Non-Linear LIF Background

Quantitative single-shot planar laser-induced fluorescence (PLIF) imaging of minor species in turbulent combusting flows is a challenging task due to the complex interplay of the various energy transfer mechanisms and the requirement of relatively high signal to noise ratio (SNR) levels. For quantitative PLIF experiments of the important combustion radical OH, the popular excitation strategy A²Σ⁺←X²Π (1,0) is typically utilised with irradiance levels near the linear irradiance limit. The requirement of high spatial resolution at large Reynolds numbers necessitates the use of thin laser sheets, when combined with low molecular concentrations operation within the linear regime may not be possible at a satisfactory SNR. The restriction of limiting irradiance levels to within the linear regime is not an experimental equipment limitation, but rather a lack of established models that accurately quantify spectrally integrated non-linear regime laser induced fluorescence (LIF). The development of such a model to quantify both linear and non-linear LIF is the motivation for this study.

The detailed reviews of Daily [180], Kohse-Höinghaus [181] and Eckbreth [182] direct significant attention to reviewing the topic of non-linear LIF both experimentally and theoretically, these reviews highlight that the existing body of work has evolved towards models for quantification of spectrally dispersed point measurements for saturated LIF.

Such a detailed review of non-linear LIF is of course not the aim of this background review, however a brief review of the material previously published that is relevant to the material presented in this section is warranted. The term saturated LIF is referred to in this thesis as the LIF regime that the LIF signal is virtually insensitive to laser irradiance, where as non-linear LIF is referred to as a broad term that encompasses the non-linear irradiance dependence regime and the saturated regime. Though termed "non-linear" in this thesis, non-linear regime LIF is not a true non-linear optical technique in the sense that it relies on a non-linear polarisation response, but rather it is termed non-linear because the fluorescence response is non-linearly related to the laser irradiance.

Saturated LIF was first proposed by Omenetto et al. [183] and Piepmeier [184] for atomic systems, then for molecular systems by Daily [185], some of the first molecular applications of this technique were applied by Pasternack et al. [186]. Saturated LIF was proposed as a method by which quantitative measurements could be made that are insensitive to variations in quenching rates and laser power under the assumption sufficiently high laser irradiance levels to achieve saturation are utilized. The claim of insensitivity to variations in quenching environment and laser power were based on steady state solutions to the governing rate equations for two-level and multi-level systems, representing the population dynamics of relevant atomic or molecular species.

The difficulties in applying the saturated LIF approach to systems requiring a greater level of description than the two-level model permits is acknowledged in Daily [185] as very challenging, this is particularly relevant when saturated schemes are applied to multi-level systems where the recorded signal is not spectrally dispersed.

One of the necessary assumptions for saturated LIF is temporal and spatial saturation. By considering a theoretical probe volume that is uniformly spatially irradiated and selectively sampling in time, the non-saturated temporal wings of the LIF signal may be rejected. One of the most challenging facets of saturated LIF is the fact that uniform spatial saturation is difficult to practically achieve, this is primarily due to less than ideal laser beam profiles, lens aberrations and diffraction limitations. Furthermore selectively sampling in time is typically not an option in most PLIF experiments due to equipment and SNR limitations. This means that it is difficult in practice to neglect the spatial and temporal wings of the laser pulse for most non-linear PLIF measurements.

The most successful studies in saturated LIF culminated in the development of the balanced cross rate model to account for the population dynamics under high irradiance conditions and the TOPLIF experimental geometry to account for non-uniform spatial irradiation. The balanced cross rate model [187, 188, 189, 190, 191, 192, 193, 194] is a four-level model that features levels for the laser coupled and bath levels for both the lower state and upper states. By assuming the rate of transfer from the laser coupled levels is equal to the rate of transfer into the two laser coupled levels it becomes possible

to directly relate the temporally integrated LIF signal from the excited state laser coupled level to the total number density using a simple saturated two-level model. By making the assumptions of the balanced cross rate model it is no longer necessary to assume complete steady state as required by the solution of Berg and Shackleford [195]. The balanced cross rate model requires the measured signal to be spectrally resolved such that the measured LIF signal is only from the laser-coupled excited-state level. This degree of spectral resolution in the measured signal is typically not possible in PLIF experiments due to the inherently spectrally integrated nature of PLIF when applied to molecular species. Despite the inherent neglect of the non-uniform spatial variation of irradiance in the balanced cross rate model, point measurement results that compare favourably with absorption measurements have been achieved in Lucht et al. [188], using a single spectrometer and appropriate spatial selection. Extensions of the balanced cross rate technique to high pressures by Carter et al. [193] and in turbulent flames Lucht et al.

[189], for absolute concentration measurements has been shown to be possible .

The issue of non-uniform spatial illumination in saturated LIF has been addressed in a number of papers [192, 196, 197, 198, 199]. Solutions that focus on extracting the centreline fluorescence, thus minimising the effects of the spatial wings of the laser beam have been proposed for the spherically focused Gaussian beam in integral form [185, 190, 191, 196] and later in a closed analytic form [199]. These results are based on either steady state or balanced cross rate solutions to the governing rate equations.

Experimentally, irrespective of the model used, spatial selection is almost always used in spectrally dispersed point or line measurements. Spatial selection is typically employed by using a horizontal spectrometer slit narrower than the laser beam waist diameter in an attempt to minimise the proportion of the signal obtained from the spatial wings. Utilising two spectrometers the TOPLIF approach of Desgroux and Cottereau [197, 198] and Carter et al. [192] is able to correct for non-uniform spatial illumination effects in a more rigorous manner at the expense of additional experimental complexity.

A comparison of wide band and narrow band saturated LIF measurements has been made by Carter and Laurendeau [194]. They obtained encouraging results considering the large

range of pressures, temperatures and quenching environments examined. As expected, limitations for the wide band collection were encountered; at saturated conditions the wideband method still displayed some sensitivity to the quenching environment. Also, different saturation curves and intensities were found for the wide and narrow band collection strategies. The concept of a balanced cross rate or any other steady state method was raised as questionable for wide or even narrow band collection strategies if population bleaching becomes a significant factor.

The aim of this study is to present and validate a comprehensive model for the quantitative interpretation of temporally and spectrally integrated LIF in both the linear and non-linear regimes of LIF. A detailed six-level transient rate equation model is developed in Section 4.2 to examine the important modelling assumptions for the quantification of OH LIF in both the linear and non-linear regimes of LIF. The assumption of steady state and quasi-steady state are examined in Section 4.3, based on the quasi-steady state results the IQSS model is developed as an analytic solution methodology. The IQSS model is numerically validated in Section 4.4 in terms of laser irradiance sensitivity, quenching sensitivity and temperature sensitivity. In Section 4.5 the IQSS is validated against PLIF experimental results in the exhaust gas of a flat flame.

A broader discussion of the results is presented in Section 4.6.