Energy & Fuels 1993, 7,817-826
817
Linear Eddy Modeling of Turbulent Combustion P. A. McMurtry’ Department of Mechanical Engineering, University of Utah, Salt Lake City, Utah 84112
S . Menon Department of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0150
A. R. Kerstein Combustion Research Facility, Sandia National Laboratories, Livermore, California 94551-0969 Received April 8, 1993. Revised Manuscript Received September 4, 199P
The use of the linear eddy mixing model in application to reacting flows is discussed for several different combustion geometries. The unique feature of this model is the explicit distinction made among the various physical processes (convection, diffusion, and reaction) at all scales of the flow. This is achieved by resolving all relevant scales of motion through a reduced, one-dimensional statistical description of the scalar field in a linear domain. The advantages of this modeling approach over other conventional modeling approaches are pointed out. Applications to both “stand-alone* formulations and subgrid formulations for use in large eddy simulation (LES)are discussed.
1. Introduction
Accurately predicting the reactant conversion rate ranks among the most challenging problems in turbulent combustion. The difficulties are a result of several factors. First, it is important to recognize that the overall combustion process depends on several distinctly different physical mechanisms. These include turbulent stirring, molecular diffusion, and the reaction kinetics. Further complicating the issue is the extreme range of length and time scales over which these processes act. Turbulent stirring acts on scales ranging from the integral scale down to the Kolmogorovscale. Molecular diffusion, on the other hand, acts most effectively at the smallest scalar length scales of the flow. These diffusion scales (Batchelor scale) are on the order of the Kolmogorov scale for gases but are at least 1 order of magnitude smaller for most liquids of practical interest. The combined effects of turbulent stirring and molecular diffusion constitute the process that bring reactants together at the molecular level where chemical reactions ultimately occur. For hydrocarbon flames, the energy-releasing reactions occur on time scales much faster than the time scales characterizing the largescale energy-containing motions. Other reactions, notably those characteristic of many of the secondary, pollutant formation processes,may occur on much slower time scales, resulting also in awide variation of length scales describing the reaction zones. In the followingdiscussion of scalar mixing and reaction, it is assumed that the statistics of the velocity field, as described by the conservation laws for momentum and mass, are known. This, of course, remains a difficult problem, but will not be discussed here. Many summaries and reviews of modeling turbulent momentum transport Abstract published in Advance ACS Abstracts, October 15, 1993.
can be consulted. However, to put some of the difficulties in modeling turbulent mixing in perspective, a few comments are in order. 1.1 Modeling Considerations: Momentum Transport vs Scalar Mixing. Although the exact form of the governing equations for momentum and scalar transport is well established, analytic solutions exist only for the most idealized configurations. Time-dependent numerical solutions of the exact governing equations (termed direct numerical simulation, or DNS)are also not possible due to the tremendous amount of computer resources required to resolve all the dynamically significant length and time scales; nor will such simulations be possible in the foreseeable future. As a result, it is necessary to introduce approximate models to describe turbulence phenomena. It is therefore the goal of the modeler to develop simplified representations of turbulent flowsthat reflect the essential physics in an acceptable and economic manner. In modeling the turbulent mixing process, one encounters difficulties not present in modeling momentum transport. These difficulties stem from the interactions among turbulent stirring, molecular diffusion, and chemical reaction at the smallest scales of the flow. A reliable model of the turbulent reaction process should therefore include and distinguish among these very different physical processes. This can only be accomplished in a rigorous manner if a detailed description of the small-scale scalar structure is given. This is fundamentally different from most approaches to modeling momentum transport in turbulent flows. In the latter case, the small scales act primarily as a dissipative mechanism for energy transferred from the large to small scales of the flow. As a result, the effects of turbulent momentum transport a t unresolved scales can be reasonably modeled in many applications by various eddy viscosity formulations. A similar parame-
0887-0624/93/2507-0817$04.00/00 1993 American Chemical Society
818 Energy &Fuels, Vol. 7, No. 6,1993
terization of the unresolved scalar fluctuations is neither sufficient nor correct since a realistic characterization of the small-scale processes is needed to predict finite rate combustion processes. 1.2 Modeling Considerations: Approaches to Scalar Mixing. In consideration of the above information, the objective in modeling turbulent mixing and reaction is then to provide a realistic representation of the underlying physical processes. This representation must be robust enough to be applicable under a wide range of fuels, flow field geometries, and operating conditions yet simple enough to be implemented using reasonable computational resources. It is clear from the extensive efforts currently directed at the modeling turbulent mixing and reaction that these goals are far from being achieved. Most numerical approaches to predicting turbulent combustion in engineering applications fall under one of two very general categories: moment methods and probability density function (pdf) methods. In the moment method approach, the flow field parameters are decomposed into their mean and fluctuating components. The objective of this approach is to solve for the mean value and some of the lower order moments of the flow field and chemical species variables. Details