Extended mean-flow analysis of periodic flows Olivier Marquet & Marco Carini Department of Aerodynamics, Aeroacoustics and Aeroelasticity 16 th European Turbulence Conference 21-24 August 2017, Stockholm, Sweden
Introduction Periodic flow resulting from the nonlinear saturation of a linear instability Instability growth Periodic flow Steady flow 2
Introduction Periodic flow resulting from the nonlinear saturation of a linear instability Mean flow distortion Zielinska et al (1997) Mean flow Base flow 3
Introduction 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) R eal Z ero I maginary F requency property – Turton et al. (2015) Self-consistent model - Mantic-Lugo et al (2015) • Reconstruction of the mean flow assuming the RZIF property. 4
Introduction 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. 5
Outlines 1 – Extended mean-flow analysis 2 – Results for laminar flows Circular-cylinder flow • Open-cavity flow • 3 – Conclusion/Perspectives 6
Periodic flow and Fourier decomposition �� �� = � � + �(�, �) Nonlinear operator (quadratic) Linear operator Periodic solutions � = 2 �/� � �, � + � = � �, � Fourier decomposition � �, � = � � � + � � � � � � � � + �. �. + � � � � � � � � � � + �. �. + ⋯ Second harmonic Mean flow First harmonic 7
Harmonic balanced equations A set of time-independent coupled nonlinear equations Mean flow equation ∗ + � � � ∗ + � � � −� � � − � � � , � � = � � � � � , � � ∗ , � � + � ( � � � , � � ∗ , � � ) First-harmonic equation ∗ + � � � ∗ , � � ! � � � − � � � − � � � , � � − � � � , � � = � � � � � , � � Second-harmonic equation 2 ! � � � − � � � − � � � , � � − � � � , � � = � � � , � � 8
Harmonic balanced equations A set of time-independent coupled nonlinear equations Base flow equation −� � � − � � � , � � = 0 First-harmonic equation ∗ + � � � ∗ , � � ! � � � − � � � − � � � , � � − � � � , � � = � � � � � , � � Second-harmonic equation 2 ! � � � − � � � − � � � , � � − � � � , � � = � � � , � � 9
Mean-flow analysis First-harmonic equation ∗ ) + ⋯ ! � � � − � # � � = � � � � (� � Mean-flow operator Second-harmonic operator ∗ + � � � ∗ ∗ , � � � � � � = � � � , � � � # � � = � � � + � � � , � � + � � � , � � Neglect the second-harmonic ! � � � − � # � � = 0 Eigenvalue analysis of the mean flow operator ($ % +! � % ) � & ' = � # � & ' $ % ∼ 0 ; � % ∼ � ; � & ' ∼ � � 10
Extended mean-flow analysis First-harmonic equation and its complex conjugate ∗ ! � � � − � # � � − � � � � � � = � + ⋯ ∗ + � # � � ∗ + � � � � ∗ � � = � + ⋯ ! � � � Extended first-harmonic equation ! � � � � � � � � � � # = � − � + ⋯ ∗ ∗ � � � � ∗ −� � � � −� # Extended mean-flow analysis � � � � ($ * +! � * ) + , � # + , = - , −� � � � ∗ - , −� # ∗ $ * ∼ 0 ; � * ∼ � ; + , ∼ � � ; - , ∼ � � 11
Outlines 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 12
Circular cylinder flow configuration . / 0 1� = . / 0 = 100 2 13
Circular cylinder flow – Base flow analysis Base flow $ 4 = 0.125 � 4 = 0.739 14
Circular cylinder flow – Mean-flow analysis � = 1.044 Base flow Mean flow $ 4 = 0.125 � 4 = 0.739 15
Circular cylinder flow – Mean-flow analysis � = 1.044 Base flow Mean flow $ 4 = 0.125 : ' = �. ��� � 4 = 0.739 ; ' = �. �<� Mean flow eigenmode - � % 16
Circular cylinder flow – Mean-flow analysis � = 1.044 Base flow Mean flow $ 4 = 0.125 : ' = �. ��� � 4 = 0.739 ; ' = �. �<� First Fourier mode - � 17
Circular cylinder flow – Extended mean-flow analysis � = 1.044 Second-Harmonic Mean flow 18
