Full-Inverse Design Routine (MDES)

Creation of seed surface speed distribution
Modification of surface speed distributions
Generation of new geometry

XFOIL’s Full-Inverse complex-mapping facility (MDES) takes as input a speed distribution “Qspec” specified over the entire airfoil surface, modifies it somewhat to satisfy the Lighthill constraints, and generates a new overall geometry. First a bit of the underlying theory…

The geometry and the surface velocities can both be computed from a set of complex mapping coefficients “Cn” in the form

 x + iy  =  z(w;Cn)
 u - iv  =  f(w;Cn,alpha)

where w= 0..2*pi is the independent parameter going around the airfoil. The z and f functions are rather complicated but this is not important here. The key to the full-inverse method is that the mapping coefficients Cn can be computed from a known contour angle theta(w) = arctan(dy/dz) OR from a surface speed q(w) = |u-iv|. The other quantity then follows. In summary, the operations and their commands are…

 a) Direct  problem:  theta  ->  Cn  ->  u-iv, q          (QSET)
 b) Inverse problem:  Qspec  ->  Cn  ->  x+iy, theta      (EXEC)

Creation of seed surface speed distribution

MDES performs QSET and sets Qspec = q automatically upon entry if Qspec does not exist. This Qspec is then the starting point for subsequent design operations.

This default initialization in effect makes MDES a redesign method in which the surface speed distribution of an existing airfoil is used as a starting point to generate a new speed distribution.
A ``pure’’ design code which requests the entire surface speed distribution every time is often less natural to use, since airfoil design is invariably an iterative process involving repeated analyze/fix cycles. The MDES menu is shown below.

   <cr>   Return to Top Level
   !      Redo previous command

   INIT   Re-initialize mapping
   QSET   Reset Qspec <== Q
   AQ r.. Show/select alpha(s) for Qspec
   CQ r.. Show/select  CL(s)   for Qspec

   Symm   Toggle symmetry flag
   TGAP r Set new TE gap
   TANG r Set new TE angle

   Modi   Modify Qspec
   MARK   Mark off target segment for smoothing
   SMOO   Smooth Qspec inside target segment
   FILT   Apply Hanning filter to entire Qspec
   SLOP   Toggle modified-Qspec slope matching flag

   eXec   Execute  full-inverse calculation

   Plot   Replot Qspec (line) and Q (symbols)
   VISC   Qvis overlay toggle
   REFL   Reflected Qspec overlay toggle
   SPEC   Plot mapping coefficient spectrum

   Blow   Blowup plot region
   Rese   Reset plot scale and origin
   Wind   Plot window adjust via cursor and keys

   SIZE r Change absolute plot-object size
  .ANNO   Annotate plot
   HARD   Hardcopy current plot

   PERT   Perturb one Cn and generate geometry

As described above, the initial Qspec distribution is taken from “Q”, the speed distribution corresponding to the current geometry at the last angle of attack employed in OPER. Qspec can be set back to this Q with QSET anytime.

Modification of surface speed distributions

Cursor input of modifications

Qspec can be modified to whatever is desired with the MODI command by specifying points with the screen cursor which are then splined.
The points can be entered in any order. The last point can be erased by clicking on the “Erase” button or simply typing “e” in the graphics window. The input sequence is terminated by clicking on the “Done” button or by typing “d” in the graphics window. The “Abort” button or typing “a” aborts the MODI command and returns to the MDES menu. The BLOW command can be used to enlarge regions of interest at any time by specifying opposite corners of the blowup region.

Modification endpoint blending

Normally, the modified piece of Qspec(s) is blended into the current Qspec(s) with matching values and slopes at the piece endpoints.
The slope matching can be turned on/off with the SLOP toggle command.
If slope matching is turned off, the modified piece will match only the existing value, but a slope discontinuity will be allowed.

Smoothing

Qspec can be smoothed with the SMOO command, which normally operates on the entire distribution, but can be confined to a target segment whose endpoints are selected with the MARK command. The smoothing acts to alleviate second derivatives in Qspec(s), so that with many consecutive SMOO commands Qspec(s) will approach a straight line over the target segment. If the slope-matching flag is set, the endpoint slopes are preserved.