of the implementation of this method are well documented in the literature (e.g., Patterson, 1981). In the pdf approach the objective is to predict the pdf of the flow field and/or chemical species variables, either from a transport equation or using more approximate methods. From the pdf, the statistical properties of interest can then be computed. For a review of these techniques, see Pope (1985). Both methods have their advantages and disadvantages as briefly outlined below. In either approach, modeling assumptions must be made to close the ensuing set of equations. It is here that fundamental difficulties in predicting reacting flows become apparent. The modeling of chemical reactions in turbulent flows can be separated into two different aspects: (1) describing the molecular and/or turbulent diffusion and (2) modeling the chemical reaction itself. Pdf methods have the advantage that the chemicalreaction terms appear in closed form. The significance of this cannot be overstated. However, modeling the molecular diffusion terms is the major stumbling block in this approach. Of the popular models used, for example, those based on different variations of Curl's (1963) coalescencedispersion model, none contain the appropriate physics to provide reliable predictions in a wide range of flow conditions. In moment methods, both the chemical reaction term and turbulent stirring must be modeled. Owing to the complexity and nonlinearity of the reaction rate term, overly simplified approaches must be taken. Reasonable approximations are possible when chemical reactions can be considered infinitely fast. These approximations hold for many of the energy-releasing hydrocarbon reactions but not for most of the important pollutant-forming reactions. The turbulent fluxes are modeled by gradient diffusion assumptions, in the same manner as the molecular diffusive fluxes. This latter modeling issue points to one of the inherent shortcomings of the gradient diffusion modeling of the turbulent fluxes. In particular, by taking this approach, it is not possible to distinguish between the effects of molecular diffusion and turbulent stirring at the smallest scales of the flow. In high Reynolds
McMurtry et al. number applications, the turbulent fluxes are assumed to dominate over the diffusive fluxes, end there is no explicit treatment of molecular diffusion. This is a common deficiency of practically all turbulent mixing models. Namely, the distinction among the various physical process acting at the small scales is lacking. As a result, our ability to accurately predict turbulent mixing and reaction processes is not satisfactory. Recently, a new approach to modeling turbulent mixing has been introduced which focuses specificallyon the smallscale mixing effects. This model, termed the linear eddy model, has several unique features which combineto render a mechanistically sound representation of the mixing process (e.g., Kerstein 1988,1991). In particular, the model explicitly distinguishes among the effects of turbulent stirring, molecular diffusion, and chemical reaction at all scales of the flow. This is achieved by reducing the description of the scalar field to one spatial dimension. This domain provides a detailed statistical representation of the scalar field. Specifics of the model and its implementation are described in the next section. The application of the model to both scalar mixing problems in simple geometries and realistic combustion configurations is described insection 3. These applications involve the use of the linear eddy model as a stand-alone mixing model to address specific issues of turbulent reactingflows and-its implementation within comprehensive predictive schemes. 2. The Linear Eddy Model
The development of the linear eddy model was motivated by the recognition that explicitly distinguishing among the different physical processes of turbulent stirring, molecular diffusion, and chemical reaction at all scales of the flow is necessary for an accurate representation of turbulent combustion. The most important feature of the linear eddy model is its capability to explictly differentiate among these different physical processes. This is achieved by a reduced one-dimensional description of the scalar field, which makes it possible to resolve all length scales of the scalar field, even for flows with relatively high Reynolds and Schmidt numbers. The physical interpretation of the one-dimensional domain is dependent on the application considered. For example, in works concerning turbulent round jets, the domain has been both treated as a radial line transverse to the jet and as an axial line oriented in the streamwise direction. In studies in homogeneous turbulence, the line has been treated as a space curve aligned with the scalar gradient. This interpretation of the one-dimensional domain is discussed further in some of the applications that follow. Along this one-dimensional domain, a wide range of statistical information about the scalar field can be computed, including both single and multipoint statistics. The key to the model performance lies in the manner in which the real physical mechanisms of turbulent mixing are represented. Below, some details of the model and ita implementation are given. For more complete discussions, the reader is referred to the growing body of literature on this modeling approach (e.g., Kerstein, 1988,1991,1992; McMurtry et al., 1993; Menon et al., 1993a). Along the one-dimensional domain, molecular diffusion and turbulent stirring are carried out as follows: Molecular diffusion is treated explicitly by the solution of the diffusion equation along the linear domain
Linear Eddy Modeling of Turbulent Combustion
C
0
X
L
Figure 1. Triplet map illustration. (a) Uniform scalar gradient represented in one dimension. (b) Scalar field distribution after asingle rearrangement by the triplet map. The mapping involves the following: (1) selecting a location and segment size for rearrangement,(2) making three identical compressedimages of the original scalar field, (3) replacing the original segment by the three compressed images, and (4) inverting the center image. (c) The rearranged scalarfield after acted on by molecular diffusion.
