Engineering. Spring Department of Fluid Mechanics, Budapest University of Technology and Economics. Large-Eddy Simulation in Mechanical

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1 Outline Department of Fluid Mechanics, Budapest University of Technology and Economics Spring 2011

2 Outline Outline

3 Part I First Lecture

4 Connection between time and ensemble average Ergodicity1 Ergodicity Time (space) and ensemble averages (moments) are equal. I.e. the statistics are independent from the initial condition. As conclusion for statistically stationary and homogeneous flows, the spatial and the temporal statistics agree.

5 Connection between time and ensemble average Ergodicity2 Experimental (DNS) evidence : Comparison of temporal and spatial results: they agree Temporal Spatial Galanti2004

6 Connection between time and ensemble average Quantification Feasible only for statistically steady flow ( t ϕ = 0) Temporal average: Let us take the ensemble average! ˆϕ (T ) def = 1 T ϕ dt (1) T 0 ˆϕ (T ) = 1 T T 0 ϕ dt = ϕ (2) Nice! The ensemble average of the temporal average is the ensemble average.

7 Connection between time and ensemble average Quantification Let us consider the deviation of the estimate! σ 2ˆϕ (T ) = ( 1 T = 1 T 2 T ϕ 0 T 0 ) 2 dt T ϕ (t 1 )dt 1 ϕ (t 2 )dt 2 (3) 0

8 Connection between time and ensemble average Quantification Introduce the temporal correlation function: ρ ϕ (τ) = ϕ (t)ϕ (t + τ) σ 2 ϕ For steady flows t ρ = 0, and the argument t can be dropped. (4)

9 Connection between time and ensemble average Quantification We can reformulate the expression! ( σ ˆϕ (T ) = σ ϕ T T T 1 τ T Let us define the integral time scale: ) ρ ϕ (τ)dτ (5) Θ = if the integral converges. ρ(τ) dτ (6)

10 Connection between time and ensemble average Quantification We end up: ( ) 1/2 Θ σ ˆϕ (T ) σ ϕ (7) T The deviation of our estimate is related to the integral time-scale, to the averaging time, and to the actual deviation of the investigated quantity.oi If the quantity ϕ is a deviation that it depends on its deviation.

11 Connection between time and ensemble average Example

12 Probability of more variable, joint probability ϕ, ψ are statistical variables. The joint probability for independent variables: f ϕψ (ϕ, ψ) = f ϕ (ϕ)f ψ (ψ) (8)

13 Probability of more variable, joint probability Some example form the channel.

14 Conditional probability f ϕ ψ (ϕ ψ) def = f ϕψ(ϕ, ψ) f ψ (ψ) (9)

15 Correlation Functions Definition The dependence of variables can be characterised by correlation as well. The covariance function: R ϕψ (x, y, z, t, δx, δy, δz, τ) = ϕ (x, y, z, t)ψ (x + δx, y + δy, z + δz, t + τ) (10) If ϕ and ψ are different quantities that it is called cross-correlation, if they are the same it is called auto-correlation. E.g.: R ϕϕ (x, y, z, t, 0, 0, 0, τ) (11) is the temporal auto-correlation function of ϕ.

16 Correlation Functions Definition In non-dimensional form we arrive at the correlation function ρ ϕψ (x, y, z, t, δx, δy, δz, τ) = R ϕψ σ ϕ(x,y,z,t) σ ψ(x+δx,y+δy,z+δz,t+τ) (12) Example1: R ui u j (x, y, z, t, 0, 0, 0, 0) is the Reynolds stress tensor. Example2: ρ(x, y, z, t, 0, 0, 0, τ) is used for the definition of time-scale

17 Scales Definition If we take an e i unity vector, we can define a length-scale in the direction of the vector: L (e) + ϕψ (x, y, z, t) = ρ ϕψ (x, y, z, t, e x s, e y s, e z s, 0) For example let us take the unity vector directed to z ds (13) + ϕψ (x, y, z, t) = ρ ϕψ (x, y, z, t, 0, 0, s, 0) ds (14) L (z) Vortex shedding behind a cylinder (DNS Re = 100)

18 Scales Time Similarly to the length scale a time scale can be defined: T ϕψ (x, y, z, t) = + ρ ϕψ (x, y, z, t, 0, 0, 0, τ) dτ (15) Previously it was denoted by Θ, since T was the averaging time.

