Consider the fully developed, turbulent flow of an incompressible, Newtonian fluid through a plane channel between two plates at \(x_{2}=0\) and \(x_{2}=h\). Use dimensionless variables based on the kinematic viscosity \(u\) and the wall shear velocity \(u_{\tau}\) (27.15).

1.1 Derive the averaged \(\mathrm{x}\)-momentum balance for this flow.

1.2 Express the divergence of the Lamb vector (2.54) in terms of velocity, flexion (2.55) and enstrophy (3.16) and average the result.

1.3 Determine the Taylor series expansions for the fluctuation velocity, vorticity and flexion components close to the lower wall. Derive the expansion for the dimensionless divergence of the Lamb vector and its average close to the lower wall. Show that the divergence of the Lamb vector must be negative at the wall.

Eq (27.15)

Eq (2.54)

Eq (2.55)

Eq (3.16)

Transcribed Image Text:

TO u*