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Energy transfer rates in turbulent channels with drag reduction at constant power input Davide Gatti, M. Quadrio, Y. Hasegawa, B. Frohnapfel and A. Cimarelli EDRFCM 2017, Villa Mondragone, Monte Porzio Catone www.kit.edu KIT – The Research University in the Helmholtz Association

The drag reduction experiment bulk velocity: 𝑉 𝑐 turbulent 𝝑 + mean 𝚾 kinetic energy dissipation rate pressure gradient: 𝑧 − d𝑞 d𝑦 = 𝜐 𝑥 𝑦 ℎ skin-friction coefficient: 𝑨 X 𝑔 = 2𝜐 𝑥 𝐷 2 𝜍𝑉 𝑐 2ℎ pumping power 𝑉 𝑧 (per unit area): 𝐐 𝒒 = − d𝑞 𝐐 𝒒 d𝑦 ℎ𝑉 𝑐 pumping power Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 2 18.04.2017

Integral energy budget Reynolds decomposition: 𝑣(𝑧) + 𝑣 ′ 𝑦, 𝑧, 𝑨, 𝑢 𝑣 𝑦, 𝑧, 𝑨, 𝑢 = 1 𝑣 2 2 𝜍 mean kinetic energy (MKE) budget: 𝑸 𝒒 = 𝑄 𝑣𝑤 + Φ 1 2 𝜍𝑣 ′2 turbulent kinetic energy (TKE) budget: 𝑄 𝑣𝑤 = 𝝑 global energy budget: 𝑸 𝒒 = 𝚾 + 𝝑 Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 3 18.04.2017

The drag reduction experiment bulk velocity: 𝑉 𝑐 turbulent 𝝑 + mean 𝚾 kinetic energy dissipation rate pressure gradient: 𝑧 − d𝑞 d𝑦 = 𝜐 𝑥 𝑦 ℎ skin-friction coefficient: 𝑨 X 𝑔 = 2𝜐 𝑥 𝐷 2 𝜍𝑉 𝑐 2ℎ pumping power control 𝑉 𝑧 (per unit area): power input 𝐐 𝒒 = − d𝑞 𝐐 𝒒 d𝑦 ℎ𝑉 𝑐 𝐐 𝒅 pumping power drag reduction rate: at (statistical) steady state: 𝐷 𝑔 𝑺 = 1 − 𝐐 𝐮 = 𝐐 p + 𝐐 𝒅 = 𝝑 + 𝚾 𝐷 𝑔,0 Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 4 18.04.2017

How does drag reduction affect energy transfer rates? a (seemingly) trivial question with a non trivial answer • Ricco et al., JFM (2012): substantial increase of 𝝑 caused by control with spanwise wall motions • Frohnapfel et al., (2007): 𝝑 needs to be reduced to achieve drag reduction • Martinelli, F., (2009): drag reduction obtained via feedback control aimed at minimizing 𝝑 Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 5 18.04.2017

Goal We investigate how skin-friction drag reduction affects energy-transfer rates in turbulent channels • do different control strategies behave similarly? • do universal relationships 𝜗 = 𝜗 𝑆 or Φ = Φ 𝑆 exist? • can we predict changes of 𝜗 or Φ ? by producing a direct numerical simulation (DNS) database of turbulent channels modified by several drag reduction techniques Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 6 18.04.2017

Comparing energy transfer rates correctly 𝐷 𝑔 successful control 𝑺 = 1 − 𝐷 𝑔,0 > 0 with control power P c − d𝑞 𝐐 𝐪 = − d𝑞 𝐐 𝐮 = 𝐐 𝐪 + 𝐐 𝐝 𝑫 𝒈 𝑉 𝑐 d𝑦 ℎ𝑉 𝑐 d𝑦 = CPG = ? CFR 𝑄 𝑞 and 𝑄 𝑢 change between controlled and natural flow!! Hasegawa et al., JFM (2014) propose alternative forcing methods: = CPI Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 7 18.04.2017

