No return to reflection symmetry in freely decaying homogeneous turbulence

We consider the large-scale structure of freely decaying incompressible homogeneous anisotropic helical turbulence, whose energy spectrum $E\left(k\right)$ is given by $E\left(k\right)=C{k}^2+o\left(k^2\right)$ at $k\to 0$. Here $k=\left|\mathit{k}\right|,\mathit{k}$ is the wave vector, and $C$ is a dynamical invariant. The helicity spectrum $H\left(k\right)$ is given by $H\left(k\right)=C_h{k}^3+o\left(k^3\right)$ at $k\to 0$, where $C_h$ is in general nonzero in helical turbulence. By generalizing Saffman's argument for nonhelical turbulence [Saffman, J. Fluid Mech. 27, 581 (1967)] to helical turbulence, it is shown that $C_h$ is another dynamical invariant. We present a theoretical analysis based on the time independence of the $O\left(k^0\right)$ term of the velocity correlation spectral tensor at $\mathit{k}\to \mathit{0}$ and a self-similarity assumption of flow evolution at large scales including the energy containing range scales. The analysis suggests that if the $O\left(k^0\right)$ term is reflection asymmetric at an initial instant, the turbulence does not relax to any reflection symmetric state at the large scales. A simple dimensional analysis yields the decay rates of the helicity and kinetic energy in the fully developed turbulence state. The theoretical results agree with results obtained by direct numerical simulation of incompressible helical turbulence in a periodic box.

Figure 1

Plot of $\mathrm{Re}$ with normalized time $t/T$.

Figure 2

Evolution of (a) kinetic energy $\langle |\mathit{u}{|}^2\rangle /2$, (b) integral length scale $L$, (c) compensated kinetic energy $t^{6/5}{\langle \left|\mathit{u}\right|}^2\rangle /2$, and (d) compensated integral length scale $t^{-2/5}L$.

Figure 3

Evolution of (a) $(\langle |{\mathit{u}}^{+}{|}^2\rangle -\langle |{\mathit{u}}^{-}{|}^2\rangle )/2$ and (b) $t^{6/5}(\langle |{\mathit{u}}^{+}{|}^2\rangle -\langle |{\mathit{u}}^{-}{|}^2\rangle )/2$.

Figure 4

Evolution of (a) helicity $\langle \mathit{u}·\mathbf{\omega }\rangle /2$ and (b) $t^{8/5}\langle \mathit{u}·\mathbf{\omega }\rangle /2$.

Figure 5

Plot of $A^{\mathcal{E}}$ and $A^{\mathcal{H}}$ vs $t/T$.

Figure 6

Plot of $(\langle |{\mathit{u}}^{+}{|}^2\rangle -\langle |{\mathit{u}}^{-}{|}^2\rangle )/{\langle \left|\mathit{u}\right|}^2\rangle$ vs $t/T$.

Figure 7

Spectra for run 1 at $t/T=0$, 20, 40, 60, 80, 100, 120, 140, and 160: (a) $E_{\mathrm{ave}}\left(k\right)=E\left(k\right)/4\pi k^2$ vs $k/k_p$ and (b) $H_{\mathrm{ave}}\left(k\right)/k=H\left(k\right)/4\pi k^3$ vs $k/k_p$.

Figure 8

Normalized spectra for run 1 at $t/T=0$, 20, 40, 60, 80, 100, 120, 140, and 160: (a) $E_{\mathrm{ave}}\left(k\right)/v^2{L}^3$ vs $kL$ and (b) $H_{\mathrm{ave}}\left(k\right)/k{v}^2{L}^3$ vs $kL$.

Figure 9

(a) Enstrophy $\langle |\mathbf{\omega }{|}^2\rangle /2$ vs $t/T$ for all runs and (b) $t^2{\langle \left|\mathbf{\omega }\right|}^2\rangle /2$ and $t^{2.3}{\langle \left|\mathbf{\omega }\right|}^2\rangle /2$ vs $t/T$ for run 1.

Figure 10

Relative helicity $\langle \mathit{u}·\mathbf{\omega }\rangle /\sqrt{{\langle \left|\mathit{u}\right|}^2{\rangle \langle \left|\mathbf{\omega }\right|}^2\rangle }$ vs $t/T$.