Experiments in Fluids

, Volume 50, Issue 6, pp 1539–1558 | Cite as

Secondary flow patterns and mixing in laminar pulsating flow through a curved pipe

  • Mojtaba Jarrahi
  • Cathy Castelain
  • Hassan Peerhossaini
Research Article


Mixing by secondary flow is studied by particle image velocimetry (PIV) in a developing laminar pulsating flow through a circular curved pipe. The pipe curvature ratio is η = r 0/r c  = 0.09, and the curvature angle is 90°. Different secondary flow patterns are formed during an oscillation period due to competition among the centrifugal, inertial, and viscous forces. These different secondary-flow structures lead to different transverse-mixing schemes in the flow. Here, transverse mixing enhancement is investigated by imposing different pulsating conditions (Dean number, velocity ratio, and frequency parameter); favorable pulsating conditions for mixing are introduced. To obviate light-refraction effects during PIV measurements, a T-shaped structure is installed downstream of the curved pipe. Experiments are carried out for the Reynolds numbers range 420 ≤ Rest ≤ 1,000 (Dean numbers 126.6 ≤ Dn ≤ 301.5) corresponding to non-oscillating flow, velocity component ratios 1 ≤ (β = U max,osc/U m,st) ≤ 4 (the ratio of velocity amplitude of oscillations to the mean velocity without oscillations), and frequency parameters 8.37 < (α = r 0(ω/ν)0.5) < 24.5, where α2 is the ratio of viscous diffusion time over the pipe radius to the characteristic oscillation time. The variations in cross-sectional average values of absolute axial vorticity (|ζ|) and transverse strain rate (|ε|) are analyzed in order to quantify mixing. The effects of each parameter (Rest, β, and α) on transverse mixing are discussed by comparing the dimensionless vorticities (|ζ P |/|ζ S |) and dimensionless transverse strain rates (|ε P |/|ε S |) during a complete oscillation period.


Vorticity Particle Image Velocimetry Secondary Flow Particle Image Velocimetry Measurement Curve Pipe 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Area of fully mixed core


Coefficient of diffusion


Concentration of fluid


Pipe cross-sectional radius


Curvature radius


Velocity in x direction

\( v \)

Velocity in y direction


Dean number, \( {\text{Dn}} = {\frac{{U_{m} (2r_{0} )}}{\upsilon }}\sqrt {{\frac{{r_{0} }}{{r_{c} }}}} \)

\( M_{a} \)

Augmentation in fuel consumption rate


Standard deviation


Reynolds number, \( \text{Re} = {\frac{{U_{m} (2r_{0} )}}{\upsilon }} \)


Relative standard deviation

Greek symbols


Womersley number, \( r_{o} \left( {\omega /\upsilon } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \)


Velocity component ratio

\( \upsilon \)

Kinematic viscosity


Curvature ratio of curved pipe, \( {{r_{o} } \mathord{\left/ {\vphantom {{r_{o} } {r_{c} }}} \right. \kern-\nulldelimiterspace} {r_{c} }} \)


Angular frequency

\( \zeta (x,y) \)

Axial vorticity at position (x, y) in the curved pipe cross section:\( \;{\frac{\partial v}{\partial x}} - {\frac{\partial u}{\partial y}} \)

\( \left| {\zeta_{P} } \right| \)

Cross-sectional average value of absolute vorticity in a pulsatile flow

\( \left| {\zeta_{S} } \right| \)

Cross-sectional average value of absolute vorticity in a steady flow

\( \varepsilon (x,y) \)

Transverse strain rate at position (x, y) in the curved pipe cross section: \( \frac{1}{2}\left( {\;{\frac{\partial v}{\partial x}} + {\frac{\partial u}{\partial y}}} \right) \)

\( \left| {\varepsilon_{P} } \right| \)

Cross-sectional average value of absolute transverse strain rate in a pulsatile flow

\( \left| {\varepsilon_{S} } \right| \)

Cross-sectional average value of absolute transverse strain rate in a steady flow





Mean value


Blue fluid


Pulsating flow


Red fluid


Steady flow


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Mojtaba Jarrahi
    • 1
  • Cathy Castelain
    • 1
  • Hassan Peerhossaini
    • 1
  1. 1.Thermofluids, Complex Flows and Energy Group, Laboratoire de Thermocinétique, UMR CNRS 6607Ecole Polytechnique, University of Nantes, La Chantrerie, BP 50609NantesFrance

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