Projection-free approximate balanced truncation of large unstable systems

In this article, we show that the projection-free, snapshot-based, balanced truncation method can be applied directly to unstable systems. We prove that even for unstable systems, the unmodified balanced proper orthogonal decomposition algorithm theoretically yields a converged transformation that balances the Gramians (including the unstable subspace). We then apply the method to a spatially developing unstable system and show that it results in reduced-order models of similar quality to the ones obtained with existing methods. Due to the unbounded growth of unstable modes, a practical restriction on the final impulse response simulation time appears, which can be adjusted depending on the desired order of the reduced-order model. Recommendations are given to further reduce the cost of the method if the system is large and to improve the performance of the method if it does not yield acceptable results in its unmodified form. Finally, the method is applied to the linearized flow around a cylinder at Re = 100 to show that it actually is able to accurately reproduce impulse responses for more realistic unstable large-scale systems in practice. The well-established approximate balanced truncation numerical framework therefore can be safely applied to unstable systems without any modifications. Additionally, balanced reduced-order models can readily be obtained even for large systems, where the computational cost of existing methods is prohibitive

Simplified models for the thermodynamic properties along a combustor and their effect on thermoacoustic instability prediction

Accurately predicting the thermoacoustic modes of a combustor depends upon knowledge of the thermodynamic properties within the combustor; flame temperature, heat release rate, speed of sound and ratio of specific heats all have a strong effect. Calculating the global equilibrium properties resulting from fuel combustion is not straightforward due to the presence of complex multi-species and multi-step reaction mechanisms. A method which decouples the calculations of species dissociations is proposed in this work: this improves the precision of calculation when using few species and reduces the computational cost and complexity to a degree that embedding within low order thermoacoustic network codes is feasible. When used to calculate the combustion product mole fractions, temperature, heat release rate, speed of sound and ratio of specific heats for hydrocarbon-air flames, the method is found to be accurate and highly efficient across different operating conditions and fuel types. The method is then combined with improved low-order wave-based network modelling, the latter employing wave-based acoustic models which account for the variation of thermodynamic properties along the combustion chamber. For a laboratory-scale combustor with a large downstream temperature variation, it is shown that accurate prediction of thermoacoustic modal frequencies and growth rates does depend on accounting for the variation in thermodynamic properties

Bi-modality in the wakes of simplified road vehicles: simulation and feedback control

Large Eddy Simulations are performed to investigate the bi-modal behavior of the flow past three-dimensional square - back bluff bodies. We consider two simplified road vehicle geometries: (i) the squareback Ahmed body and (ii) a simplified square -back truck geometry, with height greater than its width. The Reynolds numbers based on body height is chosen in the range 20,000 - 33,000, such that turbulent separation occurs for both. It is characteristic of such wakes to exhibit slow random switching between asymmetric states. The accessibility of full flow - field data allows us to extract wake flow features that offer new insights. Finally we apply a single-input single - output linear feedback control strategy to the flow. This consists of sensing the base pressure force fluctuations, and actuating a zero- net-mass-flux slot jet just ahead of separation to attenuate base pressure force fluctuations. Its effects on the symmetry of the wake and the mean pressure drag are investigated

Scattering of entropy waves into sound by isolated aerofoils

This article presents a modelling approach to predict the low-frequency sound generated by entropy fluctuations interacting with isolated aerofoils. A model of the acoustic field is obtained based on a linearisation of the compressible Euler equations about a steady, potential, compressible mean flow. Mean flow variations of velocity and density are accounted for in the source term, but are neglected in the sound propagation. Using a Lorentz-type transformation, the problem is reduced to solving a Helmholtz equation. This equation is recast in integral form and a solution is obtained using a compact Green's function method. This approach places no restrictions on the entropy wavelength, while assuming that the acoustic wavelength is large compared to the profile chord and spacing. The source term is further simplified by assuming that the steady flow is a small perturbation to a uniform flow. The model is illustrated using a symmetric aerofoil and its performance is assessed against numerical simulations of the compressible Euler equations. Good agreement is found for all the frequencies of validity of the theory and for all the range of subsonic Mach numbers. The solution for a symmetric aerofoil interacting with plane entropy waves corresponds to the combination of a dipole along the horizontal axis and a monopole. The dipole originates from the unsteady drag experienced by the aerofoil owing to the fluctuations of density and the monopole from the strong local acceleration of the flow at the leading edge. The monopole term becomes negligible for low Mach numbers

Numerical prediction of the Flame Describing Function and thermoacoustic limit cycle for a pressurised gas turbine combustor

The forced flame responses in a pressurized gas turbine combustor are predicted using numerical reacting flow simulations. Two incompressible1 large eddy simulation solvers are used, applying two combustion models and two reaction schemes (4-step and 15-step) at two operating pressures (3 and 6 bar). Although the combustor flow field is little affected by these factors, the flame length and heat release rate are found to depend on combustion model, reaction scheme, and combustor pressure. The flame responses to an upstream velocity perturbation are used to construct the flame describing functions (FDFs). The FDFs exhibit smaller dependence on the combustion model and reaction chemistry than the flame shape and mean heat release rate. The FDFs are validated by predicting combustor thermoacoustic stability at 3 and 6 bar and, for the unstable 6 bar case, also by predicting the frequency and oscillation amplitude of the resulting limit cycle oscillation. All of these numerical predictions are in very good agreement with experimental measurements