In the previous Chapter, it was shown that the viscous solution can be run entirely independent to the inviscid solution using the Dirichlet boundary condition. The ob- jective of this Section is the use of this independent solution to create a separate routine for viscous correction based on simultaneous coupling, using an initial surface inviscid velocity distribution, calculated by panel method.
divides the body into chordwise sections and not strips, which were dependant on an initial panel mesh. On each section, the tangential chordwise velocity UI is corrected
to the viscous flow, assuming that viscous flow develops just in chordwise direction. It is assumed that crossflow velocities, compared to chordwise direction are small and, consequently, viscous effects are negligible in this direction.
If the assumption above is true, streamwise viscous boundary layer approaches the real three-dimensional viscous flow and there will not be significative differences to a three-dimensional method, such as the one developed by Milewski [48], provided that freestream velocities are small (incompressible and irrotational flow).
The initial inviscid velocity can be calculated by a three-dimensional surface panel method and its results are interpolated at nodes of sections, obtaining a continuous velocity distribution. A two-dimensional boundary layer calculation is applied to each section using the interpolated velocity. The Figure 6.2 shows the diagram of the cal- culation process.
Figure 6.2: Calculation process on VIX method
6.3.1 Interpolation Scheme
The initial inviscid velocity on surface and wake can be calculated by any surface panel method. In this work, it was used PALISUPAN code [70]. According to Turnock [70], PALISUPAN is a surface piecewise constant strength panel method that uses vortex and source singularities. For its use with a method that divides surface into chord- wise sections, it needs an interpolation scheme to transform these piecewise constant velocities into a continuous distribution of velocities on panels for boundary layer calcu-
lation. Figure 6.3 shows schematicaly the velocity interpolation over piecewise constant singularity strength distribution, original from a panel method.
Figure 6.3: The piecewise constantUi interpolated by a linear scheme
The bi-linear interpolation, presented in Appendix B, was chosen because it does not present oscillations between original points, as it would be the case of other higher order interpolation methods, such as bi-cubic.
Inviscid tangential velocities Vx, Vy and Vz in Cartesian coordinates, calculated for
each panel collocation point by PALISUPAN are interpolated and transformed into surface velocities UI, in chord and, WI in span direction.
In U direction, only, it is based the viscous flow and UI is used as a first guess
for boundary layer calculation. Later, after interaction, viscous Ue and inviscid WI
are summed again, thus obtaining the three-dimensional pressure distribution on foil surface, lift and drag forces.
6.3.1.1 Special Considerations in Velocity Interpolation Scheme
The interpolation uses the velocities calculated by panel method at the panel collo- cation point, which is in the centre of panel, according to Hess and Smith [54]. The centres surrounding a panel vertex forms a cell, as shown in Figure 6.4.
It is difficult to interpolate velocities at nodes on body edges, such as trailing, tip or root edges then, there is a special interpolation scheme.
Using the trailing edge as an example, special panels are created at the edge with zero length, denominated as “ghostcells”. Velocities that were calculated initially by panel method, are extrapolated using a linear approximation from the last existing collocation point to the centre of the ghostcell, presented in Figure 6.5.
Figure 6.4: The scheme of interpolation on nodes when values are given on collocation points
• Trailing edge: upper panels have the same velocity on upper and lower surfaces (Kutta condition). The corner values are the average between the first ghostcell on chord direction, the first ghostcell on span direction and the corner panel centre as shown in Figure 6.5.
Figure 6.5: The scheme of interpolation on edges using a ghostcell scheme
• Far field wake: ghostcells have its velocities set to the previous chordwise cell value (constant velocity);
The interpolation scheme code, made in FORTRAN 95 language, uses an object structure of nodes, panels and sections. The main parts of this code are shown in Appendix E.
6.3.2 Construction of Sectional Influence Coefficients
The source influence coefficients are calculated according to sectional geometry. It is used a two-dimensional linear source distribution. The matrix construction is depen- dant on the location of the stagnation point, which defines where upper and lower surfaces begin.
In two-dimensional methods, using surface vortex distribution, the location of the stagnation point is not a difficult task as γ, the vortex strength, changes sign at stag- nation point. In three-dimensional interpolation scheme, this is not possible because data available does not showγ, but velocitiesVx, Vy and Vz. Hence, it was developed a
FORTRAN 95 subroutine, called STAGPOINT that searches velocity distribution near leading edge of each section and locates the stagnation point at the closest chordwise
UI value to zero.
For use with membranes, STAGPOINT identifies the membrane type and places the stagnation point at the very leading edge of membrane.
The influence coefficient matrix is calculated using a routine adapted from XFOIL, here named as AIJCALC. Both subroutines AIJCALC and STAGPOINT are shown in Appendix E.
In order to guarantee two-dimensional flow, neighbouring section sources were not allowed to have influence on present section, as discussed by Dvorak et al [69] and Hufford et al [45].
6.3.3 The Coupling Scheme
The derivations made to the coupling scheme for two-dimensional flow in Chapter 5 are valid for the sectional velocity method. Then, the same equation for two-dimensional flow can be repeated here.
Ue=UI+ N+Nw
X
i=1
Gij ·mdi
Different than methods developed by Milewski [48] and Hufford et al [45], the sectional velocity method does not depend on quadrilateral panels. This makes the numerical solution simpler and gives more freedom for user to increase the number of nodes per section, improving the solution for viscous flow.
The computing time will depend on the number of streamlines and the number of nodes chosen to each of them.