Permitted roughness

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Given a boundary layer, what degree of roughness will turn a laminar flow turbulent, or cause increased drag?  The spreadsheet reports on the effect of four kinds of roughness or protuberance, with formulas drawn from Hoerner's Fluid Dynamic Drag.

The question addresses two different issues.  The first issue is, how smooth must a surface be in order for laminar flow to persist as long as possible?  The second issue is, in order to deal with boundary layer separation at low Rn across a curved surface, how "unsmooth" should the surface be in order to bring about turbulent boundary layer flow so that separation is delayed if not completely prevented?

Let us have an IOM fin of 100 mm chord moving through the water at 1 m/sec.  The fin as a whole has a Rn of around 90,000.

First, how smooth should this fin be, on the run, so that the boundary layer flow remains laminar?  The spreadsheet suggests that the permitted roughness on the fin surface needs to be of the order of 0.001 to 0.01 mm.  That is, somewhere between as smooth as glass and smoothly sprayed paint.  In particular, the leading edge needs to be exceptionally smooth, ten times more so than the surface near the trailing edge.  Anything rougher than this, and the BL will inevitably transition to turbulent flow.  As an aside, figure 5-1 in Hoerner suggests that when Rn is 100,000, the coefficient of surface friction (drag) for laminar flow is about 0.005, while for turbulent flow it would be about 0.007 -- an increase of about 40%.

Second, how smooth should the fin be, on the run, so that it offers minimum drag, assuming the boundary layer is turbulent?  The BL could be turbulent because the surface just couldn't achieve sufficient smoothness to prolong laminar flow, or it could be because the critical Rn has been reached, so that at some point flow transition will occur no matter how smooth the surface.  The spreadsheet suggests the permitted roughness is at most 0.11 mm.  Such a surface would be like smooth cement, galvanised metal or poorly sprayed paint.  Interestingly, this amount of roughness is "permitted" across the whole surface, because it only depends upon flow velocity, and not upon Rn (ie not upon where along the surface the flow takes place).  Also, figure 5-1 suggests that this permitted roughness will show a coefficient of surface friction that drops to 0.005 with Rn = 1,000,000 and then stays at 0.005 no matter what the increase in flow velocity and Rn.

A trip wire or turbulence stimulator (turbulator) needs to have a certain size so that, when placed in the laminar flow of a boundary layer, it causes the flow to become turbulent.  Below a minimum size, the turbulator has no effect;  above a certain size, it causes extra drag.  So let us assume that our fin is now beating and making leeway;  that is, it is operating at an angle of attack to the oncoming flow and there is the likelihood of flow separation on its upper surface.  We would like to trip the boundary layer into turbulent flow just ahead of the potential point of separation.  Let us estimate that, for our particular fin thickness and profile, a turbulator should be placed half-way along the chord.  The spreadsheet then tells us that the turbulator should be about 0.4 mm thick.

Finally, according to the spreadsheet, this turbulator size is similar to the size of a single protuberance which would achieve the same effect.  In many wind tunnel and towing tank experiments, it is easier to put nails or pins into the model so their heads trip the BL, rather than stretch a wire on the surface.

As an aside, the value of bringing on a turbulent boundary layer, and avoiding separation, is to be found in how much drag is thus avoided due to separation.  A rough guess is that significant separation increases drag by at least 100%, while forcing turbulence (and avoiding separation) increases drag by "only" 40%. Not a bad deal, I guess... Trouble is, did you need to force turbulence at all? At low Rn, on the beat, making leeway, probably. But you've just gone and increased drag on the run as well...


2022 Lester Gilbert