Impact of Drag Reduction Control on Energy Box of a Fully Developed - - PowerPoint PPT Presentation

Impact of Drag Reduction Control

• n

Energy Box of a Fully Developed T urbulent Channel Flow

Yosuke Hasegawa Instjtute of Industrial Science, The University of Tokyo, Japan Davide Gattj, Bettjna Frohnapfel Karlsruhe Instjtute of Technology, Germany Andrea Cimarelli Universita Plitecnica delle Marche, Italy Maurizio Quadrio Politecnico di Milano, Italy

• Nov. 20, 2017

APS DFD@Denver

• A Fully Developed Channel Flow

Essentjal physics of near-wall turbulence Flow control (drag reductjon, mixing enhancement, etc.)

• Global Energy Budget

Back ckgr ground nd

Total kinetjc energy =

Mean Turbulent 1/24 2

• A Fully Developed Channel Flow

Essentjal physics of near-wall turbulence Flow control (drag reductjon, mixing enhancement, etc.)

• Global Energy Budget

Back ckgr ground nd

Total kinetjc energy =

Mean Turbulent Mean fjeld Turbulent fjeld Pumping Power Turbulent Productjon Direct dissipatjon Turbulent dissipatjon Direct dissipatjon (diss. of mean fjeld) Turbulent dissipatjon

𝜚= 1 𝑆𝑓( 𝑒 ´ 𝑣 𝑒𝑧 )

2

𝜁= 1 𝑆𝑓( ´ 𝑒𝑣′ 𝑗 𝑒 𝑦 𝑘 𝑒𝑣′𝑗 𝑒 𝑦 𝑘 )

2

2/24

Energy Box

Dir irect ect and nd Tur urbu bulent lent Dis issipa ipatio ion in in Unco Uncont ntrolle le d Flow

Direct dissipatjon Turbulent dissipatjon

𝑺𝒇𝝊 𝟔𝟏𝟏

3/24

Objec jectives ves

• Drag reductjon control is somewhat similar to make the “efgec

tjve” Re lower

• Then, the direct dissipatjon increases, while the turbulent dissipatjo

n decreases ?

• Ultjmate control: complete relaminarizatjon
• All input energy should be dissipated by the direct dissipatjon only.

Questjons Is there an unique relatjonship between the changes in direct/turbulent dissipatjon and a drag reductjon efgect ?

4/24

Flow

• w Cond

nditions ns

X

Pump

Pumping Power: Flow Rate:

Pressure Gradient: 5/24

Flow

• w Cond

nditions ns

Flow Rate: Pressure Gradient: Pumping Power:

Constant Flow Rate (CFR) Constant Constant Pressure Gradient (CPG) Constant Constant Flow Rate (CFR) Constant Constant Pressure Gradient (CPG) Constant X

Pump

Pumping Power: Flow Rate:

Pressure Gradient: 6/24

Ener ergy Bo Box Under der CP CPG

Ricco et al. JFM (2011)

Spanwise Wall Oscillatjon Control

Mean U

Pumping power input changes due to the applied control….

Mean W Turbulence

7/24

Pumping Power Control input Direct dissipatjon

• f U

Direct dissipatjon of W Turbulent productjon Turbulent dissipatjon

Flow

• w Cond

nditions ns

Flow Rate: Pressure Gradient: Pumping Power:

Constant Flow Rate (CFR)

Constant

Constant Pressure Gradient (CPG)

Constant

Constant Power Input (CPI)

Constant

Constant Flow Rate (CFR)

Constant

Constant Pressure Gradient (CPG)

Constant

Constant Power Input (CPI)

Constant

X

Pump

Pumping Power: Flow Rate:

Pressure Gradient: 8/24

Co Conce cept pt of

• f Con
• nst

stan ant Pow

• wer

er Inpu put (C (CPI PI)

X

Pump

Pumping Power : Control Input:

Dissipatjon

(Frohnapfel et al. JFM 2012, Hasegawa et al. JFM 2014)

• Total Power Input

𝚸𝐮

∗

=Π 𝑞

∗+Π 𝑑 ∗

=𝑑𝑝𝑜𝑡𝑢 .

• Max. Achievable Flow Rate

𝑉Π

∗=√

𝚸𝐮

∗𝜀 ∗

3𝜈

∗

• Fractjon of Control Power Input

𝛅=Π 𝑑

∗

/ Π t

∗

• Power-based Reynolds Number

𝑆𝑓Π=𝑽 𝜬

∗ 𝜀∗

𝜉

∗

9/24

Flow Rate:

Exa Exampl ple

X

Pump

Pumping Power :

Flow Rate:

• (CPG)
• (CFR)

Control Input:

/

Reference (NC)

0.4887 1.0

Reference (NC)

0.4887 1.0

• Control Schemes
• Oppositjon Control (Choi et al. JFM 1998)
• Spanwise Wall Oscillatjon (Jung et al. PoF 1992)

10/24

Exa Exampl ple

• Control Schemes
• Oppositjon Control (Choi et al. JFM 1998)
• Spanwise Wall Oscillatjon (Jung et al. PoF 1992)

X

Pump

Pumping Power :

Flow Rate:

• (CPG)
• (CFR)

Control Input:

/

Reference (NC)

0.4887 1.0

Wall Oscillatjon

0.5026 0.0978 1.028

Oppositjon Control

0.5345 0.0035 1.094

Reference (NC)

0.4887 1.0

Wall Oscillatjon

0.5026 0.0978 1.028

Oppositjon Control

0.5345 0.0035 1.094

11/24

12/24

En Ener ergy gy Bo Box of Unco ncontr ntrolle lled Fl

• w

En Ener ergy Box unde der Op Oppo posi siti tion Co Contr trol

Direct dissipatjon increases Turbulent dissipatjon decreases

13/24

Ene nergy Box und nder er Wall all Os Osci cilla llatio tion Co Cont ntrol

Turbulent dissipatjon increases Direct dissipatjon decreases

14/24

Ene nergy Box und nder er Wall all Os Osci cilla llatio tion Co Cont ntrol

Turbulent dissipatjon increases Direct dissipatjon decreases

Opposite trends in the changes of direct/turbulent dissipatjon for drag-reduced fmows

Can we explain it ?

