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## pdf calculations for swirl combustors PDF CALCULATIONS FOR SWIRL COMBUSTORS

### BibTeX

@MISC{Anand_pdfcalculations,

author = {M S Anand and A T Hsu and S B Pope and M S Anand and A T Hsu and S B Pope},

title = {pdf calculations for swirl combustors PDF CALCULATIONS FOR SWIRL COMBUSTORS},

year = {}

}

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### Abstract

AIAA 34th Aerospace Sciences Meeting and Exhibit, Reno, NV Jan 15-18, 1996 Calculations are reported for recirculating swirling reacting flows using the joint velocity-scalar probability density function (pdf) method. The pdf method offers significant advantages over conventional finite-volume, Reynolds-average based methods, especially for the computation of turbulent reacting flows. The pdf calculations reported here are based on a newly developed solution algorithm for elliptic flows, and on newly developed models for turbulent frequency and velocity that are simpler than those used in previously reported pdf calculations. Calculations are performed for two different gas-turbine-like swirl combustor flows for which detailed measurements are available. Calculations are reported for recirculating swirling reacting flows using the joint velocity-scalar probability density function (pdf) method. The pdf method offers significant advantages over conventional finite-volume, Reynolds-average based methods, especially for the computation of turbulent reacting flows. The pdf calculations reported here are based on a newly developed solution algorithm for elliptic flows, and on newly developed models for turbulent frequency and velocity that are simpler than those used in previously reported pdf calculations. Calculations are performed for two different gas-turbine-like swirl combustor flows for which detailed measurements are available. The computed results are in good agreement with experimental data. INTRODUCTION The main advantages offered by the joint velocity-scalar probability density function method for the computation of turbulent reacting flows are that the important processes such as convection by both mean and fluctuating velocities, the effect of turbulence fluctuations on complex multi-step finite-rate reactions, and the effects of reaction/heat release on turbulence appear in closed form and need not be modeled The present work builds on several past studies [e.g^ 3,4,6-10]. The study in Ref. 6 demonstrated the pdf method for elliptic recirculating flows. The pdf method was used in conjunction with a finite-volume method such that the finite-volume method supplied the mean pressure field and the turbulence time scale to the pdf method. The pdf method in turn supplied the Reynolds stresses to the finitevolume method so that conventional turbulence models are avoided. The coupling was needed since the velocity-scalar pdf method used did not include information about the turbulence time scale and although the mean pressure field could be determined from the mean velocity field, a robust algorithm was needed to solve the Poisson equation for pressure which involves the evaluation of second derivatives of mean velocities and other terms with minimal statistical error. Such a pressure algorithm was developed by Anand et al. [7] and demonstrated for elliptic recirculating flows such as the flow over a backward-facing step. The time scale was still supplied externally to the pdf method in that study. A model for the mean turbulence time scale, rather for the mean turbulence frequency (inverse of the turbulence time scale), was developed and solved in conjunction with the pdf method by Anand et al. [81. Subsequently a stochastic frequency model was developed by Pope et al. The present study represents the first fully self-contained pdf calculations for elliptic flows and incorporates the elliptic flow solution algorithm as well as the stochastic frequency model. However, with a view to making the method more robust, easier to implement and affordable for complex multidimensional flows, a significantly different elliptic flow algorithm (or pressure algorithm) has been developed and implemented. The models for turbulent frequency and velocity have also been considerably simplified. The newly developed method (elliptic algorithm and models) is validated against benchmark experimental data and previous pdf solutions mentioned above. of the pdf transport equation. Means of any functions of the independent variables are determined by a sophisticated ensemble averaging procedure (cloud-in-cell estimate using bi-linear basis functions) foDowed by smoothing using local linear least squares Particle Evolution Equations The increment dx* in the position of a particle over an infinitesimal time interval dt during the time step is given by the exact equation: This exact equation causes the mean and turbulent convection to be in closed form. The model used for the increment in the particle velocity is a variant of the simple Langevin model, and is described by: THE PDF METHOD -MODELING AND SOLUTION ALGORITHM The joint pdf f(V, \jr, T|; x, t) at position x and time t is defined as the probability density of the simultaneous event life, 0 = Y. 32(X, t) = j|£ and w(x., t) = T|, where II is the velocity vector, $ is a set of scalars, co is the turbulence frequency, and X, W and r\ are independent variables in the velocity-scalar-frequency space. Starting from the usual conservation equations for mass (continuity), momentum, scalar quantities and turbulent frequency, the transport equation for the joint pdf can be derived as described in Ref. 1. In this equation, the terms involving convection (mean and turbulent), reaction, body forces, and the mean pressure gradient effects (including the variable density effects in those terms) appear in closed form. The terms representing the effects of viscous dissipation, fluctuating pressure gradient, molecular mixing of scalars, and production and dissipation of turbulence frequency need to be modeled. A Lagrangian viewpoint is adopted in modeling and solving the joint pdf equation. The modeled pdf transport equation is solved by the Monte Carlo technique. In the Monte Carlo solution technique, notional particles representing fluid particles are distributed throughout the solution domain overlaid by a spatial or computational grid. Each particle is attributed with values for its spatial position (x*), velocity QJ*), scalar values ($*) and turbulence frequency (co*). These values evolve according to the equations described below which include modeled terms where needed. Starting from arbitrary initial conditions and specified boundary conditions, the particle values are marched in time-steps which are a fraction of a characteristic time scale in the flow until a steady-state solution is reached. The solution of these evolution equations constitutes the solution where angled brackets denote (density-weighted or Favre) means, <P(x.