Homogeneous isotropic turbulence in four spatial dimensions

Abstract

Direct Numerical Simulation is performed of the forced Navier-Stokes equation in four spatial dimensions. Well equilibrated, long time runs at sufficient resolution were obtained to reliably measure spectral quantities, the velocity derivative skewness and the dimensionless dissipation rate. Comparisons to corresponding two and three dimensional results are made. Energy fluctuations are measured and show a clear reduction moving from three to four dimensions. The dynamics appear to show simplifications in four dimensions with a picture of increased forward energy transfer resulting in an extended inertial range with smaller Kolmogorov scale. This enhanced forwards transfer is linked to our finding of increased dissipative anomaly and velocity derivative skewness.

Effect of spatial dimension on a model of fluid turbulence

“…In the DNS study of Berera et al (2020), the observed scaling of the energy spectrum was consistent with that predicted by K41 in both the three-and four-dimensional cases insofar 10 -8 10 -10 as within a given dimension the energy spectra were found to collapse upon scaling by Kolmogorov variables. However, the spatial dimension was found to have an influence on the shape of the energy spectrum in the near dissipative region.…”

“…A higher value was found in the EDQNM of Bos et al (2007), however, they take a different choice of λ 1 which will directly influence the value found. In four dimensions we find C 4d ε,∞ = 0.96 lower than the value of 1.26 found in DNS (Berera et al 2020). Since C ε is defined in terms of large-scale quantities, where the forcing is active, these discrepancies between DNS and EDQNM are not surprising given the differences at small k seen in figure 3.…”

“…Here, they found an increased efficiency of the energy transfer in four dimensions compared with three, as well as increased anomalous scaling of the longitudinal structure functions. More recently, in the DNS study of Berera et al (2020), which looked at the stationary case at a higher resolution, various measurements performed pointed to an increased tendency for energy to be transferred from large to small scales in four dimensions, confirming what had been seen in Gotoh et al (2007), potentially driven by an increase in vortex stretching. This interpretation was based on finding a higher velocity derivative skewness.…”

“…Motivated by the four-dimensional DNS results presented in Berera et al (2020), as well as analogies to critical phenomena, in this paper we have performed a thorough investigation into the effect of the spatial dimension in the EDQNM model of turbulence. While this is only an approximation to true fluid turbulence, we find that it is able to satisfactorily reproduce many of the dimensional effects seen in DNS.…”

“…This is known as the dissipative anomaly. In Berera et al (2020) an increased value for this asymptotic dissipation rate was observed in four dimensions when compared with three. This result once again suggests an enhancement of the forward energy cascade in four dimensions compared with three.…”