3.2
Grid Sensitivity Studies
Ahead of rotor computations, a 3D Euler solution was also used to explore the performance of an unswept wing of AR=5 with K¨uchemann-like tips. The grid was progressively refined from about 250,000 to near 3,500,000 points, until close agreement was obtained in comparison to the expected lift-curve slope and induced drag from lifting-line theory. This preliminary work not only gave confidence in the capability of HMB to predict the correct (inviscid) performance (from surface pressure integrations), but also helped to develop a feel for 3D grid requirements. These studies are reported in Appendix B.
Building on the aerofoil and wing studies, several Euler grids were generated for the example model tail rotor blade. Initial grid size was about 1.5 million points, but with further refinements to improve grid quality and obtain better resolution of the flow, the grid size gradually increased to approximately 2.5 million points. Some finer grids were used in the search for grid independency, and some coarser grids were also explored to better understand the sensitivity of the solution to grid spacing. This grid dependency study is reported in this section of the thesis.
While the model tail rotor has relatively short blade (R/c=6.402), main rotors have much longer blades (13<R/c<20), and consequently require a greater number of cells to populate the computational domain. A case in point is that of the Sea King which has a main rotor blade with R/c=20.4, and requires at least 3 million points. Some preliminary cases were reported in a seminar, Brocklehurst,57where the proximity of the
tip vortices to the preceding blade was noted. As the work 8 progressed, 5-6 million points were commonly
used for Euler hover solutions in an attempt to resolve the finer details of the flowfield arising from various design features, and a fine grid of just over 10 million points was used to show that the results were essentially grid independent.
Navier-Stokes grids were later developed, initially for 2D aerofoils, by the addition of points within the boundary layer region with exponential refinement, and a first cell height of about .00001 to obtain y+ near
unity. The quality of these grids was gradually refined as experience was gained (mainly using ICEM, but GHMB was also used for aerofoils). The 2D aerofoil profile drag results were scrutinised by comparing to test data and other prediction methods. Good agreement was reached, as shown by the validation results presented in the previous Section.
Initial attempts to run Navier-Stokes cases for hovering rotors were hampered by the need to economise on grid size which resulted in severe stretching of the grid and therefore undesirably large cells outside the boundary layer and in the blocks surrounding the blades. Eventually, as computational constraints became less of a problem, and the necessary experience and insight was gained to allow the generation of rotor grids which had a smooth transitions from the boundary layer at the surface, through the region around the blade, and out towards the far field. An adequate number of cells are also required to resolve the wake and capture the flow features around the root and tip of the blades. Moving the block-boundaries (and their support curves) nearer to the blade helped greatly in this respect, as did an extra block downstream of the trailing edge to control the spacing. These refinements for a viscous rotor grid increased the number of points from about 2.5 million up to about 8 million points for a blade-quadrant of the 4-blade model tail rotor example, as used later in this thesis.
Up to 16 million points were used to obtain a high-fidelity hover Navier-Stokes solution for the 2-blade model rotor (R/c=13.7) of Tung, Pucci, Caradonna and Morse,298which was computed recently and is presented in Appendix C.
The following sub-section describes the results of varying the grid size for Euler solutions for the example model tail rotor. The grid size relates to a single blade segment with periodic boundaries faces.
8
In parallel with this research project, several main and tail rotor cases relating to EH101, Lynx and Sea King were run by the author at Westland Helicopters Ltd (WHL). This background provided additional experience and gave confidence in generating good quality grids to obtain meaningful solutions to practical problems.
3.2 Grid Sensitivity Studies 3 VALIDATION
3.2.1 Model Tail Rotor - Hover - Euler
Most of the Euler hover simulations presented in the thesis have used a ’standard’ grid size of about 2.5 million cells for the tail rotor cases. This choice was based on experience gained through preliminary work for this thesis (as noted above), together with tail rotor and later main rotor design studies at WHL. A grid of this size gives a reasonable turn-around time (approx 24 hrs on 8 x 2.8GHz cpu’s), and provides sufficient resolution of the tip vortices to give good performance predictions. The number of grid points may also vary slightly depending on the blocking scheme necessary to accommodate the tip shape.
To ensure that the results obtained were reasonably independent of grid size, a series of different grid densities were run for the datum, rectangular (square-cut) tail rotor blade. Use of a coarse grid will clearly minimise runtime, but could lack the resolution required to yield reliable performance estimates. It is expected that a finer grid will better resolve and preserve the tip vortices and other weaker features of the wake, such as the inboard vortex sheet and the root vortex system, all of which may affect the blade loading and influence rotor performance. However, the requirement to use a really fine grid is perhaps not as necessary for an Euler solution as it would be for a Navier-Stokes case where the aim would be to simulate more details of the physics, and the grid must match the requirements of the type of turbulence model used.
