The stagnation point Karman constant

Transcription

1 . p.1/25 The stagnation point Karman constant V. Dallas & J.C. Vassilicos Imperial College London Using a code developed at the Université de Poitiers by S. Laizet & E. Lamballais

2 . p.2/25 DNS of turbulent channel flow Re τ u τ δ ν 11 to 4: too small...but never mind... 6th order compact finite difference scheme; fractional step method for incompressible N.S. using 3-stage 3d order R-K scheme; Poisson pressure equation solved in Fourier space (with staggered grid and FFT on non-uniform grid).

3 . p.3/25 DNS of turbulent channel flow Periodic boundary conditions except at the walls, where (i) either u = ; or the boundary conditions and near wall forcings are borrowed from numerical studies of flow control schemes aimed at drag reduction, i.e. (ii) either u = with forcing f = ( A sin(2πy/λ)h(λ y),, ) added to N.S. (Xu, Dong, Maxey & Karniadakis JFM 27) with A =.16U 2 c /δ 2u 2 τ /δ ν and Λ = 11δ ν ; (iii) or u(x, t) = (, a cos(α(x ct)), ) (Min, Kang, Meyer & Kim JFM 26) with a/u c =.5, α/δ =.5 and c = 2U c ; (iv) or u(x, z, t) = (, v(x, y d, z, t), ) where y d = 1δ ν (Choi, Moin & Kim JFM 1994). Notation: δ ν ν/u τ and U c is the mean centre-line velocity. We set the bulk velocity U b equal to 2/3 by varying the mean pressure gradient accordingly at all times in all simulations.

4 DNS of turbulent channel flow (a) Xu et al (b) Min et al ν d dy U < uv >= u2 τ (1 y/δ) for all y in all cases except with Xu et al forcing where it holds for y > Λ and 2δ y > Λ.. p.4/25

5 When Re τ = δ/δ ν 1 one might expect an intermediate region δ ν y δ where production balances dissipation locally (Townsend 1961), i.e. < uv > dy d U ɛ. ν dy d U < uv >= u2 τ (1 y/δ) implies < uv > u 2 τ in this intermediate region as δ/y and δ ν /y 1/. d (Assuming that, in this limit, d ln U + does not increase faster than y+ p with p 1.) It then follows that, in this equilibrium intermediate region, ɛ u3 τ κy IFF d dy U u τ κy. p.5/25

6 . p.6/25 P < uv > d dy U balances ɛ P/ε = B Case A Case A1 Case A2 Case A3 Case B Case C

7 . p.7/25 Mean flow profiles U Case A Case A1 Case A2 Case A3 Case B Case C U + = U + = 1/.33*ln U + = 1/.34*ln +. U + = 1/.39*ln U + = 1/.41*ln

8 . p.8/25 Karman constant 1/κ = y y u Case A Case A1 Case A2 Case A3 Case B Case C 1/κ = 1/

9 . p.9/25 B = U + ln κ B = u + ln /κ Case A Case A1 Case A2 Case A3 Case B Case C

10 . p.1/25 oncentrate on ɛ u3 τ κy rather than d dy U u κ Generalised Rice theorem for high Reynolds number HIT (Mazellier & V 28 PoF 2, 1412; Goto & V 28 PoF submitted): λ l s where λ is the Taylor microscale (ɛ = 15νu 2 /λ 2 ) and l s is the average distance between neighboring stagnation points defined as the 1/3 power of the number density of stagnation points. Stagnation points are points where the turbulent fluctuation velocity is. Proved in HIT under assumptions of (i) statistical independence between large and small scales and (ii) absence of small-scale intermittency effects.

11 . p.11/25 λ l s in TCF? Does λ l s hold in the intermediate equilibrium region of Turbulent Channel Flow (TCF) in the sense that λ = B 1 l s where B 1 is independent of y and Reynolds number for Reynolds number >> 1?

12 . p.12/25 Stagnation points of fluctuating velocities Points where u u Ue x = z x

13 Number of stagnation points N s total number of points where all components of the velocity fluctuations around the local mean are zero in a thin slab of dimensions L x L z δ y (δ y small, δ y δ ν ) parallel to the channel s y = wall. Observation : N s = N s ( ) y N s z +. p.13/25

14 B 1 λ/l s Calculate λ from ɛ = 2ν < s ij s ij >= ν 3 2E λ 2 E 1 2 < u 2 > and calculate l s from l s Lx L y N s. where B 1 = λ(n s /L x L z ) 1/ Case A Case A1 Case A2 Case A3 Case B Case C p.14/25

15 Concentrate on dissipation ɛ = ν 3 2E λ = ν 2 3 2E = ν 2E B1 2l2 s 3 B1 2L xl z N s = ν 3 2E δ B1 2 ν n s where the number density n s N s /(L x L z δ ν ). Combine with < uv > dy d U = B 2ɛ and dy d U = u τ κy as well as C 3<uv> 2E (the classical approach claims κ and C to be constants (κ.4, C 2) as Re τ ) It then follows that n s = C s δ 3 ν y 1 + where C s = B2 1 κb 2 C. p.15/25

