Direct Numerical Simulation of Drag Reduction with Uniform Blowing - - PowerPoint PPT Presentation

Direct Numerical Simulation of Drag Reduction with Uniform Blowing

• ver a Two-dimensional Roughness

Eisuke Mori1, Maurizio Quadrio2 and Koji Fukagata1

1 Keio University, Japan 2 Politecnico di Milano, Italy

European Drag Reduction and Flow Control Meeting Rome, Apr. 3-6, 2017

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Uniform blowing (UB)

• Drag contribution in a channel flow with UB(/US)
• Excellent performance (about 45% by 𝑾𝒙 = 𝟏. 𝟔%𝑽∞)
• Unknown over a rough wall

𝑫𝒈 = 𝟐𝟑 𝐒𝐟𝒄 + 𝟐𝟑 න

𝟏 𝟑

𝟐 − 𝒛 −𝒗′𝒘′ 𝒆𝒛

(Fukagata et al., Phys. Fluids, 2002)

Viscous Contribution (= laminar drag, const.) Turbulent contribution Convective (=UB/US) contribution On a boundary layer, White: vortex core, Colors: wall shear stress

𝑾𝒙: Blowing velocity

−𝟐𝟑𝑾𝒙 න

𝟏 𝟑

𝟐 − 𝒛 ഥ 𝒗𝒆𝒛

(Kametani & Fukagata, J. Fluid Mech., 2011)

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UB over a rough wall

Experimental results so far

• Similar to the smooth-wall cases
• Schetz and Nerney, AIAA J., 1977
• Voisinet, 1979
• Opposite behavior

(drag increased, turbulent intensity suppressed)

• Miller et al., Exp. Fluids, 2014

Contradicting remarks exist

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Goal

Investigate the interaction between roughness and UB for drag reduction using numerical simulation

• DNS of turbulent channel flow
• Drag reduction performance and mechanism
• Combined effect of UB and roughness

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Numerical procedure

• Based on FD code (for wall deformation)

(Nakanishi et al., Int. J. Heat Fluid Fl., 2012)

• Constant flow rate, 𝐒𝐟𝒄 = 𝟑𝑽𝒄𝜺/𝝃 = 𝟔𝟕𝟏𝟏
• so that 𝐒𝐟𝝊 ≈ 𝟐𝟗𝟏 in a plane channel (K.M.M.)
• ∆𝒚+ = 𝟓. 𝟓, 𝟏. 𝟘𝟒 < ∆𝒛+ < 𝟕, ∆𝒜+ = 𝟔. 𝟘
• UB magnitude:

Τ 𝑾𝒙 𝑽𝒄 = 𝟏, 𝟏. 𝟐%, 𝟏. 𝟔%, 𝟐%

SMOOTH CASE ROUGH CASE

E.Mori, M.Quadrio, K.Fukagata DNS/Drag Reduction/Uniform blowing(UB)/Rough wall

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Model of rough wall

Roughness displacement

𝜺: channel half height 𝑴𝒚: Channel length, 𝟓𝝆𝜺 𝑩𝒋: Amplitude of each sinusoid 𝑩𝒋 = ቊ 𝟐, 𝐠𝐩𝐬 𝒋 = 𝟐 𝟏, 𝟐 , 𝐠𝐩𝐬 𝒋 ≠ 𝟐 with rescaling so that 𝒆 𝒚 = 𝟏. 𝟏𝟔𝜺

𝒆 𝒚 = 𝜺 ෍

𝒋=𝟐 𝟗

𝑩𝒋 𝐭𝐣𝐨 𝟑𝒋𝝆𝒚 Τ 𝑴𝒚 𝟑

𝒆 𝒚

𝐧𝐛𝐲 = 𝟏. 𝟐𝟐𝜺

(Defined randomly)

(E. Napoli et al., J. Fluid Mech., 2008)

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The result of the base flow

∆𝑉+~6.5

“Transitionally-rough regime”

𝒍𝒕

+ = 𝟒𝟗

𝐷𝐸𝑞 = 2𝐷𝐸ave − (𝐷𝐸𝑔 + 𝐷𝐸𝑔,𝑣)

