## A consistent hybrid finite-volume/particle method for the PDF equations of turbulent reactive flows.(English)Zbl 0953.76062

Summary: The paper describes a new hybrid finite-volume (FV)/particle method developed for the solution of the PDF equations for statistically stationary turbulent reactive flows. In this approach, the conservation equations for mean mass, momentum, and energy conservation are solved by a FV method, while a particle algorithm is employed to solve the fluctuating velocity-turbulence frequency-compositions joint PDF transport equation. The mean velocity and pressure are supplied to the particle code by the FV code, which in turn obtains all Reynolds stresses, scalar fluxes, and reaction terms needed in the FV code. An important feature of the method is the complete consistency between the set of equations solved by the FV and particle methods.
The algorithmic and numerical issues arising in the development of the hybrid method are studied in the simple setting of stochastic ideal flow equations. Numerical results are obtained for one-dimensional reactive stochastic ideal flow to demonstrate numerical properties of the method. The total numerical error is identified as statistical error, bias, spatial truncation error, and temporal truncation error. In contrast to the self-contained particle method, the bias is found to be negligibly small. It is shown that all the numerical errors converge at the expected rates. Finally, the global convergence of the hybrid method is demonstrated and the optimal strategy for time-averaging that gives the best global convergence rate is investigated.

### MSC:

 76M12 Finite volume methods applied to problems in fluid mechanics 76M28 Particle methods and lattice-gas methods 76V05 Reaction effects in flows 76F25 Turbulent transport, mixing

CHEMKIN
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### References:

 [1] Pope, S.B., Computations of turbulent combustion: progress and challenges, 23rd international symposium on combustion, 591, (1990) [2] Pope, S.B., PDF methods for turbulent reactive flows, Prog. energy combust. sci., 11, 119, (1985) [3] Pope, S.B., Lagrangian PDF methods for turbulent flows, Ann. rev. fluid mech., 26, 23, (1994) · Zbl 0802.76033 [4] Pope, S.B., A Monte Carlo method for the PDF methods equations of turbulent reactive flows, Combust. sci. tech., 25, 159, (1981) [5] Welton, W.C.; Pope, S.B., PDF model calculations of compressible turbulent flows using smoothed particle hydrodynamics, J. comput. phys., 134, 150, (1997) · Zbl 0912.76065 [6] Patankar, S.V., Numerical heat transfer and fluid flow, (1980) · Zbl 0595.76001 [7] Pope, S.B.; Chen, Y.L., The velocity-dissipation probability density function model for turbulent flows, Phys. fluids A, 2, 1437, (1990) · Zbl 0709.76060 [8] Van Slooten, P.R.; Jayesh; Pope, S.B., Advances in PDF modeling for inhomogeneous turbulent flows, Phys. fluids, 10, 246, (1998) · Zbl 1185.76686 [9] S. B. Pope, PDF2DV: A FORTRAN code for solving modeled joint-pdf equations in two-dimensions, unpublished, 1994. [10] Pope, S.B., Mean field equations in PDF particle methods for turbulent reactive flows, (1997) [11] Delarue, B., Application of PDF methods to compressible turbulent reacting flows, (1997) · Zbl 1185.76749 [12] Xu, J.; Pope, S.B., Source of bias in particle-mesh methods for PDF models for turbulent flows, (1997) [13] Xu, J.; Pope, S.B., Numerical studies of PDF/Monte Carlo methods for turbulent reactive flows, J. comput. phys., 152, 192, (1999) · Zbl 0945.76069 [14] Correa, S.M.; Pope, S.B., Comparison of a Monte Carlo PDF finite-volume Mean flow model with bluff-body Raman data, Twenty-fourth symp. (international) on combust, 279, (1992) [15] Chang, G.-C., A Monte Carlo PDF/finite-volume study of turbulent flames, (1996) [16] Tsai, K.; Fox, R.O., Modeling the scalar dissipation rate for a turbulent series-parallel reaction, Chem. eng. sci., 51, 1929, (1996) [17] Pope, S.B., On the relationship between stochastic Lagrangian models of turbulence and second moment closures, Phys. fluids, 6 A2, 973, (1994) · Zbl 0827.76036 [18] Wouters, H.A.; Nooren, P.A.; Peters, T.W.J.; Roekaerts, D., Simulation of a bluff-body stabilized diffusion flame using second moment closure and Monte Carlo methods, Twenty-sixth symp. (international) on combust, 177, (1996) [19] Nau, M.; Neef, W.; Maas, U.; Gutheil, E.; Warnatz, J., Computational and experimental investigation of a turbulent non-premixed methane flame, Twenty-sixth symp. (international) on combust, 83, (1996) [20] F. A. Jaberi, P. J. Colucci, S. James, P. Givi, and, S. B. Pope, Filtered mass density function for large eddy simulation of turbulent reacting flows, submitted. · Zbl 0982.76053 [21] Anand, M.S.; Pope, S.B.; Mongia, H.C., A PDF method for turbulent recirculating flows, Lecture notes in engineering, 672, (1989) [22] Haworth, D.C.; El Tahry, S.H., Probability density function approach for multidimensional turbulent flow calculations with application to in-cylinder flows in reciprocating engines, Aiaa j., 29, 208, (1991) [23] Caughey, D.A., Diagonal implicit multigrid algorithm for the Euler equations, Aiaa j., 26, 841, (1988) · Zbl 0667.76108 [24] Muradoglu, M.; Caughey, D.A., Implicit multigrid solution of the preconditioned Euler equations, (1997) [25] Turkel, E., A review of preconditioning methods for fluid dynamics, Appl. numer. math., 12, 257, (1993) · Zbl 0770.76048 [26] Palmer, G., Improved flux-split algorithm applied to hypersonic flows in chemical equilibrium, Aiaa j., 28, 1153, (1990) [27] Bussing, T.R.A.; Murman, E.M., Numerical investigation of two-dimensional H_{2}-air flameholding over ramps and rearward-facing steps, J. propulsion, 3, 448, (1987) [28] Yang, B.; Pope, S.B., An investigation of the accuracy of manifold methods and splitting schemes in the computational implementation of combustion chemistry, Combust. flame, 112, 16, (1998) [29] Kee, R.J.; Rupley, F.M.; Miller, J.A., Chemkin-II: A Fortran chemical kinetics package for analysis of gas phase chemical kinetics, (1993) [30] Pope, S.B., Particle method for turbulent flows: integration of stochastic differential equations, J. comput. phys., 117, 332, (1995) · Zbl 0827.76063 [31] Hockney, R.W.; Eastwood, J.W., Computer simulations using particles, (1988) · Zbl 0662.76002 [32] Dreeben, T.D.; Pope, S.B., Nonparametric estimation of Mean fields with application to particle methods for turbulent flows, (1992) [33] Kloeden, P.; Platen, E., Numerical solution of stochastic differential equations, (1992) · Zbl 0752.60043
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