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Derive the expression for stagnation density and stagnation temperature.
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Derivation

Expression for Stagnation Pressure (Ps)

PsP1=[(1+k12M21)KK1]..(i)

Expression for Stagnation Density (ρs)

For Adiabatic process

(ρ1ρs)=(p1ps)1Kor(ρsρ1)=(psp1)1k

(Taking reciprocal)

ρs=ρ1[psp1]1K

Substituting the value of (psp1) from equation (i),

ρs=ρ1[(1+K12M21)KK1]1Kρs=ρ1[1+K12M21]1K1..(ii)

Above equation gives value of Stagnation Density.

Expression for Stagnation Temperature Ts

as Psρs=RT

For the stagnation point, we have equation of state as Psρs=RTs

Ts=1RPsρs

Substituting the value of Ps and ρs from equations (ii), we have

Ts=1RP1[1+(K12)M21](KK1)ρ1[1+(K12)M21]1K1=1RP1ρ1[1+(K12)M21](KK1)(1K1)=1RP1ρ1[1+(K12)M21](K1K1)=P1ρ1R[1+(K12)M21]=T1[1+(K12)M21]

(P1ρ1=RT1)

Ts=T1[1+(K12)M21]

Above equation gives value of Stagnation Temperature.

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