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

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. 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

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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

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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

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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

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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*

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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)

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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

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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

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.

.

Results: Time Spectral Method

SUmb: compressible multi-block structured URANS solver

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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)

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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

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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

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Sliding Mesh Interfaces

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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

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“ 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

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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

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.

.

Results: Harmonic Balance Method

SUmb: compressible multi-block URANS solver

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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

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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

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NASA Stage 35 Compressor

Solution inRotor blade row resolves: BPS

Solution inStator blade row resolves: BPR

K=1

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Mixing Plane Solution

.

Entropy Distribution

Pressure Distribution

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.

.

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

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NASA Stage 35 Cost Comparisons

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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

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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

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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

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Configuration D: BDF Solution

Frequency content of the periodic force Force variation through the transients

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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

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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.

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Acknowledgements