T HE DRAG - REDUCTION OSCILLATING - WALL PROBLEM : NEW INSIGHT AFTER 20 YEARS Pierre Ricco † , Claudio Ottonelli ⋆ , Yosuke Hasegawa ‡ , Maurizio Quadrio • † The University of Sheffield ⋆ ONERA, Paris ‡ The University of Tokyo • Politecnico di Milano 9th European Fluid Mechanics Conference Universitá di Roma, “Tor Vergata”, 10 September 2012 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 1-26
T URBULENT DRAG REDUCTION A CTIVE OPEN - LOOP TECHNIQUE Energy input into system Pre-determined forcing Channel flow DNS ( Re τ = u τ h /ν = 200) S PANWISE WALL OSCILLATIONS New approach: Turbulent enstrophy Transient evolution C ONSTANT DP / DX τ w is fixed in fully-developed conditions GAIN: U b increases 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 2-26
T URBULENT DRAG REDUCTION A CTIVE OPEN - LOOP TECHNIQUE Energy input into system Pre-determined forcing Channel flow DNS ( Re τ = u τ h /ν = 200) S PANWISE WALL OSCILLATIONS New approach: Turbulent enstrophy Transient evolution C ONSTANT DP / DX τ w is fixed in fully-developed conditions GAIN: U b increases 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 2-26
T URBULENT DRAG REDUCTION A CTIVE OPEN - LOOP TECHNIQUE Energy input into system Pre-determined forcing Channel flow DNS ( Re τ = u τ h /ν = 200) S PANWISE WALL OSCILLATIONS New approach: Turbulent enstrophy Transient evolution C ONSTANT DP / DX τ w is fixed in fully-developed conditions GAIN: U b increases 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 2-26
T URBULENT DRAG REDUCTION A CTIVE OPEN - LOOP TECHNIQUE Energy input into system Pre-determined forcing Channel flow DNS ( Re τ = u τ h /ν = 200) S PANWISE WALL OSCILLATIONS New approach: Turbulent enstrophy Transient evolution C ONSTANT DP / DX τ w is fixed in fully-developed conditions GAIN: U b increases 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 2-26
S PANWISE WALL OSCILLATIONS G EOMETRY Ww = A sin � T t � Lz 2 π Ly y x Lx Mean flow z U 2 b , o − U 2 C f , r − C f , o R = b , r C f , r = U 2 b , o Why does the skin-friction coefficent decrease? C f = τ w / ( 1 / 2 ρ U 2 b ) decreases → study why U b increases 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 3-26
S PANWISE WALL OSCILLATIONS G EOMETRY Ww = A sin � T t � Lz 2 π Ly y x Lx Mean flow z U 2 b , o − U 2 C f , r − C f , o R = b , r C f , r = U 2 b , o Why does the skin-friction coefficent decrease? C f = τ w / ( 1 / 2 ρ U 2 b ) decreases → study why U b increases 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 3-26
S PANWISE WALL OSCILLATIONS G EOMETRY Ww = A sin � T t � Lz 2 π Ly y x Lx Mean flow z U 2 b , o − U 2 C f , r − C f , o R = b , r C f , r = U 2 b , o Why does the skin-friction coefficent decrease? C f = τ w / ( 1 / 2 ρ U 2 b ) decreases → study why U b increases 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 3-26
S PANWISE WALL OSCILLATIONS G EOMETRY Ww = A sin � T t � Lz 2 π Ly y x Lx Mean flow z U 2 b , o − U 2 C f , r − C f , o R = b , r C f , r = U 2 b , o Why does the skin-friction coefficent decrease? C f = τ w / ( 1 / 2 ρ U 2 b ) decreases → study why U b increases 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 3-26
E NERGY BALANCE : A SCHEMATIC D U 9.4 MKE- x TKE + 2.7 + 3.5 15.9 6.5 U b τ w + 0.8 P uv + 1.1 6.5 D T + 13.2 E w + 0.3 P vw MKE- z + 12.9 D W Energy is fed through P x ( → U b τ w ) and wall motion ( → E w ) Energy is dissipated through: Mean-flow viscous effects ( → D U , D W ) Turbulent viscous effects ( → D T ) 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 4-26
E NERGY BALANCE : A SCHEMATIC D U 9.4 MKE- x TKE + 2.7 + 3.5 15.9 6.5 U b τ w + 0.8 P uv + 1.1 6.5 D T + 13.2 E w + 0.3 P vw MKE- z + 12.9 D W Energy is fed through P x ( → U b τ w ) and wall motion ( → E w ) Energy is dissipated through: Mean-flow viscous effects ( → D U , D W ) Turbulent viscous effects ( → D T ) 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 4-26
E NERGY BALANCE : A SCHEMATIC D U 9.4 MKE- x TKE + 2.7 + 3.5 15.9 6.5 U b τ w + 0.8 P uv + 1.1 6.5 D T + 13.2 E w + 0.3 P vw MKE- z + 12.9 D W Energy is fed through P x ( → U b τ w ) and wall motion ( → E w ) Energy is dissipated through: Mean-flow viscous effects ( → D U , D W ) Turbulent viscous effects ( → D T ) 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 4-26
