A Hybrid Algorithm for the Joint PDF Equation of Turbulent Reactive Flows

In this paper a new particle-finite-volume hybrid algorithm for the joint velocity-frequency-composition PDF method for turbulent reactive flows is presented. This method is a combination of a finite-volume scheme and a particle method. The finite-volume 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 velocity Ũ 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 energy ẽs), 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 piloted-jet flame test case a factor of 20 times less). Finally, grid convergence studies verify that the scheme is second-order accurate in space.