The DNS database at CPI Viscous “ + ” units: • Box size 𝑀 𝑦 , 𝑀 𝑧 , 𝑀 𝑨 = 4𝜌ℎ, 2ℎ, 2𝜌ℎ 𝑣 𝜐 = 𝜐 𝑥 /𝜍 Resolution Δ𝑦 + , Δ𝑧 + , Δ𝑨 + = 9.8, 0.47 − 2.59, 4.9 • 𝜀 𝜉 = 𝜉/𝑣 𝜐 • Average over 25000 viscous time units 2 𝑢 𝜉 = 𝜉/𝑣 𝜐 Constant total Power Input (CPI): 𝑆𝑓 Π = 𝑉 Π 𝜀 P 𝑢 ℎ = 6500 𝑉 Π = 𝜉 3𝜈 𝑄 𝑢 3 3 = 𝑸 𝒖 = 𝑄 + 𝑄 𝑑 is kept constant to 𝑞 𝑆𝑓 Π 𝜍𝑉 Π 𝛿 = 𝑄 𝑞 = 1 − 𝛿 𝑄 𝑢 = 3 1 − 𝛿 𝑑 𝑄 control power fraction , so that 𝑄 𝑢 𝑆𝑓 Π Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 8 18.04.2017

Control strategies Spanwise wall oscillations 𝜀 𝑧 𝑦 𝑨 𝑋 𝑥 = 𝐵sin(𝜕𝑢) 𝐷 𝑔 𝑆 = 1 − 𝑔,0 = 17.1% drag reduction 𝐷 P 𝑑 P control power 𝛿 = = 0.098 𝑢 fraction 𝑉 𝑐 = 1.028 𝑉 𝑐,𝑠𝑓𝑔 Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 9 18.04.2017

Control strategies Spanwise wall oscillations Opposition control 𝜀 𝑧 𝑧 𝑦 𝑦 𝑨 𝑨 𝑋 𝑥 = 𝐵sin(𝜕𝑢) 𝐷 𝑔 𝑧 𝑆 = 1 − 𝑔,0 = 17.1% drag reduction 𝐷 P 𝑑 P control power 𝛿 = = 0.098 𝑢 fraction 𝑉 𝑐 𝑨 = 1.028 𝑉 𝑐,𝑠𝑓𝑔 Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 10 18.04.2017

Control strategies Spanwise wall oscillations Opposition control 𝑧 𝑡 𝜀 𝑧 𝑧 𝑦 𝑦 𝑨 𝑨 𝑋 𝑥 = 𝐵sin(𝜕𝑢) 𝑤 𝑥 = −𝑤(𝑦, 𝑧 𝑡 , 𝑨, 𝑢) 𝐷 𝑔 𝑧 𝑆 = 1 − 𝑔,0 = 17.1% drag reduction 𝐷 P 𝑑 P control power 𝛿 = = 0.098 𝑧 𝑡 𝑢 fraction 𝑉 𝑐 𝑨 = 1.028 𝑉 𝑐,𝑠𝑓𝑔 −𝑤 𝑥 Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 11 18.04.2017

Control strategies Spanwise wall oscillations Opposition control 𝑧 𝑡 𝜀 𝑧 𝑧 𝑦 𝑦 𝑨 𝑨 𝑋 𝑥 = 𝐵sin(𝜕𝑢) 𝑤 𝑥 = −𝑤(𝑦, 𝑧 𝑡 , 𝑨, 𝑢) 𝐷 𝑔 𝑆 = 1 − 𝑔,0 = 17.1% 𝑆 = 23.9% drag reduction 𝐷 P 𝑑 P control power 𝛿 = = 0.098 𝛿 = 0.0035 𝑢 fraction 𝑉 𝑐 𝑉 𝑐 = 1.028 = 1.094 𝑉 𝑐,𝑠𝑓𝑔 𝑉 𝑐,𝑠𝑓𝑔 Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 12 18.04.2017

