The unique role of computational methods in combustion cannot be overstated. Reacting flow is a hostile environment for experimental measurements and investigations. Even with modern nonintrusive optical measurement techniques, it is very difficult, expensive, and time consuming to obtain field (vs. point) information about the flow and its intrinsic mechanisms. Due to the lack of predictive methods, extensive and very expensive experimentations are currently the primary means of developing new technology from basic concepts. Even fundamental analysis is encumbered by the fact that most methods are actually postdictive. Developing computer-aided design (CAD)/computer-aided manufacturing (CAM)/computer aided engineering (CAE) methods for reacting flow and combustion should lead to similar time and cost savings as those achieved by the proliferation of finite-element methods in structural mechanics.

Progress in the study of reacting flow and combustion has traditionally been difficult due to several factors, not the least of which are the complexity of the governing equations and the multiplicity of their governing processes. Combustion processes, in most cases, involve changes over a wide range of length and time scales (e.g., flames or detonation fronts are very thin compared with other flow scales). Strong interactions exist between these fronts and the overall flow, and solutions including variations of all scales must be obtained. Even in a homogeneous chemical reaction between a simple hydrocarbon fuel and air, hundreds of elementary reactions occurring over time scales that cover orders of magnitudes are encountered. Most frequently, chemical reactions take place while transport processes (convection, diffusion, and radiation) act to change the local composition and temperature at scales different than those corresponding to the reaction.

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10
CHEMICALLY REACTING FLOW AND COMBUSTION
The unique role of computational methods in combustion cannot be overstated. Reacting flow is a hostile environment for experimental measurements and investigations. Even with modern nonintrusive optical measurement techniques, it is very difficult, expensive, and time consuming to obtain field (vs. point) information about the flow and its intrinsic mechanisms. Due to the lack of predictive methods, extensive and very expensive experimentations are currently the primary means of developing new technology from basic concepts. Even fundamental analysis is encumbered by the fact that most methods are actually postdictive. Developing computer-aided design (CAD)/computer-aided manufacturing (CAM)/computer aided engineering (CAE) methods for reacting flow and combustion should lead to similar time and cost savings as those achieved by the proliferation of finite-element methods in structural mechanics.
BASIC MODELING DIFFICULTIES
Progress in the study of reacting flow and combustion has traditionally been difficult due to several factors, not the least of which are the complexity of the governing equations and the multiplicity of their governing processes. Combustion processes, in most cases, involve changes over a wide range of length and time scales (e.g., flames or detonation fronts are very thin compared with other flow scales). Strong interactions exist between these fronts and the overall flow, and solutions including variations of all scales must be obtained. Even in a homogeneous chemical reaction between a simple hydrocarbon fuel and air, hundreds of elementary reactions occurring over time scales that cover orders of magnitudes are encountered. Most frequently, chemical reactions take place while transport processes (convection, diffusion, and radiation) act to change the local composition and temperature at scales different than those corresponding to the reaction.

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Most practical combustion applications occur in a turbulent field. With the current lack of universal models of turbulence, even in much simpler nonreacting flows, one is faced with a unique challenge when turbulent combustion is a dominant process in the system. Reacting flows are intrinsically compressible flows with large, highly localized density and temperature variations. At high Mach number, as in supersonic combustion, strong pressure waves are also encountered. In diesel engines, jet engines, some rocket engines, and furnaces, combustion occurs in a turbulent multi-phase flow. In many manufacturing processes, turbulent reacting flow occurs on complex catalytic surfaces. It is no surprise, therefore, that computational methods have, and will continue to play, a significant role in this field. Some of the most important achievements of computational methods are described below in their order of complexity and outstanding challenges, identified by physical phenomena and numerical complexities, are listed. Recommendations are suggested.
RECENT ADVANCES
Some of the most visible achievements of the application of computational methods have been the development of off-the-shelf standard codes to perform some basic and widely used calculations. Computations of the equilibrium composition of a reacting mixture and one-dimensional reacting flow at any given thermodynamic state is one such accomplishment. It is currently used in undergraduate courses all the way to design and analysis in the automotive, aerospace, and utility industries. This was made possible by the development of efficient and accurate methods for the solution of nonlinear systems of algebraic equations. Similar progress has been made in chemical kinetics, which is concerned with homogeneous reacting systems far from thermodynamic equilibrium. In chemical kinetics it is necessary to integrate large systems of stiff, nonlinear, coupled ordinary differential equations. Though not as fast as those used in chemical equilibrium computations, codes are currently available for such applications.
In laminar combustion, where chemical kinetics is coupled with the transport of heat and mass by molecular diffusion, it is now possible, sometimes at a large cost, to compute the detailed structure and speed of propagation of laminar flames. The equations governing this problem are of the steady reaction-diffusion type, and progress in finite difference methods for one-dimensional initial and boundary value problems has allowed for accurate solution of these

