On the Evaluation of Control Performance in Drag Reducing Flows - - PowerPoint PPT Presentation

64th Annual Meeting of the APS Division of Fluid Dynamics, Baltimore Nov. 20-22, 2011

On the Evaluation of Control Performance in Drag Reducing Flows Money versus Time

• Y. Hasegawa1,2, B. Frohnapfel1 & M. Quadrio3

1Center of Smart Interfaces, TU Darmstadt, Germany

• 2Dept. Mech. Eng., The University of Tokyo, Japan
• 3Dept. Aerospace. Eng., Polytechnic Institute of Milan, Italy

Skin Friction Drag Reduction Technology

Key Aspects of Practical Fluid Transport Systems

Convenience

• flow rate in pipeline
• travel speed of vehicle

Energy Saving

• energy consumption to achieve certain “Convenience”

Evaluation of Control Performance in Fundamental Studies Constant Flow Rate (CFR): wall friction is changed by control Successful Control Reduction of wall friction (reduction of pumping power) Constant Pressure Gradient (CPG): wall friction is kept constant by design

Successful Control

Increase of flow rate (increase of pumping power)

Internal Flow

Pumping Energy Ep Flow rate Ub

Duct properties:

• Cross sectional area : A
• Wetted perimter: C
• Hydraulic diameter: D = 4A/C

Fluid travel time per unit length: Pumping energy per unit wetted area:

Volume: V Mass: M = V

1/U b

E p = wV A = MU b

2C f

2A

C f = w 1 2 U b

2

Friction coefficient

CPI line (Constant Power Input)

Energy Saving vs Convenience (Inconvenience: time) U b

1

(Pumping Energy)

E p = wV A = MU b

2C f

2A

C f U b

1

C f U b

1/4

: laminar : turbulent

E p U b

( )

7/4 Turbulent (uncontrolled)

E p U b

laminar (uncontrolled)

N A

CFR line

B

CPG line

Active Control of Internal Flow

Pumping Energy Ep Flow rate Ub Fluid travel time per unit length: Total energy consumption per unit wetted area:

Volume: V Mass: M = V

1/U b

Energy Consumption for Control: Ec

Control energy

Et = E p + Ec

Pumping energy

No flow states below the laminar curve Bewley 2009, Fukagata et al., 2009

CPI line (Constant Total Power Input)

Energy Saving vs Convenience (Inconvenience: time) U b

1

(Total Energy)

Et = E p + Ec

E p U b

( )

7/4 Turbulent (uncontrolled)

N A

CFR line

B

CPG line

Ec

B’ A’

Ec

E p U b

laminar (uncontrolled)

Example (Inconvenience: time) U b

1

(Total Energy)

E p U b

( )

7/4 Turbulent (uncontrolled)

E p U b

laminar (uncontrolled)

Et

Cost function: J = Et

2 + 1/U b

( )

2

CPI line (Constant Total Power Input)

Isoline of J Optimal Optimal in uncontrolled flow

Non-dimensionalization

Convenience (Fluid travel time per unit length) Energy Expenditure Pumping Energy

Total Energy (Pumping + Control)

Tc = 1/U b 1 U b

• D
• =
• U bD = Reb

1

E p = MUb

2C f

2A C f = E p 2A MU b

2

• C f Reb

2 = E p

2AD 2 M 2

• Effective wall friction

w

e =

Pp + Pc U b = w + Pc U b C f

e Reb 2 = Et

2AD 2 M 2

Conventional Cf -Reb Plot

turbulenceC f Rem

1/4

laminar

C f Rem

1

The value of Cf does not represent energy consumption, e.g.,

Cf decreases with increasing Re

Comparison of Cf at different Re does not make sense

New Plots

turbulence

C f Rem

1/4

laminar

C f Rem

1

C f Re2 Re1 plot C f

e Re2 Re1 plot

Application to External Flow

Convenience (traveling time per unit distance)

Propulsion energy per unit fluid-contacting area and unit distance

U

( )

1

/ U l

( ) = Rel

1

E p = 1 2 U

2 C f

C f Rel

2 = E p / 2

2l 2

• CfRe2-Re-1 plot can also be used for external flows

Conclusions

In real applications, a compromise between Convenience (Time) and Energy

expenditure (Money) has to be reached so as to accomplish a goal which in general depends on a specific application.

Based on this idea, we propose a new evaluation plane (money-time plane),

which can be viewed as an improved version of the conventional Cf-Re plot.

The new plane consists of two dimensionless parameters Re-1 and CfRe2

which represent the flow rate (convenience) and the energy expenditure required to achieve that flow rate, respectively.

The new evaluation plane is useful to seek the optimal control strategy for

minimizing the application-dependent cost function.

The above considerations can be easily extended to external flows.