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Research Papers: Fundamental Issues and Canonical Flows

# Entry Length Requirements for Two- and Three-Dimensional Laminar Couette–Poiseuille Flows

[+] Author and Article Information
Bayode E. Owolabi

Department of Mechanical Engineering,
University of Alberta,
Edmonton, AB T6G 1H9, Canada
e-mail: owolabi@ualberta.ca

David J. C. Dennis

School of Engineering,
University of Liverpool,
Liverpool L69 3GH, UK
e-mail: djcd@liverpool.ac.uk

Robert J. Poole

School of Engineering,
University of Liverpool,
Liverpool L69 3GH, UK
e-mail: robpoole@liverpool.ac.uk

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received March 11, 2019; final manuscript received June 9, 2019; published online June 27, 2019. Assoc. Editor: Pierre E. Sullivan.

J. Fluids Eng 141(12), 121204 (Jun 27, 2019) (9 pages) Paper No: FE-19-1157; doi: 10.1115/1.4043986 History: Received March 11, 2019; Revised June 09, 2019

## Abstract

In this study, we examine the development length requirements for laminar Couette–Poiseuille flows in a two-dimensional (2D) channel as well as in the three-dimensional (3D) case of flow through a square duct, using a combination of numerical and experimental approaches. The parameter space investigated covers wall to bulk velocity ratios, r, spanning from 0 (purely pressure-driven flow) to 2 (purely wall driven-flow; 4 in the case of a square duct) and a wide range of Reynolds numbers (Re). The results indicate an increase in the development length (L) with r. Consistent with the findings of Durst et al. (2005, “The Development Lengths of Laminar Pipe and Channel Flows,” ASME J. Fluids Eng., 127(6), pp. 1154–1160), L was observed to be of the order of the channel height in the limit as $Re→0$, irrespective of the condition at the inlet. This, however, changes at high Reynolds numbers, with L increasing linearly with Re. In all the cases considered, a uniform velocity profile at the inlet was found to result in longer entry lengths than in a flow developing from a parabolic inlet profile. We show that this inlet effect becomes less important as the limit of purely wall-driven flow is approached. Finally, we develop correlations for predicting L in these flows and, for the first time, also present laser Doppler velocimetry (LDV) measurements of the developing as well as fully-developed velocity profiles, and observe good agreement between experiment, analytical solution, and numerical simulation results in the 3D case.

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## References

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## Figures

Fig. 1

Computational domain and boundary conditions for the flow in a 2D channel: (a) uniform inlet velocity and (b) parabolic velocity profile at the inlet

Fig. 2

Computational domain and boundary conditions for the flow in a square duct: (a) uniform inlet velocity and (b) parabolic velocity profile at the inlet. Moving wall is shaded.

Fig. 3

Comparison between u/Ub from analytical solution and numerical simulation at r =1. Symbols represent numerical simulation, while lines represent the analytical solution. Values of u/Ub shown range from 0.3 to 1.8 with increment of 0.3. Translatory motion is introduced at the top wall.

Fig. 4

Development lengths in a 2D channel at different wall to bulk velocity ratios: (a) uniform inlet velocity and (b) parabolic velocity profile at the inlet. L is the entry length at the wall-normal location corresponding to umax.

Fig. 5

Development length computations in a 2D channel at high Reynolds numbers: (a) influence of entry length definition and (b) effect of inlet boundary condition. Lglobal is the global entry length, while Llocal is the development length computed at the wall-normal location corresponding to umax. Plots show results at Re > 50.

Fig. 6

Correlations for C1: (a) uniform inlet velocity and (b) parabolic velocity profile at the inlet

Fig. 7

Development lengths in purely pressure-driven flows. The inset shows better collapse when centerline velocity scaling is used to define Re (Rec=ρUcD/μ).

Fig. 8

Couette–Poiseuille flow development length simulations in a square duct: (a) development lengths at Re ranging from 0.5 to 200 for parabolic velocity profile at the inlet and (b) development lengths at Re → 0

Fig. 9

Experimental test section for the study of wall-driven flows: (a) side view and (b) front view. (1) Electric motor, (2) stainless steel belt, (3) pulleys, (4) tracking screw, (5) square duct, (6) water-tight casing, and (7) borosilicate glass window. Flow is in the positive x direction.

Fig. 10

Couette–Poiseuille flow development: (a) r≈1.7 and Re≈63 and (b) r≈1 and Re≈76. Measurements have been taken at y/D=0.3 in (a) and y/D=0.1 in (b) along the vertical wall bisector, corresponding to the respective maximum velocity locations. Error bars representing ±3% measurement uncertainty have been included.

Fig. 11

Velocity profiles in laminar Couette–Poiseuille flows in a square duct: (a) fully developed profiles along the wall bisectors at x/D≈4.69, Re≈63, and r≈1.7; (b) fully developed profile taken at the same streamwise location and Re as (a) at y/D=0.375 (10 mm from moving wall); (c) velocity profiles along the wall bisectors in developing flow at x/D≈2.5, Re≈63, and r≈1.7; and (d) fully developed profiles at x/D≈4.69, Re≈76, and r≈1. Unfilled symbols represent profiles along the vertical plane (from the bottom stationary wall to the top moving wall), while filled symbols represent those along the horizontal plane (from one stationary side wall to the other). Solid lines represent the laminar flow analytical solution for fully developed flow, while dash-dotted lines represent the developing flow numerical simulation results from ansysfluent.

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