American Institute of Mathematical Sciences

September  2014, 19(7): 2207-2225. doi: 10.3934/dcdsb.2014.19.2207

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

Received  April 2013 Revised  January 2014 Published  August 2014

We study the effect of the length scales $\alpha$ and $\beta$ on the performance of the Navier--Stokes-$\alpha\beta$ equations for numerical simulations of turbulence over coarse discretizations. To this end, we rely on the strained spiral vortex model and take advantage of the dimensional reduction allowed by that model. In particular, the three-dimensional energy spectrum is reformulated so that it can be calculated from solutions of the two-dimensional unstrained Navier--Stokes-$\alpha\beta$ equations. A similarity theory for the spiral vortex model shows that the Navier--Stokes-$\alpha\beta$ model is better equipped than the Navier--Stokes-$\alpha$ model to capture smaller-scale behavior. Numerical experiments performed using a pseudo-spectral discretization along with the second-order Adams--Bashforth time-stepping algorithm yield results indicating that the fidelity of the energy spectrum in both the inertial and dissipation ranges is significantly improved for $\beta<\alpha$.
Citation: Tae-Yeon Kim, Xuemei Chen, John E. Dolbow, Eliot Fried. Going to new lengths: Studying the Navier--Stokes-$\alpha\beta$ equations using the strained spiral vortex model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2207-2225. doi: 10.3934/dcdsb.2014.19.2207
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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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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