Réunion pour début de thèse de Luis Henrique Benetti Ramos Université de Bordeaux 28-29 septembre 2017
Ordre du jour • Rappel des discussions précédentes • Organisation des séjours à Bordeaux • Le contenu scientifique – « hydrodynamic resonance theory » – IBM temporal simulations – Fluid-structure instability in IBM
Hydrodynamic resonance theory Unsteady propulsion of rigid/flexible foils - What is the connexion between the wake’s structure and the hydrodynamic forces ? - Does a linear stability analysis of the wake gives information on the propulsion effcicieny?
Hydrodynamic resonance theory Triantafyllou et al. (1993): - 2D rigid pitching foils ( � : amplitude, � : frequency) - Reverse Von-Karman vortex street (2S wakes) �� � � � = - Propulsive efficiency: � �� � � : time-averaged net thrust � � �� : time-averaged power input to the fluid � - A single peak in propulsive efficiency occurs for �� 0.25 ≤ �� = � � ≤ 0.35
Hydrodynamic resonance theory Triantafyllou et al. (1993): They proposed that this peak in propulsive efficiency occurs at the frequency of maximum spatial growth rate of the instability of the jet from Moored et al. (2012)
Hydrodynamic resonance theory Lewin and Haj-Hariri (2003): - 2D rigid heaving foils ( � : amplitude, � : frequency) - Mutiple peaks in efficiency - Driving frequency match the resonant frequency (obtained by stability analysis) - Need to introduced the reduced frequency � ��� � = with � : the chord length � �
Hydrodynamic resonance theory Dewey et al. (2001) - Three-dimensional ray-like pectoral fin - Travelling wave motion of the fin - Non-dimensional wave length ∗ = " � = 4 $% 6 Muliple peaks in efficiency observed as �� is varied - Transition from 2P to 2S wakes when increasing �� -
Hydrodynamic resonance theory Moored et al. (2012): - Local spatial stability analysis: Assumption: the flow is weakly non-parallel (( ) ≪ ( + ) / 0 1, ., � = / 3 . 4 � (5)678) - . + �: frequency (real parameter) ;: complex wavenumber (unknown) −ℑ(;): spatial growth rate ℜ(;): spatial wavenumber velocity profile at a station x
Hydrodynamic resonance theory Moored et al. (2012): When the driving frequency matches Eigenvalue spectra obtained for the resonant frequency, a peak in efficiency - one velocity profile is expected and 40 values of the frequency � -
Hydrodynamic resonance theory Moored et al. (2012): Analysis of 2P wake Velocity profile Growth rate Eigenmode
Hydrodynamic resonance theory Moored et al. (2012): Analysis of 2S wake Velocity profile Growth rate Eigenmode
Hydrodynamic resonance theory Moored et al. (2012): Experimental results Spatial stability results (Propulsive efficiency)
Moored et al 2014 • Extends wake resonance theory to flexible propulsors; • Tuning the structural resonant frequency to a wake resonant frequency improves the propulsive efficiency; • The entrainment of momentum into the time- averaged jet is seen as a physical explanation to this phenomenon
Dewey et al (2013) • Re = 7200 • U = 0.06 m/s • C = 120 mm • W = 280 mm • 6 flexible panels and rigid one from previous article. • EI = 4.2 - 1100 * 10 E-4 Nm2
Efficiency EI increases
Momentum entrainment
Objectives: 1 - Extend the spatial stability analysis to non- parallel flows based on the resolvent operator 2 - With a momentum balance analysis, better understand the link between the time-averaged power injected in the fluid (fluctuation equation) and the mean drag force(mean flow equation). 3 - Reveal the role of the linear amplification of perturbation in the wake structure .
