Skin-friction drag reduction described via the Anisotropic - PowerPoint PPT Presentation

Skin-friction drag reduction described via the Anisotropic Generalised Kolmogorov Equations Alessandro Chiarini 1 , Davide Gatti 2 , Maurizio Quadrio 1 European Drag Reduction and Flow Control Meeting, 26-29 March 2019, Bad Herrenalb, Germany 1 Politecnico di Milano, 2 Karlsruhe Institute of Technology-KIT 1

How do DR techniques affect the production, transport and dissipation of turbulent stresses among scales and in space? 2

Turbulent channel flow forced via spanwise oscillating walls vs Controlled channel Reference channel At Constant Power Input 3

Constant Power Input: an alternative to CFR and CPG The input power is kept constant mean dissipation φ MKE TKE pumping power Π p turbulent dissipation ǫ production P control power Π c Gatti et al. JFM 2018 4

Starting point: OW effect on the global energy fluxes Reference channel at Re τ = 200 φ = 0 . 589 MKE TKE Π p = 1 ǫ = 0 . 410 P = 0 . 411 Π c = 0 5

Starting point: OW effect on the global energy fluxes Controlled channel via OW with A + = 4 . 5 and T + = 125 ∆ φ = +0 . 009 φ = 0 . 598 MKE TKE Π p = 1 ǫ = 0 . 499 ∆Π p = 0 ∆ ǫ = +0 . 089 P = 0 . 403 ∆ P = − 0 . 008 ∆Π c = +0 . 098 Π c = 0 . 098 5

Starting point: OW effect on the global energy fluxes Controlled channel via OW with A + = 4 . 5 and T + = 125 ∆ φ = +0 . 009 φ = 0 . 598 MKE TKE Π p = 1 ǫ = 0 . 499 ∆Π p = 0 ∆ ǫ = +0 . 089 P = 0 . 403 ∆ P = − 0 . 008 ∆Π c = +0 . 098 Π c = 0 . 098 Global variations → detailed changes 5

Anisotropic Generalised Kolmogorov Equations AGKE: Exact budget equation for � δ u i δ u j � δ u i = ( u i ( X + r / 2 , t ) − u i ( X − r / 2 , t )) k j i x 1 X x 2 u i ( X − r / 2) r j u i ( X + r / 2) r k r i X = ( x 1 + x 2 ) / 2 Dependent on: r = x 2 − x 1 6

AGKE: extension of the Generalised Kolmogorov Equation to anisotropy GKE: Exact budget equation for the scale energy   � δ u δ u � � δ u δ v � � δ u δ w �   � δ u i δ u i � = tr � δ v δ v � � δ v δ w �  = � δ u δ u � + � δ v δ v � + � δ w δ w �  � δ w δ w � sym 7

AGKE: extension of the Generalised Kolmogorov Equation to anisotropy GKE: Exact budget equation for the scale energy   � δ u δ u � � δ u δ v � � δ u δ w �   � δ u i δ u i � = tr � δ v δ v � � δ v δ w �  = � δ u δ u � + � δ v δ v � + � δ w δ w �  � δ w δ w � sym What if � δ u δ u �≫� δ v δ v � , � δ w δ w � ? The GKE does not account for anisotropy.. ..but the AGKE do! 7

AGKE: interpretation Amount of turbulent stresses at location X and scale (up to) r r • � δ u i δ u j � ( X , r ) X JFM, in preparation Production, transport and dissipation of turbulent stresses • AGKE in both the Space of scales & Physical space 8

AGKE ∂φ k , ij + ∂ψ k , ij = ξ ij ∂ r k ∂ X k 9

AGKE ∂φ k , ij + ∂ψ k , ij = ξ ij ∂ r k ∂ X k − 2 ν ∂ φ k , ij = � δ U k δ u i δ u j � + � δ u k δ u i δ u j � ∂ r k � δ u i δ u j � � �� � � �� � � �� � mean transport turbulent transport viscous diffusion flux of � δ u i δ u j � throughout scales r 9

AGKE ∂φ k , ij + ∂ψ k , ij = ξ ij ∂ r k ∂ X k + 1 ρ � δ p δ u i � δ kj + 1 − ν ∂ ψ k , ij = � U ∗ k δ u i δ u j � + � u ∗ k δ u i δ u j � ρ � δ p δ u j � δ ki ∂ X k � δ u i δ u j � 2 � �� � � �� � � �� � mean transport turbulent transport � �� � viscous diffusion pressure transport flux of � δ u i δ u j � in space X 9

AGKE ∂φ k , ij + ∂ψ k , ij = ξ ij ∂ r k ∂ X k source/sink of � δ u i δ u j � at scale r and location X � ∂ U i � � ∂ U j � � ∂ U i � ∗ � ∂ U j � ξ ij = −� u ∗ −� u ∗ k δ u j � δ k δ u i � δ −� δ u k δ u j � −� δ u k δ u i � + ∂ x k ∂ x k ∂ x k ∂ x k � �� � production � � � � +1 δ p ∂δ u i + 1 δ p ∂δ u j − 4 ǫ ∗ ij ρ ∂ X j ρ ∂ X i � �� � � �� � dissipation 9 pressure strain