*)> is the mean pressure, <Uj> is the Eulerian mean velocity, k is the turbulent kinetic energy, Q is the conditional mean turbulence frequency described below, p* is the particle density, C o is a universal constant, and dWj represents an isotropic Wiener random process. The first term in Eq. 2 exactly accounts for the acceleration due to mean pressure gradients including variable-density effects. The last two terms together model the effects of viscous dissipation and fluctuating pressure gradient. The main difference between the current model and the simple Langevin model used previously [e.g. 3, 6-8,12,13] is that the conditional mean frequency appears instead of the unconditional mean frequency <o». The conditional mean frequency is the "above-average" mean defined by i.e., it is proportional to the mean of the instantaneous frequencies that are greater than or equal to the unconditional mean frequency. In intermittent regions where both turbulent (co>0) and nonturbulent (eo=0) fluid exist, the conditional mean frequency is representative of the "frequency in the turbulent fluid" which is the appropriate quantity to be used in modeling the turbulent process. As a consequence of its definition (Eq. 3), Q is larger than <o» in such regions. This facilitates the entrainment of nonturbulent particles without requiring additional modeling. (See Ref. 14 for more details.). The constant CQ (determined in terms of incomplete gamma functions) has the value 0.6893, and is specified such that fl =«a> in homogeneous turbulence The value C 0 = 2.1 (determined in Ref. 12) has been used in previous studies for the Langevin model (Eq. 2) that uses the unconditional mean frequency [e.g. 3,6-8,12,13]. A new stochastic model for the evolution of the frequency of the particle (to*), developed by Jayesh and Pope where left-hand side is the mean rate of change following the fluid, the first term on the right represents the production and the second term represents the decay of <w>, and Sy is the mean rate of strain given by Jayesh and Pope The evolution of the oc-th species or scalar value of a computational particle is given by *> dtc <p <<Pa* where S a ((g*) is the reaction rate for the species <p a as a function of the instantaneous composition $*, and the second term represents a simple relaxation-to-mean model for molecular mixing of scalars. Therefore, given the reaction rate (determined by the thermochemistry used), the treatment of reaction and the turbulence chemistry interactions are in closed form. The value of the constant Cm for molecular mixing is typically in the range 1.5-2.0. Due to the differences in the models used in the present study compared to those used in the previous studies discussed above, the values of the model constants used in the present study differ slightly from their standard values. The time increment, At, for each step is chosen to be a fraction (=0.1) of the minimum of a) the inverse of the maximum mean turbulence frequency in the computational domain or b) the minimum characteristic time for any particle to cross a computational cell based on the mean and variance of velocity in the cell AH the particles in the computation are marched with the same time increment. The particle evolution equations are integrated over the time step with an accuracy of second-order or better. It should be noted that the models described above for velocity, frequency and scalar mixing are all being used in the joint pdf method for the first time to compute general (inhomogeneous, swirling, recirculating) turbulent reacting flows. As such the present study serves to validate the elliptic flow algorithm as well as the models used. Elliptic Flow Algorithm (Position, Velocity and Pressure Correction) The main purpose of the elliptic flow algorithm is to determine the mean pressure field to be used in the velocity equation (Eq. 2) while ensuring that the mean conservation equations for mass and momentum are satisfied. The elliptic flow algorithm newly developed by Pope [15] is used in the present study. The algorithm performs a velocity correction to satisfy mean mass conservation and determines a mean pressure correction on every step starting from arbitrary initial conditions. Variance reduction techniques are applied (i.e. turbulent processes such as mixing, viscous dissipation, etc. are performed on sub-ensembles such that the sub-ensemble means are not changed) so that mean momentum conservation is also maintained. In addition, a correction to the position of the particles are made to ensure that the consistency condition for particle methods, namely that the volume associated with a sub-ensemble of particles should equal the geometric volume occupied by the particles, is satisfied. For statistically stationary flows, a steady state is achieved in which these corrections tend to zero (in the mean, and the variance decreases as the number of particles increases). In the algorithm a velocity correction potential * is determined such that after adding the velocity correction the corrected velocity field satisfies the continuity equation. (<p> is the mean density of the fluid.) When the velocity increment is determined by Eq. 7 for a time step At, it is equivalent to the effect of a mean pressure correction = <D/At . The Poisson equation for the velocity correction potential is set up and solved using bi-linear basis function representation for calculating mean quantities. Thus, the mean pressure field is not determined directly from the solution of the Poisson equation. However, any error in the mean pressure field is compensated by the velocity correction, i.e., the potential <& is