# An Updated Portrait of Transition to Turbulence in Laminar Pipe Flows with Periodic Time Dependence (A Correlation Study)

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## 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 \(\sqrt {\omega ^\prime } >8\).

## Keywords

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

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## References

- 1.Akhavan, R., Kamm, R.D., Shapiro, A.H.: An investigation of transition to turbulence in bounded oscillatory stokes flows Part 1: Experiments. J. Fluid Mech.
**225**, 395–422 (1991)CrossRefGoogle Scholar - 2.Çarpınlıoğlu, M.Ö.: An approach for transition correlation of laminar pulsatile pipe flows via frictional field characteristics. Flow Meas. Instrum.
**14**, 233–242 (2003)CrossRefGoogle Scholar - 3.Çarpınlıoğlu, M.Ö., Gündoğdu, M.Y.: A critical review on pulsatile pipe flow studies directing towards future research topics. Flow Meas. Instrum.
**12**, 163–174 (2001)CrossRefGoogle Scholar - 4.Çarpınlıoğlu, M.Ö., Özahi, E.: Laminar flow control via utilization of pipe entrance inserts (a comment on entrance length concept). Flow Meas. Instrum.
**22**, 165–174 (2011)CrossRefGoogle Scholar - 5.Clamen, M., Minton, P.: An experimental investigation of flow in an oscillating pipe. J. Fluid Mech.
**81**, 421–431 (1977)CrossRefGoogle Scholar - 6.Das, D., Arakeri, J.H.: Transition of unsteady velocity profiles with reverse flow. J. Fluid Mech.
**374**, 251–283 (1998)MathSciNetzbMATHCrossRefGoogle Scholar - 7.Durst, F., Heim, U., Ünsal, B., Kullik, G.: Mass flow rate control system for time-dependent laminar and turbulent flow investigations. Meas. Sci. Technol.
**14**, 893–902 (2003)CrossRefGoogle Scholar - 8.Durst, F., Ray, S., Ünsal, B., Bayoumi, O.A.: The development lengths of laminar pipe and channel flows. Trans. ASME
**127**, 1154–1160 (2005)Google Scholar - 9.Eckhardt, B., Schneider, T.M., Hof, B., Westerweel, J.: Turbulence transition in pipe flow. Annu. Rev. Fluid Mech.
**39**, 447–468 (2007)MathSciNetCrossRefGoogle Scholar - 10.Eckmann, D.M., Grotberg, J.B.: Experiments on transition to turbulence in oscillatory pipe flow. J. Fluid Mech.
**222**, 329–350 (1991)CrossRefGoogle Scholar - 11.Einav, S., Sokolov, M.: An experimental study of pulsatile pipe flow in the transition range. Trans. ASME
**115**, 404–411 (1993)Google Scholar - 12.Fedele, F., Hitt, D.L., Prabhu, R.D.: Revisiting the stability of pulsatile pipe flow. Eur. J. Mech. B-Fluid
**24**, 237–254 (2005)zbMATHCrossRefGoogle Scholar - 13.Gerrard, J.H.: An experimental investigation of the pulsating turbulent water flow in a tube. J. Fluid Mech.
**46**, 43–64 (1971)CrossRefGoogle Scholar - 14.Gündoğdu, M.Y.: An experimental investigation on pulsatile pipe flows. Ph. D. Thesis, University of Gaziantep, Department of Mechanical Engineering, Turkey (2000)Google Scholar
- 15.Gündoğdu, M.Y., Çarpınlıoğlu, M.Ö.: Present state of art on pulsatile flow theory part I: laminar and transitional flow regimes. JSME Int. J.
**42**, 384–397 (1999)CrossRefGoogle Scholar - 16.Hershey, D., Im, C.S.: Critical Reynolds number for sinusoidal flow of water in rigid tubes. AIChE J.
**14**, 807–809 (1968)CrossRefGoogle Scholar - 17.Hino, M., Sawamoto, M., Takasu, S.: Experiments on transition to turbulence in an oscillatory pipe flow. J. Fluid Mech.
**75**, 193–207 (1976)CrossRefGoogle Scholar - 18.Iguchi, M., Ohmi, M.: Transition to turbulence in a pulsatile pipe flow. Part 3: flow regimes and the conditions describing the generation and decay of turbulence. Bull JSME
**27**, 1873–1880 (1984)CrossRefGoogle Scholar - 19.Ito, H.: On the pressure loss of turbulent flow through curved pipes. Rep. Inst. High Speed Mech. Tohoku Univ., Sendai Jpn.
**7**, 63–76 (1952)Google Scholar - 20.Kusama, H.: Study of pulsating flow (pulsating flow in a circular pipe). Soc. Mech. Eng. Trans.
**18**, 27 (1952)MathSciNetCrossRefGoogle Scholar - 21.Leite, R.J.: An experimental investigation of the stability of Poiseuille flow. J. Fluid Mech.
**5**, 81–96 (1959)zbMATHCrossRefGoogle Scholar - 22.Lessen, M., Singh, P.J.: The stability of axisymmetric free shear layers. J. Fluid Mech.
