CARS spectral fitting with multiple resonant species using sparse libraries

Abstract

The dual pump CARS technique is often used in the study of turbulent flames. Fast and accurate algorithms are needed for fitting dual-pump CARS spectra for temperature and multiple chemical species. This paper describes the development of such an algorithm. The algorithm employs sparse libraries, whose size grows much more slowly with number of species than a conventional library. The method was demonstrated by fitting synthetic "experimental" spectra containing 4 resonant species (N 2, Oz, H2 and CO2), both with noise and without it, and by fitting experimental spectra from a H 2-air flame produced by a Hencken burner. In both studies, wei ghted least squares fitting of signal, as opposed to least squares fitting signal or square-root signal, was shown to produce the least random error and nummize bias error in the fitted parameters.

WIDECARS spectra fitting in a premixed ethylene–air flame

“…The residual minimized in the fitting algorithm, R, the 'reduced chi-square statistic,' is given by Eqn (1): [5] R ¼ 1…”

“…The measurement precision (the spectrum noise) is estimated as described in Ref. 5. Other variables include K k ≡C k χ 2 k , where f is the square of the theoretical CARS susceptibility, and the rescaling function C k = a + bk + ck 2 , which is used to rescale the theory to match the signal as discussed in Section 'Fitting of synthetic spectra'.…”

“…5. To improve accuracy of the interpolation, the CARS susceptibility, χ k , is transformed to X k = ln (χ k /χ NR ), where χ NR is the CARS non-resonant susceptibility at spectrum temperature and composition.…”

“…Strictly speaking, shift may not always be required on a single shot basis, for example in a steady flow, but as a practical matter it is simplest to use it as the computational cost is negligible. Horizontal shift is optimized iteratively employing Eqns (4) and (5), except that the equations are rewritten replacing q i with h and setting M = 1. Derivatives for the optimization and the shifted spectrum are obtained by means of spline interpolation.…”

“…The problem becomes worse when fitting large experimental data sets associated with test campaigns in complex combustion systems, such as ethylene-fueled dual mode scramjets, although it can be mitigated by using libraries of previously calculated spectra. [5][6][7][8] In previous work by Cutler et al, [6] large experimental dual-pump CARS experimental data sets were acquired in a scramjet combustor. These spectra were simultaneously fitted for temperature and mole fractions of N 2 , O 2 , and H 2 .…”