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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. A Thesis Proposal by Ebru Usta Advisor: Dr.L.N.SANKAR. Supported by the National Rotorcraft Technology Center(NRTC). Motivation and Objectives Background

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A Thesis Proposal by Ebru Usta Advisor: Dr.L.N.SANKAR

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

  2. 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 Overview

  3. 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. MOTIVATION and OBJECTIVES

  4. Numerical dissipation Dissipationcauses 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 PROBLEMS WITH THE CFD METHODSI. DISSIPATION ERRORS

  5. Numerical dispersion Dispersioncauses 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. II. DISPERSION ERRORS

  6. 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. RECENT PROGRESS IN REDUCING DISPERSION ERRORS

  7. 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). Aside benefit of the high order accuracy LDFV and DRP schemes is their reduceddissipation or numerical viscosity. These schemes have numerical viscosity that is typicallyproportional toD5whereDis the grid spacing. RECENT PROGRESS (continued)

  8. 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 toD3. Fourth order operator compact implicit schemes (OCI) have been studied by M.Smith (GT, 1994) and Ekaterinaris (Nielsen Eng.,1999) RECENT PROGRESS INREDUCING DISSIPATION ERRORS

  9. Hariharan and Sankar have explored 5th order and 7th order upwind schemes with dissipation errors proportional to D5 and D7 respectively(GT thesis 1995). Wake studied the evaluation of a line vortex in space and time using 6th order spatially accurate scheme and have presented 9th order results in fixed wing mode(1995). RECENT PROGRESS (continued)

  10. 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) RECENT PROGRESS (cont’d)GRID CLUSTERING EFFECTS

  11. 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? SCOPE OF THE PRESENT WORK

  12. Use Yee's symmetric TVD scheme to accurately modeltip vortex structure and shock noise phenomenaof 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. SCOPE OF THE PRESENT WORK

  13. For a TVD scheme, Sum of slopes always decreases, ensuring no new maxima occur. + n l New Maxima + ¶ é ù n l u t å | | ê ú Sum of slopes = ¶ ë x û u n n t ¶ é ù u å | | Sum of slopes = ê ú ¶ ë x û x WHAT IS A TVD SCHEME?

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

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

  16. 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. STVD (cont’d) Fourth order STVD scheme: and : MUSCL interpolation with a suitable limiter. Sixth order STVD scheme:

  18. Eighthorder STVD scheme on Non-Uniform Grids: distance along the coordinate line STVD (cont’d)

  19. 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. STVD (cont’d)

  20. 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. CONSTRUCTION OF and

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

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

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

  24. 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. 2-D Problem:Pulse interacting with uniform flow and solid wall. t=0+ V CAA workshop test Problem organized by Prof. Chris Tam (FSU)

  26. 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. (No vorticity) BOUNDARY CONDITIONS To avoid entropy layers, to preserve total enthalpy, h0

  28. TIME HISTORY OF PRESSURE AT THE WALL

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

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

  31. OSCILLATIONS PRESSURE CONTOURS(cont’d) T=100 With dissipation T=100 T=100

  32. With dissipation T=150 T=150 PRESSURE CONTOURS(cont’d)

  33. TRUNCATION ERROR ASSESMENT CPU TIME:

  34. 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. RESULTS and DISCUSSION

  35. 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). SHOCK NOISE PREDICTION OF UH-1H ROTOR

  36. Shock Noise Prediction, r/R=1.111, Tip Mach =0.90, Grid Size 75x45x31

  37. Shock Noise Prediction, r/R=1.78, Tip Mach= 0.90, Grid Size 75x45x31

  38. Shock Noise Prediction,r/R=3.09, Tip Mach =0.90, Grid Size 75x45x31

  39. 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.

  40. PRESSURE DISTRIBUTION ALONG THE SURFACE OF UH-60A AT r/R=0.920

  41. PRESSURE DISTRIBUTION ALONG THE SURFACE OF UH-60A AT r/R=0.99

  42. PERFORMANCE OF THE UH-60A ROTOR

  43. PERFORMANCE OF THE UH-60A ROTOR

  44. PERFORMANCE OF THE UH-60A ROTORVISCOUS RESULTS for 149x89x61 GRID SIZE

  45. CONVERGENCE HISTORY FOR TURNS-STVD8 FOR UH-60A ROTOR

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

  47. Torque versus Blade Loading

  48. 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.

  49. 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. CONCLUDING REMARKS

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