SP-367 Introduction to the Aerodynamics of Flight

- III. FLUID FLOW -

[25] The Fluid

Viscosity.- There are basically three states of matter -
solid, liquid, and gas. H2O is commonly called "ice" in the solid state, "water" in the liquid
state, and "water vapor" in the gaseous state. Assume one has a piece of ice and
side forces are applied to it (called shearing forces). Very large forces are
needed to deform or break it. The solid has a very high internal friction or
resistance to shearing. The word for internal friction is viscosity and for a
solid its value is generally very large.

Liquids and gases are considered to be fluids since they
behave differently from a solid. Imagine two layers of water or air. If shear
forces are applied to these layers, one discovers a substantial and sustained
relative motion of the layers with the air layers sliding faster over one
another than the water layers. However, the fact that a shear force must be
applied to deform the fluids indicates that they also possess internal
friction.

Water, under normal temperatures, is about fifty times more
viscous than air. Ice is 5 x 1016 times more viscous than air. One
concludes that, in general, solids have extremely high viscosities whereas
fluids have low viscosities. Under the category of fluids, liquids generally
possess higher viscosities than gases. Air, of primary interest in aerodynamics,
has a relatively small viscosity, and in some theories, it is described as a
perfect fluid-one that has zero viscosity or is "inviscid." But it will be shown
that even this small viscosity of air (or internal friction) has important
effects on an airplane in terms of lift and drag.

Compressibility.- All fluids are compressible (that is,
density increases under increasing pressure) to some extent, but liquids are
generally highly incompressible Compared with gases. Even gases may be treated
as incompressible provided the flow speeds involved are not great. For subsonic
flow over an airplane below about 150 m/sec, air may be treated as
incompressible (that is, no change in density throughout the flow). At higher
speeds the effects of compressibility must be taken into account

The Flow

Pathlines and streamlines.- A fluid flow may be described
in two different - the Lagrangian approach and the Eulerian approach. From the
Lagrangian standpoint, one particle is chosen and it is followed as it moves
through space with time. The line traced out by that one particle is called a
particle pathline. An example is a transmitting ocean buoy shown in figure
20(a). Its position has been marked at
[26] 6-hour intervals
over a period of several days. The path observed is the particle
pathline.

In order to obtain a clearer idea of the flow- field at a
particular instant, a Eulerian approach is adopted. One is looking at a
"photograph" of the flow. Figure 20(b) shows the surface ocean currents at a particular fixed time. The
entire flow field is easily visualized. The lines comprising this flow field are
called streamlines.

(a) Particle airline.

(b) Streamlines.

Figure 20.- Particle pathline and
streamlines.

[27] It is important to note the differences between a particle
pathline and a streamline. A pathline refers to the trace of a single particle
in time and space whereas a streamline presents the line of motion of many
particles at a fixed time. The question of whether particle pathlines and
streamlines are ever the same is considered next.

Steady flow compared with unsteady flow.- Of basic
importance in understanding fluid movements about an object is the concept of a
"steady flow." On a windy day a person calls the wind steady if where he stands
it blows constantly from the same direction at a constant speed. If, however,
the speed or direction changes, the wind is "gusty" or unsteady. In a similar
manner the flow of a fluid about an object is steady if its velocity (speed and
direction) at each point in the flow remains constant - this does not
necessarily require that the velocity be the same at all points in the
fluid.

To consider this further, figure 21(a) presents the fluid flow (of air) about a house on a windy day at
one instant of time and figure 21(b) shows the flow an instant of time later. One sees that this flow
is unsteady. There are many areas where the flow pattern is different; the
streamlines are changing their position and shape with time. Particle pathlines
and streamlines for this flow are not equivalent.

(a) Streamlines at time t0.

(b) Streamlines at time t1.

Figure 21.- Unsteady flow of air about a house.

[28] Figure 22 shows a nicely "streamlined'' body (as opposed to the bluff-shaped
house) in a wind tunnel. At time to the tunnel is not running and no air is
flowing. At time t1
the tunnel is started and air begins flowing about the body; the flow develops
further at time t2
and finally reaches a constant pattern at time t3. The flow appears unchanged at time
t4 and time
t5. When the flow
starts. it passes through an unsteady transient state; that is. particle
pathlines and streamlines are not the same. From time t3 onwards a steady flow is
established. Streamlines appear fixed in position with respect to the body. A
particle P shown on a streamline at time t3 moves downstream along that
streamline as shown at times t4 and t5. The particle pathline coincides with the streamline.

Figure 22.- Unsteady and steady flout

Summarizing, this means that for a steady flow a particle
pathline and streamline are equivalent and the Lagrangian point of view is the
same as the Eulerian approach for flow visualization.

Rotational and irrotational flow.- Fluid flow can be
rotational or irrotational. Ii the elements of fluid at each point in the flow
have no net angular (spin) velocity about [29] the points, the fluid flow is said
to be irrotational. One can imagine a small paddle wheel immersed in a moving
fluid as in figure 23(a). If the wheel translates without rotating, the motion is
irrotational. If the wheel rotates in a flow, as illustrated in figure
23(b), the flow is rotational.

(a) Irrotational flow.

(b) Rotational flow.

(c) Inviscid, irrotational flow about an
airfoil.

Figure 23.- Rotational and irrotational flow.

According to a theorem of Helmholtz, assuming zero
viscosity, if a fluid flow is initially irrotational, it remains irrotational.
In figure 23(c), an observer is fixed to the airfoil section shown. The flow far
ahead of the airfoil section is uniform and of [30] constant velocity. It is
irrotational. As the airflow passes about the airfoil section, it remains
irrotational if zero viscosity is assumed. In real life, viscosity effects are
limited to a small region near the surface of the airfoil and in its wake. Most
of the flow may still be treated as irrotational.

