Question

How many characteristics pass through a point in the flow field if the Mach number is greater than 1?

(a) 1

(b) 2

(c) 4

(d) 0

This question was posed to me in homework.

I would like to ask this question from Two Dimensional Irrotational Flow in section Numerical Techniques for Steady Supersonic Flow of Aerodynamics

Answer 1

Correct option is (b) 2

The best explanation: The characteristic curve is given by the relation

\(\frac {dy}{dx_{char}} = \frac {- \frac {uv}{a^{2}} ± \sqrt {(\frac {u^2 + v^{2}}{a^{2}})-1}}{1 – \frac {u^{2}}{a^{2}}}\)

The term inside the square root \(\frac {u^2 + v^{2}}{a^{2}}\) – 1 can be written in the form of Mach number as follows:

\(\frac {u^2 + v^{2}}{a^{2}}\) – 1 = \(\frac {V^{2}}{a^{2}}\) – 1 = M^2 – 1

When the flow is supersonic i.e. Mach number is greater than 1, two characteristics pass through a point in a flow field. Hyperbolic partial differential equation is used to define it.