Effects of Forcing in Three-Dimensional Turbulent Flows

Abstract

We present the results of a numerical investigation of three-dimensional homogeneous and isotropic turbulence, stirred by a random forcing with a power law spectrum, E f (k) ∼ k 3−y . Numerical simulations are performed at different resolutions up to 512 3 . We show that at varying the spectrum slope y, small-scale turbulent fluctuations change from a forcing independent to a forcing dominated statistics. We argue that the critical value separating the two behaviours, in three dimensions, is yc = 4. When the statistics is forcing dominated, for y < yc, we find dimensional scaling, i.e. intermittency is vanishingly small. On the other hand, for y > yc, we find the same anomalous scaling measured in flows forced only at large scales. We connect these results with the issue of universality in turbulent flows.The effects of both external forcing mechanisms and boundary conditions on small-scale turbulent fluctuations have been the subject of many theoretical, numerical and experimental studies [1,2]. The 1941 theory of Kolmogorov [1] is based on the assumption of local isotropy and homogeneity, that is any turbulent flow, independently on the injection mechanism, recovers universal statistical properties, for scales small enough (and far from the boundaries). Indeed, experiments and numerical simulations give strong indications that Eulerian and Lagrangian isotropic/anisotropic small-scales velocity statistics are pretty independent of the large-scale forcing mechanisms [3][4][5][6][7]. Still, we lack a firm understanding for these evidences. From the theoretical point of view, precious hints arise from linear problems, like passive scalar or passively advected magnetic fields. For the class of Kraichnan models [8], anomalous scaling has been shown to be associated to statistically stationary solutions of the unforced equations for correlation functions [9]. Scaling exponents are consequently universal with respect of the injection mechanisms. Concerning non-linear problems, as the Navier-Stokes case, analytical results have been often pursued by means of the Renormalization Group (RG) [10,11]. In the RG framework, turbulence is stirred at all scales by a self-similar Gaussian field, with zero mean and white-noise in time. The two-point correlation function in Fourier space is given byHere 1/k 0 ∼ L is the largest length in the system (infrared cut-off), D 0 is the forcing intensity, P ij (k) is the projector assuring incompressibility and d is the spatial dimension (always assumed to be d = 3 hereafter). The influence of the stirring mechanism at small scales is governed by the value of the slope y. We go from a situation when the forcing has a strong input at all scales, y ∼ 0 originally investigated in [10], to a quasi large-scale forcing when y → ∞. Renormalization Group calculations, based on a y-expansion, predict a power-law energy spec-where η is the viscous scale of the system, and for y ≪ 1. Notice that the Kolmogorov value, E(k) ∼ k −5/3 , describing experimental turbulent flows stirred by a largescale fo...

New class of turbulence in active fluids

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Interstellar Turbulence I: Observations and Processes

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Electromagnetically controlled multi-scale flows

“…Sommeria 1986;Dolzhanskii et al 1992;Cardoso, Marteau & Tabeling 1994;Williams, Marteau & Gollub 1997;Voth, Haller & Gollub 2002;Rothstein, Henry & Gollub 1999;Boffeta et al 2005). Multi-scale forcing has been applied by Queiros-Conde & Vassilicos (2001), Staicu et al (2003) and finally Hurst & Vassilicos (2006) who used fractal grids to stir the flow over many scales at once (see also the DNS of Biferale, Lanotte &Toschi 2004). Here we combine both approaches to create a multi-scale fractal EM forcing of a quasi-two-dimensional flow in the laboratory.…”