AGKE ∂φ k , ij + ∂ψ k , ij = ξ ij ∂ r k ∂ X k − 2 ν ∂ φ k , ij = � δ U k δ u i δ u j � + � δ u k δ u i δ u j � ∂ r k � δ u i δ u j � � �� � � �� � � �� � mean transport turbulent transport viscous diffusion + 1 ρ � δ p δ u i � δ kj + 1 − ν ∂ ψ k , ij = � U ∗ k δ u i δ u j � + � u ∗ k δ u i δ u j � ρ � δ p δ u j � δ ki ∂ X k � δ u i δ u j � 2 � �� � � �� � � �� � mean transport turbulent transport � �� � viscous diffusion pressure transport � ∂ U i � � ∂ U j � � ∂ U i � ∗ � ∂ U j � ξ ij = −� u ∗ −� u ∗ k δ u j � δ k δ u i � δ −� δ u k δ u j � −� δ u k δ u i � + ∂ x k ∂ x k ∂ x k ∂ x k � �� � production � � � � +1 δ p ∂δ u i + 1 δ p ∂δ u j − 4 ǫ ∗ ij ρ ∂ X j ρ ∂ X i � �� � � �� � dissipation 9 pressure strain

AGKE tailored to channel flow � δ u i δ u j � ( X , r ) →� δ u i δ u j � ( Y , r x , r y , r z ) y U ( y ) x z U ( y ) r z x 1 x 2 Y r y r x ∂φ k , ij + ∂ψ ij ∂ Y = ξ ij ∂ r k 10

How do DR techniques affect the production, transport and dissipation of turbulent stresses among scales and in space? We investigate the changes of the AGKE terms 11

Numerical Data • Two Direct Numerical Simulations (with and without wall oscillation) at CPI Re τ = 200 for the uncontrolled case Wall oscillation parameters: A + = 4 . 5, T + = 125 . 5 Quadrio & Ricco JFM 2004 • Six smaller Direct Numerical Simulations at CPI A + ∈ (0 , 30), T + ∈ (100 , 125) 12

Source of � δ u δ u � in the r x = 0 space Ref OW r y ≤ 2Y r z x 2 r y y x 1 Y z 13

Production of � δ u δ u � Ref OW r y ≤ 2Y r z x 2 r y y x 1 Y z 13

Production of � δ u δ u � Ref OW Shift of P 11 , m (∆ Y + P m ∼ 3) 13

Production of � δ u δ u � What if A + is incremented? 20 Y + P m 18 16 14 12 0 10 20 30 A + P m increases with A + (% DR ) ∆ Y + 13

Fluxes of � δ u δ u � : φ k , 11 and ψ 11 Ref OW r y ≤ 2Y r z x 2 r y y x 1 Y z 13

Fluxes of � δ u δ u � : φ k , 11 and ψ 11 Ref OW Y = r y / 2 + K : attached to the wall plane 13

Fluxes of � δ u δ u � : φ k , 11 and ψ 11 Ref OW ∆ K + ∼ 3 13

Fluxes of � δ u δ u � : φ k , 11 and ψ 11 What if A + is incremented? 30 25 K + 20 15 10 0 10 20 30 A + ∆ K + increases with A + (% DR ) 13

Links with well-known results? Vertical shift ∆ y of the maximum of P � uu � Ref OW 0 . 4 P + � uu � , m ∆ y + ∼ 3 0 . 2 0 0 10 20 30 y + 13

Links with well-known results? Vertical shift ∆ y of φ � uu � = 0 Ref 1 OW φ + 0 � uu � ∆ y + ∼ 3 − 1 0 20 40 y + 13

Interpretation Single-point statistics Two-points statistics • Shift of � uu � m • Shift of � δ u δ u � m • Shift of P • Shift of P � uu � , m � δ u δ u � , m • Shift of φ � uu � = 0 • Shift of the attached to the wall plane OW → Virtual shift of the wall 13

Conclusion AGKE terms in turbulent channel forced via spanwise oscillating walls For the streamwise normal stress.. • Shift of the production activity of � δ u δ u � towards larger wall-distances • Shift of the main transport of � δ u δ u � towards larger wall-distances • Both shifts increase with A 14

Outlook: source of �− δ u δ v � Ref OW r z x 2 x 1 y Y z 15

Thanks for your kind attention! For questions or suggestions: alessandro.chiarini@polimi.it davide.gatti@kit.edu maurizio.quadrio@polimi.it 16

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� δ u δ u � in the r x = 0 space Ref OW r y ≤ 2Y r z x 2 r y y x 1 Y z 18

� δ u δ u � in the r x = 0 space Ref OW 18

� δ u δ u � in the r x = 0 space What if A + is incremented? 30 Y + m 25 20 15 10 0 10 20 30 A + m increases with A + (% DR ) ∆ Y + 18

� uu � Vertical shift ∆ y of φ = 0 8 Ref OW 6 � uu � + 4 2 ∆ y + ∼ 3 0 0 10 20 30 y + 19

Source of �− δ u δ v � Ref OW r z x 2 x 1 y Y z 20

Source of �− δ u δ v � = + Π 12 + ξ 12 P 12 ǫ 12 ���� ���� ���� ���� source production pressure strain dissipation P 12 > 0 in all the domain Π 12 < 0 in (almost) all the domain ǫ 12 is negligible in all the domain 20

Source of �− δ u δ v � ∼ + Π 12 + ξ 12 P 12 ǫ 12 ���� ���� ���� ���� source production pressure strain dissipation 20

Production of �− δ u δ v � Ref OW r z x 2 x 1 y Y z 20

Production of �− δ u δ v � How does P max changes with A + ? 38 22 0 . 3 Y + P + r + max P m z 36 21 0 . 25 20 34 0 . 2 19 32 0 . 15 18 30 0 10 20 30 0 10 20 30 0 10 20 30 A + A + A + 20

Pressure strain of �− δ u δ v � Ref OW r z x 2 x 1 y Y z 20