3-D Unsteady Multi-stage Turbomachinery Simulations

using the Harmonic Balance Technique

Arti K. Gopinath, Edwin van der Weide, Juan J. Alonso, Antony Jameson

Stanford University, CAAdvanced Simulation and Computing (ASC) Program – DoE

Kivanc Ekici and Kenneth C. HallDuke University, NC

interface interface

compressor combustor turbine

SUmb (URANS) CDP (LES) SUmb

Stanford ASC Project

Practical Turbomachinery: PW6000

5-stage HPC with 220 M cells => 2.4 M CPU hours

Mixing Plane Approximation

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• 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

The URANS equations are semi-discretized as

Solve in pseudo-time t* to its steady state

0)(*

nnt

n

wRwDdt

dw

Time Dependent Calculations

0)( wRwDt

Time Derivative Term

Use standard convergence acceleration techniques:

Runge-Kutta time stepping schemes with local Δt*

Multigrid in space

Time Integration Methods: Backward Difference Formula(BDF)

t

wwwwD

nnnn

t 2

43 21

• General time integration method: not specific for periodic problems

• Periodic state reached after 4-6 revolutions for high RPM cases

• Transients take up most of the resources.

• Could be very expensive for multi-stage turbomachinery

Time Integration Methods: Periodic Problems

• Time Spectral method( time-domain method) and

Frequency Domain methods.

• Fourier Representation in Time

*1

1*1* )()( EUt

EUEDEUEDUD ttt

Et

EDt

1• Full matrix => Solution at time instance n depends on the solution of all other time instances

Very expensive if high frequency unsteadiness need to be resolved

Approximations and Reduced-Order Models

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NASA Stage 35 Compressor

36 Rotors - 46 Stators

Approximations and Reduced-Order Models

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NASA Stage 35 Compressor Half Wheel

36 Rotors - 46 Stators

18 Rotors - 23 Stators

Periodic Boundary Conditions

Time Span = Time for Half Revolution

Approximations and Reduced-Order Models

. Scaled NASA Stage 35 Compressor

36 Rotors - 46 Stators

scaled to

36 Rotors - 48 Stators

reduced to periodic sector

Computational Grid: 3 Rotors - 4 Stators

Often used with BDF and Time Spectral Method

to keep costs low

Solve an Approximate Problem

Periodic Boundary Conditions

Time Span = Time for Periodic Sector

Approximations and Reduced-Order Models

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NASA Stage 35 Compressor True Geometry

36 Rotors - 46 Stators

Harmonic Balance Technique

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

Blade Passing Frequency (BPF)

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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

Savings in space: phase-lagged conditions

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Periodic Boundary Conditions

A

B

UA(t) = UB(t)

Phase-Lagged Boundary Conditions

A

B

UA(t) = UB(t-dt)

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

Sliding Mesh Interfaces

Sliding mesh interfaces

Interpolation in space in combination with phase-lagged conditions

Spectral Interpolation in time: time levels across do not match

Sliding mesh interface

Time levels

Sliding Mesh Interfaces.

AliasingDe-aliasing using longerstencil for interpolation

De-aliased solution

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Results

SUmb: compressible multi-block URANS solver

NASA Stage 35 Compressor.

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

NASA Stage 35 Compressor. Single-stage case with 1 Rotor row and 1 Stator row

Rotor blade row resolves: BPS 2*BPS 3*BPS 4*BPS

Stator blade row resolves: BPR 2*BPR 3*BPR 4*BPR

K=4

NASA Stage 35 Compressor

Rotor blade row resolves: BPS

Stator blade row resolves: BPR

K=1

Mixing Plane Solution.

Entropy Distribution

Pressure Distribution

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Three-Dimensional Effect

Entropy distribution at three different locations

Hub

Casing

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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

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

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

Configuration D: model compressor: Multi-stage case

K =2

Rotor: w1, w2

K =7

Rotor: w1,w2,w1+w2,w1-w2,2*w1,2*w1+w2,2*w1-w2

W1= BPS1, W2= BPS2

Magnitude of Force variation usingvarious amounts of temporal resolution K = 2, 4, 7 : HB

Magnitude of Force variation usingvarious amounts of temporal resolution K = 7 : HB and BDF

Configuration D: BDF Solution

Force variation through the transients Frequency content of the periodic force

Configuration D: Cost Comparisons

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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

Harmonic Balance Technique:Summary

Tremendous Savings:

• Only the Blade Passing Frequency of the neighboring blade row is resolved.

• Time Span = Time Period of the lowest frequency resolved in the current blade row.

• Phase-lagged boundary conditions on a computational grid with a single passage in each row.

• Interaction between blade rows in an unsteady manner: Space and Time Interpolation in physical space.

• Fourier representation in time: directly periodic state, no transients.