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Statistics of Passive Vectors in an Unstable Periodic Flow of Couette System

Statistics of Passive Vectors in an Unstable Periodic Flow of Couette System. S. Kida (Kyoto Univ.). with T. Watanabe, T. Taya. Couette System. U. 2 h. -U. Couette System. U. 2 h. -U. Simulation range （ 5.513 h , 2 h , 3.770 h ）. Minimal box. Jimenez & Moin 1991.

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Statistics of Passive Vectors in an Unstable Periodic Flow of Couette System

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  1. Statistics of Passive Vectors in an Unstable Periodic Flow of Couette System S. Kida (Kyoto Univ.) with T. Watanabe, T. Taya

  2. Couette System U 2h -U

  3. Couette System U 2h -U Simulation range （ 5.513h, 2h , 3.770h） Minimal box Jimenez & Moin 1991 Number of modes （16×31×16） Number of independent variables15,422 Re = Uh /ν= 400 Unstable periodic flow Kawahara & Kida 2001

  4. Couette Turbulence U Low-speed streak Streamwise vortex Streamwise vortex by G. Kawahara - 0.3U

  5. Unstable Periodic Flow U Low-speed streak Streamwise vortex Streamwise vortex by G. Kawahara - 0.3U

  6. Turbulence vs UPF Same Low-Order Statistics

  7. Unstable Periodic Flow 0.2 t/T=0 0.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9

  8. Turbulence vs UPF Chaotic Periodic Unpredictable Repeatable Same Low-Order Statistics Easier to increase the accuracy of statistics Not easy

  9. Subject We consider turbulence mixing by examining the statistics of passive vectors advected in an unstable periodic flow (UPF) for a Couette system. We focus our attention on the relation between the coherent structures (streamwise vortices, low-speed streaks, ejection regions, sweep regions) and the mixing properties of turbulence.

  10. Orientation of Passive Vectors Image Laminar flow Lyapunov exponent = 0 Aligned slowly Turbulent flow Aligned quickly Lyapunov exponent > 0

  11. Alignment of Directions A unit sphere represents the direction of passive vectors z y x Initially, we distribute many passive vectors at a same point with random orientations.

  12. Alignment of passive vectors planar linear

  13. Alignment of passive vectors 20 38 56 We distribute 5320 sets of passive vectors uniformly in space at an initial instant. Each set is composed of 1000 passive vectors with random orientations.

  14. Alignment of Passive Vectors

  15. Alignment of Passive Vectors

  16. Direction Field of Passive Vectors Passive vectors which start at a same position with random orientations will be aligned in a few periods of UPF. Then, can we expect that the direction field of passive vectors in UPF is unique irrespective of the initial condition ?

  17. Construction of Direction Field

  18. Direction distribution of passive vectors 2h 20 38 56 Divide the computational box into many cubes of side 0.1h. Track the motion of many（5320）passive vectors for a long time, and Store the position and direction of each vector fallen in the individual cubes every time phase of the UPF. Calculate the direction distribution of vectors in each cube.

  19. Classification of Directional Distribution linear planar scattered

  20. Classification of direction distribution of passive vectors Number of PV in a cube：　N Directional Vector Direction matrix

  21. Direction Matrix Number of PV in the direction of ∝ Eigenvalues of Eigenvectors:

  22. Classification of the Type of Directional Distribution A reference distribution to be compared

  23. Classification of Directional Distribution planar scattered linear

  24. Directional Distribution planar scattered linear

  25. Classification of Directional Distribution planar scattered linear

  26. Relative Population Cube size dependence

  27. Relative Population Cube size dependence

  28. Where are the linear and planar types located in the flow field ?

  29. Direction field and coherent structure linear Vortex center planar Periphery of vortices scattered

  30. Direction field and coherent structure linear Vortex center Ejection region planar Periphery of vortices Sweep region scattered

  31. Summary(1/2) Passive vectors which start at a same position with different orientations in UPF will be aligned in a few time periods. The direction field of passive vectors can be defined uniquely in UPF. It is found that in most of small cubes the passive vectors align either in a single direction (linear type) or on a plane (planar type).

  32. Summary(2/2) The linear type is observed in streamwise vortices and low-speed streaks (or the ejection region near the wall), the planar type in the periphery of vortices and in the sweep region near the wall. The dispersion of orientation of passive vectors may be used to quantify turbulent mixing.

  33. Dependence on the cube size

  34. Vortex Center

  35. Vortex Periphery

  36. αβ 線素の方向の変動と流線 t=0.5T

  37. 方位角とその変化 渦中心域 渦外縁域

  38. 向壁面域

  39. Past history 過去距離 現在方向 領域 大 大 向壁面 小 大 渦外縁 大 小 渦中心 小 小 壁面沿い 方向の偏差値 距離の偏差値

  40. A cross-section of the direction field of passive vectors

  41. 線素の方向場の断面

  42. 線素の方向の標準偏差

  43. 線素の方向の標準偏差

  44. Unstable Periodic Flow Kawahara & Kida 2001

  45. UPF vs Turbulence UPF Turbulence

  46. UPF vs Turbulence UPF Turbulence

  47. UPF vs Turbulence UPF Turbulence

  48. Alignment of passive vectors in direction Many （5320）passive vectors which start with random orientations will align in direction after a few periods of UPF planar linear random

  49. 対称性 • スパン方向に垂直な平面 z = 0に関する鏡映と主流方向への半周期(Lx/2)移動(すべり鏡映)。 (2) チャネルの中央を通るスパン方向の直線（x = y = 0）に関する 180o回転とスパン方向への半周期(Lz/2)移動。 y x 活発なよどみ点 静穏なよどみ点 z

  50. 向壁面域

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