Viscosity. There are basically three states of matter  solid, liquid, and gas. H_{2}O 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 10^{16} 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 fluidone 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
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] 6hour 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.
[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.
[28] Figure 22 shows a nicely "streamlined'' body (as opposed to the bluffshaped house) in a wind tunnel. At time to the tunnel is not running and no air is flowing. At time t_{1} the tunnel is started and air begins flowing about the body; the flow develops further at time t_{2} and finally reaches a constant pattern at time t_{3}. The flow appears unchanged at time t_{4} and time t_{5}. When the flow starts. it passes through an unsteady transient state; that is. particle pathlines and streamlines are not the same. From time t_{3} 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 t_{3} moves downstream along that streamline as shown at times t_{4} and t_{5}. The particle pathline coincides with the streamline.
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.
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.
Onedimensional flow. A simplifying argument often employed to aid in understanding basic ideas is that of a onedimensional 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.
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 crosssectional areas A_{1} and A_{2}, respectively. Let V_{1} and V_{2 }be the average flow speeds at these cross sections (onedimensional 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
p_{l }A_{l}V_{1} = p_{2}A_{2}V_{2} 
(3) 
Since the fluid is assumed to be incompressible, p [Greek letter rho] is a constant and equation (3) reduces to
A_{l}V_{1} = A_{2}V_{2} 
(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 V_{1} and V_{2} across the cross section are used.
By rearranging equation (4), one obtains
V_{2} = (A_{1}/A_{2})V_{1} 
(5) 
Since A_{1} is greater than A_{2} (see fig. 25(a)), it can be concluded that V_{2} is greater than V_{1}. 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 lowspeed flow and closely spaced streamlines indicate regions of highspeed flow.
Bernoulli's theoremthe 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.
[34] The kinetic energy per unit volume is called dynamic pressure q and is determined by q = 1/2pV^{2} 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 p_{t}.
Bernoulli's equation reduces to
Dynamic pressure + Static pressure = Total pressure 
(7) 
or
1/2pV^{2} + p = p_{t} 
(8) 
For rotational flow the total pressure p_{t } 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 totalpressure 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...
....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 staticpressure measuring device.
Figure 27(c) shows a combined pilotstatic tube. When properly connected to opposite ends of a pressure measuring readout instrument, the difference between total pressure and static pressure ismeasured. By Bernoulli's equation this difference is the dynamic pressure, defined as 1/2pV^{2}. If the fluid density p is known, the fluid flow speed can be calculated. In actual use on aircraft, the pilotstatic 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...
...on a boom extending from the airplane nose to insure its measuring, as closely as possible, the undisturbed approaching flow (also called the freestream condition).
Returning to the discussion of the venturi tube introduced earlier, the continuity and Bernoulli equations may be used to describe the staticpressure distribution along the venturi tube. The static pressure of the undisturbed freestream 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 freestream 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 "Utube manometer"  a tube having a Ushape within which is a liquid such as colored alcohol. When the static pressure measured at the static tap equals the freestream static pressure, the fluid levels in the tube are at some equal reference level. But static pressures above or below the freestream pressure are indicated by a decrease or increase in the level of fluid in the tube.
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 V_{2} is greater than that at station 1 V_{1} as seen previouslythe speed at the throat also is the highest speed achieved in the venturi tube. By Bernoulli's equation the total pressure p_{t} is constant everywhere in the flow (assuming irrotational flow). Therefore, one can express the total pressure p_{t} in terms of the static and dynamic pressures at stations 1 and 2 using equation (8), namely,
1/2p_{l}V_{1}^{2} + p_{1} = 1/2p_{2}V_{2}^{2} + p_{2} = pt 
(9) 
Since V_{2} is greater than V_{1} and p_{2} = p_{1} (fluid is incompressible) it follows that p_{2} is less than p_{1}, for as the dynamic pressure, hence speed, increases, the static pressure must decrease to maintain a constant value of total pressure p_{t}. 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 highspeed flow and increases in the region of lowspeed 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 freestream direction. The freestream velocity is denoted by and the freestream 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 freestream value is reached and the static pressure returns to freestream static pressure. These velocity and staticpressure distributions for the centerline 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 freestream direction and the drag parallel to the freestream 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 staticpressure forces parallel to the freestream 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 staticpressure 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....
....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.
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 straightline 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...
...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) 
where
p 
density of fluid, kg/m^{3 } [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/msec 
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.
[45] The boundary layer and skinfriction 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 lowspeed subsonic flight is the shear force caused by viscous flow over the surfaces of the vehicle. This shear force is referred to as the skinfriction force and is strongly dependent on the factors previously mentionedReynolds 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.
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 noslip 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 skinfriction 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 boundarylayer 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 skinfriction drag.
The airfoil in a real fluid. Figure 35 illustrates the real fluid flow over the airfoil surface originally considered in figure 29. The same freestream velocity and freestream 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 p_{t } at this point (total or stagnation pressure). From this point on along the airfoil, the picture changes.
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 staticpressure 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 staticpressure distribution at the airfoil surface and centerline 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 staticpressure 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 staticpressure force acting on the front surface parallel to the freestream 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 skinfriction drag due to the shearing forces (internal friction) in the boundary layer. Additionally, the modification of the staticpressure distribution causes a decrease in the pressure lift from the ideal fluid case.
[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 skinfriction drag and pressure drag is present. Both of these are detrimental effects.
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 skinfriction 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 = 10^{4} to 10^{5}). The fifth body is operating at a much higher Reynolds number (R = 10^{7}).
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 skinfriction drag being a relatively small component. The cylinder. operating at the same Reynolds number, has a smaller wake and the boundarylayer separation occurs, in this case, before the shoulders of the cylinder. The skinfriction 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.
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 'boundarylayer 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 skinfriction drag has been increasing only slightly as the bodies became more streamlined. One can explain that the increase in skinfriction 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 10^{4} 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 freestream 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 freestream 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

