r
Lab. y.
Scheepsot-mt
Technische Hogescheol
Seventh International Conference on Ship Hydrodyns,].951.Subjects 2 and - Skin Friction and Turbulence Stimulation
Contribution by Professor M. Yarriagata
The coefficient of frictional resi8tance for transitional flow along a smooth flat plate are generally represented by the expression
C5 = Ct--(Ct
-Cfl) =
Ct
,where R and. R are Reynolds numbers, referred to the length
of the plate and of laminar region, respectively,
Ct is the coefficient of frictional resistance for
turbulent flow, taken at R1,,
C?t and Gj are the coefficients of frictional
re-sistancesfor turbulent and laminar flows, taken at res-pectively,
and K depends on being equa]. to R,&(Ct -Ct).
Prandtl chose a value 1,700 for K to give agreement with Blasius' and Gebers' results of resistance measurements with flat rectangular plates, which was derived from 0.L5xlO' These values correspond to a comparatively late transition to turbulence, namely, the transition curve most likely to occur when the plate itself caused little disturbance, and
moved into quite undisturbed fluid. But Goldstein obtained
later the critical Reynolds numbers up to 1.1xl0' at wind-tunnel experiments.
The results of detecting chemically the laminar arid tur-bulent extents over flat rectangular plates, made at the Ex-periment Tank of the University of Tokyo, showed clearly the serious effects of the longitudinal edges of plates and
free surface upon the flow pattern over plates. As seen
iiiè4.
sketches of Fig.1, the laminar extent is not rectangular against our expectation, and, within these ex-periinental results, it has always the shape of a horizontal wedge, consisting of two approximately straight lines, start-ing from the fore part of the plate and makstart-ing an angle ofabout 10° to horizontal. 0f course, at higher velocity or
larger draught (or depth) of a plate, a vertical line may be expected as the aftermost boundary of laminar region, from which the critical Reynolds number free from
longitu-dinal edge and free surface effects will be obtained. Fig.
2 Bhow& some examples of such vertical boundary lines,
though very irregular saw-teeth shape-' measured with another rectangular plate, which would be suposed to occur at com-paratively low Reynolds numbers probably due to the worse
shape of its leading edge.
Thus, it may be said that, besides the degree of initial turbulence of the fluid coming up to the plate and the un-avoidable disturbance of leading edge of the plate, its draught (or depth) changes considerably the value of R',
and accordingly the value of K. The former ca1cu.late,
-2-over the plate of O.14-cm draught at the velocity of
3m/s
in the sketches of Fig.l, is about 2.9xlO , which is much
higher than Blasius' and Goldstein's values quoted above, but lower than 3.xlO' obtained by Schubauer and Skramstad
in l93 as R
of flat plates placed at zero incidence inthe air flow of least initial turbulence. This endorses
indirectly thtt the value of R obtained above is affected
by the longitudinal lower edge of the plate and the free surface of water.
It may be concluded that the remarkable effects of the longitudinal edge of a plate and the free surface of water on the laminar extent over a plate increase the frictional resistance for transitional flow with the increase of the ratio of its length to draught (or depth), i.e., its aspect ratio.
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