Abstract
A numerical study of the $d$ -dimensional eddy damped quasi-normal Markovian equations is performed to investigate the dependence on spatial dimension of homogeneous isotropic fluid turbulence. Relationships between structure functions and energy and transfer spectra are derived for the $d$ -dimensional case. Additionally, an equation for the $d$ -dimensional enstrophy analogue is derived and related to the velocity derivative skewness. Comparisons are made to recent four-dimensional direct numerical simulation results. Measured energy spectra show a magnified bottleneck effect which grows with dimension whilst transfer spectra show a varying peak in the nonlinear energy transfer as the dimension is increased. These results are consistent with an increased forward energy transfer at higher dimensions, further evidenced by measurements of a larger asymptotic dissipation rate with growing dimension. The enstrophy production term, related to the velocity derivative skewness, is seen to reach a maximum at around five dimensions and may reach zero in the limit of infinite dimensions, raising interesting questions about the nature of turbulence in this limit.
Highlights
Despite more than a century of concentrated effort, fluid turbulence remains steadfast as the oldest unsolved problem of classical physics
Before we investigate this behaviour in higher dimensions using the eddy damped quasi-normal Markovian (EDQNM) closure, we need to understand to what extent the model is capable of reproducing the effects seen in three and four-dimensional direct numerical simulation (DNS)
We show energy spectra from both DNS and EDQNM in three and four dimensions scaled by the Kolmogorov constant
Summary
Despite more than a century of concentrated effort, fluid turbulence remains steadfast as the oldest unsolved problem of classical physics. One of the most important predictions of the K41 theory, valid at sufficiently high Reynolds number, is the existence of a range of intermediate sized eddies in the flow referred to as the inertial range, characterised by scale invariance and a constant energy flux. This scale invariance manifests itself clearly in the power law form of the K41 energy spectrum in the inertial range E(k) = C ε 2 3 k− (1.1). Berera where ε is the constant energy flux, which for stationary turbulence will be equal to the rate of viscous energy dissipation, and C is a universal constant
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