freestream fluid density 

freestream velocity 
S 
characteristic body frontal area 
l 
characteristic body length 
a [Greek letter alpha] 
attitude of body 
µ 
coefficient of viscosity 

freestream 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 pV^{2} 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 C_{L}. Finally,

(14) 
Equation (14) is the fundamental lift formula for usual aircraft flight. C_{L} is known as the coefficient of lift. The equation states simply that the aerodynamic lift is determined by a coefficient of lift times the freestream dynamic pressure times the characteristic body area.
It is very important to realize that the lift coefficient C_{L} 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. C_{L} is generally found by windtunnel or flight experiments by measuring lift and the freestream conditions and having a knowledge of the body dimensions. Thus,

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

(16) 
or

(17) 
where C_{D} is the drag coefficient, dependent on the previously enumerated parameters.
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] C_{m} is the coefficient of moment and an additional characteristic length l is necessary for it to be dimensionally correct. To reiterate, C_{L}, C_{D}, and C_{m } 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 = 10^{5}, 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 C_{D} 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), C_{D} 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 C_{D}. At the Reynolds number of 10^{5}, the C_{D} 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 skinfriction drag and pressure drag.
The small cylinder, operating at a Reynolds number of 10^{4 } with its diameter reduced to one tenth the basic dimension of the previous examples, has a C_{D} of 1.2. From equation (16), the effect of smaller size nullifies the effect of larger C_{D} and the small cylinder and streamline shape have equivalent aerodynamic drags.
The last cylinder, operating at the higher Reynolds number of 10^{7}, has a C_{D} 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 C_{D} (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 C_{D} of cylinders tested in wind tunnels. At subcritical Reynolds numbers up to about 10^{5}, the laminar boundary layer stalls and separates upstream of the shoulders of the cylinder and produces a very broad wake...
....and high C_{D} values. At supercritical Reynolds numbers from 106 and larger, the laminar boundary layer becomes turbulent and separation is delayed; hence, the smaller C_{D} values. A rather abrupt transition occurs between Reynolds numbers of 10^{5} and 10^{6}. 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 freestream direction as well as parallel to it. With these ideas in mind, one may now study aircraft operating in a subsonic flow.