Outline: Phenomenology of Wall Bounded Turbulence: Minimal Models First Story: Large Re Neutrally-Stratified (NS) Channel Flow.

Transcription

1 Victor S. L vov: Israeli-Italian March 2007 Meeting Phenomenology of Wall Bounded Turbulence: Minimal Models In collaboration with : Anna Pomyalov, Itamar Procaccia, Oleksii Rudenko (WIS), Said Elgobashi (Irvine UCLA) and Sergej S. Zilitinkevich (Helsinki Univ.) The goal is physically transparent and as simple as possible but still adequate description of main statistical characteristics of wall-bounded turbulence, that accounts for basic physical processes and symmetries in classical examples of wall bounded turbulence: Large Re turbulent plain flow (Channel, Pipe and Couette flows, flow over incline plane in the gravity field (modeling river flows), atmospheric Turbulent Boundary Layer (TBL), etc; Time and spacial developing (TD & SD) TBL, TBL with Stable temperature and heavy-particle Stratifications,etc. 1

2 Outline: Phenomenology of Wall Bounded Turbulence: Minimal Models First Story: Large Re Neutrally-Stratified (NS) Channel Flow. Formulation and Analysis of the NS Minimal Model (NS-MM). DNS, LES & Experiment in comparison with the NS-MM. First Summary: Strength & Limitations of the NS-MM Second Story: TBL with Stable temperature Stratification. Minimal Model for Stably Stratified TBL (SS-MM). (l/l)-dependence of mean velocity & temperature profiles, of. the heat fluxes and profiles of kinetic & temperature energies. Second summary: Fail of the down-gradient approximation,. internal waves and more... Third and last Story: Time-Developing (TD) TBL. TD-MM for TD TBL and its Factorized asymptotical solution. The velocity and energy profiles in TD-TBL and the Energy. Third and last summary: What did we learn? 2

3 Introduction: our goal and approach The goal is physically transparent and s simple description of main statistical characteristics of wall-bounded turbulence, that accounts for basic physical processes and symmetries in Large Re turbulent plain flow (Channel, Pipe and Couette flows, flow over incline plane in the gravity field (modeling river flows), atmospheric Turbulent Boundary Layer (TBL), etc; in Time and spacial developing (TD & SD) TBL, TBL with stable temperature and heavy-particle stratifications,etc. Suggested Minimal Models (MMs) as a rule are versions of algebraic Reynolds stress models, designed to describe in the plain geometry the mean flow and statistics of turbulence on the level of all relevant simultaneous, one-point second-order (cross)-correlation functions of velocity, temperature, particle concentrations, etc. If required (for developing TBL, for the core of the channel flow, etc.) it includes nonlinear spacial fluxes of energy in differential approximations 3

4 First Story: Large Re Neutrally-Stratified (NS) Channel Flow Algebraic MM for Neutral and Stable temperature Stratifications Governing equations for velocity U and potential temperature Θ ( t + U )U = p ρ b g β Θ + ν U, U = 0, (Boussinesq Appr.) ( t + U )Θ = χ Θ, Θ T b {exp[(s S b )/c p ] 1}. (1) S(r, t) current entropy, S b entropy of isentropic atmosphere (BRS). Exact balance equations for 2 nd order correlators (D t t + U ) P ij C ij + R ij ε ij k T ijk = D t τ ij, τ ij u i u j, A i + B i ɛ i j T ij = D t F i, F i u i θ, (2) F Θ ε T = D t E θ, E θ 1 θ 2. 2 P ij τ ik U j x k τ jk U i x k, Energy Production tensor; (3) 4

5 C ij gβ(f i δ j z + F j δ i z ), Tensor of Conversion E K E θ ; A i A SΘ i + A SU i A SΘ Θ i τ ij, x j + A Eθ i, Heat-flux production: (4) A SU i F U i, A Eθ i 2gβ E θ δ i z ; Pressure-rate-of-strain traceless tensor: ( p ui R ij = + u ) j = R RI ij x j x + R IP ij + R IC ij (5) i Poisson s equation for fluctuating pressure: ρ b ( p = i j ui u j ) u i u j + Ui u j + U j u i + gβ z θ ( R RI ij γ RI τ ii 2E ) ( ) K δ ij γ 3 RI τ ij 1 δij, Return-to-Isotropy, ( R IP ij C IP P ij Tr {P ) ij} δ ij, Isotropization-of-Production, 3 ( R IC ij C IC C ij Tr {C ) ij} δ ij, Isotropization-of-Conversion ; 3

