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Going to new lengths: Studying the Navier--Stokes-$\alpha\beta$ equations using the strained spiral vortex model
1. | Civil Infrastructure and Environmental Engineering, Khalifa University of Science, Technology and Research, Abu Dhabi, 127788, United Arab Emirates |
2. | Department of Mechanical Engineering, University of Houston, Houston, Texas, TX77204-4006, United States |
3. | Department of Civil and Environmental Engineering, Duke University, Durham, NC 27708, United States |
4. | Mathematical Soft Matter Unit, Okinawa Institute of Science and Technology, Okinawal, 904-0495, Japan |
References:
[1] |
S. Chen, C. Foias, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341.
doi: 10.1103/PhysRevLett.81.5338. |
[2] |
S. Chen, C. Foias, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Physica D, 133 (1999), 49-65.
doi: 10.1016/S0167-2789(99)00098-6. |
[3] |
S. Chen, C. Foias, E. Olson, E. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids, 11 (1999), 2343-2353.
doi: 10.1063/1.870096. |
[4] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 80 (1998), 4173-4176. |
[5] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.
doi: 10.1006/aima.1998.1721. |
[6] |
S. Chen, D. D. Holm, L. G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model, Physica D, 133 (1999), 66-83.
doi: 10.1016/S0167-2789(99)00099-8. |
[7] |
C. Foias, D. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dyn. Diff. Eq., 14 (2002), 1-35.
doi: 10.1023/A:1012984210582. |
[8] |
E. Fried and M. E. Gurtin, Turbulence kinetic energy and a possible hierarchy of length scales in a generalization of the Navier-Stokes-$\alpha$ theory, Phys. Rev. E, 75 (2007), 056306, 10pp.
doi: 10.1103/PhysRevE.75.056306. |
[9] |
E. Fried and M. E. Gurtin, Tractions, balances, and boundary conditions for nonsimple materials with application to liquid ow at small length scales, Arch. Rational Mech. Anal., 182 (2006), 513-554.
doi: 10.1007/s00205-006-0015-7. |
[10] |
T.-Y. Kim, M. Cassiani, J. D. Albertson, J. E. Dolbow, E. Fried and M. E. Gurtin, Impact of the inherent separation of scales in the Navier-Stokes-$\alpha\beta$ equations, Phys. Rev. E., 79 (2009), 045307, 4pp.
doi: 10.1103/PhysRevE.79.045307. |
[11] |
T.-Y. Kim, M. Neda, L. G. Rebholz and E. Fried, A numerical study of the Navier-Stokes-$\alpha\beta$ model, Comp. Meth. Appl. Mech. Eng., 200 (2011), 2891-2902.
doi: 10.1016/j.cma.2011.05.011. |
[12] |
T. S. Lundgren, Strained spiral vortex model for turbulent fine structure, Phys. Fluids, 25 (1982), 2193-2203.
doi: 10.1063/1.863957. |
[13] |
A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Dokl. Akad. Nauk SSSR, 30 (1941), 301-305. |
[14] |
T. S. Lundgren, A small-scale turbulence model, Phys. Fluids A, 5 (1993), 1472-1483.
doi: 10.1063/1.858585. |
[15] |
D. I. Pullin and P. G. Saffman, On the Lungren-Townsend model of a turbulent fine scales, Phys. Fluids A, 5 (1993), 126-145.
doi: 10.1063/1.858798. |
[16] |
D. I. Pullin, J. D. Buntine and P. G. Saffman, On the spectrum of a stretched spiral vortex, Phys. Fluids, 6 (1994), 3010-3027.
doi: 10.1063/1.868127. |
[17] |
X. Chen and E. Fried, The influence of the dispersive and dissipative scales $\alpha$ and $\beta$ on the energy spectrum of the Navier-Stokes-$\alpha\beta$ model for turbulent flow, Phys. Rev. E, 78 (2008), 046317, 10pp.
doi: 10.1103/PhysRevE.78.046317. |
[18] |
C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Physica D, 152 (2001), 505-519.
doi: 10.1016/S0167-2789(01)00191-9. |
show all references
References:
[1] |
S. Chen, C. Foias, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341.
doi: 10.1103/PhysRevLett.81.5338. |
[2] |
S. Chen, C. Foias, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Physica D, 133 (1999), 49-65.
doi: 10.1016/S0167-2789(99)00098-6. |
[3] |
S. Chen, C. Foias, E. Olson, E. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids, 11 (1999), 2343-2353.
doi: 10.1063/1.870096. |
[4] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 80 (1998), 4173-4176. |
[5] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.
doi: 10.1006/aima.1998.1721. |
[6] |
S. Chen, D. D. Holm, L. G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model, Physica D, 133 (1999), 66-83.
doi: 10.1016/S0167-2789(99)00099-8. |
[7] |
C. Foias, D. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dyn. Diff. Eq., 14 (2002), 1-35.
doi: 10.1023/A:1012984210582. |
[8] |
E. Fried and M. E. Gurtin, Turbulence kinetic energy and a possible hierarchy of length scales in a generalization of the Navier-Stokes-$\alpha$ theory, Phys. Rev. E, 75 (2007), 056306, 10pp.
doi: 10.1103/PhysRevE.75.056306. |
[9] |
E. Fried and M. E. Gurtin, Tractions, balances, and boundary conditions for nonsimple materials with application to liquid ow at small length scales, Arch. Rational Mech. Anal., 182 (2006), 513-554.
doi: 10.1007/s00205-006-0015-7. |
[10] |
T.-Y. Kim, M. Cassiani, J. D. Albertson, J. E. Dolbow, E. Fried and M. E. Gurtin, Impact of the inherent separation of scales in the Navier-Stokes-$\alpha\beta$ equations, Phys. Rev. E., 79 (2009), 045307, 4pp.
doi: 10.1103/PhysRevE.79.045307. |
[11] |
T.-Y. Kim, M. Neda, L. G. Rebholz and E. Fried, A numerical study of the Navier-Stokes-$\alpha\beta$ model, Comp. Meth. Appl. Mech. Eng., 200 (2011), 2891-2902.
doi: 10.1016/j.cma.2011.05.011. |
[12] |
T. S. Lundgren, Strained spiral vortex model for turbulent fine structure, Phys. Fluids, 25 (1982), 2193-2203.
doi: 10.1063/1.863957. |
[13] |
A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Dokl. Akad. Nauk SSSR, 30 (1941), 301-305. |
[14] |
T. S. Lundgren, A small-scale turbulence model, Phys. Fluids A, 5 (1993), 1472-1483.
doi: 10.1063/1.858585. |
[15] |
D. I. Pullin and P. G. Saffman, On the Lungren-Townsend model of a turbulent fine scales, Phys. Fluids A, 5 (1993), 126-145.
doi: 10.1063/1.858798. |
[16] |
D. I. Pullin, J. D. Buntine and P. G. Saffman, On the spectrum of a stretched spiral vortex, Phys. Fluids, 6 (1994), 3010-3027.
doi: 10.1063/1.868127. |
[17] |
X. Chen and E. Fried, The influence of the dispersive and dissipative scales $\alpha$ and $\beta$ on the energy spectrum of the Navier-Stokes-$\alpha\beta$ model for turbulent flow, Phys. Rev. E, 78 (2008), 046317, 10pp.
doi: 10.1103/PhysRevE.78.046317. |
[18] |
C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Physica D, 152 (2001), 505-519.
doi: 10.1016/S0167-2789(01)00191-9. |
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