DECAYING TURBULENCE: THEORY AND EXPERIMENTS

Transcription

1 DECAYING TURBULENCE: THEORY AND EXPERIMENTS N.Mazellier, P. Valente & J.C. Vassilicos Department of Aeronautics and Imperial College London, U.K. p. 1

2 Richardson-Kolmogorov cascade Kolmogorov (1941): E(k) C K ǫ 2/3 k 5/3 for 1/L k 1/η assuming ǫ, the K.E. dissipation rate per unit mass, to be (G.I. Taylor 1935) ǫ = C ǫ u 3 /L with C K and C ǫ indep of Re λ for Re λ 1 Note: L/η Re 3/4, i.e. L/λ Re λ Richardson-Kolmogorov cascade p. 2

3 LES modelling Eddy viscosities in LES from Richardson-Kolmogorov cascade ν t = 2 3 C 3/2 K ǫ1/3 kc 4/3 where ǫ = C ǫ u 3 /L with C ǫ indep of Re λ ν t = 2 3 C 3/2 K C1/3 ǫ u L(k c L) 4/3 with universal values of C K and C ǫ. p. 3

4 Turbulent jets p. 4

5 Turbulent wakes p. 5

6 Homogeneous turbulence (from Ishihara et al, early/mid 2s, Japan, Earth Simulator calculations) p. 6

7 Some background WT HT Axisym Jet Plane Jet u U ( x x L b ) p, 1/2 < p < 3/4 U ( x x L b ) 1 U ( x x L b ) 1/2 L u L b ( x x L b ) q, < q < 1/2 x x x x Axisym Wake Plane Wake Mixing Layer u U ( x x L b ) 2/3 U ( x x L b ) 1/2 U L u L 2/3 b (x x ) 1/3 L 1/2 b (x x ) 1/2 (x x ) L b : characteristic cross-stream length-scale of inlet (e.g. mesh or nozzle or bluff body size...) U : characteristic inlet mean flow velocity or mean flow velocity cross-stream variation p. 7

8 Some background WT HT Axisym Jet Plane Jet L u /λ Re 1/2 ( x x L b ) p Re 1/2 ( x x L b ) Re 1/2 ( x x L b ) 1/4 Re λ Re 1/2 ( x x L b ) p Re 1/2 ( x x L b ) Re 1/2 ( x x L b ) 1/4 Axisym Wake Plane Wake Mixing Layer L u /λ Re 1/2 ( x x L b ) 1/6 Re 1/2 ( x x L b ) Re 1/2 ( x x L b ) 1/2 Re λ Re 1/2 ( x x L b ) 1/6 Re 1/2 ( x x L b ) Re 1/2 ( x x L b ) 1/2 L u /λ Re λ in all cases λ obtained from ǫ u 3 /L u νu 2 /λ 2 Re U L b ν : inlet Reynolds number p. 8

9 Decaying H(I)T Sedov (1944, 1982) and George (1992) found exact single-scale (self-preserving) solutions of Lin s equation E(k,t) t = T(k,t) 2νk 2 E(k,t). E.G. It admits exact solutions of the form E(k,t) = u 2 (t)l(t)e(kl(t),i.c./b.c.) and T(k,t) = d dt [u 2 (t)l(t)]τ(kl(t),i.c./b.c.). (See George 1992; George & Wang 29.) These solutions are such that L l(t) and λ l(t); hence L/λ remains constant during decay even though Re λ can decay fast. This implies L/λ Re λ to be contrasted with the Richardson-Kolmogorov cascade s L/λ Re λ. p. 9

10 Decay of two-scale cascading H(I)T The decay of homogeneous isotropic cascading turbulence is traditionally considered to obey the following constraints: d 3 1. dt 2 u 2 = ǫ where ǫ u 3 /L equivalent to L/λ Re λ. 2. Invariants of the von Kárman-Howarth equation (physical space equivalent of the Lin equation): (i) Either the Loitsyansky invariant u 2 + r 4 f(r)dr is non-zero and Const in time; (ii) Or the Birkhoff-Saffman invariant u 2 [ ] + 3r 2 f(r) + r 3 f(r) r dr is non-zero and Const in time. 3. u 2 f(r,t) < u(x,t)u(x + r,t) > is self-similar at large enough r, i.e. f(r,t) f[r/l(t)] if r not too small. p. 1

