adrien gomar, phd defense
DESCRIPTION
Multi-Frequential Harmonic Balance Approach for the Simulation of Contra-Rotating Open Rotors: Application to Aeroelasticity April 14, 2014 at CERFACS (Toulouse, France) JURY: P. FERRAND (President), LMFA, (Lyon, France) C. CORRE (Referee), ENSE3, (Grenoble, France) L. HE (Referee), University of Oxford, (Oxford, United-Kingdom) J-C. CHASSAING (Member), UPMC, (Paris, France) P. CINNELLA (Member), Università del Salento, (Lecce, Italy) F. SICOT (Member), CERFACS, (Toulouse, France) C. DEJEU (Invited), Snecma (Safran), (Villaroche, France) ABSTRACT: Computational Fluid Dynamics (CFD) has allowed the optimization of many configurations among which aircraft engines. In the aeronautical industry, CFD is mostly restricted to steady approaches due to the high computational cost of unsteady simulations. Nevertheless, the flow field across the rotating parts of aircraft engines, namely turbomachinery blades, is essentially periodic in time. Years ago, Fourier- based time methods have been developed to take advantage of this time periodicity. However, they are, for the most part, restricted to mono-frequential flow fields. This means that only a single base-frequency and its harmonics can be considered. Recently, a multi-frequential Fourier-based time method, namely the multi-frequential Harmonic Balance (HB), has been developed and implemented into the elsA CFD code, enabling new kinds of applications as, for instance, the aeroelasticity of multi-stage turbomachinery. The present PhD thesis aims at applying the HB approach to the aeroelasticity of a new type of aircraft engine: the contra-rotating open rotor. The method is first validated on analytical, linear and non-linear numerical test problems. Two issues are raised, which prevent the use of such an approach on arbitrary aeroelastic configurations: the conditioning of the multi-frequential HB source term and the convergence of the method. Original methodologies are developed to improve the condition number of the simulations and to provide a priori estimates of the number of harmonics required to achieve a given convergence level. The HB method is then validated on a standard configuration for turbomachinery aeroelasticity. The results are shown to be in fair agreement with the experimental data. The applicability of the method is finally demonstrated for aeroelastic simulations of contra-rotating open rotors.TRANSCRIPT
Multi-Frequential Harmonic Balance Approachfor the Simulation of Contra-Rotating Open Rotors:
Application to Aeroelasticity
Adrien Gomarsupervisor Paola Cinnella
co-supervisor Frédéric Sicot
in partnership with
Introduction
Warming of the climate system is unequivocal, and since the 1950s, many of the observed changes are unprecedented over decades to millennia”
‟
IPCC, WG1 Summary for Policymakers, 2013
2
Introduction
Demanding objectives are set toward the aeronautical industry
CO2
NOx
noise
Flightpath 2050 Europe’s Vision for Aviation
2000 values
3
Introduction
Demanding objectives are set toward the aeronautical industry
CO2
NOx
noise
-75%
-80%
-65%
Flightpath 2050 Europe’s Vision for Aviation
expected 2050 values 2000 values
4
Introduction
Demanding objectives are set toward the aeronautical industry
CO2
NOx
noise
-75%
-80%
-65%
Flightpath 2050 Europe’s Vision for Aviation
expected 2050 values 2000 values
One way to reduce certain pollutants is todecrease fuel consumption by increasing the bypass ratio
5
Introduction
The bypass ratio defines the mass-flow ratio of cold air to hot air
BPR =mc
mh
6
Introduction
The bypass ratio defines the mass-flow ratio of cold air to hot air
hot air
BPR =mc
mh
mh
7
Introduction
The bypass ratio defines the mass-flow ratio of cold air to hot air
mc
cold air
BPR =mc
mh
8
Introduction
Contra-Rotating Open Rotor (CROR) is a concept based on an increase of the bypass ratio
9
Introduction
Contra-Rotating Open Rotor (CROR) is a concept based on an increase of the bypass ratio
✔ In total, 25% fuel consumption reduction
✘ new challenges are raised, e.g. blade flutter
✔ transonic Mach number
9
Introduction
Flutter is a self-excited, self-sustained aeroelastic phenomenon
10
Introduction
Flutter is a self-excited, self-sustained aeroelastic phenomenon
11
Blade flutter on CROR can lead to failure of the engine. It should be considered
as early as possible in the design chain
Introduction
investigation
unsteady effects
Numerical methods for turbomachinery
Lifting line RANS URANS LES
minutes hours days weeks
design
1D results
validation
3D results
research
turbulent effects
12
Introduction
investigation
unsteady effects
Numerical methods for turbomachinery
Lifting line RANS URANS LES
minutes hours days weeks
design
1D results
validation
3D results
research
turbulent effects
required to simulate unsteady response
to flutter
12
Introduction
investigation
unsteady effects
Numerical methods for turbomachinery
Lifting line RANS URANS LES
minutes hours days weeks
design
1D results
validation
3D results
research
turbulent effects
turn-around time compatible with
industry
12
Introduction
investigation
unsteady effects
Numerical methods for turbomachinery
RANS URANS
hours days
validation
3D results
research
turbulent effects
12
Introduction
investigation
unsteady effects
Numerical methods for turbomachinery
RANS URANS
hours days
validation
3D results
research
turbulent effects
hours
investigation
unsteady effects
HB
12
Outline
Convergence of HB for wake passing problems
13
Conditioning of multi-frequential HB methods
Introduction to the harmonic balance approach
Application to Contra-Rotating Open Rotors
Dt = iE�1⌦E
Introduction to HB /
Outline
Convergence of HB for wake passing problems
14
Conditioning of multi-frequential HB methods
Introduction to the harmonic balance approach Mono-frequential formulationMulti-frequential formulationScientific bottlenecks associated with the PhD
Application to Contra-Rotating Open Rotors
Dt = iE�1⌦E
Introduction to HB / Mono-frequential formulation
The harmonic balance approach is an efficient unsteady method based on the Fourier transform
sampling W with 2N+1 time instants(Nyquist-Shannon sampling theorem)
make use of the Fourier transformW temporally periodic1
energy concentrated on finite number of harmonics N
2
hypotheses:
turbulence is fully modeled3 URANS equations
15
Introduction to HB / Mono-frequential formulation
The URANS equations are transformed ...
