Adrien Gomar, PhD defense

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


CO2

NOx

noise

-75%

-80%

-65%


expected 2050 values 2000 values

4

Introduction


CO2

NOx

noise

-75%

-80%

-65%


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


hot air

BPR =mc

mh

mh

7

Introduction


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




design

1D results

validation

3D results

research

turbulent effects

required to simulate unsteady response

to flutter

12

Introduction

investigation

unsteady effects




design

1D results

validation

3D results

research

turbulent effects

turn-around time compatible with

industry

12

Introduction

investigation

unsteady effects


RANS URANS

hours days

validation

3D results

research

turbulent effects

12

Introduction

investigation

unsteady effects


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


14


Introduction to the harmonic balance approach Mono-frequential formulationMulti-frequential formulationScientific bottlenecks associated with the PhD


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


The URANS equations are transformed ...

VdW

dt+ R(W ) = 0

finite volume, semi discrete form of the Navier-Stokes equations

16

unsteady problem



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



VdW

dt+ R(W ) = 0

Vd

dt

⇣E�1cW ?

⌘+ R(E�1cW ?) = 0

using the inverse Fourier matrixW ? = E�1cW ?

17

unsteady problem


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



VdW

dt+ R(W ) = 0

Vd

dt

�E�1

� cW ? + R(E�1cW ?) = 0

19

using the Fourier matrixcW ? = EW ?

unsteady problem




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




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 steady equations



VdW

dt+ R(W ) = 0

Vd

dt

�E�1

� cW ? + R(E�1cW ?) = 0


20

Vd

dt

�E�1

�EW ? + R(E�1EW ?) = 0

unsteady problem


| {z }identity matrix




VdW

dt+ R(W ) = 0

Vd

dt

�E�1

� cW ? + R(E�1cW ?) = 0


20

Vd

dt

�E�1

�EW ? + R(E�1EW ?) = 0

unsteady problem


| {z }identity matrixDt



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



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


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


harmonic balance

1000# samples

𝓛2-norm relative error

4th order

2nd order

1

10-161

24


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


25



100 times more samples than Shannon minimum

1000

larger than machine precision

10-8

4th order

# samples


26



100 times more samples than Shannon minimum

1000

larger than machine precision

10-8

4th order

# samples


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



involved frequenciesblade flutter

fAEL

28

aeroelastic



involved frequencies

wakes

fAEL

28

aeroelastic




potential effects

fAEL

28

aeroelastic




fAEL

29

aeroelastic




fAEL

fBPF

29

aeroelastic

blade passing


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


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




Dt = i!E�1KE

⇥E�1

⇤j ,k


⇤j ,k

= e i!j�Ntk

E


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



Dt = i!E�1KE

⇥E�1

⇤j ,k


⇤j ,k

= e i!j�Ntk

E


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


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



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



36

Lx

0

u(t) = 1 + sin(2⇡ft)


Conditioning of multi-frequential HB methodsThe analytical solution is the injected function translated by the spatial period

analytical solutionat t = L

x

/c

37


Conditioning of multi-frequential HB methodsharmonic balance solution with K(E) = 1 is superimposed with the analytical solution

(E ) = 1

38


Conditioning of multi-frequential HB methodsIncreasing the condition number deteriorates the amplitude of the solution

39

(E ) = 3


Conditioning of multi-frequential HB methodsFurther increasing the condition number deteriorates the shape of the solution ...

40

(E ) = 5


Conditioning of multi-frequential HB methods... and gets worse

41

(E ) = 7


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 )


Convergence of HB for wake passing problemsThe results of a CROR low-speed configuration ran with the HB approach ...

43


Convergence of HB for wake passing problems... are analyzed with the entropy field extracted on a radial slice at 75%

44


Convergence of HB for wake passing problemsWe expect to see a continuous wake at the rotor/rotor interface

45

rotor/rotor interface


Convergence of HB for wake passing problemsFor this configuration, using only N=1 gives spurious entropy oscillations ...

N=1

46



Convergence of HB for wake passing problems... these spurious oscillations vanish when increasing the number of harmonics

N=2

47



N=3 N=3

48

Convergence of HB for wake passing problems... these spurious oscillations vanish when increasing the number of harmonics



Convergence of HB for wake passing problemsN=4 seems to be the threshold value to obtain a continuous wake at interface

N=4

49



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


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


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


53




Dt = iE�1⌦E

Condition number /

Outline


54

Conditioning of multi-frequential HB methods High condition number possible for CROR blade flutter problemProposed cure: algorithms that choose the time instants



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


1041

104

(E )

1

10

fA

fB

56



1041

104

(E )

1

10

fA

fB

56


infinite condition number, (singular matrix)


1041

104

The symmetry of the results indicates that only the ratio of the frequencies matters

(E )

1

10

fA

fB

57


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


1041

104

The condition number is maximum when the frequencies are segregated ...

