Asymptotic exponents from low-Reynolds-number flows

Schumacher, Jöerg; Sreenivasan, Katepalli R.; Yakhot, Victor

doi:10.1088/1367-2630/9/4/089

Cited by 145 publications

(232 citation statements)

References 28 publications

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“…Also listed in Table 1 are parameters of two corresponding DNS of homogeneous isotropic (HI) box turbulence runs, HI1 and HI2, which are used for comparison with bulk turbulence in the convection cell. There, a cube V = H 3 with periodic boundary conditions in all directions and a standard pseudospectral method with fast Fourier transformations are used (26).…”

Section: Numerical Modelmentioning

confidence: 99%

Enhanced enstrophy generation for turbulent convection in low-Prandtl-number fluids

Schumacher

Götzfried

Scheel

2015

Proc. Natl. Acad. Sci. U.S.A.

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Turbulent convection is often present in liquids with a kinematic viscosity much smaller than the diffusivity of the temperature. Here we reveal why these convection flows obey a much stronger level of fluid turbulence than those in which kinematic viscosity and thermal diffusivity are the same; i.e., the Prandtl number Pr is unity. We compare turbulent convection in air at Pr = 0.7 and in liquid mercury at Pr = 0.021. In this comparison the Prandtl number at constant Grashof number Gr is varied, rather than at constant Rayleigh number Ra as usually done. Our simulations demonstrate that the turbulent Kolmogorov-like cascade is extended both at the large-and small-scale ends with decreasing Pr. The kinetic energy injection into the flow takes place over the whole cascade range. In contrast to convection in air, the kinetic energy injection rate is particularly enhanced for liquid mercury for all scales larger than the characteristic width of thermal plumes. As a consequence, mean values and fluctuations of the local strain rates are increased, which in turn results in significantly enhanced enstrophy production by vortex stretching. The normalized distributions of enstrophy production in the bulk and the ratio of the principal strain rates are found to agree for both Prs. Despite the different energy injection mechanisms, the principal strain rates also agree with those in homogeneous isotropic turbulence conducted at the same Reynolds numbers as for the convection flows. Our results have thus interesting implications for small-scale turbulence modeling of liquid metal convection in astrophysical and technological applications. T urbulent convection depends strongly on the material properties of the working fluid that are quantified by the Prandtl number, the ratio of kinematic viscosity of the fluid to thermal diffusivity of the temperature, Pr = ν=κ. Compared with the vast number of investigations at Pr ≥ 1 (1, 2), the very-low-Pr regime appears almost as a "terra incognita" despite many applications. Turbulent convection in the Sun is present at Prandtl numbers Pr < 10 −3 (3-5). The Prandtl number in the liquid metal core of the Earth is Pr ∼ 10 −2 (6). Convection in material processing (7), nuclear engineering (8), or liquid metal batteries (9) has Prandtl numbers between 3 × 10 −2 and 10 −3 . Rayleigh-Bénard convection (RBC), a fluid flow in a layer that is cooled from above and heated from below, is a paradigm for all of these examples. One reason for significantly fewer low-Pr RBC studies is that laboratory measurements have to be conducted in opaque liquid metals such as mercury or gallium at Pr = 0.021 (10-12). The lowest value for a Prandtl number that can be obtained in optically transparent fluids is Pr = 0.2 for binary gas mixtures (13), i.e., an order of magnitude larger than in liquid metals. Direct numerical simulations (DNS) are currently the only way to gain access to the full 3D convective turbulent fields in low-Pr convection (14-18). These simulations turn out to become very demanding if the...

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Section: Numerical Modelmentioning

confidence: 99%

Enhanced enstrophy generation for turbulent convection in low-Prandtl-number fluids

Schumacher

Götzfried

Scheel

2015

Proc. Natl. Acad. Sci. U.S.A.

