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 Flow past Helicopter blades Pitching airfoil/wing validation test cases Turbomachinery

4 interface compressor combustorturbine SUmb (URANS) CDP (LES) SUmb Stanford ASC Project

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

6 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 All unsteady effects lost

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

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

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

12 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 New Algorithm: Desirable Features

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

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

15 N is odd Time Spectral Method 1 zero e’value: e’vector e 1 = (1,1,1,1,1,…,1) T D t 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 : Frequency Set N time levels correspond to independent frequencies

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

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

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

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

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

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

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

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

25 Time Spectral Method Summary 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 Factor 6 reduction in CPU needs compared to BDF 50 ( 70,000 vs. 400,000 MG cycles ) with comparable accuracy. Almost time converged solution obtained with 11 time levels ( 5 frequencies ) per blade passing for the Stage 35 compressor test case

26 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 New Algorithm: Desirable Features

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 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 Rotor Stator Stator1 Stator2 Rotor 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

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

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

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

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

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

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. Entropy Distribution Pressure Distribution

40.. Magnitude of Force on Rotor Blade with various amounts of time resolution Magnitude of Force on Stator Blade with various amounts of time resolution K=3 converged to plotting accuracy 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 Magnitude of Force variation using various amounts of temporal resolution K = 7 : HB and BDF K = 2, 4, 7 : HB K = 7 K = 4 K = 2 S1 S2

44 Configuration D: BDF Solution Frequency content of the periodic force 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 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 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.

47 Acknowledgements