Many industrial applications of combustion are concerned with low speed combustion phenomena. As the fluctuations of the pressure field affect the density field accordingly with the square of the Mach number, the hyperbolic behaviour of Navier-Stokes equations is usually restricted, in theses cases, to sound propagation. Furthermore, when sound or acoustic instabilities are not the subject of investigation, numerical feasability for such combustion studies is dramatically increased if acoustics is filtered using the so-called Low Mach number approximation. As a result, CFL-type stability criterion is no longer based on speed of sound, but on flow velocitity. The framework of the presentation is centered on this low speed approach.
The first part of the communication is concerned with a particular treatment of the Navier-Stokes equations which involves a "momenta-modified pressure" formulation ("m-" formulation). This approach is of great interest because it leads to a simple Poisson equation on the modified pressure ("") , the global features of advection-diffusion balance being preserved. This procedure presents two main advantages ; firstly, pressure computation remains well-conditioned even for large variations of fluid density (as in spray combustion); secondly, such a simple form of the elliptic part (while the parabolic one is globaly unaffected) gives access to the straightforward use of high-precision packages as pseudo-spectral methods or hermician methods. An illustration is given concerning the ignition of a cryotechnic injector. More precicely, we treat of the propagation of triple flames along a mixing layer, composed of fluids of very different density and subjected to Kelvin-Helmholtz instability.
The second part of the talk deals with the problem of the simultaneous presence in the numerical modelling of several length scales, the smallest one being extremely localized. Hence, we turn towards Adaptive Mesh Refinement techniques (AMR). A self-adaptive refinement method is presented : the deflagration front is tracked with a nested system of grids which are capable to glide along each other in order to follow the smallest scale : the reaction zone. The coupling between grid and solution, a feature rather specific to combustion, is particularly delicate because refined zone is not only transported by the global flow field but also by its own dynamics which depends on the quality of the solution, hence on the grid position. A 2-D implementation, within the framework of multigrid methods, has been performed for studying flame front instabilities. Three different algorithms allowing the computation of the solution on all grids have been compared (a Dirichlet-Neumann iterative method, a method ofinterface penalty on residual and the Fast Adaptive Composite method). Illustration with the thermal-diffusive instability and application to the ploblem of flame-wall interaction are finally presented.