A combustion instability in a combustor terminated by a nozzle is analysed and modelled based on a low order Helmholtz solver. A Large Eddy Simulation (LES) of the corresponding turbulent, compressible and reacting flow is first performed and analysed based on Dynamic Mode Decomposition (DMD). The mode with the highest amplitude shares the same frequency of oscillation as the experiment (approx. 320 Hz) and shows the presence of large entropy spots generated within the combustion chamber and convected down to the exit nozzle. The lowest purely acoustic mode being in the range 700 − 750 Hz, it is postulated that the instability observed around 320 Hz stems from a mixed entropy/acoustic mode where the acoustic generation associated with entropy spots being convected throughout the choked nozzle plays a key role. The DMD analysis allows to extract from the LES results a low-order model that confirms that the mechanism of the low-frequency combustion instability indeed involves both acoustic and convected entropy waves. The Delayed Entropy Coupled Boundary Condition (Motheau et al. 2014) is implemented into a numerical Helmholtz solver where the baseline flow is assumed at rest. When fed with appropriate transfer functions to model the entropy generation and convection from the flame to the exit, the Helmholtz/DECBC solver predicts the presence of an unstable mode around 320 Hz, in agreement with both LES and experiments.

A novel and e cient algorithm is presented in this paper to deal with DNS of turbulent reacting ows under the low-Mach-number assumption, with detailed chemistry and a quasi-spectral accuracy. The temporal integration of the equations relies on an operating-split strategy, where chemical reactions are solved implicitly with a sti solver and the convection-di usion operators are solved with a Runge-Kutta-Chebyshev method. The spatial discretisation is performed with high-order compact schemes, and a FFT based constant-coe cient spectral solver is employed to solve a variable-coe cient Poisson equation. The numerical implementation takes advantage of the 2DECOMP&FFT libraries developed by Li and Laizet [1], which are based on a pencil decomposition method of the domain and are proven to be computationally very e cient. An enhanced pressure-correction method is proposed to speed-up the achievement of machine precision accuracy. It is demonstrated that a second-order accuracy is reached in time, while the spatial accuracy ranges from fourth-order to sixth-order depending on the set of imposed boundary conditions. The software developed to implement the present algorithm is called HOLOMAC, and its numerical e ciency opens the way to deal with DNS of reacting ows to understand complex turbulent and chemical phenomena in ames.

In this paper we present a fourth-order in space and time block-structured adaptive mesh refinement algorithm for the compressible multicomponent reacting Navier-Stokes equations. The algorithm uses a finite volume approach that incorporates a fourth-order discretization of the convective terms. The time stepping algorithm is based on a multi-level spectral deferred corrections method that enables explicit treatment of advection and diffusion coupled with an implicit treatment of reactions. The temporal scheme is embedded in a block-structured adaptive mesh refinement algorithm that includes subcycling in time with spectral deferred correction sweeps applied on levels. Here we present the details of the multi-level scheme paying particular attention to the treatment of coarse-fine boundaries required to maintain fourth-order accuracy in time. We then demonstrate the convergence properties of the algorithm on several test cases including both nonreacting and reacting flows. Finally we present simulations of a vitiated dimethyl ether jet in 2D and a turbulent hydrogen jet in 3D, both with detailed kinetics and transport.

a b s t r a c tThis paper presents a methodology to account for some mean-flow effects on thermoacoustic instabilities when using the zero-Mach-number assumption. It is shown that when a computational domain is represented under the M¼ 0 assumption, a nonzero-Machnumber element can simply be taken into account by imposing a proper acoustic impedance at the boundaries so as to mimic the mean flow effects in the outer, not computed flow domain. A model that accounts for the coupling between acoustic and entropy waves is presented. It relies on a "delayed entropy coupled boundary condition" (DECBC) for the Helmholtz equation satisfied by the acoustic pressure. The model proves able to capture low-frequency entropic modes even without mean-flow terms in the fluctuating-pressure equation.