1. School of Aerospace Engineering A Thesis Proposal by Ebru Usta Advisor: Dr.L.N.SANKAR APPLICATION OF A SYMMETRIC TOTAL VARIATION DIMINISHING SCHEME TO AERODYNAMICS AND AEROACOUSTICS OF ROTORS Supported by the National Rotorcraft Technology Center(NRTC)

2. School of Aerospace Engineering Overview Motivation and Objectives Background Mathematical and Numerical Formulation Symmetric TVD Scheme (STVD) Validation with 1-D and 2-D Wave Problem Results and Discussion Shock Noise Prediction for the UH-1H rotor Tip Vortex Structure and Hover Performance of the UH-60A rotor Proposed Work

3. School of Aerospace Engineering MOTIVATION and OBJECTIVES Helicopter rotor’s flowfield is dominated by compressibility effects, a complex vortex wake structure and viscous effects. Accurate prediction of the aerodynamic flowfield and aeroacoustics of a helicopter rotor is a challenging problem in rotorcraft CFD. Existing methods for tip vortex and noise prediction suffer from numerous errors. As a result, accurate aerodynamics and aeroacoustics prediction methods are urgently needed.

4. School of Aerospace Engineering PROBLEMS WITH THE CFD METHODS I. DISSIPATION ERRORS Numerical dissipation – Dissipation causes a gradual decrease in the amplitude of an acoustic wave or the magnitude of the tip vortex as it propagates away from the blade surface. – The computed vortical wake, in particular, diffuses very rapidly due to numerical dissipation

5. School of Aerospace Engineering II. DISPERSION ERRORS Numerical dispersion – Dispersion causes waves of different wavelengths originating at the blade surface to incorrectly propagate at different speeds. – Because of dispersion errors, the waves may distort in nonphysical manner as they propagate away from the blade surface.

6. School of Aerospace Engineering RECENT PROGRESS IN REDUCING DISPERSION ERRORS Tam and his coworkers recently developed a low dispersion numerical scheme called the Dispersion-Relation-Preserving (DRP) finite difference scheme(1996). Nance et. al. extended the DRP ideas to curvilinear grids(GT thesis 1997). Other works include: Carpenter, Baeder, Ekaterinaris, Smith et al. and CAA Workshops I and II.

7. School of Aerospace Engineering RECENT PROGRESS (continued) Wang, Sankar and Tadghighi implemented Nance's Low Dispersion Finite Volume (LDFV) ideas into TURNS and studied shock noise and hover performance of rotorcraft(1998). – A side benefit of the high order accuracy LDFV and DRP schemes is their reduced dissipation or numerical viscosity. – These schemes have numerical viscosity that is typically proportional to  5 where  is the grid spacing.

8. School of Aerospace Engineering RECENT PROGRESS IN REDUCING DISSIPATION ERRORS The easiest way to reduce dissipation errors is to increase the formal accuracy of the upwind scheme. – Third order schemes in TURNS and OVERFLOW generate errors proportional to  3. – Fourth order operator compact implicit schemes (OCI) have been studied by M.Smith (GT, 1994) and Ekaterinaris (Nielsen Eng.,1999)

9. School of Aerospace Engineering RECENT PROGRESS (continued) – Hariharan and Sankar have explored 5 th order and 7 th order upwind schemes with dissipation errors proportional to  5 and  7 respectively (GT thesis 1995). – Wake studied the evaluation of a line vortex in space and time using 6 th order spatially accurate scheme and have presented 9 th order results in fixed wing mode(1995).

10. School of Aerospace Engineering RECENT PROGRESS (cont’d) GRID CLUSTERING EFFECTS Numerical errors may also be reduced by use of a fine grid, and/or grid clustering. – Tang et. al. recently have developed a grid redistribution method that clusters the grid points near the tip vortices and reduces the numerical diffusion of vorticity(1999). – Strawn et. al. used high density embedded grids(CHIMERA) for improving the wake- capture (1999)

11. School of Aerospace Engineering SCOPE OF THE PRESENT WORK The main purpose of this study is to develop and validate the spatially higher order accurate methods for modeling rotors in hover and forward flight. As the formal order of accuracy increases, it becomes more and more difficult to simultaneously reduce dispersion, dissipation and truncation errors. Are there better schemes available?

