A wind of speed m s blows past a circular cooling tower of radius m

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow

Fluids and Continuum Physics, Ch Potential flow, complex potentials and lift

Problem

A wind of speed U=8.0U = 8.0 m s−1^{-1} blows past a circular cooling tower of radius a=10a = 10 m. In a horizontal plane well up the tower the flow is modelled by the complex potential W=U(ζ+a2ζ),ζ=x+iy=reiθ,W = U\left(\zeta + \frac{a^{2}}{\zeta}\right),\qquad \zeta = x + iy = re^{i\theta}, with the wind along +x+x, the tower axis at the origin, θ\theta measured anticlockwise from +x+x, and the flow taken as steady, incompressible, inviscid and irrotational. Gravity is the same everywhere in this horizontal plane. Air density ρ=1.20\rho = 1.20 kg m−3^{-3}. ((a)) Write the stream function ψ(r,θ)\psi(r,\theta) of this flow and state its value on the tower surface. ((b)) A streamline passes through the point x=0x = 0, y=25y = 25 m. Find its value of ψ\psi, and hence the volume flux of air per unit height of tower flowing between that streamline and the tower surface. ((c)) Find the surface pressure at the shoulder θ=90∘\theta = 90^\circ, expressed as a difference from the far-field pressure p∞p_\infty.

Browse the problem library