Extended mean-flow analysis of periodic flows Olivier Marquet - - PowerPoint PPT Presentation

Extended mean-flow analysis

• f

periodic flows

Olivier Marquet & Marco Carini

Department of Aerodynamics, Aeroacoustics and Aeroelasticity 16th European Turbulence Conference 21-24 August 2017, Stockholm, Sweden

Introduction

2

Steady flow Instability growth Periodic flow

Periodic flow resulting from the nonlinear saturation of a linear instability

Introduction

3

Periodic flow resulting from the nonlinear saturation of a linear instability

Base flow Mean flow distortion Zielinska et al (1997) Mean flow

Introduction

4

The nonlinear saturation is due to two mechanisms 1 – Mean flow distortion (circular cylinder flow)

• Mean flow analysis - Barkley (2002)

Eigenvalue analysis of a mean flow (computed from DNS) Real Zero Imaginary Frequency property – Turton et al. (2015)

• Self-consistent model - Mantic-Lugo et al (2015)

Reconstruction of the mean flow assuming the RZIF property.

Introduction

5

The nonlinear saturation is due to two mechanisms 2 – Interaction of higher-harmonics (open-cavity flow)

• Weakly nonlinear analysis - Sipp & Lebedev (2007)
• Second-order self-consistent model - Meliga (2017)

Extended mean-flow analysis An eigenvalue analysis that accounts for both effects.

Outlines

6

1 – Extended mean-flow analysis 2 – Results for laminar flows

• Circular-cylinder flow
• Open-cavity flow

3 – Conclusion/Perspectives

Periodic flow and Fourier decomposition

7

• = + (, )

, + = , = 2 / , = + + . . + + . . + ⋯

Periodic solutions Fourier decomposition

Mean flow Second harmonic First harmonic Linear operator Nonlinear operator (quadratic)

Harmonic balanced equations

8

− − , = ,

∗ + ∗, + ( , ∗ + ∗, )

! − − , − , = ,

∗ + ∗,

2 ! − − , − , = , Mean flow equation First-harmonic equation Second-harmonic equation A set of time-independent coupled nonlinear equations

Harmonic balanced equations

9

− − , = 0

! − − , − , = ,

∗ + ∗,

2 ! − − , − , = , Base flow equation First-harmonic equation Second-harmonic equation A set of time-independent coupled nonlinear equations

Mean-flow analysis

10

Eigenvalue analysis of the mean flow operator

! − # = (

∗) + ⋯

First-harmonic equation

($%+! %) &' = # &'

# = + , + ,

Mean-flow operator Second-harmonic operator

! − # = 0

Neglect the second-harmonic

∗

= ,

∗ + ∗,

$% ∼ 0 ; % ∼ ; &' ∼

Extended mean-flow analysis

11

! − # −

∗

= + ⋯

First-harmonic equation and its complex conjugate Extended first-harmonic equation

!

• ∗

− #

• −

∗

−#

• ∗

= + ⋯

!

∗ + # ∗ + ∗ = + ⋯

Extended mean-flow analysis

($*+! *) +,

• ,

= #

• −

∗

−# +,

• ,

$* ∼ 0 ; * ∼ ; +, ∼ ; -, ∼

∗

Outlines

12

1 – Extended mean flow analysis of periodic flows 2 – Results for laminar flows

• Circular-cylinder flow
• Open-cavity flow (with rounded corners)

3 – Conclusion/perspectives

Circular cylinder flow configuration

13

./ 1 = ./0 2 = 100

Circular cylinder flow – Base flow analysis

14

Base flow

$4 = 0.125 4 = 0.739

Circular cylinder flow – Mean-flow analysis

15

Base flow Mean flow

$4 = 0.125 4 = 0.739

= 1.044

Circular cylinder flow – Mean-flow analysis

16

Base flow Mean flow

$4 = 0.125 :' = . 4 = 0.739 ;' = . <

Mean flow eigenmode - %

= 1.044

Circular cylinder flow – Mean-flow analysis

17

Mean flow

= 1.044

$4 = 0.125 :' = . 4 = 0.739 ;' = . <

First Fourier mode - Base flow

Circular cylinder flow – Extended mean-flow analysis

18

Second-Harmonic Mean flow

= 1.044

Circular cylinder flow – Extended mean-flow analysis

19

Second-Harmonic Mean flow

= 1.044

:, = > ;,

= . ??

