We derive the governing equations for the mean and turbulent kinetic energy and discuss simplifications of the equations for several canonical flows, including channel flow and homogeneous isotropic turbulence. A classical expression for the dissipation rate in isotropic turbulence is provided. In addition, the governing equations for turbulent enstrophy and scalar variance are derived with parallels to the derivation of the turbulent kinetic energy equation. A model for turbulent kinetic energy evolution and dissipation in isotropic turbulence is introduced. Finally, we derive the governing equations for the Reynolds stress tensor components and discuss the roles of the terms in the Reynolds stress budgets in homogeneous shear and channel flows. A crucial link between pressure-strain correlations and the redistribution of turbulent kinetic energy between various velocity components is established. Quantifying how energy is transferred between the mean flow and turbulent fluctuations is crucial to understanding the generation and transport of turbulence and its accompanying Reynolds stresses, and thus properties that phenomenological turbulence models should conform to.
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