Question: At subsonic speeds, how do the lift coefficient Cl and drag coefficient Co for a wing vary with angle of attack al
Answer: As shown in Chapter 5, we know that:
1. The lift coefficient varies linearly with angle of attack, at least up to the stall; see, for example, Figure 5.22.
2. The drag coefficient is given by the drag polar, as expressed in Equation (5.63), repeated below:
Since Cl is proportional to a, then Ci> varies as the square of a.
Question: At supersonic speeds, how do Cl and Сд for a wing vary with a? Answer: In Chapter 12, we demonstrated for an airfoil at supersonic speeds that:
1. Lift coefficient varies linearly with a, as seen from Equation (12.23), repeated
2. Drag coefficient varies as the square of a, as seen from Equation (12.24) for the flat plate, repeated below:
The characteristics of a finite wing at supersonic speeds follow essentially the same functional variation with the angle of attack, namely, Cl is proportional to a and CD is proportional to a2.
Question: At hypersonic speeds, how do Cl and CD for a wing vary with al We have shown that Cl is proportional to a for both subsonic and supersonic speeds— does the same proportionality hold for hypersonic speeds? We have shown that Cq is proportional to a2 for both subsonic and supersonic speeds—does the same proportionality hold for hypersonic speeds? The purpose of the present section is to address these questions.
In an approximate fashion, the lift and drag characteristics of a wing in hypersonic flow can be modeled by a flat plate at an angle of attack, as sketched in Figure 14.10. The exact flow field over the flat plate involves a series of expansion and shock waves as shown in Figure 14.10; the exact lift- and wave-drag coefficients can be obtained from the shock-expansion method as described in Section 9.7. However, for hypersonic speeds, the lift – and wave-drag coefficients can be further approximated by the use of newtonian theory, as described in this equation.
Consider Figure 14.11. Here, a two-dimensional flat plate with chord length c is at an angle of attack a to the freestream. Since we are not including friction, and because surface pressure always acts normal to the surface, the resultant aerodynamic force is perpendicular to the plate; that is, in this case, the normal force N is the resultant aerodynamic force. (For an infinitely thin flat plate, this is a general result which is not limited to newtonian theory, or even to hypersonic flow.) In turn, N is resolved into
lift and drag, denoted by L and D, respectively, as shown in Figure 14.11. According to newtonian theory, the pressure coefficient on the lower surface is
Cpj = 2 sin2 a [ 14.8]
The upper surface of the flat plate shown in Figure 14.11, in the spirit of newtonian theory, receives no direct “impact” of the freestream particles; the upper surface is said to be in the “shadow” of the flow. Hence, consistent with the basic model of newtonian flow, only freestream pressure acts on the upper surface, and we have
Returning to the discussion of aerodynamic force coefficients in Section 1.5, we note that the normal force coefficient is given by Equation (1.15). Neglecting friction, this becomes
cn = – (CPti-CPtll)dx [14.10]
where x is the distance along the chord from the leading edge. (Please note: In this section, we treat a flat plate as an airfoil section; hence, we will use lowercase letters to denote the force coefficients, as first described in Chapter 1.) Substituting Equations (14.8) and (14.9) into (14.10), we obtain
cn = “(2 sin a)c c
= 2 sin2 a
From the geometry of Figure 14.11, we see that the lift and drag coefficients, defined as ci = L/q^S and q = D/q^S, respectively, where S = (c)(1), are given by
Substituting Equation (14.11) into Equations (14.12) and (14.13), we obtain
ci — 2 sin2 a cos a [14.14]
Cd — 2 sin3 a [14.15]
Finally, from the geometry of Figure 14.11, the lift-to-drag ratio is given by
— = cot a [14.16]
[Note: Equation (14.16) is a general result for inviscid supersonic or hypersonic flow over a flat plate. For such flows, the resultant aerodynamic force is the normal force N. From the geometry shown in Figure 14.11, the resultant aerodynamic force makes the angle a with respect to lift, and clearly, from the right triangle between L, D, and N, we have L/D = cot a. Hence, Equation (14.16) is not limited to newtonian theory.]
