Numerical measurements of scaling relations in two-dimensional conformal fluid turbulence

Abstract

We present measurements of relativistic scaling relations in (2+1)-dimensional conformal fluid turbulence from direct numerical simulations, in the weakly compressible regime. These relations were analytically derived previously in [1] for a relativistic fluid; this work is a continuation of that study, providing further analytical insights together with numerical experiments to test the scaling relations and extract other important features characterizing the turbulent behavior. We first explicitly demonstrate that the nonrelativistic limit of these scaling relations reduce to known results from the statistical theory of incompressible Navier-Stokes turbulence. In simulations of the inverse-cascade range, we find the relevant relativistic scaling relation is satisfied to a high degree of accuracy. We observe that the non-relativistic versions of this scaling relation underperform the relativistic one in both an absolute and relative sense, with a progressive degradation as the rms Mach number increases from 0.14 to 0.19. In the direct-cascade range, the two relevant relativistic scaling relations are satisfied with a lower degree of accuracy in a simulation with rms Mach number 0.11. We elucidate the poorer agreement with further simulations of an incompressible Navier-Stokes fluid. Finally, as has been observed in the incompressible Navier-Stokes case, we show that the energy spectrum in the inversecascade of the conformal fluid exhibits k −2 scaling rather than the Kolmogorov/Kraichnan expectation of k −5/3 , and that it is not necessarily associated with compressive effects. We comment on the implications for a recent calculation of the fractal dimension of a turbulent (3 + 1)-dimensional AdS black brane.

Relativistic dissipation obeys Chapman-Enskog asymptotics: Analytical and numerical evidence as a basis for accurate kinetic simulations

“…In this framework, fluid dynamic solutions in (2+1)-dimensions provide valuable information for the study of gravity in (3+1)-dimensions. For example, the development of turbulence in (3+1)-dimensional gravitational perturbations [56] has sparked a significant interest for the analysis of relativistic turbulent flows in (2+1)-dimensions ( [57][58][59]).…”

Stratified Kelvin–Helmholtz turbulence of compressible shear flows

“…A recent review which examines both hydrodynamic and magnetohydrodynamic implementations of supersonic compressible turbulence on statistical quantities can be found in Falceta-Gonçalves et al (2014). In this work, we follow the vast majority of investigations (Shivamoggi, 1992;Ottaviani, 1992;Domaradzki and Carati, 2007;Falkovich et al, 2010;Kuznetsov and Sereshchenko, 2015;Shivamoggi, 2015;Sun, 2016;Westernacher-Schneider et al, 2015;Qiu et al, 2016;Bershadskii, 2016;Sun, 2017;Westernacher-Schneider and Lehner, 2017) by utilizing the phenomenological description of turbulence in Fourier space as well as the utilization of two-point velocity structure functions for the statistical examination of our high-fidelity numerical simulations. One of our goals is to investigate scaling laws using a computational framework with moderately high resolutions.…”

Fractal dimension of turbulent black holes

“…Using the numerical code described in [15], we evolve a (2 + 1)-dimensional conformal perfect fluid with equation of state P = ρ/2 on a 2π-periodic domain with 2048 2 points. The energy momentum tensor of the fluid is T ab = (3/2)ρu a u b + (1/2)ρη ab , with u a = γ(1, v) and γ the Lorentz factor.…”

“…By applying a line transect madogram method [21] to the event horizon surface r + (x, y) = 4πT (x, y)/3 in boosted ingoing Finkelstein coordinates, we obtained a fractal dimension for spatial sections of the horizon H ∩ Σ of D = 2.584(1) and D = 2.645(4) for the cases with the boundary spectrum E(k) ∼ k −2 and E(k) ∼ k −5/3 , respectively. We argued that the former scaling, E(k) ∼ k −2 , is a more natural setting for the fluid-gravity duality since it corresponds to the regime of infinite Reynolds number without large-scale dissipation of energy [15,18].…”