend these results for parameters expected in the Earth's mantle suggests that robust channel systems could form through the RII from a homogeneous system in ∼100,000 years with dominant channel spacing of 1–200 m.

Self-Excited Oscillations in Combustors With Spray Atomizers

Combustors with fuel-spray atomizers are susceptible to a low-frequency oscillation, particularly at idle and sub-idle conditions. For aero-engine combustors, the frequency of this oscillation is typically in the range 50–120 Hz and is commonly called “rumble.” In the current work, computational fluid dynamics (CFD) is used to simulate this self-excited oscillation. The combustion model uses Monte Carlo techniques to give simultaneous solutions of the Williams’ spray equation together with the equations of turbulent reactive flow. The unsteady combustion is calculated by the laminar flamelet presumed pdf method. A quasi-steady description of fuel atomizer behavior is used to couple the inlet flow in the combustor. A choking condition is employed at turbine inlet. The effects of the atomizer and the combustor geometry on the unsteady combustion are studied. The results show that, for some atomizers, with a strong dependence of mean droplet size on air velocity, the coupled system undergoes low-frequency oscillations. The numerical results are analyzed to provide insight into the rumble phenomena. Basically, pressure variations in the combustor alter the inlet air and fuel spray characteristics, thereby changing the rate of combustion. This in turn leads to local “hot spots,” which generate pressure fluctuations as they convect through the downstream nozzle.

A gas-kinetic scheme for reactive flows

In this paper, the gas-kinetic BGK scheme for the compressible flow equations is extended to chemical reactive flow. The mass fraction of the unburnt gas is implemented into the gas kinetic formulation by assigning a new internal degree of freedom to the particle distribution function. This new variable can be also used to describe fluid trajectory for the nonreactive flows. The gas-kinetic BGK model basically solves the Navier–Stokes chemical reactive flow equations. Numerical examples validate the accuracy and robustness of the current approach.