One-dimensional flow.- A simplifying argument often
employed to aid in understanding basic ideas is that of a one-dimensional flow.
Figure 24(a) shows a bundle of streamlines of a simple flow. Each streamline
can be thought of as a stream tube since fluid flows along it as if in a tube.
In the case of steady flow, the stream tube is permanent. Taken together, the
bundle of stream tubes comprise an even larger stream tube. Fluid flows through
it as, for example, water flows through a pipe or channel. The velocity varies
across the tube, in general, according to the individual streamline velocity
variation, as shown in figure 24(b). One can easily imagine an "average" uniform value of velocity
at the cross section to represent the actual varying value as indicated in
figure 24(c). The velocity then is considered "one dimensional" since it
varies only with the particular distance along the tube where observations are
made. In addition to velocity, pressure, density, temperature, and other flow
properties must also be uniform at each cross section for the flow to be one
dimensional.

In order to understand how aerodynamic forces arise, two
basic principles must be considered. They are the laws of conservation of mass
and conservation of energy. Simply stated, they convey the facts that mass and
energy can neither be created nor destroyed.

For introductory purposes, simplifying assumptions are
made. The fluid is considered to be inviscid and incompressible (and hence,
"perfect"). The flow is considered steady and one dimensional.

[31]

(a) Stream tubes.

(b) Real velocity flow
profile.

(c) One-dimensional flow profile.

Figure 24.- Stream tubes and one-dimensional
flow.

Ideal Fluid Flow

The continuity equation.- The continuity equation is a
statement of the conservation of mass in a system. Consider a pipe which is
uniform in diameter at both ends, but has a constriction between the ends as in
figure 25(a). This is called a venturi tube. Furthermore, it is assumed that
the fluid, under the previously stated assumptions, is flowing in the direction
indicated. Stations 1 and 2 have cross-sectional areas A1 and A2, respectively. Let
V1 and
V2 be the average
flow speeds at these cross sections (one-dimensional flow). A further assumption
is that there are no leaks in the pipe nor is fluid being pumped in through the
sides. The continuity equation states that the fluid mass passing station 1 per
unit time must equal the fluid mass passing station 2 per unit time. In fact,
this "mass flow rate" must be the same value at any cross section examined or
there is an accumulation of mass- "mass creation"- and the steady flow
assumption is violated. Simply stated,

(Mass rate)1 = (Mass
rate)2

(1)

[32] where

Mass rate = Density x Area x Velocity

(2)

This equation reduces to

pl
AlV1 = p2A2V2

(3)

Since the fluid is assumed to be incompressible,
p [Greek letter rho] is
a constant and equation (3) reduces to

AlV1 = A2V2

(4)

This is the simple continuity equation for inviscid,
incompressible, steady, onedimensional flow with no leaks. If the flow were
viscous, the statement would still be valid as long as average values of
V1 and
V2 across the cross
section are used.

By rearranging equation (4), one obtains

V2 = (A1/A2)V1

(5)

Since A1 is greater than A2 (see fig. 25(a)), it can be concluded that V2 is greater than V1. This is a most important result.
It states, under the assumptions made, that the flow speed increases where the
area decreases and the flow speed decreases where the area increases.
Figure 25(b) shows this with the longer arrow at the constriction indicating
a larger flow speed than at the ends. In fact, by the continuity equation, the
highest speed is reached at the station of smallest area. This is at the
narrowest part of the constriction commonly called the throat of the venturi
tube.

The fact that the product AV remains a constant along a
tube of flow allows an interpretation of the streamline picture. Figure
25(c) shows the streamline pattern in the venturi
tube. In the area of the throat, the streamlines must crowd closer together than
in the wide part. Hence, the distance between streamlines decreases and the
fluid speed increases. The conclusion is that, relatively speaking, widely
spaced streamlines indicate regions of low-speed flow and closely spaced
streamlines indicate regions of high-speed flow.

Bernoulli's theorem-the conservation of energy.- Assume a
fluid flow which, as before, is inviscid, incompressible, steady, and one
dimensional. The energy in the flow is composed of several energies. The kinetic
energy arises because of the [33] directed motion of the fluid; the pressure energy is due to the
random motion within the fluid; and the potential energy is due to the position
of the fluid above some reference level. Bernoulli's theorem is an expression of
the conservation of the total energy; that is, the sum total of these energies
in a fluid flow remains a constant along a streamline. Expressed concisely, the
sum of the kinetic energy, pressure energy. and potential energy remains a
constant.

II it is further assumed that the fluid flow is horizontal
(as, for example, airflow approaching an aircraft in level flight), then the
potential energy of the flow is a constant. Bernoulli's theorem reduces
to

Kinetic energy + Pressure energy = Constant

(6)

where the constant includes the constant value of potential
energy. If one considers the energy per unit volume, one obtains the dimensions
of pressure and Bernoulli's theorem may be expressed in terms of
pressure.

Figure 25.- Venturi tube and continuity principle.

[34] The kinetic energy per unit volume is called dynamic pressure q
and is determined by q = 1/2pV2
where p and V are,
respectively, the fluid flow density and speed at the point in
question.

The pressure energy per unit volume (due to random motion
within the fluid) is the static pressure of the fluid and is given the symbol
p.

The constant energy per unit volume is called the total
pressure pt.

