Experiments in Fluids

, Volume 43, Issue 6, pp 949–958 | Cite as

Pulsatile entrance flow in curved pipes: effect of various parameters

  • Masaru Sumida
Research Article


This paper presents the results of an experimental study on the developing pulsatile flow in curved pipes with a long, straight pipe upstream. In order to examine the dependence of flow-field development on the governing parameters, LDV measurements were conducted systematically for six cases of flow, where the Womersley number α was varied from 5.5 to 18, the mean Dean number D m was 200 and 300, the flow rate ratio η was 0.5 and 1, and the curvature radius ratio Rc was 10 and 30. Peculiar flow phenomena, such as flow reversal for all values of α and a depression in the axial velocity profile for α = 10, were analyzed by decomposing the axial velocity into a time-mean and a varying component, as well as by obtaining the bias of their profiles. The velocity distributions abruptly change with the phase at turn angles Ω of 15–30°, corresponding to the nondimensional axial length z′ ≅ 1–2 from the bend entrance, and their development along the pipe axis is the most complicated for the flow at a moderate α of 10 and large η of 1. The entrance length in the case of pulsatile flow is shorter than that for steady flow with the same flow rate as the maximum pulsatile flow rate.


Axial Velocity Pulsatile Flow Entrance Length Curve Pipe Axial Velocity Profile 
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List of symbols


radius of pipe, a = d/2


pressure coefficient, C p  = (P − P ref)/(ρW m 2 /2)


Dean number, D = Re Rc −1/2


static pressure at pipe wall


reference value of P, i.e., P at z/d = −2


curvature radius of pipe


curvature radius ratio, R c  = R/a


Reynolds number, Re = W m d/ν

u, w

velocities in x and z directions, respectively


axial velocity averaged over cross section

x, y, z

coordinate system, see Fig. 2


nondimensional axial length, z′ = z/(aR)1/2


Womersley number, α = a(ω/ν)1/2


deflection of axial flow, see Eqs. (3) and (4)


phase difference, see Eq. (2)


flow rate ratio, η = W o /W m


phase angle, Θ = ωt (t: time, ω: angular frequency)


kinematic viscosity of fluid


density of fluid


turn angle, see Fig. 2


m, o

time-mean and amplitude values, respectively



The author would like to thank Professor Emeritus K. Sudo of Hiroshima University for his useful and helpful suggestions.


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.School of EngineeringKinki UniversityHigashi-HiroshimaJapan

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