Abstract
Velocity gradient dynamics play a pivotal role in understanding various nonlinear phenomena in turbulent flows. In the evolution of velocity gradient dynamics, the pressure Hessian and the viscous Laplacian are two mathematically unclosed terms which need separate modeling. The current study models the pressure Hessian term using the tensor basis neural network (TBNN). The network is trained on direct numerical simulation (DNS) data of stationary incompressible turbulence conditioned on local flow topologies. We compare the topology-based TBNN model performance with the DNS results as well as with the unconditioned (raw) TBNN model. The model results are evaluated in terms of the strain rate and the pressure Hessian eigenvector alignments. The model captures some of the essential alignment features of the DNS results.
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References
Batchelor GK (1952) The effect of homogeneous turbulence on material lines and surfaces. Proc R Soc Lond Ser A Math Phys Sci 213(1114):349–366
Cantwell BJ (1992) Exact solution of a restricted Euler equation for the velocity gradient tensor. Phys Fluids A Fluid Dyn 4(4):782–793
Chevillard L, Meneveau C, Biferale L, Toschi F (2008) Modeling the pressure Hessian and viscous Laplacian in turbulence: comparisons with direct numerical simulation and implications on velocity gradient dynamics. Phys Fluids 20(10):101504
Chevillard L, Meneveau C (2006) Lagrangian dynamics and statistical geometric structure of turbulence. Phys Rev Lett 97(17):174501
Chong MS, Perry AE, Cantwell BJ (1990) A general classification of three-dimensional flow fields. Phys Fluids A Fluid Dyn 2(5):765–777
Danish M, Sinha SS, Srinivasan B (2016) Influence of compressibility on the Lagrangian statistics of vorticity–strain-rate interactions. Phys Rev E 94(1):013101
Danish M, Suman S, Srinivasan B (2014) A direct numerical simulation-based investigation and modeling of pressure Hessian effects on compressible velocity gradient dynamics. Phys Fluids 26(12):126103
Girimaji SS, Pope SB (1990) Material-element deformation in isotropic turbulence. J Fluid Mech 220:427–458
Jeong E, Girimaji SS (2003) Velocity-gradient dynamics in turbulence: effect of viscosity and forcing. Theor Comput Fluid Dyn 16(6):421–432
Li Y, Meneveau C (2005) Origin of non-Gaussian statistics in hydrodynamic turbulence. Phys Rev Lett 95(16):164502
Li Y, Meneveau C (2006) Intermittency trends and Lagrangian evolution of non-Gaussian statistics in turbulent flow and scalar transport. J Fluid Mech 558:133–142
Li Y, Perlman E, Wan M, Yang Y, Meneveau C, Burns R, Chen S, Szalay A, Eyink G (2008) A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence. J Turbul 9:N31
Ling J, Kurzawski A, Templeton J (2016) Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J Fluid Mech 807:155–166
Martin J, Ooi A, Dopazo C, Chong MS, Soria J (1997) The inverse diffusion time scale of velocity gradients in homogeneous isotropic turbulence. Phys Fluids 9(4):814–816
Meneveau C (2011) Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows. Annu Rev Fluid Mech 43(1):219–245
O’Neill P, Soria J (2005) The relationship between the topological structures in turbulent flow and the distribution of a passive scalar with an imposed mean gradient. Fluid Dyn Res 36(3):107
Parashar N, Sinha SS, Srinivasan B (2019) Lagrangian investigations of velocity gradients in compressible turbulence: lifetime of flow-field topologies. J Fluid Mech 872:492–514
Parashar N, Srinivasan B, Sinha SS (2020) Modeling the pressure-Hessian tensor using deep neural networks. Phys Rev Fluids 5(11):114604
Pope SB (1975) A more general effective-viscosity hypothesis. J Fluid Mech 72(2):331–340
Pope SB (1985) PDF methods for turbulent reactive flows. Prog Energy Combust Sci 11(2):119–192
Pope SB, Pope SB (2000) Turbulent flows. Cambridge University Press
Pumir A (1994) A numerical study of the mixing of a passive scalar in three dimensions in the presence of a mean gradient. Phys Fluids 6(6):2118–2132
Suman S, Girimaji SS (2009) Homogenized Euler equation: a model for compressible velocity gradient dynamics. J Fluid Mech 620:177–194
Suman S, Girimaji SS (2011) Dynamical model for velocity-gradient evolution in compressible turbulence. J Fluid Mech 683:289–319
Suman S, Girimaji SS (2010) Velocity gradient invariants and local flow-field topology in compressible turbulence. J Turbul 11:N2
Vieillefosse P (1982) Local interaction between vorticity and shear in a perfect incompressible fluid. J Phys 43(6):837–842
Xu H, Pumir A, Bodenschatz E (2011) The pirouette effect in turbulent flows. Nat Phys 7(9):709–712
Acknowledgements
The authors acknowledge the computational support provided by the High-Performance Computing (HPC) center of the Indian Institute of Technology Delhi, India.
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Shikha, D., Sinha, S.S. (2024). Machine Learning-Assisted Modeling of Pressure Hessian Tensor. In: Singh, K.M., Dutta, S., Subudhi, S., Singh, N.K. (eds) Fluid Mechanics and Fluid Power, Volume 4. FMFP 2022. Lecture Notes in Mechanical Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-99-7177-0_78
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DOI: https://doi.org/10.1007/978-981-99-7177-0_78
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