EAS4101_S10_HW9S

# EAS4101_S10_HW9S - EAS 4101 Aerodynamics Spring 2010 HW 9 1...

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EAS 4101 – Aerodynamics – Spring 2010 – HW 9 1. wind tunnel at in which the stagnation conditions are at S.T.P. (as defined in Anderson). Assume that this magical wind tunnel permits the flow to be isentropically expanded to variable Mach numbers. Calculate and plot the static temperature, static pressure, static density, and expanded veloc 12 0 M ≤≤ . What assum re needed to derive these equations and when do they breakdown? Consider an air ity for ptions a Assumptions: Break down at T > 600K and T < 200K) Isentropic Thermally and Calorically Perfect ( 1 3.5 3.5 22 0 0 1 1 2.5 2.5 0 0 1 11 2 0 0 2 0 0 1 1 1 0.2 1 0.2 2 1 1 1 0.2 1 0.2 2 1 1 1 0.2 1 0.2 2 101325 / 1. p MM p p p T T T M T pN m γ ργ ρρ ρ ⎛⎞ ⎜⎟ −− ⎝⎠ ⎡⎤ 2 2 M M ⎤⎡ =+ →= + ⎦⎣ ⎢⎥ ⎣⎦ + + = = () ( ) 3 0 23 / 287 1.4 287 kg m TK V MV M a M R T M T a = =→= = = Page 1 of 1

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EAS 4101 – Aerodynamics – Spring 2010 – HW 9 Page 2 of 2
EAS 4101 – Aerodynamics – Spring 2010 – HW 9 : 1 2 0 1 1 2 p M p γ =+ , 1 1 2 0 1 1 2 M ρ 2. Prove the following stagnation relations . In addition to stagnation flow, what other assumptions are required? Assumptions: Stagnation Flow Isentropic Flow Thermally and Calorically Perfect Gas Define a Stagnation Enthalpy: 2 0 2 V hh For a perfect gas: Page 3 of 3

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EAS 4101 – Aerodynamics – Spring 2010 – HW 9 () 2 2 0 ; 1 1 22 p p V V TT C RC C 0 2 2 pp p p hC T C T C =→ =+ 2 2 0 11 p CR pv v C p vv p R R TC R T γ CC V T V T γγ =−→ = −→ =− = = Define , V aR T M a == 2 0 2 2 0 1 1 2 00 0 1 1 1 1 2 0 1 1 2 1 1 2 1 1 2 1 1 2 V T Ta T M T pT p M p T M T ρρ ⎛⎞ = + ⎜⎟ ⎝⎠ = + 2 1 3. Prove the following normal shock relations: 2 2 2 1 2 1 2 1 1 M M M + = , 2 1 2 2 1 2 21 2 M T M TM +− ⎡⎤ ⎢⎥ ++ ⎣⎦ , 2 2 1 1 2 1 P M P = + , and 2 1 2 2 1 M M ρ ργ + = . What assumptions are needed to derive this equation? ssumptions: Adiabatic Flow Steady Flow Inviscid Flow Neglect Body Forces One-Dimensional Flow Zero net work done on or by the system Thermally and Calorically Perfect Gas (Break Down for T<200K and T>600K) A Page 4 of 4
EAS 4101 – Aerodynamics – Spring 2010 – HW 9 Conservation of Mass: d ρ ∫∫∫ t () ˆ 0 VndA += ∫∫ i K A 11 1 2 2 2 2 1 ; VA VA A A ρρ == 11 2 2 VV = 12 2 2 1 1 1 1 2 2 a M a aa a M a =→ = Conservation of Momentum: Vd t ∀+ K 22 2 2 1 1 2 2 . sc o n s t = 1 2 1 2 2 2 1 2 2 21 1 2 2 ˆˆ 1 1 AA V V n dA pndA pp V V a p a a MM Ma γρ γ γγ ργρ ργ =− −= = ⎡⎤ + ÷ ⎢⎥ ⎣⎦ ⎛⎞ + ⎜⎟ ⎝⎠ + = + KK i Combining Conservation of Mass and Momentum: a 2 2 1 a 2 2 1 1 aM + = + 2 1 M M Conservation of Energy: Page 5 of 5

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EAS 4101 – Aerodynamics – Spring 2010 – HW 9 22 21 VV hhq += ++ w 11 1 12 2 2 2 1 2 1 ; 2 1 pp p pv hC T C T
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EAS4101_S10_HW9S - EAS 4101 Aerodynamics Spring 2010 HW 9 1...

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