*

= 1.016

:' = . ;' = . <

Two eigenvalues with zero growth-rate The frequency of one eigenvalue is the nonlinear frequency ;,

= ;

Circular cylinder flow – Extended mean-flow analysis

20

Second-Harmonic Mean flow

:, = > :' = . ;,

= . ??

;' = . <

Extended mean-flow eigenmode -

*

= 1.016

*

A

= 1.044

Circular cylinder flow – Extended mean-flow analysis

21

Second-Harmonic Mean flow

:, = > :' = . ;,

= . ??

;' = . < *

= 1.016

= 1.044

First Fourier mode -

Circular cylinder flow – Extended mean-flow analysis

22

Second-Harmonic Mean flow

:, = > :' = . ;,

= . ??

;' = . <

Extended mean-flow eigenmode -

*

= 1.016

*

• = 1.044

23

Circular cylinder flow – Extended mean-flow analysis

* = # B −B

∗

−#

= 0 = 1.547 = 1.547

Open cavity flow configuration

24

C 4400 < 1 = ./C 2 < 4600 ./ C

Open-cavity flow – Mean-flow analysis

25

Base flow Mean flow

Weaker mean-flow distortion

Growth rate

Open-cavity flow – Extended mean-flow analysis

26

Second-harmonic Mean flow Extended mean-flow

Conclusion and perspectives

27

Conclusion

• A new eigenvalue analysis of periodic flow accounting for the

two mechanisms of nonlinear saturation

• This analysis gives a Real Zero Imaginay Frequency Mode

Perspectives

• Develop a model where the second-harmonic is reconstructed
• Extension to fluid/structure problems and turbulent flows

modelled with a RANS approach

Flow configuration and turbulent flow model

28

./ = 1/15 = 1 1 = ./ 2 = 1.5 10E

• Turbulent flow modelled with Reynolds Averaged Navier Stokes equations
• Spalart-Almarras model for the turbulent eddy viscosity 2
• Frozen-viscosity approach:
• Steady equations solved with the Spalart-Almarras model
• Unsteady equations solved with frozen turbulent eddy viscosity 2

Base flow - Steady solution of RANS equations

29

Turbulent eddy viscosity 2/2 Streamwise velocity F Leading-edge recirculation regions Trailing-edge recirculation regions

Stability analysis with frozen eddy-viscosity

30

Streamwise velocity F Unstable eigenmode Eigenvalue spectrum

Stability analysis with frozen eddy-viscosity

31

Streamwise velocity F Unstable eigenmode – Zoom on trailing edge Eigenvalue spectrum

$4 = 0.463 4 = 14.351

Unsteady solution with frozen-eddy viscosity

32

Titre présentation

Second harmonic First harmonic Mean flow Instantaneous lift

= 0.427

= 14.714 2

Mean-flow analysis with frozen eddy-viscosity

33

Mean-flow eigenmode Mean-flow eigenvalue spectrum

:' = . G ;' = ?. H? $4 = 0.463 4 = 14.351 ( = 14.714)

Extended mean-flow analysis with frozen eddy-viscosity

34

Extended mean-flow eigenvalue spectrum

$% = 0.192 % = 14.604 ( = 14.714)

• Two eigenvalues characterized by zero growth rate

Extended mean-flow analysis with frozen eddy-viscosity

35

• Two eigenvalues characterized by zero growth rate
• The frequency of one eigenmode is in excellent agreement

with the non-linear frequency ;

Extended mean-flow eigenvalue spectrum

$% = 0.192 % = 14.604 (; = ?. I?) :, = > ;,

= ?. I?

*

= 14.479

1 2

Extended mean-flow analysis with frozen eddy-viscosity

36

Extended mean-flow eigenvalue spectrum

$% = 0.192 % = 14.604 (; = ?. I?) :, = > ;,

= ?. I?

*

= 14.479

1 2

Extended mean-flow eigenmode

Extended mean-flow analysis with frozen eddy-viscosity

37

Extended mean-flow eigenvalue spectrum

$% = 0.192 % = 14.604 (; = ?. I?) :, = > ;,

= ?. I?

*

= 14.479

1 2

First harmonic