In this paper a new particle-finite-volume hybrid algorithm for the joint velocityfrequency-composition PDF method for turbulent reactive flows is presented. This method is a combination of a finite-volume scheme and a particle method. The finitevolume scheme is used to solve the Reynolds averaged Navier-Stokes equations and the particle method to solve the joint PDF transport equation. The motivation is to reduce the bias and the statistical error and to have an algorithm which is more efficient than stand-alone particle-mesh methods. Therefore, in the particle method we use the smoother mean density ρ and Favre averaged velocityŨ fields computed by the finite-volume scheme: This scheme is an Euler solver for compressible flow with the turbulent fluxes and the reaction term, which are computed by the particle algorithm, as source terms. Since some of the quantities are computed twice (i.e., the mean density ρ and the Favre averaged sensible internal energyẽ s ), by the finite-volume scheme and by the particle method, the hybrid algorithm is redundant. Although the model differential equations are consistent, it was difficult to satisfy consistency numerically, and an accurate particle tracking algorithm is crucial. Therefore a new scheme to interpolate the Favre averaged velocity has been developed which is second-order accurate and quasi conservative; i.e., it is based on the fluxes at the volume interfaces. Another important issue is the coupling between the finite-volume scheme and the particle method. A new time-averaging technique adds stability to the hybrid algorithm, and it also reduces the bias and the statistical error enormously. The properties of the new algorithm are demonstrated by results for a nonpremixed piloted-jet flame test case. First it is shown that the solution becomes statistically stationary and that it is internally consistent. Studies of the asymptotic behavior show that, for a given error tolerance, the new hybrid algorithm requires much less computer time than the stand-alone particle-mesh method (for this pilotedjet flame test case a factor of 20 times less). Finally, grid convergence studies verify that the scheme is second-order accurate in space.
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 the Reynolds stresses, the scalar fluxes, and the 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 the stochastic ideal flow equations. Numerical results are obtained for 1D 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 selfcontained 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.
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