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 Jonas Larsson October 17, 2000 03:34

Minimum Drag Profiles

I need to design a 2D symmetric airfoil (zero-lift) with a minimum thickness (d). Typical Mach numbers are around 0.6 and typical Reynolds numbers are around 10^5. I want to minimize drag and choord-length (weight) and have a good robustness for incidence angle variations.

If anyone has any suggestions on good profiles I'd really appreciate some input. Is there anything much better than a standard NACA 4-digit series airfoil? Free-stream tubulence levels are fairly high (~ 4%) so I'm not sure if laminar profiles are so good.

 Johan Schulten October 17, 2000 08:29

Re: Minimum Drag Profiles

It seems to me that you are looking for a Liebeck airfoil. Although most of Liebeck's work was in the application for maximum lift, he also applied his optimization theory to zero-lift minimum drag airfoils. The basic idea is to have a laminar boundary layer as thin as possible over the forward part of the airfoil and a Stratford-distribution for the pressure recovery by the turbulent boundary layer over the aft part. The best reference I have is "On the design of subsonic airfoils for high lift", AIAA-paper 76-406, 1976. It contains an astonishing example of a 53% thick strut section with a drag coefficient of only 0.0077 at Re = 10**7.

The same airfoil can also be found in A.M.O. Smith "High-Lift Aerodynamics", J. Aircraft, Vol.12, No.6, June 1975, pp.501-530.

 Jonas Larsson October 18, 2000 03:12

Re: Minimum Drag Profiles, Non-lifting Struts

Many thanks for the feedback! I looked a bit at these "laminar" airfoils. However, I'm not sure if this is the best solution for non-lifting airfoils - I want a design which is as short as possible and which is fairly robust against incidence angle variations and variations in inlet conditions. When you try to keep laminar boundary layers as long as possible you get a long "nose" and a fairly sharp pressure-gradient at the aft part of the airfoil. I'm afraid that this will make the design less robust. Do you agree? Also, I'm not sure if these "laminar" designs are applicable to my type of application, where the freestream turbulence level is high (up to 5%).

Eppler has a few recommended airfoils for non-lifting struts in his book (E862, E863, ...) that look interesting - the profiles are quite similar to the NACA0022 profile that was my first shot. Epplers warns that these profiles are sensitive to incidence angle varitaions though.

Any other ideas? This must be a common problem for people designing wing-struts etc. on old aircrafts. The problem is that there is too much information to sort out what's relevant.

 Johan Schulten October 18, 2000 05:56

Re: Minimum Drag Profiles, Non-lifting Struts

Did you really look at the specific strut-sections? The example I mentioned has its maximum thickness at about 40% chord, which is not exceptionally far aft.

Indeed, a Stratford pressure recovery is rather steep just beyond the transition point but rapidly flattens out towards the trailing edge. The Stratford distribution represents the maximum pressure recovery possible with a turbulent boundary layer from given initial conditions, momentum thickness in particular.

Liebeck also gives an example of the thickest Joukowski airfoil with fully attached flow. That airfoil has a maximum thickness of "only" 31% at 27% chord. In view of the large leading edge radius, I expect a very smooth response to angle of attack fluctuations.

If you are going to apply some sort of CFD optimization, the Liebeck methodology might still give you a good starting configuration.

 Jonas Larsson October 18, 2000 08:57

Re: Minimum Drag Profiles, Non-lifting Struts

I managed to get the second reference (Smith). However, my library didn't have Liebecks AIAA paper from 76 easily avalable so I haven't seen that yet. I might have drawn too early conclusions after seeing a few of these "laminar" airfoils. The strut examplified in the reference is indeed very thick.

Let me describe my concerns a bit better:

This type of design requires that you have very good control over where the transition occurs (at least that is my guess). If transition occurs later than expected the start of the Stratford distribution will surely lead to a laminar separation and large losses. If transition occurs too early the turbulent boundary layer will be less "fresh" and might also separate at the start of the Stratford distribution. The last type of separation is probably less of a problem than the first "laminar" separation. The transition location can be affected by free-stream turbulence levels and incidence angle varitaions. In my application the Re-number is also one or two orders of magnitude lower than for the strut examplified in the references - do you think that it would be possible to design a similar strut for a Re of around 10^5 (might be difficult I would guess). If I were to use this type of airfoil I would like to be sure that the boundary layer is really turbulent before starting the steep Stratford distribution. The worst thing that could happen is an unsteady moving transition point around the start of the Stratford distr. - would lead to periodic shedding and all kinds of problems.

 Johan Schulten October 18, 2000 09:36

Re: Minimum Drag Profiles, Non-lifting Struts

For a Reynolds number in the order of 10^5 it will be hard to get transition on a smooth airfoil otherwise than via a laminar separation bubble. On the other hand, free stream turbulence will promote an earlier transition than in an ideal, steady flow.

You may also consider Prandtl's famous trick of tripping the laminar boundary well upstream of its natural separation point. Your Reynolds number seems still to be high enough for the resulting turbulent boundary layer to remain attached all along the rest of a reasonably thick strut-airfoil.

 Jonas Larsson October 18, 2000 10:12

Re: Minimum Drag Profiles, Non-lifting Struts

I can just imagine what the critical design review team will say if my struts have trip-wires and a Stratford pressure distrubution following that - will surely be a novel design that needs some explanation and convincing arguments! Do you forsee any stability problems with this type of airfoil (I'm thinking of moving transition/separation points causing shedding etc.)?

I think I want to avoid a laminar separation bubble.

 John C. Chien October 18, 2000 23:02

Re: Minimum Drag Profiles

(1). Based on the structure requirement, determine a circle (or cylinder) diameter. The flow over the cylinder is independent of the flow direction. (2). Set a trailing edge point at one diameter aft the original cylinder. This will make the max-thickness ratio of 50% at 25% length (chord) location. (3). create circular arcs from the cylinder to the trailing edge point,to make the final form something like a tear-drop shape. The final shape with larger cross-sectional area will have a factor of over 1.5 stronger. (4). Test or calculate the flow at several angle of attack, to determine the range of acceptable flow angles. (5). Further optimize the shape if necessary. (6). 3-D low aspect ratio case can create complex 3-D end-wall secondary flow interaction and loss.

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