16 research outputs found
The role of BKM-type theorems in Euler, Navier-Stokes and Cahn-Hilliard-Navier-Stokes analysis
The Beale-Kato-Majda theorem contains a single criterion that controls the
behaviour of solutions of the incompressible Euler equations. Versions of
this theorem are discussed in terms of the regularity issues surrounding the
incompressible Euler and Navier-Stokes equations together with a
phase-field model for the statistical mechanics of binary mixtures called the
Cahn-Hilliard-Navier-Stokes (CHNS) equations. A theorem of BKM-type is
established for the CHNS equations for the full parameter range. Moreover, for
this latter set, it is shown that there exists a Reynolds number and a bound on
the energy-dissipation rate that, remarkably, reproduces the upper
bound on the inverse Kolmogorov length normally associated with the
Navier-Stokes equations alone. An alternative length-scale is introduced and
discussed, together with a set of pseudo-spectral computations on a
grid.Comment: 3 figures and 3 table
Preferential Sampling of Elastic Chains in Turbulent Flows
A string of tracers, interacting elastically, in a turbulent flow is shown to
have a dramatically different behaviour when compared to the non-interacting
case. In particular, such an elastic chain shows strong preferential sampling
of the turbulent flow unlike the usual tracer limit: an elastic chain is
trapped in the vortical regions and not the straining ones. The degree of
preferential sampling and its dependence on the elasticity of the chain is
quantified via the Okubo-Weiss parameter. The effect of modifying the
deformability of the chain, via the number of links that form it, is also
examined.Comment: 6 pages, 4 figure
A regularity criterion for solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations and associated computations
We consider the 3D Cahn-Hilliard equations coupled to, and driven by, the
forced, incompressible 3D Navier-Stokes equations. The combination, known as
the Cahn-Hilliard-Navier-Stokes (CHNS) equations, is used in statistical
mechanics to model the motion of a binary fluid. The potential development of
singularities (blow-up) in the contours of the order parameter is an
open problem. To address this we have proved a theorem that closely mimics the
Beale-Kato-Majda theorem for the incompressible Euler equations [Beale et
al. Commun. Math. Phys., Commun. Math. Phys., , ]. By taking an norm of the energy of the full binary
system, designated as , we have shown that
governs the regularity of solutions of
the full 3D system. Our direct numerical simulations (DNSs), of the 3D CHNS
equations, for (a) a gravity-driven Rayleigh Taylor instability and (b) a
constant-energy-injection forcing, with to collocation points
and over the duration of our DNSs, confirm that remains bounded as
far as our computations allow.Comment: 11 pages, 3 figure
Barotropic tides in MPAS-Ocean (E3SM V2): impact of ice shelf cavities
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/175851/1/gmd-16-1297-2023_Pal_etal_BarotropicTides_and_IceShelves.pdfSEL
Two-dimensional Turbulence in Symmetric Binary-Fluid Mixtures: Coarsening Arrest by the Inverse Cascade
We study two-dimensional (2D) binary-fluid turbulence by carrying out an extensive direct numerical simulation (DNS) of the forced, statistically steady turbulence in the coupled Cahn-Hilliard and NavierStokes equations. In the absence of any coupling, we choose parameters that lead (a) to spinodal decomposition and domain growth, which is characterized by the spatiotemporal evolution of the Cahn-Hilliard order parameter phi, and (b) the formation of an inverse-energy-cascade regime in the energy spectrum E(k), in which energy cascades towards wave numbers k that are smaller than the energy-injection scale kin j in the turbulent fluid. We show that the Cahn-Hilliard-Navier-Stokes coupling leads to an arrest of phase separation at a length scale Lc, which we evaluate from S(k), the spectrum of the fluctuations of f. We demonstrate that (a) Lc-LH, the Hinze scale that follows from balancing inertial and interfacial-tension forces, and (b) Lc is independent, within error bars, of the diffusivity D. We elucidate how this coupling modifies E(k) by blocking the inverse energy cascade at a wavenumber kc, which we show is similar or equal to 2 pi/Lc. We compare our work with earlier studies of this problem