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dc.contributor.authorTran, Chuong Van
dc.contributor.authorDritschel, David Gerard
dc.identifier.citationTran , C V & Dritschel , D G 2010 , ' Energy dissipation and resolution of steep gradients in one-dimensional Burgers flows ' , Physics of Fluids , vol. 22 , no. 3 , 037102 .
dc.identifier.otherPURE: 5009887
dc.identifier.otherPURE UUID: 0ce87a4a-e39f-4312-988f-99e3bba1f737
dc.identifier.otherScopus: 77953597029
dc.identifier.otherORCID: /0000-0002-1790-8280/work/61133271
dc.identifier.otherORCID: /0000-0001-6489-3395/work/64697752
dc.description.abstractTraveling-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.
dc.relation.ispartofPhysics of Fluidsen
dc.rightsCopyright 2010, American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in Physics of Fluids, Vol 22, Issue 3, and may be found at:
dc.subjectVortex dynamicsen
dc.subjectNumerical solutionsen
dc.subjectTurbulent flowsen
dc.subjectReynolds stress modelingen
dc.titleEnergy dissipation and resolution of steep gradients in one-dimensional Burgers flowsen
dc.typeJournal articleen
dc.description.versionPublisher PDFen
dc.contributor.institutionUniversity of St Andrews.School of Mathematics and Statisticsen
dc.contributor.institutionUniversity of St Andrews.Scottish Oceans Instituteen
dc.contributor.institutionUniversity of St Andrews.Applied Mathematicsen
dc.contributor.institutionUniversity of St Andrews.Marine Alliance for Science & Technology Scotlanden
dc.description.statusPeer revieweden

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