Statistics of Passive Vectors in an Unstable Periodic Flow of Couette System S. Kida (Kyoto Univ.) with T. Watanabe, T. Taya Couette System U 2h -U Couette System U 2h -U Simulation range （ 5.513h, 2h , 3.770h ） Number of modes （16×31×16） Minimal box Jimenez & Moin 1991 Number of independent variables 15,422 Re = Uh /ν= 400 Unstable periodic flow Kawahara & Kida 2001 Couette Turbulence U - 0.3U by G. Kawahara Unstable Periodic Flow U - 0.3U by G. Kawahara Turbulence vs UPF Same Low-Order Statistics Unstable Periodic Flow 0.3 0.4 0.7 0.2 0.1 t/T=0 0.5 0.8 0.6 0.9 Turbulence vs UPF Chaotic Unpredictable Periodic Repeatable Same Low-Order Statistics Not easy Easier to increase the accuracy of statistics 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, lowspeed streaks, ejection regions, sweep regions) and the mixing properties of turbulence. Orientation of Passive Vectors Laminar flow Lyapunov exponent = 0 Aligned slowly Turbulent flow Aligned quickly Lyapunov exponent > 0 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. Alignment of passive vectors planar linear 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. Alignment of Passive Vectors Alignment of Passive Vectors 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 ? Construction of Direction Field 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. Classification of Directional Distribution linear planar scattered Classification of direction distribution of passive vectors Number of PV in a cube： N Directional Vector Direction matrix Direction Matrix Number of PV in the ∝ direction of Eigenvalues of Eigenvectors: Classification of the Type of Directional Distribution A reference distribution to be compared Classification of Directional Distribution linear planar scattered Directional Distribution planar scattered linear Classification of Directional Distribution linear planar scattered Relative Population Cube size dependence Relative Population Cube size dependence Where are the linear and planar types located in the flow field ? Direction field and coherent structure linear planar scattered Vortex center Periphery of vortices Direction field and coherent structure linear planar scattered Vortex center Ejection region Periphery of vortices Sweep region 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). 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. Dependence on the cube size Vortex Center Vortex Periphery 線素の方向の変動と流線 αβ t=0.5T 方位角とその変化 渦中心域 渦外縁域 向壁面域 Past history 過去 距離 現在 方向 向壁面 大 大 渦外縁 小 大 渦中心 大 小 壁面沿い 小 小 領域 方向の偏差値 距離の偏差値 A cross-section of the direction field of passive vectors 線素の方向場の断面 線素の方向の標準偏差 線素の方向の標準偏差 Unstable Periodic Flow Kawahara & Kida 2001 UPF vs Turbulence UPF Turbulence UPF vs Turbulence UPF Turbulence UPF vs Turbulence UPF Turbulence 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 random linear 対称性 (1)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 活発なよどみ点 静穏なよどみ点 x z 向壁面域 向壁面域 Alignment of Directions z y x After a while, all the test particles align in a single direction. Passive Vector Field The alignment of orientation of passive vectors in an unstable periodic flow (perhaps in a turbulent flow too) implies that their orientation (therefore their stretching rate as well) is a function of space and time determined uniquely irrespective of their history earlier than some finite time. Thus we can define the fields of various physical quantities associated with passive vectors, which are called the passive vector fields. Passive Vector Field In the case of unstable periodic flows, in particular, the passive vector field is also periodic in time. This property enables us to calculate the passive vector fields by particle-pair simulation. For example, the orientation field can be obtained by statistical average, over a sufficiently long time, of the relative position vector of many particle pairs near each simulation grid point. Orientation Field 20 38 56 Divide the computational box into many cubes. Integrate the motion of many particle pairs for a long time, and store the relative position vector of the pairs in each cube at each temporal phase of the unstable periodic flow. Estimate the orientation vector at the center of each cube by a linear interpolation. Linear Interpolation Let us assume there are N (≧ 4) particle pairs in a cube centered at (x, y, z). Let (xi, yi, zi) be the center of i-th pair (i = 1, 2, …, N)，and (pi, qi, ri) be the orientation vector (pi2+qi2+ri2=1) . Then, the orientation field at point (x, y, z) may be estimated by a, which minimizes the following function: where fi is pi, qi or ri . Stretching Field Threshold is 0.35 0.8 T Temporal Variation of Mean Stretching Rate Time average 0.8 Strain Characteristics Rate-of-strain tensor Eigenvalues Eigenvectors Strain Characteristics Rate-of-strain tensor Eigenvalues Eigenvectors Correlation bet. Stretching & Strain Rates λ1 λ2 γ γ λ3 0.8 T γ Spatial Symmetry (1) Reflection w.r.t. plane z=0 and translation along x axis by Lx/2. (2) 180o rotation around line (x = y = 0) and translation along z by Lz/2. y x Active S.P. Gentle S.P. z Alignment of Directions ある基準粒子の近傍に，等しい距離 (d) だけ離れ， 方向がランダムに分布した多数のテスト粒子を配置 する。数値シミュレーションの各ステップで基準粒子 との距離を d に戻す。しばらく（３周期程度）すると， テスト粒子の方向がそろってくる。 線素の方向の整列 時間位相 0.7 T において，ある立方体（一辺 0.1 h）に落ち こんだ粒子対の方向（左図）と中心の位置座標（右図）。 Main Results Unstable periodic flow varies periodically in time. That is, the flow state repeats in a constant period at all the spatial points. In this flow we submerge passive line elements at arbitrary time, at arbitrary position and in arbitrary direction. After a while (about three periods), the direction of line elements (therefore their stretching rate) is determined uniquely depending upon the time and place independent of their initial directions. Strain and Stretching Rates Dark blue : Strain rate (0.26) Light blue : 1-st Eigenvalue (0.55) 0.8 T
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