For starting we want to find the optimum design of a winggrid in a given case, using the special features of a wing with winggrid:

• Rectangular spanload; Clmax = Clavg

• Spanefficiency e > 2; Definition: e is the span efficiency e=( Cdi_{elliptic_wing} /Cdi_{actual wing})
  with Cdi= Cl^2/(L*p*e) , where Cl= M*g/A*q ,
  with L = aspect_ratio, M=mass of plane, g=gravity constant, A=wing area, q=dynamic head)
  (The span efficiency used in our discussions here is always understood only to reflect influence of deflected airmass, excluding any effects of viscous drag polar (which may be included, e.g. comparing gradients in Cl^2 vs Cd polars) )

• Cutoff at designed Cl_cutoff

The tool proposed is the L/D vs speed polar of the aircraft under study. Span efficiency e, span b, wetted surface Swet, reference surface Sref and average Cfavg (friction and interference) are the parameter frame entering the generalised equation L/D vs speed polar. Below we show a few examples of such a study with L/D polar

Required inputs for this step are:

Result of this performance prediction is a list of alternatives for

plus the respective L/D polars, which are the basis for subsequent winggrid designs.

Step 2: winggrid design procedure
The span-efficiency asked for is satisfied using suitable combinations of:

• L2/L winggrid span/half total span

• Nb number of blades

The diagram below gives the span-efficiency as function of these two parameters, gven by the expression reflecting the airmass deflected with rectangular spanload, relative span winggrid L2/L, N blades:

e = (1 + L2/L * (N - 1)) * 4 / p

In general it is advisable to have 2 < Nb < 5 for reasons of structural simplicity and control of additional interference drag and friction on the blades with smaller chord than main wing.

With L2/L and Nb a further choice has to be made on the type of configuration, e.g.:

• Equal blade load (at a design Cl with corresponding individual blade angle of attack corrections)

• Equal blade angle (load distribution dependent on overlap, operation identical at all speeds)

For the choosen configuration the stagger angle(s) have to be calculated for the defined Cl-cutoff value, see equation below

DA* = (alfa –alfa0) / (1 – HS*) * HS*; (alfa-alfa0) derived from Cl_cutoff/(2*p)

Checking this expression with the  tests gives for HS*, the critical value of the coupling parameter

0.6 < HS* <0.65

In order to calculate DA* we use the Cl-cutoff value for the wing with winggrid assuming rectangular spanload (note that DA* is the angle between line of grid and zero-lift direction of blade(s).
When applying the stagger calculus for single blades as in the equal angle configuration, blades are identified as part of the grid of blades and therefore again the respective Cl_cutoff values concerne average grid values (in contrast to the local values Cl-cutoff_local=Cl_cutoff/overlap).
Load distribution calculation needs definition of the overlap to be used. As overlap only influences this distribution (and not stagger angle necessary as assumed earlier), smaller overlaps giving flatter load distribution a value in the range 0.7 < overlap < 0.9 is advised, too small overlap results in high local Cl-values on the blades.
Different loads on the grid-elements result in different stagger angles for Cl_cutoff, for equal angle configuration see the typical "banana – look", whereas for equal blade load the mean stagger angle is constant for all blades, although in general the individual blade angles differ somewhat.

Step 3: blade profiles and Betz-correction
Based on the load distribution the blade profiles are optimized and the chord of the winggrid (LE first blade to TE last blade) is adjusted to obtain identical lift gradient for main profile and winggrid.
From the Cl-value for the grid-element of the blade considered the local Cl for the blade follows:

Cl-local = Cl_grid / overlap

This local value is the basis for optimizing the blade profile (Re,range Cl, CD).
We take note here, that the profile choosen for the blade in question should always have identical zero-lift orientation as the main wings profile section.
The aerodynamic loads of blades in a well designed grid reach higher values before separation (corner stall as with turbine grids) compared to the values at separation for single wings, typical values used for design would be: Clmax_blade < 2.5, Clmax_wing < 1.5.  A grid of blades does show (at least for deviation angles less than 0.4) nearly constant flow deviation for a quite important range of overlap, e.g. 1.0 > overlap > 0.6 is corresponding to a range of the factor k of 0.96 to 0.86 (Betz). In the final adjustement of winggrid total chord, this factor is compensated by magnifying the winggrid cross-section by 1/k. This stretching assures identical lift gradient of the winggrid and the main wing profile.

