Statistical properties of four-dimensional turbulence

The energy transfer and small-scale intermittency in decaying turbulence in four dimensions (4D) are studied by direct numerical simulation and by spectral theory in comparison with three dimensions (3D). The energy transfer is more efficient in 4D than in 3D, hence the exponent of energy decay is larger. The Kolmogorov constant is 1.31, which is smaller than 1.72 in 3D. The longitudinal third-order structure function is confirmed to be governed by a $1∕2$ law, $-(1∕2)\overline{\epsilon }r$, instead of a $4∕5$ law in 3D. The intermittency is weaker in 4D for the total dissipation rate $\nu {\sum }_{i,j}{(\partial u_j∕\partial x_i)}^2$ and the associated velocity difference (spherical velocity difference) on scale $r$ than in 3D, while it is slightly stronger for the surrogated dissipation rate $\nu {(\partial u_1∕\partial x_1)}^2$ and the associated longitudinal velocity difference. The scaling exponents of the spherical and longitudinal velocity differences are also evaluated, indicating that the spherical velocity difference is less intermittent in 4D, while the longitudinal difference is more intermittent in 4D. The distribution of the eigenvalues of the strain tensor is also examined. It was also found that the normalized variance of the pressure gradient ${\overline{\epsilon }}^{-3∕2}{\nu }^{-1∕2}⟨{(\nabla p)}^2⟩$ in 4D is smaller than in 3D. The roles of the incompressibility condition, the pressure gradient, and the intermittency in $d$-dimensional turbulence are examined, and the importance of the longitudinal component of turbulent velocity field in the energy transfer toward small scales are discussed. Burgers turbulence as the asymptote of turbulence in large dimension is suggested.

Figure 1

Variation of the Kolmogorov constant computed by the LRA with respect to the spatial dimension $d$. Straight line shows the asymptotic value $K_{\infty }=0.815$.

Figure 2

Relaxation of the ITS to the equilibrium energy spectrum $E\left(k\right)\propto k^2$ in 3D.

Figure 3

Relaxation of the ITS to the equilibrium energy spectrum $E\left(k\right)\propto k^3$ in 4D.

Figure 4

The decay of the total energy. The decay exponent is larger in 4D (Run 4D) than in 3D (Runs 3D and 3G). Straight lines show the slope computed by using the Loitsyanskii integral, 1.50 for 4D and 1.42 for 3D, respectively.

Figure 5

Variation the Loitsyanskii integrals normalized by the initial values for Runs 3D, 4D, and 3G.

Figure 6

Variation of the energy dissipation rate normalized by the initial value.

Figure 7

Comparison of the normalized energy transfer flux at $k_0{u}_{0}t=0.96$ and 2.4 for runs 3E and 4E.

Figure 8

Variation of the skewness $S^{\left(d\right)}$ in three and four dimensions.

Figure 9

Variation of the flatness $K^{\left(d\right)}$ in three and four dimensions.

Figure 10

A collection of the energy spectra in the Kolmogorov units for 4D and 3D. Solid lines: 3D, dashed lines: 4D. All the curves are taken soon after the enstrophy attains maximum. Runs $3\mathrm{A}(R_{\lambda }=13)$, $3\mathrm{B}(R_{\lambda }=26)$, $3\mathrm{C}(R_{\lambda }=43)$, $3\mathrm{F}(R_{\lambda }=66)$, and runs $4\mathrm{A}(R_{\lambda }=14)$, $4\mathrm{B}(R_{\lambda }=20)$, $4\mathrm{C}(R_{\lambda }=29)$, and $4\mathrm{F}(R_{\lambda }=64)$. The levels of the curves of 4D at wave numbers of the inertial effect dominant are lower than those of 3D. Straight lines stand for $K_d{\left(k\eta \right)}^{-5∕3}$, with the Kolmogorov constants $K_3=1.72$ (solid) and $K_4=1.31$ (dotted), respectively.

Figure 11

A plot of each term in KHK equation for 4D (Run 4A) divided by ${\overline{\epsilon }}^{\left(4\right)}r$ at $k_0{u}_{0}t=2.0$, where $G\left(r\right)=-\frac{3}{r^{d+1}}{\int }_0^r\frac{\partial D_{LL}\left(r^{\prime }\right)}{\partial t}{\left(r^{\prime }\right)}^{d+1}d{r}^{\prime }$.

Figure 12

A plot of each term in KHK equation for 3D (Run 3A) divided by ${\overline{\epsilon }}^{\left(3\right)}r$.

Figure 13

CDF’s of three types of the dissipation rates in 4D. The abscissa is normalized in terms of $\overline{\epsilon }$. The CDF for ${\epsilon }_1$ is the narrowest in the tail (Run 4A).

Figure 14

CDF’s of ${\epsilon }_1$ and ${\epsilon }_3$ for 4D (Run 4A) and for 3D (Run 3A) at $k_0{u}_{0}t=2.0$. The abscissa is normalized in terms of $\overline{\epsilon }$.

Figure 15

CDF’s of $t_1$ at various scales $r$ in 4D (Run 4A). The abscissa stands for the normalized variable in terms of the mean value.

Figure 16

CDF’s of $t_2$ at various scales $r$ in 4D (Run 4A). The abscissa stands for the normalized variable in terms of the mean value.

Figure 17

CDF’s of $t_3$ at various scales $r$ in 4D (Run 4A). The abscissa stands for the normalized variable in terms of the mean value.

Figure 18

CDF’s of $t_1$ and $t_3$ at $r∕\eta =8.43$ in 4D (Run 4A) and at $r∕\eta =7.44$ in 3D (Run 3A). The abscissa stands for the normalized variable in terms of the mean value.

Figure 19

Local scaling exponents ${\zeta }_q^{\left(d\right)}$ for the longitudinal structure functions in 4D (Run 4C) and 3D (Run 3C). The regions between two arrows are considered as the inertial effect dominant range.

Figure 20

Local scaling exponents ${\tilde{\zeta }}_q^{\left(d\right)}$ for the spherical structure functions in 4D (Run 4C) and 3D (Run 3C). The regions between two arrows are the inertial effect dominant range.

Figure 21

PDF of eigenvalues in 4D (Run 4A). The abscissa is normalized in terms of $\sqrt{{\sum }_i⟨{\left({\lambda }_i^{\left(4\right)}\right)}^2⟩}=\sqrt{{\overline{\epsilon }}^{\left(4\right)}∕\nu }$.

Figure 22

PDF of eigenvalues in 3D (Run 3A). The abscissa is normalized in terms of $\sqrt{{\sum }_i⟨{\left({\lambda }_i^{\left(3\right)}\right)}^2⟩}=\sqrt{{\overline{\epsilon }}^{\left(3\right)}∕\nu }$.

Figure 23

The comparison of PDF’s of the fused eigenvalue ${\lambda }_n^{\left(4\right)}+{\lambda }_{n+1}^{\left(4\right)}$ in 4D and ${\lambda }_n^{\left(3\right)}$ in 3D for $n=1,2,3$. The abscissa is normalized in terms of $\sqrt{⟨{\left({\tilde{\lambda }}_i^{\left(4\right)}\right)}^2⟩}$ and $\sqrt{⟨{\left({\lambda }_i^{\left(3\right)}\right)}^2⟩}$.