VdW
dt+ R(W ) = 0
finite volume, semi discrete form of the Navier-Stokes equations
16
unsteady problem
Introduction to HB / Mono-frequential formulation
The URANS equations are transformed ...
VdW
dt+ R(W ) = 0
finite volume, semi discrete form of the Navier-Stokes equations
VdW ?
dt+ R(W ?) = 0
discretizing the problem
16
unsteady problem
Introduction to HB / Mono-frequential formulation
The URANS equations are transformed ...
VdW
dt+ R(W ) = 0
Vd
dt
⇣E�1cW ?
⌘+ R(E�1cW ?) = 0
using the inverse Fourier matrixW ? = E�1cW ?
17
unsteady problem
Introduction to HB / Mono-frequential formulation
The URANS equations are transformed ...
VdW
dt+ R(W ) = 0
Vd
dt
⇣E�1cW ?
⌘+ R(E�1cW ?) = 0
using the inverse Fourier matrixW ? = E�1cW ?
17
unsteady problem
Introduction to HB / Mono-frequential formulation
The URANS equations are transformed into a subset of 2N+1 harmonic equations
VdW
dt+ R(W ) = 0
harmonics are time-independent
Vd
dt
�E�1
� cW ? + R(E�1cW ?) = 0
He & Ning, AIAA Journal, 1998McMullen et al., AIAA Paper, 2001
18
unsteady problem
2N+1 harmonicequations
Introduction to HB / Mono-frequential formulation
The URANS equations are transformed into a subset of 2N+1 harmonic equations
VdW
dt+ R(W ) = 0
Vd
dt
�E�1
� cW ? + R(E�1cW ?) = 0
19
using the Fourier matrixcW ? = EW ?
unsteady problem
2N+1 harmonicequations
Introduction to HB / Mono-frequential formulation
The URANS equations are transformed into a subset of 2N+1 harmonic equations
VdW
dt+ R(W ) = 0
Vd
dt
�E�1
� cW ? + R(E�1cW ?) = 0
19
using the Fourier matrixcW ? = EW ?
Vd
dt
�E�1
�EW ? + R(E�1EW ?) = 0
unsteady problem
2N+1 harmonicequations
Introduction to HB / Mono-frequential formulation
The URANS equations are transformed into a subset of 2N+1 harmonic equations
VdW
dt+ R(W ) = 0
Vd
dt
�E�1
� cW ? + R(E�1cW ?) = 0
Hall et al., AIAA Journal, 2002Gopinath & Jameson, AIAA Paper, 2005
20
Vd
dt
�E�1
�EW ? + R(E�1EW ?) = 0
unsteady problem
2N+1 harmonicequations
2N+1 steady equations
Introduction to HB / Mono-frequential formulation
The URANS equations are transformed into a subset of 2N+1 harmonic equations
VdW
dt+ R(W ) = 0
Vd
dt
�E�1
� cW ? + R(E�1cW ?) = 0
Hall et al., AIAA Journal, 2002Gopinath & Jameson, AIAA Paper, 2005
20
Vd
dt
�E�1
�EW ? + R(E�1EW ?) = 0
unsteady problem
2N+1 harmonicequations
| {z }identity matrix
2N+1 steady equations
Introduction to HB / Mono-frequential formulation
The URANS equations are transformed into a subset of 2N+1 harmonic equations
VdW
dt+ R(W ) = 0
Vd
dt
�E�1
� cW ? + R(E�1cW ?) = 0
Hall et al., AIAA Journal, 2002Gopinath & Jameson, AIAA Paper, 2005
20
Vd
dt
�E�1
�EW ? + R(E�1EW ?) = 0
unsteady problem
2N+1 harmonicequations
| {z }identity matrixDt
2N+1 steady equations
Introduction to HB / Mono-frequential formulation
The unsteady problem is made equivalent to 2N+1 steady problems coupled by a source term
VdW
dt+ R(W ) = 0
equivalent under given hypotheses
21
Dt = i!E�1KEwith
VDt
2
64W0...
W2N
3
75+ R(
2
64W0...
