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Later in this thesis, different rotor tip designs are explored with the HMB Navier-Stokes code, and while steady results were obtained at high pitch angles similar difficulties of convergence were eventually encountered as the stall was reached, and it was found difficult to settle on final values of CT and CQ. Attention then turned to the prospect of running high-pitch (tail) rotor cases as unsteady solutions, thus accepting any flow separation and vortex shedding that might occur (perhaps ultimately exceeding the bounds of RANS applications). It was therefore of interest to take a similar approach here for the 2D aerofoil case, and while unsteady ramp and oscillatory cases have been modelled with HMB in the past, eg Spentzos,265 _{it is unusual}

to find unsteady solutions for aerofoils at a steady high pitch near the separation boundary. Experience with model rotors, has also revealed a tendency for unsteady torsion loads and control moments to grow rapidly as the stall is approached in addition to the expected increase in nose-down mean pitching moments. Such effects would, of course be automatically included in an unsteady N-S simulation of forward flight, offering to capture both unsteady shock movement and shock-boundary layer interactions on the advancing blade and unsteady separation and vortex shedding from the retreating blade. On a tail rotor in low-speed flight, many disturbances, such as gyroscopic flapping and pitch-flap coupling, and rotor-fin and main rotor-tail rotor interactions could exacerbate the stall, and even in a steady hover the stalled flow will not leave the blade in an ordered fashion.

The unsteady cases were run at 8, 9, 10, 11 and 12 degrees, using a GHMB grid of similar size (104,000 pts) to that used for the steady cases described above. However, for the unsteady cases, the aerofoil was set at the desired incidence angle in the grid and the flow angle was set to zero. Various non-dimensional time-steps were tried with .002 being finally chosen and the simulations were run to 20.0 (chord lengths through the fluid) over a period of typically 2.5 days, per case, on a P4 3.2 GHz 64-bit processor. All cases employed the modified k-ω

3.1 Aerofoils 3 VALIDATION

Figure 120: Drag predictions from an unsteady HMB 2D simulation of NACA 0012 at high incidence, M=0.5 and Re=1.0 million

(3002) turbulence model. The Mach number was 0.5 and a Reynolds number of 1.0 million was chosen, to be consistent with the above steady cases and compatible with the model tail rotor simulations to follow.

As seen from visualisation of the results, separation first develops at the trailing edge and gradually moves forwards along the upper surface, until it meets the adverse gradient at the foot of the shock (near the leading edge) which causes it to suddenly separate and shed vorticity into the stream.

The unsteady lift response for the incidences considered is shown in Figure 119 and it can be seen that, as expected, the unsteadiness increases with increasing incidence, with the trends showing some signs of settling to a repeatable pattern, although the total time was not extended to fully confirm this expectation. Nonetheless, these results showed that the peak unsteady values far exceeded the initial steady values. Perhaps because of the modified k-ω(3002) model used, or perhaps due to the RANS methods itself, results for the lower incidences showed less unsteadiness than perhaps might have been expected.

Results for the unsteady drag, Figure 121, where the drag coefficient at the higher angles was at already very high values in steady flow, showed much larger values at the peaks and only values similar to the initial steady value in the troughs. As expected, these large fluctuations are caused by changes in pressure drag due to fluctuating separation, shock movement and vortex shedding. In contract the skin friction term is small and changes only slightly in sympathy with the incidence and length of attached boundary layer.

While the steady pitching moment, discussed earlier, had shown a gradually increasing nose-up trend with incidence prior to stall (due to shock formation near the leading edge at this mid-Mach number), with a reversal of this trend at stall, the unsteady results revealed large fluctuations in pitching moment (about the 1/4 chord) which reached values in the region of -0.15 due to aft movement of the centre of pressure arising from separation, recovering to -0.03 to -0.05 during re-attachment.

The steady state validations discussed earlier have demonstrated good accuracy for HMB predictions of lift, drag and pitching moment at incidence angles ranging from those for attached flow conditions to those near the stall. However, at the stall, strong vortex shedding is likely to occur which demands the use of the unsteady solver, and this then reveals large fluctuating force and moments on the aerofoil. HMB may therefore be used with confidence to predict the flow around 2D aerofoils, and the next step is to validate the code for 3D problem of the hovering rotor.

3.1 Aerofoils 3 VALIDATION

Figure 121: Pithcing moment predictions from an unsteady HMB 2D simulation of NACA 0012 at high incidence, M=0.5 and Re=1.0 million