Bernoulli's equation reduces to

Dynamic pressure + Static pressure = Total
pressure

(7)

or

1/2pV2 + p = pt

(8)

For rotational flow the total pressure pt is constant along a streamline but
may vary from streamline to streamline as shown in figure 26(a). In an irrotational flow, the usual case considered for airflow
approaching an aircraft, the total pressure is the same constant value
everywhere as shown in figure 26(b).

Bernoulli's equation states that in a streamline fluid
flow, the greater the speed of the flow, the less the static pressure; and the
less the speed of the flow, the greater the static pressure. There exists a
simple exchange between the dynamic and static pressures such that their total
remains the same. As one increases, the other must decrease.

Pressure measurement.- Let us now examine how total,
static, and dynamic pressures in a flow are measured. Figure
27(a) shows the fluid flow about a simple hollow
bent tube, called a pitot tube after its inventor, which is connected to a
pressure measurement readout instrument. The fluid dams up immediately at the
tube entrance and comes to rest at the "stagnation point" while the rest of the
fluid divides up to flow around the tube. By Bernoulli's equation the static
pressure at the stagnation point is the total pressure since the dynamic
pressure reduces to zero when the flow stagnates. The pitot tube is, therefore,
a total-pressure measuring device.

Figure 27(b) shows the fluid flow about another hollow tube except now the
end facing the flow is closed and a number of holes have been drilled into the
tube's side. This tube is called a static tube and may be connected to a
pressure measuring readout instrument as before. Except at the stagnation point,
the fluid is parallel to the tube everywhere. The static pressure of the fluid
acts normal to the tube's surface. Since...

[35]

(a) Rotational flow. Total pressure varies from
streamline to streamline.

(b) Irrotational flow. Total pressure same constant
value pt,1
everywhere in flow.

Figure 26.- Total-pressure variation.

....pressure must be continuous, the static pressure normal
to the holes is communicated into the interior of the tube. The static tube,
therefore, with the holes parallel to the flow direction, is a static-pressure
measuring device.

Figure 27(c) shows a combined pilot-static tube. When properly connected to
opposite ends of a pressure measuring readout instrument, the difference between
total pressure and static pressure is-measured. By Bernoulli's equation this
difference is the dynamic pressure, defined as 1/2pV2. If the fluid density
p is known, the fluid
flow speed can be calculated. In actual use on aircraft, the pilot-static tube
is connected directly to an airspeed indicator which, by proper gearing, will
automatically display the aircraft airspeed to the pilot. The device is
sometimes mounted forward...

[36]

Figure 27.- Pressure measuring
devices.

...on a boom extending from the airplane nose to insure its
measuring, as closely as possible, the undisturbed approaching flow (also called
the free-stream condition).

Returning to the discussion of the venturi tube introduced
earlier, the continuity and Bernoulli equations may be used to describe the
static-pressure distribution along the venturi tube. The static pressure of the
undisturbed free-stream fluid flow entering the tube may be used as a reference
value. Any variation of static pressure in the tube then is a greater or lesser
value than the free-stream static pressure. In figure 28 holes have been drilled into the walls of the venturi tube similar
to the static tube of figure 27(b) to measure the static pressure. These holes are commonly called
"static taps" and are connected to a "U-tube manometer" - a tube having a
U-shape within which is a liquid such as colored alcohol. When the static
pressure measured at the static tap equals the free-stream static pressure, the
fluid levels in the tube are at some equal reference level. But static pressures
above or below the free-stream pressure are indicated by a decrease or increase
in the level of fluid in the tube.

[37]

Figure 28.- Venturi tube flow.

Figure 28 shows the complete setup of a venturi tube and a set of manometers
and static taps to measure static pressure. By the continuity equation the speed
at station 2 V2 is
greater than that at station 1 V1 as seen previously-the speed at the
throat also is the highest speed achieved in the venturi tube. By Bernoulli's
equation the total pressure pt is constant everywhere in the flow (assuming irrotational flow).
Therefore, one can express the total pressure pt in terms of the static and dynamic
pressures at stations 1 and 2 using equation (8), namely,

1/2plV12 + p1 = 1/2p2V22 + p2 = pt

(9)

Since V2 is greater than V1 and p2 = p1
(fluid is incompressible) it follows that p2 is less than p1, for as the dynamic pressure, hence
speed, increases, the static pressure must decrease to maintain a constant value
of total pressure pt. The block diagrams below the venturi tube show this interchange of
dynamic and static pressures all along the venturi tube. The conclusion drawn
from this is that the static pressure decreases in the region of high-speed flow
and increases in the region of low-speed flow. This is also demonstrated by the
liquid levels of the manometers where as one reaches the throat the liquid level
has risen above the reference level and indicates lower than free- stream static
pressure. At the throat this is the minimum static pressure since the flow speed
is the highest.

The airfoil in an ideal fluid.- To supply a point of
reference in the discussions to follow of a real fluid, the following section
expands the previous discussion of venturi [38] flow to the ideal fluid flow past
an airfoil. Figure 29(a) shows a "symmetric" (upper and lower surfaces the same) airfoil
operating so that a line drawn through the nose and tail of the airfoil is
parallel to the free-stream direction. The free-stream velocity is denoted by
and the free-stream static pressure by . Following the
particle pathline (indicated by the dotted line and equal to a streamline in
this steady flow) which follows the airfoil contour, the velocity decreases from
the free- stream value as one approaches the airfoil nose (points 1 to 2). At
the airfoil nose, point 2, the flow comes to rest (stagnates). From Bernoulli's
equation the static pressure at the nose, point 2, is equal to the total
pressure. Moving from the nose up along the front surface of the airfoil (points
2 to 3), the velocity increases and the static pressure decreases. By the
continuity equation, as one reaches the thickest point on the airfoil, point 3,
the velocity has acquired its highest value and the static pressure its lowest
value.

