Statistical properties of an incompressible passive vector convected by isotropic turbulence

Statistical properties of an incompressible passive vector convected by homogeneous isotropic turbulence are studied by comparing to the velocity and passive scalar, in order to explore the physics behind the differences and similarities in the statistical properties between the velocity vector and passive scalar. The passive vector obeys an equation similar to the Navier-Stokes equation, with a pseudopressure to ensure the incompressibility of the vector. The von Kármán–Howarth equation for the passive vector is derived and the average of the velocity increment times the square of the passive vector increments obeys a 4/3 law in the inertial-convective range. We carried out direct numerical simulations (DNSs) of up to ${1024}^3$ grid points. The spectra of the kinetic and pseudokinetic energies and the scalar variance obey a $k^{-5/3}$ power law. The Kolmogorov constants are $C_K=1.57$ for the velocity and $C_K^w=0.99$ for the passive vector, and the Obukhov-Corrsin constant of the passive scalar is $C_{\mathrm{OC}}=0.67$. The spectral bump of the compensated spectrum of the passive vector is slightly larger than that of the velocity, but smaller than the passive scalar. It is found that the behavior of the passive vector fluctuations at large scales is close to that of the velocity due to the nonlocal effects of the pseudo-pressure, while the small-scale fluctuations resemble those of the passive scalar. The nonlinearity of the convective term is key to the differences between the velocity and passive fields at small scales.

Figure 1

Time evolution of dissipation rates of the kinetic energy, pseudokinetic energy, and scalar variance.

Figure 2

Time evolution of the skewnesses of the spatial derivatives.

Figure 3

Normalized energy spectra over three runs. The straight line indicates a gradient of $-5/3$. For ease of visibility, the curves of $E_u^{*}\left(k\right)$ and $E_w^{*}\left(k\right)$ are shifted by 100 and 10, respectively.

Figure 4

Compensated energy spectra for Run C.

Figure 5

Normalized energy flux spectra for Run C. The red line is for ${\mathrm{\Pi }}_u/\overline{\epsilon }$, blue is for ${\mathrm{\Pi }}_w/{\overline{\epsilon }}_w$, and green is for ${\mathrm{\Pi }}_{\theta }/\overline{\chi }$.

Figure 6

Normalized third-order structure functions of Run C.

Figure 7

Spectra of pressure and pseudopressure for Run C.

Figure 8

Comparison of shifted spectra $E_p\left(k\eta \right)/E_p\left(k_{*}\eta \right)$ (red) and $E_q\left(k\eta \right)/E_q\left(k_{*}\eta \right)$ (blue) of the pressure and pseudopressure for Run C, respectively. $k_{*}\eta =30$ is chosen such that the portions that are almost straight collapse well. The inset shows the ratio $E_p\left(k\eta \right)/E_q\left(k\eta \right)$ for Run C.

Figure 9

Spectra of production terms of enstrophy, pseudoenstrophy and scalar gradient variance. Red: ${\sigma }_{\omega }^{*}$; blue: ${\sigma }_{\zeta }^{*}$; green: ${\sigma }_g^{*}$. The blue dotted line denotes ${\sigma }_{\zeta ,\mathrm{str}}^{*}$ and the blue broken line represents ${\sigma }_{\zeta ,\mathrm{crs}}^{*}$.

Figure 10

Measurement of the bottleneck effect. The $+$ symbols denote the peaks of the compensated energy spectrum.

Figure 11

Band-to-band transfer for Run C. Top: kinetic energy; middle: pseudokinetic energy; bottom: passive scalar variance. Red: $Q\eta =0.005$; blue: $Q\eta =0.02$; black: $Q\eta =0.15$.

Figure 12

Comparison between absolute values of the band-to-band spectral transfer on a logarithmic scale for Run C. Lines with circles denote absolute values of negative numbers. The straight line indicates $K^{-2}$. Top: $Q\eta =0.005$; middle: $Q\eta =0.02$; bottom: $Q\eta =0.15$. Red: kinetic energy; blue: pseudokinetic energy; green: passive scalar variance.