Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-16T04:06:16.510Z Has data issue: false hasContentIssue false
  • English
  • Français

On the topology of the eigenframe of the subgrid-scale stress tensor

Published online by Cambridge University Press:  07 June 2016

Zixuan Yang  and
Bing-Chen Wang
Zixuan Yang
Affiliation:
Department of Mechanical Engineering, University of Manitoba, Winnipeg, MB, R3T 5V6, Canada
Bing-Chen Wang*
Affiliation:
Department of Mechanical Engineering, University of Manitoba, Winnipeg, MB, R3T 5V6, Canada
*
†Email address for correspondence: BingChen.Wang@umanitoba.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, the geometrical properties of the subgrid-scale (SGS) stress tensor are investigated through its eigenvalues and eigenvectors. The concepts of Euler rotation angle and axis are utilized to investigate the relative rotation of the eigenframe of the SGS stress tensor with respect to that of the resolved strain rate tensor. Both Euler rotation angle and axis are natural invariants of the rotation matrix, which uniquely describe the topological relation between the eigenframes of these two tensors. Different from the reference frame fixed to a rigid body, the eigenframe of a tensor consists of three orthonormal eigenvectors, which by their nature are subjected to directional aliasing. In order to describe the geometric relationship between the SGS stress and resolved strain rate tensors, an effective method is proposed to uniquely determine the topology of the eigenframes. The proposed method has been used for testing three SGS stress models in the context of homogeneous isotropic turbulence at three Reynolds numbers, using both a priori and a posteriori approaches.

JFM classification

Turbulent Flows: Isotropic turbulence Turbulent Flows: Turbulence modelling Turbulent Flows: Turbulence simulation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 798 , 10 July 2016 , pp. 598 - 627
Check for updates
Copyright
© 2016 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

