Drift towards isotropization during the 3D hydrodynamic turbulence onset

The incompressible three-dimensional Euler equations develop very thin pancake-like regions of exponentially increasing vorticity. The characteristic thickness of such regions decreases exponentially with time, while the other two dimensions do not change considerably, making the flow near each pancake strongly anisotropic. The pancakes emerge in increasing number with time, which may enhance the anisotropy of the flow, especially if they orient similarly in space. In the present paper, we study numerically the anisotropy by analyzing the evolution of the so-called isotropy markers [Phys. Rev. Fluids 10, L022602 (2025)]. We show that these functions drift slowly towards unity, indicating the process of slow isotropization, which takes place without the viscous scales getting exited and despite the similar orientation of the emerging pancakes.

💡 Research Summary

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The paper investigates whether the highly anisotropic structures that appear during the onset of three‑dimensional turbulence in an inviscid fluid tend to become more isotropic as the flow evolves. In the Euler equations, thin vortex‑sheet (“pancake”) regions develop whose thickness decays exponentially while their lateral extents remain essentially unchanged. When a shear component is added to the initial velocity field, these pancakes tend to align preferentially along the shear direction (the z‑axis in the authors’ setup), potentially reinforcing anisotropy. The authors address the open question of whether such alignment leads to a monotonic increase of anisotropy or whether a slow isotropization process occurs even before viscous scales are reached.

To quantify anisotropy they employ the recently introduced “isotropy markers” (Agafontsev & Kopiev, Phys. Rev. Fluids 10, L022602, 2025). The markers are constructed from the velocity‑gradient tensor A_{ij}=∂v_i/∂x_j. From A they form the Gram matrix Γ_{ij}=∂v_α/∂x_i ∂v_α/∂x_j, whose diagonal entries Γ_{ii}, determinant det Γ, and 2×2 minors m_{ij}=Γ_{ii}Γ_{jj}−Γ_{ij}^2 are used to build three dimensionless ratios:

1. ⟨D_{zz}/D_{yy}⟩ = ⟨Γ_{zz}^{1/2}(det Γ)^{1/2}/m_{xy}⟩,
2. ⟨D_{yy}/D_{xx}⟩ = ⟨m_{xy}^{1/2}/Γ_{xx}⟩,
3. ⟨D_{yy}D_{zz}/D_{xx}^2⟩ = ⟨(det Γ)^{1/2}/Γ_{xx}^{3/2}⟩.

For a perfectly isotropic flow each ratio equals unity; deviations from one measure the degree of anisotropy. The identities hold for any isotropic probability measure because they are consequences of the rotation group O(3), independent of the dynamics.

The numerical experiments are performed in a periodic box