# American Institute of Mathematical Sciences

August  2016, 36(8): 4349-4366. doi: 10.3934/dcds.2016.36.4349

## A thermodynamic study of the two-dimensional pressure-driven channel flow

 1 Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544 2 The Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, United States

Received  May 2015 Revised  November 2015 Published  March 2016

The instability of the two-dimensional Poiseuille flow in a long channel and the subsequent transition is studied using a thermodynamic approach. The idea is to view the transition process as an initial value problem with the initial condition being Poiseuille flow plus noise, which is considered as our ensemble. Using the mean energy of the velocity fluctuation and the skin friction coefficient as the macrostate variable, we analyze the transition process triggered by the initial noises with different amplitudes. A first order transition is observed at the critical Reynolds number $Re_* \sim 5772$ in the limit of zero noise. An action function, which relates the mean energy with the noise amplitude, is defined and computed. The action function depends only on the Reynolds number, and represents the cost for the noise to trigger a transition from the laminar flow. The correlation function of the spatial structure is analyzed.
Citation: Weinan E, Jianchun Wang. A thermodynamic study of the two-dimensional pressure-driven channel flow. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4349-4366. doi: 10.3934/dcds.2016.36.4349
##### References:
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Jimenez, Transition to turbulence in two-dimensional Poiseuille flow, J. Fluid Mech., 218 (1990), 265-297. [15] T. Kreilos and B. Eckhardt, Periodic orbits near onset of chaos in plane Couette flow, Chaos, 22 (2012), 047505, 8pp. doi: 10.1063/1.4757227. [16] T. Kreilos, B. Eckhardt and T. M. Schneider, Increasing Lifetimes and the Growing Saddles of Shear Flow Turbulence, Phys. Rev. Lett., 112 (2014), 044503. [17] P. Manneville, Spatiotemporal perspective on the decay of turbulence in wall-bounded flows, Phys Rev E, 79 (2009), 025301. [18] P. Manneville, On the growth of laminar-turbulent patterns in plane Couette flow, Fluid Dyn. Res., 44 (2012), 031412, 15pp. doi: 10.1088/0169-5983/44/3/031412. [19] F. Mellibovsky and B. Eckhardt, From travelling waves to mild chaos: A supercritical bifurcation cascade in pipe flow, J. Fluid Mech., 709 (2012), 149-190. doi: 10.1017/jfm.2012.326. [20] D. Moxey and D. Barkley, Distinct large-scale turbulent-laminar states in transitional pipe flow, Proc. Natl. Acad. Sci. USA, 107 (2010), 8091-8096. [21] M. Nagata, Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity, J. Fluid Mech., 217 (1990), 519-527. doi: 10.1017/S0022112090000829. [22] M. Nagata and K Deguchi, Mirror-symmetric exact coherent states in plane Poiseuille flow, J. Fluid Mech., 735 (2013), R4. [23] S. A. Orszag and A. T. Patera, Subcritical transition to turbulence in plane channel flows, Phys. Rev. Lett., 45 (1980), 989-993. [24] C. Pringle and R. R. Kerswell, Asymmetric, helical, and mirror-symmetric traveling waves in pipe flow, Phys. Rev. Lett., 99 (2007), 074502. [25] Y. Pomeau, Front motion, metastability and subcritical bifurcations in hydrodynamics, Physica D, 23 (1986), 3-11. [26] O. Reynolds, An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels, Philos. T. R. Soc. A, 174 (1883), 935-982. [27] B. L. Rozhdestvensky and I. N. Simakin, Secondary flows in a plane channel: their relationship and comparison with turbulent flows, J. Fluid. Mech., 147 (1984), 261-289. [28] T. M. Schneider, D. Marinc and B. Eckhardt, Localized edge states nucleate turbulence in extended plane Couette cells, J Fluid Mech, 646 (2010), 441-451. [29] J. Shen, Efficient spectral-Galerkin method I. Direct solvers of second-and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 1994), 1489-1505. doi: 10.1137/0915089. [30] L. Shi, M. Avila and B. Hof, Scale invariance at the onset of turbulence in couette flow, Phys. Rev. Lett., 110 (2013), 204502. [31] L. Trefethen, A. Trefethen, S. Reddy and T. Driscoll, Hydrodynamic stability without eigenvalues, Science, 261 (1993), 573-584. doi: 10.1126/science.261.5121.578. [32] L. Trefethen, Pseudospectra of linear operators, SIAM Rev., 39 (1997), 383-406. doi: 10.1137/S0036144595295284. [33] X. Wan, H. Yu and W. E, Model the nonlinear instability of wall-bounded shear flows as a rare event: A study on two-dimensional Poiseuille flow, Nonlinearity, 28 (2015), 1409-1440. doi: 10.1088/0951-7715/28/5/1409. [34] J. Wang, Q. Li and W. E, Study of the instability of the Poiseuille flow using a thermodynamic formalism, Proc. Natl. Acad. Sci. USA, 112 (2015), 9518-9523. [35] F. Waleffe, Three-dimensional coherent states in plane shear flows, Phys. Rev. Lett., 81 (1998), 4140-4143. [36] F. Waleffe, Exact coherent structures in channel flow, J. Fluid Mech., 435 (2001), 93-102. [37] H. Wedin and R. R. Kerswell, Exact coherent structures in pipe flow: Traveling wave solutions, J. Fluid Mech., 508 (2004), 333-371. doi: 10.1017/S0022112004009346. [38] A. P. Willis, P. Cvitanovi and M. Avila, Revealing the state space of turbulent pipe flow by symmetry reduction, J. Fluid Mech., 721 (2013), 514-540. doi: 10.1017/jfm.2013.75. [39] S. Zammert and B. Eckhardt, Streamwise and doubly-localised periodic orbits in plane Poiseuille flow, J. Fluid Mech., 761 (2014), 348-359. [40] S. Zammert and B. Eckhardt, Periodically bursting edge states in plane Poiseuille flow, Fluid Dyn. Res., 46 (2014), 041419, 13pp. doi: 10.1088/0169-5983/46/4/041419.

