The theory given on this page describes boundary layer effects and was
first presented by Ludwig Prandtl in the early 1900's. The general fluids equations
had been known for many years, but solutions to the equations did not properly describe
observed flow effects (like wing stalls). Prandtl was the first to realize that the relative magnitude
of the inertial and viscous forces changed from a layer very near the surface to a region far
from the surface. He first proposed the interactively coupled, two layer solution which properly
models many flow problems. The Wright brothers did not know about the existence of
boundary layers or the role of boundary layers in causing wing stalls.
As an object moves through a fluid, or as a fluid moves past an object,
the molecules of the fluid near the
object are disturbed and move around the object. Aerodynamic
forces are generated between the fluid and the object. The
magnitude of these forces depend on the shape of the object, the
speed of the object, the mass
of the fluid going by the object and on two other important properties
of the fluid; the viscosity, or stickiness, and the
compressibility, or springiness, of the fluid. To properly model
these effects, aerodynamicists use
which are ratios of these effects to other forces present in the
problem. If two experiments have the same values for the similarity
parameters, then the relative importance of the forces are being
Aerodynamic forces depend in a complex way on the viscosity of the
fluid. As the fluid moves past the object, the molecules right next
to the surface stick to the surface. The molecules just above the surface
are slowed down in their collisions with the molecules sticking to the surface.
These molecules in turn slow down the flow just above them. The
farther one moves away from the surface, the fewer the collisions affected by
the object surface. This creates a thin layer of fluid near the surface
in which the velocity changes from zero at the surface to the free stream
value away from the surface. Engineers call this layer the
boundary layer because it occurs on the boundary of the fluid.
The details of the flow within the boundary layer are very important
for many problems in aerodynamics, including the development of a
and the skin friction
drag of an object.
Unfortunately, the physical and mathematical details of boundary layer
theory are beyond the scope of this
beginner's guide and are usually studied in late undergraduate
or graduate school in college. We will only present some of the effects
of the boundary layer.
On the slide we show the streamwise velocity variation from free stream
to the surface. In reality, the effects are three dimensional. From the
conservation of mass
in three dimensions, a change in velocity in the streamwise direction causes
a change in velocity in the other directions as well. There is a small
component of velocity perpendicular to the surface which displaces or moves
the flow above it. One can define the thickness of the boundary layer to be
the amount of this displacement. The displacement thickness depends
Reynolds number which is the ratio of
inertial (resistant to change or motion) forces to viscous
(heavy and gluey) forces and is given by the equation : Reynolds number (Re) equals
velocity (V) times density (r) times a characteristic length (l) divided
by the viscosity coefficient (mu).
Re = V * r * l / mu
Boundary layers may be either laminar (layered), or turbulent (disordered)
depending on the value of the Reynolds number.
For lower Reynolds numbers, the boundary layer is laminar
and the streamwise velocity changes uniformly
as one moves away from the wall, as shown on the left side of the figure.
For higher Reynolds numbers, the boundary layer is turbulent
and the streamwise velocity is characterized by unsteady (changing with time)
swirling flows inside the boundary layer.
The external flow reacts to the edge of the boundary layer just as it
would to the physical surface of an object. So the boundary layer gives any object
an "effective" shape which is usually slightly different from the physical shape.
To make things more confusing,
the boundary layer may lift off or "separate" from the
body and create an effective shape much different from the physical
shape. This happens because the flow in the boundary has very low energy (relative to the
free stream) and is more easily driven by changes in pressure. Flow separation is the
reason for wing stall at high
angle of attack.
The effects of the boundary layer on lift are contained
in the lift coefficient and the effects on
drag are contained in the drag coefficient.
- Re-Living the Wright Way
- Beginner's Guide to Aeronautics
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