Encounters With The Golden Ratio In Fluid Dynamics
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This paper suggests that the golden ratio, prominent in nature and art, has also its presence in fluid dynamics. The first example draws from the investigation of the resonance in wind tunnels with ventilated walls. Using acoustic wave theory, the reciprocal golden ratio is shown to determine the critical Mach number below which refraction is possible and above which total reflection takes place. The second example concerns the vortex merger, such as observed in aircraft turbulence and large-scale atmospheric or oceanic flows. Based on a numerical simulation and available experimental data, a conjecture is made that the distance below which two identical Rankine vortices merge and above which they do not is the product of the vortex diameter and golden ratio. Keywords: golden ratio, wind tunnel resonance, vortex merger. 1 Introduction The ratio b a = Φ , 0 > > b a is termed \“golden” if a b a b a + = , implying that the ratio between the greater part, a , and the smaller part, b , is equal to the ratio between the whole, b a + , and the greater part, a . Combining the above equations yields 1 1 − Φ + = Φ , (1)
golden ratio, wind tunnel resonance, vortex merger.