19 Scales Example for channel The velocities are uncorrelated at domain length (x).

20 Scales Example for channel The velocities are uncorrelated at domain width (z).

21 Many scales of turbulence Density variation visualise the different scales of turbulence in a mixing layer Goal: Try to find some rules about the properties of turbulence at different scales

22 Kinetic energy Kinetic energy: Its Reynolds decomposition: Its Reynolds average E def = 1 2 u iu i (16) E = 1 2 u iu i = 1 2 (u i u i + 2u iu i + u iu i) (17) E = 1 2 (u i u i ) + 1 }{{} 2 (u i u i ) = Ê + k (18) }{{} k Ê The kinetic energy of the mean flow: Ê The kinetic energy of the turbulence: k (Turbulent Kinetic Energy, TKE)

23 Richardson energy cascade Vortex scales High Re flow Typical velocity of the flow U Typical length scale of the flow L Corresponding Reynolds number (Re = UL ν ) is high is made of vortices of different sizes Each class of vortex has: length scale: l velocity scale: u(l) time scale: τ(l) = l/u(l)

24 Richardson energy cascade The big scales Biggest vortices size l 0 L velocity u 0 = u 0 (l 0 ) u = 2/3k U Re = u 0l 0 ν is also high Fragmentation of the big vortices High Re corresponds to low viscous stabilisation Big vortices are unstable Big vortices break up into smaller ones

25 Richardson energy cascade To the small scales Inertial cascade As long as Re(l) is high, inertial forces dominate, the break up continue At small scales Re(l) 1 viscosity start to be important The kinetic energy of the vortices dissipates into heat

26 Richardson energy cascade The poem The poem of Richardson Big whorls have little whorls that feed on their velocity, and little whorls have smaller whorls and so on to viscosity. Lewis Fry Richardson F.R.S.

27 Richardson energy cascade Connection between small and large scales Dissipation equals production Dissipation is denoted by ε Because of the cascade can be characterised by large scale motion kin. energy Dissipation: ε the large scales By formula: ε = u2 0 l 0/u 0 = u3 0 l 0

28 Kolmogorov hypotheses Intro Questions remains: Size of the dissipative vortices? What is the dependence of velocity (u(l)) and time (τ(l)) scale on the length scale (l). Can be answered by the three hypotheses of Kolmogorov.

29 Kolmogorov hypotheses 1st: local isotropy Beside that the large scales are anisotropic, the anisotropy is decreasing when considering smaller scales. At a certain scale isotropy prevails. Local isotropy hypothesis of Kolmogorov At high enough Reynolds number the turbulent motions at small scales are statistically isotropic. Let us define a scale l EI where the isotropic property is justified. If l > l EI the big vortices are anisotropic, and if l < l EI the vortices are isotropic.

30 Kolmogorov hypotheses 1st similarity hyp. Beside that anisotropy disappears at small scales it can be assumed that any dependence on the large scales disappears and the statistics are universal. What is the parameter characterising this range of scales? The cascade is governed by the energy rate T EI coming from the large scales adn the dissipation characterised by the viscosity ν. Since we know: ε T EI, the hypothesis sounds: 1st similarity hypothesis of Kolmogorov In any flow of high enough Reynolds number the small scale motions (l < l EI )) have and universal form, which only depends on ε and ν.

31 Kolmogorov hypotheses Kolmogorov scales Using ε and ν the following scales can be defined: η = (ν 3 /ε) 1/4 (19) u η = (εν) 1/4 (20) τ η = (ν/ε) 1/2 (21) This scales characterise the dissipative vortices. Its Reynolds (ηu η /ν = 1) is unity, i.e the vortices are stabilised at this scale.

32 Kolmogorov hypotheses Kolmogorov scales The ratio of the large and the Kolomogorov scales can be estimated using ε u 3 0 /l 0. η/l 0 Re 3/4 (22) u η /u 0 Re 1/4 (23) τ η /τ 0 Re 1/2 (24)

33 Kolmogorov hypotheses 2nd similarity hyp. 2nd similarity hypothesis of Kolmogorov In any turbulent flow at high enough Reynolds number the motions of scale l independent of ν the statistics are are only dependent on ε, if l is in the l 0 l η range When introducing l DI to write the 1st hyp. as l 0 > l > l DI this scale divide the universal range (l < l EI ). The inertial range is: l EI > l > l DI The dissipation range is: l < l DI I is for Intertial, E for Energy, and D is for Dissipation.

34 Kolmogorov hypotheses Inertial range scaling Using only ε no length, velocity and time-scale can be defined, but the corresponding velocity and time-scale can be determined to a given length scale l. u(l) = (εl) 1/3 = u η (l/η) 1/3 u 0 (l/l 0 ) 1/3 (25) τ(l) = (l 2 /ε) 1/3 = τ η (l/η) 2/3 τ 0 (l/l 0 ) 2/3 (26) See the decreasing trend!

35 Kolmogorov hypotheses The energy spectrum The spatial correlation function and its spectral form: R ij (x l, r m, t) = u i (x l )u j (x l + r m, t) (27) 1 + Φ ij (κ l, t) = (2π) 3 e ıκ l r m R ij (r m, t) dr m (28) The energy spectrum is defined: + 1 E(κ, t) = 2 Φ ii(κ m, t)δ( κ m κ) dκ m (29) Recall the relationship between wave number and length scale: κ = 2π/l (30)

36 Kolmogorov hypotheses The energy spectrum The energy between wave number κ a and κ b : k κa,κ b = κb κ a E(κ) dκ (31) The dissipation between wave number κ a and κ b can be written: ε κa,κ b = κb κ a 2νκ 2 E(κ) dκ (32) Because of 1st hyp. it can be only function of ε and ν. Becasue of 2nd hyp. in the inertial range it can be only function of ε. E(κ) = Cε 2/3 κ 5/3 (33)