Roll waves in an annular channel
The goal of this study is to analytically construct periodic wind perturbations in an annular channel that are numerically produced by a regularized model. R. Dressler’s technique in the shallow water approximation is used to prove the nonexistence of smooth periodic solutions, and discontinuous solutions related to roll waves on inclined surfaces are constructed. The constraints on the accelerating and dissipating forces are obtained under which periodic solutions can exist. A numerical analysis of the problem is carried out, and a qualitative comparison of the numerical and theoretical results is presented.
Keywordsannular channel shallow water equations solitary waves regularized model existence of periodic solutions
Unable to display preview. Download preview PDF.
- 6.K. O. Friedrichs, “On the derivation of the shallow water theory,” Commun. Appl. Math. Inst. Math. Mech. 1, 81–85 (1948).Google Scholar
- 8.B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics (Nauka, Moscow, 1968; Am. Math. Soc., Providence, 1983).Google Scholar
- 9.H. A. Thomas, “The propagation of waves in steep prismatic conduits,” Iowa Univ. Hydraulic Conf. Proc. Eng. Studies Bull. 20, 214–229 (1940).Google Scholar
- 10.R. F. Dressler, “Stability of uniform flow and roll-wave formation,” Proceedings of the NBS Semicentennial Symposium on Gravity Waves, NBS Circular (Department of Commerce, USA, 1952).Google Scholar
- 14.V. A. Il’in and E. G. Poznyak, Fundamentals of Calculus (Fizmatlit, Moscow, 2005), Part 2.Google Scholar
- 17.T. G. Elizarova, A. A. Zlotnik, and O. V. Nikitina, Preprint Nos. 33, 36, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2011).Google Scholar
- 20.N. N. Kalitkin, Numerical Methods (Nauka, Moscow, 1978).Google Scholar