Reactive flow calculations

R. M. C. So, H. C. Mongia, and J. H. Whitelaw, Turbulent Reactive Flow Calculations, special issue of Combustion Science and Technology, Gordon and Breach Science Pubhshers, Inc., Montreux, Swit2erland, 1988. 

Of course, for reactive flow calculations a new model would have to be constructed based on these techniques which used instead the equations governing compressible fluids and which contained the added chemical reactions and diffusive transport effects. 

The algorithm consists of the judicious application of one of two integration formulas to each equation in the system and the choice of formula is based on the time constant for each equation evaluated at the beginning of each chemical time step. Species with time constants too small are treated by the stiff method and the remaining species are treated by a classical second order method. The algorithm is characterized by a high degree of stability, moderate accuracy and low overhead which are very desirable features when applied to reactive flow calculations. 

In reactive flow calculations, the transport and chemical reaction parts of the equations are separated by time step splitting techniques (6) and solved separately and sequentially for each transport time step. Therefore, when combined with transport, the choice of solution formula is made for each equation at each chemical time step at each mesh point for each hydro time step. The integrator, for stability, is allowed to reduce the time step independently at each mesh point to appropriate values less than the transport time step. 

For none reactive flow calculations there are no mass exchange between the phases, the continuity equation was reduced to ... 

In our experience, on the final approach to the steady state in an Eulerian type of time-dependent reactive flow calculation (see Section 4), when the properties at a point in the flow field vary slowly with time and fresh calculations of the transport coefficients are only required at infrequent intervals, use of the detailed transport model presents little problem. A sound basis then exists for studies of the effect of variations in basic transport parameters on the steady-state flow. Unsteady flows, on the other hand, are most economically treated by less demanding, though more approximate, methods. Some approaches along these lines are discussed below and in Section 3.4. 

The incorporation of partial equilibrium (or quasi-steady-state) conditions into a reactive flow calculation thus necessitates consideration specifically of the overall size of the radical pool and the distribution of species within it. The ... 

In turbulent reactive flows, the chemical species and temperature fluctuate in time and space. As a result, any variable can be decomposed in its mean and fluctuation. In Reynolds-averaged Navier-Stokes (RANS) simulations, only the means of the variables are computed. Therefore, a method to obtain a turbulent database (containing the means of species, temperature, etc.) from the laminar data is needed. In this work, the mean variables are calculated by PDF-averaging their laminar values with an assumed shape PDF function. For details the reader is referred to Refs. [16, 17]. In the combustion model, transport equations for the mean and variances of the mixture fraction and the progress variable and the mean mass fraction of NO are solved. More details about this turbulent implementation of the flamelet combustion model can also be found in Ref. [20],... 

The center manifold approach of Mercer and Roberts (see the article Mercer and Roberts, 1990 and the subsequent article by Rosencrans, 1997) allowed to calculate approximations at any order for the original Taylor s model. Even if the error estimate was not obtained, it gives a very plausible argument for the validity of the effective model. This approach was applied to reactive flows in the article by Balakotaiah and Chang (1995). A number of effective models for different Damkohler numbers were obtained. Some generalizations to reactive flows through porous media are in Mauri (1991) and the preliminary results on their mathematical justification are in Allaire and Raphael (2007). 

Chemical kinetics and thermochemistry are important components in reacting flow simulations. Reaction mechanisms for combustion systems typically involve scores of chemical species and hundreds of reactions. The reaction rates (kinetics) govern how fast the combustion proceeds, while the thermochemistry governs heat release. In many cases the analyst can use a reaction mechanism that has been developed and tested by others. In other situations a particular chemical system may not have been studied before, and through coordinated experiments and simulation the goal is to determine the key reaction pathways and mechanism. Spanning this spectrum in reactive flow modeling is the need for some familiarity with topics from physical chemistry to understand the inputs to the simulation, as well as the calculated results. 

The Surface Chemkin formalism [73] was developed to provide a general, flexible framework for describing complex reactions between gas-phase, surface, and bulk phase species. The range of kinetic and transport processes that can take place at a reactive surface are shown schematically in Fig. 11.1. Heterogeneous reactions are fundamental in describing mass and energy balances that form boundary conditions in reacting flow calculations. 

The techniques just described have been extensively used in modeling reactive flow problems at NRL. Efficient solution of the coupled ordinary differential equations associated with these problems has enabled us to perform a wide variety of calculations on H2 °2 anC Ha/Oo mixtures which have greatly extended our understanding of tne combustion and detonation behavior of these systems. In addition numerous atmospheric problems have been studied. Details on these investigations are provided in references (7) and (9). 

The intent of this was to start with a useful calculation, which could not be done using brute force techniques, and demonstrate the importance of optimizing the numerical implementation of a reactive flow model to run on a vector computer. As similar problems in combustion become more extensive and intricate, it behooves us to utilize computers in the most efficient manner possible. It is no longer feasible to continue to "ask the computer to do more and more work, without thought as to how a particular problem is to be implemented. The number of problems for which one would like to use a computer, as well as the complexity of these problems, is increasing at an astronomical rate. The other side of the coin, of course is that computers, and especially central processors (CPU s) are becoming cheaper. 