𝐷𝐸ave: Overall drag coefficient 𝐷𝐸𝑔: 𝐷

𝑔 of the rough wall side

𝐷𝐸𝑔,𝑣: 𝐷

𝑔 of the smooth wall side

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The result of UB

ROUGH CASE SMOOTH CASE

Total, 𝑆 𝟐𝟐% 𝟒𝟖% 𝟔𝟘% 𝟖% 𝟑𝟕% 𝟓𝟒% 𝑆 = 1 − 𝐷𝐸,ctr 𝐷𝐸,nc

𝐷𝐸,ctr: controlled 𝐷𝐸,nc: no control

E.Mori, M.Quadrio, K.Fukagata DNS/Drag Reduction/Uniform blowing(UB)/Rough wall

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The result of UB

ROUGH CASE SMOOTH CASE

Total, 𝑆 𝟐𝟐% 𝟒𝟖% 𝟔𝟘% 𝟖% 𝟑𝟕% 𝟓𝟒% Friction, 𝑆𝐺 𝟐𝟐% 𝟒𝟖% 𝟔𝟘% 𝟘% 𝟒𝟓% 𝟔𝟖% Pressure, 𝑆𝑄

• 𝟔% 𝟐𝟘% 𝟒𝟑%

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Bulk mean streamwise velocity

Friction drag reduction mechanism

Black: Τ 𝑽𝒄 𝑾𝒙 = 0 Green: Τ 𝑽𝒄 𝑾𝒙 = 0.1% Red: Τ 𝑽𝒄 𝑾𝒙 = 0.5% Blue: Τ 𝑽𝒄 𝑾𝒙 = 1%

+nc: normalization with no control case 𝒗𝝊

SMOOTH CASE ROUGH CASE

𝒆 𝒚 𝐧𝐛𝐲

+

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How does pressure drag decrease?

Pressure contours

averaged in the spanwise and time dashed lines: zero contour

𝑞+nc Τ 𝑽𝒄 𝑾𝒙 = 𝟐% Τ 𝑽𝒄 𝑾𝒙 = 𝟏

𝒆 𝒚 𝐧𝐛𝐲

+

𝒆 𝒚 𝐧𝐣𝐨

+

E.Mori, M.Quadrio, K.Fukagata DNS/Drag Reduction/Uniform blowing(UB)/Rough wall

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“Smoothing effect”

Wall-normal velocity contours

averaged in the spanwise and time dashed lines: zero contour

𝑤+nc

𝒆 𝒚 𝐧𝐛𝐲

+

𝒆 𝒚 𝐧𝐣𝐨

+

Τ 𝑽𝒄 𝑾𝒙 = 𝟐% Τ 𝑽𝒄 𝑾𝒙 = 𝟏 𝑙𝑡

+ = 38

𝑙𝑡

+ = 20

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Velocity defect

Outer layer similarity with UB

Base flow (No controlled) of

• ne-side rough wall

1% UB case of

• ne-side rough wall

Smooth side Rough side 𝜺𝒖: distance from a wall to the minimum RMS location (K. Bhaganagar et al., Flow, Turbul. Combust., 2004)

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Comparison with smooth case

Velocity defect

1% UB case of both-side smooth wall 1% UB case of

• ne-side rough wall

Smooth side Rough side Suction side Blowing side

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Comparison with smooth case

Velocity defect

1% UB case of both-side smooth wall 1% UB case of

• ne-side rough wall

Same tendency, but quantitatively weakened

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Stevenson’s law of the wall

Plots using Stevenson’s law of the wall (Stevenson, 1963)

2 𝑊

𝑥 +

1 + 𝑊

𝑥 +𝑉+ − 1

= 1 𝜆 ln 𝑧+ + 𝐶

𝑉+𝑇

Modified law is suggested: 2 𝑊

𝑥 +

1 + 𝑊

𝑥 +𝑉+ − 1

= 1 𝜆 ln 𝑧+ + 𝐶 − ∆𝑉+ Roughness function ∆𝑉+

E.Mori, M.Quadrio, K.Fukagata DNS/Drag Reduction/Uniform blowing(UB)/Rough wall

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Normalization by 𝒗𝝊

𝐨𝐝+

Drag reduction, ∆𝑫𝑬 = 𝑫𝑬,𝐨𝐝 − 𝑫𝑬,𝐝𝐮𝐬 Drag reduction rate, 𝑆

nc: no control ctr: controlled

𝑆 becomes similar when plotted with 𝑊

𝑥 +

E.Mori, M.Quadrio, K.Fukagata DNS/Drag Reduction/Uniform blowing(UB)/Rough wall

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Concluding remarks

DNS of turbulent channel flow is performed

• ver a rough wall with UB
• UB is effective over a rough wall
• Almost same in drag reduction rate, but larger in drag

reduction amount (when normalized by 𝑣𝜐

+nc)

• Drag reduction mechanisms are considered
• Friction drag is reduced by wall-normal convection
• Pressure drag is reduced by “smoothing effect”
• Combined effect (UB + roughness) slightly exists
• Modified Stevenson’s law of the wall is suggested

Thank you for your kind attention

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Background

Turbulence

• Huge drag
• Environmental problems
• High operation cost
• How to control?

