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The understanding of the dynamics of the velocity gradients in turbulentflows is critical to understanding various non-linear turbulent processes. Thepressure-Hessian and the viscous-Laplacian govern the evolution of thevelocity-gradients and are known to be non-local in nature. Over the years,several simplified dynamical models have been proposed that models theviscous-Laplacian and the pressure-Hessian primarily in terms of local velocitygradients information. These models can also serve as closure models for theLagrangian PDF methods. The recent fluid deformation closure model (RFDM) hasbeen shown to retrieve excellent one-time statistics of the viscous process.However, the pressure-Hessian modelled by the RFDM has various physicallimitations. In this work, we first demonstrate the limitations of the RFDM inestimating the pressure-Hessian. Further, we employ a tensor basis neuralnetwork (TBNN) to model the pressure-Hessian from the velocity gradient tensoritself. The neural network is trained on high-resolution data obtained fromdirect numerical simulation (DNS) of isotropic turbulence at Reynolds number of433 (JHU turbulence database, JHTD). The predictions made by the TBNN aretested against two different isotropic turbulence datasets at Reynolds numberof 433 (JHTD) and 315 (UP Madrid turbulence database, UPMTD) and channel flowdataset at Reynolds number of 1000 (UT Texas and JHTD). The evaluation of theneural network output is made in terms of the alignment statistics of thepredicted pressure-Hessian eigenvectors with the strain-rate eigenvectors forturbulent isotropic flow as well as channel flow. Our analysis of the predictedsolution leads to the discovery of ten unique coefficients of the tensor basisof strain-rate and rotation-rate tensors, the linear combination over whichaccurately captures key alignment statistics of the pressure-Hessian tensor.

Modelling pressure-Hessian from local velocity gradients information in an incompressible turbulent flow field using deep neural networks