Introduction and Motivation
Modern gas turbines use lean premixed combustion to achieve the best compromise between pollutant NOx emissions and efficiency. This type of combustion is very sensitive to combustion instabilities. Thermoacoustic instabilities can lead to very high levels of pressure fluctuations, resulting in structural damage of the combustor. Therefore it is important to be able to predict the stability of the combustor on early design stage.
The flame dynamics plays an important role in establishing thermoacoustic instabilities. As the result, the correct description of the flame dynamics has strong technological importance. The established procedure to describe the flame dynamics is through determination of its Flame Transfer Function (FTF), namely the flame heat release response to velocity or pressure perturbations , . In this context, the widespread assumption is to treat the heat release as a function of a single variable (velocity perturbations in gas turbines applications). However this might be an over-simplified model, since in most practical situations, premixed flames are ducted and, accordingly, subjected to externally imposed pressure gradients (steady and oscillating). Compared to "free" flames, i.e. flames without externally imposed pressure gradients, the combination of the external pressure gradients with the large density changes found in premixed flames may lead to strong modifications of the flame structure. This is mainly due to the differential buoyancy effects between cold heavy reactants and hot light products. The impact of the pressure gradient on the flame dynamics is ignored at the moment.
The effect caused by the combination of non-parallel pressure and density gradients is known as a baroclinic torque. It causes generation of vorticity at the flame front and, as a result, changes the flame surface, which has a direct impact on the reacting area and heat release. Figure 1 depicts a sketch of the vorticity generated on the flame surface due to baroclinic torque effect.
Figure 1: Sketch of the vorticity generated due to baroclinic torque. The combination of misaligned pressure and density gradients causes generation of vorticity at the flame front.In 2010 Duchaine and Poinsot  studied sensitivity of FTF for a simple laminar premix flame. Authors concluded that the most important parameters controlling the delay of the FTF (i.e. thermoacoustic stability) are the flame speed and the temperature of the inlet duct. The same author back in 1997  studied the effect of the imposed steady pressure gradient on turbulent flame dynamics and shown that for the imposed pressure decreasing from unburned to burned gases, the flame speed decreases and vice versa. It is natural to connect the imposed pressure gradient directely to FTF definition. However this connection has not been done up to present.
A laminar premixed flame previously studied experimentally  and numerically  is adopted as a reference. For each forcing frequency we design two setups in order to create a different pressure gradient field at the flame location with the same velocity excitation. For the first setup we design negative imposed pressure gradient, for the second setup - positive pressure gradient.
Forcing signal is imposed on the left boundary (fully non-reflecting) and it is reflected from the right "open-end" boundary forming a pressure field at flame position, see Figure 2. Established procedures would predict the same value of FTF for both configurations, however we expect to obtain different results with the discrepancy of about 10%. Comparisons with available results  and  will be presented.
Figure 2: Sketch of the pressure field at the flame. Forcing signal is imposed on the left boundary (fully non-reflecting) and it is reflected from the right "open-end" boundary forming a pressure field at flame position.
For numerical simulation it is essential to resolve both acoustics and hydrodynamics scales. The flow is laminar therefore no turbulent models for simplification are accessible. Consequently a large computational domain must be resolved with a very fine mesh. The task is computationally demanding. Numerical simulations will be performed on the new supercomputer SuperMUC, Leibniz Supercomputing Centre in Garching.
With the support of the Technische Universität München. The project is financed within EU FP-7 Marie Curie Initial Training Network PINT-GA-2008210781 Massively Parallel Computations of Combustion and Emission Simulations (http://www.cerfacs.fr/myplanet/)