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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer Prediction of Mixing and Reacting Flow Inside a Combustor A. L. De Bortoli * Department of Pure and Applied Mathematics, UFRGS, Porto Alegre, PO Box 15080, Brazil Email: [email protected] Abstract The aim of this work is the numerical solution of mixing and reacting flow inside a combustor where the fuel and the air enter it in separate streams. The model approximates the overall and irreversible reaction between two species and the reaction-rate is controlled by temperature-dependent Arrhenius kinetics. Numerical tests, for governing equations discretized by the finite difference explicit Runge-Kutta five-stage scheme based on fourth order space and third order time approximations, are carried out for Reynolds 10000, Damk¨ohler 300, Zel'dovich 10, Heat Release 10 and Prandtl and Schmidt values both equal to 0.7. The results contribute to obtain a better understanding of the mixture-reaction behaviour which is common in combustors. Key words: diffusion flame, combustor, finite difference, Runge-Kutta INTRODUCTION Chemical-reacting flows are characterised by rapid and exothermic reactions. Such rapid, exothermic and complex convective-diffusive-reactive process presents many scales which are related to velocity, length, time, energy and vorticity. While big scales are geometry dependent, the small scales depend on the energy dissipation process [1]. The field of combustion is usually divided into non-premixed and premixed combustion. The majority applications of technical interest are classified as non-premixed and turbulent. Therefore, in practice, the fuel and the oxidiser are not perfectly mixed before burning, then the combustion process turns more pollutant formation (less efficient). Mixing is intensified by flame-vortex interactions [2]; the heat release distribution exerts a significant influence on the flame evolution and on turbulence. To understand the turbulent field it is necessary an accurate prediction of the turbulent velocity field. Turbulent mixing plays an important role in non-premixed combustion. It changes the density, temperature, heat capacity, molar mass and also the mixture transport properties [3]. The mixing can be divided in, at least, three levels: The first can be considered that where dispersion and mixing are driven by the turbulent flow, so correct mixing is not required to describe the flow dynamics. In the intermediate level the mixing is coupled to the fluid dynamics; therefore, the mixing must be captured correctly. The third level occurs when the mixing produces changes to the fluid: at its composition, density, enthalpy and pressure, like in the combustion phenomena; therefore, at this level the mixing must be captured correctly [3]. Turbulence effects, for moderate to high Reynolds values, seems to be best modelled using Large-Eddy Simulation (LES), where the eddy resolution is determined directly by the grid. Reynolds Averaged Navier-Stokes (RANS) is employed to steady flows, while LES and Direct Numerical Simulation (DNS) are preferred to unsteady flows; the last technique is the most expensive and the grid resolution is usually not fine enough to resolve the internal structures of the flame. Using LES the big scales are captured while the unresolved subgrid scales need to be modelled. Usually RANS combustion models are adapted to LES modelling. Moreover, combustion models may turn very complex. The complete reaction mechanism of the methane-air, for example, involves over 30 chemical species and more than 300 elementary reactions [4]. For iso-octane oxidation there appear 3600 elementary reactions and 860 chemical species [5]. Therefore, reduced reaction mechanisms are usually employed to analyse combustion due to difficulties to establish general chemical kinetics models. This work develops a numerical technique for the solution of mixing and reacting flow inside a combustor where the fuel and the air enter it in separate streams, as shown in Fig. 1. The numerical model is based on the finite difference ⎯ 284 ⎯ explicit Runge-Kutta five-stage scheme for third order time and fourth order space approximations. Numerical tests are carried out for Reynolds Re = 10000, Damköhler Da = 300, Heat Release He = 10, Zel'dovich Z = 10 and Prandtl and Schmidt both equal to 0.7. Such non-dimensional values are assumed to be the same and constant for all species. Figure 1: Burner geometry GOVERNING EQUATIONS AND SOLUTION PROCEDURE The model considers the following hypothesis: the Mach number is low, the pressure remains almost constant and the heat losses to the walls are negligible [6, 1]. Simulations are based on the two-dimensional set of reacting Navier-Stokes equations. Although the flow is threedimensional in nature, much insight is gained when analysing a two-dimensional situation, which allows to obtain a better understanding of this complex mixing and reacting flow with reduced computational costs. Moreover, the model assumes the finite rate Arrhenius kinetics hypothesis Contribution arising from radiation is considered negligible; it is more important for large flames as in furnaces or in building or wildland fires [7]. For large density variations it is convenient [8] to introduce the density weighted filtered quantities, which are defined . The Favre filtered reacting Navier-Stokes equations are, for the problem of interest, given by [9]: by Mass Conservation Momentum Energy Chemical Species where Re is the Reynolds, Da the Damköhler, Sc the Schmidt, Pr the Prandtl, Z the Zel'dovich and He the heat release is the averaged or filtered reaction rate of species, the filtered parameter; ⎯ 285 ⎯ stress tensor and velocities and the temperature comes from the state relation the mean resolved strain rate. The incognits are the mixture density, , the fluid , the chemical specie mass fractions and and the pressure which Viscosity is assumed to be temperature dependent and is approximated as [8, 10]. The sub-grid terms are conveniently modelled with an eddy viscosity as where is the eddy viscosity and C =0:1 the Smagorinsky constant. There are many subgrid models, but it seems that there is no established model for all mixing and reacting flow situations. The Smagorinsky model with a Van Driest Damping function has some drawbacks, but