Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-17T14:56:14.179Z Has data issue: false hasContentIssue false
  • English
  • Français

Statistical properties of pressure-Hessian tensor in a turbulent channel flow

Published online by Cambridge University Press:  18 January 2022

Jiu-Peng Tang ,
Zhen-Hua Wan Open the ORCID record for Zhen-Hua Wan [Opens in a new window] ,
Nan-Sheng Liu Open the ORCID record for Nan-Sheng Liu [Opens in a new window]  and
Xi-Yun Lu Open the ORCID record for Xi-Yun Lu [Opens in a new window]
Jiu-Peng Tang
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, PR China Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang 621900, PR China
Zhen-Hua Wan*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, PR China
Nan-Sheng Liu
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, PR China
Xi-Yun Lu*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, PR China
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A direct numerical simulation database of a weakly compressible turbulent channel flow with bulk Mach number 1.56 is studied in detail, including the geometrical relationships between the pressure-Hessian tensor and the vorticity/strain-rate tensor, as well as the mechanism of the pressure-Hessian tensor contributing to the evolution of invariants of the velocity gradient tensor. The results show that the geometrical relationships between the pressure-Hessian tensor and the vorticity/strain-rate tensor in the central region of the channel are consistent with that of isotropic turbulence. However, in the buffer layer with relatively stronger inhomogeneity and anisotropy, the vorticity tends to be aligned with the first or second eigenvector of the pressure-Hessian tensor in the unstable focus/compressing topological region, and tends to be aligned with the first eigenvector of the pressure-Hessian tensor in the stable focus/stretching topological region. In the unstable node/saddle/saddle and stable node/saddle/saddle topological regions, the vorticity prefers to lie in the plane of the first and second eigenvectors of the pressure-Hessian tensor. The strain-rate and the pressure-Hessian tensors tend to share their second principal direction. Moreover, for the coupling between the pressure-Hessian tensor and the principal strain rates, we clarify the influence on dissipation, the nonlinear generation of dissipation and the enstrophy generation. The decomposition of the pressure-Hessian tensor further shows that the slow pressure-related term dominates the pressure-Hessian tensor's contribution, and the influence of inhomogeneity and anisotropy mainly originates from the inhomogeneity and anisotropy of the fluctuating velocity. These statistical properties would be instructive in formulating dynamical models of the velocity gradient tensor for wall turbulence.