6 Pressure-temperature-gradient vector: p θ B i = B RD i + B SU i ρ b x i + B Eθ i, (6) B RD i γ RD F i, Renormalization of Dissipation ɛ i, B SU i (1 C SU )(F )U i, Renormalization of Production A SU i, B Eθ i 2 gβ (C Eθ + 1)E θ δ i z, Renormalization of Production A Eθ i. The dissipation rates of ui u j ε ij 2 ν x k x k θ u ɛ i (ν + i χ) x k x k ( χ a2 2 ( ν a2 l 2 + γ uu [ ) ) (ν + χ) a2 1 l [ νã2 E K δ ij + l τ ij (1 δ ij ), Reynolds-stress; ] ] F i, Heat-flux; (7) ε χ θ 2 l 2 + γ θθ E θ, Temperature-energy ; Nonlinear dissipation frequencies γ s can be estimated as: u u u θ θ u γ uu γ θ θ, l- outer scale, a s - constants; (8) l u u l θ θ

7 T ijk, T ij, T describe spacial transport, which currently will be neglected. K-41 closure for the relaxation frequencies: γ uu = c uu EK l, γ RI = C RI γ uu, γ RI = C RI γ RI, γ θθ = C θθ γ uu, γ RD = C uθ γ uu, E K kinetic energy, l outer scale of turbulence (distance to the wall, c uu, C, C fitting constants NS-MM: Minimal Model for Neutral Stratification Θ, θ 0 has five fitting constants: c uu, C RI, C RI a and ã τ xy (y) + ν 0 S(y) = P(y), [Γ uu + 3 γ RI ]τ xx = γ RI τ xy 2Sτ xy, [Γ uu + 3 γ RI ]τ yy = γ RI τ xy, [Γ uu + 3 γ RI ]τ zz = γ RI τ xy, Γ RI τ xy = Sτ yy. a 2 Here Γ uu γ uu + ν 0 y 2 and Γ ã RI γ 2 RI + ν 0 y 2. These Eqs. were solved analytically with accuracy of few %

8 Reminder: Wall units and characteristic regions of TBL u τ P 0 (t), l τ ν u τ, τ ν P, V + V u τ, K + K u 2 τ, z + z l τ, t + t τ,... Friction Reynolds # : Re λ L + (t), L -half width of the channel. Von-Karman law, red dash straight line ( ), V + (y + ) = 1 κ ln y+ +B,. κ 0.436, B

9 Anisotropy of the outer layer: DNS, LES & Experiment vs MM result: Asymptotic values of the relative kinetic energies τ ii in the log-law region y > 200 taken from DNS by Moser, R. G., Kim, J., and Mansour, N. N.: (1999) with Re λ = 395, 590) LES by Carlo M. Casciola (2004) and experiment by A. Agrawal, L. Djenidi and R.A. Antonia (2004), in a water channel with Re λ = The last column presents the predictions of the minimal model. τ ii,, ii DNS LES Water channel Minimal Model xx ± /2 yy ± /4 zz ± /4 6

10 W Sum MM Root MM DNS y + MM comparison with DNS for the total energy profile, Re λ = 590 7

11 0.8 DNS Root MM W xy y + Root-MM comparison with DNS for the Reynolds-stress, Re λ = 395, 590 8

12 8 DNS, W xx DNS, W yy 6 DNS, W yy Root MM, W xx Root MM, W yy =W zz W ii x z y y + Root-MM comparison with DNS for the diagonal W + ii, Re λ = 590 9

13 30 flatness z x y y + Profiles of the flatness of the u, v and w components of the turbulent velocity fluctuations, DNS data for Re τ = 590. Horizontal dotted line shows Gaussian value for the flatness, equal to three. 10

