Note on the stability of viscous roll waves

Blake Barker; Mathew A. Johnson; Pascal Noble; Luis Miguel Rodrigues; Kevin Zumbrun

doi:10.1016/j.crme.2016.11.001

Note on the stability of viscous roll waves
[Note sur la stabilité des roll waves visqueuses]

Blake Barker ¹ ; Mathew A. Johnson ² ; Pascal Noble ³ ; Luis Miguel Rodrigues ⁴ ; Kevin Zumbrun ⁵

¹ Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
² Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA
³ Institut de mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, INSA, 31077 Toulouse, France
⁴ IRMAR – UMR CNRS 6625, Université de Rennes-1, 35042 Rennes, France
⁵ Department of Mathematics, Indiana University Bloomington, IN 47405, USA

Comptes Rendus. Mécanique, Volume 345 (2017) no. 2, pp. 125-129.

Résumé
Abstract

Les roll waves sont des ondes progressives périodiques hydrodynamiques, modélisées comme des solutions des équations de Saint-Venant. Dans cette note, nous annonçons une classification complète des roll waves stables de leur apparition à F (le nombre de Froude) proche de 2 à $F \to \infty$ . Pour les nombres de Froude intermédiaires, nous avons mené une étude numérique des critères de stabilité spectrale. Dans le régime asymptotique $F \to 2$ , nous donnons une expression analytique des limites de stabilité, alors que pour $F \to \infty$ , nous montrons que les roll waves sont toujours instables.

In this note, we announce a complete classification of the stability of periodic roll-wave solutions of the viscous shallow water equations, from their onset at Froude number $F \approx 2$ up to the infinite Froude limit. For intermediate Froude numbers, we obtain numerically a particularly simple power-law relation between F and the boundaries of the region of stable periods, which appears potentially useful in hydraulic engineering applications. In the asymptotic regime $F \to 2$ (onset), we provide an analytic expression of the stability boundaries, whereas in the limit $F \to \infty$ , we show that roll waves are always unstable.

Reçu le : 2016-06-07
Accepté le : 2016-11-07
Publié le : 2016-12-03

DOI : 10.1016/j.crme.2016.11.001

Keywords: Fluid mechanics, Shallow water flows, Roll waves
Mot clés : Mécanique des fluides, Écoulements peu profonds, Roll waves

Affiliations des auteurs :

Blake Barker ¹ ; Mathew A. Johnson ² ; Pascal Noble ³ ; Luis Miguel Rodrigues ⁴ ; Kevin Zumbrun ⁵

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     author = {Blake Barker and Mathew A. Johnson and Pascal Noble and Luis Miguel Rodrigues and Kevin Zumbrun},
     title = {Note on the stability of viscous roll waves},
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     pages = {125--129},
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     doi = {10.1016/j.crme.2016.11.001},
     language = {en},
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Blake Barker; Mathew A. Johnson; Pascal Noble; Luis Miguel Rodrigues; Kevin Zumbrun. Note on the stability of viscous roll waves. Comptes Rendus. Mécanique, Volume 345 (2017) no. 2, pp. 125-129. doi : 10.1016/j.crme.2016.11.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2016.11.001/

Bibliographie
Cité par

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[2] M.A. Johnson; P. Noble; L.M. Rodrigues; K. Zumbrun Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations, Invent. Math., Volume 197 (2014) no. 1, pp. 115-213

[3] L.M. Rodrigues; K. Zumbrun Periodic-coefficient damping estimates, and stability of large-amplitude roll waves in inclined thin film flow, SIAM J. Math. Anal., Volume 48 (2016) no. 1, pp. 268-280

[4] R.R. Brock Development of roll-wave trains in open channels, J. Hydraul. Div., Volume 95 (1969) no. HY4, pp. 1401-1427

[5] M.A. Johnson; P. Noble; K. Zumbrun Nonlinear stability of viscous roll waves, SIAM J. Math. Anal., Volume 43 (2011) no. 2, pp. 557-611

[6] A. Boudlal; V.Y. Liapidevskii Stabilité de trains d'ondes dans un canal découvert, C. R. Mecanique, Volume 330 (2002), pp. 291-295

[7] S. Benzoni-Gavage; C. Mietka; L.M. Rodrigues Co-periodic stability of periodic waves in some Hamiltonian PDEs, Nonlinearity, Volume 29 (2016) no. 11, pp. 3241-3308

[8] G.L. Richard; S. Gavrilyuk A new model of roll waves: comparison with Brock's experiments, J. Fluid Mech., Volume 698 (2012), pp. 374-405

[9] B. Barker; J. Humpherys; J. Lytle; K. Zumbrun STABLAB: a MATLAB-based numerical library for Evans function computation, 2009 https://github.com/nonlinear-waves/stablab.git (available at)

[10] J. Yu; J. Kevorkian; R. Haberman Weak nonlinear waves in channel flow with internal dissipation, Stud. Appl. Math., Volume 105 (2000), p. 143

[11] M.A. Johnson; P. Noble; L.M. Rodrigues; K. Zumbrun Spectral stability of periodic wave trains of the Korteweg–de Vries/Kuramoto–Sivashinsky equation in the Korteweg–de Vries limit, Trans. Amer. Math. Soc., Volume 367 (2015) no. 3, pp. 2159-2212

[12] B. Barker Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flow, J. Differ. Equ., Volume 257 (2014) no. 8, pp. 2950-2983

[13] C. Kranenburg On the evolution of roll waves, J. Fluid Mech., Volume 245 (1992), pp. 249-261

[14] N.J. Balmforth; S. Mandre Dynamics of roll waves, J. Fluid Mech., Volume 514 (2004), pp. 1-33

[15] J. Liu; J.B. Schneider; J.P. Gollub Three dimensional instabilities of film flows, Phys. Fluids A, Volume 7 (1995), pp. 55-67

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