Question

If we know the value of θ1,ν1 at point 1 and θ2,ν2 in a flow, then what is the flow field condition at an internal point 3 lying at the intersection of characteristic lines passing from points 1 and 2?

(a) θ3 = \(\frac {(K_- )_1 + (K_+ )_2}{2}\)

(b) θ3 = \(\frac {(K_- )_1 + (K_+ )_3}{2}\)

(c) θ3 = \(\frac {(K_+ )_1 + (K_+ )_2 + (K_+ )_3}{2}\)

(d) θ3 = \(\frac {(K_- )_1 + (K_- )_2 + (K_- )_3}{2}\)

I got this question by my school teacher while I was bunking the class.

Enquiry is from Determination of Compatibility Equations topic in chapter Numerical Techniques for Steady Supersonic Flow of Aerodynamics

Answer 1

Right answer is (a) θ3 = \(\frac {(K_- )_1 + (K_+ )_2}{2}\)

Easiest explanation: The value of K+ is constant along left running Mach wave and K– is constant along right running Mach wave. Point 3 lies at the intersection of characteristic lines passing through point 1 and 2 thus, considering the characteristic curve passing through point 1,

(K–)1 = (K–)3

θ1 + ν1 = θ3 + ν3

θ3 + ν3 = (K–)1 (equation 1)

Similarly along the characteric curve through point 2,

(K+)2 = (K+)3

θ2 – ν2 = θ3 – ν3

θ3 – ν3 = (K+)2 (equation 2)

Solving equation 1 and 2 we get

θ3 = \(\frac {(K_- )_1 + (K_+ )_2}{2}\)