2.0 Stagger angle, high speed and low speed (approach to landing) performance
As borne out by our tests, in order to operate as induced drag reduction, the WINGGRID has to have a minimum stagger angle relative to chord of main wing it is attached to. This stagger angle should as a rule be twice the angle of attack at the lowest speed (speed-limit) for full reduction of induced drag. This speed-limit separates two distinct regimes of speed:

3.0 Lift distribution and angle of attack adjustements of winglets in WINGGRID
There are basically three different designs possible regarding lift distribution (for details pls ask):

4.0 Design for minimum interference drag of winggrid
A winggrid does cause additional drag, which if not controlled will offset partially the gain made on reduction of induced drag, especially so at the high speed end of an aircrafts L/D polar. In this contribution a few propositions are listed to aid a designer, how to acually control or even nullify parts of this additional drag components.
Background:
Quantitative confirmation of this additional drag was actually the result of the high precision L/D tests of idaflieg 99 made on the first full scale testbed with winggrid. In the tested designs configuration no consideration was given to optimize this drag components and around maximum L/D of the actual polar its amount of around 20% of Cd0 was also verified by detailed drag calculations.

Drag analysis of winggrid
The winggrid exhibits the following additional drag components excluding induced drag compared to drag of the wing considered without winggrid:

• Additional profile drag of the blades due to smaller Re-number because of reduced chord. Since total chord of the blades (i.e. sum of blade chords) is due to overlap smaller than unity less than chord of main wing profile, this additional profile drag is partly compensated. In practical cases the resultant specific profile drag of a blade is about 150% of profile drag per span of main wing. This gives for example an increase of resultant profile drag for a relative span of the winggrid of 0.2 of total halfspan and with an overlap of 0.6 very near the original profile drag of the main wing profile alone.

• Interference drag due to endplates carrying the winggrid blades. If not designed well the interference drag of the different contributions such as:
  - Connection main wing to inner endplate
  - Connection of blades to inner endplate and outer endplate
  Can get quite important as the example of the testbed performance has demonstrated. On the other hand today it is well known how to minimize or even nullify this drag component successfully. 
  - Friction drag on the additional endplates and endstripes

The next paragraph is meant to enumerate and explain a few current methods for this.

Minimizing interference drag for lifting profiles ending in a plate
Interference drag for such cases is caused by development of a horseshoe-vortex at the basis of the profile. This vortex carrying away kinetic energy causes the interference drag. Its size is directly a consequence of the boundary layer thickness on the plate near the leading edge of the profile. So we have two methods dealing with this drag components:
Minimizing the boundary layer thickness at profile leading edge on the plate.

This is conveniently achieved by
a)       having a bump  in the plate before the leading edge of the profile, see for information also the bump on modern transport aircraft housing the wheels at the   
          base of the wings.
b)       Placing a NACA ventilation entry port in front of the leading edge of the profile, in order to reset the thickness of the boundary layer to near zero at the    
          profile leading edge junction with the plate
c)       Having the plate length extending in front of the leading edge of the profile as short as possible (applies to the connection of main wing with inner endplate    
          winggrid also)

Avoiding the horseshoe-vortex by adding to the profile a suitable forebody, which helps the streamlines within the boundary layer upstream to surmount the difference in total pressure to the stagnation line on the profile outside the boundary layer on the wall. As the available literature on this method shows, such forebodies inside the boundary layer thickness have to be carefully adapted to the operational characteristics of the profile and the wall boundary layer however. Control of additional friction drag of endplates. To achieve this the main considerations are two:
- Use minimal area to cover blades and main profile sections
- Use careful rounding design at entry leading edge in order to achieve reasonable yaw angle tolerance without partial separations developping.

Summary
Additional drag components of the winggrid consist essentially of three contributions.

• Additional profile drag due to lower Re-number on the blades, which have smaller chord length than the main wing profile. This is at least partly compensated by using an overlap (ratio of blade chord to distance leading edge – leading edge) smaller than about 0.7 for practical purpose.

• Interference drag due to horseshoe-vortex beeing created at the intersection of the winggrid endplates/endstripes with the main wing and with the blades. This drag if not treated can be quite important. It can however be almost nullified by using appropriate surface design near the leading edges of the profiles involved.

• Friction drag on the additional endplates and endstripes enclosing the winggrid. Here it is essential to use minimum area and insensitivity ( no separation) to limited yaw angles.

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