We present high-resolution numerical simulations of the Euler and NavierStokes equations for a pair of colliding dipoles. We study the possible approach to a finite-time singularity for the Euler equations, and contrast it with the formation of developed turbulence for the NavierStokes equations. We present numerical evidence that seems to suggest the existence of a blow-up of the inviscid velocity field at a finite time ((t s) with scaling u∞∼(t s-t) -1/2, ω∞∼(t s-t) -1. This blow-up is associated with the formation of a k -3 spectral range, at least for the finite range of wavenumbers that are resolved by our computation. In the evolution toward t s, the total enstrophy is observed to increase at a slower rate, Ω∼(t s-t) -3/4, than would naively be expected given the behaviour of the maximum vorticity, ω∞∼(t s-t) -1. This indicates that the blow-up would be concentrated in narrow regions of the flow field. We show that these regions have sheet-like structure. Viscous simulations, performed at various R e, support the conclusion that any non-zero viscosity prevents blow-up in finite time and results in the formation of a dissipative exponential range in a time interval around the estimated inviscid t s. In this case the total enstrophy saturates, and the energy spectrum becomes less steep, approaching k -5/3. The simulations show that the peak value of the enstrophy scales as R e 3/2, which is in accord with Kolmogorov phenomenology. During the short time interval leading to the formation of an inertial range, the total energy dissipation rate shows a clear tendency to become independent of Re, supporting the validity of Kolmogorovs law of finite energy dissipation. At later times the kinetic energy shows a t -1.2 decay for all Re, in agreement with experimental results for grid turbulence. Visualization of the vortical structures associated with the stages of vorticity amplification and saturation show that, prior to t s, large-scale and the small-scale vortical structures are well separated. This suggests that, during this stage, the energy transfer mechanism is non-local both in wavenumber and in physical space. On the other hand, as the spectrum becomes shallower and a k -5/3 range appears, the energy-containing eddies and the small-scale vortices tend to be concentrated in the same regions, and structures with a wide range of sizes are observed, suggesting that the formation of an inertial range is accompanied by transfer of energy that is local in both physical and spectral space. © 2011 Cambridge University Press.
Vortex events in Euler and NavierStokes simulations with smooth initial conditions / Orlandi, Paolo; Pirozzoli, Sergio; Carnevale, George. - In: JOURNAL OF FLUID MECHANICS. - ISSN 0022-1120. - STAMPA. - 690:(2012), pp. 288-320. [10.1017/jfm.2011.430]
Vortex events in Euler and NavierStokes simulations with smooth initial conditions
ORLANDI, Paolo;PIROZZOLI, Sergio;CARNEVALE, GEORGE
2012
Abstract
We present high-resolution numerical simulations of the Euler and NavierStokes equations for a pair of colliding dipoles. We study the possible approach to a finite-time singularity for the Euler equations, and contrast it with the formation of developed turbulence for the NavierStokes equations. We present numerical evidence that seems to suggest the existence of a blow-up of the inviscid velocity field at a finite time ((t s) with scaling u∞∼(t s-t) -1/2, ω∞∼(t s-t) -1. This blow-up is associated with the formation of a k -3 spectral range, at least for the finite range of wavenumbers that are resolved by our computation. In the evolution toward t s, the total enstrophy is observed to increase at a slower rate, Ω∼(t s-t) -3/4, than would naively be expected given the behaviour of the maximum vorticity, ω∞∼(t s-t) -1. This indicates that the blow-up would be concentrated in narrow regions of the flow field. We show that these regions have sheet-like structure. Viscous simulations, performed at various R e, support the conclusion that any non-zero viscosity prevents blow-up in finite time and results in the formation of a dissipative exponential range in a time interval around the estimated inviscid t s. In this case the total enstrophy saturates, and the energy spectrum becomes less steep, approaching k -5/3. The simulations show that the peak value of the enstrophy scales as R e 3/2, which is in accord with Kolmogorov phenomenology. During the short time interval leading to the formation of an inertial range, the total energy dissipation rate shows a clear tendency to become independent of Re, supporting the validity of Kolmogorovs law of finite energy dissipation. At later times the kinetic energy shows a t -1.2 decay for all Re, in agreement with experimental results for grid turbulence. Visualization of the vortical structures associated with the stages of vorticity amplification and saturation show that, prior to t s, large-scale and the small-scale vortical structures are well separated. This suggests that, during this stage, the energy transfer mechanism is non-local both in wavenumber and in physical space. On the other hand, as the spectrum becomes shallower and a k -5/3 range appears, the energy-containing eddies and the small-scale vortices tend to be concentrated in the same regions, and structures with a wide range of sizes are observed, suggesting that the formation of an inertial range is accompanied by transfer of energy that is local in both physical and spectral space. © 2011 Cambridge University Press.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.