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systems. Extensions to laminar-reacting boundary layer flows have also been achieved using adaptive one-dimensional grids in two-point boundary value problems with Newton iteration. Attempts to extend such computations to study the interaction between a laminar flame and the field of a single vortex, modeling one component of a reacting flow, are beginning to yield interesting results. However, in this case, models have been limited to simplified chemical kinetics schemes of one or a few reactions.
The picture is not as bright for turbulent combustion, the most frequently encountered in practice. In this case, solutions of the full Navier-Stokes equations coupled with the equations governing chemical kinetics and state are, while mostly unattainable, necessary. However, one cannot ignore the enormous progress in the computation of high Reynolds number nonreacting shear and separating flows. Both flows are fundamental to combustion since they are used in the mixing of reactants prior to their burning or in combustion stabilization in premixed streams. The results of these computations are called simulations since they are based on the unsteady, unaveraged Navier-Stokes equations. They have been instrumental in shedding light on some important instability modes of shear flows, the large-scale structure of turbulent flow, processes that lead to the transfer of energy from the mean flow to turbulence, and the decay of turbulence.
These simulations were made possible by the development of numerical schemes that are highly accurate in temporal and spatial discretizations and stable at high Reynolds numbers. In particular, spectral methods and vortex methods have been particularly successful in flows dominated by highly concentrated regions of vorticity that change their shapes as the flow evolves between different transition modes. Spectral methods have mostly been applied to flows with simple boundary conditions. Recent progress in domain decomposition and global finite element discretization promises to lead to the application of spectral simulation to flows with complex boundaries. Vortex methods, on the other hand, conform to complex domains at the cost of increasing the number of computational elements. The formulation of fast particle solvers and multipole expansion methods has contributed to overcoming the cost problem in vortex methods.

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OPEN PROBLEMS AND RESEARCH NEEDS
Applying computational methods for the simulation of turbulent reacting flows has been limited to cases in which the flow's time and length scales and the chemical reaction are of the same order of magnitude. Extensive research is required to extend their applicability to more interesting problems where the length scales vary over a wide range. Methods that rely on multidimensional adaptive gridding, such as adaptive projection methods; Lagrangian methods based on resolving the gradients of the gas dynamic variable, such as vortex and transport element methods; methods based on numerical asymptotic expansions; and hybrid methods that combine the convenience of kernel function discretization with spectrally accurate core functions should all be pursued for this purpose. It is important to mention that the problem of turbulence-combustion interactions—the crux of turbulent combustion— even in its most rudimentary form involving flame-strain and flame-vortex interactions, has not been satisfactorily explored due to the lack of powerful numerical schemes.
Increasing the accuracy and efficiency of time-dependent, multidimensional, adaptive numerical methods is one of the major tasks in the next decade. This includes further exploration of methods based on the Lagrangian formulation of the conservation equations as a natural means of achieving spatial adaptivity. Both grid-based and grid-free Lagrangian methods are natural candidates for this development. Methods that combine the convenience of grids with the accuracy of grid-free schemes, such as those that reduce the dimensionality of the grid by utilizing Green function solutions in the direction where the steepest gradients exit, should be explored further. Finite difference schemes based on embedding solutions of the governing equations at the elementary cell level, such as Godonov methods in which the solution of the Reimann problem is implemented within each cell, offer higher accuracy than conventional methods and should be pursued further.
It is unlikely that all the spatial scales of a turbulent flow, let alone turbulent reacting flow, can be represented in a practical system using stationary or moving grids. Thus, progress in modeling subgrid or subscale effects in a form compatible with the discretization of the resolvable scales will be instrumental in continued development in flow simulation. This includes methods that consider structures of scales smaller than the smallest grid size (e.g., flame fronts) as discontinuities. It is not yet clear how to embed all the physics

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of the phenomena that occur within the discontinuity into the formulation of the equations governing its motion or how to implement all the interactions between the discontinuity and the "external" flow (e.g., how the turbulent flow wrinkles a discontinuity and how changes within the discontinuity affect the turbulent flow). Methods based on the fractal representation of these discontinuities have been proposed, but work is still needed to determine the relationship between the fractal properties of turbulized surfaces and the physics of turbulent combustion.
The physics and numerical implementation of small structures of turbulence that cannot be resolved using discretization schemes (e.g., folding hairpin vortices encountered in the interial subrange of turbulence) are currently under investigation using vortex methods. Also, methods are being pursued for representation of the effects of high wavenumber structures in spectral simulations using methods of renormalization group theory. Research on the relationship between the two discretization methods, vortex and spectral methods (particle vs. wavelike representation of the dynamics), is also needed to show ways of optimizing the application of hybrid methods. Wavelet methods offer an opportunity in this regard. Using such statistical methods as probability density function formulation to model high-order moments has also been proposed in connection with solutions of the averaged equations. However, these methods rely heavily on experimental input, and their theoretical foundations are not yet clear.
Other phenomena, which have emerged from engineering applications of combustion and present research opportunities in the coming decade, include supersonic combustion involving shock wave reaction/front turbulence interaction (which is the mode of propulsion in the hypersonic transport vehicles); different modes of combustion instability that encumber the optimization of energy conversion processes (knock in internal combustion engines, chucking in rocket engines, screech in turbojet and ramjet engines); flow and combustion of sprays, slurries, and two-phase media (coal-air, coal-water); combustion of non-Newtonian fluids (which arises in incineration); effects of radiation on combustion dynamics (encountered in, e.g., the formation and destruction of ozone in the atmosphere); and coagulation of heavy particulates (such as carbon) in reacting streams.
Numerical methods in computational fluid dynamics have evolved in two directions: (1) general methods that can be applied to "any" partial differential equation irrespective of its type, such as finite difference and finite element methods, and (2) special methods formulated to obtain solutions for

