fluid_mechanics

472 Chapter 9 ■ Flow over Immersed Bodies
μ = 0
y a
u = 0.99 U
u = U b
U U U
δ
Equal
areas
μ ≠ 0
u = u(y)
δ*
U – u
a
b
(a)
(b)
F I G U R E 9.8 Boundary layer thickness: (a) standard boundary
layer thickness, (b) boundary layer displacement thickness.
F l u i d s i n t h e N e w s
The Albatross: Nature’s Aerodynamic Solution for Long
Flights The albatross is a phenomenal seabird that soars just
above ocean waves, taking advantage of the local boundary layer
to travel incredible distances with little to no wing flapping. This
limited physical exertion results in minimal energy consumption
and, combined with aerodynamic optimization, allows the albatross
to easily travel 1000 km (620 miles) per day, with some
tracking data showing almost double that amount. The albatross
has high aspect ratio wings (up to 11 ft in wingspan) and a liftto-drag
ratio (l/d) of approximately 27, both similar to highperformance
sailplanes. With this aerodynamic configuration,
the albatross then makes use of a technique called “dynamic
soaring” to take advantage of the wind profile over the ocean surface.
Based on the boundary layer profile, the albatross uses the
rule of dynamic soaring, which is to climb when pointed upwind
and dive when pointed downwind, thus constantly exchanging
kinetic and potential energy. Though the albatross loses energy
to drag, it can periodically regain energy due to vertical and directional
motions within the boundary layer by changing local
airspeed and direction. This is not a direct line of travel, but it
does provide the most fuel-efficient method of long-distance
flight.
In actuality (both mathematically and physically), there is no sharp “edge” to the boundary
layer; that is, u S U as we get farther from the plate. We define the boundary layer thickness, ,
as that distance from the plate at which the fluid velocity is within some arbitrary value of the
upstream velocity. Typically, as indicated in Fig. 9.8a,
y
where u 0.99U
The boundary layer
displacement thickness
is defined in
terms of volumetric
flowrate.
To remove this arbitrariness 1i.e., what is so special about 99%; why not 98%?2, the following
definitions are introduced. Shown in Fig. 9.8b are two velocity profiles for flow past
a flat plate—one if there were no viscosity 1a uniform profile2 and the other if there are viscosity
and zero slip at the wall 1the boundary layer profile2. Because of the velocity deficit,
U u, within the boundary layer, the flowrate across section b–b is less than that across section
a–a. However, if we displace the plate at section a–a by an appropriate amount d*, the
boundary layer displacement thickness, the flowrates across each section will be identical. This
is true if
d*b U 1U u2b dy
where b is the plate width. Thus,
d* a1 u U b dy
0
0
The displacement thickness represents the amount that the thickness of the body must be
increased so that the fictitious uniform inviscid flow has the same mass flowrate properties as
the actual viscous flow. It represents the outward displacement of the streamlines caused by the
(9.3)