Understanding the scaling exponents in transitional pipe flows

The subcritical nature of the transition to turbulence in pipe flows is manifested as a scaling of the minimal triggering amplitude with Reynolds number. Previous studies have shown the existence of a power law relating the minimal amplitude A to Re^β. The exponent β is reported to be in around a range between -1 to -1.5. However, an investigation into the details of the initial conditions that can give a specific exponent is still not clear. Through a series of direct numerical simulations, we implement different initial conditions of finite width to investigate the variation of this scaling exponent.

Our study found that the exponents are related to the spatial distribution of the initial disturbance. Specifically, the scaling exponent is influenced by the radial distribution of the initial disturbance. A higher (in magnitude) exponent is mainly observed when the wall region is disturbed with a finite-width disturbance, while a lower exponent is mainly observed when disturbed elsewhere. This hints at the existence of two different routes to turbulence in pipe flows. We compare our results with the existing experimental works and suggest methods to implement disturbances to obtain a specific scaling exponent. This will be discussed in detail, and the physical mechanism will be elucidated through the identification of flow structures evolving during the transition process.