W2N
3
75) = 0
unsteady problem
2N+1 steady equations
Introduction to HB / Mono-frequential formulation
A pure five-harmonic signal is assessed
u(t) = cos(!t) + sin(2!t) + cos(3!t) + sin(4!t) + cos(5!t)
du
dt= ! [� sin(!t) + 2 cos(2!t)� 3 sin(3!t) + 4 cos(4!t)� 5 sin(5!t)]
du
dt= ! [� sin(!t) + 2 cos(2!t)� 3 sin(3!t) + 4 cos(4!t)� 5 sin(5!t)]
u(t) = cos(!t) + sin(2!t) + cos(3!t) + sin(4!t) + cos(5!t)
22
Introduction to HB / Mono-frequential formulation
du
dt(t = tq) =
�uq+2 + 8uq+1 � 8uq�1 + uq�2
12�t+O(�t4)
Three time derivative operators are tested
du
dt= i!E�1KE
2nd order
4th order
harmonic balance
du
dt(t = tq) =
uq+1 � uq�1
2�t+O(�t2)
23
Introduction to HB / Mono-frequential formulation
harmonic balance
1000# samples
𝓛2-norm relative error
4th order
2nd order
1
10-161
24
Introduction to HB / Mono-frequential formulation
The HB time operator is very efficient on thin spectrum time periodic signals
Shannon minimum sampling to capture a five-harmonic signal
11
machine precision 10-14
harmonic balance
# samples
𝓛2-norm relative error
25
Introduction to HB / Mono-frequential formulation
The HB time operator is very efficient on thin spectrum time periodic signals
100 times more samples than Shannon minimum
1000
larger than machine precision
10-8
4th order
# samples
𝓛2-norm relative error
26
Introduction to HB / Mono-frequential formulation
The HB time operator is very efficient on thin spectrum time periodic signals
100 times more samples than Shannon minimum
1000
larger than machine precision
10-8
4th order
# samples
𝓛2-norm relative error
The harmonic balance time operator is very efficient(i.e. spectral accurate) on discrete spectrum signals
27
Introduction to HB / Multi-frequential formulation
The final goal of the PhD is to compute blade flutter of CROR which is a multi-frequential problem
28
Introduction to HB / Multi-frequential formulation
The final goal of the PhD is to compute blade flutter of CROR which is a multi-frequential problem
involved frequenciesblade flutter
fAEL
28
aeroelastic
Introduction to HB / Multi-frequential formulation
The final goal of the PhD is to compute blade flutter of CROR which is a multi-frequential problem
involved frequencies
wakes
fAEL
28
aeroelastic
Introduction to HB / Multi-frequential formulation
The final goal of the PhD is to compute blade flutter of CROR which is a multi-frequential problem
involved frequencies
potential effects
fAEL
28
aeroelastic
Introduction to HB / Multi-frequential formulation
The final goal of the PhD is to compute blade flutter of CROR which is a multi-frequential problem
involved frequencies
fAEL
29
aeroelastic
Introduction to HB / Multi-frequential formulation
The final goal of the PhD is to compute blade flutter of CROR which is a multi-frequential problem
involved frequencies
fAEL
fBPF
29
aeroelastic
blade passing
Introduction to HB / Multi-frequential formulation
The previously presented harmonic balance time operator is mono-frequential
recall
with
Dt = i!E�1KE
⇥E�1
⇤j ,k
= e i!(j�N)tk
one angular frequency is considered with its
harmonics
30
Introduction to HB / Multi-frequential formulation
In the almost-periodic function framework, a multi-frequential HB time operator can be defined
Besicovitch, Almost Periodic Functions, 1932Ekici & Hall, AIAA Journal, 2007Gopinath et al., AIAA Paper, 2007Guédeney et al., JCP, 2013
Dt = i!E�1KE
⇥E�1
⇤j ,k
= e i!(j�N)tk⇥E�1
⇤j ,k
= e i!j�Ntk
multiple frequencies can be used
mono-frequential multi-frequential
Dt = iE�1⌦E
31
Dt is real
⇥E
⇤k,j
=e�i!(j�N)tk
2N + 1
Introduction to HB / Multi-frequential formulation
In the almost-periodic function framework, a multi-frequential HB time operator can be defined
Besicovitch, Almost Periodic Functions, 1932Ekici & Hall, AIAA Journal, 2007Gopinath et al., AIAA Paper, 2007Guédeney et al., JCP, 2013
Dt = i!E�1KE
⇥E�1
⇤j ,k
= e i!(j�N)tk⇥E�1
⇤j ,k
= e i!j�Ntk
E
mono-frequential multi-frequential
has to be inverted numerically ( is not orthogonal)
Dt = iE�1⌦E
31
Dt is real
Dt is shown numerically to be real at machine precision
E�1
⇥E
⇤k,j
=e�i!(j�N)tk
2N + 1
Introduction to HB /
⇥E
⇤k,j
=e�i!(j�N)tk
2N + 1
Multi-frequential formulation
In the almost-periodic function framework, a multi-frequential HB time operator can be defined
Besicovitch, Almost Periodic Functions, 1932Ekici & Hall, AIAA Journal, 2007Gopinath et al., AIAA Paper, 2007Guédeney et al., JCP, 2013
Dt = i!E�1KE
⇥E�1
⇤j ,k
= e i!(j�N)tk⇥E�1
⇤j ,k
= e i!j�Ntk
E
mono-frequential multi-frequential
has to be inverted numerically ( is not orthogonal)
Dt = iE�1⌦E
32
Dt is real
Dt is shown numerically to be real at machine precision
E�1
The objective of the PhD is to use the multi-frequential harmonic balance approach to
compute the flutter boundary of CROR
Introduction to HB / Multi-frequential formulation
2009
2010
2011
2012
2013
2014
The implementation of harmonic balance into CFD code is a team work
33
2009
2010
2013
2014
F. Sicot, mono-frequential implementation
G. Dufour, extension to decoupled aeroelasticity
T. Guédeney, multi-frequential implementationB. François, extension to CROR computations
A. Gomar, extension to decoupled aeroelasticity of CROR
Introduction to HB / Scientific bottlenecks
Conditioning of multi-frequential HB methodsTest case: periodic injection of a sine function into an advection equation code
34
Ax = b
matrix equation
k�xkkxk (A)
k�AkkAk +
k�bkkbk
�(A)
�A�x �b
numerical errors
condition number
A, b inputs of the problemx
unknown
(A) = kAk · kA�1kby definition
Condition number measures the error amplification during resolution of matrix equation
Introduction to HB / Scientific bottlenecks
Conditioning of multi-frequential HB methodsTest case: periodic injection of a sine function into an advection equation code
35
VDt(W?) + R(W ?) = 0 with Dt = iE�1⌦E
(Dt) = (E�1) · (⌦) · (E )
the equation that is solved is a matrix equation
by definition
condition number depends on the input frequencies, which cannot be changed
(E�1) = (E )
⇥E�1
⇤j ,k
= e i!j�Ntk
looking at the definition of the inverse Fourier transform, the only degree of freedom left is the choice of time samples
we choose to focus on the condition number of E
Introduction to HB / Scientific bottlenecks
Conditioning of multi-frequential HB methodsTest case: periodic injection of a sine function into an advection equation code
36
Lx
0
u(t) = 1 + sin(2⇡ft)
Introduction to HB / Scientific bottlenecks
Conditioning of multi-frequential HB methodsThe analytical solution is the injected function translated by the spatial period
analytical solutionat t = L
x
/c
37
Introduction to HB / Scientific bottlenecks
Conditioning of multi-frequential HB methodsharmonic balance solution with K(E) = 1 is superimposed with the analytical solution
(E ) = 1
38
Introduction to HB / Scientific bottlenecks
Conditioning of multi-frequential HB methodsIncreasing the condition number deteriorates the amplitude of the solution
39
(E ) = 3
Introduction to HB / Scientific bottlenecks
Conditioning of multi-frequential HB methodsFurther increasing the condition number deteriorates the shape of the solution ...
