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Skin-Friction and Mixed Convection from an Isothermal Rough Plate


Abstract

Presented are new correlations for skin-friction and forced convection from an isothermal plate having RMS height-of-roughness ε. Corresponding to Prandtl and Schlichting's "fully-developed roughness" region, the skin-friction correlation is independent of Reynolds number; the convection correlation is linear in Reynolds number.

Measurements taken with a new apparatus match the convection correlation within ±4% at Reynolds numbers from 4000 to 40000. The full mixed convection model extends that range to 2400<Re<95000.

Natural convection measurements made at plate angles from −90° to +90° are close to values predicted by established (smooth plate) formulas from Fujii and Imura, Churchill and Chu, and Schulenberg; and showed no significant dependence on surface roughness.

Mixing natural convection with horizontal forced flow is successfully modeled by the L2-norm for vertical and upward-facing plates and the L4-norm for downward-facing plates. With vertical flow on a vertical plate, mixing transitions between the L4-norm and the L2-norm for both upward and downward flows.

The mixed convection model is generalized to any inclination of a rectangular plate having at least one horizontal edge. Comprehensive formulas are presented along with their graphs and corresponding measurements.

Full Article

Skin-Friction and Mixed Convection from an Isothermal Rough Plate

Supplementary Data

In the temperature versus time graphs in the supplementary files, the green, blue, and black traces are the plate, (insulated) back, and ambient temperatures respectively. The upper red trace is a simulation of the plate temperature with the back and average ambient temperatures as inputs. The middle red trace is a simulation of the back temperature with the plate and ambient temperatures as inputs. The lower red line is the ambient temperature averaged over the final 3/4 of the measurement period.

The diamonds on the plate temperature trace mark the beginning and end of the measurement period, the ending temperature difference with ambient being at most half of the peak temperature difference with ambient. The simulated plate temperature is reset to the real plate temperature at the first diamond.

Underneath each temperature graph is a graph of the air velocity versus time for mixed convection runs and Raleigh number versus time for natural convection runs.

Natural Convection
natural.pdfNatural Convection at a dozen Raleigh numbers in vertical and horizontal orientations.
angles.pdfNatural Convection at angles from −90 to +90
Mixed Convection
mixed-up.pdf3mm roughness face up; horizontal forced flow
mixed-aid.pdf3mm roughness face vertical; upward forced flow
mixed-vt.pdf3mm roughness face vertical; horizontal forced flow
mixed-opp.pdf3mm roughness face vertical; downward forced flow
mixed-dn.pdf3mm roughness face down; horizontal forced flow
mixed-dnnt.pdf1mm roughness face down; horizontal forced flow
mixed-aid+84.pdf1mm roughness face down inclined +84
mixed-opp+95.pdf1mm roughness face down inclined +95
Zip of Supplementary Files
supplementary.zipZip Archive of All Files

Bibliography

  1. L. Prandtl and H. Schlichting.
    The Resistance Law for Rough Plates.
    Navy Department, the David W. Taylor Model Basin, 1934.
    Translated 1955 by P.S. Granville.

  2. Hermann Schlichting, Klaus Gersten, Egon Collaborateur. Krause, Herbert Collaborateur. Oertel, and Katherine Mayes.
    Boundary-layer theory.
    Springer, Berlin, Heidelberg, Paris, 2000. Corrected printing 2003.

  3. A. F. Mills and Xu Hang.
    On the skin friction coefficient for a fully rough flat plate.
    J. Fluids Eng,
    105(3):364–365, 1983.

  4. M. M. Pimenta, R. J. Moffat, and W. M. Kays.
    The Turbulent Boundary Layer: An Experimental Study of the Transport of Momentum and Heat with the Effect of Roughness.
    Report No. HMT-21, Thermosciences Division,
    Department of Mechanical Engineering, Stanford University, 1975.

  5. S. W. Churchill and R. Usagi.
    A general expression for the correlation of rates of transfer and other phenomena.
    AIChE Journal,
    18(6):1121–1128, 1972.

  6. HT Lin, WS Yu, and CC Chen.
    Comprehensive correlations for laminar mixed convection on vertical and horizontal flat plates.
    Wärme-und Stoffübertragung,
    25(6):353–359, 1990.

  7. Blake Lance.
    Experimental Validation Data for CFD of Steady and Transient Mixed Convection on a Vertical Flat Plate.
    All Graduate Theses and Dissertations, 2015.

  8. H. Babaee, P. Perdikaris, C. Chryssostomidis, and G. E. Karniadakis.
    Multi-fidelity modelling of mixed convection based on experimental correlations and numerical simulations.
    Journal of Fluid Mechanics,
    809:895917, 2016.

  9. 2009 ASHRAE Fundamentals Handbook (SI)
    American Society of Heating Refrigerating and Air-conditioning Engineers Inc.,
    ISBN: 978-1-931862-70-7 (I-P); 978-1-931862-71-4 (SI)
    ISSN: 1523-7222 (I-P); 1523-7230 (SI)

  10. John H. Lienhard IV and John H. Lienhard V,
    A heat transfer textbook, 4th edition, 2016 Version 2.04,
    Phlogiston Press, Cambridge, Massachusetts, US
    TJ260.L445

  11. Noor Afzal, Abu Seena, and A. Bushra.
    Turbulent flow in a machine honed rough pipe for large reynolds numbers: General roughness scaling laws.
    Journal of Hydro-environment Research,
    7(1):81–90, 2013.

  12. Karen A Flack, Michael P Schultz, Julio M Barros, and Yechan C Kim.
    Skin-friction behavior in the transitionally-rough regime.
    International Journal of Heat and Fluid Flow,
    61:21–30, 2016.
    doi: 10.1115/1.3241008

  13. Rice R.W.
    Emittance factors for infrared thermometers used for wood products.
    Wood and Fiber Science, 36:520–526, 2004.

  14. Tetsu Fujii, Hideaki Imura,
    Natural-convection heat transfer from a plate with arbitrary inclination,
    International Journal of Heat and Mass Transfer,
    Volume 15, Issue 4, April 1972, Pages 755-764,
    IN5-IN6, 765-767, ISSN 0017-9310, DOI: 10.1016/0017-9310(72)90118-4.

  15. J. R. Lloyd and W. R. Moran,
    Natural Convection Adjacent to Horizontal Surface of Various Planforms,
    J. Heat Transfer 96, 443-447 (1974),
    DOI:10.1115/1.3450224

  16. Stuart W. Churchill, Humbert H. S. Chu,
    Correlating equations for laminar and turbulent free convection from a vertical plate,
    International Journal of Heat and Mass Transfer,
    Volume 18, Issue 11, November 1975, Pages 1323-1329,
    ISSN 0017-9310,
    DOI: 10.1016/0017-9310(75)90243-4

  17. T. Schulenberg,
    Natural convection heat transfer below downward facing horizontal surfaces,
    International Journal of Heat and Mass Transfer,
    Volume 28, Issue 2, February 1985, Pages 467-477,
    ISSN 0017-9310, DOI: 10.1016/0017-9310(85)90080-8.

  18. W.M. Rohsenow, J.P. Hartnett, and Y.I. Cho.,
    Handbook of heat transfer,
    McGraw-Hill handbooks.,
    McGraw-Hill, 1998.

Copyright © 2016, 2017 Aubrey Jaffer

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