where 4 is the particular scalar under consideration and D is its diffusion coefficient. Thus, molecular diffusion is treated exactly subject to the interpretation that the statistics of a three-dimensional mixing process can be representedwithinthe reduced dimensionalityof the linear eddy model. When chemical reactions are considered, the chemical source term can also be treated explicitly by solution of -a4 =w
at where w is the reaction rate. Note that since the flow field is resolved in the one-dimensional domain, no modeling is required of the processes described by eqs 1 and 2. The influence of turbulent convection is modeled in a stochastic manner which is carried out by random rearrangements (subject to certain rules which are derived below) of the scalar field along the domain. The effect of these rearrangements on the scalar field is to induce an apparent diffusivity of the scalar field, analogous to a random walk analysis. The rules by which these rearrangement processes are carried out are then established such that the diffusivities induced by the process on the scalar field result in a diffusivity that a fluid particle would experience in a real turbulent flow. In particular, high Reynolds number scalings based on the Kolmogorov cascadeare used in determiningthe parameters that govern the stirring (rearrangement) process. Specifically, two parameters govern the stochasticstirring process: A, which is a frequency parameter determining the rate of occurrence of therearrangement events (stirring),and f ( l ) , which is a pdf describing the size distribution (eddy size) of the
Energy & Fuels, Vol. 7, No. 6, 1993 819 segments of the flow which are rearranged. In general, both parameters are functions of space z and time t . For steady homogeneous turbulence, they are independent of x and t . To derive the explicit expressions for the size and frequency of rearrangement events, a particular rearrangement mapping must be chosen and appropriate scalinglaws for this process must be invoked. In the linear eddy references cited in the Introduction,it has been shown that the triplet map has several features that suggest its choice in these applications. The mapping is illustrated and explained in Figure 1. Some features of this mapping pertinent to the turbulent stirring process should be noted. First, the triplet map results in a tripling of the scalar gradients within a selected segment, analogous to the effects of compressive strain. Furthermore, a multiplicative increase in level crossings of a single scalar value results. This is analogous to the increase in surface area of a specified scalar value-a characteristic feature of turbulent mixing processes. Thus, two main features of turbulent mixing are accounted for with this mappingthe increase in surface area and the associated increase in the scalar gradient. As mentioned above, the stirring events induced by this mapping introduce a random walk of a marker particle on the linear domain. In general, the diffusivity associated with a random walk in one dimension is given by Dt = N(x2)/2, where ( x 2 ) is the mean square displacement of a particle and Nis the frequency of events. For the triplet map, the mean square displacement of particlesassociated with a rearrangement of a size 1 eddy can be shown to be equal to ( x 2 ) = (4/27)12(Kerstein, 1991). Using the fact that N = A1 results in a marker particle diffusivity of D, = (2/27)XP. Turbulent mixing is not a single length scale process but consists of a wide range of dynamically active scales (eddies) ranging from the integral scale, L , down to the Kolmogorov scale, q. If f(1)dl is the fraction of segments (eddies) in the range ( I , 1 + dl) then the diffusivity associated with eddies in the size range (I, 1 + dl) is D,= ~2X l ~ f ( l ) d l (3) and the total diffusivity associated with all eddies of size 1 and smaller will then be given by (4)
Equations 3 and 4 describe the diffusivity associated with the random rearrangement process itself. To relate this process to mixing in a turbulent flow the parameters f ( 1 ) and X must be chosen appropriately. This is accomplished by equating the diffusivity derived in eq 4 with accepted scalings for the turbulent diffusivity. Dimensional analysis yields Dt(L) = L2/Tt where Tt is a characteristic time scale and L is the integral scale. This time scale of the turbulence is Tt = L fu', where u' is a characteristic velocity fluctuation, giving D,= u'L = vRe, (5) where Y is the kinematic viscosity. The 1 dependence of eq 4 thus scales as the Reynolds number based on 1. Application of Kolmogorov scalings based on high Reynolds number flow then yields Since X is not a function of 1, eq 6 implies that f(1) = 1-813.
820 Energy & Fuels, Vol. 7, No. 6,1993
McMurtry et al.
The fact that the probability that an eddy has a size between the Kolmogorov scale and the integral scale is, by definition, unity gives the final expression for f(1): (7)
The value of X can now be determined by substituting eq 7 into eq 4 and equating with eq 5. Solving for X yields
For high Reynolds number flow,L >> g. The leading order approximation to eq 8 is
Note that all order one constants that appear in the previous scaling relations used in the above derivation are set equal to unity. To obtain quantitative comparisons with experiments, model parameters must be calibrated by adjusting order unity coefficients. Although this introduces a level of empiricism in the model, the representation of the underlying mixing processes is physically sound. With these parameters specified, a model simulation is carried out as follows. The scalar field is initialized along the linear domain in a manner consistent with the configuration under study. Along this domain, the effects of molecular diffusion and chemical reaction are implemented as a continuous process as described by eqs 1and 2. At randomly selected times governed by the rate parameter A, diffusion and reaction processes are interrupted by rearrangement events. The size of the domain to be rearranged is randomly selected from the pdf f(1). For homogeneous flow, the location of the event can be chosen randomly. For nonhomogeneous flows, X will be a function of position so that the location of an event will be selected by a random sampling of the X distribution. This process continues until a specified time has elapsed. Because stirring is implemented as a random process (subject to the constraints derived above), the statistical properties of the flow are different for different realizations based on the same initial configuration. This is a feature characteristic of all stochastic models. Statistical properties of the scalar field are therefore obtained by averaging results over many separate individual realizations of the scalar field. Because of the one-dimensional formulation of the linear eddy model, the physical parameter range that can be accurately studied is well beyond that of current direct numerical simulation (DNS) capabilities, which require full resolution of the scalar and hydrodynamic fields in three spatial dimensions. The model therefore provides a unique tool to investigate important mixing and reaction phenomena. Simulations can be conducted for a fraction of the cost of full DNS while at the same time maintaining the integrity of high Reynolds number turbulence. It is also important to reiterate that the results obtained using the linear eddy model are most appropriate for highReynolds-number flows, as the model is built upon scaling laws which assume just that. In particular, some caution must be exercised in comparing linear eddy model results with low-Reynolds number results, for example, DNS. Despite this, the linear eddy model has acceptably
reproduced many features of scalar mixing predicted by DNS (McMurtry et al., 1993)and has been used to further interpret DNS results (Cremer et al., 1993). 3. Applications
