Square root wing loading conversion method

[Andrew Ouellet]

In many of Johnson's flight test reports, he refers to something like 'the square root of the wing loading' method to unify glide polars taken from different gliders at different wing loadings down to a common wing loading in order to be effectively compared. Can someone explain what this is and how to do this?

I have a strong engineering background so no need to mince words.

Thanks!

[Bruce Hoult[_2_]]

On 2014-08-22 06:08:12 +0000, Andrew Ouellet said:

> In many of Johnson's flight test reports, he refers to something like
> 'the square root of the wing loading' method to unify glide polars
> taken from different gliders at different wing loadings down to a
> common wing loading in order to be effectively compared. Can someone
> explain what this is and how to do this?
>
> I have a strong engineering background so no need to mince words.
> Thanks!

Which part of that description do you not understand?

I'll assume you understand "square root". So, "wing loading" is the
total mass of the aircraft and occupents divided by the area of the
wings. Generally you'll find it in kg/m^2, in which case a modern
racing glider can generally be adjusted in range typically something
like 30 - 60, with older wood/fabric gliders having lower numbers.

Work out the square root of it, and use it to normalise all horizontal
and vertical speeds in the polar diagram.

[[email protected]]

On Thursday, August 21, 2014 11:08:12 PM UTC-7, Andrew Ouellet wrote:
> In many of Johnson's flight test reports, he refers to something like 'the square root of the wing loading' method to unify glide polars taken from different gliders at different wing loadings down to a common wing loading in order to be effectively compared. Can someone explain what this is and how to do this?
>
>
>
> I have a strong engineering background so no need to mince words.
>
> Thanks!

First you need to have the polar in the form of a quadratic: ax^2+bx+c. If you don't have that you can take any graphical depiction of the polar and use any standard mathematical technique to pick three points and fit a quadratic (I use matrix algebra for three equations with three unknowns). You need to have the reference weight for the polar: W. To get the polar for any other weight (W') you need three new coefficients for a new quadratic representation of the polar: a'x^2+b'x+c', where a'=a/(W'/W)^0.5, b'=b and c'=c*(W'/W)^0.5. It's that simple. (Pardon the Excel notation for multiplication and exponents).

There are a bunch of aerodynamic reasons why this is true that as to do with the relationship between weight and lift (need to be equal), the relationship between lift and lift coefficient and velocity and the relationship between induced drag and lift coefficient and velocity. You don't need to derive all the aerodynamic formulae to translate the polar so I won't belabor it.

9B

[Andrew Ouellet]

Thanks guys for your replies.

[Andrew[_14_]]

On Friday, August 22, 2014 2:08:12 AM UTC-4, Andrew Ouellet wrote:
> In many of Johnson's flight test reports, he refers to something like 'the square root of the wing loading' method to unify glide polars taken from different gliders at different wing loadings down to a common wing loading in order to be effectively compared. Can someone explain what this is and how to do this?
>
>
>
> I have a strong engineering background so no need to mince words.
>
>
>
> Thanks!

It all comes from the lift equation which can be used to determine what airspeed (and sink rate) is needed to achieve the same glide ratio following a weight change. When I wrote this I forgot your original question was about wing loading. So, pretty much what you'd do is figure out what weight each glider needs to be to achieve the same wing loading and correct each of their polars to that weight and plot them together.

Here's the write up explaining how the weight correction can be derived...
https://dl.dropboxusercontent.com/u/1695013/glide_polar_weight_change.pdf

[Eurofighter]

"the theory says each point on
the dry polar should move up and to the right by a factor
equal to the square root of the sailplane’s ballasted weight
divided by its dry weight"

This is taken from one of Dick Johnson's flight test evaluations. It will not unify glider polars, but will provide the new L/D and speed to fly in order to obtain that L/D

"In this case the ballasted test gross
weight was about 957 lbs., compared to a 707 lb. dry value,
which results in a √957/707 = 1.163 ballasting factor. The measured sink rates with ballast fall fairly close to those computed by multiplying the unballasted values by the theoretical 1.163 factor. The maximum L/D still appears to be about 40.5, but it occurs at about 63 knots. This is almost 4 knots higher airspeed than one would normally expect by multiplying the dry polar’s 51-knot best L/D airspeed by the 1.163 ballasting factor."