12. School of Aerospace Engineering SCOPE OF THE PRESENT WORK Use Yee's symmetric TVD scheme to accurately model tip vortex structure and shock noise phenomena of rotors. Yee’s idea: High order central difference schemes can be coupled to lower order dissipation terms to yield accurate results. For this purpose, a version of the NASA Ames code TURNS, referred to here as TURNS- STVDx (x=4,6,8), has been developed.

13. School of Aerospace Engineering WHAT IS A TVD SCHEME? For a TVD scheme, Sum of slopes always decreases, ensuring no new maxima occur. n t x u Sum of slopes = n x u          | | New Maxima ln t  Sum of slopes = ln x u           ||

14. School of Aerospace Engineering Symmetric TVD Scheme The semi-discrete form at a typical node 'i' is:

15. School of Aerospace Engineering Symmetric TVD Scheme (continued) Dr. Helen Yee recommends the following second order form: where computed using “Roe averages” of q at adjacent points.

16. School of Aerospace Engineering STVD (cont’d) Second order STVD scheme: This part is used to control dispersion and truncation errors This part is used to control dissipation errors Dispersion and dissipation errors may be independently controlled.

17. School of Aerospace Engineering Fourth order STVD scheme: STVD (cont’d) and : MUSCL interpolation with a suitable limiter. Sixth order STVD scheme:

18. School of Aerospace Engineering STVD (cont’d) Eighth order STVD scheme on Non-Uniform Grids: distance along the coordinate line

19. School of Aerospace Engineering STVD (cont’d) –Where a,b,c,d,e,f, g,h are coefficients of the related fluxes. Note that this scheme also accounts for the non-uniform grid spacing.

20. School of Aerospace Engineering CONSTRUCTION OF and and were found using third order MUSCL interpolations. Koren Limiter, and a LDFV Limiter were explored. In some sample bench mark cases, and were found using higher order (4th, 6th and 8th) dissipation terms with no limiters.

21. School of Aerospace Engineering 1-D WAVE PROBLEM The initial solution at t=0 is given by The exact solution is

22. School of Aerospace Engineering 1-D WAVE PROBLEM (continued) The accuracy of the schemes is assessed by computing the of the error calculated as: IMAX : The maximum number of grid points

23. School of Aerospace Engineering 1-D WAVE PROBLEM (cont’d) 1-D wave equation is solved explicitly using second order Runge Kutta method as follows: : Formal accuracy of the scheme

24. School of Aerospace Engineering Higher order schemes, e.g. STVD8, consistently produces lowest errors on all grids. For STVD8, the slope is the steepest, indicating that the errors decrease quickly with refinement.

25. School of Aerospace Engineering 2-D Problem: Pulse interacting with uniform flow and solid wall. CAA workshop test Problem organized by Prof. Chris Tam (FSU) t=0 + VV

26. School of Aerospace Engineering Several baseline solutions (6th order MacCormack, 3rd order Upwind) are available for comparison. Exact solutions are also available for comparison(Nance, Ph.D Dissertation) At boundaries, non-reflective boundary conditions were used. In this study,STVD4, STVD6 and STVD8 solutions were obtained. Only the 8th order results are shown here. Approach:

27. School of Aerospace Engineering BOUNDARY CONDITIONS To avoid entropy layers, to preserve total enthalpy, h 0 (No vorticity)

28. School of Aerospace Engineering TIME HISTORY OF PRESSURE AT THE WALL

29. School of Aerospace Engineering

30. PRESSURE CONTOURS T=75 T=100 T=150 Oscillations due to no dissipation term T=75

31. School of Aerospace Engineering PRESSURE CONTOURS Oscillations due to no dissipation term With dissipation term T=75

32. School of Aerospace Engineering PRESSURE CONTOURS(cont’d) T=100 With dissipation OSCILLATIONS T=100

33. School of Aerospace Engineering PRESSURE CONTOURS(cont’d) T=150 With dissipation

34. School of Aerospace Engineering TRUNCATION ERROR ASSESMENT CPU TIME:

35. School of Aerospace Engineering RESULTS and DISCUSSION 4th,6th and 8th order Symmetric TVD schemes have been applied to model helicopter rotor shock noise for UH-1H rotor and tip vortex structure of UH-60A rotor. The following results are presented: – Original TURNS code (3rd order MUSCL scheme) – Modified flow solver TURNS-STVDx (x=4,6,8) – Comparison with experimental data for UH-60A and UH-1H rotor. All rotor calculations were done on identical grids, to eliminate grid differences from skewing the interpretation of results.