The energy box reference flow 𝑆𝑓 𝑐 = 3177 𝑆𝑓 𝜐 = 199.7 Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 13 18.04.2017

The energy box 𝑉 𝑐 = 1.094 opposition control 𝑆𝑓 𝑐 = 3474 𝑆𝑓 𝜐 = 190.5 𝑉 𝑐.0 MKE dissipation rate Φ increases TKE production rate 𝑄 𝑣𝑤 and dissipation rate 𝜗 decrease Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 14 18.04.2017

The energy box 𝑉 𝑐 = 1.028 oscillating wall 𝑆𝑓 𝑐 = 3267 𝑆𝑓 𝜐 = 186.9 𝑉 𝑐.0 Both MKE dissipation Φ and TKE production 𝑄 𝑣𝑤 rates decrease, 𝑉 𝑐 increases! TKE dissipation rate 𝜗 increases Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 15 18.04.2017

The energy box: lesson Drag reduction reduction of TKE production rate 𝑄 𝑣𝑤 Drag reduction ≠ increase of MKE dissipation rate Φ 𝑄 𝑑 surprisingly good alternative to pumping with wall oscillations! By accounting for the physics of the control and separating the contribution of 𝑄 𝑑 to 𝜗 , it is also true that: Drag reduction reduction of TKE dissipation rate 𝜗 Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 16 18.04.2017

Predicting 𝝑 𝑺 for 𝑺 ≈ 𝟏 (1) The dissipation 𝜗 in power units is linked to 𝜗 + in viscous units by the following: 3 𝑆𝑓 𝜐 𝜗 = 𝜗 + 𝑆𝑓 Π 𝑆𝑓 𝜐 can be substituted with 𝑆𝑓 𝑐 with the following relationship: 𝑞 = − d𝑞 2 2 𝑆𝑓 𝑐 = 3 1 − 𝛿 𝑆𝑓 Π 𝑆𝑓 𝜐 , which in nondimensional form reads P d𝑦 ℎ𝑉 𝑐 this yields 3/2 𝜗 = 𝜗 + 3 1 − 𝛿 𝑆𝑓 b Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 17 18.04.2017

Predicting 𝝑 𝑺 for 𝑺 ≈ 𝟏 (2) The following relation holds for both controlled and reference flow 3/2 𝜗 = 𝜗 + 3 1 − 𝛿 𝑆𝑓 b by taking the ratio in the controlled and reference channel we obtain 3/2 = 𝜗 + 𝜗 1 − 𝛿 𝑆𝑓 𝑐.0 + 𝜗 0 𝜗 0 𝑆𝑓 𝑐 for a reference channel flow it is known that the 𝜗 + is a mild function of 𝑆𝑓 𝜐 𝜗 + = 2.54 ln 𝑆𝑓 𝜐 − 6.72 Abe & Antonia, JFM (2016) 𝜗 + Hypothesis: if 𝑆 ≈ 0 then 𝑆𝑓 𝜐 ≈ 𝑆𝑓 𝜐,0 , so we assume ≈ 1 + 𝜗 0 Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 18 18.04.2017

Predicting 𝝑 𝑺 for 𝑺 ≈ 𝟏 (3) The relation reduces eventually to: no general statement on 𝜗 + 3/2 𝜗 1 − 𝛿 𝑆𝑓 𝑐.0 = without considering 𝜗 0 𝑆𝑓 𝑐 the physics of the control! opposition control wall oscillation Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 19 18.04.2017

Conclusions • CPI approach is essential to assess energy transfer rates in drag- reduced flows • Energy box analysis yields two statements Drag reduction reduction of TKE dissipation rate 𝜗 Drag reduction ≠ increase of MKE dissipation rate Φ • No universal relationship between 𝑆 and 𝜗 could be found without considering the physics of the control Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction 20 18.04.2017