MIAReX and numerical optimization


   Matlab Optimization toolbox is a very powerful set of numerical optimization routines. The output of Miarex calculation was redirected to those optimization routines, for precise design cases.

Here are some results.

Optimal taper location on sailplane wing

Presentation of the problem

The problem to be optimized was the following :

  • High aspect ration wing (around AR=21, typical for sailplane).
  • Two taper over the wing (Taper1=0.9, Taper2=0.55).
  • Degree of freedom position of the taper. 0 is root, 1 is tip.
  • No washout.

An optimal value has to be optimized. At first, only simple condition were set :

  1. Maximize L/D ratio.
  2. Minimize sink.
Then the optimum taper position was searched, for different operating point (CL).
Nota : I intend to work on a more general optimization function, as a sum of different operating point over the speed polar.
Illustration of the degree of freedom

Click !

The optimization process was repeated for two cases, with different Reynolds numbers and airfoils :
  1. Full size sailplane case (Re = 1 100 000 at root)
  2. Model sailplane case (Re = 150 000 at root)

The interest of the optimization is to see which contribution to drag is predominant in directing the solution : Airfoil drag or Induced drag.

The influence of the degree of freedom is the following (see the illustration):

  1. When the taper location goes close to the root (x=0)
    • The AR is greater.
    • Since the wing is more tapered the airfoils works with smaller Re number over the wing
  2. When the taper location goes close to the tip (x=1)
    • The AR is smaller.
    • Since the wing is more "squared" the airfoils works with higher Re number over the wing
Some results

Here are displayed some result of the optimization process.
   First the optimal taper location for the 4 cases (model or full size, L/D or sink optimization) are plotted according to CL.
   Some conclusion can be drawn from those plots :
  • The higher the CL, the closer from the root (x=0) the optimal taper position.
  • There is a "shift" of optimal taper position between high CL and small CL, whatever optimization condition (L/D or sink).
  • The "shift" occurs at lower CL for L/D condition than for sink condition.
    Indeed, the higher the CL, the greater the induced drag : higher AR allowed by x close to 0 helps !.
  • For the model, the contribution of the airfoil drag is higher. Then optimal wing has a rather large chord.

Click !
   Then the same result about optimal taper location are plotted according to a non-dimensionnal version of the speed polar. This allow to see were on the speed polar we need a precise value for taper location.
   Then the relative importance of each area is quite changed.

   We can see that the statistical value for taper location found among existing sailplane (between x=0.6 and 0.7) is an optimum for cruising part of the polar.

Click !
So, what shall we think about all this ?
    There is a compromise to be found, between high speed and low speed, as usual. But this numerical work does not help much in choosing the accurate location for taper. Some work should be done on the optimization function.

    But anyway, this was fun and interesting to investigate this !

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