helps to simplify the analysis. Observe that the governing equations and the boundary conditions need to be high order approximated; it is common practice to employ fourth to six order for doing that. When the central finite difference scheme is employed, for a general finite difference grid point (i, j) in the computational domain, results the conventional fourth order space derivative approximations for a generic variable and similarly for other derivatives [11]. Integration in time is performed using a third order low storage Runge-Kutta time-stepping scheme. After these approximations results for the species mass fraction (Eq. 5), for example, with and in a similar manner for the other terms and derivatives. The use of fourth order space approximation is normally sufficient for many mixing and reacting flow situations. Certainly, such order is required not only for the interior domain, but also for the boundary conditions. ⎯ 286 ⎯ BOUNDARY CONDITIONS For the geometry definition the virtual boundary technique is employed. The great advantage of using this technique is its ability to model geometries without the necessity of coordinates transformation, using simple Cartesian grids [12]. Although the uniform Cartesian grid allow considerable simplification compared to the boundary fitted technique, there remains the necessity to find the best quantity of grid nodes to represent a great variety of length and time scales present in the flow near to the walls of the combustor. Consider the burner as shown in Fig. 1. The fuel and the air enter the domain in separate streams and the heat releasing reactions are considered to be fast. Observe that the physical boundary conditions can only be related to incoming waves while waves travelling from the inside of the domain to the outside are completely determined by the interior field [13]. LES computations have already shown that the flow can be sensitive to the outflow conditions [14]. Therefore, the boundary conditions can be summarised in non-dimensional form as follows: For south and north boundaries (for n the normal to the combustor surface): For west boundary: except at fuel injection place where = 1 (parabolic), and . For east boundary: ; the pressure is set equal to 1. Perfect where uc is the convective velocity and non-reflecting boundary ctonditions at the outlet are not possible because a reflected wave is produced to bring p to at the outlet [1]. when p is different from NUMERICAL RESULTS A jet is formed by admitting that a fuel stream enters an aired environment with a parabolic velocity profile, as indicated in Fig. 1. The computational domain corresponds to a rectangle and the grid contains 133×69 points; the time-step was chosen to be 10−6. Before solving the flow inside this combustor, some results were obtained for a simplified combustor geometry [9]. Fig. 2 left shows the velocity map based on Cp formation for Z = 10, Da = 300, He = 10 and Re = 10000. As it is known, the structure of the diffusion flame depends on the time needed to consume the reactants [15], that means on the Damköhler value. This flow turns rapidly asymmetric; such turns evident when analysing the velocity profiles for each combustor cross section. After 0:4s travelling waves are seen reflecting inside the combustor; between 1.6 and 2.8 s the central jet diffuses and oscillates, increasing the mixing, and after 4 s the flow turns turbulent. Fig. 2 right shows the product formation contours till 4 seconds. Observe that the central jet turns unstable, increasing the mixing rapidly. This figure confirms that the inflow and outflow boundary conditions do not introduce any significant spurious perturbation to the interior domain, as desired. Fig. 3 left presents the velocity profiles inside the combustor at position x =3L=4 and Fig. 3 right at position x =L =1. Observe that after 2s the jet of fuel reaches the combustor position x = 3L=4 and the component u remains growing till 3.0 s; after it the velocity profile tends to oscillate considerably. At position x = L, on the other hand, the gas expansion is best seen. During the first 1.0 s the gas expand almost 50% and till 4s this mass expand more than 100%; the velocity profile changes considerably from 1.0 to 4.0 s. Such behaviour turns interesting, but time-consuming to be predicted with the increase of the Reynolds value. It is clear that, after 1.0s, the principal jet distorts inside the combustor because the vortices are able to change the mean flow configuration. Finally, Fig. 4 shows the velocity components fluctuations at position (x = L=2; y = H=4) inside the combustor. This picture clearly indicate the flow unpredicability, as because such components change value not obviously in a short period of time. ⎯ 287 ⎯ Figure 2: Velocity map (left) and product contours (right) for the flow inside the combustor till 4s Re = 10000, Da = 300, Z = 10 and He = 10 Figure 3: Velocity component u at position x = 3L=4 (left), at position x = L (right) till 4 s Figure 4: Velocity component fluctuations at position x = L=2, y = H=4 ⎯ 288 ⎯ CONCLUSIONS A computational model is developed for coupling fluid dynamics, chemistry and heat transfer for the mixing and reacting flow inside a combustor. Computations are performed to obtain a better physical understanding of this mixture/reaction behaviour. Consistent results are obtained showing that the model is able to follow the non linear behaviour of the mixing and reacting progress for reasonable values for gaseous hydrocarbon chemistry. Besides, it is shown that the mixture increases when starting the reaction. After around 1 s the jet turns unstable and the recirculations diffuse the central jet. These vortices are small and they decrease size with the increase of the Reynolds value. It is clear that the asymmetry of the axial velocity profile along the duct centerline is a consequence of the diffusion which is by the flow expansion accelerated. In conclusion, the numerical results obtained here indicate that there are many interesting physical phenomena associated with mixing and reacting flows which are untill now not totally understood. Such understanding is essential to have real progress in the solution of turbulent combustion whose applications in aircraft engines or jet engines are obvious. 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