Consider stationary incompressible flow around a cylinder of radius a with sufficiently large Reynolds number that viscosity may be ignored except in a thin boundary layer, which is assumed to extend all the way around the cylinder. The velocity is assumed to have the uniform value V at large distances from the cylinder.

(a) Show that the velocity field outside the boundary layer can be derived from a scalar velocity potential v = ∇ψ, that satisfies Laplace’s equation: ∇²ψ = 0.

(b) Write down suitable boundary conditions for ψ.

(c) Write the velocity potential in the form

and solve for f . Sketch the streamlines and equipotentials.

(d) Use Bernoulli’s theorem to compute the pressure distribution over the surface and the net drag force given by this solution. Does your drag force seem reasonable? It did not seem reasonable to d’Alembert in 1752, and it came to be called d’Alembert’s paradox.

(e) Finally, consider the effect of the pressure distribution on the boundary layer. How do you think this will make the real flow different from the potential solution? How will the drag change?

Transcribed Image Text:

↓ =V x + f(x),