Fig. 6: Lift vs. drag polars for the MH 42, with different sag factors applied, at two Reynolds numbers. The dotted line corresponds to a turbulator at 25% chord, placed on the upper surface of the original (0% sag) MH 42 airfoil.

Case 1: Re=100'000, angle of attack=3° (Cl=0.55)

• The cd-cl polar for the true airfoil shape (0% sag) shows the influence of a laminar separation bubble at medium lift coefficients. This bubble is also present at lower cl values, but it is quite thin, and thus causes only a small amount of additional drag.
• When the sag factor is increased to 20%, the additional drag due to the separation bubble is decreased and the transition moves slightly forward. This is caused by the small disturbances introduced by the discontinuity between the leading edge box and the covered area and also by the slightly flattened shape in the covered area.
• When the sag factor is increased to 40%, the drag is very similar to the 20% case, the slight improvements in the low cl range probably are the result of a more concave pressure distribution due to the modified airfoil shape.
• For a 60% sag factor, which occurs on the center line between two ribs, the drag at low cl values is decreased still more dramatically, but the stronger pressure gradient leads to a laminar separation close to the leading edge box. This results in a separation bubble and an early transition, which both cause additional drag at medium cl levels.

For the case of a medium lift coefficient of 0.55 at a Reynolds number of 100'000 the junction between D-nose and the covered area does not introduce enough disturbances to act as an efficient turbulator. The main effects of the sag between the ribs seem to be a forward shift and a thinning of the laminar separation bubble, which has a relatively small impact on the drag coefficient. Experimental results in [30] also show a drag reduction between the ribs, but the effect is much stronger there, despite the smaller sagging between the ribs.

Fig. 7: Location of separation and transition for the MH 42, with different sag factors. The lift coefficient is approximately 0.55.

Combining the two dimensional results into a three dimensional view shows the complex separation bubble more clearly (figure 8).

Fig. 8: Sketch of the bubble structure developing on a covered rib structure at low Reynolds numbers.

Despite the fact, that the laminar separation bubble moves by nearly 20% of the chord length, the variation of the drag coefficient between two ribs is relatively small. The drag of the true shape (0% sag) is decreasing, when we move away from the rib. This is because the bubble moves forward and gets thinner due to the slight disturbances introduced at the end of the D-box. When we approach the center between two ribs, the bubble moves still further forward, but the drag increases. This is caused by the substantially longer length of turbulent flow, which adds more to the drag than the reduction of the bubble height. Also the pressure distribution shows a more concave pressure raise due to the flatter surface, which may contribute to the bubble height.

At higher lift coefficients, the polar for the large sag factor of 60% shows a drag increase, which is the result of a larger, further forward shifted, separation bubble due to the steeper pressure gradient. All the other polars show similar drag values as the one with a turbulator at 25% chord.

Case 2: Re=100'000, angle of attack=-2° (Cl=0.05)

When the angle of attack is reduced, the separation bubble moves to the rear part of the airfoil (figure 9). Increasing the sag factor seems to have a beneficial effect on laminar separation, which does even vanish for sag factors above 20%. For the 40% case, the thick, laminar boundary layer is close to separation, when it arrives at the trailing edge.

Fig. 9: Location of separation and transition for the MH 42, with different sag factors. The lift coefficient is close to zero.

The overall drag is reduced for all sag factors, most noticeable for the 60% case. Here the concave pressure distribution seems to be responsible for the rather thin, laminar boundary layer, which extends to the trailing edge. This is also supported by the fact, that the drag is considerably lower that the fully turbulent case (turbulator at 25% chord).

Fig. 10: Polars of the MH 42 for the true shape (0% sag) and for the covered rib structure, integrated along the span (compare with figure 1).

Cases at Re=200'000

At higher Reynolds numbers, the original airfoil (0% sag) shows only a very small laminar separation bubble. The dependencies between drag and sag are more straightforward than in the Re=100'000 case. At medium and higher lift coefficients, an increase of the sag factor creates a steeper, more concave pressure distribution on the covered panel, which also increases the height of the separation bubble and thus its drag. However, when compared against the turbulent case (T.U. = 25%, 0% sag), the drag of all airfoils is lower, except for a small region at higher lift coefficients, where the 60% sag airfoil develops some additional drag.

Fig. 11: Location of separation and transition for the MH 42, with different sag factors. The lift coefficient is approximately 0.55.

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