11 Invariants of von Kárman-Howarth t (u 2 f) = u 3 ( k r + 4k r ) + 2νu 2 ( 2 f r r where u 3 k(r,t) < u 2 (x,t)u(x + r,t) >, f r ) I Mnn u 2 + r M+n n f(r) r n dr+c Mnn u 2 + r M+n n f(r) r n dr are all invariants of the von Kárman-Howarth equation provided that M > 1, lim r (r M k) = and lim r (r M 1 f) = and that I Mnn is well-defined. M = 4 is the Loitsyansky and M = 2 is the Birkhoff-Saffman inv. The von Kárman-Howarth equation admits an infinity of possible finite integral invariants depending on conditions at infinity. p. 11

12 Consequences of these invariants When M > 1 and M 4, assuming (i) f(r,t) a M+1 (t)(l(t)/r) M+1 to leading order when r and (ii) lim r (r M k) =, then I Mn is finite for all n 1 and its invariance leads to d dt (a M+1L M+1 u 2 ) =. This proves a precise version of the principle of permanence of large eddies. p. 12

13 None or one or two invariants For conditions at infinity such that the Birkhoff-Saffman invariant is not infinite, either none or only one or only two invariants are finite. Assuming that there exists a number M f 2 for which lim r (r M f+1 f) = a Mf +1L M f+1 and a number M g for which lim r (r M k) = for any M in the interval 2 M < M g but lim r (r M k) for any M M g, then we have the following four possibilities. (1) M f /M g > 1, min(m f,m g ) < 4: no finite invariants (2) M f /M g > 1, min(m f,m g ) 4: Loitsyansky invariant (3) M f /M g 1, min(m f,m g ) < 4: d dt (a Mf +1L M f+1 u 2 ) = (4) M f /M g 1, min(m f,m g ) > 4: d dt (a Mf +1L M f+1 u 2 ) = and Loitsyansky. If min(m f,m g ) = 4 only Loitsyansky. p. 13

14 None or one or two invariants p. 14

15 Implications for self-preserving decay George (1992) exact single-scale solutions of the von Kárman-Howarth equation: f(r,t) = f[r/l(t)] and k(r,t) = b(ν,u,l,t t )κ[r/l(t)] Solvability conditions (α >, c > ): [ 2α/c u 2 (t) = u cν (t t l 2 )] and l 2 (t) = l 2 + cν(t t ) (1) M f /M g > 1, min(m f,m g ) < 4: any α and c. (2) M f /M g > 1, min(m f,m g ) 4: 2α/c = 5/2 (3) M f /M g 1, min(m f,m g ) < 4: 2α/c = (M f + 1)/2 and lies between 3/2 and 5/2 as 2 M f < 4 (4) M f /M g 1, min(m f,m g ) > 4: self-preserving George solutions impossible. If M f = 4 then 2α/c = 5/2 p. 15

16 Implications for cascading decay Decay of homogeneous isotropic cascading turbulence: (i) d dt 3 2 u 2 = ǫ where ǫ u 3 /L equivalent to L/λ Re λ. (ii) f(r,t) f[r/l(t)] if r not too small. (iii) Implications of von Kárman-Howarth invariants: (1) M f /M g > 1, min(m f,m g ) < 4: open. (2) M f /M g > 1, min(m f,m g ) 4: u 2 (t) = u 2 [1 + c(t t )] 1/7 and L(t) = L 2 [1 + c(t t )] 2/7 (3) M f /M g 1, min(m f,m g ) < 4: u 2 (t) = u 2 [1 + c(t t )] n where n = 2(M f + 1)/(M f + 3) lies between 6/5 and 1/7 as 2 M f < 4. L(t) = L [1 + c(t t )] 2/(M f+3). (4) M f /M g 1, min(m f,m g ) > 4: large-scale self-similarity impossible. If M f = 4 then (2). p. 16

17 Summary Decay of two-scale (inner and outer) homogeneous isotropic cascading turbulence: u 2 (t) = u 2 [1 + c(t t )] n where n lies between 6/5 = 1.2 and 1/7 = Decay of self-preserving single-scale homogeneous isotropic turbulence: ] 2α/c [ u 2 (t) = u cν (t t l 2 ) where 2α/c lies between 3/2 = 1.5 and 5/2 = 2.5. p. 17

18 Summary There exist asymptotic behaviours at infinity of the double and triple velocity correlation functions which are a priori possible and for which no finite invariant of the von Kárman-Howarth equation exists. In this case, it is unknown what sets the exponents n and 2α/c. There exist asymptotic behaviours at infinity of the double and triple velocity correlation functions which are a priori possible and for which no self-preserving and no large-scale self-similar decays of homogeneous isotropic turbulence are possible. p. 18

