The aerodynamic forces and moments on a body, and the corresponding force and moment coefficients, have been defined and discussed in Section 1.5. Question: What physical quantities determine the variation of these forces and moments? The answer can be found from the powerful method of dimensional analysis, which is introduced in this section.3
Consider a body of given shape at a given angle of attack, e. g., the airfoil sketched in Figure 1.10. The resultant aerodynamic force is R. On a physical, intuitive basis, we expect R to depend on:
1. Freestream velocity Voo.
2. Freestream density Poo.
3. Viscosity of the fluid. We have seen that shear stress r contributes to the aerodynamic forces and moments. In turn, in Chapter 15, we will see that r is proportional to the velocity gradients in the flow. For example, if the velocity gradient is given by du/dy, then r = /іди/ду. The constant of proportionality is the viscosity coefficient fi. Hence, let us represent the influence of viscosity on aerodynamic forces and moments by the freestream viscosity coefficient fi<*,.
4. The size of the body, represented by some chosen reference length. In Figure 1.10, the convenient reference length is the chord length c.
5. The compressibility of the fluid. The technical definition of compressibility is given in Chapter 7. For our present purposes, let us just say that compressibility is related to the variation of density throughout the flow field, and certainly the aerodynamic forces and moments should be sensitive to any such variation. In turn, compressibility is related to the speed of sound a in the fluid, as shown in Chapter 8.4 Therefore, let us represent the influence of compressibility on aerodynamic forces and moments by the freestream speed of sound, ax.
3 For a more elementary treatment of dimensional analysis, see Chapter 5 of Reference 2.
4 Common experience tells us that sound waves propagate through air at some finite velocity, much slower than the speed of light; you see a flash of lightning in the distance, and hear the thunder moments later. The speed of sound is an important physical quantity in aerodynamics and is discussed in detail in Section 8.3.
In light of the above, and without any a priori knowledge about the variation of R, we can use common sense to write
R — f (Pco, Fqo, C, fJ, oc, r^oc)
Equation (1.23) is a general functional relation, and as such is not very practical for the direct calculation of R. In principle, we could mount the given body in a wind tunnel, incline it at the given angle of attack, and then systematically measure the variation of R due to variations of px, V^, с, /xTO, and ax, taken one at a time. By crossplotting the vast bulk of data thus obtained, we might be able to extract a precise functional relation for Equation (1.23). However, it would be hard work, and it would certainly be costly in terms of a huge amount of required windtunnel time. Fortunately, we can simplify the problem and considerably reduce our time and effort by first employing the method of dimensional analysis. This method will define a set of dimensionless parameters which governs the aerodynamic forces and moments; this set will considerably reduce the number of independent variables as presently occurs in Equation (1.23).
Dimensional analysis is based on the obvious fact that in an equation dealing with the real physical world, each term must have the same dimensions. For example, if
ф + г} + ї =Ф
is a physical relation, then ifr, t], £, and ф must have the same dimensions. Otherwise we would be adding apples and oranges. The above equation can be made dimensionless by dividing by any one of the terms, say, ф:
These ideas are formally embodied in the Buckingham pi theorem, stated below without derivation. (See Reference 3, pages 2128, for such a derivation.)
Buckingham pi theorem. Let К equal the number of fundamental dimensions required to describe the physical variables. (In mechanics, all physical variables can be expressed in terms of the dimensions of mass, length, and time’, hence, К = 3.)
Let P, P2, ■ ■ ■, PN represent N physical variables in the physical relation
fi(Pi, P2,…,PN)= 0
Then, the physical relation Equation (1.24) may be reexpressed as a relation of (N — K) dimensionless products (called П products),
/2(П,,П2……….. Пм) = 0
where each П product is a dimensionless product of a set of К physical variables plus one other physical variable. Let P, P2, …, Pk be the selected set of К physical variables. Then
n, = h(Pi, P2,…,PK, PK+,) П2 = f4(Pi, P2,…, PK, PK+2)
П/1r — fs{P, P2, ■ ■ ■, Pk, Pn)
The choice of the repeating variables, P, P2, ■ ■ ■, Pk should be such that they include all the К dimensions used in the problem. Also, the dependent variable [such as R in Equation (1.23)] should appear in only one of the П products.
Returning to our consideration of the aerodynamic force on a given body at a given angle of attack, Equation (1.23) can be written in the form of Equation (1.24):
§(R> Росі koo> C, oo, ttoo) — 0 [1 *27]
Following the Buckingham pi theorem, the fundamental dimensions are
m = dimensions of mass l = dimension of length t = dimension of time
Hence, К — 3. The physical variables and their dimensions are
[R] = mlt~2 iPod = ml~3
[Vco]=/r1
[c] = l
[доо] = ml~xt~l
[floo] = If1
Hence, N = 6. In the above, the dimensions of the force R are obtained from Newton’s second law, force = mass x acceleration; hence, [RJ = mlt~2. The dimensions of p. x are obtained from its definition, e. g., i =■ т/(ди/ду), and from Newton’s second law. (Show for yourself that [доо] = ml~lt~x.) Choose px, and c as the arbitrarily selected sets of К physical variables. Then Equation (1.27) can be reexpressed in terms of N — К = 6 — 3 = 3 dimensionless П products in the form of Equation (1.25);