show all references

##### References:
 [1] K. T. Allhoff and B. Eckhardt, Directed percolation model for turbulence transition in shear flows, Fluid Dyn. Res., 44 (2012), 031201. [2] K. Avila, D. Moxey, A. de Lozar, M. Avila, D. Barkley and B. Hof, The onset of turbulence in pipe flow, Science, 333 (2011), 192-196. [3] M. Avila, F. Mellibovsky, N. Roland and B. Hof, Streamwise-localized solutions at the onset of turbulence in pipe Flow, Phys. Rev. Lett., 110 (2013), 224502. [4] D. Barkley, Simplifying the complexity of pipe flow, Phys Rev E, 84 (2011), 016309. [5] D. Barkley, Modeling the transition to turbulence in shear flows, J. Phys.: Conf. Ser., 318 (2011), 032001. [6] P. S. Casasa and À. Jorbab, Hopf bifurcations to quasi-periodic solutions for the two-dimensional plane Poiseuille flow, Comm. Nonlinear. Sci. Numer. Simulat., 17 (2012), 2864-2882. doi: 10.1016/j.cnsns.2011.11.008. [7] M. Chantry, A. P. Willis and R. R. Kerswell, Genesis of streamwise-localized solutions from globally periodic traveling waves in pipe flow, Phys. Rev. Lett., 112 (2014), 164501. [8] B. Eckhardt, T. M. Schneider, B. Hof and J. Westerweel, Turbulence transition in pipe flow, Annu. Rev. Fluid Mech., 39 (2007), 447-468. doi: 10.1146/annurev.fluid.39.050905.110308. [9] H. Faisst and B. Eckhardt, Traveling waves in pipe flow, Phys. Rev. Lett., 91 (2003), 224502. [10] M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, 2nd edn (New York: Springer), 1998. doi: 10.1007/978-1-4612-0611-8. [11] T. Herbert, Secondary instability of boundary layers, Annu. Rev. Fluid Mech., 20 (1988), 487-526. [12] B. Hof, J. Westerweel, T. M. Schneider and B. Eckhardt, Finite lifetime of turbulence in shear flows, Nature, 443 (2006), 59-62. [13] J. Jimenez, Bifurcations and bursting in two-dimensional Poiseuille flow, Phys. Fluids, 30 (1987), 3644-3646. [14] J. Jimenez, Transition to turbulence in two-dimensional Poiseuille flow, J. Fluid Mech., 218 (1990), 265-297. [15] T. Kreilos and B. Eckhardt, Periodic orbits near onset of chaos in plane Couette flow, Chaos, 22 (2012), 047505, 8pp. doi: 10.1063/1.4757227. [16] T. Kreilos, B. Eckhardt and T. M. Schneider, Increasing Lifetimes and the Growing Saddles of Shear Flow Turbulence, Phys. Rev. Lett., 112 (2014), 044503. [17] P. Manneville, Spatiotemporal perspective on the decay of turbulence in wall-bounded flows, Phys Rev E, 79 (2009), 025301. [18] P. Manneville, On the growth of laminar-turbulent patterns in plane Couette flow, Fluid Dyn. Res., 44 (2012), 031412, 15pp. doi: 10.1088/0169-5983/44/3/031412. [19] F. Mellibovsky and B. Eckhardt, From travelling waves to mild chaos: A supercritical bifurcation cascade in pipe flow, J. Fluid Mech., 709 (2012), 149-190. doi: 10.1017/jfm.2012.326. [20] D. Moxey and D. Barkley, Distinct large-scale turbulent-laminar states in transitional pipe flow, Proc. Natl. Acad. Sci. USA, 107 (2010), 8091-8096. [21] M. Nagata, Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity, J. Fluid