A 1:10 scale model of an airplane propeller is to be tested in a laboratory using a wind tunnel. The prototype on which we are to estimate the thrust moves at V = 200 km/hr at 8000 m altitude. Assuming the prototype and model have geometric similarity, use the result from Problem 6.28

Question

A 1:10 scale model of an airplane propeller is to be tested in a laboratory using a wind tunnel. The prototype on which we are to estimate the thrust moves at V = 200 km/hr at 8000 m altitude. Assuming the prototype and model have geometric similarity, use the result from Problem 6.28 to achieve dynamic similarity. (a) What air speed should be used in the wind tunnel using standard air? (b) What is the corresponding thrust for the prototype, if a thrust of 10 N is measured on the model?

Accepted Answer

Properties of the atmosphere at 8 km altitude: T = −37ºC +273 = 236 K and pressure = 35.7 kPa, density = 0.526 kg/[latex]m^3[/latex], and viscosity = 1.527 × [latex]10^{−5} N·s/m^2[/latex].

Properties of air at standard atmosphere: T = 20ºC, p = 101.325 kPa, density = 1.204 kg/[latex]m^3[/latex], dynamic viscosity = 1.82 × [latex]10^{−5} N·s/m^2[/latex].

Use the Reynolds number to achieve dynamic similarity, [latex]\left\lgroup\frac{ ho VD}{\mu } ight group _m =\left\lgroup\frac{ ho VD}{\mu } ight group _p[/latex]

Then [latex]V_m =V_p \left\lgroup\frac{ ho D}{\mu } ight group _p \left\lgroup\frac{\mu }{ ho D } ight group _m[/latex]

and [latex]V_m =V_p \left\lgroup\frac{ ho _p \mu _m}{ ho _m \mu _p} ight group \left\lgroup\frac{D_p }{ D_m } ight group =200\left\lgroup\frac{0.526}{1.204} ight group \left\lgroup\frac{1.82 imes 10^{-5}}{1.527 imes 10^{-5}} ight group \left\lgroup\frac{10}{1} ight group =1041 Km/hr[/latex]

To calculate the thrust apply: [latex] \left\lgroup\frac{T}{ ho V^2D^2} ight group _p=\left\lgroup\frac{T}{ ho V^2D^2} ight group _m[/latex]

Then [latex]T_p=T_m \left\lgroup \frac{ ho _pV ^2_pD^2_p}{ ho _mV ^2_mD _m^2} ight group =10 \left\lgroup\frac{0.526 imes 1041^2 imes 10^2}{1.204 imes 200^2 imes 1^2} ight group =\underline{1184 N}[/latex]

Question 6.50: A 1:10 scale model of an airplane propeller is to be tested ...

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