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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

2 18.04.2017

The drag reduction experiment

𝑉 𝑧 2ℎ X

𝐐𝒒 pumping power

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

pumping power (per unit area): 𝐐𝒒 = − d𝑞 d𝑦 ℎ𝑉𝑐 bulk velocity: 𝑉𝑐 pressure gradient: − d𝑞 d𝑦 = 𝜐𝑥 ℎ skin-friction coefficient: 𝐷

𝑔 = 2𝜐𝑥

𝜍𝑉𝑐

2

𝑦 𝑧 𝑨

turbulent 𝝑 + mean 𝚾 kinetic energy dissipation rate

3 18.04.2017

Integral energy budget

mean kinetic energy (MKE) budget: 1 2 𝜍 𝑣2

𝑸𝒒 = 𝑄

𝑣𝑤 + Φ

turbulent kinetic energy (TKE) budget: 1 2 𝜍𝑣′2

𝑄

𝑣𝑤 = 𝝑

global energy budget: 𝑸𝒒 = 𝚾 + 𝝑

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

𝑣 𝑦, 𝑧, 𝑨, 𝑢 = 𝑣(𝑧) + 𝑣′ 𝑦, 𝑧, 𝑨, 𝑢 Reynolds decomposition:

4 18.04.2017

𝐐𝐮 = 𝐐p + 𝐐𝒅 = 𝝑 + 𝚾

The drag reduction experiment

𝑉 𝑧 2ℎ

control power input

X

𝐐𝒅 𝐐𝒒 pumping power at (statistical) steady state: drag reduction rate: 𝑺 = 1 −

𝐷𝑔 𝐷𝑔,0

pumping power (per unit area): 𝐐𝒒 = − d𝑞 d𝑦 ℎ𝑉𝑐 turbulent 𝝑 + mean 𝚾 kinetic energy dissipation rate bulk velocity: 𝑉𝑐 pressure gradient: − d𝑞 d𝑦 = 𝜐𝑥 ℎ skin-friction coefficient: 𝐷

𝑔 = 2𝜐𝑥

𝜍𝑉𝑐

2

𝑦 𝑧 𝑨

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

5 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

6 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

• f turbulent channels

modified by several drag reduction techniques

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

7 18.04.2017

CFR

= ?

𝑉𝑐 − d𝑞 d𝑦 𝐐𝐪 = − d𝑞 d𝑦 ℎ𝑉𝑐 𝐐𝐮 = 𝐐𝐪 + 𝐐𝐝 𝑫𝒈 CPG

=

successful control 𝑺 = 1 −

𝐷𝑔 𝐷𝑔,0 > 0 with control power P c

Comparing energy transfer rates correctly

𝑄

𝑞 and 𝑄𝑢 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

8 18.04.2017

control power fraction , so that

The DNS database at CPI

• Box size 𝑀𝑦, 𝑀𝑧, 𝑀𝑨 = 4𝜌ℎ, 2ℎ, 2𝜌ℎ
• Resolution Δ𝑦+, Δ𝑧+, Δ𝑨+ = 9.8, 0.47 − 2.59, 4.9
• Average over 25000 viscous time units

Constant total Power Input (CPI):

Viscous “+” units: 𝑣𝜐 = 𝜐𝑥/𝜍 𝜀𝜉 = 𝜉/𝑣𝜐 𝑢𝜉 = 𝜉/𝑣𝜐

2

𝑆𝑓Π = 𝑉Π𝜀 𝜉 = 6500 𝑉Π = P

𝑢ℎ

3𝜈 𝑸𝒖 = 𝑄

𝑞

+ 𝑄

𝑑 is kept constant to

𝛿 = 𝑄

𝑑

𝑄𝑢 𝑄𝑢 𝜍𝑉Π

3 =

3 𝑆𝑓Π 𝑄

𝑞 = 1 − 𝛿 𝑄𝑢 = 3 1 − 𝛿

𝑆𝑓Π

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

9 18.04.2017

Control strategies

Spanwise wall oscillations 𝑆 = 1 − 𝐷

𝑔

𝐷

𝑔,0 = 17.1%

drag reduction control power fraction

𝑉𝑐 𝑉𝑐,𝑠𝑓𝑔 = 1.028

𝜀 𝑋

𝑥 = 𝐵sin(𝜕𝑢)

𝑦 𝑧 𝑨

𝛿 = P

𝑑 P 𝑢

= 0.098

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

10 18.04.2017

Control strategies

Spanwise wall oscillations Opposition control 𝑆 = 1 − 𝐷

𝑔

𝐷

𝑔,0 = 17.1%

drag reduction control power fraction

𝑉𝑐 𝑉𝑐,𝑠𝑓𝑔 = 1.028

𝜀 𝑋

𝑥 = 𝐵sin(𝜕𝑢)

𝑦 𝑧 𝑨 𝑦 𝑧 𝑨 𝑧 𝑨

𝛿 = P

𝑑 P 𝑢

= 0.098

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

11 18.04.2017

Control strategies

Spanwise wall oscillations Opposition control 𝑆 = 1 − 𝐷

𝑔

𝐷

𝑔,0 = 17.1%

𝛿 = P

𝑑 P 𝑢

= 0.098

drag reduction control power fraction

𝑉𝑐 𝑉𝑐,𝑠𝑓𝑔 = 1.028

𝜀 𝑧𝑡 𝑋

𝑥 = 𝐵sin(𝜕𝑢)

