A Consistent Hybrid Finite-Volume/Particle Method for the PDF Equations of Turbulent Reactive Flows

Abstract

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.

Langevin PDF simulation of particle deposition in a turbulent pipe flow

“…(d) It has been found that it is important to ensure that the turbulence Eulerian model and the Lagrangian one are as consistent as possible 38,49 . The lack of consistency may also lead to unphysical results, at least for the limit case of very small particles 38 .…”

A relaxation scheme for hybrid modelling of gas-particle flows

“…However, the two most popular approaches (Eulerian and Lagrangian) for two-phase flow modelling have disadvantages which limit their capacities. Hybrid methods try to gather the advantages of the Eulerian approach (expected values free from statistical error and low calculation costs) with those of the Lagrangian approach (polydispersity and non-linear local source terms are treated without approximations) [31,32]. At the moment, most hybrid methods use only one description (Eulerian or Lagrangian) for each phase.…”

“…This approach, which will be referred to as A1, relies on the combined use of relaxation-type techniques [1,4,5,9,12,13,28,35,36,38] and upwinding schemes [14,19,22,23,29,30] and was first introduced in [25] in order to predict single-phase turbulent flows [32]. In the latter approach, an extended hyperbolic system was introduced in a natural way, following what is done for previous associated Eulerian models (see [2,3,6,21]).…”

A Hybrid Finite-Volume/Transported PDF Model for Simulations of Turbulent Flames on Vector Machines

“…A good overview of the different models can be found in [17]. In the work presented here a joint PDF of velocity and composition vector is employed.…”

“…One possibility is to couple the particle method with an ordinary finite-volume or finite-difference solver to obtain the mean pressure field from the Navier-Stokes equations. These so called hybrid PDF/CFD methods are widely used by different authors for many types of flames [15,16,17,18,19,20]. In the presented paper a hybrid scheme is presented and used as well.…”