Beyond this point as one moves along the rear surface of
the airfoil, points 3 to 4, the velocity decreases and the static pressure
increases until at the trailing edge, point 4, the flow comes to rest with the
static pressure equal to the total pressure. Beyond the trailing edge the flow
speed increases until the free-stream value is reached and the static pressure
returns to free-stream static pressure. These velocity and static-pressure
distributions for the center-line streamline are shown in figures
29(b) and 29(c).

Note particularly that on the front surfaces of the airfoil
(up to the station of maximum thickness), one has decreasing pressures (a
negative pressure gradient) whereas on the rear surfaces one has increasing
pressures (a positive pressure gradient). This relationship will be of
importance in the real fluid case.

The lift is defined as the force normal to the free-stream
direction and the drag parallel to the free-stream direction. For a planar
airfoil section operating in a perfect fluid, the drag is always zero no matter
what the orientation of the airfoil is. This seemingly defies physical intuition
and is known as D'Alembert's paradox. It is the result of assuming a fluid of
zero viscosity. The components of the static-pressure forces parallel to the
free-stream direction on the front surface of the airfoil always exactly balance
the components of the pressure forces on the rear surface of the airfoil. The
lift is determined by the static-pressure difference between the upper and lower
surfaces and is zero for this particular case since the pressure distribution is
symmetrical. If, however, the airfoil is tilted at an angle to the free stream,
the pressure distribution symmetry between the upper and lower surfaces no
longer exists and a lift force results. This is very desirable and the main
function of the airfoil section.

Air is not a perfect fluid. It possesses viscosity. With
slight modification, the continuity and Bernoulli principles still apply in the
real world. The airflow over an....

[39]

Figure 29.- Ideal fluid flow about an airfoil.

....airfoil will appear to be slightly different with an
accompanying reduction in lift and the existence of drag in several forms. The
discussions of the past few pages represent basic principles. From this point
on, the inviscid assumption is dropped and a real, viscous flow of air is
allowed to exist.

Real Fluid Flow

Laminar and turbulent flow.- There are two different types
of real fluid flow: laminar and turbulent. In laminar flow the fluid moves in
layers called laminas. Figure 30(a) shows a laminar flow, the uniform rectilinear flow, consisting
of air moving in straight-line layers (laminas) from left to right. The laminas
may be considered the adjacent streamtubes and then the streamlines indicate the
direction of movement of these fluid layers. Laminar flow need not be in a
straight line. Figure 30(b) shows...

[40]

Figure 30.- Laminar and turbulent flow.

...a small segment of the surface of a curved airfoil. For
an ideal fluid the flow follows the curved surface smoothly, in laminas.
Figure 30(c) shows the more complex flow case for a real fluid to be
discussed later. The closer the fluid layers are to the airfoil surface, the
slower they move. However, here also, as indicated by the streamlines, the fluid
layers slide over one another without fluid being exchanged between
layers.

In turbulent flow, secondary random motions are
superimposed on the principal flow. Figure 30(d) shows a disorganized number of streamlines. They are evidently
[41] not fluid layers
and there is an exchange of fluid from one adjacent sector to another. More
importantly, there is an exchange of momentum such that slow moving fluid
particles speed up and fast moving particles give up their momentum to the
slower moving particles and slow down themselves. Consider figure
30(e) which shows the smoke rising from a
cigarette. For some distance the smoke rises in smooth filaments which may wave
around but do not lose their identity; this flow is laminar. The filaments (or
streamtubes) suddenly break up into a confused eddying motion some distance
above the cigarette; this flow is turbulent. The transition between laminar and
turbulent flow moves closer to the cigarette when the air in the room is
disturbed.

"Another example of a common occurrence of laminar and
turbulent flow is the water faucet. Opened slightly, at low speeds the water
flows out in a clear column - laminar flow. But open the faucet fully and the
flow speeds out in a cloudy turbulent column. In a mountain brook the water may
slide over smooth rocks in laminas. In the Colorado River the flow churns
downstream in the confused turbulent rapids. It will be seen that the flow over
airfoil surfaces may assume both a laminar and turbulent characteristic
depending upon a number of factors.

In some cases, turbulent flow will appear "naturally" in a
laminar flow as in the smoke rising in the air. In other cases, by causing a
disturbance, a laminar flow can be changed to a turbulent flow. The question
arises as to how one can tell whether a flow is to be laminar or turbulent. In
1883, Osborne Reynolds introduced a dimensionless parameter which gave a
quantitative indication of the laminar to turbulent transition.

Reynolds number effects on the flow w field.- In his
experiments, Reynolds demonstrated the fact that under certain circumstances the
flow in a tube changes from laminar to turbulent over a given region of the
tube. The experimental setup is illustrated in figure 31(a). A large water tank had a long tube outlet with a stopcock at
the end of the tube to control the flow speed. The tube was faired smoothly into
the tank. A thin filament of colored fluid was injected into the flow at the
mouth.

When the speed of the water flowing through the tube was
low, the filament of colored fluid maintained its identity for the entire length
of the tube. (See fig. 31(b).) However, when the flow speed was high, the filament broke up
into the turbulent flow that 'existed through the cross section. (See
fig. 31(c).)

Reynolds defined a dimensionless parameter, which has since
been known as the Reynolds number, to give a quantitative description of the
flow. In equation form the Reynolds number R is..