Circular cylinder flow – Extended mean-flow analysis � = 1.044 Second-Harmonic Mean flow : , = �� >�� : ' = �. ��� � = �. �?? ; , ; ' = �. �<� � = 1.016 � * Two eigenvalues with zero growth-rate The frequency of one eigenvalue is the nonlinear frequency � = ; ; , 19
Circular cylinder flow – Extended mean-flow analysis � = 1.044 Second-Harmonic Mean flow : , = �� >�� : ' = �. ��� � = �. �?? ; , ; ' = �. �<� � = 1.016 � * A � * Extended mean-flow eigenmode - 20
Circular cylinder flow – Extended mean-flow analysis � = 1.044 Second-Harmonic Mean flow : , = �� >�� : ' = �. ��� � = �. �?? ; , ; ' = �. �<� � = 1.016 � * First Fourier mode - � 21
Circular cylinder flow – Extended mean-flow analysis � = 1.044 Second-Harmonic Mean flow : , = �� >�� : ' = �. ��� � = �. �?? ; , ; ' = �. �<� � = 1.016 � * � � * Extended mean-flow eigenmode - 22
Circular cylinder flow – Extended mean-flow analysis B � � � � # � * = ∗ −B � � � −� # � = 1.547 � = 0 � = 1.547 23
Open cavity flow configuration . / � C C 4400 < 1� = . / C < 4600 2 24
Open-cavity flow – Mean-flow analysis Base flow Mean flow Weaker mean-flow distortion Growth rate 25
Open-cavity flow – Extended mean-flow analysis Mean flow Second-harmonic Extended mean-flow 26
Conclusion and perspectives Conclusion A new eigenvalue analysis of periodic flow accounting for the • two mechanisms of nonlinear saturation This analysis gives a R eal Z ero I maginay F requency Mode • Perspectives Develop a model where the second-harmonic is reconstructed • Extension to fluid/structure problems and turbulent flows • modelled with a RANS approach 27
Flow configuration and turbulent flow model � = 1 . / � = 1/15 1� = . / � = 1.5 10 E 2 Turbulent flow modelled with R eynolds A veraged N avier S tokes 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 � 28
Base flow - Steady solution of RANS equations Streamwise velocity F Trailing-edge recirculation regions Leading-edge recirculation regions Turbulent eddy viscosity 2 � /2 29
Stability analysis with frozen eddy-viscosity Eigenvalue spectrum Unstable eigenmode Streamwise velocity F 30
Stability analysis with frozen eddy-viscosity Eigenvalue spectrum $ 4 = 0.463 � 4 = 14.351 Unstable eigenmode – Zoom on trailing edge Streamwise velocity F 31
Unsteady solution with frozen-eddy viscosity Instantaneous lift � = 0.427 0 Mean flow � = 14.714 First harmonic Second harmonic 2 � 32 Titre présentation
Mean-flow analysis with frozen eddy-viscosity Mean-flow eigenvalue spectrum : ' = �. �G� $ 4 = 0.463 ; ' = �?. H�? � 4 = 14.351 (� = 14.714) Mean-flow eigenmode 33
Extended mean-flow analysis with frozen eddy-viscosity Extended mean-flow eigenvalue spectrum $ % = 0.192 � % = 14.604 (� = 14.714) Two eigenvalues characterized by zero growth rate • 34
Extended mean-flow analysis with frozen eddy-viscosity Extended mean-flow eigenvalue spectrum 1 : , = �� >�� $ % = 0.192 2 � = �?. I�? ; , � % = 14.604 � = 14.479 (; = �?. I�?) � * Two eigenvalues characterized by zero growth rate • The frequency of one eigenmode is in excellent agreement • with the non-linear frequency ; 35
Extended mean-flow analysis with frozen eddy-viscosity Extended mean-flow eigenvalue spectrum 1 : , = �� >�� $ % = 0.192 2 � = �?. I�? ; , � % = 14.604 � = 14.479 (; = �?. I�?) � * Extended mean-flow eigenmode 36
Extended mean-flow analysis with frozen eddy-viscosity Extended mean-flow eigenvalue spectrum 1 : , = �� >�� $ % = 0.192 2 � = �?. I�? ; , � % = 14.604 � = 14.479 (; = �?. I�?) � * First harmonic 37
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