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“…In particular, we examine the alignment at every point x of the vorticity ω relative to the eigenvectors of the total strain rate tensor field S ij (x) and those of the background strain field data S B ij (x). This analysis uses data from a highly-resolved, three-dimensional, direct numerical simulation (DNS) of statistically stationary, forced, homogeneous, isotropic turbulence [17,18]. The simulations correspond to a periodic cube with sides of length of 2π resolved by 2048 3 grid points.…”

Section: Vorticity Alignment In Turbulent Flowsmentioning

confidence: 99%

“…More details on the numerical simulations are given in Refs. [17,18]. Figure 6 gives a representative sample of the DNS data, where the instantaneous shear component S 12 of the total strain rate tensor field S ij (x) is shown in a typical twodimensional intersection through the 2048 3 cube.…”

Section: Vorticity Alignment In Turbulent Flowsmentioning

confidence: 99%

“…Higher-order evaluation of the background strain rate is not feasible, as the results in Ref. [17] show that only spatial derivatives of the velocity field up to order six can be accurately obtained from these highresolution DNS data. For n = 4, the expansion in (20) involves seventh-order derivatives of the velocity field, and the background strain evaluation becomes limited due to the grid resolution.…”

Section: Vorticity Alignment In Turbulent Flowsmentioning

confidence: 99%

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Local and nonlocal strain rate fields and vorticity alignment in turbulent flows

2008

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Local and nonlocal contributions to the total strain rate tensor Sij at any point x in a flow are formulated from an expansion of the vorticity field in a local spherical neighborhood of radius R centered on x. The resulting exact expression allows the nonlocal (background) strain rate tensor S B ij (x) to be obtained from Sij (x). In turbulent flows, where the vorticity naturally concentrates into relatively compact structures, this allows the local alignment of vorticity with the most extensional principal axis of the background strain rate tensor to be evaluated. In the vicinity of any vortical structure, the required radius R and corresponding order n to which the expansion must be carried are determined by the viscous lengthscale λν. We demonstrate the convergence to the background strain rate field with increasing R and n for an equilibrium Burgers vortex, and show that this resolves the anomalous alignment of vorticity with the intermediate eigenvector of the total strain rate tensor. We then evaluate the background strain field S B ij (x) in DNS of homogeneous isotropic turbulence where, even for the limited R and n corresponding to the truncated series expansion, the results show an increase in the expected equilibrium alignment of vorticity with the most extensional principal axis of the background strain rate tensor.

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“…However, recent investigations of how dissipative structures such as shocks, tubes and sheets enter the dissipation range [14][15][16] suggest a more conservative choice which was realized in Run2 and Run3.…”

Section: Navier-stokes Turbulencementioning

confidence: 99%

Lagrangian statistics of Navier–Stokes and MHD turbulence

Homann¹,

Grauer²,

Busse³

et al. 2007

J. Plasma Phys.

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Abstract. We report on a comparison of high-resolution numerical simulations of Lagrangian particles advected by incompressible turbulent hydro-and magnetohydrodynamic (MHD) flows. Numerical simulations were performed with up to 1024 3 collocation points and 10 million particles in the Navier- ) whereas in MHD turbulence the predictions from the multifractal approach are more intermittent than observed in our simulations. In addition, our simulations reveal that Lagrangian Navier-Stokes turbulence is more intermittent than MHD turbulence, whereas the situation is reversed in the Eulerian case. Those findings can not consistently be described by the multifractal modeling. The crucial point is that the geometry of the dissipative structures have different implications for Lagrangian and Eulerian intermittency. Application of the multifractal approach for the modeling of the acceleration probability density functions works well for the Navier-Stokes case but in the MHD case just the tails are well described.

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Asymptotic exponents from low-Reynolds-number flows

Cited by 145 publications

References 28 publications

Enhanced enstrophy generation for turbulent convection in low-Prandtl-number fluids

Enhanced enstrophy generation for turbulent convection in low-Prandtl-number fluids

Local and nonlocal strain rate fields and vorticity alignment in turbulent flows

Lagrangian statistics of Navier–Stokes and MHD turbulence

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