Accuracy Considerations

How accurate is newtonian theory in the prediction of pressure distributions over hy­personic bodies? The comparison shown in Figure 14.9 indicates that Equation (14.7) leads to a reasonably accurate pressure distribution over the surface of a blunt body. Indeed, for “back-of-the-envelope” estimates of the pressure distributions over blunt bodies at hypersonic speeds, modified newtonian is quite satisfactory. However, what about relatively thin bodies at small angles of attack? We can provide an answer by using the newtonian flat-plate relations derived in the present section, and compare these results with exact shock-expansion theory (Section 9.7), for flat plates at small angles of attack. This is the purpose of the following worked example.

Consider an infinitely thin flat plate at an angle of attack of 15° in a Mach 8 flow. Calculate | Example 1 4.1 the pressure coefficients on the top and bottom surface, the lift and drag coefficients, and the lift-to-drag ratio using (a) exact shock-expansion theory, and (b) newtonian theory. Compare the results.

Solution

(a) Using the diagram in Figure 9.26 showing a flat plate at angle of attack, and following the shock-expansion technique given in Example 9.8, we have for the upper surface, for A?, =8 and V{ = 95.62°,

v2 = V, +0 = 95.62 + 15 = 110.62° From Appendix C, interpolating between entries,

M2 = 14.32

From Appendix A, for M = 8, poJp = 0.9763 x 104, and for M2 = 14.32, po-,/p2 = 0.4808 x 106. Since p0l = /+,,

The pressure coefficient is given by Equation (11.22), and the freestream static pressure in Figure 9.26 is denoted by p. Hence

To obtain the pressure coefficient on the bottom surface from the oblique shock theory, we have from the в-fi-M for Mt = 8 and в = 15°, /і = 21°:

Af„,i = M sin p = 8 sin 21° = 2.87

Interpolating from Appendix B, for Mn = 2.87, p3/pt = 9.443. Hence the pressure coeffi­cient on the bottom surface is

The lift coefficient can be obtained from the pressure coefficients via Equations (1.15), (1.16), and (1.18).

c„ = – f (СрЛ – Єр,,,) dx = Cm – CP2 = 0.1885 – (-0.0219) = 0.2104

c Jo

The axial force on the plate is zero, because the pressure acts only perpendicular to the plate. On a formal basis, dy/dx in Equation (1.16) is zero for a flat plate. Hence, from Equation (1.18),

C„, = 2 sin2 a = 2 sin2 15° =

From Equation (14.9), we have for the upper surface

r —

^ pi

Discussion. From the above worked example, we see that newtonian theory underpredicts the pressure coefficient on the bottom surface by 29 percent, and of course predicts a value of zero for the pressure coefficient on the upper surface in comparison to —0.0219 from exact theory—an error of 100 percent. Also, newtonian theory underpredicts q and c, j by 36.6 percent. However, the value of L/D from newtonian theory is exactly correct. This is no surprise, for two reasons. First, the

newtonian values of ct and q are both underpredicted by the same amount, hence their ratio is not affected. Second, the value of L/D for supersonic or hypersonic inviscid flow over a flat plate, no matter what theory is used to obtain the pressures on the top and bottom surfaces, is simply a matter of geometry. Because the pressure acts normal to the surface, the resultant aerodynamic force is perpendicular to the plate (i. e., the resultant force is the normal force N). Examining Figure 1.10, when this is the case, the vectors R and N are the same vectors, and L/D is geometrically given by

L

— = cot a

D

For the above worked example, where a = 15°, we have

L

— = cot 15° = 3.73 D

which agrees with the above calculations where q and c, i were first obtained, and L/D is found from the ratio, L/D = cy/cj. So, Equation (14.16), derived in our discussion of newtonian theory applied to a flat plate, is not unique to newtonian theory; it is a general result when the resultant aerodynamic force is perpendicular to the plate.

We induce from Example 14.1 the general fact that the newtonian sine-squared law, Equation (14.4), does not accurately predict the hypersonic pressure distribution on the surface of two-dimensional bodies with local tangent lines that are at small or moderate angles to the flow, such as the bi-convex airfoil shape shown in Figure 12.3. On the other hand, it generally turns out that the newtonian prediction of the lift-to-drag ratio for slender shapes at small to moderate angles of attack is reasonably accurate. These statements apply to a gas with the ratio of specific heats substantially greater than one, such as the case of air with у = 1.4 treated in Example 14.1. In the next section, we will see that newtonian theory becomes more accurate as Mx —» со and у —>■ 1. For more information on the accuracy of newtonian theory applied to two-dimensional slender shapes, see Reference 77 which is a study of this specific matter.

Finally, we note that newtonian theory does a better job of predicting the pressure on axisymmetric slender bodies, such as the 15° half-angle cone shown in Figure 14.13.