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Efficient Fourier-Based Algorithms for Time-Periodic Unsteady Problems

Efficient Fourier-Based Algorithms for Time-Periodic Unsteady Problems. Arti K. Gopinath Aeronautics and Astronautics Stanford University Ph.D. Oral Defense Presentation April 16, 2007. What? Why? and How? Outline. What…. are time-periodic unsteady problems, and why are they important?.

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Efficient Fourier-Based Algorithms for Time-Periodic Unsteady Problems

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  1. Efficient Fourier-Based Algorithms for Time-Periodic Unsteady Problems Arti K. Gopinath Aeronautics and Astronautics Stanford University Ph.D. Oral Defense Presentation April 16, 2007

  2. What? Why? and How? Outline What… are time-periodic unsteady problems, and why are they important? Why… do we need specialized algorithms to solve them? Aren’t the current algorithms good enough? How… are we going to develop these algorithms? Why are they more efficient? What are their pros and cons?

  3. Time-Periodic Unsteady Problems • Wind Turbines • Pitching airfoil/wing • validation test cases • Turbomachinery • Flow past • Helicopter blades

  4. interface interface SUmb (URANS) CDP (LES) SUmb Stanford ASC Project combustor turbine compressor

  5. Practical Turbomachinery: PW6000 5-stage HPC with 220 M cells => 2.4 M CPU hours ( Using the dual-time stepping second-order Backward Difference Formula )

  6. All unsteady effects lost Mixing Plane Approximation • Steady computation in each blade row . • Computational grid spanning one blade passage per blade row • Circumferentially averaged quantities passed between blade rows

  7. . NASA Stage 35 Compressor 36 Rotors - 46 Stators

  8. . NASA Stage 35 Compressor Half Wheel 36 Rotors - 46 Stators 18 Rotors - 23 Stators Periodic Boundary Conditions Time Span = Time for Half Revolution

  9. 36 Rotors - 48 Stators reduced to periodic sector Computational Grid: 3 Rotors - 4 Stators Periodic Boundary Conditions Time Span = Time for Periodic Sector Approximation: Scaled Geometry Scaled NASA Stage 35 Compressor . 36 Rotors - 46 Stators Often used with BDF to keep costs low Solve an Approximate Problem

  10. Time Derivative Term Second-order implicit BDF Solve in pseudo-time t* to its steady state Time-Accurate Method: Backward Difference Formula (BDF) The URANS equations are semi-discretized as

  11. Turbomachinery: 50-100 physical time steps per blade passing 25-50 inner iterations in pseudo-time 4-6 revolutions to reach periodic state Time-Accurate Method: Backward Difference Formula (BDF) Divide the time period into N time levels  varies sinusoidally

  12. New Algorithm: Desirable Features Directly solve for final periodic solution, not resolve transients Time Domain algorithm => existing solver can be readily used Solve for the true geometry of the problem Computational domain as small as possible Time Span of computation as small as possible

  13. or Fourier Representation in Time The discrete Fourier transform of U* using N time intervals or Time derivative of U* (Frequency Domain Methods) at each time n at each wavenumber k

  14. Time Spectral Method N is even Analytical Expression for elements of (Even and Odd N) Matrix Operator 2 zero e’values; e’vectors e1 = (1,1,1,1,1,…,1)T, e2 = (1,0,1,0,…..,1,0)T

  15. Time Spectral Method N is odd 1 zero e’value: e’vector e1 = (1,1,1,1,1,…,1)T Dt is a central difference full matrix operator connecting all time levels, yielding an integrated space-time formulation which requires a simultaneous solution of the equations at all time levels

  16. Time Spectral Method One fundamental frequency : independent frequencies N time levels correspond to Frequency Set

  17. . . Results: Time Spectral Method SUmb: compressible multi-block structured URANS solver