/2(ПьП2,Пз) = 0 [1.28]
From Equation (1.26), these П products are
Пі =f3(p0O, V00,c, R) [1.29a]
П2 — /4(Roo 1 кос5 f, Доо) [ 1.298]
П3 = /5(Poo, ^ocChOoc) [1.29c]
For the time being, concentrate on Пь from Equation (1.29a). Assume that
Пі =piV^ceR [1.30]
where d, b, and e are exponents to be found. In dimensional terms, Equation (1.30) is
Because П, is dimensionless, the right side of Equation (1.31) must also be dimensionless. This means that the exponents of m must add to zero, and similarly for the exponents of / and t. Hence,
Form: d. + 1=0
For/: 3d + b + e + 1 = 0
For t: – b – 2 = 0
Solving the above equations, we find that d — – l, b — 2, ande = 2. Substituting these values into Equation (1.30), we have
Пі = fip^V"2^2 [1.32]
_ R ~ PooVfr*
The quantity R/px F^c2 is a dimensionless parameter in which c2 has the dimensions of an area. We can replace c2 with any reference area we wish (such as the planform area of a wing S), and Пі will still be dimensionless. Moreover, we can multiply Пі by a pure number, and it will still be dimensionless. Thus, from Equation (1.32), П! can be redefined as
Hence, Пі is a force coefficient CR, as defined in Section 1.5. In Equation (1.33), S is a reference area germane to the given body shape.
The remaining П products can be found as follows. From Equation (1.29b), assume
n2 = /OooV£c’V [1.34]
Paralleling the above analysis, we obtain
[П21 = (тГ3)(Іг’У’(іУ(тГ’г1У
Hence,
Form: 1 + j — 0
For/: —3 + h + i— j = 0
For t: – h – j — 0
Thus, j = — 1, h = 1, and і = 1. Substitution into Equation (1.34) gives
The dimensionless combination in Equation (1.35) is defined as the freestream Reynolds number Re = Poo Vooc/Pao – The Reynolds number is physically a measure of the ratio of inertia forces to viscous forces in a flow and is one of the most powerful parameters in fluid dynamics. Its importance is emphasized in Chapters 15 to 20.
Returning to Equation (1.29c), assume
n3 = Voo p^c’a^
[П3] = (ІГ1)(тГ3)к(.іУ(ІГ1У
For m: к — О
For 1: 1— 3fc + r + s = 0
For P. — 1 — s = 0
Hence, к = 0, s = — 1, and r = 0. Substituting Equation (1.36), we have
Пе = — [1.37]
floo
The dimensionless combination in Equation (1.37) is defined as the freestream Mach number M = Vqo/Ooo – The Mach number is the ratio of the flow velocity to the speed of sound; it is a powerful parameter in the study of gas dynamics. Its importance is emphasized in subsequent chapters.
The results of our dimensionless analysis may be organized as follows. Inserting Equations (1.33), (1.35), and (1.37) into (1.28), we have
fi(CR, Re, Moo) = 0
This is an important result! Compare Equations (1.23) and (1.38). In Equation (1.23), R is expressed as a general function of five independent variables. However, our dimensional analysis has shown that:
1. R can be expressed in terms of a dimensionless force coefficient,
cR = R/Poovls.
2. CR is a function of only Re and M„, from Equation (1.38).
Therefore, by using the Buckingham pi theorem, we have reduced the number of independent variables from five in Equation (1.23) to two in Equation (1.38). Now, if we wish to run a series of windtunnel tests for a given body at a given angle of attack, we need only to vary the Reynolds and Mach numbers in order to obtain data for the direct formulation of R through Equation (1.38). With a small amount of analysis, we have saved a huge amount of effort and windtunnel time. More importantly, we have defined two dimensionless parameters, Re and М^, which govern the flow. They are called similarity parameters, for reasons to be discussed in the following section. Other similarity parameters are introduced as our aerodynamic discussions progress.
Since the lift and drag are components of the resultant force, corollaries to Equation (1.38) are
Cl = /7 (Re, Mx) [1.39]
CD = MRe, Moo) [1.40]
Moreover, a relation similar to Equation (1.23) holds for the aerodynamic moments, and dimensional analysis yields
CM = /9(Re, Mx) [1.41]
Keep in mind that the above analysis was for a given body shape at a given angle of attack a. If a is allowed to vary, then CL, CD, and CM will in general depend on the value of a. Hence, Equations (1.39) to (1.41) can be generalized to
Equations (1.42) to (1.44) assume a given body shape. Much of theoretical and experimental aerodynamics is focused on obtaining explicit expressions for Equations (1.42) to (1.44) for specific body shapes. This is one of the practical applications of aerodynamics mentioned in Section 1.2, and it is one of the major thrusts of this book.
For mechanical problems that also involve thermodynamics and heat transfer, the temperature, specific heat, and thermal conductivity of the fluid, as well as the temperature of the body surface (wall temperature), must be added to the list of physical variables, and the unit of temperature (say, kelvin or degree Rankine) must be added to the list of fundamental dimensions. For such cases, dimensional analysis yields additional dimensionless products such as heat transfer coefficients, and additional similarity parameters such as the ratio of specific heat at constant pressure to that at constant volume cp/cv, the ratio of wall temperature to freestream temperature Tw/Too, and the Prandtl number Pr = k^_. where кж is the thermal conduc
tivity of the freestream.[3] Thermodynamics is essential to the study of compressible flow (Chapters 7 to 14), and heat transfer is part of the study of viscous flow (Chapters 15 to 20). Hence, these additional similarity parameters will be emphasized when they appear logically in our subsequent discussions. For the time being, however, the Mach and Reynolds numbers will suffice as the dominant similarity parameters for our present considerations.

