## Abstract

Transition to turbulence in axially symmetrical laminar pipe flows with periodic time dependence classified as pure oscillating and pulsatile (pulsating) ones is the concern of the paper. The current state of art on the transitional characteristics of pulsatile and oscillating pipe flows is introduced with a particular attention to the utilized terminology and methodology. Transition from laminar to turbulent regime is usually described by the presence of the disturbed flow with small amplitude perturbations followed by the growth of turbulent bursts. The visual treatment of velocity waveforms is therefore a preferred inspection method. The observation of turbulent bursts first in the decelerating phase and covering the whole cycle of oscillation are used to define the critical states of the start and end of transition, respectively. A correlation study referring to the available experimental data of the literature particularly at the start of transition are presented in terms of the governing periodic flow parameters. In this respect critical oscillating and time averaged Reynolds numbers at the start of transition; Re_{
os,crit
} and Re_{
ta,crit
} are expressed as a major function of Womersley number, \(\sqrt {\omega ^\prime } \) defined as dimensionless frequency of oscillation, *f*. The correlation study indicates that in oscillating flows, an increase in Re_{
os,crit
} with increasing magnitudes of \(\sqrt {\omega ^\prime } \) is observed in the covered range of \(1<\sqrt {\omega ^\prime } <72\). The proposed equation (Eq. 7), \({\rm{Re}}_{os,crit} ={\rm{Re}}_{os,crit} \left( {\sqrt {\omega ^\prime } } \right)\), can be utilized to estimate the critical magnitude of \(\sqrt {\omega ^\prime }\) at the start of transition with an accuracy of ±12 % in the range of \(\sqrt {\omega ^\prime } <41\). However in pulsatile flows, the influence of \(\sqrt {\omega ^\prime }\) on Re_{
ta,crit
} seems to be different in the ranges of \(\sqrt {\omega ^\prime } <8\) and \(\sqrt {\omega ^\prime } >8\). Furthermore there is rather insufficient experimental data in pulsatile flows considering interactive influences of \(\sqrt {\omega ^\prime } \) and velocity amplitude ratio, *A*
_{1}. For the purpose, the measurements conducted at the start of transition of a laminar sinusoidal pulsatile pipe flow test case covering the range of 0.21< *A*
_{1} <0.95 with \(\sqrt {\omega ^\prime } <8\) are evaluated. In conformity with the literature, the start of transition corresponds to the observation of first turbulent bursts in the decelerating phase of oscillation. The measured data indicate that increase in \(\sqrt {\omega ^\prime } \) is associated with an increase in Re_{
ta,crit
} up to \(\sqrt {\omega ^\prime } =3.85\) while a decrease in Re_{
ta,crit
} is observed with an increase in \(\sqrt {\omega ^\prime } \) for\(\sqrt {{\omega }'} >3.85\). Eventually updated portrait is pointing out the need for further measurements on i) the end of transition both in oscillating and pulsatile flows with the ranges of \(\sqrt {\omega ^\prime } <8\) and \(\sqrt {\omega ^\prime } >8\), and ii) the interactive influences of \(\sqrt {\omega ^\prime } \) and *A*
_{1} on Re_{
ta,crit
} in pulsatile flows with the range of \(\sqrt {\omega ^\prime } >8\).

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Çarpinlioğlu, M.Ö., Özahi, E. An Updated Portrait of Transition to Turbulence in Laminar Pipe Flows with Periodic Time Dependence (A Correlation Study).
*Flow Turbulence Combust* **89, **691–711 (2012). https://doi.org/10.1007/s10494-012-9420-1

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### Keywords

- Periodic flow
- Transition to turbulence
- Time averaged Reynolds number
- Womersley number
- Velocity amplitude ratio
- Oscillation Reynolds number