Shell-crossing in quasi-one-dimensional flow

Blow-up of solutions for the cosmological fluid equations, often dubbed shell-crossing or orbit crossing, denotes the breakdown of the single-stream regime of the cold-dark-matter fluid. At this instant, the velocity becomes multi-valued and the density singular. Shell-crossing is well understood in one dimension (1D), but not in higher dimensions. This paper is about quasi-one-dimensional (Q1D) flow that depends on all three coordinates but differs only slightly from a strictly 1D flow, thereby allowing a perturbative treatment of shell-crossing using the Euler--Poisson equations written in Lagrangian coordinates. The signature of shell-crossing is then just the vanishing of the Jacobian of the Lagrangian map, a regular perturbation problem. In essence the problem of the first shell-crossing, which is highly singular in Eulerian coordinates, has been desingularized by switching to Lagrangian coordinates, and can then be handled by perturbation theory. Here, all-order recursion relations are obtained for the time-Taylor coefficients of the displacement field, and it is shown that the Taylor series has an infinite radius of convergence. This allows the determination of the time and location of the first shell-crossing, which is generically shown to be taking place earlier than for the unperturbed 1D flow. The time variable used for these statements is not the cosmic time $t$ but the linear growth time $\tau \sim t^{2/3}$. For simplicity, calculations are restricted to an Einstein--de Sitter universe in the Newtonian approximation, and tailored initial data are used. However it is straightforward to relax these limitations, if needed.Comment: 9 pages; received 2017 May 24, and accepted 2017 June 21 at MNRA

A constructive approach to regularity of Lagrangian trajectories for incompressible Euler flow in a bounded domain

The 3D incompressible Euler equation is an important research topic in the mathematical study of fluid dynamics. Not only is the global regularity for smooth initial data an open issue, but the behaviour may also depend on the presence or absence of boundaries. For a good understanding, it is crucial to carry out, besides mathematical studies, high-accuracy and well-resolved numerical exploration. Such studies can be very demanding in computational resources, but recently it has been shown that very substantial gains can be achieved first, by using Cauchy's Lagrangian formulation of the Euler equations and second, by taking advantages of analyticity results of the Lagrangian trajectories for flows whose initial vorticity is H\"older-continuous. The latter has been known for about twenty years (Serfati, 1995), but the combination of the two, which makes use of recursion relations among time-Taylor coefficients to obtain constructively the time-Taylor series of the Lagrangian map, has been achieved only recently (Frisch and Zheligovsky, 2014; Podvigina {\em et al.}, 2016 and references therein). Here we extend this methodology to incompressible Euler flow in an impermeable bounded domain whose boundary may be either analytic or have a regularity between indefinite differentiability and analyticity. Non-constructive regularity results for these cases have already been obtained by Glass {\em et al.} (2012). Using the invariance of the boundary under the Lagrangian flow, we establish novel recursion relations that include contributions from the boundary. This leads to a constructive proof of time-analyticity of the Lagrangian trajectories with analytic boundaries, which can then be used subsequently for the design of a very high-order Cauchy--Lagrangian method.Comment: 18 pages, no figure

Cauchy's almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow

Two prized papers, one by Augustin Cauchy in 1815, presented to the French Academy and the other by Hermann Hankel in 1861, presented to G\"ottingen University, contain major discoveries on vorticity dynamics whose impact is now quickly increasing. Cauchy found a Lagrangian formulation of 3D ideal incompressible flow in terms of three invariants that generalize to three dimensions the now well-known law of conservation of vorticity along fluid particle trajectories for two-dimensional flow. This has very recently been used to prove analyticity in time of fluid particle trajectories for 3D incompressible Euler flow and can be extended to compressible flow, in particular to cosmological dark matter. Hankel showed that Cauchy's formulation gives a very simple Lagrangian derivation of the Helmholtz vorticity-flux invariants and, in the middle of the proof, derived an intermediate result which is the conservation of the circulation of the velocity around a closed contour moving with the fluid. This circulation theorem was to be rediscovered independently by William Thomson (Kelvin) in 1869. Cauchy's invariants were only occasionally cited in the 19th century --- besides Hankel, foremost by George Stokes and Maurice L\'evy --- and even less so in the 20th until they were rediscovered via Emmy Noether's theorem in the late 1960, but reattributed to Cauchy only at the end of the 20th century by Russian scientists.Comment: 23 pages, 6 figures, EPJ H (history), in pres

How smooth are particle trajectories in a Λ\LambdaCDM Universe?

