Exercises Ch12

Question 1

Define the Mach angle and relate it to the local Mach number

Answer 1

sinμ=1/M​
It’s obtained by taking a supersonic disturbance (V>a) and by using geometry: sinμ=at/Vt=a/V=1/M

Question 2

What is a weak shock wave, both normal and oblique?

Answer 2

A weak shock (normal or oblique) is one in which the change across the shock is very small: the shock strength parameter F−1=du/u is small, the turning angle 𝜃 is small, and the flow can be treated approximately isentropically.

Question 3

In the following graph representing the turning angle versus the shock angle in an oblique shock at various Mach numbers, each of the lines shown hits the x axis at two different shock angles beta. Explain the physical meaning if these two intersections.

Answer 3

tan 𝜃 =(1-F)tanβ/1+Ftan^2β) shows that 𝜃 depends on β, and the turning angle becomes zero when the flow is not turned by the shock.
β can correspond to different physical branches:

In the weak-shock limit, β approaches the Mach angle. Larger β corresponds to a stronger (larger-F) shock.
Thus, the two intersections with θ = 0 represent two possible shock angles at which the flow experiences zero net turning:

A weak (small-β) solution — corresponds to a weak oblique shock close to the Mach angle.

A strong (large-β) solution — corresponds to a strong oblique shock with a large shock angle.

Both satisfy the condition of zero turning, but they represent distinct physical shock strengths.

Question 4

Give an example of a situation where it would be appropriate to describe the flow field based on the Prandtl-Meyer function ν(M).

Answer 4

An isentropic expansion of a supersonic flow, where the turning angle determines the change in Mach number — such as in a supersonic expansion fan.

Any situation in which the flow turns through an expansion (acceleration) and remains isentropic can be described using ν(M).

Question 5

In problems of supersonic aerodynamic involving small angles of attack and thin airfoils, what is the basis of so-called linearized theory implemented in dp/p=dθ γ M^2/(M^2-1)^1/2, where M is the fixed Mach number upstream of the airfoil? Why could this approximation be acceptable even at incident Mach numbers quite different from unity, such as M1=2?

Answer 5

• For thin airfoils and small angle of attack, “the changes in θ will be small.”
• Under these conditions, “M will not depart appreciably from its free-stream value.”
• Thus, in dp/p=dθ γ M^2/(M^2-1)^1/2, M can be treated as a constant.

The approximation relies on small turning angles, not on low Mach numbers, allowing linearized theory to remain valid even when the incoming Mach number is significantly greater than 1.
Treating M as const provides pressure at every point of the airfoil where theta is given from the airfoil geometry and then the lift is simply related ot the geometry!