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Finite difference analysis of plane Poiseuille and Couette flow developments

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Summary

The problem of flow development from an initially flat velocity profile in the plane Poiseuille and Couette flow geometry is investigated for a viscous fluid. The basic governing momentum and continuity equations are expressed in finite difference form and solved numerically on a high speed digital computer for a mesh network superimposed on the flow field. Results are obtained for the variations of velocity, pressure and resistance coefficient throughout the development region. A characteristic development length is defined and evaluated for both types of flow.

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Abbreviations

h :

width of channel

L :

ratio of development length to channel width

p :

fluid pressure

p 0 :

pressure at channel mouth

P :

dimensionless pressure, p/ρū 2

P 0 :

dimensionless pressure at channel mouth

ΔP :

pressure defect, P 0−P

(ΔP)0 :

pressure defect neglecting inertia

Re :

Reynolds number, ρuh/μ

u :

fluid velocity in x-direction

ū :

mean u velocity across channel

u 0 :

wall velocity

U :

dimensionles u velocity u/ū

U c :

dimensionless centreline velocity

U 0 :

dimensionless wall velocity

v :

fluid velocity in y-direction

V :

dimensionless v velocity, hvρ/μ

x :

coordinate along channel

X :

dimensionless x-coordinate, μx/ρh 2 ū

y :

coordinate across channel

Y :

dimensionless y-coordinate, y/h

λ :

resistance coefficient, \(\frac{{p_0 - p}}{{pu^{ - 2} }}\frac{{2h}}{x}\)

λ 0 :

resistance coefficient neglecting inertia

ρ :

fluid density

μ :

fluid viscosity

References

  1. Schlichting, H., Boundary Layer Theory, McGraw-Hill Book Co., Inc. 1955, 146.

  2. Rouleau, W., and F. Osterle, J. Aero. Sci. 22 (1955) 249.

    MATH  MathSciNet  Google Scholar 

  3. Bodoia, J. R., Ph. D. Thesis, Carnegie Institute of Technology, July 1959.

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Authors and Affiliations

  1. Bettis Atomic Power Division, Westinghouse Electric corporation, Pittsburgh, Pennsylvania, U.S.A.

    J. R. Bodoia

  2. Department of Mechanical Engineering, Carnegie Institute of Technology Pittsburgh, Pennsylvania, U.S.A.

    J. F. Osterle

Authors
  1. J. R. Bodoia
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  2. J. F. Osterle
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Bodoia, J.R., Osterle, J.F. Finite difference analysis of plane Poiseuille and Couette flow developments. Appl. sci. Res. 10, 265 (1961). https://doi.org/10.1007/BF00411919

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  • DOI: https://doi.org/10.1007/BF00411919

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