Energy transfer rates in turbulent channels with drag reduction at constant power input

Transcription

1 Energ transfer rates in turbulent channels with drag reduction at constant power input Davide Gatti, M. Quadrio, Y. Hasegawa, B. Frohnapfel and A. Cimarelli EDRFCM 2017, Villa Mondragone, Monte Porio Catone KIT The Research Universit in the Helmholt Association

2 The drag reduction experiment turbulent ε + mean Φ kinetic energ dissipation rate 2h x P p pumping power X U bulk velocit: U b pressure gradient: dp dx = τ w h skin-friction coefficient: C f = 2τ w ρu b 2 pumping power (per unit area): P p = dp dx hu b

3 Integral energ budget Renolds decomposition: u x,,, t = u() + u x,,, t 1 mean kinetic energ (MKE) budget: 2 ρ u2 P p = P uv + Φ 1 turbulent kinetic energ (TKE) budget: 2 ρu 2 P uv = ε global energ budget: P p = Φ + ε

4 The drag reduction experiment turbulent ε + mean Φ kinetic energ dissipation rate 2h x P p pumping power X U P t = P p + P c = ε + Φ control power input P c at (statistical) stead state: bulk velocit: U b pressure gradient: dp dx = τ w h skin-friction coefficient: C f = 2τ w ρu b 2 pumping power (per unit area): P p = dp dx hu b drag reduction rate: R = 1 C f C f,

5 How does drag reduction affect energ transfer rates? a (seemingl) trivial question with a non trivial answer Ricco et al., JFM (2012): substantial increase of ε caused b control with spanwise wall motions Frohnapfel et al., (2007): ε needs to be reduced to achieve drag reduction Martinelli, F., (2009): drag reduction obtained via feedback control aimed at minimiing ε

6 Goal We investigate how skin-friction drag reduction affects energ-transfer rates in turbulent channels do different control strategies behave similarl? do universal relationships ε = ε R or Φ = Φ R exist? can we predict changes of ε or Φ? b producing a direct numerical simulation (DNS) database of turbulent channels modified b several drag reduction techniques

7 Comparing energ transfer rates correctl successful control R = 1 C f C f,0 > 0 with control power P c U b dp dx CPG = P p = dp dx hu b P t = P p + P c C f CFR =? P p and P t change between controlled and natural flow!! Hasegawa et al., JFM (2014) propose alternative forcing methods: CPI =

8 The DNS database at CPI Box sie L x, L, L = 4πh, 2h, 2πh Resolution Δx +, Δ +, Δ + = 9.8, , 4.9 Average over viscous time units Viscous + units: u τ = τ w /ρ δ ν = ν/u τ t ν = ν/u τ 2 Constant total Power Input (CPI): Re Π = U Πδ ν = 6500 U Π = P th 3μ P t = P p + P c is kept constant to control power fraction γ = P c P t, so that P t 3 ρu = 3 Π Re Π P p = 1 γ P t = 3 1 γ Re Π

9 Control strategies Spanwise wall oscillations δ x W w = Asin(ωt) drag reduction R = 1 C f Cf,0 = 17.1% control power fraction γ = P c Pt = U b U b,ref =

10 Control strategies Spanwise wall oscillations Opposition control δ x x W w = Asin(ωt) drag reduction R = 1 C f Cf,0 = 17.1% control power fraction γ = P c Pt = U b U b,ref =

11 Control strategies Spanwise wall oscillations Opposition control δ s x x W w = Asin(ωt) v w = v(x, s,, t) drag reduction R = 1 C f Cf,0 = 17.1% control power fraction γ = P c Pt = U b U b,ref = v w s

12 Control strategies Spanwise wall oscillations Opposition control δ s x x W w = Asin(ωt) v w = v(x, s,, t) drag reduction R = 1 C f Cf,0 = 17.1% R = 23.9% control power fraction γ = P c Pt = γ = U b U b,ref = U b U b,ref =

13 The energ box reference flow Re b = 3177 Re τ =

14 The energ box opposition control Re b = 3474 Re τ = U b U b.0 = MKE dissipation rate Φ increases TKE production rate P uv and dissipation rate ε decrease

15 The energ box oscillating wall Re b = 3267 Re τ = U b U b.0 = Both MKE dissipation Φ and TKE production P uv rates decrease, U b increases! TKE dissipation rate ε increases

16 The energ box: lesson Drag reduction reduction of TKE production rate P uv Drag reduction increase of MKE dissipation rate Φ P c surprisingl good alternative to pumping with wall oscillations! B accounting for the phsics of the control and separating the contribution of P c to ε, it is also true that: Drag reduction reduction of TKE dissipation rate ε

17 Predicting ε R for R 0 (1) The dissipation ε in power units is linked to ε + in viscous units b the following: ε = ε + Re τ Re Π 3 Re τ can be substituted with Re b with the following relationship: P p = dp dx hu b this ields, which in nondimensional form reads Re τ 2 Re b = 3 1 γ Re Π 2 ε = ε γ Re b 3/

18 Predicting ε R for R 0 (2) The following relation holds for both controlled and reference flow ε = ε γ Re b 3/2 b taking the ratio in the controlled and reference channel we obtain ε ε 0 = ε+ ε γ Re b.0 Re b 3/2 for a reference channel flow it is known that the ε + is a mild function of Re τ ε + = 2.54 ln Re τ 6.72 Abe & Antonia, JFM (2016) Hpothesis: if R 0 then Re τ Re τ,0, so we assume ε + ε

19 Predicting ε R for R 0 (3) The relation reduces eventuall to: ε ε 0 = 1 γ Re b.0 Re b 3/2 no general statement on ε + without considering the phsics of the control! opposition control wall oscillation

20 Conclusions CPI approach is essential to assess energ transfer rates in dragreduced flows Energ box analsis ields two statements Drag reduction reduction of TKE dissipation rate ε Drag reduction increase of MKE dissipation rate Φ No universal relationship between R and ε could be found without considering the phsics of the control

21 The drag reduction experiment turbulent ε + mean Φ kinetic energ dissipation rate 2h x P p pumping power X U control power input P c bulk velocit: U b pressure gradient: dp dx = τ w h skin-friction coefficient: C f = 2τ w ρu b 2 pumping power (per unit area): P p = dp dx hu b drag reduction rate: R = 1 C f C f,

22 THANKS for our kind attention! for questions, complaints, ideas: