A model for the Reynolds number dependence of the dimensionless dissipation rate C ε was derived from the dimensionless Kármán-Howarth equation, resulting in, where R L is the integral scale Reynolds number. The coefficients C and C ε,∞ arise from asymptotic expansions of the dimensionless second-and third-order structure functions. This theoretical work was supplemented by direct numerical simulations (DNSs) of forced isotropic turbulence for integral scale Reynolds numbers up to R L = 5875 (R λ = 435), which were used to establish that the decay of dimensionless dissipation with increasing Reynolds number took the form of a power law R n L with exponent value n = −1.000 ± 0.009, and that this decay of C ε was actually due to the increase in the Taylor surrogate U 3 /L. The model equation was fitted to data from the DNS which resulted in the value C = 18.9 ± 1.3 and in an asymptotic value for C ε in the infinite Reynolds number limit of C ε,∞ = 0.468 ± 0.006.

No abstract

This review is concerned with modem theoretical approaches to turbulence, in which the problem can be seen as a branch of statistical field theory, and where the treatment has been strongly influenced by analogies with the quantum many-body problem. The dominant themes treated are the development (since the 1950s) of renormalized perturbation theories (RPT) and, more recently, of renormalization group (RG) methods.As fluid dynamics is rarely part of the physics curriculum, in section 1 we introduce some background concepts in fluid dynamics, followed by a skeleton treatment of the phenomenology of turbulence in section 2, taking flow through a straight pipe or a plane channel as a representative example. In section 3, the general statistical formulation of the problem is given, leading to a moment closure problem, which is analogous to the well known BBGKY hierarchy, and to the Kolmogorov -5/3 power law, which is a consequence of dimensional analysis. In section 4, we show how RPT have been used to tackle the moment closure problem, distinguishing between those which are compatible with the Kolmogorov spectrum and those which are not. In section 5, we discuss the use of RG to reduce the number of degrees of freedom in the numerical simulation of the turbulent equations of motion, while giving a clear statement of the technical problems which lie in the way of doing this. Lastly, the theories are discussed in section 6, in terms of their ability to meet the stated goals, as assessed by numerical computation and comparison with experiment.

The effect of polyethyleneoxide (Polyox grade WSR 301) on grid-generated turbulence was investigated using laser anemometry and flow-visualization techniques. It was found that the polymer additive reduced both the turbulent intensity and the rate of decay behind the grid. At typical drag-reducing concentrations, turbulent energy spectra were qualitatively the same as those in water, in agreement with the results of other investigations. However, at higher additive concentrations, the dissipation-range spectra showed noticeable attenuation. This seemed to be a threshold effect with onset at a polymer concentration between 100 and 250 ppm. This result was supported by photographs of dye-injection tracer but in this case the onset concentration for small-eddy suppression was between 50 and 100 ppm.

A model equation for the Reynolds number dependence of the dimensionless dissipation rate in freely decaying homogeneous magnetohydrodynamic turbulence in the absence of a mean magnetic field is derived from the real-space energy balance equation, leading to Cε = Cε,∞ + C/R− + O(1/R 2 − )), where R− is a generalized Reynolds number. The constant Cε,∞ describes the total energy transfer flux. This flux depends on magnetic and cross helicities, because these affect the nonlinear transfer of energy, suggesting that the value of Cε,∞ is not universal. Direct numerical simulations were conducted on up to 2048 3 grid points, showing good agreement between data and the model. The model suggests that the magnitude of cosmological-scale magnetic fields is controlled by the values of the vector field correlations. The ideas introduced here can be used to derive similar model equations for other turbulent systems.PACS numbers: 47.65. 52.30.Cv, 47.27.Jv, 47.27.Gs Magnetohydrodynamic (MHD) turbulence is present in many areas of physics, ranging from industrial applications such as liquid metal technology to nuclear fusion and plasma physics, geo-, astro-and solar physics, and even cosmology. The numerous different MHD flow types that arise in different settings due to anisotropy, alignment, different values of the diffusivities, to name only a few, lead to the question of universality in MHD turbulence, which has been the subject of intensive research by many groups [1][2][3][4][5][6][7][8][9][10][11][12]. The behavior of the (dimensionless) dissipation rate is connected to this problem, in the sense that correlation (alignment) of the different vector fields could influence the energy transfer across the scales [2,13,14], and thus possibly the amount of energy that is eventually dissipated at the small scales.For neutral fluids it has been known for a long time that the dimensionless dissipation rate in forced and freely decaying homogeneous isotropic turbulence tends to a constant with increasing Reynolds number. The first evidence for this was reported by Batchelor [15] in 1953, while the experimental results reviewed by Sreenivasan in 1984 [16], and subsequent experimental and numerical work by many groups, established the now wellknown characteristic curve of the dimensionless dissipation rate against Reynolds number: see [17][18][19][20] and references therein. For statistically steady isotropic turbulence, the theoretical explanation of this curve was recently found to be connected to the energy balance equation for forced turbulent flows [19], where the asymptote describes the maximal inertial transfer flux in the limit of infinite Reynolds number.For freely decaying MHD, recent results suggest that the temporal maximum of the total dissipation tends to a constant value with increasing Reynolds number. The first evidence for this behavior in MHD was put forward in 2009 by Mininni and Pouquet [21] using results from direct numerical simulations (DNSs) of isotropic MHD turbulence. The temporal maximum of the total ...