19 Wind tunnels.91 2 m 2 width; test section 4.8m; max speed 45m/s; background turbulence.25% m 2 width; test section 4.m; max speed 33m/s; background turbulence.4%. p. 19

20 D f = 2, σ = 25% fractal square grids and equal M eff 2.6cm, L max 24cm, L min 3cm, N = 4, T =.46m. BUT t r = 2.5, 5., 8.5, 13., 17. p. 2

21 omparison with regular grid turbulence ln( (λ / T) 2 ) ln( (x x ) / M ) eff R λ (x x ) / M eff ln ( <u 2 > / U 2 ) ln( (x x ) / M ) eff ln ( <u 2 > / U 2 ) (x x ) / M eff p. 21

22 Streamwise turbulence intensity 12 1 SFG8 SFG13 SFG17 BFG17 u C (x) / U C (x) [%] x / M eff (b) p. 22

23 Wake-interaction length-scale (u c (x) / U c (x)) / (u c (x peak ) / U c (x peak )) SFG8 SFG13 SFG17 BFG17 (b) x / x * (u /U)/(u /U) peak versus x/x where L = t x x peak.5x u /U exp( Bx/x ) where x > x peak ; B 2.6. In agreement with Seoud & V 19, 1518 (27). p. 23

24 Homogeneity where x > x peak (b) x / x * U c /U d and u c/u d versus x/x p. 24

25 From inhomogeneity to homogeneity 1 9 (a) 1 9 (b) 8 8 (y 2 + z 2 ) 1/2 / M eff (y 2 + z 2 ) 1/2 / M eff U(x, y=z) / U c (x) u (x, y=z) / U(x, y=z) [%] Mean flow and turbulence intensity profiles at x.2x (x 7M eff ) and x.15x (x 53M eff ). p. 25

26 From inhomogeneity to homogeneity.25 centerline corner.2.15 u / U x / x * p. 26

27 Statistical homogeneity at x > x peak U local / U inf U local / U inf v y / cm U = 1.5 m/s, 18 x/cm 37 inf Grid =tr y / cm From Seoud & V PoF, 27. p. 27

28 Statistical homogeneity at x > x peak u rms / U local u rms / U local v y / cm U inf = 16.2 m/s, 18 x / cm 37 Grid = tr v rms / U local v / U v y / cm rms local U = 16.2 m/s, 18 x / cm 37 inf Grid = tr y / cm y / cm From Seoud & V PoF, 27. p. 28

29 From non-isotropy to near-isotropy x peak helps collapse u /v as fct of x u /v t = 2.5 r 1 t r = 5. t = 8.5 r.9 t = 13. r t = 17. r x (m) 3 4 u /v t = 2.5 r 1 t r = 5. t = 8.5 r.9 t = 13. r t = 17. r x/x peak From Hurst & V PoF, 27; T =.46m tunnel with U = 1m/s. p. 29

30 Statistical local isotropy at x > x peak K1 v Re λ U inf = 1.5 m/s and 16.2 m/s Grid = tr17, 28 x / cm 37 cm 1.5 m/s 3 cm 1.5 m/s 6 cm 1.5 m/s cm 16.2 m/s 3 cm 16.2 m/s 6 cm 16.2 m/s 1.1 K Re λ From Seoud & V PoF, 27: K 1 2 < ( u x )2 > / < ( v as function of Re λ at locations (x,y) downstream from t r = 17 fractal grid where x 2x peak and y =, 3, 6cm. Local isotropy implies K = 1. x )2 > p. 3

31 From non-gaussianity to gaussianity u / u x/x *.2 x/x *.7 u / u 1 (a) (b) x [m] Prob(u / u ) u / u (c) Note that S u and F u are close to and 3 respectively at x > x peak. p. 31

32 L u /λ and Re λ 12 1 SFG8 SFG13 SFG17 BFG U = 5.2 m/s U = 1 m/s U = 15 m/s L u / λ 6 R λ (a) (b) x / x * x / x * p. 32

33 L u /λ versus Re λ and Re 2 18 U = 5.2 m/s U = 1 m/s U = 15 m/s L u / λ 12 1 L u / λ 6 4 U = 5.2 m/s U = 1 m/s U = 15 m/s 8 2 (a) R λ (b) x / x * L u /λ indep of Re λ but L u /λ Re 1/2? p. 33

