Hypersonic Aerodynamics

According to Newtonian impact theory, which fails to explain the classical lift generation, the pressure exerted by the air particles impinging on a surface is equal to the time rate of change of momentum vertical to the wall. Using this principle we can find the pressure exerted by the air particles on the wall which is inclined with free stream with angle 0w. Since the velocity, as shown in Fig. 1.8, normal to the wall is Un the time rate of change of momentum becomes p = q Ц).

If we write Un = U sin 0w, the surface pressure coefficient reads as

Fig. 1.8 Velocity compo­nents for the impact theory

The free stream Mach number M is always high for hypersonic flows. There­fore, its square M2 ^ 1 is always true. If the wall inclination under consideration is sufficiently large i. e. hw is greater than 35o-40o, the second term in Eq. 1.26 becomes negligible compared to the first term. This allows us to obtain a simple expression for the surface pressure at hypersonic speeds as follows

Cp ffi 2 sin2 0W (1-27)

Now, we can find the lift and the drag force coefficients for hypersonic aero­dynamics based on the impact theory. According to Fig. 1.8 the sectional lift coefficient reads as

cl = 2 sin2 hw cos 0w, (1-28)

and the sectional drag coefficient becomes

cd = 2 sin3 hw (1-29)

Starting with Newton until the beginning of twentieth century, the lifting force was unsuccessfully explained by the impact theory. Because of sin2 term in Eq. 1.28 there was never sufficient lift force to be generated in small angles of attack. For this reason, even though Eq. 1.28 has been known since Newton’s time, it is only valid at hypersonic speeds and at high angles of attack. The drag coef­ficient expressed with Eq. 1.29, gives reasonable values at high angles of attack but gives small values at low angles of attack. We have to keep in mind that these formulae are obtained with perfect gas assumption.

The real gas effects at upper levels of atmosphere at hypersonic speeds play a very important role in physics of the external flows. At high speeds, the heat generated because of high skin friction excites the nitrogen and oxygen molecules of air to release their vibrational energy which increases the internal energy. This internal energy increase makes the air no longer a calorically perfect gas and therefore, the specific heat ratio of the air becomes a function of temperature. At higher speeds, when the temperature of air rises to the level of disassociation of nitrogen and oxygen molecules into their atoms, new species become present in the mixture of air. Even at higher speeds and temperatures, the nitrogen and oxygen atoms react with the other species to create new species in the air. Another real gas effect is the diffusion of species into each other. The rate of this diffusion becomes the measure of the catalyticity of the wall. At the catalytic walls, since the chemical reactions take place with infinite speeds the chemical equilibrium is established immediately. Because of this reason, the heat transfer at the catalytic walls is much higher compared to that of non-catalytic walls.

For a hypersonically cruising aerospace vehicle, there exists a heating prob­lem if it is slender, and low lift/drag ratio problem if it has a blunt body. The solution to this dilemma lies in the concept of ‘wave rider’. The wave rider
concept is based on a delta shaped wing which is immersed in a weak conic shock of relevant to the cruising Mach number. Necessary details will be given in following chapters.