Impact of the Acoustic Forcing Level on the Transfer Matrix of a Turbulent Swirling Combustor with and Without Flame

Abstract

Thermoacoustic instabilities are a major issue for industrial and domestic burners. One possible framework to study these instabilities is to represent the system by a network of Dimensionless Acoustic Transfer Matrices (DATM) that link pressure and velocity fluctuations upstream and downstream each element of the network. In this article, the DATM coefficients of a turbulent swirling combustor are determined for a thermoacoustically stable configuration using harmonic acoustic forcing. Since the dynamics of the whole system is controlled by nonlinearities, the impact of the forcing level needs to be considered. The four DATM coefficients are thus measured for reactive operating conditions (premixed flame) and cold flow conditions for increasing acoustic excitation levels. The velocity level is controlled by a hot wire located inside the injector, in a region with a laminar top-hat velocity profile. The upstream and downstream specific acoustic impedances are also measured. Results for the acoustic response under cold flow conditions are first presented. In this case, the DATM coefficients are found to be independent of the forcing level except for the modulus of the coefficients linking the downstream velocity fluctuations to the upstream pressure and velocity fluctuations. This behavior is linked to the nonlinear response of the injector but is not entirely captured by the acoustic network model developed in this work. For reactive operating conditions, measurements indicate that all DATM coefficients depend on the forcing level to a certain extent. The Flame Describing Function, linking heat release rate fluctuations to velocity fluctuations, is used to reconstruct the transfer matrix through an acoustic network model. This network model accurately predicts the trend of the measured coefficients but the impact of the forcing level is not reproduced. Saturation for reactive operating conditions is shown to be not only related to the nonlinear flame response but also to the nonlinear injector dynamics. Finally, a data-driven reconstruction of the FDF using the acoustic network model along with the hot wire and microphone measurements is performed. This data-driven acoustic reconstruction is subsequently compared with the FDF determined with an optical technique.

The flame displacement speed: A key quantity for turbulent combustion and combustion instability

“…The first equation of the system Eqs. 21 Expression for modulated stabilized flames. In this section, the dynamic component (the fluctuation field) is made of an harmonic modulation and turbulent fluctuations and the amplitude of the harmonic modulation is dominating over the turbulent fluctuation.…”

Acoustic Energy Balance During the Onset, Growth and Saturation of Thermoacoustic Instabilities

“…(5)-(8) along with Eqs. 14- (17), the expression for the acoustic absorption coefficient of a generic non-compact element under the previous assumptions is then given by [35,36]:…”

“…(30) then provide the frequencies ω r and damping rates ω i of the thermoacoustic modes of the combustor. Limit cycle oscillations, which are inherently nonlinear, can be predicted with (linear) acoustic network models by using a weakly nonlinear description of the flame frequency response [15][16][17]25].…”

“…Such instabilities are undesirable as they lead to increased heat fluxes [10,11], structural vibration and failure [5,10,11], flame extinction [5,10] and significant sound levels at certain frequencies [7,9]. Accordingly, a large research effort has been devoted to predicting the onset, growth and saturation of such instabilities since the 1950s [7,[12][13][14][15][16][17].…”

“…For instance, each element can be represented by a 2x2 matrix called Scattering matrix or S-matrix, which relates the ingoing and outgoing propagating waves across the element [20,21]. Alternatively, Transfer Matrices relating the acoustic pressure and acoustic velocity across the element may be used [13,14,17,20,21]. For given acoustic boundary conditions, the combustor's thermoacoustic modes are then determined by searching for the complex roots of a characteristic equation [22,23].…”