In several of our previous discussions, we made use of the concept of a fluid element moving through the flow field. In this section, we examine this motion more closely, paying particular attention to the orientation of the element and its change in shape as it moves along a streamline. In the process, we introduce the concept of vorticity, one of the most powerful quantities in theoretical aerodynamics.
Consider an infinitesimal fluid element moving in a flow field. As it translates along a streamline, it may also rotate, and in addition its shape may become distorted as sketched in Figure 2.30. The amount of rotation and distortion depends on the velocity field; the purpose of this section is to quantify this dependency.
Element at time 11
Figure 2.30 The motion of a fluid element along a


streamline is a combination of translation and rotation; in addition, the shape of the element can become distorted.
Consider a twodimensional flow in the xy plane. Also, consider an infinitesimal fluid element in this flow. Assume that at time t the shape of this fluid element is rectangular, as shown at the left of Figure 2.31. Assume that the fluid element is moving upward and to the right; its position and shape at time t + At are shown at the right in Figure 2.31. Note that during the time increment At, the sides AB and AC have rotated through the angular displacements — Дві and Аві, respectively. (Counterclockwise rotations by convention are considered positive; since line A В is shown with a clockwise rotation in Figure 2.31, the angular displacement is negative, — Дві.) At present, consider just the line AC. It has rotated because during the time increment At, point C has moved differently from point A. Consider the velocity in the у direction. At point A at time t, this velocity is v, as shown in Figure 2.31. Point C is a distance dx from point A; hence, at time t the vertical component of velocity of point C is given by v + (dv/dx) dx. Hence,
Distance in у direction that A moves
= vAt
during time increment At
Distance in у direction that C moves
This net displacement is shown at the right of Figure 2.31. From the geometry of Figure 2.31,
[2.119]
Figure 2.31 Rotation and distortion of a fluid element.