R = pVl/µ

(10)

[42]

Figure 31.- Dependence of flow on reynolds number. R =
(pVl)/µ

where

p

density of fluid, kg/m3
[Greek letter rho]

V

mean velocity of fluid, m/sec

l

characteristic length, m

µ

coefficient of viscosity (called simply "viscosity"
in the earlier discussion), kg/m-sec

For this setup, Reynolds found, by using water, that below
R = 2100 the flow in the pipe was laminar as evidenced by the distinct colored
filament. This value was true regardless of his varying combinations of values
of p , V,
l , or µ. A transition
between laminar and turbulent flow occurred for Reynolds numbers between 2100
[43] and 40 000
depending upon how smooth the tube junction was and how carefully the flow
entered the tube. Above R = 40 000 the flow was always turbulent, as evidenced
by the colored fluid filament breaking up quickly. The fact that the transition
Reynolds number (between 2100 and 40 000) was variable indicates the effect that
induced turbulence has on the flow.

The numerical values given for the transition are for this
particular experiment since the characteristic length chosen l is the diameter of the pipe. For an
airfoil, l would be the
distance between the leading and trailing edge called the chord length.
Additionally, water was used in the Reynolds experiment whereas air flows about
an airfoil. Thus, the transition number between laminar and turbulent flow would
be far different for the case of an airfoil. Typically, airfoils operate at
Reynolds numbers of several million. The general trends, however, are evident.
For a particular body, low Reynolds number flows are laminar and high Reynolds
number flows are mostly turbulent.

The Reynolds number may be viewed another way:

(11)

The viscous forces arise because of the internal friction
of the fluid. The inertia forces represent the fluid's natural resistance to
acceleration. In a low Reynolds number flow the inertia forces are negligible
compared with the viscous forces whereas in high Reynolds number flows the
viscous forces are small relative to the inertia forces. An example of a low
Reynolds number flow (called Stoke's flow) is a small steel ball dropped into a
container of heavy silicon oil. The ball falls slowly through the liquid;
viscous forces are large. Dust particles settling through the air are another
case of a low Reynolds number flow. These flows are laminar. In a high Reynolds
number flow, such as typically experienced in the flight of aircraft, laminar
and turbulent flows are present. Some very interesting contrasts between the
results of low Reynolds number flow and high Reynolds number flow will be
demonstrated shortly.

Surface roughness effects on the flow field.- The effect of
surface roughness of a body immersed in a flow field is that it causes the flow
near the body to go from laminar to turbulent. As the surface roughness
increases, the point of first occurrence of turbulent flow will move upstream
along the airfoil. Figure 32 illustrates this point. An airfoil surface is shown. In each
succeeding case the degree of surface roughness is increased and the Reynolds
number is held fixed. The flow is seen to go turbulent further upstream in each
case. The Reynolds number and [44] surface roughness are not independent of each other and both
contribute to the determination of the laminar to turbulent transition. A very
low Reynolds number flow will be laminar even on a rough surface and a very high
Reynolds number flow will be turbulent even though the surface of a body is
highly polished.

Pressure gradient effects on the flow field.- Another
important factor in the transition from laminar to turbulent flow is the
pressure gradient in the flow field. If the static pressure increases with
downstream distance, disturbances in a laminar flow will be amplified and
turbulent flow will result. If the static pressure decreases with downstream
distance, disturbances in a laminar flow will damp out and the flow will tend to
remain laminar. Recall that over an airfoil the static pressure decreased up to
the point of maximum thickness. A laminar flow will be encouraged in this
region. Beyond the point of maximum thickness (or shoulder of the airfoil) the
static pressure increased. The laminar flow now is hindered and may go turbulent
before the trailing edge.

Figure 32.- Surface roughness and flow field. All
cases at same Reynolds number.

[45] The boundary layer and skin-friction drag.- The foregoing
discussion has provided the background needed to show how drag is produced on a
body immersed in a real fluid flow. An important aerodynamic force during
low-speed subsonic flight is the shear force caused by viscous flow over the
surfaces of the vehicle. This shear force is referred to as the skin-friction
force and is strongly dependent on the factors previously mentioned-Reynolds
number, surface roughness, and pressure gradients. Figure 33 shows that in addition to the pressure forces that act everywhere
normal to a body immersed in a moving fluid, viscous forces are also present. It
is these viscous forces which modify the ideal fluid lift and help create the
real fluid drag.

Figure 33.- Pressure and viscous forces.

Consider figure 34 which shows a very thin, smooth plate parallel to the approaching
flow; the flow ahead of the leading edge of the plate is a uniform free stream.
If the fluid were ideal, that is, inviscid, the fluid would simply slip over the
surface with velocity as shown in
figure 34(a). At all points along the surface of the plate, the velocity
distribution (that is, the variation of velocity as one moves [46] perpendicularly away from the
surface) would be a uniform constant value of. No drag would
result if the fluid were frictionless (inviscid).

In a real fIuid, however, a very thin film of fluid adheres
to the surface. (See fig. 34(b).) This is the very important no-slip condition. It states that
at the surface of a body, point B, the flow velocity is zero. As one moves away
from the body the velocity of the fluid gradually increases until at some point
A the velocity becomes a constant value; in the case of a flat plate this value
is . The layer of fluid where the
velocity is changing from zero to a constant value is known as the boundary
layer. Within the boundary layer there are relative velocities between the
particle layers and an internal friction is present. This internal friction
extends to the surface of the body. The cumulative effect of all these friction
forces is to produce a drag force on the plate. This drag force is referred to
as skin-friction drag.