(E )

1

10

fA � fB

fB � fA

fA

fB

59


1041

104

... or too close from one another

(E )

1

10

fB ⇡ fA

fA

fB

60


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


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


... 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


... 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

?



66


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



67


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


The OPT algorithm retrieves a condition number close to 1 for all choice of frequencies ...

1041

104

(E )

1

10

fA

fB

68


... and zooming on the colormap emphasizes this analysis

1041

104

(E )

1

1.2

fA

fB

69


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


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


72




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




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


Convergence /

C2 for which N=8 provides a continuous wake at interface

76

N=8N=7

N=6N=5


Convergence /

C3 for which N=10 is not sufficient to converge

77

N=10N=9

N=8N=7


Convergence /


78

C2converged at N=8


C3not converged at N=10

AI-PX7 CROR at cruisedesign by Airbus

C1converged at N=4




Convergence /


C2converged at N=8



AI-PX7 CROR at cruisedesign by Airbus

79

C1converged at N=4



Two questions:1) why do these entropy oscillations appear ?2) how to estimate the number of harmonics ?


Convergence /

The wake is steady in the front rotor frame of reference


80A priori estimate of the required number of harmonics

Convergence /



front rotor frame of reference


Convergence /



wakes are seen steadyby the front rotor frame

of reference


Convergence /

But becomes unsteady when crossing the rotor/rotor interface


rear rotor frame of reference


Convergence /



front rotor wakes become unsteady due to the relative velocity difference


Convergence /



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


Convergence /

The tangential distortion at rotor/rotor interface is available with a steady mixing-plane computation


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


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


0 %

100 %

E (N)

100

120

00 25

R [%]

# harmonics

Convergence /

For C1, the energy seems to decay fast ...


Convergence /

... but what is the threshold of energy required to properly reconstruct a wake ?

full energy wake


Convergence /

using 50% of the energy gives a wake whose width is doubled and deficit divided by two

wake represented with 50% of energy


Convergence /

using 90% improves the reconstruction as the width is correctly estimated but not the deficit



Convergence /

With 99%, both the wake width and deficit seem to be correctly represented



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%


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


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


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


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


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


prediction N=17

Gomar et al., JCP minor revisions, 2014


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


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


101




Dt = iE�1⌦E

Application to CROR /

Outline


102



Application to Contra-Rotating Open RotorsSteady resultsUnsteady rigid resultsAeroelastic results

Dt = iE�1⌦E


A low-speed configuration is studied

M = 0.2

103steady results


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


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


The radial distribution of two variables is analyzed at six locations

106

P2P1 P4P3 P6P5

steady results


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


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


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


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


radius


110steady results

Mrel


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


radius


111steady results

Mrel


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


radius

... with the leaving of the tip vortices on the rear rotor

112steady results

Mrel


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


radius


113steady results

Mrel


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


radius


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


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


The harmonic balance approach is able to compute the main unsteady effect seen in a CROR ...

116unsteady rigid results


... and also the rotor/rotor interactions

117

M 0

unsteady rigid results



118

We look at Mach number fluctuations as perceived by the

front rotorM 0


M 0(t) = M(t)�M



119


front rotor

potential effects

M 0


M 0(t) = M(t)�M



120

M 0


rear rotor


M 0(t) = M(t)�M



121

M 0


rear rotor

velocity deficits attributed to wake

passing


M 0(t) = M(t)�M



122

M 0



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


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


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


Modes nodal lines are also nodal lines for the damping

126

mode 2F mode 1T

stableunstable

aeroelastic results



Aerodynamic variation lines are also nodal lines for the damping

127

mode 2F mode 1T

stableunstable

aeroelastic results


attributed to the end of acceleration at

suction side


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


Comparison of phaselag HB and phaselag DTS, elsA HB implementation facts

129

Rotor/Stator configuration


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


Comparison of phaselag HB and phaselag DTS, elsA HB implementation facts

130

Rotor/Stator configuration


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


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)


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


133



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


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


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


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


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

Adrien Gomar, PhD defense

Presentations & Public Speaking

Transcript of Adrien Gomar, PhD defense