Joint Probability Distribution Functions for the Filtered Velocity Gradient Invariants in Homogeneous Isotropic Turbulence

Waleed Abdel Kareem1, *, Mahmoud Abdel Aty2, Zafer M. Asker1
1 Department of Mathematics and Computer Science, Faculty of Science, Suez University, Suez, Egypt
2 Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt

© 2018 Abdel Kareem et al.

open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: ( This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

* Address correspondence to this author at the Department of Mathematics and Computer Science, Faculty of Science, Suez University, Suez, Egypt; Tel: +2-062-3493623; E-mail:



Turbulent flow is characterized by vortices with different scales. Extraction of various scales and filtering the turbulent field into coherent and incoherent parts are important processes that improve our understanding of turbulent characteristics.


Joint probability distribution functions (JPDFs) for the filtered velocity gradient invariants are extensively studied for different scales as well as for the coherent and incoherent parts of each scale.


The Fourier decomposition and the anisotropic diffusion model are used in the investigation. The extraction process is performed by employing the Fourier decomposition at different cutoff wavenumbers for the velocity field and three distinct scales (large, medium and fine scale) are identified. The velocity gradient invariants such as the second invariant Q and the third invariant R for the different scales are extracted. Then other important invariants such as the rate of rotation tensor QW and the rate of deformation QS are also identified for each scale. The anisotropic diffusion model is used to extract the coherent and incoherent parts of each invariant at each scale. Then the JPDFs of the coherent and incoherent invariants are compared. The scale decomposition and the filtering process are applied for turbulent flow fields that are simulated using the lattice Boltzmann method with resolution of 1283.


Results show that the (R-Q) space has a universal topological pear-like shape for the different scales as well as their coherent field. However, the (R-Q)-space for the incoherent fields are found different and no general shape can be observed. The (Qw-QS)-space results show self-similar shapes for coherent fields and for the incoherent fields no specific shape can be observed since the noise distributed as separated points everywhere.


Two different methods for extraction and filtering of forced isotropic turbulence and the JPDFs of the velocity gradient invariants are studied. Some universal characteristics for the coherent parts were found. However, for the incoherent parts, no universal JPDFs were found.

Keywords: Lattice Boltzmann, Turbulent scales, JPDFs, Coherent and incoherent parts, Velocity invariants, Extraction and filtering.