Figure 5. Isoconversion contours in two-dimensional reactive flow at two different mass diffusivities, Galerkin calculations.

Figure 6. Isothermal contours in two-dimensional reactive flow, Galerkin calculation. Heat generation by viscous dissipation and reaction heating.

Optimization of internal engine combustion in respect of fuel efficiency and pollutant minimization requires detailed insight in the microscopic processes in which complex chemical kinetics is coupled with transport phenomena. Due to the development of various pulsed high power laser sources, experimental possibilities have expanded quite dramatically in recent years. Laser spectroscopic techniques allow nonintrusive measurements with high temporal, spectral and spatial resolution. New in situ detection techniques with high sensitivity allow the measurement of multidimensional temperature and species distributions required for the validation of reactive flow modeling calculations. The validated models are then used to And optimal conditions for the various combustion parameters in order to reduce pollutant formation and fuel consumption. 

Figure 15 Porosity structure of a high-resolution single-channel calculation for an upwelling system undergoing melting by both adiabatic decompression and reactive flow (see Spiegelman and Kelemen, 2003). Colors show the porosity field at late times in the run where the porosity is quasi steady-state. The maximum porosity at the top of the column (red) is 0.8% while the minimum porosity at the bottom (dark blue) is 10 times smaller. Axis ticks are height and width relative to the overall height of the box. In the absence of channels this problem is identical to the equilibrium one-porosity transport model of Spiegelman and Elliott (1993). Introduction of channels, however, produces interesting new chemical effects similar to the two porosity models.

If the chemical reactions are very fast compared to the mixing rate, it may be assumed that any mixed reactants are immediately reacted. No rate expression is therefore necessary. The simplest model to represent such cases is called the eddy break up (EBU) model (Spalding, 1970 Magnussen and Hjertager, 1976). In the EBU model, the effective rate of chemical reactions is equated to the smaller of rate calculated based on kinetic model and that based on the eddy break-up rate. The eddy breakup rate is defined as the inverse of a characteristic time scale kle. Therefore, for fast reactions, the rate of consumption or formation is proportional to the product of density, mass fraction and the eddy break-up rate elk). The model is useful for the prediction of premixed and partially premixed fast reactive flows. EBU, however, was originally developed for single-step chemical reactions. Its extension to multiple step reactive systems should be made with caution. 

Jones, W.P. and Whitelaw, J.H. (1982), Calculation methods for turbulent reactive flows A review, Combust. Flame, 48, 1-26. 

This form of the mixture model is called the drift flux model. In particular cases the flow calculations is significantly simplified when the problem is described in terms of drift velocities, as for example when ad is constant or time dependent only. However, in reactor technology this model formulation is restricted to multiphase cold flow studies as the drift-flux model cannot be adopted simulating reactive systems in which the densities are not constants and interfacial mass transfer is required. 

Future work might consider extensions of these interfacial transfer concepts to ameliorate the simulation accuracy by utilizing the local information provided by the multi-fluid models. For multiphase reactive systems these processes can be rate determining, in such cases there are no use for advanced flow calculations unless these fluxes can be determined with appropriate accuracy. 

For reactive flows the governing equations used by Lindborg et al [92] resemble those in sect 3.4.3, but the solid phase momentum equation contains several additional terms derived from kinetic theory and a frictional stress closure for slow quasi-static flow conditions based on concepts developed in soil mechanics. Moreover, to close the kinetic theory model the granular temperature is calculated from a separate transport equation. To avoid misconception the model equations are given below (in which the averaging symbols are disregarded for convenience) ... 

A proper convergence criterion is important, from both the accuracy and efficiency points of view, because it is deciding when to stop the iterative process. Research codes are generally iterating until the machine accuracy is reached, whereas the commercial codes are less accurate as efficiency is commonly desired by the customers. In commercial CFD codes, a convergence criterion defined by the reduction of the normalized residual, as calculated from the initial guess variable values, by a factor of 10 is frequently considered sufficient by contract research- and salespersons. However, for complex multiphase reactive flows this approach may easily lead to unphysical solutions. 

For most multiphase reactive flow problems, it is not possible to analyze all the operators in the complete solution method simultaneously. Instead the different operators of the method are analyzed separately one by one. The working hypothesis is that if the operators do not possess the desired properties solely, neither will the complete method. Unfortunately, the reverse is not necessarily true. In practical calculation we can only use a finite grid resolution, and the numerical results will only be physically realistic when the discretization schemes have certain fundamental properties. The usual numerical terminology employed in the CFD literature is outlined in this section [141, 202, 49]. 

All reactive flow models require as a minimum two equations of state, one for the unreacted explosive and one for its reaction products a reaction rate law for the conversion of explosive to products and a mixture rule to calculate partially reacted states in which both explosive and products are present. The Ignition and Growth reactive flow model [60] uses two Jones-Wilkins-Lee (JWL) equations of state, one for the unreacted explosive and another one for the reaction products, in the temperature dependent form ... 

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