Invariants of the velocity-gradient tensor in a spatially developing inhomogeneous turbulent flow.

*(English)*Zbl 1383.76179Summary: Tomographic particle image velocimetry experiments were performed in the near field of the turbulent flow past a square cylinder. A classical Reynolds decomposition was performed on the resulting velocity fields into a time invariant mean flow and a fluctuating velocity field. This fluctuating velocity field was then further decomposed into coherent and residual/stochastic fluctuations. The statistical distributions of the second and third invariants of the velocity-gradient tensor were then computed at various streamwise locations, along the centreline of the flow and within the shear layers. These invariants were calculated from both the Reynolds-decomposed fluctuating velocity fields and the coherent and stochastic fluctuating velocity fields. The range of spatial locations probed incorporates regions of contrasting flow physics, including a mean recirculation region and separated shear layers, both upstream and downstream of the location of peak turbulence intensity along the centreline. These different flow physics are also reflected in the velocity gradients themselves with different topologies, as characterised by the statistical distributions of the constituent enstrophy and strain-rate invariants, for the three different fluctuating velocity fields. Despite these differing flow physics the ubiquitous self-similar ’tear drop’-shaped joint probability density function between the second and third invariants of the velocity-gradient tensor is observed along the centreline and shear layer when calculated from both the Reynolds decomposed and the stochastic velocity fluctuations. These ’tear drop’-shaped joint probability density functions are not, however, observed when calculated from the coherent velocity fluctuations. This ’tear drop’ shape is classically associated with the statistical distribution of the velocity-gradient tensor invariants in fully developed turbulent flows in which there is no coherent dynamics present, and hence spectral peaks at low wavenumbers. The results presented in this manuscript, however, show that such ’tear drops’ also exist in spatially developing inhomogeneous turbulent flows. This suggests that the ’tear drop’ shape may not just be a universal feature of fully developed turbulence but of turbulent flows in general.

##### MSC:

76F05 | Isotropic turbulence; homogeneous turbulence |

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\textit{O. R. H. Buxton} et al., J. Fluid Mech. 817, 1--20 (2017; Zbl 1383.76179)

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##### References:

[1] | Baj, P.; Bruce, P. J. K.; Buxton, O. R. H., The triple decomposition of a fluctuating velocity field in a multiscale flow, Phys. Fluids, 27, 7, (2015) |

[2] | Batchelor, G., The Theory of Homogeneous Turbulence, (1953), Cambridge University Press · Zbl 0053.14404 |

[3] | Berkooz, G.; Holmes, P.; Lumley, J. L., The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech., 25, 1, 539-575, (1993) |

[4] | Blackburn, H. M.; Mansour, N. N.; Cantwell, B. J., Topology of fine-scale motions in turbulent channel flow, J. Fluid Mech., 310, 269-292, (1996) · Zbl 0864.76036 |

[5] | Brereton, G. J.; Kodal, A., A frequency-domain filtering technique for triple decomposition of unsteady turbulent flow, Trans. ASME J. Fluids Engng, 114, 1, 45-51, (1992) |

[6] | Buxton, O., Modulation of the velocity gradient tensor by concurrent large-scale velocity fluctuations in a turbulent mixing layer, J. Fluid Mech., 777, R1, 1-12, (2015) |

[7] | Buxton, O.; Ganapathisubramani, B., Amplification of enstrophy in the far field of an axisymmetric turbulent jet, J. Fluid Mech., 651, 483-502, (2010) · Zbl 1189.76016 |

[8] | Buxton, O.; Laizet, S.; Ganapathisubramani, B., The effects of resolution and noise on kinematic features of fine-scale turbulence, Exp. Fluids, 51, 5, 1417-1437, (2011) |

[9] | Cantwell, B., Exact solution of a restricted Euler equation for the velocity gradient tensor, Phys. Fluids A, 4, 4, 782-793, (1992) · Zbl 0754.76004 |

[10] | Cantwell, B. J., On the behavior of velocity gradient tensor invariants in direct numerical simulations of turbulence, Phys. Fluids A, 5, 8, 2008-2013, (1993) · Zbl 0794.76044 |

[11] | Cantwell, B.; Coles, D., An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder, J. Fluid Mech., 136, 1, 321-374, (1983) |

[12] | Chacin, J.; Cantwell, B., Dynamics of a low Reynolds number turbulent boundary layer, J. Fluid Mech., 404, 87-115, (2000) · Zbl 0985.76038 |

[13] | Elsinga, G.; Marusic, I., Universal aspects of small-scale motions in turbulence, J. Fluid Mech., 662, 514-539, (2010) · Zbl 1205.76123 |

[14] | Elsinga, G.; Scarano, F.; Wieneke, B.; Van Oudheusden, B., Tomographic particle image velocimetry, Exp. Fluids, 41, 6, 933-947, (2006) |

