Skip to content

Research at St Andrews

Energy dissipation and resolution of steep gradients in one-dimensional Burgers flows

Research output: Contribution to journalArticle

DOI

Open Access permissions

Open

Abstract

Traveling-wave solutions of the inviscid Burgers equation having smooth initial wave profiles of suitable shapes are known to develop shocks (infinite gradients) in finite times. Such singular solutions are characterized by energy spectra that scale with the wave number k as k−2. In the presence of viscosity ν>0, no shocks can develop, and smooth solutions remain so for all times t>0, eventually decaying to zero as t→∞. At peak energy dissipation, say t = t∗, the spectrum of such a smooth solution extends to a finite dissipation wave number kν and falls off more rapidly, presumably exponentially, for k>kν. The number N of Fourier modes within the so-called inertial range is proportional to kν. This represents the number of modes necessary to resolve the dissipation scale and can be thought of as the system’s number of degrees of freedom. The peak energy dissipation rate ϵ remains positive and becomes independent of ν in the inviscid limit. In this study, we carry out an analysis which verifies the dynamical features described above and derive upper bounds for ϵ and N. It is found that ϵ satisfies ϵ ≤ ν2α−1‖u∗‖∞2(1−α)‖(−Δ)α/2u∗‖2, where α<1 and u∗ = u(x,t∗) is the velocity field at t = t∗. Given ϵ>0 in the limit ν→0, this implies that the energy spectrum remains no steeper than k−2 in that limit. For the critical k−2 scaling, the bound for ϵ reduces to ϵ ≤ k0‖u0‖∞‖u0‖2, where k0 marks the lower end of the inertial range and u0 = u(x,0). This implies N ≤ L‖u0‖∞/ν, where L is the domain size, which is shown to coincide with a rigorous estimate for the number of degrees of freedom defined in terms of local Lyapunov exponents. We demonstrate both analytically and numerically an instance, where the k−2 scaling is uniquely realizable. The numerics also return ϵ and t∗, consistent with analytic values derived from the corresponding limiting weak solution.
Close

Details

Original languageEnglish
Article number037102
Number of pages7
JournalPhysics of Fluids
Volume22
Issue number3
Early online date19 Mar 2010
DOIs
Publication statusPublished - Mar 2010

    Research areas

  • Viscosity, Vortex dynamics, Numerical solutions, Turbulent flows, Reynolds stress modeling

Discover related content
Find related publications, people, projects and more using interactive charts.

View graph of relations

Related by author

  1. Late time evolution of unforced inviscid two-dimensional turbulence

    Dritschel, D. G., Scott, R. K., Macaskill, C., Gottwald, G. & Tran, C. V., 2009, In : Journal of Fluid Mechanics. 640, p. 215-233 19 p.

    Research output: Contribution to journalArticle

  2. Unifying scaling theory for vortex dynamics in two-dimensional turbulence

    Dritschel, D. G., Scott, R. K., Macaskill, C., Gottwald, G. A. & Tran, C. V., 29 Aug 2008, In : Physical Review Letters. 101, 9, 4 p., 094501.

    Research output: Contribution to journalArticle

  3. Revisiting Batchelor's theory of two-dimensional turbulence

    Dritschel, D. G., Tran, C. V. & Scott, R. K., 25 Nov 2007, In : Journal of Fluid Mechanics. 591, p. 379-391 13 p.

    Research output: Contribution to journalArticle

  4. Large-scale dynamics in two-dimensional Euler and surface quasigeostrophic flows

    Tran, C. V. & Dritschel, D. G., Dec 2006, In : Physics of Fluids. 18, 12, p. 121703 3 p.

    Research output: Contribution to journalArticle

Related by journal

  1. Entrapping of a vortex pair interacting with a fixed point vortex revisited. I. Point vortices

    Koshel, K. V., Reinaud, J. N., Riccardi, G. & Ryzhov, E. A., 28 Sep 2018, In : Physics of Fluids. 30, 9, 096603.

    Research output: Contribution to journalArticle

  2. Entrapping of a vortex pair interacting with a fixed point vortex revisited. II. Finite size vortices and the effect of deformation

    Reinaud, J. N., Koshel, K. V. & Ryzhov, E. A., 28 Sep 2018, In : Physics of Fluids. 30, 9, 10 p., 096604.

    Research output: Contribution to journalArticle

  3. Hetonic quartets in a two-layer quasi-geostrophic flow: V-states and stability

    Reinaud, J. N., Sokolovskiy, M. & Carton, X., 11 May 2018, In : Physics of Fluids. 30, 21 p., 056602.

    Research output: Contribution to journalArticle

  4. Geostrophic tripolar vortices in a two-layer fluid: linear stability and nonlinear evolution of equilibria

    Reinaud, J. N., Sokolovskiy, M. & Carton, X., Mar 2017, In : Physics of Fluids. 29, 3, 16 p., 036601.

    Research output: Contribution to journalArticle

  5. Interaction between a surface quasi-geostrophic buoyancy anomaly jet and internal vortices

    Reinaud, J. N., Dritschel, D. G. & Carton, X., Aug 2017, In : Physics of Fluids. 29, 8, 16 p., 086603.

    Research output: Contribution to journalArticle

Related by journal

  1. Physics of Fluids (Journal)

    David Gerard Dritschel (Editor)
    2005 → …

    Activity: Publication peer-review and editorial work typesEditor of research journal

ID: 5009887