40
(E ) = 5
Introduction to HB / Scientific bottlenecks
Conditioning of multi-frequential HB methods... and gets worse
41
(E ) = 7
Introduction to HB / Scientific bottlenecks
Conditioning of multi-frequential HB methodsThe literature overview shows that the problem is still open
42
preliminary studies show that CROR blade flutter yields a large condition number (> 100)
Kundert et al. (1988) provide APFT to automatically choose time samplesEkici & Hall (2007) oversample with 3N+1 time samples instead of 2N+1Gopinath et al. (2007) do not mention the problemGuédeney (2013) uses the algorithm of Kundert
Ekici & Hall (2007)
Gopinath et al. (2007)
Guédeney (2013)
observed 3.84 16.66 4.57max(E )
Introduction to HB / Scientific bottlenecks
Convergence of HB for wake passing problemsThe results of a CROR low-speed configuration ran with the HB approach ...
43
Introduction to HB / Scientific bottlenecks
Convergence of HB for wake passing problems... are analyzed with the entropy field extracted on a radial slice at 75%
44
Introduction to HB / Scientific bottlenecks
Convergence of HB for wake passing problemsWe expect to see a continuous wake at the rotor/rotor interface
45
rotor/rotor interface
Introduction to HB / Scientific bottlenecks
Convergence of HB for wake passing problemsFor this configuration, using only N=1 gives spurious entropy oscillations ...
N=1
46
rotor/rotor interface
Introduction to HB / Scientific bottlenecks
Convergence of HB for wake passing problemsFor this configuration, using only N=1 gives spurious entropy oscillations ...
N=1
46
rotor/rotor interface
Introduction to HB / Scientific bottlenecks
Convergence of HB for wake passing problems... these spurious oscillations vanish when increasing the number of harmonics
N=2
47
rotor/rotor interface
Introduction to HB / Scientific bottlenecks
N=3 N=3
48
Convergence of HB for wake passing problems... these spurious oscillations vanish when increasing the number of harmonics
rotor/rotor interface
Introduction to HB / Scientific bottlenecks
Convergence of HB for wake passing problemsN=4 seems to be the threshold value to obtain a continuous wake at interface
N=4
49
rotor/rotor interface
Introduction to HB / Scientific bottlenecks
Convergence of HB for wake passing problemsbest practice is not relevant considering the scattered results found in the literature
50
N application
Vilmin et al. (2006) 5 compressor
Ekici (2010) 7 compressor forced vibration
Sicot et al. (2012) 4 compressor
Introduction to HB / Scientific bottlenecks
Convergence of HB for wake passing problemsA naive approach is to increase N until convergence is reached, which is not efficient ...
N=1 N=2 N=3 N=4
51
Introduction to HB / Scientific bottlenecks
Convergence of HB for wake passing problems... as more than 60% of CPU time is wasted for N=4 computation
N=1 N=2 N=3 N=4
62.5 % 37.5 %
52
Outline
Convergence of HB for wake passing problems
53
Conditioning of multi-frequential HB methods
Introduction to the harmonic balance approach Mono-frequential formulationMulti-frequential formulationScientific bottlenecks associated with the PhD
Application to Contra-Rotating Open Rotors
Dt = iE�1⌦E
Condition number /
Outline
Convergence of HB for wake passing problems
54
Conditioning of multi-frequential HB methods High condition number possible for CROR blade flutter problemProposed cure: algorithms that choose the time instants
Introduction to the harmonic balance approach Mono-frequential formulationMulti-frequential formulationScientific bottlenecks associated with the PhD
Application to Contra-Rotating Open Rotors
Dt = iE�1⌦E
Condition number / High condition number for arbitrary frequencies
1041
104
The condition number is computed for arbitrary couples of frequencies
(E )
lower bound1
10observed stability limit
fA
fB
55
Condition number / High condition number for arbitrary frequencies
1041
104
(E )
1
10
fA
fB
56
The condition number is computed for arbitrary couples of frequencies
Condition number / High condition number for arbitrary frequencies
1041
104
(E )
1
10
fA
fB
56
The condition number is computed for arbitrary couples of frequencies
infinite condition number, (singular matrix)
Condition number / High condition number for arbitrary frequencies
1041
104
The symmetry of the results indicates that only the ratio of the frequencies matters
(E )
1
10
fA
fB
57
Condition number / High condition number for arbitrary frequencies
1041
104
(E )
1
10
The condition number is minimum for harmonically related frequencies
fA
fB
fB/fA = 4
fB/fA = 3
fB/fA = 2
58
Condition number / High condition number for arbitrary frequencies
1041
104
The condition number is maximum when the frequencies are segregated ...
(E )
1
10
fA � fB
fB � fA
fA
fB
59
Condition number / High condition number for arbitrary frequencies
1041
104
... or too close from one another
(E )
1
10
fB ⇡ fA
fA
fB
60
Condition number / High condition number for arbitrary frequencies
The statistics for the uniform time sampling are bad since the frequencies are unknown a priori
min max mean std
UNI 2N+1 1.0 9.4 x 1016 1.5 x 1014 2.8 x 1015
61
Condition number / High condition number for arbitrary frequencies
oversampling improves the results (here 3N+1 time instants are used) ...
