Theoretical investigation of frictional effects for laminar compressible flow in a tube entry
An approximate integral method is used for calculating the longitudinal pressure distribution of a laminar compressible flow in the entrance region of a duct. The method is based on the observation that when the boundary layer is thin, its growth along the duct is approximated by that for laminar flow over a flat plate. The velocity and temperature profiles in the boundary layer of the tube flow are assumed to be identical with those pertaining to a plate flow having a free-stream Mach number equal to the entrance Mach number of the corresponding tube flow.
This integral method has been used to compute the longitudinal pressure distribution, the mean pressure-gradient coefficient and the mean apparent friction factor for a laminar compressible flow in the entrance region of a round tube, for several entrance Mach numbers and thermal conditions at the wall.
In the case of constant values of specific heat, absolute viscosity and thermal conductivity of the compressible fluid, the computed approximate results are compared with those obtained by solving exactly the partial differential equations of motion. In the region where the integral method is considered to be valid, the agreement is within 5%, thus indicating the reliability of the procedure.
In the case of temperature-dependent viscosity and thermal conductivity, the computed results are compared with those obtained experimentally at Mach number 2.8, and the agreement is found to be within 8%.
For the incompressible case, the computed results are compared with those obtained by more exact integral methods and with experimental data. The agreement is good, except that at the limit of the region of validity of the present method, the discrepancy is increased to about 12%.
A plot is included showing, as a function of entrance Mach number, the ratio of the mean pressure-gradient coefficient for compressible flow to that for incompressible flow. With this information, the compressible-flow pressure distribution may be readily computed.
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- F. E. Neumann: Einleitung in die theoretische Physik, Leipzig, 1883, p. 262;Google Scholar
- M. Couette: Études sur le frottement des liquides, Annales de Chimie et de Physique (6), vol. 21, 1890, pp. 494–510.Google Scholar
- H. L. Langhaar: Steady Flow in the Transition Length of a Straight Tube, Journal of Applied Mechanics, Trans. ASME, vol. 64, 1942, pp. A-55-A-58.Google Scholar
- A. H. Shapiro, R. Siegel and S. J. Kline: Friction Factor in the Laminar Entry Region of a Smooth Tube, to be published in Proceedings of Second US National Congress for Applied Mechanics, 1954.Google Scholar
- T. Y. Toong: The Laminar Boundary Layer of a Steady, Compressible Flow in the Entrance Region of a Tube,. Sc. D. Thesis, Mechanical Engineering Department, Mass. Institute of Technology, Jan. 1952.Google Scholar
- S. J. Kline and A. H. Shapiro: The Effect of Cooling on Boundary Layer Transition in a Gas, Final Report on DIC Project 3–6927 at Mass. Institute of Technology, Sept. 1952.Google Scholar
- J. Kaye, T. Y. Toong and R. H. Shoulberg: Measurement of Recovery Factors and Friction Coefficients for Supersonic Flow of Air in a Tube, Journal of Applied Mechanics, Trans. ASME, vol. 74, 1952, pp. A-185-A-194.Google Scholar
- A. H. Shapiro: The Dynamics and Thermodynamics of Compressible Fluid Flow, vol. I, 1953, Ronald Press Co., N. Y., Chapt. 4.Google Scholar
- G. M. Edelmann and A. H. Shapiro: Tables for Numerical Solution of Problems in the Mechanics and Thermodynamics of Steady One-Dimensional Gas Flow Without Discontinuities, Journal of Applied Mechanics, Trans. ASME, vol. 69, 1947, pp. A-344 — A-351.Google Scholar
- A. H. Shapiro: The Dynamics and Thermodynamics of Compressible Fluid Flow, vol. I, 1953, Ronald Press Company, N. Y., pp. 159–173.Google Scholar
- J. Kaye and T. Y. Tqong: Measurements of Friction Coefficients for Supersonic Flow of Air in the Entrance Region of a Tube with and without Heat Transfer, Proceedings of the Iowa Thermodynamics Symposium, State University of Iowa, April 27—28, 1953.Google Scholar
- E. R. van Driest. Investigation of Laminar Boundary Layer in Compressible Fluids Using the Crocco Method, NACA TN 2597, Jan. 1952f.Google Scholar