Formulae for local skin-friction coefficient and drag

Although it is not valid in the outer part of the boundary layer, Eqn (7.117) can be used to obtain the following more accurate semi-empirical formulae for the local skin-friction coefficient and the corresponding drag coefficient for turbulent bound­ary layers over flat plates.

cf =r^- = (2 log10 Rex – 0.65)-2’3 jPUSo

_ Df 0.455

m-PUlaBL (log10 Ref)25*

where В and L are the breadth and length of the flat plate. The Prandtl-Schlichting formula (7.122) is more accurate than Eqn (7.88) when Rel > Ю7.

Effects of wall roughness

Turbulent boundary layers, especially at high Reynolds numbers, are very sensitive to wall roughness. This is because any roughness element that protrudes through the viscous sub-layer will modify the law of the wall. The effect of wall roughness on the boundary layer depends on the size, shape and spacing of the elements. To bring a semblance of order Nikuradze matched each ‘type’ of roughness against an equivalent sand-grain roughness having roughness of height, ks. Three regimes of wall roughness, corresponding to the three regions of the near-wall region, can be defined as follows:

Hydraulically smooth If ks Vt/v < 5 the roughness elements lie wholly within the viscous sublayer, the roughness therefore has no effect on the velocity profile or on the value of skin friction or drag.

Completely rough If ks V*jv > 50 the roughness elements protrude into the region of fully developed turbulence. This has the effect of displacing the logarithmic profile downwards, i. e. reducing the value of Cj in Eqn (7.117). In such cases the local skin-friction and drag coefficients are independent of Reynolds number and are given by

Transitional roughness If 5 < ksV, ju < 50 the effect of roughness is more complex and the local skin-friction and drag coefficients depend both on Reynolds number and relative roughness, ks/6.

The relative roughness plainly varies along the surface. But the viscous sub-layer increases slowly and, although its maximum thickness is located at the trailing edge, the trailing-edge value is representative of most of the rest of the surface. The degree of roughness that is considered admissible in engineering practice is one for which the surface remains hydraulically smooth throughout, i. e. the roughness elements remain within the viscous sub-layer all the way to the trailing edge. Thus

In the case of a flat plate it is found that Eqn (7.125) is approximately equivalent to

kadm ОСOC(7.126) * oo лЄь

Thus for plates of similar length the admissible roughness diminishes with increasing Rbl – In the case of ships’ hulls admissible roughness ranges from 7 pm (large fast ships) to 20 pm (small slow ships); such values are utterly impossible to achieve in practice, and it is always neccessary to allow for a considerable increase in drag due to roughness. For aircraft admissible roughness ranges from 10 pm to 25 pm and that is just about attainable in practice. Model aircraft and compressor blades require the same order of admissible roughness and hydraulically smooth surfaces can be obtained without undue difficulty. At the other extreme there are steam-turbine blades that combine a small chord (L) with a fairly high Reynolds number (5 x 106) owing to the high velocities involved and to the comparatively high pres­sures. In this cases admissible roughness values are consequently very small, ranging from 0.2 pm to 2 pm. This degree of smoothness can barely be achieved on newly manufactured blades and certainly the admissible roughness would be exceeded after a period of operation owing to corrosion and the formation of scaling.

The description of the aerodynamic effects of surface roughness given above has been in terms of equivalent sand-grain roughness. It is important to remember that the aerodynamic effects of a particular type of roughness may differ greatly from that of sand-grain roughness of the same size. It is even possible (see Section 8.5.3) for special forms of wall ‘roughness’, such as riblets, to lead to a reduction in drag.[43]