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
.
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)
.
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
.
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
.
.
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
.
.
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.