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equations of certain forms that describe a narrow range of phenomena, such as the method of characteristics, vortex, and particle methods. Each class of methods offers some advantages and suffers from certain drawbacks. Progress in both directions has recently led to the formulation of some hybrid methods, especially the coupling of Eulerian and Lagrangian methods, which combine the best of the two approaches; the results have been encouraging. Sample results obtained from such methods will be shown at the end of this chapter. Continued research in the construction and application of hybrid schemes is necessary for the investigation of complex phenomena in reacting flow (e.g., shock/flame-flow/boundary layer interactions that have not yet been satisfactorily simulated). For these problems it is potentially desirable to couple interface-tracking algorithms with grid-based or grid-free solutions of the Navier-Stokes equations.
Hybrid methods, such as combined domain decomposition and interacting particle methods, also permit effective utilization of modern computer architecture (e.g., medium and fine-grain parallel processing in which accurate, but limited, interaction between various parts of the solution domain is necessary in order to take advantage of distributed computing). The availability of many computing environments with different capabilities and characteristics in terms of speed and memory combination, vector length, and number of processors has necessitated, and will continue to necessitate, the development of new algorithms, as well as ways of combining different schemes to take advantage of particular hardware technology. In this regard, advanced graphics software used in pre-and postprocessing of multidimensional, time-dependent solutions represents another important research area that promises to make computational fluid and combustion dynamics more accessible to design engineers. Modern CAD/CAM/CAE systems will have to incorporate such software. Both pre-and postprocessing software will have to be interactive, and the graphics interface may have to be run in real time (i.e., exhibit the solution as it is being computed). Here also, the coupling between the algorithms used to generate the solution of the governing equations (Eulerian or Lagrangian) and those used to interpret them in a way compatible with the intuition of the design engineer (particle trajectories or streamlines) poses another challenge.

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POTENTIAL PAYOFF
The need to continue the development of computational methods in combustion becomes more critical when considering the wide range of applications of turbulent reacting flow. The availability of predictive methods will reduce the design and testing time and cost of new products in the automotive, aerospace, electric utilities, and many manufacturing industries. Predictive methods allow one to perform "computer experiments" to investigate the fundamentals and outcome of new processes, thus realizing how clean, efficient, and fast these processes can be without actually building the hardware and running laboratory experiments. Computer experiments can also be used to test different concepts to exercise more effective control over reacting flow systems if they operate at off-design conditions. The continuous improvement of existing products and the introduction of new and innovative products are expected to play a major role in global industrial competitiveness. This can only be achieved via the development of tools to model and analyze the fundamental thermofluid processes that govern the operation of products utilizing combustion processes. For example, as new materials are used in the production of stationary and mobile engines, it will be possible to increase the combustion temperature attained in these engines, which will call for analysis of high-temperature turbulent combustion stabilization and efficiency. Computational methods can substantially limit the time and cost required for such an analysis.
The utilization of computational models to study combustion dynamics is not limited to situations in which there is a need to economize the time and cost. There are cases when one cannot afford to perform experiments or to wait for results in order to reach decisions based on a trial-and-error strategy. Such cases include the aquatic and atmospheric transport of pollutants, their effect on climatic change, and the ecological and biological systems of the earth. Complex chemical reactions accompany this transport and may lead to the formation or destruction of compounds detrimental or necessary for the critical balance that has been maintained over the decades for life to continue and flourish. The depletion of ozone in the upper atmosphere, the formation of acid rain, the release and buildup of formaldehydes, the chemical contamination of soil, ground water supplies and harbors, etc., have been recognized only after years of damage caused partly by a lack of understanding of reacting flow physics. The current effort to reverse the trend by eliminating the sources and cleaning up some of the afflicted areas using newly discovered

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processes has been supported by, and should continue to witness, extensive research activities in the field of reacting flow. It is expected, as mentioned above, that this research will benefit extensively and directly from continued developments in computational fluid mechanics and combustion.
Figure 10.1 shows results obtained by a method based on combining the better features of Lagrangian vortex methods and Eulerian finite element methods to study the evolution of a reacting shear layer in three-space dimensions.9 Besides being a canonical problem in turbulent combustion physics, reacting shear layers pose a severe challenge to computational methods since they possess all the properties that cause the breakdown of simple schemes: time-dependence, changing spatial length scales and overall topology, and the generation of strong strain fields and multiplicity of physical instabilities. Figure 10.1 shows the distortion of the computational grid, which also reveals the evolution of the reaction front as different modes of flow instability grow into their nonlinear stages, and confirms the adaptivity of the method. This adaptivity is crucial for the long time stability and accuracy of the computation at the lowest possible cost.
9
Knio and Ghoniem, 1990, Journal of Computational Physics, Vol. 86, p. 75.

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Figure 10.1