A chimney 100 f t high is being forced to vibrate at a frequency f in a wind of 20 f t/s by vortices that are shed in its wake. This phenomenon depends on the fluid density ρ and viscosity µ and the chimney material modulus of elasticity E, where E = stress/strain. A model is constructed which is geometrically similar to the chimney in every way and is 10 ft high. The mass per unit length, m, of chimney model can be adjusted by attaching dummy masses inside without affecting its elastic behavior. Gravity is not involved in this problem. (a) Using dimensional analysis derive all relevant nondimensional groups. Try to use physically meaningful nondimensional groups wherever possible. Make sure these groups are independent. (b) If the full-scale chimney is made from steel with a modulus of elasticity E = 30 × 106 lbf /in2 , find the necessary modulus of elasticity so as to simulate the correct conditions at the model scale. (c) At what frequency would you expect the full scale chimney to vibrate if the model vibrates at 5 Hz?

Question

A chimney 100 f t high is being forced to vibrate at a frequency f in a wind of 20 f t/s by vortices that are shed in its wake. This phenomenon depends on the fluid density ρ and viscosity µ and the chimney material modulus of elasticity E, where E = stress/strain. A model is constructed which is geometrically similar to the chimney in every way and is 10 ft high. The mass per unit length, m, of chimney model can be adjusted by attaching dummy masses inside without affecting its elastic behavior. Gravity is not involved in this problem.

(a) Using dimensional analysis derive all relevant nondimensional groups. Try to use physically meaningful nondimensional groups wherever possible. Make sure these groups are independent.

(b) If the full-scale chimney is made from steel with a modulus of elasticity E = 30 × 10⁶ lbf /in² , find the necessary modulus of elasticity so as to simulate the correct conditions at the model scale.

(c) At what frequency would you expect the full scale chimney to vibrate if the model vibrates at 5 Hz?