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Koopmanism’s World Wide Quest to Rule
Shervin Bagheri and Predrag Cvitanović
(a very early draft of chapter for ChaosBook.org - any comments/edits much appreciated)

Paper that introduces the notion of spectral expansion of the Koopman operator for model reduction and connects to Reynolds triple decomposition
Mezić, I. (2005). Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dynamics, 41(1-3), 309-325.

Paper in JFM that utilizes the above idea to study jet in cross flow and show there are 2 dominant modes:
Rowley, C.W., Mezić, I., Bagheri, S., Schlatter, P. and Henningson, D.S., 2009. Spectral analysis of nonlinear flows. Journal of fluid mechanics, 641, pp.115-127.

Paper that discusses the Koopman Mode Expansion concept and, in section .... discusses its use in analysis of the role of the mean flow :
Mezic, I., 2013. Analysis of fluid flows via spectral properties of the Koopman operator. Annual Review of Fluid Mechanics, 45, pp.357-378.

Paper that discusses Dynamic Mode Decomposition. See section 4.1 for the connection with the Koopman operator:
Tu, JH, Rowley, CW, Luchtenburg, DM, Brunton, SL, and Kutz, JN. On dynamic mode decomposition: theory and applications. Journal of Computational Dynamics, 1(2):391-421, 2014.

Review article on reduced-order models for fluids. See sections 4.2-4.3 for a discussion of the Koopman operator and DMD, and their use in developing reduced-order models.
Rowley, CW and Dawson, STM. Model reduction for flow analysis and control. Annual Review of Fluid Mechanics 49:387-417, 2017.