Local-friction and local-dissipation methods

To determine the physical basis of the form factor, we can compare the wetted-area drag formula (4.110) with the exact profile drag formula (4.40) based on the integral momentum equation.

Sref CDp = Swet Cf Kf (4.111)

SrefCbp = JJ^CfdSwet + Jj6*^dSwet + jj8* ё^аке (4.112)

Evidently the form factor Kf accounts for the larger local dynamic pressures via the ratio qe/qTO inside the first friction integral in (4.112), and also accounts for the remaining two surface and wake integrals which represent roughly the pressure drag. Clearly, Kf represents fairly complex flow physics, and consequently has been resistant to being reliably estimated from only potential-flow quantities via first principles. For example, one might attempt including a local dynamic pressure in the wetted area integral

but this will significantly under-predict the drag for most bodies of finite thickness.

Sato [13] has made some progress in simple profile drag estimation by employing the profile drag formula as related to the kinetic energy equation (4.49), which can be written as follows.

Noting that the dissipation coefficient cv is very insensitive to pressure gradients (much less so than Cf), we can interpret Kf as a measure of the average peu)i/pTO V3 ratio over the surface and the wake. This leads to a fairly reliable profile drag estimation formula

3

-^%dSwet (4.115)

which has the same form as (4.113), but with a local peu^ weighting rather than peu^. This still has the great simplicity of requiring only potential-flow velocities to be integrated over the surface. The ratio between cv and Cf, and the additional contribution of the wake dissipation, have all been lumped into the Cf factor, by the requirement that the formula give the correct result for the flat plate. Additional refinements can be made by splitting the integral between the laminar and turbulent portions as appropriate.

Sato [13] has shown that the profile drag predictions of formula (4.115) are reliable to within a few per­cent for flows which do not have trailing edge separation. He has also introduced modifications for wall roughness, and also for compressibility which gives good results up to weakly transonic flow.