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Entry and exit flows in curved pipes

Published online by Cambridge University Press:  23 February 2017

Jesse T. Ault Open the ORCID record for Jesse T. Ault [Opens in a new window] ,
Bhargav Rallabandi Open the ORCID record for Bhargav Rallabandi [Opens in a new window] ,
Orest Shardt ,
Kevin K. Chen Open the ORCID record for Kevin K. Chen [Opens in a new window]  and
Howard A. Stone
Jesse T. Ault
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Bhargav Rallabandi
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Orest Shardt
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Kevin K. Chen
Affiliation:
Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089, USA
Howard A. Stone*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email address for correspondence: hastone@princeton.edu
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Abstract

Solutions are presented for both laminar developing flow in a curved pipe with a parabolic inlet velocity and laminar transitional flow downstream of a curved pipe into a straight outlet. Scalings and linearized analyses about appropriate base states are used to show that both cases obey the same governing equations and boundary conditions. In particular, the governing equations in the two cases are linearized about fully developed Poiseuille flow in cylindrical coordinates and about Dean’s velocity profile for curved pipe flow in toroidal coordinates respectively. Subsequently, we identify appropriate scalings of the axial coordinate and disturbance velocities that eliminate dependence on the Reynolds number $Re$ and dimensionless pipe curvature $\unicode[STIX]{x1D6FC}$ from the governing equations and boundary conditions in the limit of small $\unicode[STIX]{x1D6FC}$ and large $Re$. Direct numerical simulations confirm the scaling arguments and theoretical solutions for a range of $Re$ and $\unicode[STIX]{x1D6FC}$. Maximum values of the axial velocity, secondary velocity and pressure perturbations are determined along the curved pipe section. Results collapse when the scalings are applied, and the theoretical solutions are shown to be valid up to Dean numbers of $D=Re^{2}\unicode[STIX]{x1D6FC}=O(100)$. The developing flows are shown numerically and analytically to contain spatial oscillations. The numerically determined decay of the velocity perturbations is also used to determine entrance/development lengths for both flows, which are shown to scale linearly with the Reynolds number, but with a prefactor ${\sim}60\,\%$ larger than the textbook case of developing flow in a straight pipe.

JFM classification

Mathematical Foundations: General fluid mechanics Mathematical Foundations: Mathematical Foundations
Type
Papers
Information
Journal of Fluid Mechanics , Volume 815 , 25 March 2017 , pp. 570 - 591
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Copyright
© 2017 Cambridge University Press 

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