Symmetries and novel universal properties of turbulent hydrodynamics in a symmetric binary fluid mixture

Abstract

We elucidate the universal properties of the nonequilibrium steady states (NESS) in a driven symmetric binary fluid mixture, an example of active advection, in its miscible phase. We use the symmetries of the equations of motion to establish the appropriate form of the structure functions which characterise the statistical properties of the NESS of a driven symmetric binary fluid mixture. We elucidate the universal properties described by the scaling exponents and the amplitude ratios. Our results suggest that these exponents and amplitude ratios vary continuously with the degree of crosscorrelations between the velocity and the gradient of the concentration fields. Furthermore, we demonstrate, in agreement with Celani et al, [Phys. Rev. Lett., 89, 234502 (2002)], that the conventional structure functions as used in passive scalar turbulence studies exhibit only simple scaling in the problem of symmetric binary fluid mixture even in the weak concentration limit. We also discuss possible experimental verifications of our results. PACS numbers:Critical dynamics of equilibrium systems show universal scaling properties for thermodynamic functions and correlation functions characterised by universal scaling exponents that depend on the spatial dimension d and the symmetry of the order parameter (e.g., Ising, XY etc.) and conservation laws [1, 2], but not on the parameters that specify the details of the Hamiltonian. A different situation arises in driven, dissipative, nonequilibrium systems with nonequilibrium statistical steady states (NESS). In this letter we demonstrate that in the NESS of externally driven turbulent homogeneous and isotropic symmetric binary fluid mixtures (50-50 concentration of the two components) universal properties are characterised by certain structure functions, determined by the symmetries of the equations of motion, and which are different from those used in pasive scalar turbulence studies. We find that the universal properties vary continuously with the degree of crosscorrelations between the velocity and the gradient of the concentration fields, a situation which has no analogue in equilibrium.The turbulent hydrodynamics of an incompressible symmetric binary fluid is described by the coupled evolution of the velocity field u(x, t) and the concentration field gradient ∇ψ(x, t) with ψ(x, t) = [ρ A (x, t) − ρ B (x, t)]/ρ 0 where ρ 0 is the mean density, and ρ A and ρ B are the densities of the two components. In a symmetric binary fluid ψ(x, t) = 0. Experiments performed on such systems have been typically concerned with the measurements of effective transport coefficients [3]. In a symmetric binary mixture, ψ is not simply advected passively by the u-field, but is active, i.e., concentration gradient ∇ψ reacts back on u to produce flows. In ordinary fluid turbulence the NESS is characterised by the multiscaling exponents of the structure functions of the velocity fields (see below) [4,5]. Compared to the studies of ordinary fluid turbulence the study of turbulent binary mixtures is in it...

Universality of scaling and multiscaling in turbulent symmetric binary fluids

“…We show for the first time that although the exponents associated with b are multiscaling (like the exponents for u), the equal-time exponents for ψ show simple-scaling. Our results are similar to the numerical quasi-Lagrangian (in two-dimensional flows) [13] and in agreement with the predictions of one-loop field theoretical [14] studies of the SBF system.…”

“…Our results may be explained from the analytical framework based on symmetry arguments developed in Ref. [14], where it has been shown that the presence of an additional continuous symmetry (kind of a gauge symmetry), not present in the passive scalar turbulence model, is responsible for the simple scaling behaviour of ζ ψ p . It would be interesting to investigate the properties of the turbulent NESS of SBF at low temperature, below the consolute point, when instabilities leading to phase separation competes with turbulent mixing.…”

“…Ref. [14] used symmetry arguments to show that ζ ψ p = p/3 and suggested that ζ b p should show multiscaling akin to ζ u p . It is thus expected that the multiscaling behaviour of the NESS in homogeneous and isotropic SBF turbulence is characterised by ζ u p and ζ b p .…”

“…where k is a wavenumber, t time, i, j Cartesian components in d-dimensions, A f and A ψ are a constant amplitude and P ij (k) = δ ij − k i k j /k 2 is the transverse projector which enforces the incompressibility condition [14]. One-loop renormalisation group studies [14,16] of this model yield K41 energy spectra for u and b fields:…”

“…The functions f and f ψ are forcing terms which drive the system to a statistically steady state. Galilean invariance of the system enforces λ 1 = λ 3 = 1 [14,16]. Further, λ 2 may be set to unity by appropriately choosing the unit of ψ (equivalently, by exploiting the rescaling invariance of ψ) [14].…”