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

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Abstract

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.

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Abbreviations

Amixed :

Area of fully mixed core

D:

Coefficient of diffusion

K:

Concentration of fluid

r o :

Pipe cross-sectional radius

r c :

Curvature radius

u :

Velocity in x direction

\( v \) :

Velocity in y direction

Dn:

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

\( M_{a} \) :

Augmentation in fuel consumption rate

SD:

Standard deviation

Re:

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

RSD:

Relative standard deviation

α:

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

β:

Velocity component ratio

\( \upsilon \) :

Kinematic viscosity

η:

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

ω:

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

Γ:

Circulation

m:

Mean value

B:

Blue fluid

P:

Pulsating flow

R:

Red fluid

S:

Steady flow

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Authors and Affiliations

  1. Thermofluids, Complex Flows and Energy Group, Laboratoire de Thermocinétique, UMR CNRS 6607, Ecole Polytechnique, University of Nantes, La Chantrerie, BP 50609, 44306, Nantes, France

    Mojtaba Jarrahi, Cathy Castelain & Hassan Peerhossaini

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  1. Mojtaba Jarrahi
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  2. Cathy Castelain
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  3. Hassan Peerhossaini
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Correspondence to Hassan Peerhossaini.

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Jarrahi, M., Castelain, C. & Peerhossaini, H. Secondary flow patterns and mixing in laminar pulsating flow through a curved pipe. Exp Fluids 50, 1539–1558 (2011). https://doi.org/10.1007/s00348-010-1012-z

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