This is just an example, but it demonstrates that the theory does not always apply in real life.

[2G]

On Friday, August 22, 2014 2:22:52 AM UTC-7, Bruce Hoult wrote:
> On 2014-08-22 06:08:12 +0000, Andrew Ouellet said:
>
>
>
> > In many of Johnson's flight test reports, he refers to something like
>
> > 'the square root of the wing loading' method to unify glide polars
>
> > taken from different gliders at different wing loadings down to a
>
> > common wing loading in order to be effectively compared. Can someone
>
> > explain what this is and how to do this?
>
> >
>
> > I have a strong engineering background so no need to mince words.
>
> > Thanks!
>
>
>
> Which part of that description do you not understand?
>
>
>
> I'll assume you understand "square root". So, "wing loading" is the
>
> total mass of the aircraft and occupents divided by the area of the
>
> wings. Generally you'll find it in kg/m^2, in which case a modern
>
> racing glider can generally be adjusted in range typically something
>
> like 30 - 60, with older wood/fabric gliders having lower numbers.
>
>
>
> Work out the square root of it, and use it to normalise all horizontal
>
> and vertical speeds in the polar diagram.

I found your reply to be condescending and did not explain how to "normalize" whereas Andrew's reply did.
Tom

[Andrew[_14_]]

On Friday, August 22, 2014 11:22:55 AM UTC-4, Andrew wrote:
> On Friday, August 22, 2014 2:08:12 AM UTC-4, Andrew Ouellet wrote:
>
> > In many of Johnson's flight test reports, he refers to something like 'the square root of the wing loading' method to unify glide polars taken from different gliders at different wing loadings down to a common wing loading in order to be effectively compared. Can someone explain what this is and how to do this?
>
> >
>
> >
>
> >
>
> > I have a strong engineering background so no need to mince words.
>
> >
>
> >
>
> >
>
> > Thanks!
>
>
>
> It all comes from the lift equation which can be used to determine what airspeed (and sink rate) is needed to achieve the same glide ratio following a weight change. When I wrote this I forgot your original question was about wing loading. So, pretty much what you'd do is figure out what weight each glider needs to be to achieve the same wing loading and correct each of their polars to that weight and plot them together.
>
>
>
> Here's the write up explaining how the weight correction can be derived....
>
> https://dl.dropboxusercontent.com/u/1695013/glide_polar_weight_change.pdf

After thinking about it again I realized the final formula I gave doesn't necessarily have to use just weight or airspeed. It's possible to use weight or wing loading to suit ones goal. Also, one could use sink rates instead of airspeed. This is because the final formula deals in ratios of speeds and the square root of the weight or wing loading ratio.

V could be the airspeed or sink rate, and W could be the weight or wing loading...

V_light / V_heavy = sqrt( W_light / W_heavy )

[[email protected]]

What about total weight of glider? For example, in dolphin-style flight.

50kg/m2 for Diana2 and ASW27 doesn't mean finaly the same L/D

Regards
Andrzej

[[email protected]]

On Tuesday, August 26, 2014 12:50:01 PM UTC-4, wrote:
> What about total weight of glider? For example, in dolphin-style flight.
>
>
>
> 50kg/m2 for Diana2 and ASW27 doesn't mean finaly the same L/D
>
>
>
> Regards
>
> Andrzej

L/D is driven by geometry and any differences between the 2 gliders described would be determined by that.
There is a small change that does not follow the classic quadratic due to the effect of Reynolds number. Many(most?) of the airfoils in use have lower Cd as Reynolds number increases, other variables being constant. Most analysis ignores this because it is very hard to measure. It is reasonable to expect that they may respond differently, especially given the smaller chord of the Diana 2 wing.
Just to confuse things a bit more.
UH

[[email protected]]

The other issue with the original question is that normalizing for wing loading is a theoretical comparison that gives a relative sense for how the aerodynamic designs perform, but most gliders have different min/max wing loading ranges so in the real world you might not end up apples to apples in terms of wing loading anyway. Also, the different glider designs will reflect optimizing for a different wing loading ranges. Load a V2 up to the min wing loading of a -27 and it's a very different comparison than both gliders with a 175 lb pilot dry.

Kind of depends what you intend to do with the analysis.

9B