15/24

Fl Flow Rate te Und nder er CP CPI

• FIK Identjty (Fukagata et al. PoF 2002)

𝑽 𝒄=𝜷 𝑺𝒇𝚸 𝟑 {−𝟐+√ 𝟐+ 𝟓 (𝟐−𝜹 )

(𝜷 𝑺𝒇𝚸)

𝟑}

(Hasegawa et al. JFM 2014)

𝜷=∫

𝟏 𝟐

(𝟐−𝒛 ) (

´ −𝒗

′ 𝒘 ′ ) 𝒆𝒛

• Weighted Reynolds Shear Stress

𝛅=Π 𝑑

∗

/ Π t

∗

• Fractjon of Control Power Input

The fmow rate is determined by and only.

{

𝜷 →𝟏 𝜹 →𝟏 𝑽 𝒄→𝟐

𝜷 →∞𝑽 𝒄→𝟏

Two Limitjng Cases

16/24

Ener nergy Flu lux Unde der CP CPI

• Triple Decompositjon

𝑣=𝑉+𝑣′

17/24

Ener nergy Flu lux Unde der CP CPI

• Triple Decompositjon

𝑣=𝑉+𝑣′=𝑉 𝑚𝑏𝑛+ Δ𝑉 +𝑣′

𝑽 𝑉 𝑚𝑏𝑛

Δ𝑉

: parabolic profjle with the same fmow rate : deviatjon from parabola

(Echhardt et al., JFM 2007) 18/24

Ener nergy Flu lux Unde der CP CPI

• Productjon

𝑄=𝑄𝑚𝑏𝑛+ 𝑄Δ=∫

1

− ´ 𝑣

′ 𝑤′ 𝑒𝑉 𝑚𝑏𝑛

𝑒𝑧 𝑒𝑧+∫

1

− ´ 𝑣

′𝑤 ′ 𝑒 Δ𝑉

𝑒𝑧 𝑒𝑧

• Direct Dissipatjon

𝜚= 1 𝑆𝑓Π∫

1

(

𝑒𝑉 𝑒𝑧 )

2

𝑒𝑧= 1 𝑆𝑓Π∫

1

{

𝑒 𝑒𝑧 (𝑉 𝑚𝑏𝑛+ Δ𝑉 )}

2

𝑒𝑧

¿ 1 𝑆𝑓Π∫

1

{(

𝑒𝑉 𝑚𝑏𝑛 𝑒𝑧 )

2

+( 𝑒 Δ𝑉 𝑒𝑧 )

2

+( 𝑒𝑉 𝑚𝑏𝑛 𝑒𝑧 )( 𝑒 Δ 𝑉 𝑒𝑧 )}𝑒𝑧

¿ 𝜚𝑚𝑏𝑛+𝜚Δ

In additjon, it can be shown that

19/24 Only true in the present triple decompositjon

Mod

• difj

ifjed ed En Energy Box

20/24

Mod Modifje fjed Energy Box

𝜚𝑚𝑏𝑛= 3 𝑆𝑓 Π{

(𝜷 𝑆𝑓Π )

2

2

(1−√

1+ 4 (1−𝛿) 𝜷 𝑆𝑓Π

2)+ (1−𝛿 )}

𝜚 Δ=𝑆𝑓Π ( 𝜸 −3 𝜷2)

• Direct Dissipatjon ()
• Turbulent Dissipatjon

𝜁= 3 𝑆𝑓Π{

(𝜷 𝑆𝑓Π )

2

2

(1+√

1+ 4 (1−𝛿 ) 𝜷 𝑆𝑓Π

2)− 𝜸 𝑆𝑓𝛲 2

3 +𝛿}

𝜷=∫

𝟏 𝟐

(𝟐−𝒛 ) (

´ −𝒗

′ 𝒘 ′ ) 𝒆𝒛

𝜸=∫

𝟏 𝟐

(

´ −𝒗′𝒘 ′ )

𝟑 𝒆𝒛

The following two quantjtjes dictates all the fmuxes in Energy Box !!!

21/24

Con

• nclusions

ns

• Constant Power Input (CPI) is benefjcial to analyze global energy

budgets of turbulent fmows w/wo control

• There exists no unique relatjonship between the changes in direc

t/turbulent dissipatjon and DR efgect

• DR efgect under CPI is determined only by : the integral of weight

ed RSS

• Triple decompositjon of the velocity fjeld reveals that global ener

gy fmuxes are expressed by two quantjtjes: and .

• Total dissipatjon:
• “Wind” dissipatjon can be considered to be a loss from energetjc

viewpoint.

• A target quantjty to be minimized
• 𝜷=∫

𝟏 𝟐

(𝟐−𝒛 ) (

´ −𝒗

′ 𝒘 ′ ) 𝒆𝒛

𝜸=∫

𝟏 𝟐

(

´ −𝒗′𝒘 ′ )

𝟑 𝒆𝒛

23/24

Th Thank you for yo your kin ind atten tten tio tion Questions ?