1041
104
(E )
1
10
fA
fB
Ekici and Hall, AIAA Journal, 2008
62
Condition number / High condition number for arbitrary frequencies
... the statistics are better but still not satisfying
min max mean std
UNI 2N+1 1.0 9.4 x 1016 1.5 x 1014 2.8 x 1015
UNI 3N+1 = / 2.5 / 3.2 / 2.9
2N+1 3N+1
63
Condition number / High condition number for arbitrary frequencies
... the statistics are better but still not satisfying
min max mean std
UNI 2N+1 1.0 9.4 x 1016 1.5 x 1014 2.8 x 1015
UNI 3N+1 = / 2.5 / 3.2 / 2.9
2N+1 3N+1
64
Oversampling gives a negligible improvement considering the additional CPU and memory cost that it requires
Condition number / Proposed cure: algorithms that choose the time instants
Using an optimization approach, time instants are chosen to minimize the condition number
65
TOPT = minl�bfgs�b
((E [T ]), Tini )
Byrd et al., SIAM Journal on Scientific Computing, 1995Zhu et al., ACM Transactions on Mathematical Software, 1997
Tini
take smallest frequency to compute largest period1
fmin = min(F ) Tmax
= 2⇡/fmin
0
oversample largest period with M sub-periods (typically M=1000)2
T1 T2 TM�1 TM = Tmax
?
Condition number / Proposed cure: algorithms that choose the time instants
Using an optimization approach, time instants are chosen to minimize the condition number
66
Byrd et al., SIAM Journal on Scientific Computing, 1995Zhu et al., ACM Transactions on Mathematical Software, 1997
compute the condition number associated to each set of time instances
4
i = (E [Ti ])
take the set that gives the minimum condition number as starting point
5
min(i ) = (E [Tini ])
uniformly sample each period with 2N+1 time instants3
Ti =0,
1
2N + 1, · · · , 2N
2N + 1
�Ti
Condition number / Proposed cure: algorithms that choose the time instants
Using an optimization approach, time instants are chosen to minimize the condition number
67
Byrd et al., SIAM Journal on Scientific Computing, 1995Zhu et al., ACM Transactions on Mathematical Software, 1997
compute the condition number associated to each set of time instances
4
i = (E [Ti ])
take the set that gives the minimum condition number as starting point
5
min(i ) = (E [Tini ])
uniformly sample each period with 2N+1 time instants3
Ti =0,
1
2N + 1, · · · , 2N
2N + 1
�TiA good starting point combined with a
gradient-based optimization algorithm on the condition number gives a set of time instants
Condition number / Proposed cure: algorithms that choose the time instants
The OPT algorithm retrieves a condition number close to 1 for all choice of frequencies ...
1041
104
(E )
1
10
fA
fB
68
Condition number / Proposed cure: algorithms that choose the time instants
... and zooming on the colormap emphasizes this analysis
1041
104
(E )
1
1.2
fA
fB
69
Condition number / Proposed cure: algorithms that choose the time instants
The OPT algorithm provides results compatible with an industrial context
min max mean std
UNI 2N+1 1.0 9.4 x 1016 1.5 x 1014 2.8 x 1015
OPT 1.0 2.6 1.1 0.077
Guédeney et al., JCP, 2013
70
Condition number / Proposed cure: algorithms that choose the time instants
The OPT algorithm provides results compatible with an industrial context
min max mean std
UNI 2N+1 1.0 9.4 x 1016 1.5 x 1014 2.8 x 1015
OPT 1.0 2.6 1.1 0.077
OPT algorithm alleviates the stability issuesencountered with an arbitrary choice of frequencies.
This is a pre-processing step that takes less than a minute
71Guédeney et al., JCP, 2013
Outline
Convergence of HB for wake passing problems
72
Conditioning of multi-frequential HB methods High condition number possible for CROR blade flutter problemProposed cure: algorithms that choose the time instants
Introduction to the harmonic balance approach Mono-frequential formulationMulti-frequential formulationScientific bottlenecks associated with the PhD
Application to Contra-Rotating Open Rotors
Dt = iE�1⌦E
Condition number /
Outline
Convergence of HB for wake passing problemsScattered convergence observed on CROR configurationsA priori estimate of the required number of harmonics
73
Conditioning of multi-frequential HB methods High condition number possible for CROR blade flutter problemProposed cure: algorithms that choose the time instants
Introduction to the harmonic balance approach Mono-frequential formulationMulti-frequential formulationScientific bottlenecks associated with the PhD
Application to Contra-Rotating Open Rotors
Dt = iE�1⌦E
Convergence /
The convergence of three configurations is assessed ...
74
C2
same mockup CROR at cruise design by Safran-Snecma
C3AI-PX7 CROR at cruise
design by Airbus
C1
mockup CROR at take-off design by Safran-Snecma
Negulescu, SAE, 2013François, PhD thesis, 2013
Scattered convergence on CROR
Convergence /
C1 for which N=4 seems to be the threshold value to obtain a continuous wake at interface
75
N=4N=3
N=2N=1
Scattered convergence on CROR
Convergence /
C2 for which N=8 provides a continuous wake at interface
76
N=8N=7
N=6N=5
Scattered convergence on CROR
Convergence /
C3 for which N=10 is not sufficient to converge
77
N=10N=9
N=8N=7
Scattered convergence on CROR
Convergence /
The convergence of three configurations is assessed ...
78
C2converged at N=8
same mockup CROR at cruise design by Safran-Snecma
C3not converged at N=10
AI-PX7 CROR at cruisedesign by Airbus
C1converged at N=4
mockup CROR at take-off design by Safran-Snecma
Negulescu, SAE, 2013François, PhD thesis, 2013
Scattered convergence on CROR
Convergence /
The convergence of three configurations is assessed ...
