# Pulsatile entrance flow in curved pipes: effect of various parameters

## Abstract

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.

## Keywords

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

*a*radius of pipe,

*a*=*d*/2*C*_{p}pressure coefficient,

*C*_{ p }= (*P*−*P*_{ref})/(*ρW*_{ m }^{2}/2)*D*Dean number,

*D*=*Re**Rc*^{−1/2}*P*static pressure at pipe wall

*P*_{ref}reference value of

*P*, i.e.,*P*at*z*/*d*= −2*R*curvature radius of pipe

*R*_{c}curvature radius ratio,

*R*_{ c }=*R*/*a**Re*Reynolds number,

*Re*=*W*_{ m }*d*/*ν**u, w*velocities in

*x*and*z*directions, respectively*W*axial velocity averaged over cross section

*x, y, z*coordinate system, see Fig. 2

*z*′nondimensional axial length,

*z′*=*z*/(*aR*)^{1/2}*α*Womersley number, α =

*a*(ω/*ν*)^{1/2}*β**ϕ*phase difference, see Eq. (2)

*η*flow rate ratio,

*η*=*W*_{ o }/*W*_{ m }*Θ*phase angle,

*Θ*=*ωt*(*t*: time,*ω*: angular frequency)*ν*kinematic viscosity of fluid

*ρ*density of fluid

*Ω*turn angle, see Fig. 2

## Subscripts

*m*,*o*time-mean and amplitude values, respectively

## Notes

### Acknowledgment

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

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