  18. 1-1 Scaled NASA Stage 35 Compressor • Original blade count 36 (rotor), 46 (stator) • Scaled the stator to 36, such that a 1-1 configuration can be used; primary focus is verification of the Time Spectral Method • 17,119 RPM • 7 block mesh; 773,184 cells

  19. 0 0 1 1 2 2 3 3 4 4 1-1 scaled Stage 35 2nd order BDF Results 50 physical time steps per blade passing; 50 inner iterations per time step Periodic convergence of the torque torque plotted every 50 time steps ( first time instance in each time period ) Stator Rotor Number of Revs Number of Revs

  20. 1-1 scaled Stage 35, 2nd order BDF Periodic state has NOT been fully reached after 4.5 revolutions, which corresponds to 400,000 multigrid cycles. Entropy Pressure

  21. 1-1 scaled Stage 35, Time Spectral Method Results Time Spectral Method with various amounts of temporal resolution • Time converged using 7 time intervals ( 3 frequencies ) N = 3, 5, 7, 9, 11, 13 Variation of Torque on the rotor blade during one blade passing

  22. 1-1 scaled Stage 35, Time Spectral Method Results Comparison of Time Spectral Results with BDF results Variation of Torque on the Rotor blade during one blade passing 50 time steps per blade passing for BDF not good enough, 100 time steps better

  23. 1-1 scaled Stage 35, Time Spectral Method Results Time Spectral Method with various amounts of temporal resolution • Time converged using 11 time intervals ( 5 frequencies ) - 70,000 MG cycles N = 3, 5, 7, 9, 11, 13 Variation of Torque on the stator blade during one blade passing

  24. 1-1 scaled Stage 35, Time Spectral Method Results Comparison of Time Spectral Results with BDF results Variation of Torque on the Stator blade during one blade passing 13 time levels for TS not good enough at high frequencies

  25. Conclusions • Very good algorithm to predict time-periodic unsteady • problems where the frequency of unsteadiness is known and has • narrow frequency spectrum • For turbomachinery problems, TS compares favorably • to time-accurate schemes on a small domain and short time span Time Spectral Method Summary • Almost time converged solution obtained with 11 time levels ( 5 frequencies ) per blade passing for the Stage 35 compressor test case • Factor 6 reduction in CPU needs compared to BDF 50 • ( 70,000 vs. 400,000 MG cycles ) with comparable accuracy.

  26. New Algorithm: Desirable Features Directly solve for final periodic solution, not resolve transients Time Domain algorithm => existing solver can be readily used Solve for the true geometry of the problem Computational domain as small as possible Time Span of computation as small as possible

  27. Reduced-Order Harmonic Balance Method NASA Stage 35 Compressor True Geometry . 36 Rotors - 46 Stators Computational Grid: 1 Rotor - 1 Stator Modified Periodic Boundary Conditions Time Span such that only dominant frequencies are resolved • Fraction of the cost of a BDF/Time Spectral Computation on the true geometry

  28. Multi-Stage Case: Combinations of BPF of Stator1 and Stator2 resolved in the Rotor row Only BPF of Rotor resolved in Stator1 and Stator2 No one fundamental frequency resolved by the rotor row Rotor Stator Rotor Stator2 Stator1 Blade Passing Frequency (BPF) Single-Stage Case: BPF of the Stator and its higher harmonics resolved in the Rotor row . BPF of the Rotor and its higher harmonics resolved in the Stator row Only One Fundamental Frequency in each blade row

  29. Savings in space: phase-lagged conditions . Phase-Lagged Boundary Conditions Periodic Boundary Conditions A A B B UA(t) = UB(t) UA(t) = UB(t-dt)

  30. 1 Freq => 3 time levels 2 Freq => 5 time levels 1 Frequency => 3 time levels Savings in time:Smaller Time Span and only Dominant Frequencies . Harmonic Balance Method Time Spectral Method 5 Frequencies => 11 time levels