Initially, near the leading edge of the plate, one has a
laminar flow and the boundary layer also is steady and layered- hence, a laminar
boundary layer. As one moves further downstream, viscosity continues to act and
the laminar boundary layer thickens as more and more fluid is slowed down by
internal friction. Eventually, a point is reached on the plate where the laminar
boundary layer undergoes transition and becomes a turbulent boundary layer. (See
fig. 34(b).) As is usual for turbulent flow, there is a random motion in
the boundary layer as well as the downstream directed motion. There is no slip
at the surface of the plate. Another important difference from the laminar
boundary layer is the fact that the velocity builds up more quickly as one moves
away from the wall, although the total boundary-layer thickness is greater. This
condition can be seen by comparing the two profiles as shown in figure
34(c). This tendency in a turbulent boundary layer
of the fluid further away from the wall to reenergize the slower moving fluid
near the wall will be shown to produce important consequences.

The Reynolds number has an important effect on the boundary
layer. As the Reynolds number increases (caused by increasing the flow speed
and/or decreasing the viscosity), the boundary layer thickens more slowly.
However, even though the Reynolds number becomes large, the velocity at the
surface of the body must be zero. Thus, the boundary layer never
disappears.

It is interesting to note that a typical thickness of the
boundary layer on an aircraft wing is generally less than a centimeter. Yet, the
velocity must vary from zero at the surface of the wing to hundreds of m/sec at
the outer edge of the boundary layer. It is clearly evident that tremendous
shearing forces (internal friction) must be acting in this region. This gives
rise to the skin-friction drag.

[47]

(a) Inviscid flow along a flat
plate.

(b) Viscous flow along a flat plate.

(c) Comparison of laminar and turbulent
flow.

Figure 34.- Boundary-layer flow in a real fluid.

The airfoil in a real fluid.- Figure 35 illustrates the real fluid flow over the airfoil surface originally
considered in figure 29. The same free-stream velocity and free-stream
static pressure apply. The flow field ahead of the airfoil is only
slightly modified and for all practical purposes the velocities and static
pressures are the same as for the ideal fluid case. Again a stagnation point
occurs at the leading edge of the airfoil and the pressure reaches its maximum
value of pt at this
point (total or stagnation pressure). From this point on along the airfoil, the
picture changes.

[48]

Figure 35.- Real fluid flow about an airfoil.
Thickness of boundary layers and wake greatly exaggerated. Bottom flow along
lower surface is the same as on the upper surface.

As noted earlier in the example of the flat plate, a
boundary layer begins to form because of viscosity. This boundary layer is very
thin and outside of it the flow acts very much like that of an ideal fluid.
Also, the static pressure acting on the surface of the airfoil is determined by
the static pressure outside the boundary layer. This pressure is transmitted
through the boundary layer to the surface and thus acts as if the boundary layer
were not present at all. But the boundary layer feels this static pressure and
will respond to it.

Over the front surface of the airfoil up to the shoulder,
an assisting favorable pressure gradient exists (pressure decreasing with
distance downstream). The flow speeds up along the airfoil. The flow is laminar
and a laminar boundary layer is present. This laminar boundary layer grows in
thickness along the airfoil. When the shoulder is reached, however, the fluid
particles are moving slower than in the ideal fluid case. This is an unfavorable
condition because the previous ideal flow just came to rest at the trailing
edge. It would appear now, with viscosity present, that the flow will come to
rest at some distance before the trailing edge is reached.

As the flow moves from the shoulder to the rear surface,
the static-pressure gradient is unfavorable (increasing pressure with downstream
distance). The fluid particles must push against both this unfavorable pressure
gradient and the viscous forces. At the transition point, the character of the
flow changes and the laminar boundary layer quickly becomes a turbulent boundary
layer. This turbulent boundary layer continues to thicken downstream. Pushing
against an unfavorable pressure gradient and viscosity is too much for the flow,
and at some point, the flow stops completely. The boundary layer has stalled
short of reaching the trailing edge. (Remember that the flow reached the
trailing edge before stopping in the ideal fluid case.)

[49] This stall point is known as the separation point. All along a
line starting from this point outward into the flow, the flow is stalling.
Beyond this line the flow is actually moving back, upstream toward the nose
before turning around. This is a region of eddies and whirlpools and represents
"dead,, air which is disrupting the flow field away from the airfoil. Thus, flow
outside the dead air region is forced to flow away and around it. The region of
eddies as shown in figure 35 is called the wake behind the airfoil.

Figure 36(a) compares the ideal fluid case static-pressure distribution at
the airfoil surface and center-line streamline with; the real fluid case. Note
that up to the separation point, the differences are not very large but once
separation occurs the pressure field is greatly modified. In the ideal fluid
case the net static-pressure force acting on the front surface of the airfoil
(up to the shoulder) parallel to the free stream exactly opposed and canceled
that acting on the rear surfaces of the airfoil. (See fig.
36(b).) Now, however, in the real fluid case this
symmetry and cancellation of forces is destroyed. The net static-pressure force
acting on the front surface parallel to the free-stream direction now exceeds
that acting on the rear surface. The net result is a drag force due to the
asymmetric pressure distribution called pressure drag. (See fig.
36(c).) This is a drag in addition to the
skin-friction drag due to the shearing forces (internal friction) in the
boundary layer. Additionally, the modification of the static-pressure
distribution causes a decrease in the pressure lift from the ideal fluid
case.

(a) Airfoil upper surface static-pressure
distributions.

(b) Ideal fluid airfoil (no pressure drag).

(c) Real fluid airfoil (net pressure
drag).

Figure 36.- Real fluid effects on an airfoil.

[50] Figure 36(d) shows figuratively the lift and drag for an airfoil producing
lift in both an ideal and real fluid case. One sees the effects of viscosity -
the lift is reduced and a total drag composed of skin-friction drag and pressure
drag is present. Both of these are detrimental effects.