**60**, 433–457 (1973)zbMATHCrossRefGoogle Scholar - 23.Mackrodt, P.A.: Stability of Hagen-Poiseuille flow with superimposed rigid rotation. J. Fluid Mech.
**73**, 153–164 (1976)zbMATHCrossRefGoogle Scholar - 24.Merkli, P., Thomann, H.: Transition to turbulence in oscillating pipe flow. J. Fluid Mech.
**68**, 567–575 (1975)CrossRefGoogle Scholar - 25.Mizushina, T., Maruyama, T., Shiozaki, Y.: Pulsating turbulent flow in a tube. J. Chem. Eng. Jpn.
**6**, 487–494 (1973)CrossRefGoogle Scholar - 26.Nerem, R.M., Seed, W.A., Wood, N.B.: An experimental study of the velocity distribution and transition to turbulence in the aorta. J. Fluid Mech.
**52**, 137–160 (1972)CrossRefGoogle Scholar - 27.Ohmi M., et al.: Preprint of Jpn. Soc. Mech. Engrs. (in Japanese) 795-15, 106 (1979-10)Google Scholar
- 28.Ohmi, M., Iguchi, M.: Critical Reynolds number in an oscillating pipe flow. Bull. JSME
**25**, 165–172 (1982)CrossRefGoogle Scholar - 29.Ohmi, M., Iguchi, M., Usui, T.: Flow pattern and frictional losses in pulsating pipe flow, Part 5: Wall shear stress and flow pattern in a laminar flow. Bull. JSME
**24**, 75–81 (1981)CrossRefGoogle Scholar - 30.Ohmi, M., Iguchi, M., Kakehashi, K., Masuda, T.: Transition to turbulence and velocity distribution in an oscillating pipe flow. Bull. JSME
**25**, 365–371 (1982)CrossRefGoogle Scholar - 31.Ohmi, M., Iguchi, M., Urahata, I.: Transition to turbulence in a pulsatile pipe flow. Part 1: Wave forms and distribution of pulsatile velocities near transition region. Bull. JSME
**25**, 182–189 (1982)CrossRefGoogle Scholar - 32.Özahi, E.: Analysis of laminar-turbulent transition in time-dependent pipe flows. Ph.D. thesis, University of Gaziantep, Turkey (2011)Google Scholar
- 33.Özahi, E., Çarpınlıoğlu, M.Ö., Gündoğdu, M.Y.: Simple methods for low speed calibration of hot-wire anemometers. Flow Meas. Instrum.
**21**, 166–170 (2010)CrossRefGoogle Scholar - 34.Peacock, J., Jones, T., Tock, C., Lutz, R.: The onset of turbulence in physiological pulsatile flow in a straight tube. Exp. Fluids
**24**, 1–9 (1998)CrossRefGoogle Scholar - 35.Ramaprian, B., Tu, W.W.: An experimental study of oscillatory pipe flow at transitional Reynolds numbers. J. Fluid Mech.
**100**, 513–544 (1980)CrossRefGoogle Scholar - 36.Reynolds, O.: An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous and the law of resistance in parallel channels. Phil. Trans. R. Soc.
**174**, 935–982 (1883)zbMATHCrossRefGoogle Scholar - 37.Salwen, H., Grosch, C.E.: Stability of Poiseuille flow in a pipe of circular cross section. J. Fluid Mech.
**54**, 93–112 (1972)zbMATHCrossRefGoogle Scholar - 38.Sarpkaya, T.: Experimental determination of the critical Reynolds number for pulsating poiseuille flow. Trans. ASME D, J. Basic Eng.
**88**, 589–598 (1966)CrossRefGoogle Scholar - 39.Sarpkaya, T.: A note on the stability of developing laminar pipe flow subjected to axisymmetric and non-axisymmetric disturbances. J. Fluid Mech.
**68**, 345–351 (1975)CrossRefGoogle Scholar - 40.Sergeev, S.I.: Fluid oscillations in pipes at moderate Reynolds numbers. Fluid Dyn.
**1**, 121–122 (1966)CrossRefGoogle Scholar - 41.Sexl, T.: On the annular effect discovered by E.G. Richardson. Z. Physik.
**61**, 349–362 (1930)zbMATHCrossRefGoogle Scholar - 42.Shemer, L.: Laminar-turbulent transition in a slowly pulsating pipe flow. Phys. Fluids
**28**, 3506–3509 (1985)CrossRefGoogle Scholar - 43.Stettler, J.C., Hussain, K.M.F.: On transition of the pulsatile pipe flow. J. Fluid Mech.
**170**, 169–197 (1986)CrossRefGoogle Scholar - 44.Szymanski, P.: Some exact solution of the hydrodynamic equations of a viscous fluid in the case of a cylindrical. J. Math. Pure Appl.
**11**, 67–107 (1932)zbMATHGoogle Scholar - 45.Ünsal, B., Durst, F.: Pulsating flows: experimental equipment and its application. JSME
**49**, 980–987 (2006)Google Scholar - 46.Womersley, J.R.: Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol.
**127**, 553–563 (1955)Google Scholar - 47.Yang, W.H., Yih, C.-S.: Stability of time-periodic flows in a circular pipe. J. Fluid Mech.
**82**, 497–505 (1977)zbMATHCrossRefGoogle Scholar - 48.Yellin, E.L.: Laminar-turbulent transition process in pulsatile flow. Circ. Res.
**19**, 791–804 (1966)CrossRefGoogle Scholar