http://people.csail.mit.edu/jaffer/convect | |
Skin-Friction and Forced Convection from an Isothermal Rough Plate |
Presented are new correlations for skin-friction and forced convection from an isothermal plate having root-mean-squared height-of-roughness ε_{q}. Corresponding to Prandtl and Schlichting's "fully-developed roughness" region, the skin-friction correlation is independent of Reynolds number; the convection correlation is a linear function of Reynolds number (versus an exponent of 4/5 for turbulent convection from a smooth plate).
f_{c} = | 1
16 [log_{10}(ε/L)]^{2} |
Nu = | Re Pr^{1/3}
32 [log_{10}(ε/L)]^{2} |
Measurements taken of a plate having precisely 3 mm of roughness match the convection correlation within 3% at Reynolds numbers from 5000 to 50000. A formula addressing the transition from fully-rough turbulent to smooth turbulent convection for this bi-level test plate matches those measurements within 2% from Re=5000 to 80000.
Convection measurements of a bi-level plate having 1.03 mm of roughness are within the expected measurement uncertainties of the transition formula.
An analysis of sand-roughness finds that its relation to root-mean-squared height-of-roughness is non-linear. The resulting scale error explains discrepancies between the Prandtl and Schlichting formula, two recent papers, and the results presented here.
These new correlations and measurements imply a threefold tighter upper bound for the height of admissible-roughness than is given in Schlichting's book "Boundary-layer theory."
Skin-Friction and Forced Convection from an Isothermal Rough Plate
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 | |
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natural.pdf | Natural Convection at a dozen Raleigh numbers in vertical and horizontal orientations. |
angles.pdf | Natural Convection at angles from −90 to +90 |
Mixed Convection | |
mixed-up.pdf | 3mm roughness face up; horizontal forced flow |
mixed-aid.pdf | 3mm roughness face vertical; upward forced flow |
mixed-vt.pdf | 3mm roughness face vertical; horizontal forced flow |
mixed-opp.pdf | 3mm roughness face vertical; downward forced flow |
mixed-dn.pdf | 3mm roughness face down; horizontal forced flow |
mixed-dn.pdf | 1mm roughness face down; horizontal forced flow |
mixed-aid+84.pdf | 1mm roughness face down inclined aiding +84.5° |
mixed-opp+85.pdf | 1mm roughness face down inclined opposing +84.5° |
Zip of Supplementary Files | |
supplementary.zip | Zip Archive of All Files |
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SimRoof | ||
agj @ alum.mit.edu | Go Figure! |