𝑤𝑥 = −𝑤(𝑦, 𝑧𝑡, 𝑨, 𝑢) 𝑦 𝑧 𝑨 𝑦 𝑧 𝑨 𝑧 𝑨 𝑧𝑡 −𝑤𝑥

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

12 18.04.2017

Control strategies

Spanwise wall oscillations Opposition control 𝑆 = 1 − 𝐷

𝑔

𝐷

𝑔,0 = 17.1%

𝑆 = 23.9% 𝛿 = 0.0035

drag reduction control power fraction

𝑉𝑐 𝑉𝑐,𝑠𝑓𝑔 = 1.028 𝑉𝑐 𝑉𝑐,𝑠𝑓𝑔 = 1.094

𝜀 𝑧𝑡 𝑋

𝑥 = 𝐵sin(𝜕𝑢)

𝑤𝑥 = −𝑤(𝑦, 𝑧𝑡, 𝑨, 𝑢) 𝑦 𝑧 𝑨 𝑦 𝑧 𝑨

𝛿 = P

𝑑 P 𝑢

= 0.098

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

13 18.04.2017

The energy box

reference flow 𝑆𝑓𝑐 = 3177 𝑆𝑓𝜐 = 199.7

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

14 18.04.2017

MKE dissipation rate Φ increases TKE production rate 𝑄

𝑣𝑤 and dissipation rate 𝜗 decrease

The energy box

• pposition control

𝑆𝑓𝑐 = 3474 𝑆𝑓𝜐 = 190.5 𝑉𝑐 𝑉𝑐.0 = 1.094

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

15 18.04.2017

The energy box

• scillating wall

𝑆𝑓𝑐 = 3267 𝑆𝑓𝜐 = 186.9 𝑉𝑐 𝑉𝑐.0 = 1.028 Both MKE dissipation Φ and TKE production 𝑄

𝑣𝑤 rates decrease, 𝑉𝑐 increases!

TKE dissipation rate 𝜗 increases

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

16 18.04.2017

The energy box: lesson

Drag reduction reduction of TKE production rate 𝑄

𝑣𝑤

Drag reduction ≠ increase of MKE dissipation rate Φ 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 𝑄

𝑑 surprisingly good alternative to pumping with wall oscillations!

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

17 18.04.2017

Predicting 𝝑 𝑺 for 𝑺 ≈ 𝟏 (1)

𝑆𝑓𝜐

2𝑆𝑓𝑐 = 3 1 − 𝛿 𝑆𝑓Π 2

𝜗 = 𝜗+ 𝑆𝑓𝜐 𝑆𝑓Π

3

The dissipation 𝜗 in power units is linked to 𝜗+ in viscous units by the following: 𝑆𝑓𝜐 can be substituted with 𝑆𝑓𝑐 with the following relationship: P

𝑞 = − d𝑞

d𝑦 ℎ𝑉𝑐 , which in nondimensional form reads this yields 𝜗 = 𝜗+ 3 1 − 𝛿 𝑆𝑓b

3/2

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

18 18.04.2017

𝜗+ = 2.54 ln 𝑆𝑓𝜐 − 6.72 𝜗 = 𝜗+ 3 1 − 𝛿 𝑆𝑓b

3/2

The following relation holds for both controlled and reference flow by taking the ratio in the controlled and reference channel we obtain 𝜗 𝜗0 = 𝜗+ 𝜗0

+

1 − 𝛿 𝑆𝑓𝑐.0 𝑆𝑓𝑐

3/2

for a reference channel flow it is known that the 𝜗+ is a mild function of 𝑆𝑓𝜐

Abe & Antonia, JFM (2016)

Hypothesis: if 𝑆 ≈ 0 then 𝑆𝑓𝜐 ≈ 𝑆𝑓𝜐,0, so we assume

Predicting 𝝑 𝑺 for 𝑺 ≈ 𝟏 (2)

𝜗+ 𝜗0

+

≈ 1

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

19 18.04.2017

The relation reduces eventually to: 𝜗 𝜗0 = 1 − 𝛿 𝑆𝑓𝑐.0 𝑆𝑓𝑐

3/2

Predicting 𝝑 𝑺 for 𝑺 ≈ 𝟏 (3)

• pposition control

wall oscillation

no general statement on 𝜗+ without considering the physics of the control!

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

20 18.04.2017

Conclusions

• CPI approach is essential to assess energy transfer rates in drag-

reduced flows

• Energy box analysis yields two statements
• No universal relationship between 𝑆 and 𝜗 could be found

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

without considering the physics of the control Drag reduction reduction of TKE dissipation rate 𝜗 Drag reduction ≠ increase of MKE dissipation rate Φ

21 18.04.2017

The drag reduction experiment

𝑉 𝑧 2ℎ

control power input

X

P

𝑑

P

𝑞

pumping power

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction

drag reduction rate: 𝑆 = 1 −

𝐷𝑔 𝐷𝑔,0

pumping power (per unit area): P

𝑞 = − d𝑞

d𝑦 ℎ𝑉𝑐 bulk velocity: 𝑉𝑐 pressure gradient: − d𝑞 d𝑦 = 𝜐𝑥 ℎ skin-friction coefficient: 𝐷

𝑔 = 2𝜐𝑥

𝜍𝑉𝑐

2

𝑦 𝑧 𝑨

turbulent 𝝑 + mean 𝚾 kinetic energy dissipation rate

22 18.04.2017

THANKS

for your kind attention! for questions, complaints, ideas: davide.gatti@kit.edu

Dr.-Ing. Davide Gatti – Energy transfer rates in turbulent channels with drag reduction