34 Self-preserving single-scale spectra In homogeneous region x > x peak, E(k,t) t = T(k,t) 2νk 2 E(k,t). Admits exact solutions of the form E(k,t) = u 2 (t)l(t)f(kl(t),re, ) and T(k,t) = d dt (u 2 (t)l(t))g(kl(t),re, ). (See George PoF 4, 2192, 1992; George & Wang PoF 21, 2518, 29.) These solutions are all such that L = α(re, )l(t) and λ = β(re, )l(t), hence L/λ remains constant during decay but can nevertheless depend on Re. One particular such solution is such that l is itself constant during decay and u 2 decays exponentially, close to what is observed. Such solutions are such that the ratio of outer (L) to inner (λ) length-scales is constant during decay even though Re λ decays fast. As observed! p. 34

35 Self-preserving energy spectrum E u (k x,x) = u 2 (x)l u f(k x L u,re ) 1 x/x *.53 x/x *.71 x/x *.89 1 x/x *.53 x/x *.71 x/x * (k x λ) 5/3 E u / (u 2 λ) (k x L u ) 5/3 E u / (u 2 L u ) (a) k x λ 1 4 (b) k x L u One fractal square grid and three different x positions p. 35

36 Self-preserving energy spectrum 1 SFG8 SFG13 SFG17 1 SFG8 SFG13 SFG (k x λ) 5/3 E u / (u 2 λ) (k x L u ) 5/3 E u / (u 2 L u ) (a) k x λ 1 3 (b) k x L u One x/x position and three different fractal square grids p. 36

37 Energy spectrum s Re dependence E u (k x,x) = u 2 (x)l u (k x L u ) p for 1 < k x L u < Re 3/ (k x L u ) p E u / (u 2 L u ) U = 5m/s U = 1m/s U = 15m/s k x L u One x/x position, one fractal square grid, three different Re p increases with Re, perhaps towards 5/3 p. 37

38 Longer streamwise fetch From.5 < x/x < 1. to.5 < x/x < RG 1ms 1 RG 15ms 1 RG 2ms 1 SFG 1ms 1 SFG 15ms 1 AG 6.7ms 1 u 2 /U 2 m (x x )/T Decay exponents: from 1.21 to 1.72 for RG and AG (Mydlarski & Warhaft 96); from 2.36 to 2.57 for FSG by p. 38

39 No universality of model constants Consider for the fun of it the k-epsilon model equation for decaying homogeneous isotropic turbulence: d dt ǫ = 2 3 C ǫ 2 ǫ 2 /u 2 where the decay exponent is equal to 1/(C ǫ2 1). Hence, C ǫ2 1.8 (close to standard value) for RG and AG, but C ǫ2 1.4 for FSG. However, the more important point is the suggestion that there may be at least two different classes of decaying homogeneous turbulence: a two-scale cascading type of decaying turbulence and a single-scale self-preserving type of decaying turbulence. This is a more serious hit on universality...but the possibility exists at this stage of considering universality classes... p. 39

40 Two classes of small-scale turbulence? 1 1 (klu) 5 3 F11/u 2 Lu 1 1 (klu) 5 3 F11/u 2 Lu Re λ = 116 Re λ = Re λ = 231 Re λ = kl u kl u p. 4

41 Two classes of small-scale turbulence? A self-preserving/single-scale class where (assuming 5/3) (i) E u (k x ) ( u 3 L u ) 2/3 kx 5/3 for 1 k x L u Re 3/4 (ii) L u /λ Re 1/2 but independent of x in the decay region where Re λ decays fast. Decoupling between L u /λ and Re λ. (iii) Fast turbulence decay, decay exponents around 2.5 And the K41 class where (assuming asymptotic 5/3) (i) E u (k x ) ( u 3 L u ) 2/3 kx 5/3 ǫ 2/3 kx 5/3 for 1 k x L u Re 3/2 λ. (iii) L u /λ Re λ locally at every x in the decay region. (iv) Slow turbulence decay, decay exponents around 1.3. What does this mean for LES modeling? p. 41

42 And two final thoughts Possibilities to passively design/manage bespoke small-scale turbulence for various applications? 2. What if turbulence in various cases in nature and engineering appears as a mixture of such different classes? How do we model it then? This talk in papers JCV, Phys. Lett A 375, 11 (211) N. Mazellier & JCV, PoF 2, 7511 (21) P. Valente & JCV, preprint submitted December 21 p. 42