In this paper, we perform a numerical analysis of a turbulent hydrogen flame in a stream of hot co-flowing oxidizer. As shown in the literature, compared to classical circular jets, the jets issuing from non-circular nozzles with sharp corners and flat edges characterise increased entrainment of the surrounding fluid. In reacting flows, using such nozzles causes mixing intensification and thus a more dynamic combustion process. In this study, we analyse the dynamics of a flame developed downstream of the jet issuing from triangular and square nozzles, and compare them to the reference case with a circular nozzle. Additionally, we investigate to what extent the mixing process and flame behaviour can be altered by applying a low-energy harmonic excitation of the fuel stream. Note, that the applied excitation can have a positive or negative impact on flame dynamics (position, size, temperature), it can intensify the mixing process and thus facilitates the combustion, or lead to local high-strain regions causing flame extinctions. The research is performed with the help of the large eddy simulation (LES) method to model a turbulent flow and the Eulerian PDF approach for combustion modelling. The simulations are conducted by applying ANSYS-CFD software and an in-house high-order code SAILOR. The basic configuration reflects an experimental stand known as the lifted Cabra flame in which hydrogen-diluted nitrogen at 300K issues into a co-flow of hot oxidizer (1045K). The experimental results allow us to validate the simulation results and tune parameters of the numerical model, e.g., a mesh density, simulation and time-averaging time, LES and combustion model settings. The test results characterise good agreement between the simulation and experimental data. The applied excitation leads to qualitative and quantitative flame dynamics alteration. Changes in the excitation impact the amount of the oxidiser transported to the fuel jet region and its mixing with the fuel. In effect, the shape of the flame and its lift-off height alter. Also, the localization of temperature maximum, species mass fractions, and amplitudes of their variations change. We observe that the impact of the excitation on the flame depends on whether the triangular and rectangular nozzle is used. In the former case, the initial shape of injected fuel is preserved over a longer distance and this causes a more irregular flame structure.