Back to Navier-Stokes equations Γ � (@ (� + A ⋅ @ ⊗ @ + D @, E = 0 ; A ⋅ @ = 0 D @, E = −E G + 1 I4 (J@ + J@ K ) @ ., � = @ L ., � on O L
Reynolds decomposition of the wake � P + @′(P, �) @ P, � = @ @ � (., �) = @ L . + @ L ′(., �) E P, � = E̅ P + E′(P, �) Time-averaged (mean) flow equations = − A ⋅ @ 0 ⊗ @ 0 A ⋅ @ � ⊗ @ � + D @ �, E̅ � ., � = @ � L ., � on O L @ Fluctuating flow (momentum) equations (@′ (� + A ⋅ @ 0 ⊗ @ � ⊗ @ 0 + D @ 0 , E 0 = −A ⋅ (@ 0 ⊗ @ 0 ) + A ⋅ @ 0 ⊗ @ 0 � + @ 0 ., � on O L @′ ., � = @ L
Nonlinear fluctuation equation (@′ (� + A ⋅ @ 0 ⊗ @ � ⊗ @ 0 + D @ 0 , E 0 = −A ⋅ (@ 0 ⊗ @ 0 ) + A ⋅ @ 0 ⊗ @ 0 � + @ � @ � : linear operator (around the mean flow) (@′ @ 0 , E 0 = S P, � ; A ⋅ @ 0 = 0 (� + � @ � @ � ., � = @ � L ., � on O L
Nonlinear fluctuation equation U 0 = @ 0 , E 0 K V (U′ � U 0 = W (� + T @ � S P, � @′ ., � = @′ L ., � on O L � : Jacobian operator (around the mean flow) T @ � S = S, 0 K ) W � : Prolongation operator ( W
Fourier decomposition U 0 = U X P 4 � 78 + Z. Z. + U [ P 4 � �78 + Z. Z. + ⋯ S 0 = U X P 4 � 78 + Z. Z. + U [ P 4 � �78 + Z. Z. + ⋯ 0 = @ LX P 4 � 78 + Z. Z. + @ L[ P 4 � �78 + Z. Z. + ⋯ @ L @ X . = @ LX (Y) on O L ] � V + T @ � U X = W � S ^ @ [ . = @ L[ (Y) on O L � 2] � V + T @ U [ = W � S [
Decompostion into forced and inlet problems @ _ X . = @ LX (Y) on O L ] � V + T @ � U X = W � S X We assume that S ^ and @ LX (Y) are unknowns and independent of the solution � + U ^ � We decompose the solution as U ^ = U ^ Forced problem S = W S . = ` on O L ] � V + T @ � U X � S ^ @ X Inlet problem L = ` L . = @ LX (Y) on O L � ] � V + T @ U X @ X
Forced problem and resolvent modes 6^ W _ � = I � W a = ] � V + T @ a � U � S � S Resolvent operator with boundary condition _ . = ` on O L @ _ �e V U _ � a ( U max ) ? a e S a S S a e V I W _ � = I W a = S a g W a _ �e V U K I e V I W U � S � S � S �
Forced problem and resolvent modes Eigenvalue problem a h = i a h K I e V I W � S W � S � � ≥ ^ � ≥ ⋯ ) � : positive real eigenvalue ( j i a h : eigenvector S Associated flow solution S = I � W a h with @ _ h . = ` on O L _ h U � S Sg l U S � = U _ h _ h i a h = energetic ratio a h g S S
Projection of the forcing Projection of the harmonic forcing onto the basis a X + ; � S a [ + ⋯ S X = ; ^ S The flow solution writes then S = I � W a X + ; � S a [ + ⋯ U X � S ^ = I � W � ; ^ S S + ; � U S + = ; ^ U _ X _ [ S = ; ^ ^ � + ; � � � + ⋯ S Sg V U X m n U X = U X