E NERGY BALANCE : A SCHEMATIC D U 9.4 MKE- x TKE + 2.7 + 3.5 15.9 6.5 U b τ w + 0.8 P uv + 1.1 6.5 D T + 13.2 E w + 0.3 P vw MKE- z + 12.9 D W Energy is fed through P x ( → U b τ w ) and wall motion ( → E w ) Energy is dissipated through: Mean-flow viscous effects ( → D U , D W ) Turbulent viscous effects ( → D T ) 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 4-26
K EY QUESTIONS S TILL TO BE ANSWERED Why does TKE decrease? Why does U b increase? D OES W ACT ON TURBULENT DISSIPATION ? Stokes-layer-type flow is generated by the wall oscillation Stokes layer’s direct action on D T = � ω i ω i dV V � Study the transport of turbulent enstrophy � ω i ω i The term enstrophy was coined by G. Nickel and is from Greek στρ o φ ´ η → turn 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 5-26
K EY QUESTIONS S TILL TO BE ANSWERED Why does TKE decrease? Why does U b increase? D OES W ACT ON TURBULENT DISSIPATION ? Stokes-layer-type flow is generated by the wall oscillation Stokes layer’s direct action on D T = � ω i ω i dV V � Study the transport of turbulent enstrophy � ω i ω i The term enstrophy was coined by G. Nickel and is from Greek στρ o φ ´ η → turn 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 5-26
K EY QUESTIONS S TILL TO BE ANSWERED Why does TKE decrease? Why does U b increase? D OES W ACT ON TURBULENT DISSIPATION ? Stokes-layer-type flow is generated by the wall oscillation Stokes layer’s direct action on D T = � ω i ω i dV V � Study the transport of turbulent enstrophy � ω i ω i The term enstrophy was coined by G. Nickel and is from Greek στρ o φ ´ η → turn 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 5-26
K EY QUESTIONS S TILL TO BE ANSWERED Why does TKE decrease? Why does U b increase? D OES W ACT ON TURBULENT DISSIPATION ? Stokes-layer-type flow is generated by the wall oscillation Stokes layer’s direct action on D T = � ω i ω i dV V � Study the transport of turbulent enstrophy � ω i ω i The term enstrophy was coined by G. Nickel and is from Greek στρ o φ ´ η → turn 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 5-26
T URBULENT ENSTROPHY EQUATION ∂ � ∂ � ∂ � ∂ � U W � ∂ u W � ∂ w U ∂ � 1 ω i ω i = � + � ω x ω y ω z ω y + ω j − ω j ∂ y ∂ y ∂ x j ∂ y ∂ x j ∂ y 2 ∂τ � �� � � �� � � �� � � �� � � �� � 1 2 5 3 4 ∂ 2 � ∂ 2 � � � W U � ∂ u i − 1 ∂ v ω x v ω z v ω i ω i − � + � � + ω i ω j ∂ y 2 ∂ y 2 ∂ x j ∂ y 2 � �� � � �� � � �� � � �� � 7 9 8 6 � ∂ 2 � + 1 ω i ω i ∂ω i ∂ω i − . ∂ y 2 ∂ x j ∂ x j 2 � �� � � �� � 10 11 Terms scaled in viscous units Stokes layer influences dynamics of turbulent enstrophy Three terms: which is the dominating one? → Let’s look at the terms of the equation 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 6-26
T URBULENT ENSTROPHY EQUATION ∂ � ∂ � ∂ � ∂ � U W � ∂ u W � ∂ w U ∂ � 1 ω i ω i = � + � ω x ω y ω z ω y + ω j − ω j ∂ y ∂ y ∂ x j ∂ y ∂ x j ∂ y 2 ∂τ � �� � � �� � � �� � � �� � � �� � 1 2 5 3 4 ∂ 2 � ∂ 2 � � � W U � ∂ u i − 1 ∂ v ω x v ω z v ω i ω i − � + � � + ω i ω j ∂ y 2 ∂ y 2 ∂ x j ∂ y 2 � �� � � �� � � �� � � �� � 7 9 8 6 � ∂ 2 � + 1 ω i ω i ∂ω i ∂ω i − . ∂ y 2 ∂ x j ∂ x j 2 � �� � � �� � 10 11 Terms scaled in viscous units Stokes layer influences dynamics of turbulent enstrophy Three terms: which is the dominating one? → Let’s look at the terms of the equation 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 6-26
T URBULENT ENSTROPHY EQUATION ∂ � ∂ � ∂ � ∂ � U W � ∂ u W � ∂ w U ∂ � 1 ω i ω i = � + � ω x ω y ω z ω y + ω j − ω j ∂ y ∂ y ∂ x j ∂ y ∂ x j ∂ y 2 ∂τ � �� � � �� � � �� � � �� � � �� � 1 2 5 3 4 ∂ 2 � ∂ 2 � � � W U � ∂ u i − 1 ∂ v ω x v ω z v ω i ω i − � + � � + ω i ω j ∂ y 2 ∂ y 2 ∂ x j ∂ y 2 � �� � � �� � � �� � � �� � 7 9 8 6 � ∂ 2 � + 1 ω i ω i ∂ω i ∂ω i − . ∂ y 2 ∂ x j ∂ x j 2 � �� � � �� � 10 11 Terms scaled in viscous units Stokes layer influences dynamics of turbulent enstrophy Three terms: which is the dominating one? → Let’s look at the terms of the equation 10 S EPTEMBER 2012 WALL - OSCILLATION DRAG - REDUCTION PROBLEM 6-26
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