14 First summary: Strength & Limitations of the NS Minimal Model ns-mm with the given set of five parameters describes five profiles: V (y) with accuracy of 1% almost throughout the channel the Reynolds stress τ xy (y) with accuracy of few percents the kinetic energy 1 2W(y) with reasonable, semi-quantitative accuracy, including the position and width of its peak in the buffer sub-layer The profiles of τ xx (y), τ yy (y) & τ zz (y), including distribution. NS-MM cannot pretend to describe all the aspects of the turbulent statistics: The MM ignores the quasi-two dimensional character of turbulence, coherent structures in the very vicinity of the wall, etc., etc. The NS MM takes into account the essential physics of wall-bounded turbulence almost throughout the channel flow. With a proper generalization, the MM will be useful in studies of more complicated turbulent flows with temperature stratifications, developing TBL, etc. 11

15 Second Story: Temperature stratified TBL Balance Eq. outside of the viscous & buffer layers: (E K kinetic energy, F heat flux, E θ θ 2 /2. τ + xx = E+ K l F + z 2 γ uu, τ + yy = E+ K 2, τ+ zz = E+ K + l F + z 2 γ uu, l l L, (9) γ uu E+ K = l F + z τ+ xz S U S U l, τ + xz 1, S U S+ U l+, 4 C RI γ uu τ+ xz = l F x + τ+ zz S, U F+ z 1, S Θ S+ Θ l+, C θθ γ uu E+ θ = F z + S Θ S, Θ γ uu = c uu E +, (10) K C uθ γ uuf x + = ( τ xzs + + C Θ SU F+ z S ) U S + C Θ SU S. U C uθ γ uu F+ z = ( τ + zz S Θ + 2 C EΘ l E + θ Here z-vertical and y-cross-stream directions; l is the outer scale (distance to the wall for neutral stratification); ). Task: Find five functions of l l/l: S U, S Θ, E+ K, E+ θ and F + x. 12

16 Analysis of the l/l-dependence Strong stratification, l l/l 1: E + K (l ) = l 2/3 c uu ( S Θ (l ) = S U (l ) E C EΘ C θθ ) ( 2 CEΘ C uθ C θθ (, S U (l ) = 2 l 1 2 C EΘ C θθ + C SU ) S + Θ ) S + U = 2 L (...) = 2 (...)(...), (11) L θ (l ) = S Θ (l ) l 2/3, F + l 1/3 x (l ) = 2 C EΘ S U l 2/3. c uu C θθ l 1/3 c uu C θθ Solution for all l interpolation formulai: E + K (l ) 3/2 E K (l ) 3/2 + E + K (0)2 E K (l ) + E + K (0), (12) S Θ (l ) S Θ (l ) + S Θ (0) + 6(c uuα) 4/3 β l ( 1 + α l ) 4/3, (13) where α and β are known combinations of constants C s. 13

17 Mean velocity shear S + U and S U S+ U l+ (blue) Left panel and Mean temperature shear S + Θ and S Θ S+ Θ l+ (red) Right panel vs. l/l. Black dashed lines: represent analytical approximate solutions Region l L may not be realized in the Nature. If so, it has eristic, methodological meaning. 14

18 Mean velocity U + (blue) and temperature Θ + (red) vs z + for L + = & : Log-linear appr : Improved approx. - : Numer. Integr. 1 l = 1 z L 2 15

19 Analytical approximate solutions for Turbulent kinetic energy E + K (Left).and normalized by E K partial kinetic energies τ ii Anisotropy (Right) Black dashed lines: represent exact numerical solutions Region l L may not be realized in the Nature. If so, it has eristic, methodological meaning. 16

20 Analytical approximate solutions for Turbulent potential energy E + Θ (Left) and streamwise component of the heat flux F + x (Right) Black dashed lines: represent exact numerical solutions Region l L may not be realized in the Nature. If so, it has eristic, methodological meaning. 17

21 Fail of down-gradient approximations for the momentum & heat fluxes Define: τ xz = ν T du x /dz, ν T C ν l z τzz, (14) F z = χ T dθ/dz, χ T C χ l z τzz, (15) Conclusion: C ν & C χ cannot be considered as constants in any reasonable approximation 18

