Question 8.6: Assume that you are given the task of determining the aerody......
Assume that you are given the task of determining the aerodynamic forces and moments acting on a slender missile which flies at a Mach number of 3.5 at an altitude of 27,432 \mathrm{~m}(90,000~ \mathrm{ft}). Aerodynamic coefficients for the missile, which is 20.0 \mathrm{~cm} ( 7.874~ \mathrm{in} ) in diameter and 10 diameters long, are required for angles of attack from 0^{\circ} to 55^{\circ}. The decision is made to obtain experimental values of the required coefficients in the Vought High-Speed Wind Tunnel (in Dallas, Texas). Upstream of the model shock system, the flow in the wind tunnel is isentropic and air, at these conditions, behaves as a perfect gas. Thus, the relations developed in this section can be used to calculate the wind-tunnel test conditions.
1. Flight conditions. Using the properties for a standard atmosphere (such as were presented in Chapter 1), the relevant parameters for the flight condition include
\begin{aligned} U_{\infty} & =1050 \mathrm{~m} / \mathrm{s} \\ p_{\infty} & =1.7379 \times 10^{-2} p_{\mathrm{SL}}=1760.9 \mathrm{~N} / \mathrm{m}^{2} \\ T_{\infty} & =224 \mathrm{~K} \\ \operatorname{Re}_{\infty, d} & =\frac{\rho_{\infty} U_{\infty} d}{\mu_{\infty}}=3.936 \times 10^{5} \\ M_{\infty} & =3.5 \end{aligned}
2. Wind-tunnel conditions. Information about the operational characteristics of the Vought High-Speed Wind Tunnel is contained in the tunnel hand-book [Arnold (1968)]. To ensure that the model is not so large that its presence alters the flow in the tunnel (i.e., the model dimensions are within the allowable blockage area), the diameter of the wind-tunnel model, d_{\mathrm{wt}}, will be 4.183 \mathrm{~cm} (1.6468 in.).
Based on the discussion in Chapter 2, the Mach number and the Reynolds number are two parameters which we should try to simulate in the wind tunnel. The free-stream unit Reynolds number \left(U_{\infty} / \nu_{\infty}\right) is presented as a function of the free-stream Mach number and of the stagnation pressure for a stagnation temperature of 311 \mathrm{~K}\left(100^{\circ} \mathrm{F}\right) in Fig. 8.20 (which has been taken from Arnold (1968) and has been left in English units). The student can use the equations of this section to verify the value for the unit Reynolds number given the conditions in the stagnation chamber and the free-stream Mach number. In order to match the flight Reynolds number of 3.936 \times 10^{5} in the wind tunnel,
\left(\frac{U_{\infty}}{\nu_{\infty}}\right)_{\mathrm{wt}}=\frac{3.936 \times 10^{5}}{d_{\mathrm{wt}}}=2.868 \times 10^{6} / \mathrm{ft}
But as indicated in Fig. 8.20, the lowest unit Reynolds number possible in this tunnel at M_{\infty}=3.5 is approximately 9.0 \times 10^{6} / \mathrm{ft}. Thus, if the model is 4.183 \mathrm{~cm} in diameter, the lowest possible tunnel value of \operatorname{Re}_{\infty, d} is 1.235 \times 10^{6}, which is greater than the flight value. This is much different than the typical subsonic flow, where (as discussed in Chapter 5) the maximum wind-tunnel Reynolds number is usually much less than the flight value. To obtain the appropriate Reynolds number for the current supersonic flow, we can choose to use a smaller model. Using a model that is 1.333 \mathrm{~cm} in diameter would yield a Reynolds number of 3.936 \times 10^{5} (as desired). If this model is too small, we cannot establish a tunnel condition which matches both the flight Mach number and the Reynolds number. In this case, the authors would choose to simulate the Mach number exactly rather than seek a compromise Mach-number/Reynolds-number test condition, since the pressure coefficients and the shock interaction phenomena on the control surfaces are Mach-number-dependent in this range of free-stream Mach number.
The conditions in the stagnation chamber of the tunnel are T_{t}=311 \mathrm{~K} \left(560^{\circ} \mathrm{R}\right) and p_{t 1}=5.516 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}(80 ~\mathrm{psia}). Thus, using either equations (8.34) and (8.36)
\frac{T_t}{T}=1+\frac{U^2}{2[\gamma R /(\gamma-1)] T}=1+\frac{\gamma-1}{2} M^2 (8.34)
\frac{p_{t 1}}{p}=\left(1+\frac{\gamma-1}{2} M^2\right)^{\gamma /(\gamma-1)} (8.36)
or the values presented in Table 8.1, one finds that T_{\infty}=90.18 \mathrm{~K}\left(162.32^{\circ} \mathrm{R}\right) and p_{\infty}=7.231 \times 10^{3} \mathrm{~N} / \mathrm{m}^{2}(1.049 ~\mathrm{psia}). The cold free-stream temperature is typical of supersonic tunnels (e.g., most highspeed wind tunnels operate at temperatures near liquefaction of oxygen). Thus, the free-stream speed of sound is relatively low and, even though the free-stream Mach number is 3.5 , the velocity in the test section U_{\infty} is only 665 \mathrm{m} / \mathrm{s}(2185 \mathrm{ft} / \mathrm{s}). In summary, the relevant parameters for the wind-tunnel conditions include
\begin{aligned} U_{\infty} & =665 \mathrm{~m} / \mathrm{s} \\ p_{\infty} & =7.231 \times 10^{3} \mathrm{~N} / \mathrm{m}^{2} \\ T_{\infty} & =90.18 \mathrm{~K} \end{aligned}
\begin{aligned} \operatorname{Re}_{\infty, d}=3.936 \times 10^5 \text { if } d & =1.333 \mathrm{~cm} \text { or } 1.235 \times 10^6 \text { if } d=4.183 \mathrm{~cm} \\ M_{\infty} & =3.5 \end{aligned}
Because of the significant differences in the dimensional values of the flow parameters (such as U_{\infty}, p_{\infty}, and T_{\infty} ), we must again nondimensionalize the parameters so that correlations of wind-tunnel measurements can be related to the theoretical solutions or to the design flight conditions.