Mech., 217 (1990), 519-527. doi: 10.1017/S0022112090000829. [22] M. Nagata and K Deguchi, Mirror-symmetric exact coherent states in plane Poiseuille flow, J. Fluid Mech., 735 (2013), R4. [23] S. A. Orszag and A. T. Patera, Subcritical transition to turbulence in plane channel flows, Phys. Rev. Lett., 45 (1980), 989-993. [24] C. Pringle and R. R. Kerswell, Asymmetric, helical, and mirror-symmetric traveling waves in pipe flow, Phys. Rev. Lett., 99 (2007), 074502. [25] Y. Pomeau, Front motion, metastability and subcritical bifurcations in hydrodynamics, Physica D, 23 (1986), 3-11. [26] O. Reynolds, An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels, Philos. T. R. Soc. A, 174 (1883), 935-982. [27] B. L. Rozhdestvensky and I. N. Simakin, Secondary flows in a plane channel: their relationship and comparison with turbulent flows, J. Fluid. Mech., 147 (1984), 261-289. [28] T. M. Schneider, D. Marinc and B. Eckhardt, Localized edge states nucleate turbulence in extended plane Couette cells, J Fluid Mech, 646 (2010), 441-451. [29] J. Shen, Efficient spectral-Galerkin method I. Direct solvers of second-and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 1994), 1489-1505. doi: 10.1137/0915089. [30] L. Shi, M. Avila and B. Hof, Scale invariance at the onset of turbulence in couette flow, Phys. Rev. Lett., 110 (2013), 204502. [31] L. Trefethen, A. Trefethen, S. Reddy and T. Driscoll, Hydrodynamic stability without eigenvalues, Science, 261 (1993), 573-584. doi: 10.1126/science.261.5121.578. [32] L. Trefethen, Pseudospectra of linear operators, SIAM Rev., 39 (1997), 383-406. doi: 10.1137/S0036144595295284. [33] X. Wan, H. Yu and W. E, Model the nonlinear instability of wall-bounded shear flows as a rare event: A study on two-dimensional Poiseuille flow, Nonlinearity, 28 (2015), 1409-1440. doi: 10.1088/0951-7715/28/5/1409. [34] J. Wang, Q. Li and W. E, Study of the instability of the Poiseuille flow using a thermodynamic formalism, Proc. Natl. Acad. Sci. USA, 112 (2015), 9518-9523. [35] F. Waleffe, Three-dimensional coherent states in plane shear flows, Phys. Rev. Lett., 81 (1998), 4140-4143. [36] F. Waleffe, Exact coherent structures in channel flow, J. Fluid Mech., 435 (2001), 93-102. [37] H. Wedin and R. R. Kerswell, Exact coherent structures in pipe flow: Traveling wave solutions, J. Fluid Mech., 508 (2004), 333-371. doi: 10.1017/S0022112004009346. [38] A. P. Willis, P. Cvitanovi and M. Avila, Revealing the state space of turbulent pipe flow by symmetry reduction, J. Fluid Mech., 721 (2013), 514-540. doi: 10.1017/jfm.2013.75. [39] S. Zammert and B. Eckhardt, Streamwise and doubly-localised periodic orbits in plane Poiseuille flow, J. Fluid Mech., 761 (2014), 348-359. [40] S. Zammert and B. Eckhardt, Periodically bursting edge states in plane Poiseuille flow, Fluid Dyn. Res., 46 (2014), 041419, 13pp. doi: 10.1088/0169-5983/46/4/041419.
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