The ’standard’ case quoted here comprises 2,875,680 points, and the geometry is that of the datum rect- angular (square-cut) tail rotor blade with no twist and NACA0012 aerofoil. The radius of the cylindrical hub was 0.25 chords for this particular case (TRB-000-00). All cases are compared with a coning appropriate to the model rotor tests and the tip Mach number was 0.448.
To check on grid sensitivity a coarse grid of about 1.0 million points and a fine grid of 8.5 million points were generated. Subsequently, further cases of 0.75 million and 8.0 million points were also added in order to confirm the various trends seen in the results.
In modifying the ICEM grids, the node spacing was re-adjusted in each case to give a smooth variation of grid points in regions of greatest interest, and the spacings were matched as closely as possible at block boundaries. In all cases an intermediate blocking plane was used at 0.75R below the rotor to give fine control of the grid spacing just below the blocks attached to the blades. For these Euler solutions, the blade-attached blocks started about 1-chord above and below the blade surface, whereas for a Navier-Stokes grid this would be reduced to about 0.25 chords, or less, to avoid stretching.
The results for the ’coarser’, ’coarse’, ’standard’, ’fine’ and ’finer’ grids are compared at pitch angles of 4, 6 and 10 degrees impressed pitch in Figures 122 to 126, for grid sizes of 746,372, 1,041,788, 2,875,680, 7,900,456 and 8,479,876 respectively, where the standard size grid of 2,875,680 points is that used for TRB-000-00 for the model rotor twist study, as described later in this validation section. All grids contained 170 blocks.
Figure 122 shows that the thrust for a given pitch is not very sensitive to changes in the grid size, provided of course that the grid is fine enough to capture suction peaks on the blades. The coarser grids considered here produced an artificially high thrust (perhaps because the mean induced downwash in the wake was under- resolved), while the finer grids gave a thrust comparable to that of the standard grid (when adequate wake resolution had been established). When the thrust variation is plotted against gridsize, as in Figure 123, this trend becomes clearer. It is seen that the differences become greater at higher pitch and thrust, and this is a point worth bearing in mind for future work, particularly where it is desired to apply very high pitch angles to achieve maximum thrust (for which purpose a full Navier-Stokes analysis would be necessary).
Greater effects of changing the grid size in these Euler cases was observed in the torque results, Figure 124, where it is clear that the coarser grids result in a significantly higher torque value, while the finer grids give reasonable agreement, with the torque perhaps decreasing as the grid is refined. This trend is confirmed and shown in greater clarity in Figure 125.
The combined effect of variations in the thrust and torque due to use of the various grids is illustrated in Figure 126, where a significant impact on the (inviscid, Euler) Figure of Merit is found. A grid that is too coarse will result in an artifically low Figure of Merit, while too fine a grid would waste valuable computer resources. The overall trend with grid refinement is fairly flat, and any small variations for the finer grids are
3.2 Grid Sensitivity Studies 3 VALIDATION
Figure 122: Effect of Grid Size on Thrust-Pitch Characteristics: Euler Solution for Configuration TBR-000-00 at a Mach Number of 0.448
Figure 123: Effect of Grid Size on Thrust Coefficient: Euler Solution for Configuration TBR-000-00 at a Mach Number of 0.448
3.2 Grid Sensitivity Studies 3 VALIDATION
Figure 124: Effect of Grid Size on Torque-Thrust Characteristics: Euler Solution for Configuration TBR-000-00 at a Mach Number of 0.448
Figure 125: Effect of Grid Size on Torque Coefficient: Euler Solution for Configuration TBR-000-00 at a Mach Number of 0.448
3.2 Grid Sensitivity Studies 3 VALIDATION
Figure 126: Effect of Grid Size on Figure of Merit: Euler Solution for Configuration TBR-000-00 at a Mach Number of 0.448
almost completely masked by the addition of a notional ’profile’ drag (the addition of an estimated profile drag to the Euler solution to make the Figure of Merit more realistic is discussed further in Section 5).
Conclusions for Tail Rotor Grid Sensitivity Study
While a relatively coarse grid may offer some economy during initial trials, it is likely that pessimistic performance predictions would result due to under-resolved flow features. The results of this grid sensitivity study confirm that a grid of about 2.8 million cells is sufficient to obtain good resolution of the vortex wake structure, and has been found to preserve vortices to an age of about 360 degrees. Use of a finer Euler grid did not appear to significantly affect the vortex trajectories. Therefore correctly predicted values of the hover thrust and induced power should be obtained for the example model tail rotor blade(s) considered in this research project.
For subsequent Navier-Stokes analysis, additional points must be added to capture the boundary layer and resolve the flow near the tip, and near the vortex sheets and cores in the wake, so that grids of up to about 8-10 million cells are anticipated for viscous simulations using one quadrant of the example model tail rotor in hover.