C2converged at N=8
same mockup CROR at cruise design by Safran-Snecma
C3not converged at N=10
AI-PX7 CROR at cruisedesign by Airbus
79
C1converged at N=4
mockup CROR at take-off design by Safran-Snecma
Negulescu, SAE, 2013François, PhD thesis, 2013
Two questions:1) why do these entropy oscillations appear ?2) how to estimate the number of harmonics ?
Scattered convergence on CROR
Convergence /
The wake is steady in the front rotor frame of reference
rotor/rotor interface
80A priori estimate of the required number of harmonics
Convergence /
The wake is steady in the front rotor frame of reference
rotor/rotor interface
front rotor frame of reference
81A priori estimate of the required number of harmonics
Convergence /
The wake is steady in the front rotor frame of reference
rotor/rotor interface
wakes are seen steadyby the front rotor frame
of reference
81A priori estimate of the required number of harmonics
Convergence /
But becomes unsteady when crossing the rotor/rotor interface
rotor/rotor interface
rear rotor frame of reference
82A priori estimate of the required number of harmonics
Convergence /
But becomes unsteady when crossing the rotor/rotor interface
rotor/rotor interface
front rotor wakes become unsteady due to the relative velocity difference
82A priori estimate of the required number of harmonics
Convergence /
But becomes unsteady when crossing the rotor/rotor interface
rotor/rotor interface
front rotor wakes become unsteady due to the relative velocity difference
at rotor/rotor interface, steady tangential distortions in front rotor frame of reference are seen unsteady by rear rotor
83A priori estimate of the required number of harmonics
Convergence /
The tangential distortion at rotor/rotor interface is available with a steady mixing-plane computation
84A priori estimate of the required number of harmonics
Convergence /
One can then estimate the content of the temporal spectrum using a steady result
extract tangential distortion at interface
1
for each radii
compute azimuthal spectrum = temporal
spectrum
2
85
spec
trum
of uTangential Fourier
transform
frequency
A priori estimate of the required number of harmonics
Convergence /
We define the accumulated energy up to a given number of harmonics
spec
trum
of u
frequency
resolved part
unresolved part
N
86A priori estimate of the required number of harmonics
Convergence /
We define the accumulated energy up to a given number of harmonics
N
E (N) =
� �2�
+�2
87
"(N) =
� ��
+�
truncation error
accumulated energy
E (N) = 1� "2(N)
spec
trum
of u
frequency
A priori estimate of the required number of harmonics
0 %
100 %
E (N)
100
120
00 25
R [%]
# harmonics
Convergence /
front rotor blade tip
front rotor hub
88A priori estimate of the required number of harmonics
0 %
100 %
E (N)
100
120
00 25
R [%]
# harmonics
Convergence /
For C1, the energy seems to decay fast ...
89A priori estimate of the required number of harmonics
Convergence /
... but what is the threshold of energy required to properly reconstruct a wake ?
full energy wake
90A priori estimate of the required number of harmonics
Convergence /
using 50% of the energy gives a wake whose width is doubled and deficit divided by two
wake represented with 50% of energy
91A priori estimate of the required number of harmonics
Convergence /
using 90% improves the reconstruction as the width is correctly estimated but not the deficit
wake represented with 90% of energy
92A priori estimate of the required number of harmonics
Convergence /
With 99%, both the wake width and deficit seem to be correctly represented
wake represented with 99% of energy
93A priori estimate of the required number of harmonics
0 %
100 %
E (N)
100
120
00 25
R [%]
# harmonics
Convergence / 94A priori estimate of the required number of harmonics
0 %
100 %
E (N)
100
120
00 25
R [%]
# harmonics
Convergence /
E (N) = 99%
95A priori estimate of the required number of harmonics
0 %
100 %
E (N)
100
120
00 25
R [%]
# harmonics
Convergence /
For C1, N=4 is needed to capture 99% of energy on the whole blade radius
E (N) = 99%
N=4
95A priori estimate of the required number of harmonics
0 %
100 %
E (N)
100
120
00 25
R [%]
# harmonics
Convergence /
For C2, N=7 is needed to capture 99% of energy on the whole blade radius
N=7
96A priori estimate of the required number of harmonics
0 %
100 %
E (N)
100
120
00 25
R [%]
# harmonics
Convergence /
For C3, N=17 is needed to capture 99% of energy on almost all blade radius
N=17
97A priori estimate of the required number of harmonics
0 %
100 %
E (N)
100
120
00 25
R [%]
# harmonics
Convergence /
Actually, N=22 is needed to capture the front rotor tip vortex
N=22
98A priori estimate of the required number of harmonics
Convergence /
The prediction given by the a priori estimator is consistent with the observed convergence
C1converged at N=4
prediction N=4
C2converged at N=8
prediction N=7
C3not converged at N=10
prediction N=17
Gomar et al., JCP minor revisions, 2014
99A priori estimate of the required number of harmonics
Convergence /
The prediction given by the a priori estimator is consistent with the observed convergence
C1converged at N=4
prediction N=4
C2converged at N=8
prediction N=7
C3not converged at N=10
prediction N=17
100
Using the proposed a priori estimator, one can estimate the number of harmonics required
to compute a given configuration
A priori estimate of the required number of harmonicsGomar et al., JCP minor revisions, 2014
Outline
Convergence of HB for wake passing problemsScattered convergence observed on CROR configurationsA priori estimate of the required number of harmonics
101
Conditioning of multi-frequential HB methods High condition number possible for CROR blade flutter problemProposed cure: algorithms that choose the time instants
Introduction to the harmonic balance approach Mono-frequential formulationMulti-frequential formulationScientific bottlenecks associated with the PhD
Application to Contra-Rotating Open Rotors
Dt = iE�1⌦E
Application to CROR /
Outline
Convergence of HB for wake passing problemsScattered convergence observed on CROR configurationsA priori estimate of the required number of harmonics
102
Conditioning of multi-frequential HB methods High condition number possible for CROR blade flutter problemProposed cure: algorithms that choose the time instants
Introduction to the harmonic balance approach Mono-frequential formulationMulti-frequential formulationScientific bottlenecks associated with the PhD
Application to Contra-Rotating Open RotorsSteady resultsUnsteady rigid resultsAeroelastic results
Dt = iE�1⌦E
Application to CROR /
A low-speed configuration is studied
M = 0.2
103steady results
Application to CROR /
A steady mixing plane computation is first launched with
- Roe scheme with a second order MUSCL extrapolation- Spalart & Allmaras one equation turbulence model- maximum CFL set to 10- 04H topology with 129 grid points around the blade, 45
on the pitch and 181 in the radial extent
104
Roe, JCP, 1981Spalart & Allmaras, AIAA Paper, 1992
steady results
Application to CROR /
The computation is converged starting at 500 iterations
105steady results
0
0.5
1
1.5
2
2.5
3
0 500 1000 1500 2000
front rotorrear rotorglobal
101
102
103
104
105
106
107
108
0 500 1000 1500 2000
iteration
tract
ion
coef
ficien
t
dens
ity re
sidua
l
iteration
Application to CROR /
The radial distribution of two variables is analyzed at six locations
106
P2P1 P4P3 P6P5
steady results
Application to CROR /
The Mach number goes from 0.2 up to 0.4 which yields the thrust
107
0.2
0.4
0.6
0.8
1
0.1
0.2
0.3
0.4
0.5
R/R
f[!
