In cross flow of real fluids over a single tube, a laminar boundary layer is developed from the front stagnation point and separates at some point around the circumference. This gives rise to a symmetric steady-state pair of vortices and a recirculation zone in the rear. When the Reynolds number^† is greater than about 40, the flow in the recirculation zone becomes unstable and periodic vortex shedding commences in the rear, with separation of the laminar boundary layer at φ = 82°, where φ is the angle subtended at the tube axis by the stagnation and separation points. With a further increase of Re, a critical flow regime is reached at Re > 2 × 10⁵, characterized by a laminar-turbulent transition in the boundary layer preceding the separation point, which is shifted downstream to φ = 140°. The frequency of vortex shedding is determined from the Strouhal number Sr = fd /u, where f is the frequency of vortex shedding and d is the tube diameter. For practical purposes, Sr for a single tube may be taken as 0.2 in the Reynolds number range 300 to 2 × 10⁵. In the critical region, the value of Sr increases to 0.46 and then decreases to 0.27 (Žukauskas, 1972) at Re = 3.5 × 10⁶. For an incompressible fluid, velocity and pressure variations outside the boundary layer are conveniently described by the Bernoulli equation:

\[\label{eq1} p_{b}+\frac{\rho {u}^2_b}{2}=p_{x}+\frac{\rho{u}^2_x}{2}=\mbox{const} \tag{1}\]

^† The Reynolds number it defined as ud /ν, where d is thi tube diameter, ν is the kinematic viscosity, and u is the upstream velocity ( = u_b ) for single tubes ami the maximum intertube velocity for tube banks.

The quantities are defined in Figure 1; u_x is the peripheral local velocity on the tube, p_x and p_b are the local and main flow pressures, and ρ is the fluid mass density. Thus,

\[\label{eq2} u_x=u_b\sqrt{1-\frac{p_x-p_b}{\rho u^2_b/2}} \tag{2}\]