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, CA
Advanced Simulation and Computing (ASC) Program – DoE
Kivanc Ekici and Kenneth C. Hall
Duke University, NC
Stanford ASC Project
combustor
CDP (LES)
interface
SUmb (URANS)
interface
compressor
turbine
SUmb
Practical Turbomachinery: PW6000
5-stage HPC with 220 M cells => 2.4 M CPU hours
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
Time Dependent Calculations
The URANS equations are semi-discretized as
Dt w  R(w)  0
Time Derivative Term
Solve in pseudo-time t* to its steady state
dwn
n
n

D
w

R
(
w
)0
t
*
dt
Use standard convergence acceleration techniques:
Runge-Kutta time stepping schemes with local Δt*
Multigrid in space
Time Integration Methods: Backward
Difference Formula(BDF)
• 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
n
n 1
n2


3
w

4
w

w
n
Dt w  

2

t


Time Integration Methods:
Periodic Problems
•
Time Spectral method( time-domain method) and
Frequency Domain methods.
• Fourier Representation in Time
1

E
DtU *  Dt ( E 1EU * )  Dt ( E 1 U ) 
EU *
t

E 1
Dt 
E
t
Very expensive if high
frequency unsteadiness
need to be resolved
• Full matrix => Solution at time instance n depends
on the solution of all other time instances
Approximations and
Reduced-Order Models
.
NASA Stage 35 Compressor
36 Rotors - 46 Stators
Approximations and
Reduced-Order Models
.
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
Periodic Boundary Conditions
Time Span = Time for Periodic Sector
Often used with
BDF and Time Spectral Method
to keep costs low
Solve an Approximate Problem
Approximations and
Reduced-Order Models
.
Harmonic Balance Technique
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
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
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
Stator
Rotor
Stator1
Stator2
Savings in space:
phase-lagged conditions
.
Periodic Boundary Conditions
A
Phase-Lagged Boundary Conditions
A
B
B
UA(t) = UB(t)
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
Spectral Interpolation in time:
time levels across do not match
Sliding mesh interfaces
Interpolation in space in combination with
phase-lagged conditions
Time levels
Sliding mesh interface
Sliding Mesh Interfaces
.
Aliasing
De-aliasing using longer
stencil for interpolation
De-aliased solution
.
.
Results
SUmb: compressible multi-block URANS solver
NASA Stage 35 Compressor
.
3-D Single-stage test case
36 Rotors at 17,119 RPM
46 Stators
8 blocks with 1.8 M cells
Viscous test case: Turbulence modeled
using Spalart-Allmaras model
NASA Stage 35 Compressor
.
Single-stage case with 1 Rotor row and 1 Stator row
Rotor blade row
resolves:
Stator blade row
resolves:
BPS
2*BPS
3*BPS
4*BPS
BPR
2*BPR
3*BPR
4*BPR
K=4
NASA Stage 35 Compressor
Rotor blade row
resolves:
Stator blade row
resolves:
BPS
BPR
K=1
Mixing Plane Solution
.
Pressure Distribution
Entropy Distribution
.
Three-Dimensional Effect
Entropy distribution at
three different locations
Hub
Casing
.
Magnitude of Force on Rotor Blade
with various amounts of time resolution
.
K=3 converged to
plotting accuracy
Magnitude of Force on Stator Blade
with various amounts of time resolution
K=4 converged to
plotting accuracy
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
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
W1= BPS1, W2= BPS2
K =7
Rotor: w1,w2,w1+w2,w1-w2,2*w1,2*w1+w2,2*w1-w2
Magnitude of Force variation using
various amounts of temporal resolution
K = 2, 4, 7 : HB
Magnitude of Force variation using
various 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
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.