]
Ma [!]
P1 P2 P3 P4 P5 P6
steady results
Application to CROR /
The flow is straighten up which is the main advantage of CROR compared to a propeller
108
0.2
0.4
0.6
0.8
1
-10.0
0.0
10.0
20.0
R/R
f[!
]
! [!]
P1 P2 P3 P4 P5 P6
steady results
Application to CROR /
x/c
Kp
Kp
-10
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1
-10
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1
radial slice 25% front blade
radius
The relative Mach number contours shows a smooth subsonic flow field
109
Mrel
steady results
Application to CROR /
x/c
Kp
Kp
-10
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1
-10
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1
radial slice 50% front blade
radius
The relative Mach number contours shows a smooth subsonic flow field
110steady results
Mrel
Application to CROR /
x/c
Kp
Kp
-10
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1
-10
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1
radial slice 75% front blade
radius
The relative Mach number contours shows a smooth subsonic flow field
111steady results
Mrel
Application to CROR /
x/c
Kp
Kp
-10
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1
-10
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1
radial slice 90% front blade
radius
... with the leaving of the tip vortices on the rear rotor
112steady results
Mrel
Application to CROR /
x/c
Kp
Kp
-10
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1
-10
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1
radial slice 95% front blade
radius
... with the leaving of the tip vortices on the rear rotor
113steady results
Mrel
Application to CROR /
x/c
Kp
Kp
-10
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1
-10
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1
radial slice 95% front blade
radius
... with the leaving of the tip vortices on the rear rotor
114steady results
Mrel
The a priori estimator can then be used on the steady results to estimate the
number of harmonics needed for the HB
Application to CROR /
Using the a priori estimator, N=4 harmonics should be sufficient
E (N) = 99%
N=4
0 %
100 %
E (N)
100
120
00 25
R [%]
# harmonics
115unsteady rigid results
Application to CROR /
The harmonic balance approach is able to compute the main unsteady effect seen in a CROR ...
116unsteady rigid results
Application to CROR /
... and also the rotor/rotor interactions
117
M 0
unsteady rigid results
Application to CROR /
... and also the rotor/rotor interactions
118
We look at Mach number fluctuations as perceived by the
front rotorM 0
unsteady rigid results
M 0(t) = M(t)�M
Application to CROR /
... and also the rotor/rotor interactions
119
We look at Mach number fluctuations as perceived by the
front rotor
potential effects
M 0
unsteady rigid results
M 0(t) = M(t)�M
Application to CROR /
... and also the rotor/rotor interactions
120
M 0
We look at Mach number fluctuations as perceived by the
rear rotor
unsteady rigid results
M 0(t) = M(t)�M
Application to CROR /
... and also the rotor/rotor interactions
121
M 0
We look at Mach number fluctuations as perceived by the
rear rotor
velocity deficits attributed to wake
passing
unsteady rigid results
M 0(t) = M(t)�M
Application to CROR /
... and also the rotor/rotor interactions
122
M 0
unsteady rigid results
Application to CROR /
We can now compute the flutter boundary of the front rotor
mode 2Ffront rotor
123
mode 1Tfront rotor
aeroelastic resultsmodes amplified by a factor 600
Application to CROR /
fBPF2fBPF3fBPF4fBPF5fBPF
1.0 (E ) 1.4
We compute five frequencies using the multi-frequential HB
124
fAEL
(E ) = 19.2 (E ) 1313
using OPT
uniform sampling
since the frequencies are harmonically related
aeroelastic results
for the two modes, the condition number varies:
fBPF2fBPF3fBPF4fBPF
compute BPF associated to front rotor
compute BPF associated to
rear rotor
Application to CROR /
Both modes are shown to be cleared from flutter (positive damping)
125
mode 2F mode 1T
stableunstable
aeroelastic results
suction side pressure side suction side pressure side
Application to CROR /
Modes nodal lines are also nodal lines for the damping
126
mode 2F mode 1T
stableunstable
aeroelastic results
suction side pressure side suction side pressure side
Application to CROR /
Aerodynamic variation lines are also nodal lines for the damping
127
mode 2F mode 1T
stableunstable
aeroelastic results
suction side pressure side suction side pressure side
attributed to the end of acceleration at
suction side
Application to CROR /
The HB requires less degrees of freedom meaning that convergence is easily ensured
‣ time-stepcan capture up to
‣ number of sub-iterations‣ number of time-steps
(i.e. number of time periods to bypass transient)
‣ phaselag (mono-frequential) or 360°
128
‣ number of harmonics‣ number of iterations
steady convergence monitoring‣ always phaselag
(mono or multi-frequential)
DTS degrees of freedom HB degrees of freedom
tentative cost comparison
1/2�t
Application to CROR /
Comparison of phaselag HB and phaselag DTS, elsA HB implementation facts
129
Rotor/Stator configuration
tentative cost comparison
Isolated aeroelastic configuration
N=4 is needed to capture the unsteady outlet mass-flow rate.
