In most situations it is inevitable that the boundary layer becomes detached
from a solid body. This *boundary layer separation* results in a large
increase in the drag on the body. We can understand this by returning to the
flow of a nonviscous fluid around a cylinder. The pressure distribution is
the same on the downstream side of the cylinder as on the upstream side;
thus, there were no unbalanced forces on the cylinder and therefore no drag
(d'Alembert's paradox again). If the flow of a viscous fluid about a body is
such that the boundary layer remains attached, then we have almost the same
result--we'll just have a small drag due to the skin friction. However, if
the boundary layer separates from the cylinder, then the pressure on the
downstream side of the cylinder is essentially constant, and equal to the
low pressure on the top and bottom points of the cylinder. This pressure is
much lower than the large pressure which occurs at the stagnation point on
the upstream side of the cylinder, leading to a pressure imbalance and a
large *pressure drag* on the cylinder. For instance, for a cylinder in a
flow with a Reynolds number in the range , the boundary
layer separates and the coefficient of drag is , much larger
that the coefficient of drag due to skin friction, which we would estimate
to be about .

A Reynolds number-independent drag coefficient leads to a drag force . More importantly, the power *P* required to maintain a
constant speed in the presence of this drag is , so
that it increases with the *cube* of the speed. To see the importance of
this dependence, suppose that you ride a bicycle at 15 mph, which is a
respectable speed. Most of the resistance at this speed is due to
aerodynamic drag (there are other sources, such as mechanical friction,
rolling friction, and so on, but I don't think they dominate at this speed).
Now suppose you want to get to class in half the time in the morning, so you
decide to ride at 30 mph. This requires 8 times as much power!

Boundary layers tend to separate from a solid body when there is an
increasing fluid pressure in the direction of the flow--this is known as an
*adverse pressure gradient* in the jargon of fluid mechanics.
Increasing the fluid pressure is akin to increasing the potential energy of
the fluid, leading to a reduced kinetic energy and a deceleration of the
fluid. When this happens the boundary layer thickens (recall that ), leading to a reduced gradient of the velocity profile ( decreases), with a concomitant decrease in the wall
shear stress . For a large enough pressure gradient the shear
stress can be reduced to zero, and separation often occurs. The fluid is no
longer ``pulling'' on the wall, and opposing flow can develop which
effectively pushes the boundary layer off of the wall. *Separation is
bound to occur in a sufficiently large adverse pressure gradient.* On the
other hand, boundary layers like decreasing pressure gradients, which
accelerate the fluid and cause the boundary layer to thin.

Given these considerations, we see that minimizing the pressure drag amounts
to preventing or delaying boundary layer separation. Since adverse pressure
gradients are the cause of separation, we want to avoid these or at least
make the gradients small. Trailing stagnation points are bound to cause
problems, so separation can often be delayed by placing the trailing
stagnation point at a cusp, so that the fluid leaves the body smoothly. This
is known as *streamlining* , and is the preferred shape for airfoils, cars, and
fish! Another way of delaying separation is by forcing the boundary layer to
become turbulent. The more efficient mixing which occurs in a turbulent
boundary layer reduces the boundary layer thickness and increases the wall
shear stress, often preventing the separation which would occur for a
laminar boundary layer under the same conditions. You can see that there is
a trade-off here--the turbulent boundary layer produces a greater drag due
to skin friction, but can often *reduce* the pressure drag by
preventing, or reducing, boundary layer separation. Since the latter is
usually dominant at high Reynolds numbers, various schemes have been
invented for producing turbulent boundary layers. The dimples on a golf
ball, the fuzz on a tennis ball, or the seams on a baseball are good
examples of this. Apparently a dimpled golf ball has one-fifth the drag of a
smooth golf ball of the same size! Also, airplane wings are often engineered
with vortex generators on the upper surface to produce a turbulent boundary
layer.

Sun Sep 28 22:13:11 EDT 1997