Background scalar-level anisotropy caused by low-wave-number truncation in turbulent flows

We rigorously show that the truncation at low wave numbers always leads to background scalar-level anisotropy at large scales. Neither the resolution nor the spectral low-pass filter is dominant for this anisotropy, while the shape of the energy spectrum at low wave numbers is an important influence factor. Quantitative results are shown to provide references to the statistics in future postprocessing studies. Also, a simplified analytical model is introduced to explain the single-mode effects for this anisotropy.

Figure 1

$R_{ii}\left(\vec{r}\right)$ with $r_3=0$ fixed. $R_{ii}\left(\vec{0}\right)$ is used for normalization. The energy spectrum is Eq. (10). The half-grid number is $n=35$. The dashed and the dash-dotted curves correspond to $r=3$ and $r=5$, respectively.

Figure 2

$R_{ii}\left(\vec{r}\right)$ with $\vec{r}$ in axis direction, face diagonal direction, and cube diagonal direction, respectively. $R_{ii}\left(\vec{0}\right)$ is used for normalization. The energy spectrum is Eq. (10). The half-grid number is $n=35$. The dash-dotted curve is the spherical average value of $R_{ii}\left(\vec{r}\right)$ over the sphere with radius $r$. (a) Normal view; (b) log view.

Figure 3

The standard variation $\mathcal{A}\left(r\right)$. $R_{ii}\left(\vec{0}\right)$ is used for normalization. The energy spectrum is Eq. (10). The half-grid number is $n=35$. The dash-dotted lines show the positions of 1‰ and $1\%$ variations, respectively.

Figure 4

Comparison of different resolutions. (a) Spherical average value $\mu \left(r\right)$ (subfigure: log view); (b) standard variation $\mathcal{A}\left(r\right)$. $R_{ii}\left(\vec{0}\right)$ is used for normalization. The energy spectrum is Eq. (10).

Figure 5

Comparison of using isotropic filter in spectral space. (a) Spherical average value $\mu \left(r\right)$ (subfigure: log view); (b) standard variation $\mathcal{A}\left(r\right)$. $R_{ii}\left(\vec{0}\right)$ is used for normalization. The half-grid number is $n=35$. The energy spectrum is Eq. (10).

Figure 6

Sketch of model spectra.

Figure 7

Comparison of using different energy spectra. (a) Spherical average value $\mu \left(r\right)$ (subfigure: log view); (b) standard variation $\mathcal{A}\left(r\right)$. $R_{ii}\left(\vec{0}\right)$ is used for normalization. The half-grid number is $n=35$.

Figure 8

$R_{ii}\left(\vec{r}\right)$ with $\vec{r}$ in axis direction, face diagonal direction, and cube diagonal direction, respectively. $R_{ii}\left(\vec{0}\right)$ is used for normalization. The simplified analytical model is used. (a) $m=1$; (b) $m=3$.