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Technical Brief

Mixed Boundary Layer Skin Friction and Heat Transfer With Abrupt Transition

[+] Author and Article Information
M. Q. Brewster

Department of Mechanical Science and Engineering,
University of Illinois,
Urbana, IL 61801
e-mail: brewster@illinois.edu

The assumption of constant momentum or enthalpy thickness is usually mentioned explicitly or at least implied in the integration of local shear stress or heat flux; displacement thickness is usually not mentioned explicitly, but the implication, by no discussion of mass conservation over the transition zone, seems to be that it would also be constant.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received June 16, 2014; final manuscript received August 7, 2014; published online September 16, 2014. Assoc. Editor: Zhixiong Guo.

J. Heat Transfer 136(11), 114501 (Sep 16, 2014) (5 pages) Paper No: HT-14-1407; doi: 10.1115/1.4028442 History: Received June 16, 2014; Revised August 07, 2014

In the form that is commonly published in introductory textbooks, the classical problem of skin friction and heat transfer for a mixed laminar–turbulent boundary-layer flow on a flat-plate with an abrupt transition is nonconservative in mass, momentum, and energy. By forcing continuity in momentum and enthalpy thicknesses, the textbook problem takes on the appearance of conserving momentum and energy. But, by doing so while retaining a turbulent virtual origin at the plate's leading edge, the textbook example omits necessary jumps in these quantities and violates conservation of mass, momentum, and energy in the flow. Here we modify this classical problem to satisfy conservation principles through the introduction of either concentrated mass/momentum/energy fluxes, at the top of the boundary layer and/or concentrated surface shear stress/heat fluxes at the bottom. Out of this simple analysis comes the intriguing idea of an entrainment flux or inflow at the top of the boundary layer over the transition region.

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References

Bergman, T. L., Lavine, A. S., Incropera, F. P., and DeWitt, D. P., 2011, Fundamentals of Heat and Mass Transfer, 7th ed., Wiley, New York, pp. 389–417.
Mills, A. F., 1999, Heat Transfer, 2nd ed., Prentice-Hall, Upper Saddle River, NJ, pp. 312–316.
White, F. M., 1988, Heat and Mass Transfer, Addison-Wesley, Reading, MA, pp. 325–327.
Holman, J. P., 1990, Heat Transfer, 7th ed., McGraw-Hill, New York, pp. 252–253. [PubMed] [PubMed]
Fox, R. W., and McDonald, A. T., 1992, Introduction to Fluid Mechanics, 4th ed., Wiley, New York, pp. 439–440.
Kreith, F. M., Manglik, R. M., and Bohn, M. S., 2011, Heat Transfer, 7th ed., Cengage Learning, Stamford, CT, pp. 281–282.
White, F. M., 1986, Fluid Mechanics, 2nd ed., McGraw-Hill, New York, p. 403.

Figures

Grahic Jump Location
Fig. 1

Boundary layer control volumes (0 < y < Y), local and integrated in x, for incompressible, constant-speed flow over a uniform temperature, smooth flat plate with a mixed laminar–turbulent boundary layer abrupt transition at x = xc. Local: − < x < xc+; integrated: 0 < x < L. Turbulent boundary layer behavior is extrapolated backward to a virtual origin at x = xc − xo.

Grahic Jump Location
Fig. 2

Qualitative boundary layer growth (Pr = 1) behavior of three conservative cases

Grahic Jump Location
Fig. 3

Integrated or average nondimensional wall shear stress or heat flux (Pr = 1) for transition Reynolds number 500,000

Grahic Jump Location
Fig. 4

Integrated or average nondimensional mass flux at boundary layer top for transition Reynolds number 500,000

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