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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?. - PowerPoint PPT Presentation
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Efficient Fourier-Based Algorithms for
Time-Periodic Unsteady Problems
Arti K. GopinathAeronautics and Astronautics
Stanford University
Ph.D. Oral Defense PresentationApril 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 interface
compressor combustor turbine
SUmb (URANS) CDP (LES) SUmb
Stanford ASC Project
5
Practical Turbomachinery: PW60005-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
0)(*
nnt
n
wRwVDdt
dwV
0)( wRwVDt
Time Derivative Term
0)(2
43 21
n
nnn
wRt
wwwV
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:50-100 physical time steps
per blade passing
25-50 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
*U U
*ˆ EUU niktN
nnk eU
NU
1
0
*1ˆ or
nikt
kkn eUU ˆ* UEU ˆ1* or
k
iktknt
neUikUD ˆ*
0ˆˆ kk RUVik0)( ** nnt URUVDat each time n at each wavenumber k
(Frequency Domain Methods)
Time derivative of U*
14
Time Spectral Method
00
00
00
1221
2211
11
21
2
1
dddd
dddd
dddd
D
NN
event
DEEDt1Matrix Operator
2 zero e’values; e’vectors e1 = (1,1,1,1,1,…,1)T, e2 = (1,0,1,0,…..,1,0)T
*111* ˆ)()ˆ( DEUEUEDUEDUD ttt
ikDkk
1
2
12
N
Nm
mnevenm
nevent wdwD
dmeven
1
2( 1)m1 cot(
m
N) : m 0
0 : m 0
N is even
Analytical Expression for elements of tD (Even and Odd N)
15
Dtodd wn
2T
dmodd wnm
m1 N
2
N 1
2
0:0
0:)(cos)1(2
1 1
m
mN
mecd
moddm
N is odd
Time Spectral Method
0
0
0
1221
2211
1
2
1
2
11
dddd
dddd
dddd
D
NN
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
0)( ***
nntn URUVD
d
dUV
16
Time Spectral Method
111101
111101
101000
1
N
N
N
tiftiftif
tiftiftif
tiftiftif
eee
eee
eee
NE
TKK fffffff ],,,,,,[ 1210 00 f kk ff
,2 12 ff ,3 13 ff 1kffk
One fundamental frequency : 1f
Frequency Set
DEEDt1
N time levels T
Ntttt ],,,,[ 1210 correspond to 2
1
NK independent
frequencies
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
1-1 scaled Stage 35 2nd order BDF Results
Periodic convergence of the torque
RotorStator
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 )
0 21 3 4 0 21 3 4
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.
Pressure Entropy
21
1-1 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
22
1-1 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
23
1-1 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
24
1-1 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 harmonicsresolved 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
UA(t) = UB(t)
Phase-Lagged Boundary Conditions
A
B
UA(t) = UB(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
.
AliasingDe-aliased solution
unfilteredfiltered EUEU **
12 K
Filter High frequencies captured on this longer stencil
)12(2 K
De-aliasing using longerstencil 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 RPM46 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 inRotor blade row resolves: BPS 2*BPS 3*BPS 4*BPS
Solution inStator blade row resolves: BPR 2*BPR 3*BPR 4*BPR
K=4
38
NASA Stage 35 Compressor
Solution inRotor blade row resolves: BPS
Solution inStator 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 usingvarious amounts of temporal resolution
K = 7 : HB and BDF
1 0
0 1
1 1
1 -1
2 0
2 -1
2 1
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