such the total effect of the correct pressure should be felt In contrast, the pressure algorithm developed and used by Anand et al. [7] solves for the Poisson equation for pressure as well as for the velocity correction potential. However, since the Poisson equation involves second derivatives of mean velocities it is necessary to determine the mean velocity field to a high degree of accuracy. Hence bi-directional cross-validated cubic splines are used to determine means in that algorithm which can be computationally expensive. The current algorithm is expected to be less expensive and more robust. The more important advantage is that it is easier to extend the current algorithm to irregular geometries (body-fitted grids) and to three-dimensional flow calculations. Thermochemistry Hydrogen and methane flames are studied in the present work. A fast equilibrium chemistry model is used for the hydrogen flame calculations because the time scale for hydrogen-air reaction is very small compared to the turbulent time scale. For the hydrogen case, the only scalar variable in the calculations is the mixture fraction. Temperature is also included but is needed for output only. The mixture fraction is a conserved variable (reaction rate is zero). The density and temperature are determined as equilibrium properties from the mixture fraction. In addition to mixture fraction, two more scalar variables--namely the mass fractions of carbon dioxide and water-are included in the pdf calculations. The temperature and density are determined as functions of these three scalar variables. For both the fast chemistry and the 2-step chemistry models, lookup tables were created to reduce the CPU requirements of the calculations. In the case of the fast chemistry, a onedimensional table is created, and for the 2-step chemistry, a three-dimensional table is generated. For the 2-step chemistry calculations, the table is generated for a given specific time increment, At, used by the flow calculations (At = 2.5x10"* s in the present calculations). In the table generation processes, the NASA CEC thermal data were used to calculate the variable specific heats and the temperature. RESULTS AND DISCUSSION The present pdf method was applied to the (constantdensity) flow over a backward-facing step previously calculated by Anand et al. Results are presented for two laboratory swirl combustor configurations which have the essential flow features of gas turbine combustors, namely swirl, recirculation, large velocity gradients, turbulence and combustion. The experiments were conducted by researchers at the University of Dayton Research Institute at the Wright Patterson Air Force Base, Dayton. The velocity measurements were made using a three-component laser Doppler velocimeter (LDV) and temperature measurements were made using coherent antiStokes Raman spectroscopy (CARS). ing a joint pdf method. Since the flow is primarily parabolic in nature (with no recirculation), the pdf solution algorithm was based on boundary-layer assumptions with extensions for swirling flows. The method also used more sophisticated models, namely the stochastic frequency model of Pope [10] in conjunction with the refined Langevin model for velocity Swirling Hydrogen Diffusion Flame The present computations were performed, on an IBM RS6000/370, using a nonuniform grid (31 in x by 61 along the radius r) with about 190 particles per cell. Increasing the nominal number of particles per cell to 290 produced nearly the same results. The inlet boundary conditions were taken from experimental data as described in Ref. 4. Calculations were performed for time 2000 steps. In the figures to be presented for this case the measured mean axial velocity on the centerline at the nozzle exit, <U> OC , which is 130.3 m/s in this case, is used to normalize the velocity statistics. The axial distances and radius are normalized by the nozzle diameter, D (= 9.45 mm) and nozzle radius R (= D/2) respectively. The temperature results are normalized by the stoichiometric temperature, T st (= 2377 K). For the experimental data presented, the open squares represent data conditioned on the inner fuel jet, the solid circles represent data conditioned on the swirling air jet, and the inverted triangles represent the data conditioned on the outer coflow air. Profiles for turbulent kinetic energy presented in Overall, the results are in very good agreement with data, and are as good or better than those obtained with the boundary-layer calculations. Methane Step-Swirl Combustor The step-swirl combustor shown in The computations were performed using a nonuniform 41 The profiles of mean axial velocity, <U>, presented in The profiles of mean radial velocity, <V>, in The mean temperature profiles presented in The profiles of turbulent kinetic energy and a fourth order turbulent correlation shown in The results for the hydrogen and methane cases have validated the new models and the elliptic flow algorithm used. The calculations represent the first quantitative results from the new code incorporating the algorithm and models. The results compare very well with the detailed data from practical combustors. CONCLUDING REMARKS Computations using the joint pdf approach have been reported for two swirl combustor configurations. The study uses a newly developed solution algorithm for elliptic flows and new simplified models for velocity and turbulence frequency. The methane combustor calculations represent the first fully self-contained joint pdf calculations for elliptic reacting flows. The results for both combustors are in good agreement with data. The study serves to further validate the joint pdf method and the models, and is a significant step in the development of a pdf-based combustor design system. The ability of the joint pdf method to accurately calculate the mean and turbulent velocity fields, scalar transport, and temperature using multi-step finite-rate chemistry offers significant advantages for its use in the design of current and future high performance and low emissions gas turbine combustors. The present results are compared against calculations using the scalar pdf method (in which the joint pdf of only the scalars is considered) and other conventional turbulent combustion models in an accompanying paper ACKNOWLEDGMENTS