The linear eddy model has been employed as both a stand-alone model to address specific issues of turbulent mixing and as a submodel in more comprehensive applications. In stand-alone applications, the velocity field statistics are an assumed input to the model, and the scalar field evolves subject to the description of the flow (velocity) field. In other words, the flow field is completely specified prior to computing the scalar field statistics. The application of the model to several different flow field configurations and geometries will be discussed. As an element in a more comprehensive predictive scheme, the model is used in conjunction with the solution of the flow field. This aspect of the model is still in its early stages of development, but progress has been made, with encouraging results. Examples of this application will also follow. 3.1 Stand-Alone Application of Linear Eddy to Scalar Mixing. The purpose of this paper is to illustrate examples and potential applications of the linear eddy model to turbulent reacting flows. However, it is instructive to first discuss some of the applications of the model to passive scalar mixing. 3.la. Homogeneous Turbulence. One of the most significant applications of the linear eddy model in establishing its capabilities as a viable tool for studying turbulent mixing is its application to scalar mixing in a homogeneous turbulent flow with an assumed scalar gradient (Kerstein, 1991). The linear eddy domain in this application is interpreted as a space-curve aligned with the local scalar gradient. A uniform scalar gradient along the one-dimensional domain was maintained by enforcing jump periodic boundary conditions. In this study, several features of the scalar field were considered. Computed pdf s of the scalar agreed well with experimental observations, including the appearance of exponential tails in the scalar pdf. The interplay between molecular diffusion and turbulent stirring was also shown to produce the correct scaling properties and dependence on Schmidt number. In particular, computed scalar power spectra obtained from the simulation reproduced the known behavior in the various wave number regimes, and scaling exponents governing the higher-order fluctuation statistics agreed well with measured values. Although it has been noted that the model is not free of artifacts (e.g., the representation of eddies by instantaneous events, onedimensional formulation, appearance of discontinuous derivatives), it captured many of the features of turbulent mixing in a very economical manner. 3.lb. Mixing in a Two-Stream Shear Layer. In application to shear layer mixing, the linear eddy model was used to study effects of Schmidt, Reynolds, and Damkohler number on the scalar statistics. The physical and model configurations are shown in Figure 2. In this application, the computational domain represents a transverse line moving with the mean flow. To carry out the model processes some experimental observations were noted. In particular, it is observed that the vorticity thickness of the shear layer is approximately half of the visual thickness. In the computations, the growth of the vorticity thickness is an empirical input determined from experimental observations. In this application the centers
Energy &Fuels, Vol. 7, No. 6, 1993 821
Linear Eddy Modeling of Turbulent Combustion EXPERIMENT
Air Entrainment
Primaly Fuel Stream
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Figure 3. Linear eddy domain for spatially developing jet. The domain is taken as a line down the axis of the jet. A volume, V(z), and diameter, L ( z ) ,are associated with each cylindrical cell. The first cell is taken as a fluid reservoir where fluid is injected into the flow.
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Figure 2. Application of the linear eddy model to the shear
layer confiiation. (Top) physical configuration. (Bottom) linear eddy representation. The linear eddy domain is taken to be a transverse line. The values 1 and 0 mark the initially segregated fluid in the two incomingstreams. Molecular diffusion acts to smooth out gradients. The rearrangement events model the turbulent stirring which acts to redistribute fluid elements without changing their internal state. of rearrangement events were confined to the vorticity layer. Within this region, the location for each event was selected randomly, and the segment sizes and event frequency were selected from the eddy size distribution described in eq 3 and rate parameter given by eq 4. This work addressed several different issues related to shear layer mixing. These included the dynamical origins for the observed structure of the scalar pdf in shear layers, the evolution of chemical product parametrized by Damkiihler number for reacting flows, and the dependence of Reynolds and Schmidt number on the amount of molecular mixing. Furthermore, this work provided additional evidence that the small-scale mixing processes can be modeled with principles that are independent of the geometrical configuration, while the application to a specific flow configuration can be achieved by specifying configuration-dependent inputs. Other studies concerned with mixing in round jets (Kerstein, 1990), the prediction of scalar statistics in multiple stream mixing configurations (Kerstein, 19921, the effects of Schmidt and Reynolds number on a decaying scalar in steady turbulence (McMurtry et al., 19931, and scalar fluctuation statistics in turbulent plumes (Kerstein, 1988,1992)have demonstrated that the linear eddy model is a powerful tool for studying turbulent mixing phenomena. The detailed information obtainable from this model in terms of both single- and multipoint statistics and its ability to represent mixing in a wide range of geometrical configurations is unmatched by any other mixing model. 3.2 Combustion Processes in Turbulent Jets. A further illustration of the ability of the model to treat different geometrical configurations plus the added complication of combustion is the application of the linear eddy model to turbulentjet flames (Kerstein, 1992;Menon et al., 1993a). In these two applications the linear eddy domain was taken to be a line down the axis of the jet. The discretization of the flow field is shown in Figure 3. Note that a volume and diameter, and hence a width Ax, is associated with each cell of the domain. This reflects the
a
7
x
X'
n X'
x'+L
Figure 4. Air entrainment process in linear eddy jet configuration. (a) The scalar distributiondown the centerlineof the jet prior to entrainment. (b) Entrainment is treated by inserting free stream fluid of uniform concentration into the domain at
random locations with a length scale L and at a frequency determinedfrom empirical entrainmentlaws. The injected fluid induces a downstream displacement to conserve mass. growth of the jet (a configuration-specific input) due to entrainment of ambient air. Fuel is introduced into the domain by inserting reservoir fluid into the first cell at a rate determined by the input mass flow rate. To maintain a mass balance, this process displaces the fluid in the jet to downstream cells. Air entrainment, on the other hand, occurs throughout the length of the jet. The entrainment process is governed by specified entrainment laws, in particular that of Becker and Yamazaki (19781, based on their experimental study of turbulent diffusion flames. It is interesting to note that by specifying the appropriate entrainment relationships in both the inertial- (near field) and buoyancy-dominated (far field) regimes, the effects of buoyancy are incorporated. This particular aspect is not rigorously accounted for in other existing mixing models. The length and time scales of the physical entrainment process are modeled by entraining air in finite parcels, with a size characteristic of the local jet diameter. This mass entrainment is accompanied by a subsequent displacement of all downstream fluid (Figure 4). The frequency of entrainment is determined from the entrainment laws mentioned above. In the model, this is implemented as statistically independent events by sampling an exponential distribution with a mean taken as the mean time between entrainment events. Two different combustion processes were studied: propane-air and hydrogen-argon-air. The chemistry was implemented assuming equilibrium so that the local
McMurtry et al.