  31. Blade-Row Interactions: Sliding Mesh Interfaces Spectral Interpolation in time: time levels across do not match Sliding mesh interfaces Sliding mesh interface Interpolation in space in combination with phase-lagged conditions

  32. donor receiver De-aliasing using longer stencil for interpolation Filter High frequencies captured on this longer stencil De-aliased solution Sliding Mesh Interfaces . Aliasing

  33. “ Blow up of an aliased, non-energy-conserving model is God’s way of protecting you from believing in a bad simulation.” - J. P. Boyd

  34. Harmonic Balance Method: Features Fourier Representation in Time: take advantage of periodicity Directly solve for the periodic state: avoid transients Time Domain algorithm: acceleration techniques like Multigrid, local time stepping used Solution at all time levels computed simultaneously Interaction between blade rows: Unsteady Only Dominant Frequencies (Blade Passing Frequencies) are resolved Smaller Time Span = Time Period of lowest frequency Computational Domain: One Blade Passage per blade row

  35. . . Results: Harmonic Balance Method SUmb: compressible multi-block URANS solver

  36. NASA Stage 35 Compressor: True Geometry . 3-D Single-stage test case 36 Rotors at 17,119 RPM 46 Stators 8 blocks with 1.8 M cells Viscous test case: Turbulence modeled using Spalart-Allmaras model

  37. NASA Stage 35 Compressor Single-stage case with 1 Rotor row and 1 Stator row . Solution in Rotor blade row resolves: BPS 2*BPS 3*BPS 4*BPS Solution in Stator blade row resolves: BPR 2*BPR 3*BPR 4*BPR K=4

  38. NASA Stage 35 Compressor Solution in Rotor blade row resolves: BPS Solution in Stator blade row resolves: BPR K=1

  39. Mixing Plane Solution . Pressure Distribution Entropy Distribution

  40. . . Magnitude of Force on Rotor Blade with various amounts of time resolution K=3 converged to plotting accuracy Magnitude of Force on Stator Blade with various amounts of time resolution K=4 converged to plotting accuracy

  41. NASA Stage 35 Cost Comparisons Harmonic Balance Technique: Computational Grid : 1 Rotor, 1 Stator 4 frequencies in each blade row => 9 time levels for time convergence 1400 CPU hours . Backward Difference Formula (BDF): (Estimated Cost) Computational Grid : 18 Rotors, 23 Stators 50 time steps per blade passing, 50 inner multigrid iterations, 3-4 revolutions for periodic state 150,000 CPU hours

  42. Configuration D: Model Compressor 2-D Multi-stage test case 3 blocks with 18,000 cells Pitch ratio: 1.0:0.8:0.64 Inviscid test case

  43. K = 2 K = 4 K = 7 Magnitude of Force variation using various amounts of temporal resolution K = 2, 4, 7 : HB K = 7 : HB and BDF S1 S2

  44. Frequency content of the periodic force Configuration D: BDF Solution Force variation through the transients

  45. Configuration D: Cost Comparisons Harmonic Balance Technique: Computational Grid : 1 Stator1, 1 Rotor, 1 Stator2 7 frequencies in each blade row => 15 time levels for reasonable accuracy 33 CPU hours . Backward Difference Formula (BDF): Computational Grid : 16 Stator1, 20 Rotor, 25 Stator2 50 time steps per blade passing, 25 inner multigrid iterations, 3 revolutions for periodic state 290 CPU hours

  46. Conclusions Excellent reduced-order model for multi-stage turbomachinery problems where the designer can choose the frequency set based on a trade-off between accuracy and cost If the frequency set cannot be predicted a priori, a quick calculation using small amounts of temporal resolution can be used to initiate the time-accurate computation so numerical transients are avoided. Harmonic Balance Technique Summary For the 3D single stage viscous test case: estimated 2 orders of magnitude savings in CPU requirements For the 2D 1.5 stage inviscid test case: about 1 order of magnitude savings in CPU requirements

  47. Acknowledgements

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