(d) Viscosity effects on an airfoil.

Figure 36.- Concluded.

It should be noted, very strongly, that although the
previous discussion was limited to an airfoil section, similar processes are
occurring on all the other components of the aircraft to one degree or another.
It is beyond the scope of this text to treat these in detail but the effects
will be noted when the total airplane drag is discussed.

In summarizing this discussion, one observes that the
effects of a real fluid flow are the result of the viscosity of the fluid. The
viscosity causes a boundary layer and, hence, a skin-friction drag. The flow
field is disrupted because of viscosity to the extent that a pressure drag
arises. Also, the net pressure lift is reduced. The next section considers the
effects of "streamlining.''

Effects of streamlining.- Figure 37 shows five bodies placed in a real fluid flow of air and the
resultant flow field. Four of the bodies are operating at Reynolds numbers
normally encountered in the flight of subsonic aircraft (R = 104 to 105). The fifth body is operating at a
much higher Reynolds number (R = 107).

The flat plate placed broadside to the flow has a large
wake with separation points at the plate edge. A large pressure drag is the
result, the skin-friction drag being a relatively small component. The cylinder.
operating at the same Reynolds number, has a smaller wake and the boundary-layer
separation occurs, in this case, before the shoulders of the cylinder. The
skin-friction drag is a little larger in this case than for the plate. but is
still smaller than the pressure drag. Overall, the total drag has been reduced
from that of the flat plate; some effects of streamlining are already
evident.

[51]

Figure 37.- Effects of streamlining at various
Reynolds numbers.

Also, at the same Reynolds number is a streamlined shape.
There is almost no boundary- layer separation and the wake is very small. One
may assume then that a streamline shape may be defined as the absence of
'boundary-layer separation. Operating in the condition shown, the skin- friction
drag now is the dominant component and the pressure drag is very small. Even
more noticeable is the very large reduction in overall drag compared with the
cylinder or plate. This has been accomplished by eliminating the pressure drag
since the skin-friction drag has been increasing only slightly as the bodies
became more streamlined. One can explain that the increase in skin-friction drag
is due to the simple fact that the streamlined body has more area exposed to the
flow and thus has a greater area over which the boundary layer may
act.

[52] Finally, in figure 37 at a Reynolds number of 104 is a cylinder approximately 1/10
the diameter of the streamline shape thickness. Surprisingly it has the same
drag as the much larger streamlined shape. The pressure drag is large because of
the turbulent wake. It is not hard to imagine the reason for the slow speeds of
the early biplanes when all the wire bracing used is considered. A considerably
reduced drag could have been realized if the wire were streamlined. However, the
introduction of the monoplane and better structures eliminated the need for this
wire bracing.

The fifth body shown is a cylinder operating in the flow at
a much higher Reynolds number (accomplished by increasing the free-stream
velocity). The separation points are downstream of the shoulders of the cylinder
and a much smaller wake is evidenced. This result would lead one to expect a
smaller pressure drag than for the previously discussed cylinder of the same
size. However, the flow speed is higher and the actual drag is much larger.
These contradictory facts may be explained by realizing that actual drag values,
under different flow speeds, have been compared. A better measure of the
performance is needed. This measure is demonstrated in the next section to be
the nondimensional drag coefficient.

Aerodynamic coefficients.- From everyday experience,
consider the factors that determine the aerodynamic resistance on a body. If one
places his hand broadside to a flow outside a car window at 20 km/hr, little
resistance is felt, but if one speeds along at 100 km/hr, the force felt is
considerable. Velocity is one factor that determines the resistance. In fact,
considering the flow problems of subsonic flight (high Reynolds number under
relatively small viscosities), the resistance depends directly on (velocity)
times (velocity) or (velocity)2. In the preceding example, although the velocity of 100 km/hr is
five times that of 20 km/hr, the aerodynamic resistance is about 25 times as
great at the higher velocity.

If one walks along a beach, there is little aerodynamic
resistance to doing so. But try to wade in the water at the same speed. It is
considerably more difficult, if not impossible. The density of water is much
greater than the density of air. Density of the fluid represents another
determining factor in the resistance felt by a body.

One more experiment: hold a small piece of cardboard up
against a stiff wind. Little resistance is experienced. Now hold a much larger,
similarly shaped sheet of cardboard up against the same stiff wind. A
considerable resistance is felt. Area (or length times length) exposed to the
airflow is another determining factor of resistance.

It is now possible to generalize the discussion by stating
that, in the flow of the real fluid, air, about a body, the aerodynamic
resistance is dependent on the size, shape, and attitude of the body, the
properties of the fluid, and the relative velocity between the body and the
fluid (air). To illustrate, consider the lift force defined as the
[53] aerodynamic
reaction perpendicular to the free-stream velocity direction. From the previous
discussion, lift depends on (size shape attitude fluid properties. and
velocity). For an ideal fluid, the fluid properties (except for density) did not
influence the lift force. For a real fluid, however. viscous, elastic, and
turbulent properties are also important. In addition to the shape and attitude
of the body the surface roughness has an effect on the force. Based on the
introductory discussion of this section, it may be demonstrated that

(12)

where

free-stream fluid density

free-stream velocity

S

characteristic body frontal area

l

characteristic body length

a [Greek letter alpha]

attitude of body

µ

coefficient of viscosity

free-stream speed of sound of fluid

(S is a characteristic body frontal area that is usually
chosen to be consistent with a series of comparison experiments. For a cylinder
it would be the diameter of the cylinder times its length. For a wing, however,
it is usually taken to be the planform area (chord length times wing span for a
rectangular wing). Thus. it is necessary to check the particular definition of S
used for a body.)

