Numerical simulation of premixed flames interacting with obstacles

In this work the modeling of the interaction of a premixed flame with one ore more obstacles of different shape is considered. The challenge of this work was to design a fast numerical tool suitable for a standard personal computer. A tool able to use a simplified chemical model that removes the need to solve a large system of conservation equations. At the same time a stable numerical scheme was needed to model high variations of density generated between the fresh mixture and the burnt gases during the propagation of the flame. Moreover, an efficient strategy was required to model the complex geometry. Many combustion applications are characterized by low speeds, both for the flow and the flame propagation. In this study we consider systems with typical velocity of the order O(10^0-10^1)m/s. Therefore, the flow is described by the Navier-Stokes equations in the low-Mach number limit. This implies that the velocities are much smaller than the speed of sound, so that density variations due to pressure variations can be neglected. In other words, the terms containing the acoustic time scales can be removed from the governing equations. The reacting nature of the flow is modeled by using a source term in the energy equation which depends on the position of the flame. Energy is released only at the flame front. Hence it is assumed that the combustion takes place in the flamelet regime, i.e. the thickness of the flame is thin enough to be considered a geometric interface. In this case a level set approach can be used to track the position of the flame by using the G-equation formulation (Peters 2000). The source term in the energy equation is modeled as function of the zero level of the G-equation. This approach removes the need for solving the detailed chemistry because the source term can be thought as the contribution of a single step chemical reaction. The spatial discretization of the momentum and the continuity equations is a second order finite volume method (Hirsch, 1988) and the time integration is based on a third order Adams-Bashforth scheme (AB3). For the G-equation the space discretization is a local third order WENO scheme (Jiang and Peng, 2000), while for its time integration AB3 is used. The same WENO scheme is used for the spatial discretization of the convective term in the temperature equation while the discretization of the diffusive term is carried out using a central difference scheme. An IMEX scheme ('implicit' integration of the source and 'explicit' integration of the advection-diffusion terms) is used for the time integration of the temperature equation. The IMEX scheme used in this approach was proposed by Pareschi (2001). The computation of the Navier-Stokes equations is based on a pressure correction algorithm. The main difference between previous pressure correction schemes (i.e. Najm, 1998 or Treurniet, 2002) and the scheme we have introduced here is that in the first cases the time derivative of the density is calculated with a backward discretization whilst in the second case it is computed using the temperature equation and the equation of state. Another important difference in the second case is that the updated value of the density is found by integration of the continuity equation. The modeling of the complex geometries has been done with an Immersed Boundary Method (IBM) that retains the advantages of numerical accuracy and computational efficiency associated with simple orthogonal grids. On the contrary, conventional numerical models generally use a complex (non-orthogonal) grid structure which requires a substantial computational effort. The IBM can simulate the shape of the part of the computational domain accessible to fluid by locally adding extra forces to the momentum equations. The square or rectangular obstacles considered in this study are aligned with the Cartesian mesh and this allows to apply exactly the forcing at their boundaries. In this method the shear stress on the boundary of the simulated obstacles is replaced in such a way that the no-slip velocity condition for the tangential component is applied at the wall. In conjunction, a non-penetration condition is also applied for the perpendicular velocity components at the boundary. The heat flux between the boundaries of the obstacles and the flow is well represented with a procedure similar to the stress replacement method used for the momentum equation. This approach keeps the internal region of an obstacle well isolated under different conditions, while the correct heat flux is imposed at the surface of the body. We have considered several experiments regarding the interaction of a premixed flame with obstacles. In particular, we have simulated the experiment performed by Ibrahim and Masri (2001). This case consists of the evolution of a flame front during deflagration of a air-gas mixture in a rectangular domain. The cases with constant and variable viscosity have been considered. The results obtained are comparable with the experimental data. The flame tip speed and the interaction of the front in the wake are well predicted. We note that the thermal thickness is reduced due to the interaction with the body. This interaction produces also an overpressure. In the case of variable viscosity a higher interaction of the flame front in the recirculation zone beyond the obstacle was found