Gain 5 compared to phaselag DTS
N=1 is enough to capture the damping.
Gain 3 to10 compared to phaselag DTS, depending on the case considered (operating point, IBPA ...)
Sicot et al., JTM, 2012Sicot et al., AIAA, 2014
Application to CROR /
Comparison of phaselag HB and phaselag DTS, elsA HB implementation facts
130
Rotor/Stator configuration
tentative cost comparison
Isolated aeroelastic configuration
N=4 is needed to capture the unsteady outlet mass-flow rate.
Gain 5 compared to phaselag DTS
N=1 is needed to capture the damping enough.
Gain 3 to10 compared to phaselag DTS, depending on the case considered (operating point, IBPA ...)
Sicot et al., JTM, 2012Sicot et al., AIAA, 2014
This can be explained as the DTS requires 20 pts per period to converge while
HB requires only 3 pts which leads to gain for HB of 6.6
Application to CROR /
Tentative cost comparison between HB and 360 DTS for CROR blade flutter
- 360 DTS required as phaselag DTS is not applicable with multiple frequencies non harmonically related
- In opposite, HB multi-frequential can be used with multiple frequencies and the cost scales with the number of additional frequencies
- Finally:
Gain = 7 x (Mean number of blades)
131tentative cost comparison
Temporal operator comparison
Mesh size reduction (multi-frequential
phaselag)
Application to CROR /
Tentative cost comparison between HB and 360 DTS for CROR blade flutter
- 360 DTS required as phaselag DTS is not applicable with multiple frequencies non harmonically related
- In opposite, HB multi-frequential can be used with multiple frequencies and the cost scales with the number of additional frequencies
- Finally:
Gain = 7 x (Mean number of blades)
132tentative cost comparison
Temporal operator comparison
Mesh size reduction (multi-frequential
phaselag)
For our CROR configurations, this gives an expected gain of 70,
roughly 2 order of magnitude
Outline
Convergence of HB for wake passing problemsScattered convergence observed on CROR configurationsA priori estimate of the required number of harmonics
133
Conditioning of multi-frequential HB methods High condition number possible for CROR blade flutter problemProposed cure: algorithms that choose the time instants
Introduction to the harmonic balance approach Mono-frequential formulationMulti-frequential formulationScientific bottlenecks associated with the PhD
Application to Contra-Rotating Open RotorsSteady resultsUnsteady rigid resultsAeroelastic results
Dt = iE�1⌦E
Conclusions & Perspectives
On the conditioning of multi-frequential HB methods
134
‣ The presented OPT algorithm allows to minimize the condition number of the multi-frequential HB approach by properly choosing the time instants. It shows to be both robust and accurate as it gives a condition number close to the theoretical lower bound for almost all choices of frequencies. This work has been published in:
Guédeney, Gomar, Gallard, Sicot, Dufour and PuigtNon-Uniform Time Sampling for Multiple-Frequency Harmonic Balance Journal of Computational Physics, 2013
Conclusions & Perspectives
On the convergence of HB approaches for wake passing problems
135
‣ The number of harmonics required to compute a given CROR configuration has been estimated. This a priori estimator is based on an affordable mixing plane simulation. The prediction for three CROR configurations shows to be accurate, allowing to skip the naive iterative approach. This work has been accepted with minor revisions in:
Gomar, Bouvy, Sicot, Dufour, Cinnella and FrançoisConvergence of Fourier-based time methods for turbomachinery wake passing problemsJournal of Computational Physics, minor revisions in April 2014
Conclusions & Perspectives
On the aeroelasticity of CROR‣ The multi-frequential HB method with both the OPT algorithm
and the a priori estimator allowed to compute the blade flutter response of a low-speed CROR. This proved the maturity and robustness of the multi-frequential HB approach
136
Conclusions & Perspectives
Toward installed CROR configurations‣ Using the a priori estimator on a mixing plane simulation of an
installed CROR showed that 300 harmonics are required to capture 99% of the energy. This highlight the power of the tool as it helps the decision making, e.g. here the HB computation on such a configuration was not launched.
137
400
0
R [%]
# harmonicsN=300
Conclusions & Perspectives
Toward accurate aeroelastic simulations of CROR
‣ In this PhD, only the front rotor forced vibration has been assessed, the rear rotor remains to be done. Second, the choice of frequencies for the multi-frequential HB approach has been partially assessed and further studies need to be conducted: the influence of the vibration on the aerodynamic of the opposite blade should be taken into account and also its influence back to the vibrating blade.
138
Multi-Frequential Harmonic Balance Approachfor the Simulation of Contra-Rotating Open Rotors:
Application to Aeroelasticity
Adrien Gomarsupervisor Paola Cinnella
co-supervisor Frédéric Sicot
in partnership with
List of publicationsPeer-reviewed Journals
‣ Guédeney, Gomar, Gallard, Sicot, Dufour and PuigtNon-Uniform Time Sampling for Multiple-Frequency Harmonic Balance Journal of Computational Physics, 2013
‣ Sicot, Gomar, Dufour and DugeaiTime-Domain Harmonic Balance Method for Turbomachinery Aeroelasticity AIAA Journal, 2014
‣ Gomar, Bouvy, Sicot, Dufour, Cinnella and FrançoisConvergence of Fourier-based time methods for turbomachinery wake passing problemsJournal of Computational Physics, minor revision in April 2014
Conference contributions‣ Guédeney, Gomar, and Sicot
Multi-Frequential Harmonic Balance Approach for the Computation of Unsteadiness in Multi-Stage TurbomachinesCongrès Français de Mécanique, August 2013