Since A92 is a small angle, tan A92 ~ A92. Hence, Equation (2.119) reduces to
Now consider line AB. The x component of the velocity at point A at time t is u, as shown in Figure 2.31. Because point В is a distance dy from point A, the horizontal component of velocity of point В at time t is и + (du/dy)dy. By reasoning similar to that above, the net displacement in the x direction of В relative to A over the time increment At is [(du/dy) dy] At, as shown in Figure 2.31. Hence,




Consider the angular velocities of lines AB and AC, defined as d9/dt and d92/dt, respectively. From Equation (2.122), we have


d9 j. A9 du dt At—>o At dy
From Equation (2.120), we have
d92 A92 dv
dt Ar>o At dx




By definition, the angular velocity of the fluid element as seen in the xy plane is the average of the angular velocities of lines AB and AC. Let a>z denote this angular velocity. Therefore, by definition,
Combining Equations (2.123) to (2.125) yields
1 /Эи Эи
z 2 Эх ду
In the above discussion, we have considered motion in the xy plane only. However, the fluid element is generally moving in threedimensional space, and its angular velocity is a vector ш that is oriented in some general direction, as shown in Figure 2.32. In Equation (2.126), we have obtained only the component of ш in the z direction; this explains the subscript z in Equations (2.125) and (2.126). The x and у components of <о can be obtained in a similar fashion. The resulting angular velocity of the fluid element in threedimensional space is
Equation (2.127) is the desired result; it expresses the angular velocity of the fluid element in terms of the velocity field, or more precisely, in terms of derivatives of the velocity field.
The angular velocity of a fluid element plays an important role in theoretical aerodynamics, as we shall soon see. However, the expression 2со appears frequently, and therefore we define a new quantity, vorticity, which is simply twice the angular velocity. Denote vorticity by the vector :
£ = 2<u
Hence, from Equation (2.127),
Recall Equation (2.22) for V x V in cartesian coordinates. Since u, v, and w denote the x, y, and z components of velocity, respectively, note that the right sides of Equations (2.22) and (2.128) are identical. Hence, we have the important result that
[2.129]
In a velocity field, the curl of the velocity is equal to the vorticity.
The above leads to two important definitions:
1. If V x V ф 0 at every point in a flow, the flow is called rotational. This implies that the fluid elements have a finite angular velocity.
2. If V x V = 0 at every point in a flow, the flow is called irrotational. This implies that the fluid elements have no angular velocity; rather, their motion through space is a pure translation.
The case of rotational flow is illustrated in Figure 2.33. Here, fluid elements moving along two different streamlines are shown in various modes of rotation. In contrast, the case of irrotational flow is illustrated in Figure 2.34. Here, the upper streamline shows a fluid element where the angular velocities of its sides are zero. The lower streamline shows a fluid element where the angular velocities of two intersecting sides are finite but equal and opposite to each other, and so their sum is identically zero. In both cases, the angular velocity of the fluid element is zero; i. e., the flow is irrotational.
If the flow is twodimensional (say, in the xy plane), then from Equation (2.128),
[2.130]
Figure 2.33 Fluid elements in a rotational flow.


Figure 2.34 Fluid elements in an irrotational flow.