Inlet problem and resolvent modes _ L = I � W � @ U _ L where W � : operateur de relèvement de la condition aux limites a P = ` in Ω and the forcing term vanishes S _ �e V U _ � _ L ( U max ) ? g @ @ _ L _ L @ _ � = I W � @ g W K I e V I W � @ _ �e V U _ L e V I W � @ U _ L = @ _ L _ L �
Inlet problem and resolvent modes Eigenvalue problem K I e V I W � @ W � @ _ Lh = q i _ Lh � � ≥ q ^ � ≥ ⋯ ) � : positive real eigenvalue ( q j q i @ _ Lh : eigenvector Associated flow solution L = I � W a h = ` in Ω _ h _ Lh with S U � @ Lg l U L � = U _ h _ h q i _ Lh = energetic ratio g @ @ _ Lh
Projection of the inlet Projection of the harmonic inlet velocity profile onto the basis _ LX + r � @ _ L[ + ⋯ @ LX = r ^ @ The flow solution then writes L = I � W � @ �^ = I � W � r ^ @ _ LX + r � @ _ L[ + ⋯ U X L + r � U L + = r ^ U _ X _ [ L = r ^ q ^ � + r � q � � + ⋯ L Lg V U X m n U X = U X
New resolvent problems - 1 Γ Γ � s _ L = ` ] � V + T @ � U _ L 1, . = @ _ L 1, . = @ @ _ L Y ; 1 ∈ O L _ u Y ; 1 ∈ O u @ g @ _ u _ u _ L ( @ max ) ? g @ _ L _ L @ @
New resolvent problems - 1 Γ Γ � s _ L = I � W � @ K U _ L _ L U _ u = W @ s _ L = @ _ L g W e ( W e I � e W g @ K U K I � W � ) @ @ _ u _ u = U s W _ L s W _ L s s � e I � e W K I � W � @ _ L = � @ W � s W _ L s
Effect on the mean flow Γ Γ � s _ ∗ ⊗ @ � = − A ⋅ @ _ ∗ U _ = I � W � @ _ L v _ + @ _ ⊗ @ Examine the mean flow force (in the streamwise direction) induced by the optimized harmonic response � 3 ∗ ( ) / 3 ∗ ( + / w ) = −2 ℜ( / 3 + x 3)
Reynolds stress tensor 3 ∗ / 3 ∗ x 2 / 3 2ℜ / 3 _ ∗ ⊗ @ _ ∗ = J ⋅ � = − A ⋅ @ _ + @ _ ⊗ @ v 3 ∗ / 3 ∗ x 2 ℜ x 3 2 x 3 _ ∗ ⊗ @ _ ∗ ⋅ } ~� � dΩ = y w | 2 @ _ + @ _ ⊗ @ { •{
Reynolds stress tensor 3 ∗ / 3 ∗ x 2 / 3 2ℜ / 3 _ ∗ ⊗ @ _ ∗ = J ⋅ � = − A ⋅ @ _ + @ _ ⊗ @ v 3 ∗ x 3 ∗ x 2 ℜ / 3 2 x 3 • Γ Γ Γ € � s 6 Γ € � dΩ = y 3 ∗ / 3 ∗ / 3 ∗ x 3 ∗ x y w 2/ 3 ~. − y 2/ 3 ~. + y 2ℜ / 3 ~1 − y 2ℜ / 3 ~1 ) … † { ‚ ƒ ‚ � ‚ „ ‚ „
Reynolds stress tensor 3 ∗ / 3 ∗ x 2 / 3 2ℜ / 3 _ ∗ ⊗ @ _ ∗ = J ⋅ � = − A ⋅ @ _ + @ _ ⊗ @ v 3 ∗ x 3 ∗ x 2 ℜ / 3 2 x 3 • Γ Γ Γ € � s 6 Γ € � dΩ = y 2ℜ / 3 ∗ x 3 ∗ x 3 ∗ x 3 ∗ x y w 3 ~. − y 2ℜ / 3 ~. + y 2 x 3 ~1 − y 2 x 3 ~1 + … † { ‚ ƒ ‚ � ‚ „ ‚ „
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