The FILT command is an alternative smoothing procedure which acts on the Fourier coefficients of Qspec directly, and is global in its effect. It is useful for “cleaning up” the entire Qspec(s) distribution if noise is present from some geometric glitch on the airfoil surface. Also, unintended noise might be introduced into Qspec from a poor modification via the cursor.

FILT acts by multiplying the Fourier coefficients by a Hanning window filter function raised to the power of a filter parameter “F”. This tapers off the high frequencies of Qspec to varying degrees. A value of F = 0.0 gives no filtering, F = 1.0 gives the standard Hanning filter, F = 2.0 applies the Hanning filter twice, etc. The standard Hanning filter appears to be a bit too drastic, so a filter parameter of F = 0.2 is currently used. Hence, issuing FILT five times corresponds to the standard Hanning filter. The SPEC command displays the mapping coefficient spectrum at any time.

Symmetry forcing

The symmetry-forcing option (SYMM toggle) is useful when a symmetric airfoil is being designed. If active, this option zeroes out all antisymmetric (camber) Qspec changes, and doubles all symmetric (thickness) changes. This unfortunately has the annoying side effect of also doubling the numerical roundoff noise in Qspec every time a MODI operation is performed. This noise sooner or later becomes visible as high-frequency wiggles which double with each MODI command. Issuing FILT occasionally keeps this parasitic noise growth under control.

Adjustment for Lighthill constraints

The MODI, BLOW, MARK, SMOO, SLOP, FILT commands can be issued repeatedly in any order until Qspec is modified to have the desired distribution. In general, the surface speed distributions actually plotted will not exactly match what was input with the cursor, since corrections are automatically added to maintain the specified trailing edge gap and to enforce consistency with the freestream speed. These are known as the Lighthill constraints on the surface speed.

The trailing edge gap is initialized from the initial airfoil and can be changed with TGAP. To reduce the “corrupting” effect of the constraint-driven corrections, a good rule of thumb is that the Qspec distribution should be modified so as to preserve the total CL.
The CL is simply twice the area under the Qspec(s) curve (= 2 x circulation), so that this area should be preserved.

Multipoint surface speed display

A very useful feature of the MDES facility is the ability to display and modify a number of Qspec distributions corresponding to different alpha or inviscid CL values. These values are displayed and/or selected via the AQ or CQ commands. When any one Qspec distribution is modified, the result of modification is also displayed on all the other distributions. This allows rapid design at multiple operating points. When the Qspec curves correspond to specified CL values, the alpha for each curve will be adjusted after each Qspec modification so as to preserve that curve’s CL.
The resulting Qspec will therefore not match the input cursor points exactly because of this alpha correction.

Generation of new geometry

The EXEC command generates a new buffer airfoil corresponding to the current Qspec distribution. If subsequent operations on this airfoil are to be performed (SAVE, OPER, etc.), it is necessary to first generate a current airfoil from this buffer airfoil using PANE at the top level menu. This seemingly complicated sequence is necessary because the airfoil points generated by EXEC are uniformly spaced in the circle plane, which gives a rather poor point (panel node) spacing distribution on the physical airfoil. This sequence also prevents the current airfoil from being overwritten immediately when EXEC is issued.
Once the new current airfoil is generated with PANE, it can then be analyzed in OPER, modified in GDES, or whatever.

The PERT command allows manual input of the complex mapping coefficients Cn which determine the geometry. These coefficients are normally determined from Qspec(s) (this is the essence of the inverse method). The PERT command is provided simply as a means of allowing generation of geometric perturbation modes, possibly for external optimization or whatever.

The manually-changed Cn values result in changes in geometry as well as the current Qspec(s) distributions. The QSET command will restore everything to its unperturbed state.

The Full-Inverse facility is very fast, after an initialization calculation of several seconds (on a RISC workstation), it requires only a fraction of a second to generate the new buffer airfoil.