Characteristics of Flow Past a Finite Flat Plate Subject to Different Reynolds Numbers:
(a) Re = 0.1, (b) Re = 10 and (c) Re = 107
Boundary Layer Thickness
When fluid flows past an immersed body, a thin boundary layer will be
developed near the solid body due to the no-slip condition (i.e., fluid
is stuck to the solid boundary). The flow can be treated as inviscid flow outside
of this boundary layer, while viscous effects are important inside of
this boundary layer.
Take flow past a flat plate for example. The characteristics of flow
past a flat plate with finite length L subject to different
Reynolds numbers (Re = ρUL/μ)
are shown in the figures. At a low Reynolds number (Re = 0.1),
the presence of the flat plate is felt in a relatively large area where the
viscous effects are important .
At a moderate Reynolds number (Re = 10),
the viscous layer region becomes smaller.
Viscous effects are only important inside of this region,
and streamlines are deflected as fluid enters it.
As the Reynolds number is increased further (Re = 107),
only a thin boundary layer develops near the flat plate, and the fluid
forms a narrow wake region behind the flat plate. Hence, the flow can
be considered as inviscid flow everywhere except the boundary
Now consider flow past an infinite long flat plate with neglegible thickness.
In this case, the Reynolds number is defined using the local distance
x (i.e., the distance from the leading edge along the flat plate as
the characteristic length). The local Reynolds number is then given by
Rex = ρUx/μ
As the fluid flows past the long flat plate, the flow will become turbulent at a critical distance
xcr downstream from the leading edge. For flow past a flat plate, the transition from
laminar to turbulent begins when the critical Reynolds number (Rexcr) reaches
The boundary layer changes from laminar to turbulent at this point.
The concept of a boundary layer was introduced and formulated
by Prandtl for steady, two-dimensional laminar flow past a flat plate
using the Navier-Stokes equations. Prandtl's student, Blasius, was able
analytically for large Reynolds number flows. The details of
the derivation are omitted for simplicity, and the results are summarized
Based on Blasius' analytical solutions, the boundary layer thickness
(δ) for the laminar region is given by
where δ is defined as the boundary layer
thickness in which the velocity is 99% of the free stream velocity (i.e.,
y = δ, u = 0.99U).
The wall shear stress is determined by
If this shear stress is integrated over the surface of the plate ares, the
drag coefficient for laminar flow can be obtained for the flat plate with finite length as
If the flow is turbulent, then the equations for boundary layer and drag coefficient is