Adrien Gomar, PhD defense

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

Introduction

Demanding objectives are set toward the aeronautical industry

Flightpath 2050 Europe’s Vision for Aviation

2000 values

Introduction

expected 2050 values 2000 values

Introduction

expected 2050 values 2000 values

One way to reduce certain pollutants is todecrease fuel consumption by increasing the bypass ratio

Introduction

The bypass ratio defines the mass-flow ratio of cold air to hot air

BPR =mc

Introduction

hot air

BPR =mc

Introduction

cold air

BPR =mc

Introduction

Contra-Rotating Open Rotor (CROR) is a concept based on an increase of the bypass ratio

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

Introduction

Flutter is a self-excited, self-sustained aeroelastic phenomenon

Introduction

Flutter is a self-excited, self-sustained aeroelastic phenomenon

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

Introduction

investigation

unsteady effects

design

1D results

validation

3D results

research

turbulent effects

required to simulate unsteady response

to flutter

Introduction

investigation

unsteady effects

design

1D results

validation

3D results

research

turbulent effects

turn-around time compatible with

industry

Introduction

investigation

unsteady effects

RANS URANS

hours days

validation

3D results

research

turbulent effects

Introduction

investigation

unsteady effects

RANS URANS

hours days

validation

3D results

research

turbulent effects

investigation

unsteady effects

Outline

Convergence of HB for wake passing problems

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

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

hypotheses:

turbulence is fully modeled3 URANS equations

The URANS equations are transformed ...

dt+ R(W ) = 0

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

unsteady problem

dt+ R(W ) = 0

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

dt+ R(W ?) = 0

discretizing the problem

unsteady problem

dt+ R(W ) = 0

⇣E�1cW ?

⌘+ R(E�1cW ?) = 0

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

unsteady problem

dt+ R(W ) = 0

⇣E�1cW ?

⌘+ R(E�1cW ?) = 0

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

unsteady problem

The URANS equations are transformed into a subset of 2N+1 harmonic equations

dt+ R(W ) = 0

harmonics are time-independent

�E�1

� cW ? + R(E�1cW ?) = 0

He & Ning, AIAA Journal, 1998McMullen et al., AIAA Paper, 2001

unsteady problem

2N+1 harmonicequations

dt+ R(W ) = 0

�E�1

� cW ? + R(E�1cW ?) = 0

using the Fourier matrixcW ? = EW ?

unsteady problem

dt+ R(W ) = 0

�E�1

� cW ? + R(E�1cW ?) = 0

using the Fourier matrixcW ? = EW ?

�E�1

�EW ? + R(E�1EW ?) = 0

unsteady problem

dt+ R(W ) = 0

�E�1

� cW ? + R(E�1cW ?) = 0

Hall et al., AIAA Journal, 2002Gopinath & Jameson, AIAA Paper, 2005

�E�1

�EW ? + R(E�1EW ?) = 0

unsteady problem

2N+1 steady equations

dt+ R(W ) = 0

�E�1

� cW ? + R(E�1cW ?) = 0

�E�1

�EW ? + R(E�1EW ?) = 0

unsteady problem

| {z }identity matrix

dt+ R(W ) = 0

�E�1

� cW ? + R(E�1cW ?) = 0

�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

dt+ R(W ) = 0

equivalent under given hypotheses

Dt = i!E�1KEwith

64W0...

75+ R(

64W0...

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)

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)

dt(t = tq) =

�uq+2 + 8uq+1 � 8uq�1 + uq�2

12�t+O(�t4)

Three time derivative operators are tested

dt= i!E�1KE

2nd order

4th order

harmonic balance

dt(t = tq) =

uq+1 � uq�1

2�t+O(�t2)

harmonic balance

1000# samples

𝓛2-norm relative error

4th order

2nd order

10-161

The HB time operator is very efficient on thin spectrum time periodic signals

Shannon minimum sampling to capture a five-harmonic signal

machine precision 10-14

harmonic balance

# samples

100 times more samples than Shannon minimum

larger than machine precision

4th order

# samples

100 times more samples than Shannon minimum

larger than machine precision

4th order

# samples

The harmonic balance time operator is very efficient(i.e. spectral accurate) on discrete spectrum signals

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

aeroelastic

involved frequencies

aeroelastic

potential effects

aeroelastic

blade passing

The previously presented harmonic balance time operator is mono-frequential

recall

Dt = i!E�1KE

⇥E�1

⇤j ,k

= e i!(j�N)tk

one angular frequency is considered with its

harmonics

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

Dt is real

⇤k,j

=e�i!(j�N)tk

2N + 1

Dt = i!E�1KE

⇥E�1

⇤j ,k

= e i!j�Ntk

has to be inverted numerically ( is not orthogonal)

Dt = iE�1⌦E

Dt is real

Dt is shown numerically to be real at machine precision

⇤k,j

=e�i!(j�N)tk

2N + 1

Introduction to HB /

⇤k,j

=e�i!(j�N)tk

2N + 1

Multi-frequential formulation

Dt = i!E�1KE

⇥E�1

⇤j ,k

= e i!j�Ntk

has to be inverted numerically ( is not orthogonal)

Dt = iE�1⌦E

Dt is real

Dt is shown numerically to be real at machine precision

The objective of the PhD is to use the multi-frequential harmonic balance approach to

compute the flutter boundary of CROR

The implementation of harmonic balance into CFD code is a team work

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

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

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

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

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

(E ) = 1

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

(E ) = 3

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

(E ) = 5

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

(E ) = 7

Conditioning of multi-frequential HB methodsThe literature overview shows that the problem is still open

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 ...