[15] | Ganapathisubramani, B.; Lakshminarasimhan, K.; Clemens, N., Determination of complete velocity gradient tensor by using cinematographic stereoscopic PIV in a turbulent jet, Exp. Fluids, 42, 923-939, (2007) |

[16] | Ganapathisubramani, B.; Lakshminarasimhan, K.; Clemens, N., Investigation of three-dimensional structure of fine-scales in a turbulent jet by using cinematographic stereoscopic PIV, J. Fluid Mech., 598, 141-175, (2008) · Zbl 1151.76323 |

[17] | Gomes-Fernandes, R.; Ganapathisubramani, B.; Vassilicos, J., Evolution of the velocity-gradient tensor in a spatially developing turbulent flow, J. Fluid Mech., 756, 252-292, (2014) |

[18] | Gomes-Fernandes, R.; Ganapathisubramani, B.; Vassilicos, J., The energy cascade in near-field non-homogeneous non-isotropic turbulence, J. Fluid Mech., 771, 676-705, (2015) |

[19] | Herpin, S.; Wong, C.; Stanislas, M.; Soria, J., Stereoscopic PIV measurements of a turbulent boundary layer with a large spatial dynamic range, Exp. Fluids, 45, 4, 745-763, (2008) |

[20] | Hussain, A. K. M. F.; Reynolds, W. C., The mechanics of an organized wave in turbulent shear flow, J. Fluid Mech., 41, 2, 241-258, (1970) |

[21] | Jiménez, J.; Wray, A.; Saffman, P.; Rogallo, R., The structure of intense vorticity in isotropic turbulence, J. Fluid Mech., 255, 65-90, (1993) · Zbl 0800.76156 |

[22] | Kolmogorov, A., The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, C. R. Acad. Sci. URSS, 30, 4, 301-305, (1941) · JFM 67.0850.06 |

[23] | Kraichnan, R. H., On Kolmogorov’s inertial-range theories, J. Fluid Mech., 62, 2, 305-330, (1974) · Zbl 0273.76037 |

[24] | Laizet, S.; Nedić, J.; Vassilicos, J. C., The spatial origin of 5/3 spectra in grid-generated turbulence, Phys. Fluids, 27, 6, (2015) |

[25] | Van Oudheusden, B.; Scarano, F.; Van Hinsberg, N.; Watt, D., Phase-resolved characterization of vortex shedding in the near wake of a square-section cylinder at incidence, Exp. Fluids, 39, 1, 86-98, (2005) |

[26] | Perrin, R.; Braza, M.; Cid, E.; Cazin, S.; Barthet, A.; Sevrain, A.; Mockett, C.; Thiele, F., Obtaining phase averaged turbulence properties in the near wake of a circular cylinder at high Reynolds number using POD, Exp. Fluids, 43, 2, 341-355, (2007) |

[27] | Perrin, R.; Braza, M.; Cid, E.; Cazin, S.; Chassaing, P.; Mockett, C.; Reimann, T.; Thiele, F., Coherent and turbulent process analysis in the flow past a circular cylinder at high Reynolds number, J. Fluids Struct., 24, 1313-1325, (2008) |

[28] | Pope, S., Turbulent Flows, (2000), Cambridge University Press · Zbl 0966.76002 |

[29] | Ruetsch, G.; Maxey, M., Small-scale features of vorticity and passive scalar fields in homogeneous isotropic turbulence, Phys. Fluids A, 3, 6, 1587-1597, (1991) |

[30] | De Silva, C. M.; Philip, J.; Marusic, I., Minimization of divergence error in volumetric velocity measurements and implications for turbulence statistics, Exp. Fluids, 54, 7, 1-17, (2013) |

[31] | Soria, J.; Sondergaard, R.; Cantwell, B. J.; Chong, M. S.; Perry, A. E., A study of the fine-scale motions of incompressible time-developing mixing layers, Phys. Fluids, 6, 2, 871-884, (1994) · Zbl 0827.76031 |

[32] | Taylor, G., Production and dissipation of vorticity in a turbulent fluid, Proc. R. Soc. Lond. A, 164, 916, 15-23, (1938) · JFM 64.1453.04 |

[33] | Tsinober, A., An Informal Conceptual Introduction to Turbulence, (2009), Springer · Zbl 1177.76001 |

[34] | Valente, P. C.; Vassilicos, J. C., Universal dissipation scaling for nonequilibrium turbulence, Phys. Rev. Lett., 108, 21, (2012) · Zbl 1255.76032 |

[35] | Vieillefosse, P., Local interaction between vorticity and shear in a perfect incompressible fluid, J. Phys. (Paris), 43, 6, 837-842, (1982) |

[36] | Wieneke, B., Volume self-calibration for 3d particle image velocimetry, Exp. Fluids, 45, 4, 549-556, (2008) |

[37] | Worth, N.; Nickels, T.; Swaminathan, N., A tomographic PIV resolution study based on homogeneous isotropic turbulence DNS data, Exp. Fluids, 49, 3, 637-656, (2010) |

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