JFM classification

Turbulent Flows: Shear layer turbulence Turbulent Flows: Turbulence modelling
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 934 , 10 March 2022 , A22
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Ashurst, W.T., Kerstein, A.R., Kerr, R.M. & Gibson, C.H. 1987 Alignment of vorticity and scalar gradient with strain rate in simulated Navier–Stokes turbulence. Phys. Fluids 30 (8), 2343–2353.CrossRefGoogle Scholar
Atkinson, C., Chumakov, S., Bermejo-Moreno, I. & Soria, J. 2012 Lagrangian evolution of the invariants of the velocity gradient tensor in a turbulent boundary layer. Phys. Fluids 24 (10), 105104.CrossRefGoogle Scholar
Balint, J.-L., Wallace, J.M. & Vukoslavčević, P. 1991 The velocity and vorticity vector fields of a turbulent boundary layer. Part 2. Statistical properties. J. Fluid Mech. 228, 53–86.Google Scholar
Bechlars, P. & Sandberg, R.D. 2017 Evolution of the velocity gradient tensor invariant dynamics in a turbulent boundary layer. J. Fluid Mech. 815, 223–242.CrossRefGoogle Scholar
Buxton, O.R.H. & Ganapathisubramani, B. 2010 Amplification of enstrophy in the far field of an axisymmetric turbulent jet. J. Fluid Mech. 651, 483–502.CrossRefGoogle Scholar
Cantwell, B.J. 1992 Exact solution of a restricted euler equation for the velocity gradient tensor. Phys. Fluids A 4 (4), 782–793.CrossRefGoogle Scholar
Cantwell, B.J. 1993 On the behavior of velocity gradient tensor invariants in direct numerical simulations of turbulence. Phys. Fluids 5, 2008–2013.CrossRefGoogle Scholar
Chacin, J.M. & Cantwell, B.J. 2000 Dynamics of a low Reynolds number turbulent boundary layer. J. Fluid Mech. 404, 87–115.CrossRefGoogle Scholar
Chevillard, L., Lévêque, E., Taddia, F., Meneveau, C., Yu, H. & Rosales, C. 2011 Local and nonlocal pressure Hessian effects in real and synthetic fluid turbulence. Phys. Fluids 23 (9), 095108.CrossRefGoogle Scholar
Chevillard, L. & Meneveau, C. 2006 Lagrangian dynamics and statistical geometric structure of turbulence. Phys. Rev. Lett. 97, 174501.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Chong, M.S., Perry, A.E. & Cantwell, B.J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids A 2 (5), 765–777.CrossRefGoogle Scholar
Chu, Y., Wang, L. & Lu, X. 2014 Interaction between strain and vorticity in compressible turbulent boundary layer. Sci. China Phys. Mech. 57 (12), 2316–2329.CrossRefGoogle Scholar
Chu, Y.-B. & Lu, X.-Y. 2013 Topological evolution in compressible turbulent boundary layers. J. Fluid Mech. 733, 414–438.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Danish, M., Suman, S. & Srinivasan, B. 2017 Pressure Hessian evolution in compressible turbulence. In Fluid Mechanics and Fluid Power–Contemporary Research (ed. A.K. Saha et al. ), pp. 649–658. Springer.CrossRefGoogle Scholar
Duan, L., Beekman, I. & Martín, M.P. 2010 Direct numerical simulation of hypersonic turbulent boundary layers. Part 2. Effect of wall temperature. J. Fluid Mech. 655, 419–445.CrossRefGoogle Scholar
Duan, L., Beekman, I. & Martín, M.P. 2011 Direct numerical simulation of hypersonic turbulent boundary layers. Part 3. Effect of Mach number. J. Fluid Mech. 672, 245–267.CrossRefGoogle Scholar
Elsinga, G.E. & Marusic, I. 2010 Evolution and lifetimes of flow topology in a turbulent boundary layer. Phys. Fluids 22 (1), 015102.CrossRefGoogle Scholar
Foysi, H., Sarkar, S. & Friedrich, R. 2004 Compressibility effects and turbulence scalings in supersonic channel flow. J. Fluid Mech. 509, 207–216.CrossRefGoogle Scholar
Gerolymos, G.A., Sénéchal, D. & Vallet, I. 2013 Wall effects on pressure fluctuations in turbulent channel flow. J. Fluid Mech. 720, 15–65.CrossRefGoogle Scholar
Gerolymos, G.A. & Vallet, I. 2014 Pressure, density, temperature and entropy fluctuations in compressible turbulent plane channel flow. J. Fluid Mech. 757, 701–746.CrossRefGoogle Scholar
Gerolymos, G.A. & Vallet, I. 2016 The dissipation tensor in wall turbulence. J. Fluid Mech. 807, 386–418.CrossRefGoogle Scholar
Girimaji, S.S. & Pope, S.B. 1990 A diffusion model for velocity gradients in turbulence. Phys. Fluids A 2 (2), 242–256.CrossRefGoogle Scholar
Holzner, M., Lüthi, B., Tsinober, A. & Kinzelbach, W. 2009 Acceleration, pressure and related quantities in the proximity of the turbulent/non-turbulent interface. J. Fluid Mech. 639, 153–165.CrossRefGoogle Scholar
Jiang, G.S. & Shu, C.W. 1996 Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228.CrossRefGoogle Scholar
Johnson, P.L. & Meneveau, C. 2016 A closure for lagrangian velocity gradient evolution in turbulence using recent-deformation mapping of initially Gaussian fields. J. Fluid Mech. 804, 387–419.CrossRefGoogle Scholar
Johnson, P.L. & Meneveau, C. 2017 Turbulence intermittency in a multiple-time-scale Navier–Stokes-based reduced model. Phys. Rev. Fluids 2 (7), 072601.CrossRefGoogle Scholar
Kalelkar, C. 2006 Statistics of pressure fluctuations in decaying isotropic turbulence. Phys. Rev. E 73 (4), 046301.CrossRefGoogle ScholarPubMed
Lawson, J.M. & Dawson, J.R. 2015 On velocity gradient dynamics and turbulent structure. J. Fluid Mech. 780, 60–98.CrossRefGoogle Scholar