822 Energy & Fuels, Vol. 7,No.6,1993
temperature and density were determined from a specified functional relationship of the mixture fraction. The implementation of combustion chemistry introduces another issue in the mixing and reaction process-thermal expansion resulting from heat release. This also induces streamwise fluid displacements which were accounted for by computing the appropriate volume change and transferring portions of the cell contents to downstream neighbors, with corresponding displacement of fluid farther downstream. The chemical process was taken a step further in the work of Menon et al. (1993a)who studied effects of mixing on NO formation in turbulent hydrogen-air and hydrogenargon-air diffusion flames. A reduced H2-air mechanism with a thermal NO mechanism was implemented within the linear eddy formulation. The advantage of the reduced mechanisms is that the number of scalars that need to be tracked during the simulation is minimized. The reduced mechanisms can be conveniently incorporated into the linear eddy subgrid model formulation. In the hydrogenair problem treated there, 10 elementary reactions were considered. The total number of scalars in the H2-air mechanism is 10,of which seven are active chemical species (H2,02, H20,0, H, OH, H o d , plus temperature, density, and pressure. In addition, a single-step Zeldovich mechanism for the production of thermal NO was incorporated. By using the reduced mechanism (developed by Chen and Kollmann (1990)),only four scalars are tracked throughout the simulation: the mixture fraction, 4, a progress variable describing the extent of reaction, n,the enthalpy, h, and the NO concentration. Therefore, four separate and parallel linear eddy calculations corresponding to each of the four scalars were carried out in the simulation. This generalization is a straightforward extension of the previous applications. Each of the cell contents undergoes the above-mentioned stirring, diffusion, and reaction processes, where the source terms for the progress variable, enthalpy, and NO concentration were obtained from the reduced mechanism. Simulations were carried out to study the NO concentration downstream of the jet. The instantaneous mixture fraction and NO concentration down the axis of the jet at one instant in time are shown in Figures 5 and 6. From a comparison of these two figures it is seen that most of the NO is formed at mixture fractions near stiochiometric. (The stoichiometric mixture fraction for the 78% H222% argon air combustion case shown in the figure is 4 = 0.1636). The decrease in NO concentration further downstream indicates that the combustion is completed and the jet has become depleted with air. Also of interest is the NO emission index (defined here as grams of NO per kg of fuel) shown in Figure 7 for two cases with different inlet jet velocities. The index increases sharply at about xJd, = 15 for the low speed jet and xfd, = 25 in the higher speed jet. Of interest is the significant decrease in the NO emission index in the far field with increasing jet velocity, in agreement with experimental observations (Chen and Driscoll, 1990). The advantage of the linear eddy model in the applications shown above is that it allows for detailed parametric studies to be conducted while maintaining a rigorous treatment of the small scale turbulent fluctuations. Efforts are currently underway to study effects of controlling and modifying the air entrainment on the NO production. Calculations in which the air entrainment (or injection)
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Figure 6. Instantaneousaxial distributionof NO concentration for same case as Figure 5. Units are kmol NO/kg mixture, ubt = 75 m/sand jet diameter do = 5.2 mm. mass fraction is varied, the fuel types are modified, the physical parameters defining the flow field (e.g., Reynolds, Schmidt numbers) and combustion chemistry (DamkBhler number) are varied, and the geometrical configuration of the flame is controlled are all economically feasible with this approach. 3.3 Application to Large Eddy Simulation. The uses of the linear eddy model described above have all been "stand-alone" applications in which the statistics of the velocity field were assumed inputs which defined the mixing model parameters (i.e., A, f ( l ) , and the domain interpretation). These applications have been extremely useful in addressing specific issues concerning the turbulent mixing and reaction and can be used as an aid in specific combustion design applications. However, there is also an existing need to improve the mixing and reaction modeling approaches that are used in more comprehensive numerical treatments of combustion processes in industrial applications. These applications involve predictions of
Linear Eddy Modeling of Turbulent Combustion 45
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Energy & Fuels, Vol. 7, No. 6,1993 823 /
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Figure 7. NO emission index: (a) ubt = 75 m/s; (b) uht = 225 m/e.
