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. Hasegawa 1,2 , B. Frohnapfel 1 & M. Quadrio 3 � 1 Center of Smart Interfaces, TU Darmstadt, Germany � 2 Dept. Mech. Eng., The University of Tokyo, Japan � 3 Dept. Aerospace. Eng., Polytechnic Institute of Milan, Italy �

Skin Friction Drag Reduction Technology � � � Key Aspects of Practical Fluid Transport Systems � � � Convenience � - fl ow 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 fl ow rate (increase of pumping power) �

Internal Flow � Pumping Energy E p � Flow rate U b � Duct properties: Volume: V � - Cross sectional area : A � - Wetted perimter: C � Mass: M = � V � - Hydraulic diameter: D = 4 A/C � � � Fluid travel time per unit length: 1/ U b Friction coefficient � � � Pumping energy per unit wetted area: � w C f = 1 2 C f MU b 2 E p = � w V 2 � U b = A 2 A

Energy Saving vs Convenience � 2 C f MU b E p = � w V Turbulent (uncontrolled) � = A 2 A 7/4 ( ) E p � U b (Pumping Energy) � CPI line (Constant Power Input) � B � N � CPG line � � 1 C f � U b : laminar � 1/4 C f � U b : turbulent � A � laminar (uncontrolled) � E p � U b CFR line � (Inconvenience: time) � U b � 1

Active Control of Internal Flow � Pumping Energy E p � Flow rate U b � Energy Consumption Volume: V � Mass: M = � V � for Control: E c � 1/ U b � � Fluid travel time per unit length: � � Total energy consumption per unit wetted area: E t = E p + E c Control energy � Pumping energy �

Energy Saving vs Convenience � Turbulent (uncontrolled) � E t = E p + E c 7/4 ( ) E p � U b B’ � (Total Energy) � CPI line E c (Constant Total N � CPG line � Power Input) � B � A’ � E c No flow states below the laminar curve A � Bewley 2009, Fukagata et al., 2009 laminar (uncontrolled) � E p � U b CFR line � (Inconvenience: time) � U b � 1

Example � 2 + 1/ U b 2 ( ) Cost function: � J = E t E t Optimal in uncontrolled fl ow � Isoline of J � (Total Energy) � CPI line (Constant Total Power Input) � Optimal � Turbulent (uncontrolled) � 7/4 ( ) E p � U b laminar E p � U b (uncontrolled) � (Inconvenience: time) � U b � 1

Non-dimensionalization � � � Convenience (Fluid travel time per unit length) � � � 1 � � � � � 1 T c = 1/ U b U b D = Re b � = � � � � � U b D � � � � Energy Expenditure � � � Pumping Energy � 2 C f MU b 2 AD 2 � � � � 2 A 2 = E p E p = C f Re b C f = E p � � � � 2 A M � 2 2 MU b � � � � � � Total Energy (Pumping + Control) � Effective wall friction � 2 AD 2 � � P p + P c e Re b = � w + P c 2 = E t e = C f � w � � M � 2 � � U b U b

Conventional C f - Re b Plot � � 1/4 turbulence � C f � Re m laminar � � � The value of C f does not represent energy consumption, e.g., � 1 C f � Re m C f decreases with increasing Re � � Comparison of C f at different Re does not make sense

New Plots � e Re 2 � Re � 1 plot � C f Re 2 � Re � 1 plot � C f turbulence � � 1/4 C f � Re m laminar � � 1 C f � Re m

Application to External Flow � � � Convenience (traveling time per unit distance) � � 1 ( ) ( ) = Re l � 1 U � � / U � l � � Propulsion energy per unit fl uid-contacting area and unit distance � 2 = E p / �� 2 E p = 1 � � 2 C f 2 � U � C f Re l � � 2 l 2 � � C f Re 2 - R e -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 speci fi c 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 C f Re 2 which represent the fl ow rate (convenience) and the energy expenditure required to achieve that fl ow 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 fl ows. �