# zbMATH — the first resource for mathematics

Geometry of enstrophy and dissipation, grid resolution effects and proximity issues in turbulence. (English) Zbl 1156.76406
Summary: We perform a multi-scale non-local geometrical analysis of the structures extracted from the enstrophy and kinetic energy dissipation-rate, instantaneous fields of a numerical database of incompressible homogeneous isotropic turbulence decaying in time obtained by DNS in a periodic box. Three different resolutions are considered: 256$$^3$$, 512$$^3$$ and 1024$$^3$$ grid points, with $$k_{\max}\overline \eta$$ approximately 1, 2 and 4, respectively, the same initial conditions and $$Re_\lambda\approx 77$$. This allows a comparison of the geometry of the structures obtained for different resolutions. For the highest resolution, structures of enstrophy and dissipation evolve in a continuous distribution from blob-like and moderately stretched tube-like shapes at the large scales to highly stretched sheet-like structures at the small scales. The intermediate scales show a predominance of tube-like structures for both fields, much more pronounced for the enstrophy field. The dissipation field shows a tendency towards structures with lower curvedness than those of the enstrophy, for intermediate and small scales. The 256$$^3$$ grid resolution case $$k_{\max}\overline \eta\approx 1$$ was unable to detect the predominance of highly stretched sheet-like structures at the smaller scales in both fields. The same non-local methodology for the study of the geometry of structures, but without the multi-scale decomposition, is applied to two scalar fields used by existing local criteria for the eduction of tube- and sheet-like structures in turbulence, $$Q$$ and $$[A_{ij}]_+$$, respectively, obtained from invariants of the velocity-gradient tensor and alike in the 1024$$^3$$ case. This adds the non-local geometrical characterization and classification to those local criteria, assessing their validity in educing particular geometries. Finally, we introduce a new methodology for the study of proximity issues among structures of different fields, based on geometrical considerations and non-local analysis, by taking into account the spatial extent of the structures. We apply it to the four fields previously studied. Tube-like structures of $$Q$$ are predominantly surrounded by sheet-like structures of $$[A_{ij}]_+$$, which appear at closer distances. For the enstrophy, tube-like structures at an intermediate scale are primarily surrounded by sheets of smaller scales of the enstrophy and structures of dissipation at the same and smaller scales. A secondary contribution results from tubes of enstrophy at smaller scales appearing at farther distances. Different configurations of composite structures are presented.

##### MSC:
 76F65 Direct numerical and large eddy simulation of turbulence
Full Text:
##### References:
 [1] DOI: 10.1017/S0022112005004040 · Zbl 1070.76033 · doi:10.1017/S0022112005004040 [2] DOI: 10.1023/B:APPL.0000044408.46141.26 · Zbl 1081.76564 · doi:10.1023/B:APPL.0000044408.46141.26 [3] DOI: 10.1063/1.858333 · doi:10.1063/1.858333 [4] DOI: 10.1017/S0022112008001511 · Zbl 1145.76024 · doi:10.1017/S0022112008001511 [5] DOI: 10.1103/PhysRevLett.79.1253 · doi:10.1103/PhysRevLett.79.1253 [6] DOI: 10.1098/rspa.1926.0043 · doi:10.1098/rspa.1926.0043 [7] DOI: 10.1137/05064182X · Zbl 1122.65134 · doi:10.1137/05064182X [8] Richardson, Weather Prediction by Numerical Process. (1922) · JFM 48.0629.07 [9] DOI: 10.1063/1.858441 · Zbl 0775.76078 · doi:10.1063/1.858441 [10] DOI: 10.1146/annurev.fluid.30.1.31 · Zbl 1398.76084 · doi:10.1146/annurev.fluid.30.1.31 [11] DOI: 10.1063/1.863442 · doi:10.1063/1.863442 [12] Onsager, Phys. Rev. 68 pp 286– (1945) [13] DOI: 10.1017/S002211200800092X · Zbl 1151.76522 · doi:10.1017/S002211200800092X [14] DOI: 10.1063/1.2771661 · Zbl 1182.76566 · doi:10.1063/1.2771661 [15] Batchelor, Proc. R. Soc. Lond. A 199 pp 238– (1949) [16] DOI: 10.1017/S0022112098003024 · Zbl 0933.76035 · doi:10.1017/S0022112098003024 [17] DOI: 10.1063/1.866513 · doi:10.1063/1.866513 [18] DOI: 10.1017/S0022112004009802 · Zbl 1107.76328 · doi:10.1017/S0022112004009802 [19] DOI: 10.1063/1.869361 · Zbl 1185.76770 · doi:10.1063/1.869361 [20] DOI: 10.1017/S0022112091003786 · Zbl 0749.76033 · doi:10.1017/S0022112091003786 [21] DOI: 10.1063/1.863957 · Zbl 0536.76034 · doi:10.1063/1.863957 [22] Landau, Fluid Mechanics. (1959) [23] DOI: 10.1017/S002211207400070X · Zbl 0273.76037 · doi:10.1017/S002211207400070X [24] DOI: 10.1017/S0022112062000518 · Zbl 0112.42003 · doi:10.1017/S0022112062000518 [25] Kolmogorov, Dokl. Nauk. SSSR. 30 pp 301– (1941) [26] Kolmogorov, Dokl. Nauk. SSSR. 32 pp 16– (1941) [27] DOI: 10.1016/0262-8856(92)90076-F · doi:10.1016/0262-8856(92)90076-F [28] DOI: 10.1017/S0022112085001136 · Zbl 0587.76080 · doi:10.1017/S0022112085001136 [29] DOI: 10.1017/S0022112061000974 · Zbl 0103.19903 · doi:10.1017/S0022112061000974 [30] DOI: 10.1017/S0022112093002393 · Zbl 0800.76156 · doi:10.1017/S0022112093002393 [31] DOI: 10.1063/1.858282 · doi:10.1063/1.858282 [32] DOI: 10.1038/nature01334 · doi:10.1038/nature01334 [33] DOI: 10.1017/S0022112095000462 · Zbl 0847.76007 · doi:10.1017/S0022112095000462 [34] DOI: 10.1063/1.869866 · Zbl 1147.76537 · doi:10.1063/1.869866 [35] DOI: 10.1143/JPSJ.72.983 · doi:10.1143/JPSJ.72.983 [36] DOI: 10.1017/S0022112006009128 · Zbl 1156.76405 · doi:10.1017/S0022112006009128 [37] DOI: 10.1017/S0022112094003319 · Zbl 0800.76157 · doi:10.1017/S0022112094003319 [38] DOI: 10.1063/1.2147610 · Zbl 1188.76063 · doi:10.1063/1.2147610 [39] DOI: 10.1017/S002211208100181X · Zbl 0476.76051 · doi:10.1017/S002211208100181X [40] DOI: 10.1017/S0022112092002325 · doi:10.1017/S0022112092002325 [41] DOI: 10.1017/S0022112007009251 · Zbl 1159.76334 · doi:10.1017/S0022112007009251 [42] DOI: 10.1063/1.870348 · Zbl 1149.76561 · doi:10.1063/1.870348 [43] DOI: 10.1063/1.1410981 · Zbl 1184.76230 · doi:10.1063/1.1410981 [44] DOI: 10.1063/1.858546 · doi:10.1063/1.858546 [45] He, Phys. Rev. Lett. 81 pp 4639– (1998) [46] DOI: 10.1103/PhysRevA.38.6287 · doi:10.1103/PhysRevA.38.6287 [47] DOI: 10.1103/PhysRevE.77.026303 · doi:10.1103/PhysRevE.77.026303
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.