combustion and pollutant formation in large-scale industrial burners, propulsion devices, and waste incineration, to mention only a few. The current models used to describe mixing and reaction processes in combustion codes intended for industrial application all exhibit important omissions (as discussed in the introduction) in their descriptions of turbulent combustion. Below, we discuss some of the current research intended to extend the linear eddy model as an element of comprehensive codes. Among the numerical approaches to studying turbulent flow phenomena, including turbulent combustion, DNS has proven to be a particularly useful tool. The basic idea of this approach is to provide a detailed description of the flow by numerically solving the exact, unsteady governing equations of mass, momentum, energy, and chemical species. (For an excellent and detailed review of the application of DNS in combustion applications, see Givi (1989).) The appeals of this approach are also the source of ita limitations. By resolving all relevant length and time scales of the flow and species fields, no modeling assumptions are necessary. This implies that the output of a well designed simulation can be confidently treated as accurate data for that particular configuration. Unfortunately, resolution requirements restrict the relevant
parameter range that can be studied to a narrow range of values, well outside the range of realistic combustion applications. Although DNS has provided much insight into the physics of turbulent flow phenomena, its impact on the design and analysis process of practical combustion processes has been limited. A related approach, but one with more potential for application in real combustion systems, is termed large eddy simulation (LES). The basic idea of LES is to solve explicitly for the unsteady time development of only the largest scales of the flow and to model the effects of the small scales and their influence on the development of the large scales. A collection of papers on the most recent developments in LES for both reacting and nonreacting flows can be consulted for details on the methodology (Galperin and Orszag, 1993). The philosophy behind this approach is that the small-scale structures of the flow have a more universal behavior than the large scale structures and should thus be easier to model. Modeling of the small scales is referred to as subgrid modeling. This idea has worked respectably well in applications to nonreacting flows. In applications involving turbulent mixing and chemical reaction processes, the intricate details of the small-scale processes and how they affect the mean flow properties cannot be conveniently described in terms of the large-scale structure of the flow. This reflects the importance of realistically describing the interactions among the different physical processes acting at the smallest scales of the flow, i.e., turbulent stirring, molecular diffusion, and chemical reactions. Because of these difficulties, LES has not yet proven to be an effective tool for studying turbulent combustion problems. Recently, a few new subgrid modeling approaches have been suggested. These involve using ideas based on probability density function methods (Madnia and Givi, 1993) and also some approaches utilizing the linear eddy model (Menon et al., 1993b,c; McMurtry et al., 1992). In keeping with the focus of this paper, only the use of the linear eddy model as a subgrid model in a large eddy simulation is discussed here. 3.3a. The Linear Eddy Subgrid Model Formulation. The basic idea of the linear eddy subgrid model is to describe the subgrid mixing and reaction processes with an individual linear eddy simulation in each grid cell of the discretized computational domain. Within each computational grid cell, the linear eddy simulation represents the mixing, diffusion, and reaction that occur at the small scales of the flow. The advantage of this approach is that an explicit description of the small-scale mixing mechanisms is achieved. This is a result of the one-dimensional formulation which makes it feasible to resolve the small length scales below the grid size. Specifically, if the ratio of the smallest scalar length scale to the grid size is 1/N, then the scalar information within a grid cell is described with a one-dimensional array of length N, regardless of the spatial dimensionality of the problem. Fully resolved direct simulations would require an array of dimensionN3. The economy of using the linear eddy as a subgrid model is clearly apparent. Furthermore, the model provides a detailed description of the smallscale structure which is lacking in other parametrizations of mixing and reaction a t unresolved scales. The main element of the linear eddy subgrid formulation is the implementation of a separate linear eddy calculation in each grid cell. The model processes are parametrized
McMurtry et al.
824 Energy & Fuels, Vol. 7, No. 6, 1993
I
I$=
Figure 8. Schematic illustration of linear eddy subgrid model. The one-dimensional elements within each grid cell represent the ongoing linear eddy calculationsthat are performed in each cell. At each large eddy time step, linear eddy elements are transferred across neighboring grid cells, accounting for largescale transport. The arrows indicate the components of convective flux across the cell boundaries, which determine the amount of material transferred at each large eddy time step.
Figure 9. Contour of grid-averaged mixture fraction. Within each grid cell, the linear eddy elements are averaged to give the local scalar average value for eachgrid cell, t = 6. Contour interval
by the Reynolds number based on grid size. The processes of turbulent mixing (described by eqs 3 and 41, molecular diffusion (eq l),and chemical reaction (eq 2) are treated as described previously in the standard implementation. However, one other process is needed to couple the subgrid mixing process to the large-scale transport processes responsible for convection across grid cell surfaces. In implementations to date, this is achieved by splicing events, in which portions of the linear eddy domains are transferred to neighboring grid cells. The amount of material transferred across cell boundaries is determined based on the flux across each cell boundary, as computed from the resolvable grid scale velocity. (The computation of the grid scale velocity is a separate modeling problem which is not discussed here.) These splicing events occur at a frequency determined by the large eddy time step. This time step is muchgreater than the time step governing the convection-diffusion-reaction process in the subgrid. Implications of this with respect to the numerical implementation of the subgrid model will be mentioned later. In addition to the resolved velocity field, mixing across cell boundaries will occur due to turbulent fluctuations. To account for this, the splicing process also incorporates equal-and-opposite exchanges of material across neighboring boundaries. The size of these exchanges is a function of the turbulence intensity, and because the exchanges are equal and opposite, the mean flux is not affected by this fluctuating component. These model aspects are illustrated schematically in Figure 8. 3.3b. Application to Nonpremixed Flames: H z A i r Combustion. The first application of linear eddy as a subgrid model addressed Hz-air combustion (McMurtry et al., 1992). The flow field was taken to be a diffusing, two-dimensional vortex which could be represented in analytical form. This flow field was specified on a 32 X 32 grid. In each grid cell, a separate linear eddy calculation was performed between time steps in the large eddy simulation. Although the flow field was specified analytically and not solved by its own transport equation, this had no impact on the implementation of the subgrid model and provides a simple configuration in which to evaluate the performance of the model.