Also, if the flow is irrotational, £ = 0. Hence, from Equation (2.130),
dv 3 и dx dy
Equation (2.131) is the condition of irrotationality for twodimensional flow. We will have frequent occasion to use Equation (2.131).
Why is it so important to make a distinction between rotational and irrotational flows? The answer becomes blatantly obvious as we progress in our study of aerodynamics; we find that irrotational flows are much easier to analyze than rotational flows. However, irrotational flow may at first glance appear to be so special that its applications are limited. Amazingly enough, such is not the case. There are a large number of practical aerodynamic problems where the flow field is essentially irrotational, e. g., the subsonic flow over airfoils, the supersonic flow over slender bodies at small angle of attack, and the subsonicsupersonic flow through nozzles. For such cases, there is generally a thin boundary layer of viscous flow immediately adjacent to the surface; in this viscous region the flow is highly rotational. However, outside this boundary layer, the flow is frequently irrotational. As a result, the study of irrotational flow is an important aspect of aerodynamics.
Return to the fluid element shown in Figure 2.31. Let the angle between sides A В and AC be denoted by к. As the fluid element moves through the flow field, к will change. In Figure 2.31, at time t, к is initially 90°. At time t + At, к has changed by the amount А к, where
А к = – Д02 – (Д0О
By definition, the strain of the fluid element as seen in the xy plane is the change in к, where positive strain corresponds to a decreasing к. Hence, from Equation (2.132),
Strain = —Лк = Л02 — A#i [2.133]
In viscous flows (to be discussed in Chapters 15 to 20), the time rate of strain is an important quantity. Denote the time rate of strain by єху, where in conjunction with
" 3 и

du

du

dx

dy

3z

dv

dv

dv

dx

dy

dz

dw

dw

dw

_ dx

dy

~dz

The sum of the diagonal terms is simply equal to V • V, which from Section 2.3 is equal to the time rate of change of volume of a fluid element; hence, the diagonal terms represent the dilatation of a fluid element. The offdiagonal terms are cross derivatives which appear in Equations (2.127), (2.128), and (2.135a to c). Hence, the offdiagonal terms are associated with rotation and strain of a fluid element.
In summary, in this section, we have examined the rotation and deformation of a fluid element moving in a flow field. The angular velocity of a fluid element and the corresponding vorticity at a point in the flow are concepts which are useful in the analysis of both inviscid and viscous flows; in particular, the absence of vorticity— irrotational flow—greatly simplifies the analysis of the flow, as we will see. We take advantage of this simplification in much of our treatment of inviscid flows in subsequent chapters. On the other hand, we do not make use of the time rate of strain until we discuss viscous flow, beginning with Chapter 15.

For the velocity field given in Example 2.3, calculate the vorticity. Solution





The flow field is irrotational at every point except at the origin, where x2 + y2 = 0.


Consider the boundarylayer velocity profile used in Example 2.2, namely, u/V, x = (y/S)025. Is this flow rotational or irrotational?
Solution
For a twodimensional flow, the irrotationality condition is given by Equation (2.131), namely




Does this relation hold for the viscous boundarylayer flow in Example 2.2? Let us examine this question. From the boundarylayer velocity profile given by






In Example 2.5, we demonstrated a basic result which holds in general for viscous flows, namely, viscous flows are rotational. This is almost intuitive. For example, consider an infinitesimally small fluid element moving along a streamline, as sketched in Figure 2.35. If this is a viscous flow, and assuming that the velocity increases in the upward direction (i. e., the velocity is higher on the neighboring streamline above and lower on the neighboring streamline below), then the shear stresses on the upper and lower faces of the fluid element will be in the directions shown. Such shear stresses will be discussed at length in Chapter 15. Examining Figure 2.35, we see clearly that the shear stresses exert a rotational moment about the center of the element, thus providing a mechanism for setting the fluid element into rotation. Although this picture is overly simplistic, it serves to emphasize that viscous flows are rotational flows. On the other hand, as stated earlier in this section, there are numerous
Figure 2.35 Shear stress and the consequent rotation of a fluid element.