22 Second summary: Fail of the down-gradient approximation, internal waves and more... Presented approach takes into full account all the second-order statistics conserving the total mechanical energy. It results in the analytic solution of the profiles of all mean quantities and all second-order correlations as a function of l/l, including (l/l)-dependence of effective turbulent diffusion coefficients C ν and C χ. This removes the unphysical predictions of previous theories, based on the assumption C... const, and thus violating the conservation of energy. Role of internal gravity waves in the upper wave-dominated TBL with large Ri grad 19

23 Third Story: Time-Developing (TD) TBL Minimal model for TD TBL NSE νs(z, t) + τ(z, t) = P(z, t), P(z, t) P 0 (t) + t z 0 dz V (z, t). P(z, t) is z, t-dependent momentum flux toward the wall. Integration constant P 0 (t) is the wall shear stress the momentum flux at the wall. Exact Balance of the Kinetic energy: NSE K t + E + T z = τs, Our standard Closure for the rate of energy dissipation: [ ( ) a 2 b ] K E ν s ij s ij ν + K b K z z z New Turbulent diffusion approximation for the spacial energy flux: T (z) D(z) z K(z, t), D(z, t) = d z K(z, t).. New Closure for the Reynolds stress: τ(z, t) K(z, t) c2 (z) c 2 c2. 20

24 Self-consistent factorized asymptotical solution of the MM Suggested asymptotical factorization K(z, t) = P 0(t) c 2 k(x), x z S(z, t) V (z, t) z = P 0 (t) κ Z(t) s(x), Z(t), P(x, t) = P 0(t) k(x), (Fac) V (z, t) = P 0 [ B + 1 κ Factorization of the mechanical and energy balance x=z/z 1/Re s(x )dx ], With the suggested factorization (Fac) partial differential balance Eqs. for S(z, t) & K(z, t) turn to the ordinary differential Eqs. for k(x) & s(x) with explicit form of P 0 (t) & Z(t): d k(x) d x = α κ x s(x), (MecBal); P 0(t) = b k(x)s(x) = α c x dk(x) dx + b[k(x)]3/2 x [ κv ln(t/τ) d d dx x k(x) dk(x) dx ] 2, Z(t) = α t. (EnBal) P 0 (t), 21

25 Approximate Asymptotical Analytical solution for TD-TBL k(x) = (1 x) 2 (1 + β 1 x + β 2 x 2 + β 3 x 3 + β 4 x 4 ) 2, (Guess) [ ] 1 1 α = κ xs(x)dx, β 1 = 1 β 0 0 2, β 2 = b + β 0[(β 0 2(2 + β 3 + β 4 )] d, 2β 0 d β 3 = 3 β 0 2 2β 4 5d, β 4 = b3 + b 2 β0 2d + 104bβ (β 0 8)β0 3d3 4β 0 48β0 3, d3 α 0.431, β , β , β , β k(x), Eq.(Guess), blue line x s(x), (EnBal) with k(x), red dashed line, k(x), from (MecBal) with x s(x), black dots small x asymptote k(x) = 1 x, dash- dotted line

26 Discussion of the predictions. DNS for a channel flow (Right) Mean velocity profile with wake V (Left) Similarity for the developing TBL and channel flow: P(x) 1 x in both cases 23

27 Energy balance: Production term, blue solid lines, Dissipation, black dots ( ), turbulent diffusion, dashed line ( ), temporal term K/ t green dash-dotted lines, ( ), the energy input (difference between the energy production and energy dissipation) black solid line

28 Third Summary: What do we learn For the temporal developing TBL we have analytical predictions for the shear and the mean velocity profile with wake contribution; for the profile of the Reynolds stress and kinetic energy, which in the first half of TBL is close to that in the channel flow; time dependence of the friction at the wall, vτ 2 P 0 (t) [κv /ln(t/t)] 2 ; time dependence of the turbulent front velocity dz(t)/dt = κ 3/2 / V ln(t/t). For the spacial developing TBL one should replace t with x/v where x is the distance from the front edge. T HE EN D 26