Several other simplifications were adopted in this study. Both the grid cell size and the subgrid root mean square velocity fluctuations, u’, were specified as constants. Transport properties were also specified as constants. The chemical mechanism used was the reduced Hz-air combustion mechanism described in section 3.2. The spatial domain was [-7r,7r3 in both the x and y directions. A1inea.r eddy simulation with a maximum array size of N = 100 elements was implemented for each of the 32 X 32 grid cells, giving an effective Reynolds number of 5000 for the simulations. (This allowed for six linear eddy elements to resolve the Kolmogorov scale.) The computer code for this application was written in dimensional units. For details on the velocity field specification and boundary conditions, see McMurtry et al. (1992). The initial scalar field consisted of a mixture fraction 4 = 0 (pure air) for allx andy > 0 and r#~ = 1(pure hydrogen) for all x and y I0. The structure of the flame as indicated by mixture fraction contours is shown in Figure 9. This figure is based on a representative scalar concentration for each grid cell obtained by averaging over the linear eddy array for that cell. The characteristic roll-up structure is reproduced by the exchange of scalar elements across grid surfaces. These exchange (splicing) events are seen to accurately represent the large scale roll-up of the shear layer. The temperature field at t = 6 is shown in Figure 10, and the NO concentration at the same time is shown in Figure 11. The stoichiometric temperature of 2300 K occurs at the flame front. The flame is located at the outer regions of the vortex, extending into the air side of the shear layer. The penetration of the flame into the air side of the stream is a result of the fuel rich initial condition. (The free stream reactants for this test were equal moles of hydrogen and ambient air in each of the two streams.) In this application, effects of combustion heat release on the velocity field and on transport properties was not addressed, although such effects can be accounted for within the linear eddy model (Kerstein 1992; Menon et al., 1993a,c). 3 . 3 ~ Application . to Premixed Flames in a 3 - 0 NavierStokes Solver. The linear eddy subgrid model has also
1
x
= 0.1.
Energy & Fuels, Vol. 7, No. 6, 1993 825
Linear Eddy Modeling of Turbulent Combustion
T=300K
.Y 4
Figure 10. Temperature concentration contours for the hydrogen-air flame a t time t = 6. Maximum contour = 2400 K. Contour interval = 300 K.
the application of the linear eddy model is the same as in the nonpremixed flame computations with one exception: the molecular diffusion mechanism is replaced by a flame propagation mechanism. Preliminary results of the fully coupled LES-linear eddy model have been reported (Menon et al., 1993~).Analysis of this coupled formulation is currently underway. Further simulations are required to quantitatively determine the overall accuracy of the method. Also underway are evaluations of the model under more realistic conditions. 3.3d. Application to Steady-State Combustion Simulations. A final note is in order regarding the use of the linear eddy model as an element of comprehensive steadystate combustion simulations. The mixing and reaction models currently used to predict species concentration in industrial-scale applications have not demonstrated the degree of reliability or robustness needed to provide accurate predictions. This is a consequence of the inability of the models (e.g., eddy diffusivity, coalescence-dispersion) to describe the underlying physical processes. The linear eddy model is potentially a realistic alternative to conventional modeling approaches in such configurations. Several formulations are under development. One application under development involves a combined Lagrangian Eulerian approach (Pit, 1993). In this approach, the velocity field is solved in an Eulerian manner using second-order moment closures. The scalar variables are treated in a Lagrangian manner by following fluid elements through the flow field. As opposed to similar conventional approaches, each Lagrangian fluid particle is now taken to have an internal structure described by the linear eddy model. As a result mixing and diffusion processes are accurately represented at the small scales. A detailed evaluation of this approach is forthcoming. Purely Eulerian approaches are also under development, but results have yet to be reported. 4. Conclusion
1
.
x
Figure 11. NO concentration (kmol NO/kg mixture), t = 6. Maximum contour = 0.0005, contour interval = 0.0001.
been implemented in a full three-dimensional NavierStokes solver in a premixed flame application (Menon et al., 1993~).In this work, the problems associated with employing a finite rate kinetics model were circumvented by using a thinflame model. This model involves defining a progress variable, G , that is governed by the equation (Kerstein et al., 1988) (5)
The flame is located along a specified isosurface G =.Go. The physics described by this equation indicate that the flame is convected by the velocity field u and propagates . advantage locallynormal to its surface at a velocity u ~The of the linear eddy model in this application is that the flame speed can be taken as the laminar flame speed since the model resolves the thin flamelets which are assumed to propagate at the laminar flame speed. Note also that
The inability to accurately predict finite rate chemical processes in turbulent combustion has highlighted the need for improved models of the turbulent mixing and reaction process. In this paper an alternative approach, the linear eddy model, has been outlined and discussed. Its application to problems of scalar mixing has demonstrated a unique capability to predict scalar field statistics in turbulent flows. Furthermore, the wide range of problems that have been studied using this formulation has demonstrated the robustness of the linear eddy modeling assumptions. In combustion applications, the linear eddy model provides an ideal tool to perform parametric studies. By extending the previous work to different fuel types and by specifying different flow field scenarios in the model formulation, the linear eddy model also promises to be a useful design tool. To date, most uses of the linear eddy model have been in a stand-alone formulation in which velocity field statistics are specified inputs to the mixing model. For the near future at least, this use of the linear eddy model will continue to be the most productive application. Work in progress or recently completed based on applications of the stand-alone linear eddy formulation includes studies of multistep chemical reactions, premixed flames, and combustion in plug flow reactors.
