Direct Numerical Simulation of Turbulent Wall Flows at Constant - PowerPoint PPT Presentation

14th European Turbulence Conference, Sep. 1-4 2013, Lyon, France Direct Numerical Simulation of Turbulent Wall Flows at Constant Power Input Y. Hasegawa1, B. Frohnapfel2 & M. Quadrio3 1Institute of Industrial Science, The University of Tokyo 2Institute of Fluid Mechanics, Karlsruhe Institute of Technology 3Dept. Aerospace. Eng., Polytechnic Institute of Milan

Flow Condition in Numerical Simulation wall friction Example: Channel flow ∆ P = τ w L / δ Pressure drop: Length: L Depth: 2 δ Bulk velocity: U B control, X X obstacles, Pump roughness etc. Conventional approaches  Constant Flow Rate (CFR) : pressure drop ( wall friction ) fluctuates in time Successful Control Reduction of pressure drop  Constant Pressure Gradient (CPG) : The flow rate fluctuates in time Successful Control Increase of flow rate No ! Are they the only available options ?

Money versus Time (Frohnapfel, Hasegawa & Quadrio, JFM 2012) Flow control problem compromise between convenience and energy consumption 7/4 E p ∝ U b Turbulent (uncontrolled) ( ) Pumping Energy CPI line (Constant Power Input) per Unit Mass B N CPG line − 1 C f ∝ U b : laminar C f ∝ U b − 1/4 : turbulent A laminar (uncontrolled) E p ∝ U b CFR line (Inconvenience: time) U b − 1

Practical Problems Unsteady flow in piping system Stenosis of arteries Most flow conditions in real systems should be neither CFR nor CPG !

laminar flow in pipe w/wo orifice Flow rate: Q R Πρεσσυρε γραδιεντ: ∆ p/ ∆ x Power input: Pp R CFR Q=const R CPI Pp=const R CPG ∆ p/ ∆ x=const color code corresponds to pressure gradient

Comparison between Different Flow Conditions Successful control ∝ τ ) Pumping power Ub ∆ P ( w ∝ Δ ) ( Ub P CFR Const. CPG Const.

Comparison between Different Flow Conditions Successful control ∝ τ ) Pumping power Ub ∆ P ( w ∝ Δ ) ( Ub P CFR Const. CPG Const. CPI Const. Advantage of CPI  Close to real operational condition (mechanical pump, heart, ……)  Constant power input = constant dissipation = constant energy transfer rate  Optimal ratio of total power Ptotal and control power input Pc γ = control power input = P P c c = total power input P P p + P total c

Introduction to CPI concept

Problem Setting Control power input Pc Channel flow Depth: 2 δ X X Pumping power Pp Prescribed quantities  Channel half depth δ  Fluid physical properties (kinetic viscosity: ) ν  Total power input: Ptotal = Pp + Pc = const.

Velocity Scale based on Power Input “ The lower-limit of power consumption under CFR is achieved in the Stokes flow ” Bewley (JFM, 2009), Fukagata et al. (Physica D, 2009) The flow rate becomes maximum under CPI in the Stokes flow . Stokes (laminar) flow  Pumping power per unit wetted area ⎛ ⎞ P p = − dp ⋅ U b P p ⎠ δ ⎝ dx  Bulk velocity in the Stokes flow ⎛ ⎞ P p δ U b = 1 3 µ − dp 2 = ⎠ δ ⎝ dx 3 µ  The upper-limit of the bulk mean velocity under CPI P t δ Velocity scale based on the total power consumption U p = 3 µ

Non-dimensionalization All quantities are Channel flow Total power input: Ptotal normalized by  Up = ( Ptotal δ /3 µ )1/2  δ Depth: 2 δ Power-based Reynolds number Re p = U p δ ν ≅ 6500 Navier-Stokes & Continuity Equations: ∂ u i ∂ u i u j 2 u i Re τ ,0 = 200 ( ) = − ∂ p ∂ ∂ u i 1 ( ) ∂ t + + , = 0 ∂ x j ∂ x i ∂ x j ∂ x j ∂ x i Re p 3 Total power input: P total = ( = const . ) Re p Gain in flow rate Evaluation of control performance U b / U p ≤ 1 ( )

Uncontrolled flow under CPI

Relationship between Different Reynolds Numbers in Uncontrolled Flow

Time Trace of Ub & dp/dx Time

Fundamental Flow Statistics Mean Velocity Velocity fluctuation Results in CFR, CPG & CPI converge to the identical flow state in uncontrolled flow if Reb, Re , Rep are adjusted properly. τ

Controlled flow under CPI (Spanwise wall oscillation) y Rep = 6500 ( Re , 0 = 200 ) Pumping τ x power Ppump T+ = 125 z Control power input Pcontrol P total = P pump + P control = const .

Optimal Power Input γ = P c total ~ 0.1 leads to the maximum bulk mean velocity P

Conclusions  Constant power input (CPI) condition is proposed as a flow condition alternative to conventional CFR and CPG  close to real operational condition  power input (= energy transfer rate = dissipation) is kept constant  optimal ratio of total power input and control power input  CPI condition is first implemented in DNS of wall turbulence  Power-based velocity scale: Up  dimensionless total power input: 3/Rep  CPI simulation successfully run for the uncontrolled and controlled flows.  Uncontrolled flow under CPI is essentially same as those under CFR and CPG.  In the controlled flow, the maximum Ub is obtained when is γ around 10%.

Turbulent Intensity in Spanwise Wall Oscillation Control : uncontrolled urms wrms vrms

Turbulent Intensity in Spanwise Wall Oscillation Control : uncontrolled : CFR normalized by reference u τ ,0

Turbulent Intensity in Spanwise Wall Oscillation Control : uncontrolled : CFR normalized by reference u τ ,0 : CFR normalized by actual u τ

Turbulent Intensity in Spanwise Wall Oscillation Control : uncontrolled : CFR normalized by reference u τ ,0 : CFR normalized by actual u τ : CPG Interpretation can be changed depending on flow conditions and normalization !

Fundamental Flow Statistics Mean Velocity Velocity fluctuation controlled uncontrolled

Numerical Implementation Time step: Advance Flow Simulation n → n +1 Advance Flow Simulation n +1 = P n total − P P pump Calculate control Power Input Pc Calculate control Power Input Pc c Calculate bulk mean velocity Calculate bulk mean velocity n +1 n +1 ⎛ ⎞ = P pump − dp Calculate pressure gradient Calculate pressure gradient ⎝ ⎠ dx U b

Energy Box Ricco, Ottonelli, Hasegawa & Quadrio. (JFM, 2012)