チャネル乱流における流体線の伸長

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いとはみしま
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1 69 d(l/l )/dt y + = 15 Re τ = Kolmogorov τ η.1.18 Kolmogorov.65τ η,min 1 Stretching Rate of Material Lines in Turbulent Channel Flow Takahiro TSUKAHARA, Faculty of Science and Technology, Tokyo Univ. of Science Hiroshi KAWAMURA, Faculty of Science and Technology, Tokyo Univ. of Science (Received 27 January, 25; in revised form 22 September, 25) The evolution of passive material lines in turbulent channel flow is studied by the direct numerical simulation. The Reynolds numbers based on the friction velocity, the channel half width and the kinematic viscosity are 18 and 395. The deformation of a line for the higher Re τ is more rapid especially at y + = 15. The line length increases exponentially in time as exp[γt], with the stretching rate γ =.1.18τ 1 η in the range of the present Reynolds numbers. This is reasonable because the lines are deformed efficiently by the Kolmogorov-scale eddies with the Kolmogorov time-scale (τ η ). The obtained value of γ is close to that of homogeneous isotropic turbulence. KEY WORDS : Material Line, Direct Numerical Simulation, Turbulent Channel Flow 1 1) Batchelor 1) e de/dt a759919@rs.noda.tus.ac.jp kawa@rs.noda.tus.ac.jp u i, j de i dt = u i, j e j (1) DNS 2, 3) Kida 4, 5)

61 1 Numerical parameters; number of grid points N i, grid width i, sampling time period T, time step t, threshold length of line segment L, Kolmogorov scale η and time-scale τ η.

2 61 1 Numerical parameters; number of grid points N i, grid width i, sampling time period T, time step t, threshold length of line segment L, Kolmogorov scale η and time-scale τ η. Re τ N x N y N z x +, y + min-max, z+ T + t + L + η +,min τ + η,min , , , , Batchelor Batchelor.13τ 1 η ( τ η Kolmogorov ) 2) 1.17τ η Kida 4, 5) DNS Kim 6) Re τ = 18 DNS 7-1) 11, 12) DNS x 1 Navier-Stokes 1 Configuration u i x i =, (2) u + i t + u + u+ i j x j = p+ x i u + i Re 2 τ x j (3) + u τ ν δ Re τ = u τ δ/ν = N i i + T + Fractional step (y ) 2 Crank-Nicolson 2 Adams-Bashforth (z ) Morinishi 13) x p (t) u(x p (t), t)

dx p (t) dt 2 Particle Boundary treatment Imaginary radius Wall = u(x p (t), t) (4) y min 2 2 L 2 Kolmogorov η t Kolmogorov τ η t τ η Kolmogorov η τ η η = ν 3 4 ε 1 4, τη = ν 1 2 ε 1 2 (5) ε ε = ν u

3 dx p (t) dt 2 Particle Boundary treatment Imaginary radius Wall = u(x p (t), t) (4) y min 2 2 L 2 Kolmogorov η t Kolmogorov τ η t τ η Kolmogorov η τ η η = ν 3 4 ε 1 4, τη = ν 1 2 ε 1 2 (5) ε ε = ν u i x j u i x j (6) 1 Kolmogorov η τ η 2 L t 611 u + Turbulent kinetic energy k u + =(1/.41)ln(y + )+5.6 Present Re τ =18 y + =15 u + =y + Iwamoto et al. 8) Re τ =18 Wei & Willmarth et al. 14) Re τ = y + Mean veloctiy profile Present Reτ =18 Iwamoto et al. 8) Re τ = y + y /δ 4 3 Turbulent kinetic energy 3.1 wall-unit 3 DNS 8) Wei-Willmarth 14) 4 k + = (u + i u + i )/2 8) Re τ = 395 y + = Re τ = L 3.2δ Re τ = 18

612 (a ) 5 Material lines at t=; the lines are

6δ Re τ = 395 y + s = 5y + s = 15 mid-height y s =.

1 6 8 3 6 z + 1 y + s = 15 ejection burst sweep tu

8 15) (c) (d) (b ) (c ) (d ) (e) (e ) 6 Temporal

turbulent channel flow for Re τ =18 (a-e) and 395

The straight lines are released from y + s =5.

4 612 (a ) 5 Material lines at t=; the lines are generated along lines parallel to the z-axis. 1.6δ Re τ = 395 y + s = 5y + s = 15 mid-height y s =.5δ y s = 1.δ Re τ = 18 8 Re τ = z + 1 y + s = 15 ejection burst sweep tu τ /δ =.8 15) (c) (d) (b ) (c ) (d ) (e) (e ) 6 Temporal evolution of passive material lines in the turbulent channel flow for Re τ =18 (a-e) and 395 (a -e ). The straight lines are released from y + s =5. The color of the line indicates a height from the lower wall (blue: bottom, red: center). The mean flow direcion is from bottom-left to top-right. (a,a ) tu τ /δ=.8, (b,b ).16, (c,c ).32, (d,d ).48 and (e,e ).64. 6, 1, 16) y + 2 d + 15 ( 1η + ) y + s = 15 9

613 (c) (c) (d) (d) (e) (e) 7 Temporal evolution of passive

The straight lines are released from y + s =15, y s=.