36. School of Aerospace Engineering SHOCK NOISE PREDICTION OF UH-1H ROTOR Calculations have been performed for a two- bladed UH-1H rotor in hover. The blades are untwisted and have a rectangular planform with NACA 0012 airfoil sections and an aspect ratio of 13.7133. The sound pressure levels have been compared to the experimental data for a 1/7 scale model (Purcell,1989).

37. School of Aerospace Engineering Shock Noise Prediction, r/R=1.111, Tip Mach =0.90, Grid Size 75x45x31

38. School of Aerospace Engineering Shock Noise Prediction, r/R=1.78, Tip Mach= 0.90, Grid Size 75x45x31

39. School of Aerospace Engineering Shock Noise Prediction,r/R=3.09, Tip Mach =0.90, Grid Size 75x45x31

40. School of Aerospace Engineering PLANFORM OF THE UH-60A MODEL ROTOR Four blades, a non-linear twist, and no taper. 20 degrees of rearward sweep that begins at r/R=0.93. The aspect ratio and Solidity Factor 15.3 and 0.0825.

41. School of Aerospace Engineering PRESSURE DISTRIBUTION ALONG THE SURFACE OF UH-60A AT r/R=0.920

42. School of Aerospace Engineering PRESSURE DISTRIBUTION ALONG THE SURFACE OF UH-60A AT r/R=0.99

43. School of Aerospace Engineering PERFORMANCE OF THE UH-60A ROTOR

44. School of Aerospace Engineering PERFORMANCE OF THE UH-60A ROTOR

45. School of Aerospace Engineering PERFORMANCE OF THE UH-60A ROTOR VISCOUS RESULTS for 149x89x61 GRID SIZE

46. School of Aerospace Engineering CONVERGENCE HISTORY FOR TURNS-STVD8 FOR UH-60A ROTOR

47. School of Aerospace Engineering VISCOUS CALCULATIONS DONE IN COLLABORATION WITH UTRC AT UTRC ON A 181x75 x49 FINER GRID OF UH-60A ROTOR Blade Loading vs. collective pitch

48. School of Aerospace Engineering Torque versus Blade Loading

49. School of Aerospace Engineering Figure of Merit versus Blade Loading Error of 0.01-0.02 in FM; well within 100 lb. or 200 lb. error in thrust; considered very good by industry.

50. School of Aerospace Engineering CONCLUDING REMARKS The accuracy characteristics of the STVDx schemes have been systematically investigated in 1-D and 2-D problems where exact solutions exist. Several high order Symmetric TVD schemes have been implemented in the TURNS code. The tip vortex structure of UH-60A rotor and shock noise phenomena for UH-1H rotor are accurately modeled with these high order schemes compared to the baseline third order MUSCL scheme.

51. School of Aerospace Engineering CONCLUDING REMARKS(cont’d) The eighth order STVD scheme is found to give the best thrust predictions for the UH- 60A rotor, even on a coarse grid. The shock noise predictions were also, in general, better with the higher order schemes in spite of having loss in accuracy when a high scheme is used on a very coarse grid, 3 radii away. The STVDx schemes require little or no additional computational time, compared to the MUSCL scheme.

52. School of Aerospace Engineering CONCLUDING REMARKS(cont’d) Many existing CFD solvers may easily be retrofitted with the symmetric TVD scheme. UTRC Viscous results compare very well with the model test. The Figure of Merit is generally 1-2 points under the experimental data which is considered very good. These results are much better than using baseline TURNS.

53. School of Aerospace Engineering PROPOSED WORK Perfecting the Hover Code: Increase formal accuracy of metrics, Jacobian, time, boundary conditions, load integration schemes. Additional validations for another rotor, to be chosen in consultation with industry and thesis committee. Study of Vortex Ring State and climb using GT experimental data

54. School of Aerospace Engineering PROPOSED WORK (continued) IF TIME PERMITS IF TIME PERMITS, Use embedded adaptive grid for improved wake capturing Use of Spalart-Allmaras turbulence model for hover prediction.