You are to conduct wind tunnel testing of a new football design that has a smaller lace height than previous designs (see the Videos a and b). It is known that you will need to maintain Re and St similarity for the testing. Based on standard college quarterbacks, the prototype parameters are set at V = 40 mph and ω = 300 rpm. The prototype football has a 7-in. diameter. Due to instrumentation required to measure pressure and shear stress on the surface of the football, the model will require a length scale of 2.1:1 (the model will be larger than the prototype). Determine the required (a) model freestream velocity and (b) model angular velocity.

Respuesta :

Answer:

[tex]{V_m}=26.66\ mph[/tex]

[tex]\omega_m=133.33\ rpm[/tex]

Explanation:

Given that

[tex]V_P=40\ mph[/tex]

We know that

1 mph = 0.44 m/s

[tex]V_P=17.88\ m/s[/tex]

[tex]\omega_p=300\ rpm[/tex]

[tex]D_P=7 \ in[/tex]

1 in  = 0.0254 m

[tex]D_P=0.1778 \ in[/tex]

[tex]\dfrac{d_m}{d_p}=1.5[/tex]

For Reynolds similarity

[tex]Re=\dfrac{Vd}{\nu }[/tex]

[tex]\left(\dfrac{Vd}{\nu }\right)_m=\left(\dfrac{Vd}{\nu }\right)_p[/tex]

ν is same for model and prototype .

[tex]\left({Vd}\right)_m=\left({Vd}\right)_p[/tex]

[tex]{V_m}=V_p\dfrac{d_p}{d_m}[/tex]

[tex]{V_m}=\dfrac{40}{1.5}[/tex]

[tex]{V_m}=26.66\ mph[/tex]

[tex]St=\dfrac{\omega d}{V}[/tex]

[tex]\left(\dfrac{\omega d}{V}\right)_m=\left(\dfrac{\omega d}{V}\right)_p[/tex]

[tex]\omega_m=\omega_p\dfrac{d_p}{d_m}\dfrac{V_m}{V_p}[/tex]

[tex]\omega_m=300\times \dfrac{1}{1.5}\times \dfrac{1}{1.5}[/tex]

[tex]\omega_m=133.33\ rpm[/tex]