A thin flat plate is held at rest on the centreplane of a two-dimensional water tunnel

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity · Viscous flow and dynamical similarity

Fluid Dynamics, Ch The boundary layer: Prandtl's equations and the Blasius solution

Problem

A thin flat plate is held at rest on the centreplane of a two-dimensional water tunnel, aligned with the stream, and water (ρ=998 kg m−3\rho=998\,\mathrm{kg\,m^{-3}}, μ=1.00×10−3 Pa s\mu=1.00\times10^{-3}\,\mathrm{Pa\,s}) flows past it at U=2.0 m s−1U=2.0\,\mathrm{m\,s^{-1}}, so that a laminar boundary layer develops from its leading edge. Consider the station x=0.10 mx=0.10\,\mathrm{m} from the leading edge. ((a)) Estimate the boundary-layer thickness at this station by balancing the time for vorticity to diffuse across the layer against the time for the fluid to convect past it. ((b)) Using the Blasius solution, calculate δ99\delta_{99} and the displacement thickness δ1\delta_1 at this station. ((c)) The tunnel has full height 40 mm40\,\mathrm{mm} (it is wide enough for its side walls to be ignored), and its top and bottom walls begin at the same station as the plate's leading edge, so laminar layers grow on them in the same way as on the plate. Estimate the fractional blockage of the tunnel cross-section at x=0.10 mx=0.10\,\mathrm{m} due to the four layers, one on each wall and one on each face of the plate. ((d)) Compute the Reynolds number RexRe_x at this station and state whether the layer there is laminar or turbulent.

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