inviscid flow problems that are irrotational, with the attendant simplifications to be explained later. Some inviscid flows are rotational, but there exists such a large number of practical aerodynamic problems described by inviscid, irrotational flows that the distinction between rotational and irrotational flow is an important consideration.
Before discussing the theoretical calculation of airfoil properties, let us examine some typical results. During the 1930s and 1940s, the NACA carried out numerous measurements of the lift, drag, and moment coefficients on the standard NACA airfoils. These experiments were performed at low speeds in a wind tunnel where the constantchord wing spanned the entire test section from one sidewall to the other. In this fashion, the flow “sees” a wing without wing tips—a socalled infinite wing,
which theoretically stretches to infinity along the span (in the у direction in Figure 4.1). Because the airfoil section is the same at any spanwise location along the infinite wing, the properties of the airfoil and the infinite wing are identical. Hence, airfoil data are frequently called infinite wing data. (In contrast, we see in Chapter 5 that the properties of a finite wing are somewhat different from its airfoil properties.)
The typical variation of lift coefficient with angle of attack for an airfoil is sketched in Figure 4.4. At lowtomoderate angles of attack, с/ varies linearly with a; the slope of this straight line is denoted by ao and is called the lift slope. In this region, the flow moves smoothly over the airfoil and is attached over most of the surface, as shown in the streamline picture at the left of Figure 4.4. However, as a becomes large, the flow tends to separate from the top surface of the airfoil, creating a large wake of relatively “dead air” behind the airfoil as shown at the right of Figure 4.4. Inside this separated region, the flow is recirculating, and part of the flow is actually moving in a direction opposite to the freestream—socalled reversed flow. (Refer also to Figure 1.36.) This separated flow is due to viscous effects and is discussed in Chapter 15. The consequence of this separated flow at high a is a precipitous decrease in lift and a large increase in drag; under such conditions the airfoil is said to be stalled. The maximum value of С/, which occurs just prior to the stall, is denoted by ci, max; it is one of the most important aspects of airfoil performance, because it determines the stalling speed of an airplane. The higher is c/>max, the lower is the stalling speed. A great deal of modem airfoil research has been directed toward increasing c/,max. Again examining Figure 4.4, we see that c; increases linearly with а until flow separation begins to have an effect. Then the curve becomes nonlinear, q reaches a maximum value, and finally the airfoil stalls. At the other extreme of the curve, noting Figure 4.4, the lift at a = 0 is finite; indeed, the lift goes to zero only when the airfoil is pitched to some negative angle of attack. The value of a when lift equals zero is called the zerolift angle of attack and is denoted by chl=o For a
Stall due to flow separation
Figure 4.4 Schematic of liftcoefficient variation with angle of attack for an airfoil.


symmetric airfoil, aL=0 — 0, whereas for all airfoils with positive camber (camber above the chord line), aL=0 is a negative value, usually on the order of —2 or —3°.
The inviscid flow airfoil theory discussed in this chapter allows us to predict the lift slope ao and aL=0 for a given airfoil. It does not allow us to calculate c/.max, which is a difficult viscous flow problem, to be discussed in Chapters 15 to 20.
Experimental results for lift and moment coefficients for the NACA 2412 airfoil are given in Figure 4.5. Here, the moment coefficient is taken about the quarterchord point. Recall from Section 1.6 that the forceandmoment system on an airfoil can be transferred to any convenient point; however, the quarterchord point is commonly used. (Refresh your mind on this concept by reviewing Section 1.6, especially Figure 1.19.) Also shown in Figure 4.5 are theoretical results to be discussed later. Note that the experimental data are given for two different Reynolds numbers. The lift slope ao is not influenced by Re; however, c/,max is dependent upon Re. This makes sense, because c/,max is governed by viscous effects, and Re is a similarity parameter that governs the strength of inertia forces relative to viscous forces in the flow. [See
Figure 4.5 Experimental data for lift coefficient and moment
coefficient about the quarterchord point for an NACA 2412 airfoil. (Source: Data obtained from Abbott and von Doenhoff, Reference 1 1.) Also shown is a comparison with theory described in Section 4.8.

Section 1.7 and Equation (1.35).] The moment coefficient is also insensitive to Re except at large a. The NACA 2412 airfoil is a commonly used airfoil, and the results given in Figure 4.5 are quite typical of airfoil characteristics. For example, note from Figure 4.5 that (Xl=о = —2.1°, c/ max ~ 1 6, and the stall occurs at a ~ 16°.
This chapter deals with airfoil theory for an inviscid, incompressible flow; such theory is incapable of predicting airfoil drag, as noted earlier. However, for the sake of completeness, experimental data for the drag coefficient cd for the NACA 2412 airfoil are given in Figure 4.6 as a function of the angle of attack.[14] The physical source of this drag coefficient is both skin friction drag and pressure drag due to flow separation (socalled form drag). The sum of these two effects yields the profile drag coefficient cd for the airfoil, which is plotted in Figure 4.6. Note that cd is sensitive to Re, which is to be expected since both skin friction and flow separation are viscous effects. Again, we must wait until Chapters 15 to 20 to obtain some tools for theoretically predicting cd.
