 Solution

ID: past-exam-of-the-topics-in-fluid-mechanics-course-of-the-university-of-oxford/2023/3/a/i/solution

u_{x} + v_{y} ρ (u_{t} + u u_{x} + v u_{y}) ρ (v_{t} + u v_{x} + v v_{y}) = 0, = - p_{x} + μ (u_{xx} + u_{yy}), = - p_{y} + μ (v_{xx} + v_{yy}) .

A conservative body force, if retained, may be incorporated into a modified pressure. Using scales

L, U_{0}, L / U_{0}

and

ρ U_{0}^{2}

gives

Re = \frac{ρ U _{0} L}{μ} = \frac{U _{0} L}{ν}, u_{t} + u \cdot \nabla u = - \nabla p + Re^{- 1} \nabla^{2} u .

Here

ν = μ / ρ

is the molecular kinematic viscosity. At high Reynolds number, inertia and the pressure gradient dominate on the outer length scale, while molecular viscous diffusion is small there. Thin boundary layers, narrow shear flows and fine turbulent scales can still have sufficiently large velocity gradients for viscous effects to matter. High Reynolds number permits instability and turbulence, but does not alone prove that every flow is turbulent. Molecular viscous dissipation remains essential at the small end of the turbulent cascade.

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