It is shown here that in a flat, cold dark matter (CDM) dominated Universe with positive cosmological constant ($\Lambda$), modelled in terms of a Newtonian and collisionless fluid, particle trajectories are analytical in time (representable by a convergent Taylor series) until at least a finite time after decoupling. The time variable used for this statement is the cosmic scale factor, i.e., the "$a$-time", and not the cosmic time. For this, a Lagrangian-coordinates formulation of the Euler-Poisson equations is employed, originally used by Cauchy for 3-D incompressible flow. Temporal analyticity for $\Lambda$CDM is found to be a consequence of novel explicit all-order recursion relations for the $a$-time Taylor coefficients of the Lagrangian displacement field, from which we derive the convergence of the $a$-time Taylor series. A lower bound for the $a$-time where analyticity is guaranteed and shell-crossing is ruled out is obtained, whose value depends only on $\Lambda$ and on the initial spatial smoothness of the density field. The largest time interval is achieved when $\Lambda$ vanishes, i.e., for an Einstein-de Sitter universe. Analyticity holds also if, instead of the $a$-time, one uses the linear structure growth $D$-time, but no simple recursion relations are then obtained. The analyticity result also holds when a curvature term is included in the Friedmann equation for the background, but inclusion of a radiation term arising from the primordial era spoils analyticity.Comment: 16 pages, 4 figures, published in MNRAS, this paper introduces a convergent formulation of Lagrangian perturbation theory for LCD

Genesis of d'Alembert's paradox and analytical elaboration of the drag problem

We show that the issue of the drag exerted by an incompressible fluid on a body in uniform motion has played a major role in the early development of fluid dynamics. In 1745 Euler came close, technically, to proving the vanishing of the drag for a body of arbitrary shape; for this he exploited and significantly extended existing ideas on decomposing the flow into thin fillets; he did not however have a correct picture of the global structure of the flow around a body. Borda in 1766 showed that the principle of live forces implied the vanishing of the drag and should thus be inapplicable to the problem. After having at first refused the possibility of a vanishing drag, d'Alembert in 1768 established the paradox, but only for bodies with a head-tail symmetry. A full understanding of the paradox, as due to the neglect of viscous forces, had to wait until the work of Saint-Venant in 1846.Comment: 10 pages, 4 figures, Physica D, in pres

Extended Self Similarity works for the Burgers equation and why

Extended Self-Similarity (ESS), a procedure that remarkably extends the range of scaling for structure functions in Navier--Stokes turbulence and thus allows improved determination of intermittency exponents, has never been fully explained. We show that ESS applies to Burgers turbulence at high Reynolds numbers and we give the theoretical explanation of the numerically observed improved scaling at both the infrared and ultraviolet end, in total a gain of about three quarters of a decade: there is a reduction of subdominant contributions to scaling when going from the standard structure function representation to the ESS representation. We conjecture that a similar situation holds for three-dimensional incompressible turbulence and suggest ways of capturing subdominant contributions to scaling.Comment: 10 pages, 1 figure, submitted to J. Fluid Mech. (fasttrack

Nelkin scaling for the Burgers equation and the role of high-precision calculations

Nelkin scaling, the scaling of moments of velocity gradients in terms of the Reynolds number, is an alternative way of obtaining inertial-range information. It is shown numerically and theoretically for the Burgers equation that this procedure works already for Reynolds numbers of the order of 100 (or even lower when combined with a suitable extended self-similarity technique). At moderate Reynolds numbers, for the accurate determination of scaling exponents, it is crucial to use higher than double precision. Similar issues are likely to arise for three-dimensional Navier--Stokes simulations.Comment: 5 pages, 2 figures, Published in Phys. Rev E (Rapid Communications