The aerodynamic characteristics of a flat plate based on newtonian theory are shown in Figure 14.12. Although an infinitely thin flat plate, by itself, is not a practical aerodynamic configuration, its aerodynamic behavior at hypersonic speeds is consistent with some of the basic characteristics of other hypersonic shapes. For example, consider the variation of a shown in Figure 14.12. First, note that, at a small angle of attack, say, in the range of a from 0 to 15°, с/ varies in a nonlinear fashion; that is, the slope of the lift curve is not constant. This is in direct contrast to the subsonic case we studied in Chapters 4 and 5, where the lift coefficient for an airfoil or a finite wing was shown to vary linearly with a at small angles of attack, up to the stalling angle. This is also in contrast with the results from linearized supersonic theory as itemized in Section 12.3, leading to Equation (12.23) where a linear variation of q with a for a flat plate is indicated. However, the nonlinear lift curve shown in Figure 14.12 is totally consistent with the results discussed in Section 11.3, where hypersonic flow was shown to be governed by the nonlinear velocity potential equation, not by the linear equation expressed by Equation (11.18). In that section, we noted that both transonic and hypersonic flow cannot be described by a linear theory—both these flows are inherently nonlinear regimes, even for low angles of attack. Once again, the flat-plate lift curve shown in Figure 14.12 certainly demonstrates the nonlinearity of hypersonic flow.
Also, note from the lift curve in Figure 14.12 that q first increases as a increases, reaches a maximum value at an angle of attack of about 55° (54.7° to be exact), and then decreases, reaching zero at a = 90°. However, the attainment of q max (point A) in Figure 14.12 is not due to any viscous, separated flow phenomenon analogous to that which occurs in subsonic flow. Rather, in Figure 14.12, the attainment of a maximum ci is purely a geometric effect. To understand this better, return to Figure 14.11. Note that, as a increases, Cp continues to increase via the newtonian expression
Cp = 2 sin2 a
That is, Cp reaches a maximum value at a = 90°. In turn, the normal force N shown in Figure 14.11 continues to increase as a increases, also reaching a maximum value at a = 90°. However, recall from Equation (14.12) that the vertical component of
the aerodynamic force, namely, the lift, is given by
L = N cos a [14.17]
Hence, as a increases to 90°, although N continues to increase monotonically, the value of L reaches a maximum value around a = 55°, and then begins to decrease at higher a due to the effect of the cosine variation shown in Equation (14.17)— strictly a geometric effect. In other words, in Figure 14.11, although N is increasing with a, it eventually becomes inclined enough relative to the vertical that its vertical component (lift) begins to decrease gradually. It is interesting to note that a large number of practical hypersonic configurations achieve a maximum CL at an angle of attack in the neighborhood of that shown in Figure 14.12, namely, around 55°.
The maximum lift coefficient for a hypersonic flat plate, and the angle at which it occurs, is easily quantified using newtonian theory. Differentiating Equation (14.14) with respect to a, and setting the derivative equal to zero (for the condition of maximum ci), we have
This is the angle of attack at which q is a maximum. The maximum value of q is obtained by substituting the above result for a into Equation (14.14):
Q. max = 2sin2(54.7°) cos(54.7°) = 0.77
Note, although С/ increases over a wide latitude in the angle of attack (q increases in the range from a = 0 to a = 54.7°), its rate of increase is small (that is, the effective lift slope is small). In turn, the resulting value for the maximum lift coefficient is relatively small—at least in comparison to the much higher q. max values associated with low-speed flows (see Figures 4.20 and 4.22). Returning to Figure 14.12, we now note the precise values associated with the peak of the lift curve (point Л), namely, the peak value of q is 0.77, and it occurs at an angle of attack of 54.7°.