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

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

rotor/rotor interface

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

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

N=3 N=3

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

Convergence of HB for wake passing problemsbest practice is not relevant considering the scattered results found in the literature

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

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 %

Outline

Dt = iE�1⌦E

Condition number /

Outline

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

The condition number is computed for arbitrary couples of frequencies

lower bound1

10observed stability limit

infinite condition number, (singular matrix)

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

The condition number is minimum for harmonically related frequencies

fB/fA = 4

fB/fA = 3

fB/fA = 2

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

fA � fB

fB � fA

... or too close from one another

fB ⇡ fA

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

oversampling improves the results (here 3N+1 time instants are used) ...

Ekici and Hall, AIAA Journal, 2008

... 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

... 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

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

TOPT = minl�bfgs�b

((E [T ]), Tini )

Byrd et al., SIAM Journal on Scientific Computing, 1995Zhu et al., ACM Transactions on Mathematical Software, 1997

take smallest frequency to compute largest period1

fmin = min(F ) Tmax

= 2⇡/fmin

oversample largest period with M sub-periods (typically M=1000)2

T1 T2 TM�1 TM = Tmax

compute the condition number associated to each set of time instances

i = (E [Ti ])

take the set that gives the minimum condition number as starting point

min(i ) = (E [Tini ])

uniformly sample each period with 2N+1 time instants3

Ti =0,

2N + 1, · · · , 2N

2N + 1

compute the condition number associated to each set of time instances

i = (E [Ti ])

take the set that gives the minimum condition number as starting point

min(i ) = (E [Tini ])

uniformly sample each period with 2N+1 time instants3

Ti =0,

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 ...

... and zooming on the colormap emphasizes this analysis

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

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

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

Dt = iE�1⌦E

Convergence /

The convergence of three configurations is assessed ...

same mockup CROR at cruise design by Safran-Snecma

C3AI-PX7 CROR at cruise

design by Airbus

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

N=4N=3

N=2N=1

Convergence /

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

N=8N=7

N=6N=5

Convergence /

C3 for which N=10 is not sufficient to converge

N=10N=9

N=8N=7

Convergence /

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

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

for each radii

compute azimuthal spectrum = temporal

spectrum

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

frequency

resolved part

unresolved part

Convergence /

We define the accumulated energy up to a given number of harmonics

E (N) =

� �2�

"(N) =

� ��

truncation error

accumulated energy

E (N) = 1� "2(N)

frequency

A priori estimate of the required number of harmonics

# harmonics

Convergence /

front rotor blade tip

front rotor hub

# 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

# harmonics

Convergence / 94A priori estimate of the required number of harmonics

# harmonics

Convergence /

E (N) = 99%

# harmonics

Convergence /

For C1, N=4 is needed to capture 99% of energy on the whole blade radius

E (N) = 99%

# harmonics

Convergence /

For C2, N=7 is needed to capture 99% of energy on the whole blade radius

# harmonics

Convergence /

For C3, N=17 is needed to capture 99% of energy on almost all blade radius

# harmonics

Convergence /

Actually, N=22 is needed to capture the front rotor tip vortex

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

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

Dt = iE�1⌦E

Application to CROR /

Outline

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

Roe, JCP, 1981Spalart & Allmaras, AIAA Paper, 1992

steady results

The computation is converged starting at 500 iterations

105steady results

0 500 1000 1500 2000

front rotorrear rotorglobal

0 500 1000 1500 2000

iteration

ficien

ity re

iteration

The radial distribution of two variables is analyzed at six locations

P2P1 P4P3 P6P5

steady results

The Mach number goes from 0.2 up to 0.4 which yields the thrust

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

P1 P2 P3 P4 P5 P6

steady results

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

steady results

0 0.2 0.4 0.6 0.8 1

radius

110steady results

0 0.2 0.4 0.6 0.8 1

radius

111steady results

0 0.2 0.4 0.6 0.8 1

radius

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

112steady results

0 0.2 0.4 0.6 0.8 1

radius

113steady results

0 0.2 0.4 0.6 0.8 1

radius

114steady results

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%

# 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

unsteady rigid results

We look at Mach number fluctuations as perceived by the

front rotorM 0

M 0(t) = M(t)�M

front rotor

potential effects

M 0(t) = M(t)�M

rear rotor

M 0(t) = M(t)�M

rear rotor

velocity deficits attributed to wake

passing

M 0(t) = M(t)�M

We can now compute the flutter boundary of the front rotor

mode 2Ffront rotor

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

(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)

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

mode 2F mode 1T

stableunstable

aeroelastic results

Aerodynamic variation lines are also nodal lines for the damping

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°

‣ 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

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

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

Application to Contra-Rotating Open RotorsSteady resultsUnsteady rigid resultsAeroelastic results

Dt = iE�1⌦E

Conclusions & Perspectives

On the conditioning of multi-frequential HB methods

‣ 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

‣ 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

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.

# 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.

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