On the (multi)scale nature of fluid turbulence Kolmogorov axiomatics Laurent Chevillard Laboratoire de Physique, ENS Lyon, CNRS, France Laurent Chevillard, Laboratoire de Physique, ENS Lyon, CNRS, France p.1/9
Wind tunnel at Modane 3 2 u(x)/σ u 1 0 1 2 3 10 20 30 x/l Laurent Chevillard, Laboratoire de Physique, ENS Lyon, CNRS, France p.2/9
Axiomatics of Kolmogorov phenomenology Consider (as observed) a homogeneous, isotropic stationary solution of the (forced over L) Navier and Stokes equations: call it u ν (x,t), with x R 3. u ν t +(u ν. )u ν = p ν +ν u ν + large-scale L forcing {}}{ f Homogeneous, isotropic velocity field, of finite variance σ 2 (independent on viscosity ν): lim ν 0 E uν 2 = σ 2 <. Finiteness of average dissipation lim ν 0 εν = lim νe u ν 2 σ3 ν 0 L At infinite Reynolds number, the velocity field is rough, with (consider longitudinal velocity increments) δ l u = u(x+l) u(x) E(δ l u) q l 0 C q l ζ q, with ζ q a universal nonlinear function of the order q. Note γ 2 = ζ 0 the intermittency coefficient, ζ 3 = 1 and C 3 being exact. Laurent Chevillard, Laboratoire de Physique, ENS Lyon, CNRS, France p.3/9
Two-point statistical structure of turbulence Define the energy spectrum (Fourier transform of the correlation) as E(k) = e 2iπkl u(x)u(x+l) dl Kolmogorov energy spectrum 10 2 k 5/3 Air Jet R =380 λ 10 0 E(k) 10 2 10 4 Injection Inertial Dissipative 10 6 10 3 10 2 10 1 10 0 10 1 k/k η Laurent Chevillard, Laboratoire de Physique, ENS Lyon, CNRS, France p.4/9
Fractional Gaussian fields Consider u ǫ (x) = R ϕ L (x y) 1 x y 1 2 H ǫ W(dy), where: dw is a Gaussian white measure 0 < H < 1 being the Hurst (i.e. Holder) exponent (H = 1/3 for turbulence) ǫ a regularizing length-scale (that eventually depends on viscosity) ϕl a large-scale (i.e. L) cut-off function (ensuring finite variance) This field fulfills some aspects of the axiomatics of Kolmogorov: Convergence of the variance (whatever the regularization over ǫ) lim ǫ 0 Eu2 ǫ < +. For a proper parametrization of ǫ with viscosity, reproduces finiteness of dissipation. Asymptotically rough with E(δlu) q l 0 C ql qh Laurent Chevillard, Laboratoire de Physique, ENS Lyon, CNRS, France p.5/9
Intermittency in Eulerian fluctuations Eulerian longitudinal velocity increments: δ l u(x) = u(x+l) u(x) Flatness F = (δ lu) 4 (δ l u) 2 2 (a) ln(p(δ l u)) 10 20 30 (b) log(f/3)/ln(re/r*) 0.2 0.15 0.1 0.05 0 slope 0.1 R λ 140 208 380 463 703 929 2500 15 10 5 0 5 10 15 δu 1.2 0.9 0.6 0.3 0 0.3 ln(l/l)/ln(re/r*) Laurent Chevillard, Laboratoire de Physique, ENS Lyon, CNRS, France p.6/9
Temporal structure of turbulence Consider (as observed) a homogeneous, isotropic stationary solution of the (forced over L) Navier and Stokes equations: call it u ν (x,t), with x R 3. Velocity variance σ 2 is finite and independent on viscosity ν, i.e. lim E( u ν 2 ) = σ 2 < + ν 0 Consider the time evolution of the velocity field uν (x 0,t) at a fixed position x 0. To ensure a bounded velocity variance, the flow will develop small scales: lim E[ u ν (x 0,t+τ) u ν (x 0,t) 2] τ 2/3 ν 0 τ 0 Laurent Chevillard, Laboratoire de Physique, ENS Lyon, CNRS, France p.7/9
Regularized fractional Ornstein-Uhlenbeck processes Consider the following linear stochastic differential equation (α > 0) du ǫ,h (t) = αu ǫ,h (t)dt+dw ǫ,h (t) where dw ǫ,h (t) = ω ǫ,h (t)dt+ǫ H 1 2dW(t) and ω ǫ,h (t) = ( H 1 ) t 2 1 dw(s) (t s+ǫ) 2 3 H uǫ,h is a regularized fractional Ornstein-Uhlenbeck process of Hurst exponent H. 4 cases need to be discussed: (i)1/2 < H < 1, (ii) H = 1/2, (iii) 0 < H < 1/2 and (iv) H = 0. (Recall that H = 1/3 for turbulence) the differential form is especially well suited for numerical simulations Laurent Chevillard, Laboratoire de Physique, ENS Lyon, CNRS, France p.8/9
Regularized fractional Ornstein-Uhlenbeck processes For α > 0 and a given H ]0,1[, the Gaussian process u ǫ,h (t) reaches a stationary regime. It is zero-average and the variance remains bounded when ǫ 0. We note Eu 2 H = lim ǫ 0 lim t E [ (uǫ,h (t) ) 2 ] [ = α 2H Γ ( H + 1 2 2sin(πH) )] 2 <, where enters the Gamma function Γ(z) = 0 xz 1 e x dx defined z > 0. Let us call then δ τ u H the corresponding increment over τ, note its variance as E(δ τ u H ) 2 = lim ǫ 0 lim t E [ (uǫ,h (t+τ) u ǫ,h (t) ) 2 ]. For H ]0,1[, we have the following behavior of the increment variance at small scales E(δ τ u H ) 2 τ 0 1 sin(πh) [ ( )] Γ H + 1 2 2 τ 2H. Γ(2H +1) see: Phys. Rev. E 96, 033111 (2017). Laurent Chevillard, Laboratoire de Physique, ENS Lyon, CNRS, France p.9/9