Lozano-Durán, A., Holzner, M. & Jiménez, J. 2015 Numerically accurate computation of the conditional trajectories of the topological invariants in turbulent flows. J. Comput. Phys. 295, 805–814.CrossRefGoogle Scholar
Lozano-Durán, A., Holzner, M. & Jiménez, J. 2016 Multiscale analysis of the topological invariants in the logarithmic region of turbulent channels at a friction Reynolds number of 932. J. Fluid Mech. 803, 356–394.CrossRefGoogle Scholar
Lüthi, B., Holzner, M. & Tsinober, A. 2009 Expanding the Q–R space to three dimensions. J. Fluid Mech. 641, 497–507.CrossRefGoogle Scholar
Martín, J., Dopazo, C. & Valino, L. 1998 a Dynamics of velocity gradient invariants in turbulence: restricted Euler and linear diffusion models. Phys. Fluids 10 (8), 2012–2025.CrossRefGoogle Scholar
Martín, J., Ooi, A., Chong, M.S. & Soria, J. 1998 b Dynamics of the velocity gradient tensor invariants in isotropic turbulence. Phys. Fluids 10 (9), 2336–2346.CrossRefGoogle Scholar
Meneveau, C. 2011 Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows. Annu. Rev. Fluid Mech. 43 (1), 219–245.CrossRefGoogle Scholar
Ohkitani, K. 1993 Eigenvalue problems in three-dimensional Euler flows. Phys. Fluids A 5 (10), 2570–2572.CrossRefGoogle Scholar
Ohkitani, K. & Kishiba, S. 1995 Nonlocal nature of vortex stretching in an inviscid fluid. Phys. Fluids 7 (2), 411–421.CrossRefGoogle Scholar
Ooi, A., Martin, J., Soria, J. & Chong, M.S. 1999 A study of the evolution and characteristics of the invariants of the velocity-gradient tensor in isotropic turbulence. J. Fluid Mech. 381, 141–174.CrossRefGoogle Scholar
Pereira, R.M., Moriconi, L. & Chevillard, L. 2018 A multifractal model for the velocity gradient dynamics in turbulent flows. J. Fluid Mech. 839, 430–467.CrossRefGoogle Scholar
Pirozzoli, S., Bernardini, M. & Grasso, F. 2008 Characterization of coherent vortical structures in a supersonic turbulent boundary layer. J. Fluid Mech. 613, 205–231.CrossRefGoogle Scholar
Pirozzoli, S. & Grasso, F. 2004 Direct numerical simulations of isotropic compressible turbulence: influence of compressibility on dynamics and structures. Phys. Fluids 16 (12), 4386–4407.CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Sarkar, S. 1992 The pressure–dilatation correlation in compressible flows. Phys. Fluids A 4 (12), 2674–2682.CrossRefGoogle Scholar
She, Z.-S., Jackson, E. & Orszag, S.A. 1991 Structure and dynamics of homogeneous turbulence: models and simulations. Proc. R. Soc. Lond. 434 (1890), 101–124.Google Scholar
Suman, S. & Girimaji, S.S. 2009 Homogenized Euler equation: a model for compressible velocity gradient dynamics. J. Fluid Mech. 620, 177–194.CrossRefGoogle Scholar
Suman, S. & Girimaji, S.S. 2010 Velocity gradient invariants and local flow-field topology in compressible turbulence. J. Turbul. 11, N2.CrossRefGoogle Scholar
Suman, S. & Girimaji, S.S. 2011 Dynamical model for velocity-gradient evolution in compressible turbulence. J. Fluid Mech. 683, 289–319.CrossRefGoogle Scholar
Suman, S. & Girimaji, S.S. 2013 Velocity gradient dynamics in compressible turbulence: characterization of pressure-Hessian tensor. Phys. Fluids 25 (12), 125103.CrossRefGoogle Scholar
Tang, J., Zhao, Z., Wan, Z.-H. & Liu, N.-S. 2020 On the near-wall structures and statistics of fluctuating pressure in compressible turbulent channel flows. Phys. Fluids 32 (11), 115121.CrossRefGoogle Scholar
Tsinober, A. 2009 An Informal Conceptual Introduction to Turbulence. Springer.CrossRefGoogle Scholar
Tsinober, A., Kit, E. & Dracos, T. 1992 Experimental investigation of the field of velocity gradients in turbulent flows. J. Fluid Mech. 242, 169–192.CrossRefGoogle Scholar
Vieillefosse, P. 1982 Local interaction between vorticity and shear in a perfect incompressible fluid. J. Phys. 43 (6), 837–842.CrossRefGoogle Scholar
Vreman, A.W. & Kuerten, J.G.M. 2014 Statistics of spatial derivatives of velocity and pressure in turbulent channel flow. Phys. Fluids 26 (8), 085103.CrossRefGoogle Scholar
Wallace, J.M. 2009 Twenty years of experimental and direct numerical simulation access to the velocity gradient tensor: what have we learned about turbulence? Phys. Fluids 21 (2), 021301.CrossRefGoogle Scholar
Wallace, J.M. & Vukoslavčević, P.V. 2010 Measurement of the velocity gradient tensor in turbulent flows. Annu. Rev. Fluid Mech. 42 (1), 157–181.CrossRefGoogle Scholar
Wang, J., Wan, M., Chen, S., Xie, C., Zheng, Q., Wang, L.-P. & Chen, S. 2020 Effect of flow topology on the kinetic energy flux in compressible isotropic turbulence. J. Fluid Mech. 883, A11.CrossRefGoogle Scholar
Wang, L. & Lu, X.-Y. 2012 Flow topology in compressible turbulent boundary layer. J. Fluid Mech. 703, 255–278.CrossRefGoogle Scholar
Wilczek, M. & Meneveau, C. 2014 Pressure Hessian and viscous contributions to velocity gradient statistics based on Gaussian random fields. J. Fluid Mech. 756, 191–225.CrossRefGoogle Scholar
Yu, J.-L. & Lu, X.-Y. 2020 Subgrid effects on the filtered velocity gradient dynamics in compressible turbulence. J. Fluid Mech. 892, A24.CrossRefGoogle Scholar
Zamansky, R., Vinkovic, I. & Gorokhovski, M. 2011 Acceleration statistics of solid particles in turbulent channel flow. Phys. Fluids 23 (11), 113304.CrossRefGoogle Scholar