Angeles, J. 1988 Rational Kinematics. Springer.Google Scholar
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.Google Scholar
Battin, R. H. 1999 An Introduction to the Mathematics and Methods of Astrodynamics, rev. edn. Springer.Google Scholar
Betchov, R. 1956 An inequality concerning the production of vorticity in isotropic turbulence. J. Fluid Mech. 1, 497–504.Google Scholar
Blackburn, H. M., Mansour, N. N. & Cantwell, B. J. 1996 Topology of fine-scale motions in turbulent channel flow. J. Fluid Mech. 310, 269–292.Google Scholar
van der Bos, F., Tao, B., Meneveau, C. & Katz, J. 2002 Effects of small-scale turbulent motions on the filtered velocity gradient tensor as deduced from holographic particle image velocitmetry measuremets. Phys. Fluids 14 (7), 2456–2474.CrossRefGoogle Scholar
Carati, D., Ghosal, S. & Moin, P. 1995 On the representation of backscatter in dynamic localization models. Phys. Fluids 7 (3), 606–616.Google Scholar
Chen, Q., Otte, M. J., Sullivan, P. P. & Tong, C. 2009 A posteriori subgrid-scale model tests based on the conditional means of subgrid-scale stress and its production rate. J. Fluid Mech. 626, 149–181.Google Scholar
Chen, Q. & Tong, C. 2006 Investigation of the subgrid-scale stress and its production rate in a convective atmospheric boundary layer using measurement data. J. Fluid Mech. 547, 65–104.Google Scholar
Chen, Q., Zhang, H., Wang, D. & Tong, C. 2003 Subgrid-scale stress and its production rate: Conditions for the resolvable-scale velocity probability density function. J. Turbul. 4, N27.Google Scholar
Eswaran, V. & Pope, S. B. 1988 An examination of forcing in direct numerical simulations of turbulence. Comput. Fluids 16 (3), 257–278.CrossRefGoogle Scholar
Gotoh, T. & Fukayama, D. 2001 Pressure spectrum in homogeneous turbulence. Phys. Rev. Lett. 86 (17), 3775–3778.Google Scholar
Higgins, C. W., Parlange, M. B. & Meneveau, C. 2003 Alignment trends of velocity gradients and subgrid-scale fluxes in the turbulent atmospheric boundary layer. Boundary-Layer Meteorol. 109, 59–83.Google Scholar
Hughes, P. C. 1986 Spacecraft Attitude Dynamics. Wiley.Google Scholar
Ishihara, T., Gotoh, T. & Kaneda, Y. 2009 Study of high-Reynolds number isotropic turbulence by direct numerical simulation. Annu. Rev. Fluid Mech. 41, 165–180.Google Scholar
Jiménez, J., Wray, A. A., Saffman, P. G. & Rogallo, R. S. 1993 The structure of intense vorticity in isotropic turbulence. J. Fluid Mech. 255, 65–90.Google Scholar
Kaneda, Y., Ishihara, T., Yokokawa, M., Itakura, K. & Uno, A. 2003 Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box. Phys. Fluids 15 (2), L21–L24.Google Scholar
Kang, H. S. & Meneveau, C. 2005 Effect of large-scale coherent structures on subgrid-scale stress and strain-rate eigenvector alignments in turbulent shear flow. Phys. Fluids 17 (5), 055103.Google Scholar
Kerr, R. M. 1987 Histograms of helicity and strain in numerical turbulence. Phys. Rev. Lett. 59 (7), 783–786.Google Scholar
Kerr, R. M. 1990 Velocity, scalar and transfer spectra in numerical turbulence. J. Fluid Mech. 211, 309–332.Google Scholar
Lilly, D. K. 1966 On the Application of the Eddy Viscosity Concept in the Inertial Sub-Range of Turbulence. National Center for Atmospheric Research.Google Scholar
Lilly, D. K. 1967 The representation of small scale turbulence in numerical simulation experiments. In IBM Scientific Computing Symposium on Environmental Sciences, pp. 195–210.Google Scholar
Lilly, D. K. 1992 A proposed modification of the Germano subgrid scale closure method. Phys. Fluids A 4 (3), 633–635.Google Scholar
Liu, S., Meneveau, C. & Katz, J. 1994 On the properties of similarity subgrid-scale models as deduced from measurements in a turbulent jet. J. Fluid Mech. 275, 83–119.Google Scholar
Lund, T. & Rogers, M. 1994 An improved measure of strain state probability in turbulent flows. Phys. Fluids 6 (5), 1838–1847.Google Scholar
Misra, A. & Pullin, D. I. 1997 A vortex-based subgrid stress model for large-eddy simulation. Phys. Fluids 9 (8), 2443–2454.Google Scholar
Morinishi, Y. & Vasilyev, O. V. 2001 A recommended modification to the dynamic two-parameter mixed subgrid scale model for large eddy simulation of wall bounded turbulent flow. Phys. Fluids 13 (11), 3400–3410.Google Scholar
Murray, R. M., Li, Z. & Sastry, S. S. 1994 A Mathematical Introduction to Robotic Manipulation. CRC Press.Google Scholar
Park, N., Yoo, J. Y. & Choi, H. 2005 Toward improved consistency of a priori tests with a posteriori tests in large eddy simulation. Phys. Fluids 17 (1), 015103.Google Scholar
Perry, A. E. & Chong, M. S. 1987 A description of eddying motions and flow patterns using critical-point concepts. Annu. Rev. Fluid Mech. 19, 125–155.Google Scholar
Pullin, D. I. 2000 A vortex-based model for the subgrid flux of a passive scalar. Phys. Fluids 12 (9), 2311–2319.Google Scholar
Pullin, D. I. & Saffman, P. G. 1993 On the Lundgren–Townsend model of turbulent fine scales. Phys. Fluids A 5 (1), 126–145.Google Scholar
Pullin, D. I. & Saffman, P. G. 1994 Reynolds stresses and one-dimensional spectra for a vortex model of homogeneous anisotropic turbulence. Phys. Fluids 6 (5), 1787–1796.Google Scholar
Saffman, P. G. & Pullin, D. I. 1994 Anisotropy of the Lundgren–Townsend model of fine-scale turbulence. Phys. Fluids 6 (2), 802–807.Google Scholar
Saffman, P. G. & Pullin, D. I. 1996 Calculation of velocity structure functions for vortex models of isotropic turbulence. Phys. Fluids 8 (11), 3072–3084.Google Scholar
Sagaut, P. & Grohens, R. 1999 Discrete filters for large eddy simulation. Intl J. Numer. Meth. Fluids 31, 1195–1220.Google Scholar
Shuster, M. D. 1993 A survey of attitude representations. J. Astronaut. Sci. 41 (4), 439–517.Google Scholar
Smagorinsky, J. 1963 General circulation experiments with the primitive equations: I: the basic experiment. Mon. Weath. Rev. 91, 99–164.Google Scholar
Sreenivasan, K. R. 1995 On the universality of the Kolmogorov constant. Phys. Fluids 7 (11), 2778–2784.Google Scholar
Tao, B., Katz, J. & Meneveau, C. 2002 Statistical geometry of subgrid-scale stresses determined from holographic particle image velocimetry measurements. J. Fluid Mech. 457, 35–78.Google Scholar
Voelkl, T., Pullin, D. I. & Chan, D. C. 2000 A physical-space version of the stretched-vortex subgrid-stress model for large-eddy simulation. Phys. Fluids 12 (7), 1810–1825.Google Scholar
Wang, B.-C. & Bergstrom, D. J. 2005 A dynamic nonlinear subgrid-scale stress model. Phys. Fluids 17 (3), 035109.Google Scholar
Wang, B.-C., Bergstrom, D. J. & Yee, E. 2006a Turbulence topologies predicted using large eddy simulations. J. Turbul. 7, N34.Google Scholar
Wang, B.-C., Yee, E. & Bergstrom, D. J. 2006b Geometrical description of subgrid-scale stress tensor based on Euler axis/angle. AIAA J. 44, 1106–1110.Google Scholar
Yeung, P. K. & Zhou, Y. 1997 Universality of the Kolmogorov constant in numerical simulations of turbulence. Phys. Rev. E 56, 1746–1752.Google Scholar