This lesson covers the derivation of the Reynolds Average Navier Stokes (RANS) equations, which are used for turbulent flows. The lesson begins with an explanation of the decomposition of velocities into mean and fluctuating components, leading to additional stresses in the governing equations. The instructor then applies the rules of averaging to turbulent quantities and demonstrates how to derive the RANS equations using these rules. The lesson also introduces the concept of Reynolds stress and explains how to express it in terms of velocity gradients and turbulent kinetic energy. The lesson concludes with a discussion of the closure problem in turbulence modeling and the concept of turbulence intensity.

- The Reynolds Average Navier Stokes (RANS) equations are derived from the decomposition of velocities into mean and fluctuating components.
- The rules of averaging are applied to turbulent quantities to derive the RANS equations.
- Reynolds stress, which arises due to fluctuating components, can be expressed in terms of velocity gradients and turbulent kinetic energy.
- The closure problem in turbulence modeling arises due to the presence of more unknowns than governing equations.
- Turbulence intensity, which describes the relative magnitude of the root mean square value of the fluctuating components with respect to the time average mean velocity, is a key concept in turbulence modeling.

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