Experiments in Fluids

, 56:22 | Cite as

Length and time for development of laminar flow in tubes following a step increase of volume flux

  • Rafeed A. Chaudhury
  • Marcus Herrmann
  • David H. Frakes
  • Ronald J. Adrian
Research Article

Abstract

Laminar flows starting up from rest in round tubes are relevant to numerous industrial and biomedical applications. The two most common types are flows driven by an abruptly imposed constant pressure gradient or by an abruptly imposed constant volume flux. Analytical solutions are available for transient, fully developed flows, wherein streamwise development over the entrance length is absent (Szymanski in J de Mathématiques Pures et Appliquées 11:67–107, 1932; Andersson and Tiseth in Chem Eng Commun 112(1):121–133, 1992, respectively). They represent the transient responses of flows in tubes that are very long compared with the entrance length, a condition that is seldom satisfied in biomedical tube networks. This study establishes the entrance (development) length and development time of starting laminar flow in a round tube of finite length driven by a piston pump that produces a step change from zero flow to a constant volume flux for Reynolds numbers between 500 and 3,000. The flows are examined experimentally, using stereographic particle image velocimetry and computationally using computational fluid dynamics, and are then compared with the known analytical solutions for fully developed flow conditions in infinitely long tubes. Results show that step function volume flux start-up flows reach steady state and fully developed flow five times more quickly than those driven by a step function pressure gradient, a 500 % change when compared with existing estimates. Based on these results, we present new, simple guidelines for achieving experimental flows that are fully developed in space and time in realistic (finite) tube geometries. To a first approximation, the time to achieve steady spatially developing flow is nearly equal to the time needed to achieve steady, fully developed flow. Conversely, the entrance length needed to achieve fully developed transient flow is approximately equal to the length needed to achieve fully developed steady flow. Beyond this level of description, the numerical results reveal interaction between the effects of space and time development and nonlinear Reynolds number effects.

List of symbols

\(a\)

Tube radius (m)

\(D\)

Tube diameter (2a) (m)

\(L\)

Tube length (m)

\(r\)

Radial coordinate (m)

\(r^{*}\)

Dimensionless radial coordinate (\(r/a\))

\(Re_D\)

Diameter Reynolds number (\(UD/\nu\))

\(t\)

Time (s)

\(t^{*}\)

Dimensionless time (\(t\nu /a^2\))

\(t_d\)

Development time for fully developed flow

\(T\)

Dimensionless function of time

\(u\)

Axial velocity component (m/s)

\(u^{*}\)

Dimensionless axial velocity component (\(u/U\))

\(U\)

Velocity scale (m/s)

\(\tilde{U}\)

Approximate centerline velocity at any (\(x,t\)) (m/s)

\(U_B\)

Bulk velocity (m/s)

\(x\)

Axial coordinate (m)

\(x_d\)

Development length for fully developed flow (m)

Greek letters

\(\nu\)

Kinematic viscosity (\(\hbox {m}^2/\hbox {s}\))

\(\rho\)

Density (\(\hbox {kg}/\hbox {m}^3\))

\(\tau\)

Time constant (s)

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Rafeed A. Chaudhury
    • 1
  • Marcus Herrmann
    • 2
  • David H. Frakes
    • 1
    • 3
  • Ronald J. Adrian
    • 2
  1. 1.School of Biological and Health Systems EngineeringArizona State UniversityTempeUSA
  2. 2.School for Engineering of Matter, Transport and EnergyArizona State UniversityTempeUSA
  3. 3.School of Electrical, Computer, and Energy EngineeringArizona State UniversityTempeUSA

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