Effect of pressure on joint cascade of kinetic energy and helicity in compressible helical turbulence

Direct numerical simulations of three-dimensional compressible helical turbulence are carried out at a grid resolution of ${1024}^3$ to investigate the effect of pressure, which is important for the joint cascade of kinetic energy and helicity in compressible helical turbulence. The principal finding is that the pressure term of the helicity equation [defined as ${\mathrm{\Phi }}^H=p{\partial }_i({\omega }_i/\rho )$] has a smaller effect on the helicity cascade in the aspect of amplitude and a smaller effective range, which leads to a longer inertial subrange of the helicity cascade, in contrast to a kinetic energy cascade. In addition, we also find that the effective range of ${\mathrm{\Phi }}^H$ is concentrated only in large scales statistically, which is similar to the effect of the pressure term of the kinetic energy equation (defined as ${\mathrm{\Phi }}^E=p{\partial }_i{u}_i$). From the overall sense of the effect of ${\mathrm{\Phi }}^E$ and ${\mathrm{\Phi }}^H$ on the kinetic energy and the helicity, respectively, both of them play a role of dissipation especially in the compression region. We propose that high enough helicity can affect the process of energy transformation between kinetic energy and internal energy, which means that the absolute local helicity hinders the process of kinetic energy transferring to internal energy, and promotes internal energy transferring to kinetic energy. In addition, ${\mathrm{\Phi }}^H$ plays a source role both for positive and negative helicity. We also study the mechanism of cancellations between compression and rarefaction regions, and we find that the impact of a shocklet on the helicity cascade can be ignored statistically.

Figure 1

(a) Compensated spectra of right-chirality kinetic energy (solid red line), free-chirality kinetic energy (dashed green line), left-chirality kinetic energy (blue dot-dashed line), and total kinetic energy (double dot-dashed pink line). (b) Compensated spectra of right-chirality helicity (dashed red line), left-chirality helicity (dot-dashed green line), and total helicity (solid blue line).

Figure 2

Isosurface of the velocity divergence at $\theta =-3{\theta }^{\prime }$ of the ${1024}^3$ simulation at $M_t=0.71$.

Figure 3

Compensated pressure-dilatation cospectrum (dashed green line) and compensated pressure-vorticity per mass (solid blue line).

Figure 4

(a) Ensemble-averaged subgrid kinetic energy flux ${\mathrm{\Pi }}_l^E$ (solid red line), and ensemble-averaged pressure term ${\mathrm{\Phi }}_l^E$ (dashed blue line) in the kinetic energy equation. (b) Ensemble-averaged negative subgrid helicity flux $-{\mathrm{\Pi }}_l^H$ (solid red line), and ensemble-averaged pressure term ${\mathrm{\Phi }}_l^H$ (dashed blue line) in the helicity equation, where $l$ is the filter width, and $\eta$ is the Kolmogorov dissipation length scale.

Figure 5

(a) Normalized pressure term for kinetic energy of different filter width. (b) Normalized pressure term for helicity of different filter width.

Figure 6

(a) Ensemble-averaged ${\mathrm{\Phi }}_l^E$ conditioned on local velocity divergence with a different filter width. (b) Ensemble-averaged ${\mathrm{\Phi }}_l^E$ conditioned on absolute local helicity with a different filter width.

Figure 7

(a) Ensemble-averaged negative ${\mathrm{\Phi }}_l^E$ conditioned on absolute local helicity with a different filter width. (b) Ensemble-averaged positive ${\mathrm{\Phi }}_l^E$ conditioned on absolute local helicity with a different filter width.

Figure 8

(a) Ensemble-averaged ${\mathrm{\Phi }}_l^H$ conditioned on local velocity divergence with a different filter width. (b) Ensemble-averaged ${\mathrm{\Phi }}_l^H$ conditioned on local helicity with a different filter width.

Figure 9

Top left: two-dimensional slice of pointwise ${\mathrm{\Phi }}_l^E$. Bottom left: normalized distribution curve for ${\mathrm{\Phi }}_l^E$ (red solid line) and the corresponding Gaussian distribution (blue dashed line). Top right: two-dimensional slice of residual pointwise ${\mathrm{\Phi }}_l^E$. Bottom right: normalized distribution curve for residual ${\mathrm{\Phi }}_l^E$ (red solid line) and the corresponding Gaussian distribution (blue dashed line). Here ${\mu }_1$ is the mean value of ${\mathrm{\Phi }}_l^E, {\sigma }_1$ is the standard deviation of ${\mathrm{\Phi }}_l^E, {\mu }_2$ is the mean value of the residual of ${\mathrm{\Phi }}_l^E$, and ${\sigma }_2$ is the standard deviation of the residual of ${\mathrm{\Phi }}_l^E$.

Figure 10

Top left: two-dimensional slice of pointwise ${\mathrm{\Phi }}_l^H$. Bottom left: normalized distribution curve for ${\mathrm{\Phi }}_l^H$ (red solid line) and the corresponding Gaussian distribution (blue dashed line). Top right: two-dimensional slice of residual pointwise ${\mathrm{\Phi }}_l^H$. Bottom right: normalized distribution curve for residual ${\mathrm{\Phi }}_l^H$ (red solid line) and the corresponding Gaussian distribution (blue dashed line). Here ${\mu }_1$ is the mean value of ${\mathrm{\Phi }}_l^H, {\sigma }_1$ is the standard deviation of ${\mathrm{\Phi }}_l^H, {\mu }_2$ is the mean value of the residual of ${\mathrm{\Phi }}_l^H$, and ${\sigma }_2$ is the standard deviation of the residual of ${\mathrm{\Phi }}_l^H$.