826 Energy & Fuels, Vol. 7,No. 6, 1993 More ambitious uses of the model involve its application as a subgrid closure in unsteady LES and conventional steady-state solvers. Although this application of the model is still in the early stages of development, it has the potential to be a valuable tool in engineering design of combustion processes.
Acknowledgment. This work of was funded by the National Science Foundation through a grant to the Advanced Combustion Engineering Research Center at the University of Utah and Brigham Young University and by the Division of Engineering and Geosciences,Office of Basic Energy Sciences, U.S. Department of Energy. References Becker, H. A,; Yamazaki, S. Entrainment, momentum flux, and temperature in vertical free turbulent diffusion flames. Combust. Flame 1978, 33, 123. Chen, J.-Y.; Kollmann, W. Chemical models of pdf modeling of hydrogen-air non-premixed turbulent flames. Combust. Flame 1990, 79, 75. Chen, J.-Y.; Driscoll, J. F. Nitric oxide levels of jet diffusion flames: Effects of coaxial air and other mixing parameters. 23rd. Symp. (Zntl.) on Combustion, the Combustion Institute (1991) 1990, 281. Cremer, M. A.; McMurtry, P. A.; Kerstein, A. R. Effects of turbulent length scale distribution on scalar mixing in homogeneous turbulent flow. Submitted to Phys. Fluids A. Curl, R. L. Dispersed phase mixing: I. Theory and effects in simple reactors. AZChE J. 1963, 9, 175. Galperin, B.; Orszag, S., Eds. Large Eddy Simulations in Complex Engineering and Geophysical Flows; Cambridge University Press, Cambridge, 1993. Givi, P. Model free simulations of turbulent reactive flows. Prog. Energy Combust. Sci. 1989, 15, 119. Kerstein, A. R. A Linear eddy model of turbulent scalar transport and mixing. Combust. Sci. Tech. 1988,60,391. Kerstein, A. R. Linear eddy modeling of turbulent transport. 11. Application to shear layer mixing. Combust. Flame 1989, 75, 397. Kerstein, A. R. Linear eddy modeling of turbulent transport. Part 6. Microstructure of diffusive scalar mixing fields. J. Fluid Mech. 1991, 216, 361.
McMurtry et al. Kerstein, A. R. Linear eddy modeling of turbulent transport. Part 4. Structure of diffusion flames. Combust. Sci. Tech. 1992,81, 75. Kerstein, A. R.; Ashurst, W. T.; Williams, F. A. Field equation for interface propagation in an unsteady homogeneous flow field. Phys. Rev. A 1988, 37, 2728. Madnia, C. K.; Givi, P. Direct numerical simulation and large eddy simulation of homogeneous turbulence. In Large Eddy Simulations in Complex Engineering and Geophysical Flows; Galperin, B., Orszag, S., Eds.; Cambridge University Press: Cambridge, 1993. McMurtry, P. A,; Gansauge, T. C.; Kerstein, A. R.; Krueger, S. K. Linear eddy simulation of mixing in a homogeneous turbulent flow. Phys. Fluids A 1993a, 5, 1023. McMurtry, P. A.; Menon, S.; Kerstein, A. R. A linear eddy subgrid model for turbulent mixing and reaction: Applicationto hydrogen& combustion. Proc. 24th Symp. (Zntn.) on Combustion,the Combustion Institute 199313, 271. Menon, S.; McMurtry, P. A.; Kerstein, A. R.; Chen, J.Y. A new unsteady mixing model to predict NOx production during rapid mixing in a duel stage combustor. To be published in J. Propul. Power, 1993a. Menon, S., McMurtry, P. A.; Kerstein, A. R. A linear eddy mixing model for large eddy simulation of turbulent combustion. In Large Eddy Simulations in Complex Engineering and GeophysicalFlows;Galperin, B., Orszag, S., Eds.; Cambridge University Press: Cambridge, 1993b. Menon, S., McMurtry, P. A.; Kerstein, A. R. A linear eddy sub-grid model for turbulent combustion: Application to premixed flames. AIAA 31btAerospace Sciences Meeting, AIAA Paper no. 93-0107, 1993c. Patterson, G. K. Application of turbulence fundamentals to reactor modeling and scaleup. Chem. Eng. Commun. 1981,8, 25. Pit, F. Modeling of Mixing for the Simulation of Turbulent Reactive Flows: Tests of Probabilistic Eulerian-Lagrangian Models (English Translation). Ph.D. thesis, University of Rouen, 1993. Pope, S. B. PDF methods for turbulent reactive flows. Prog. Energy Combust. Sci. 1985, 11, 119.