Examining the variation of drag coefficient q/ in Figure 14.12, we note that it monotonically increases from zero at a = 0 to a maximum of 2 at a = 90°. The newtonian result for drag is essentially wave drag at hypersonic speeds because we are dealing with an inviscid flow, hence no friction drag. The variation of cj with a for the low angle of attack in Figure 14.12 is essentially a cubic variation, in contrast to the result from linearized supersonic flow, namely, Equation (12.24), which shows that c, i varies as the square angle of attack. The hypersonic result that q varies as a3 is easily obtained from Equation (14.15), which for small a becomes
The variation of the lift-to-drag ratio as predicted by newtonian theory is also shown in Figure 14.12. The solid curve is the pure newtonian result; it shows that L/D is infinitely large at a = 0 and monotonically decreases to zero at a = 90°. The infinite value of L/ D at a = 0 is purely fictional—it is due to the neglect of skin friction. When skin friction is added to the picture, denoted by the dashed curve in Figure 14.12, L/D reaches a maximum value at a small angle of attack (point В in Figure 14.12) and is equal to zero at a = 0. (At a — 0, no lift is produced, but there is a finite drag due to friction; hence, L/D = 0 at a — 0.)
Let us examine the conditions associated with (L/D)nm more closely. The value of (L/D)mM and the angle of attack at which it occurs (i. e., the coordinates of point В in Figure 14.12) are strictly a function of the zero-lift drag coefficient, denoted by Q o – The zero-lift drag coefficient is simply due to the integrated effect of skin friction over the plate surface at zero angle of attack. At small angles of attack, the skin friction exerted on the plate should be essentially that at zero angle of attack; hence, we can write the total drag coefficient [referring to Eq. (14.15)] as
Furthermore, when a is small, we can write Equations (14.14) and (14.19) as
Q = 2a2 [14.20]
and cd — 2a3 + Cd, o 114.21 ]
Dividing Equation (14.20) by (14.21), we have
Cd 2 a3 + Cd, о
The conditions associated with maximum lift-to-drag ratio can be found by differentiating Equation (14.22) and setting the result equal to zero:
d(ci/cd) _ (:2a3 + Cd, o)4a – 2q;2(6q!2) _ da (2a3 + cd, o)
or 8a4 + 4ac^o — 12a4 = 0
Substituting Equation (14.23) into Equation (14.21), we obtain
Q = 2(c^0)2/3 = 2/3
Q/max 2cd, o + cd, o (Q, o)1/3
Equations (14.23) and (14.24) are important results. They clearly state that the coordinates of the maximum L/D point in Figure 14.12, when friction is included (point В in Figure 14.12), are strictly a function of cd, o- In particular, note the expected trend that (L/D)max decreases as cd, о increases—the higher the friction drag, the lower is L/D. Also, the angle of attack at which maximum L/D occurs increases as Cd, o increases. There is yet another interesting aerodynamic condition that holds at (L/D)max, derived as follows. Substituting Equation (14.23) into (14.21), we have
Cd — 2с^о + Cd, о = 3q о
Since the total drag coefficient is the sum of the wave-drag coefficient cd, w and the friction drag coefficient cd, o we can write
Cd ~cd, w+ cd, о [14.26]
However, at the point of maximum L/D (point В in Figure 14.12), we know from Equation (14.25) that cd = 3q. o. Substituting this result into Equation (14.26), we obtain
3Cd,0 — Cd, w З – са, о
This clearly shows that, for the hypersonic flat plate using newtonian theory, at the flight condition associated with maximum lift-to-drag ratio, wave drag is twice the friction drag.
This brings to an end our short discussion of the lift and drag of wings at hypersonic speeds as modeled by the newtonian flat-plate problem. The quantitative and qualitative results presented here are reasonable representations of the hypersonic aerodynamic characteristics of a number of practical hypersonic vehicles; the flat – plate problem is simply a straightforward way of demonstrating these characteristics.