The mean flow direcion is from left to right. tu τ /δ=.8,.

8 Temporal evolution of passive material lines in the

5 613 (c) (c) (d) (d) (e) (e) 7 Temporal evolution of passive material lines in the turbulent channel flow for Re τ =18. The straight lines are released from y + s =15, y s=.5δ and y s =δ. The color of the line indicates as Fig. 6. The mean flow direcion is from left to right. tu τ /δ=.8,.16, (c).32, (d).48 and (e) Temporal evolution of passive material lines in the turbulent channel flow for. The straight lines are released from y + s =15, y s=.5δ and y s =δ. The color of the line indicates as Fig. 6. The mean flow direcion is from left to right. tu τ /δ=.8,.16, (c).32, (d).48 and (e).64.

614 9 Dual-view (top-front) of material line and finescale vortices in the near-wall region of the turbulent channel flow,. The deformed line is a part of the line shown in Fig.

Iso-surface indicates the second invariant of deformation tensor II + -.4, which corresponds to a vortical position. y + s = 15

6 614 9 Dual-view (top-front) of material line and finescale vortices in the near-wall region of the turbulent channel flow,. The deformed line is a part of the line shown in Fig. 8 (y + s =15, tu τ /δ=.8). The color of the line indicates a stretching rate (blue: not stretched, red: strongly stretched part). Iso-surface indicates the second invariant of deformation tensor II + -.4, which corresponds to a vortical position. y + s = 15 9 z =.1.2 y + s = 5 3 y + s = 5 y + s = 15 1 f (y p ) Deformed material line from y + s =15 in the turbulent channel flow for Re τ =18. The color of the line indicates as Fig. 6. The length of the line is L t 1 4 L at tu τ /δ=1.6. Imaginary radius Re τ =18 (y s + =15) Imaginary radius (y s + =15) y + s =15 tu τ /δ=.8 tu τ /δ= y min tu τ /δ=.16 tu τ /δ=.64 y min tu τ /δ=.16 tu τ /δ=.32 tu τ /δ=1.2 tu τ /δ=.48 tu τ /δ=1.6 tu τ /δ= y 1 2 p y 1 2 p y -1 p /δ y -1 p /δ 1 11 Probability density function of the height of the particles on the material line (y + s =15) in the tubulent channel flow. The height of the lowest grid point y min and the imaginary radius as the wall-boundary condition are also shown for comparison, cf. Table (t = 1.6) y + s = y p 11 y + s = 15 Re τ = 395 tu τ /δ = y + p 3 5 y + p <.3 f (y p ) = 11

615 13 Deformed the material line from y s =δ in the turbulent channel flow for Re τ =18.

2 L at the same time as Fig. 1. 12 Deformed material line from y + s =15 in the turbulent channel flow for.

top view; bird s eye view. 3.2.2 (y s =.5δ, 1.δ) 3 13 7 8 3 Antonia 17) y s =.5δ, 1.δ 3 Re τ = 18 Re τ = 395 y + 8 17) y + s = 5 3.

7 Deformed the material line from y s =δ in the turbulent channel flow for Re τ =18. The color of the line indicates as Fig. 6. The length of the line is L t 1 2 L at the same time as Fig Deformed material line from y + s =15 in the turbulent channel flow for. The color of the line indicates as Fig. 7. The length of the line is L t 1 4 L at tu τ /δ=.65. top view; bird s eye view (y s =.5δ, 1.δ) Antonia 17) y s =.5δ, 1.δ 3 Re τ = 18 Re τ = 395 y ) y + s = L y + s = 5, 15, y s =.5δ, δ t L L t 14 4 (y + = 15) Re τ = 18 Re τ = 395 Re τ = 18 tu τ /δ > 1.5 Re τ = 395 tu τ /δ = Re τ = 395

8 616 L t /L Re τ =18 γ =(γδ/u τ ) 15 1 Re τ = ys + = 5 y s + =15 y s =.5δ y s =1.δ 1 1 y + s = 5 y + s =15 y s =.5δ y s =1.δ tu τ /δ L t /L y s + = 5 y s + =15 y s =.5δ y s =1.δ γ + =(γν/u τ 2 ) tu 1.5 τ /δ Re τ = ys + = 5 y s + =15 y s =.5δ y s =1.δ t L t /L 1 2 Homogeneous isotropic turbulence (Re λ =84, Kida et al. 4) ) 1 1 y + s = 5 y + s =15 y s =.5δ y s =1.δ (c) t /τ η (y s ) 14 Average total length of the material lines. The time t is normalized by the outer-scale [in ], by the wall-unit [in ] or by the mean Kolmogorov time τ η of the each height [in (c)]. Black line: Re τ =18, gray line:, red line: Kida et al. 4). tu τ /δ > (c) t outer-scale wall-unit Kolmogorov ε 5 Kolmogorov η Kolmogorov τ η t Re τ = ys = 5 y + s =15.2 y s =.5δ y s =1.δ γ τ η (y s ) (c) 15.1 Homogeneous isotropic turbulence (Re λ =84, Kida et al. 4) ) t /τ η (y s ) Average stretching rate of material lines. The time t and the stretching rate γ are normalized by the outer-scale [in ], by the wall-unit [in ] or by the mean Kolmogorov time τ η of the each height [in (c)]. Legend as Fig. 14. Kolmogorov 14 L t t L t = L exp[γt] (7)