16 . p.16/25 n s = C s δ 3 ν y 1 + with C s about constant even though κ and C are not constant. ( N s δ ν 2 / (L x L z ) = C s )* Case A Case A1 Case A2 Case A3 Case B Case C

17 . p.17/25 κ and C 2E 3<uv> 1/κ = y y u Case A Case A1 Case A2 Case A3 Case B Case C 1/κ = 1/ C Case A Case A1 Case A2 Case A3 Case B Case C

18 . p.18/25 Meaning of B 1 λ/l s constant The eddy turnover time τ is given by ɛ = E/τ. Hence, from ɛ = 2ν 3 E λ 2, 3λ 2 = 2ντ. Therefore, B 1 constant means l 2 s/ν τ: this means that in the equilibrium layer, the time it takes for viscous diffusion to spread over neighboring stagnation points is the same proportion of the eddy turnover time (i.e. the time it takes to cascade the energy to the smallest scales) at all locations and all Reynolds numbers.

19 Meaning of C s constant n s = C s δν 3 y 1 + implies l2 s = Cs 1 δ ν y From ɛ = ν 3 2E λ 2 and B 1 = λ/l s it then follows that ɛ = 2 3 Eu τ κ y with κ B 2 1/C s instead of ɛ = u3 τ κy Therefore, C s constant means τ y/u τ in the equilibrium layer where the constant of proportionality 3κ /2 is the same at all locations and all Reynolds numbers. Indeed κ is related to the stagnation point constants B 1 and C s and is constant if they are constant. κ IS THE STAGNATION POINT KARMAN CONSTANT. p.19/25

20 Start from B 1 and C s constants B 1 constant and C s constant as Re τ in the equilibrium layer mean that ɛ = 2 3 Eu τ κ y, κ = B 2 1 /C s in that layer and that limit. In the equilibrium layer < uv > dy d U ɛ, in fact < uv > dy d U = B 2ɛ with B 2 1 as Re τ. This means that < uv > dy d U = B Eu τ κ y In the equilibrium layer and in the limit Re τ, < uv > u 2 τ and even B 2 1 (Brouwers, PoF 27): hence d dy U = E + u τ κ y with κ = B 2 1/C s. p.2/25

21 C s, B 2 1, 3 2 P y Eu τ and 1/κ ( N s δ ν 2 / (L x L z ) = C s )* Case A Case A1 Case A2 Case A3 Case B Case C B 1 2 = λ 2 N s /L x L z 1.6 x Case A Case A1 Case A2 Case A3 Case B Case C /κ * Case A Case A1 Case A2 Case A3 Case B Case C 1/κ = y y u Case A Case A1 Case A2 Case A3 Case B Case C 1/κ = 1/ p.21/25

22 . p.22/25 Concluding remarks I 1. The DNS suggest that B 1 = λ/l s and C s = n s δ 3 ν are constants in the limit Re τ and in the region δ ν y < δ. Their asymptotic values may be reached at Re τ as low as a few hundred. 2. This is equivalent to stating that in the limit Re τ and in the region δ ν y < δ, λ/l s is a constant and the eddy turnover time equals 3 2 κ y/u τ with κ = B 2 1 /C s. 3. Either of these equivalent statements implies that in the d u limit Re τ, dy U = E τ + κ y in the equilibrium region δ ν y δ where we may expect production to balance dissipation.

23 . p.23/25 Concluding remarks II 4. According to classical similarity scalings, as Re τ, E u 2 τ independently of y in the equilibrium range δ ν y δ. If this is true, then the log-law will be recovered but with a Karman constant that is proportional to κ which is inversely proportional to C s, the number of stagnation points (number of eddies?) within a volume δν 3 at the upper edge of the buffer layer. Why would anyone expect this number to be universal? If it is not, then the Karman constant might not be universal either. from d dy U = E + u τ κ y

24 Concluding remarks III 5. However, various DNS and experiments, as well as Townsend s (1976) idea of inactive motions, seem to suggest that E does not scale as u 2 τ in the equilibrium region as Re τ. If this is the case, then there is, strictly speaking, no log-law and mean flow data fitted by a log-law may yield non-universal Karman constants as a result of κ = B1 2/C s but also as a result of the small effect that inactive motions have on E, and thereby on dy d U = E + u τ κ y (see Davidson 26 on this last point). Note also, that if E does not scale with u 2 τ, then ɛ Eu τ /y instead of ɛ u 3 τ /y. Assuming independence of d dy U on ν and δ where δ ν y δ neglects the small effect of inactive motions whereas assuming λ l s and τ y/u τ may not.. p.24/25

25 . p.25/25 Concluding remarks IV 6. A few points of caution: (i) B 1 may have its own weak (logarithmic?) dependencies on Re τ and if small-scale intermittency effects are taken into account (see Mazellier & V PoF 2, 1412). These dependencies will cause weak dependencies of κ on Re τ and. (ii) For dy d U = E + u τ κ y to hold, the Reynolds number must be so large that < uv > u 2 τ and P ɛ. If some small cross-stream diffusion of turbulent kinetic energy remains and, for example, P.9ɛ as often seems to be the case, then the value of the measured 1/κ will be 9 % of C s /B1 2.