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Universal aspects of small-scale motions in turbulence. (English) Zbl 1205.76123
Summary: Two aspects of small-scale turbulence are currently regarded universal, as they have been reported for a wide variety of turbulent flows. Firstly, the vorticity vector has been found to display a preferential alignment with the eigenvector corresponding to the intermediate eigenvalue of the strain rate tensor; and secondly, the joint probability density function (p.d.f.) of the second and third invariant of the velocity gradient tensor, \(Q\) and \(R\), has a characteristic teardrop shape. This paper provides an explanation for these universal aspects in terms of a spatial organization of coherent structures, which is based on an evaluation of the average flow pattern in the local coordinate system defined by the eigenvectors of the strain rate tensor. The approach contrasts with previous investigations, which have relied on assumed model flows. The present average flow patterns have been calculated for existing experimental (particle image velocimetry) or numerical (direct numerical simulation) datasets of a turbulent boundary layer (TBL), a turbulent channel flow and for homogeneous isotropic turbulence. All results show a shear-layer structure consisting of aligned vortical motions, separating two larger-scale regions of relatively uniform flow. Because the directions of maximum and minimum strain in a shear layer are in the plane normal to the vorticity vector, this vector aligns with the remaining strain direction, i.e. the intermediate eigenvector of the strain rate tensor. Further, the \(QR\) joint p.d.f. for these average flow patterns reveals a shape reminiscent of the teardrop, as seen in many turbulent flows. The above-mentioned organization of the small-scale motions is not only found in the average patterns, but is also frequently observed in the instantaneous velocity fields of the different turbulent flows. It may, therefore, be considered relevant and universal.

MSC:
76F05 Isotropic turbulence; homogeneous turbulence
76F40 Turbulent boundary layers
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References:
[1] DOI: 10.1017/S002211200700777X · Zbl 1141.76316
[2] DOI: 10.1103/PhysRevE.65.046304 · Zbl 1244.76018
[3] DOI: 10.1103/PhysRevLett.98.214501
[4] DOI: 10.1063/1.869361 · Zbl 1185.76770
[5] DOI: 10.1017/S0022112056000317 · Zbl 0071.40603
[6] DOI: 10.1098/rspa.1949.0136 · Zbl 0036.25602
[7] DOI: 10.1063/1.868594
[8] DOI: 10.1063/1.866513
[9] DOI: 10.1017/S0022112004003283 · Zbl 1154.76303
[10] DOI: 10.1063/1.869228 · Zbl 1185.76735
[11] DOI: 10.1017/S002211209400203X · Zbl 0814.76049
[12] DOI: 10.1017/S0022112006000814 · Zbl 1157.76346
[13] DOI: 10.1063/1.1328358 · Zbl 1184.76281
[14] DOI: 10.1017/S002211200300733X · Zbl 1059.76031
[15] DOI: 10.1017/S0022112097004941 · Zbl 0892.76037
[16] DOI: 10.1007/BF03182598
[17] DOI: 10.1017/S0022112085001136 · Zbl 0587.76080
[18] DOI: 10.1017/S0022112000001580 · Zbl 0959.76503
[19] DOI: 10.1017/S0022112093002393 · Zbl 0800.76156
[20] DOI: 10.1063/1.858282
[21] DOI: 10.1017/S0022112006003946 · Zbl 1113.76004
[22] DOI: 10.1063/1.868993
[23] DOI: 10.1017/S0022112007009251 · Zbl 1159.76334
[24] DOI: 10.1017/S002211209900467X · Zbl 0946.76030
[25] DOI: 10.1017/S0022112005004362 · Zbl 1114.76301
[26] DOI: 10.1017/S0022112009006600 · Zbl 1181.76015
[27] DOI: 10.1017/S0022112008001511 · Zbl 1145.76024
[28] DOI: 10.1017/S0022112094003319 · Zbl 0800.76157
[29] Gere, Mechanics of Materials (1991) · Zbl 0058.19201
[30] DOI: 10.1016/0378-4371(84)90008-6 · Zbl 0599.76040
[31] DOI: 10.1017/S0022112002003270 · Zbl 1032.76500
[32] DOI: 10.1017/S0022112092002325
[33] DOI: 10.1017/S0022112007009706 · Zbl 1151.76323
[34] Tsinober, An Informal Introduction to Turbulence (2001)
[35] DOI: 10.1007/s00348-006-0212-z
[36] DOI: 10.1017/S0022112003005251 · Zbl 1063.76514
[37] DOI: 10.1017/S0022112087002337
[38] DOI: 10.1063/1.3291070 · Zbl 1183.76191
[39] DOI: 10.1063/1.868323 · Zbl 0827.76031
[40] DOI: 10.1017/S0022112009992047 · Zbl 1189.76023
[41] DOI: 10.1038/344226a0
[42] DOI: 10.1017/S0022112097008057 · Zbl 0908.76039
[43] DOI: 10.1063/1.857730
[44] DOI: 10.1063/1.858333
[45] DOI: 10.1017/S002211209900720X · Zbl 0985.76038
[46] DOI: 10.1063/1.857938
[47] DOI: 10.1063/1.858828 · Zbl 0794.76044
[48] DOI: 10.1063/1.858295 · Zbl 0754.76004
[49] DOI: 10.1017/S0022112007009020 · Zbl 1159.76342
[50] DOI: 10.1017/S0022112004000941 · Zbl 1131.76326
[51] DOI: 10.1017/S0022112095000152 · Zbl 0831.76026
[52] DOI: 10.1017/S0022112096001802 · Zbl 0864.76036
[53] DOI: 10.1017/S0022112098003681 · Zbl 0965.76031
[54] DOI: 10.1017/S0022112004009802 · Zbl 1107.76328
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