9 617 η Re τ = y y/δ 16 τ η y + Re τ = y/δ Kolmogorov scale and time-scale γ γ = d dt log L t. (8) γ (c) 15(c) Taylor Re λ = 84 4, 5) (tu τ /δ > 1.5) 2 Outer-scale wall-unit Kolmogorov τ η (y s ) 14(c) 15(c) y s Kolmogorov t/τ η < 5 (y + s = 5) Kolmogorov 2 Estimated value of the stretching rate for the fully developed line in the turbulent channel flow Re τ γ γ + γτ η,min (du/dy) γτ η (y s ) 5 < t/τ η < 1 γ = τ 1 η γ =.17τ 1 η t/τ η > 1 Kolmogorov γτ η τ η (y s ) y + s = 5, (c) γ.1 ( τ η (y s ) ) 1 γ =.17τ 1 η Kolmogorov 16 Kolmogorov Kolmogorov γ.65 ( ) τ 1 η,min 2 4 Re τ = DNS y + = 15

10 618 y y + = 5, 15 Kolmogorov τ η γ.1 ( ) 1 τ η 1.17τ η Kolmogorov Kolmogorov Kolmogorov γτ η t < 5τ η 2-5) : 1) Batchelor, G. K.: The effect of homogeneous turbulence on material lines and surfaces, Proc. Roy. Soc. London Ser. A 213 (1952) ) Girimaji, S. S. & Pope, S. B.: Material-element deformation in isotropic turbulence, J. Fluid Mech. 22 (199) ) Huang, M.-J.: Correlations of vorticity and material line elements with strain decatying turbulence, Phys. Fluids 8 (1996) ) Kida, S. & Goto, S.: Line statistics: Stretching rate of passive lines in turbulence, Phys. Fluids 14 (22) ) Goto, S. & Kida, S.: A multiplicative process of material line stretching by turbulence, J. Turbulence 3 (22). 6) Kim, J., Moin, P. & Moser, R.: Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech. 177 (1987) ) Abe, H., Kawamura, H. & Matsuo, Y.: Direct numerical simulation of a fully developed turbulent channel flow with respect to the Reynolds number dependence, Trans. ASME J. Fluids Eng. 123 (22) ) Iwamoto, K., Suzuki, Y. & Kasagi, N: Reynolds number effect on wall turbulence: toward effective feedback control, Int. J. Heat and Fluid Flow 23 (22) ) del Álamo, J. C. & Jiménez, J: Spectra of the very large anisotropic scales in turbulent channels, Phys. Fluids 15 (23) L41-L44. 1) Tanahashi, M., Kang. S.-J., Miyamoto, T., Shiokawa, S. & Miyauchi, T.: Scaling law of fine scale eddies in turbulent channel flows up to Re τ =8, Int. J. Heat and Fluid Flow 25 (24) ) Kawamura, H., Abe, H. & Matsuo, Y.: DNS of turbulent heat transfer in channel flow with respect to Reynolds number and Prandtl number effects, Int. J. Heat and Fluid Flow 19 (1999) ) Abe, H., Kawamura, H. & Matsuo, Y.: Surface heat-flux fluctuations in a turbulent channel flow up to Re τ =12 with Pr=.25 and.71, Int. J. Heat and Fluid Flow 25 (24) ) Morinishi, Y.: Conservative properties of finite difference scheme for incompressible flow, CTR Annual Reserch Briefs (1995) ) Wei, T. & Willmarth, W. W.: Reynolds-number effects on the structure of a turbulent channel flow, J. Fluid Mech. 24 (1989) ) Smith, C. R. & Schwartz, S. P.: Observation of streamwise rotation in the near-wall region of a turbulent boudary layer, Phys. Fluids 26 (1983) ) Robinson, S. K.: Coherent motions in the turbulent boundary layer, Annu. Rev. Fluid Mech. 23 (1991) ) Antonia, R.A., Kim, J. & Browne, L. W. B.: Some characteristics of small-scale turbulence in a turbulent duct flow, J. Fluid Mech. 233 (1991)

Fig. 1 Experimental apparatus.

Fig. 1 Experimental apparatus. Effects of Concentration of Surfactant Solutions on Drag-Reducing Turbulent Boundary Layer In this study, the influence of a drag-reducing surfactant on the turbulent boundary layer was extensively investigated