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Flow control classification

(M. Gad-el-Hak, J. Aircraft, 2001)

Flow control strategies Passive Feedback Feedforward Active

• Uniform blowing

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Governing equations

(S. Kang & H. Choi, Phys. Fluids, 2000)

Incompressible Continuity and Navier-Stokes in 𝝄𝒋 coordinate 𝛜𝒗𝒋 𝛜𝝄𝒋 = −𝑻 𝛜𝒗𝒋 𝝐𝒖 = − 𝛜 𝒗𝒋𝒗𝒌 𝛜𝝄𝒌 − 𝝐𝒒 𝛜𝝄𝒋 + 𝟐 𝐒𝐟𝒄 𝝐𝟑𝒗𝒋 𝛜𝝄𝒌𝝄𝒌 − 𝒆𝑸 𝐞𝝄𝟐 𝜺𝒋𝟐 + 𝑻𝒋

𝑇𝑗 = −𝜒𝑢 𝜖𝑣𝑗 𝜖𝜊2 − 𝜚𝑘 𝜖 𝑣𝑗𝑣𝑘 𝜖𝜊2 − 𝜚𝑘 𝑒𝑞 𝑒𝜊2 𝜀𝑗𝑘 + 1 𝑆𝑓 2𝜚𝑘 𝜖2𝑣𝑗 𝜖𝜊𝑘𝜊2 + 𝜚𝑘𝜚𝑘 𝜖2𝑣𝑗 𝜖𝜊2

2 + 1

2 𝜖 𝜚𝑘𝜚𝑘 𝜖𝜊2 𝜖𝑣𝑗 𝜖𝜊2 𝜒𝑘 = − 1 1 + 𝜃 𝜊2 𝜖𝜃 𝜖𝜊𝑗 + 𝜖𝜃0 𝜖𝜊𝑗 , for j = 1,3 1 1 + 𝜃 , for j = 2 𝜚𝑘 = 𝜒𝑘 − 𝜀𝑘2 𝑇 = 𝜚𝑘 𝜖𝑣𝑗 𝜖𝜊2

where

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Coordinate transformation

(S. Kang & H. Choi, Phys. Fluids, 2000)

Calculation grids: 𝝄𝒋 (Cartesian with extra force)

Actual grid points allocation

wall

ቐ 𝒚 = 𝝄𝟐 𝒛 = 𝛐𝟑 𝟐 + 𝛉 + 𝛉𝐞 𝒜 = 𝛐𝟒

(𝑦, 𝑧, 𝑨: physical coordinate) 𝛉 ≡ Τ 𝛉𝐯 − 𝛉𝒆 𝟑 = − Τ 𝒆 𝐲 𝟑 𝛉𝐞 = 𝐬 𝐲 , 𝛉𝒗 = 𝟏 𝛉𝐞, 𝛉𝐯: displacement of lower/upper wall

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Post processing

• Drag coefficient decomposition for rough case
• Drag reduction rate

𝑆𝐸𝑚 = ∆𝐷𝐸 𝐷𝐸,𝑁=0 × 100 [%] 𝐷𝐸𝑣𝑔 = 8 Re𝑐 ቤ 𝑒ത 𝑣 𝑒𝑧 𝜊2=2 𝐷𝐸𝑚𝑔 = 8 Re𝑐 ቤ 𝑒𝑣 𝑒𝑧 𝜊2=0 + ቤ 𝑒𝑤 𝑒𝑦 𝜊2=0 𝐷𝐸𝑚𝑞 = −16 𝑒𝑄 𝑒𝜊1 − 𝐷𝐸𝑚𝑔 + 𝐷𝐸𝑣𝑔

Only focusing on lower side, subscript “𝑚” omitted hereafter

(Friction component) (Pressure component)

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Discretization methods

• Energy-conservative second-order finite difference

schemes (In space)

• Low-storage third-order Runge-Kutta / Crank-

Nicolson scheme (In time) + SMAC method for pressure correction Discretized in the staggered grid system

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Validation & Verification

Bulk mean streamwise velocity Time trace of instantaneous 𝐷𝐸 Less than 2% of difference from the most resolved case

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Stevenson’s law of the wall

Plots using Stevenson’s law of the wall (Stevenson, 1963)

2 𝑊

𝑥 +

1 + 𝑊

𝑥 +𝑉+ − 1

= 1 𝜆 ln 𝑧+ + 𝐶

𝑉+𝑇 Normalized by local 𝑣𝜐 Normalized by Stevenson’s law