It has previously been shown that the quantity is the Reynolds number or R. Also, is defined to be the Mach number or M. The Reynolds number is the
dimensionless quantity associated with the fluid viscosity whereas the Mach
number is associated with the fluid compressibility. Surface roughness was shown
to have influenced the transition from a laminar to a turbulent flow. Air
turbulence represents the degree of the wake formed past the separation points.
Furthermore, the effects of [54] attitude and shape of a body are lumped together into the factor.
Letting the factor be called K, then,

(13)

The dynamic pressure of a fluid flow was previously defined
as 1/2 pV2 so
if a value of 1/2 is included in equation (13) and the value of K is doubled to
keep the equation the same, 2K may be replaced by CL. Finally,

(14)

Equation (14) is the fundamental lift formula for usual
aircraft flight. CL
is known as the coefficient of lift. The equation states simply that the
aerodynamic lift is determined by a coefficient of lift times the free-stream
dynamic pressure times the characteristic body area.

It is very important to realize that the lift coefficient
CL is a number
dependent upon the Reynolds number, Mach number, surface roughness, air
turbulence, attitude, and body shape. It is not by any means a constant.
CL is generally
found by wind-tunnel or flight experiments by measuring lift and the free-stream
conditions and having a knowledge of the body dimensions. Thus,

(15)

The aerodynamic drag is the aerodynamic resistance parallel
to the free-stream direction. One obtains analogous equations to equations (14)
and (15), namely,

(16)

or

(17)

where CD is the drag coefficient, dependent on the previously enumerated
parameters.

[55]

Figure 38.- Drag coefficients of various bodies.

The moment acting on a body is a measure of the body's
tendency to turn about its center of gravity. This moment represents the
resultant aerodynamic force times a moment distance. Let it be stated that a
similar derivation may be applied to the moment equation as used for the lift
and drag equations (14) and (16) such that,

(18)

or

(19)

[56] Cm is
the coefficient of moment and an additional characteristic length
l is necessary for it
to be dimensionally correct. To reiterate, CL, CD, and Cm are dependent on the Reynolds
number, Mach number surface roughness, air turbulence, attitude, and body
shape.

It is now possible to return to the discussion associated
with figure 37 and compare the five bodies by using the force coefficient as a
measure of the resistance. The first three bodies demonstrated the effects of
progressively more streamlining. All had the same basic body dimension d , the
same Reynolds number R = 105, the same Mach number, and were assumed to be smooth and alined
symmetrically with the flow. The aerodynamic resistance was, therefore, entirely
drag and the drag coefficient CD is a measure of this resistance. By assuming a unit length for the
bodies, the frontal area S is the same for all the bodies as is the dynamic
pressure. By equation (17), CD is then directly proportional to the measured drag. Figure
38 repeats the results of figure
37 except that now the relative drag force has been
replaced by the drag coefficients CD. At the Reynolds number of
105, the
CD values for the
flat plate, cylinder, and streamline shape are, respectively, 2.0, 1.2, and
0.12. These values include the combined effects of the skin-friction drag and
pressure drag.

The small cylinder, operating at a Reynolds number of
104 with its
diameter reduced to one- tenth the basic dimension of the previous examples, has
a CD of 1.2. From
equation (16), the effect of smaller size nullifies the effect of larger
CD and the small
cylinder and streamline shape have equivalent aerodynamic drags.

The last cylinder, operating at the higher Reynolds number
of 107, has a
CD of 0.6, that is,
half as large as the cylinders discussed previously. Its aerodynamic drag in
figure 37 is large because has been
increased to obtain the higher Reynolds number. The smaller drag coefficient
indicates the effect of the smaller wake and, hence, smaller pressure drag
coefficient component. At high Reynolds numbers, the boundary layer becomes
turbulent further upstream along the cylinder. The turbulence in the boundary
layer reenergizes the flow close to the surface and the fluid drives further
along the cylinder against viscous forces and the unfavorable pressure gradient
before stalling. Separation occurs downstream of the shoulders and a smaller
wake results. Compare this condition with the separation and wake at the lower
Reynolds number.

Figure 39 is a plot of drag coefficient CD (based on frontal area)
against Reynolds number. The values for each body are shown. Also, the solid
line is an experimentally determined curve of the CD of cylinders tested in wind
tunnels. At subcritical Reynolds numbers up to about 105, the laminar boundary layer stalls
and separates upstream of the shoulders of the cylinder and produces a very
broad wake...

[57]

Figure 39.- Drag coefficients as function of Reynolds
number.

....and high CD values. At supercritical Reynolds
numbers from 106 and larger, the laminar boundary layer becomes turbulent and
separation is delayed; hence, the smaller CD values. A rather abrupt transition
occurs between Reynolds numbers of 105 and 106. These values are the critical
Reynolds numbers.

It is interesting to note that spheres exhibit behavior
very similar to that of cylinders. Golf balls of today are dimpled rather than
smooth as they once were, to induce a turbulent boundary layer and thus decrease
their drag coefficient. Much improved driving distances are the
result.

The discussion thus far has been rather general and has
introduced many important ideas and principles. Fluid flow behavior has been
demonstrated. Numerous references to airfoil or streamline shapes have been
made. Viscous flow of the boundary layer and unsteady flow in the turbulent wake
have been examined. The flow is twodimensional since velocity and other flow
parameters vary normal to the free-stream direction as well as parallel to it.
With these ideas in mind, one may now study aircraft operating in a subsonic
flow.