Narrow Results

We study steady vortex sheet solutions of the Navier–Stokes in the limit of vanishing viscosity at fixed energy flow. We refer to this as the turbulent limit. These steady flows correspond to a minimum of the Euler Hamiltonian as a functional of the tangent discontinuity of the local velocity parametrized as $\mathrm{\Delta }{\overrightarrow{v}}_t=\overrightarrow{\nabla }\mathrm{\Gamma }$. This observation means that the steady flow represents the low-temperature limit of the Gibbs distribution for vortex sheet dynamics with the normal displacement $\delta r_{\perp }$ of the vortex sheet as a Hamiltonian coordinate and $\mathrm{\Gamma }$ as a conjugate momentum. An infinite number of Euler conservation laws lead to a degenerate vacuum of this system, which explains the complexity of turbulence statistics and provides the relevant degrees of freedom (random surfaces). The simplest example of a steady solution of the Navier–Stokes equation in the turbulent limit is a spherical vortex sheet whose flow outside is equivalent to a potential flow past a sphere, while the velocity is constant inside the sphere. Potential flow past other bodies provide other steady solutions. The new ingredient we add is a calculable gap in tangent velocity, leading to anomalous dissipation. This family of steady solutions provides an example of the Euler instanton advocated in our recent work, which is supposed to be responsible for the dissipation of the Navier–Stokes equation in the turbulent limit. We further conclude that one can obtain turbulent statistics from the Gibbs statistics of vortex sheets by adding Lagrange multipliers for the conserved volume inside closed surfaces, the rate of energy pumping, and energy dissipation. The effective temperature in our Gibbs distribution goes to zero as ${\mathrm{\text{Re}}}^{-\frac{1}{3}}$ with Reynolds number $\mathrm{\text{Re}}\sim {\nu }^{-\frac{6}{5}}$ in the turbulent limit. The Gibbs statistics in this limit reduces to the solvable string theory in two dimensions (so-called $c=1$ critical matrix model). This opens the way for nonperturbative calculations in the Vortex Sheet Turbulence, some of which we report here.

We revise the steady vortex surface theory following the recent finding of asymmetric vortex sheets (Migdal, 2021). These surfaces avoid the Kelvin–Helmholtz instability by adjusting their discontinuity and shape. The vorticity collapses to the sheet only in an exceptional case considered long ago by Burgers and Townsend, where it decays as a Gaussian on both sides of the sheet. In generic asymmetric vortex sheets (Shariff, 2021), vorticity leaks to one side or another, making such sheets inadequate for vortex sheet statistics and anomalous dissipation. We conjecture that the vorticity in a turbulent flow collapses on a special kind of surface (confined vortex surface or CVS), satisfying some equations involving the tangent components of the local strain tensor.

The most important qualitative observation is that the inequality needed for this solution’s stability breaks the Euler dynamics’ time reversibility. We interpret this as dynamic irreversibility. We have also represented the enstrophy as a surface integral, conserved in the Navier–Stokes equation in the turbulent limit, with vortex stretching and viscous diffusion terms exactly canceling each other on the CVS surfaces.

We have studied the CVS equations for the cylindrical vortex surface for an arbitrary constant background strain with two different eigenvalues. This equation reduces to a particular version of the stationary Birkhoff–Rott equation for the 2D flow with an extra nonanalytic term. We study some general properties of this equation and reduce its solution to a fixed point of a map on a sphere, guaranteed to exist by the Brouwer theorem.

We continue the study of Confined Vortex Surfaces (CVS) that we introduced in the previous paper. We classify the solutions of the CVS equation and find the analytical formula for the velocity field for arbitrary background strain eigenvalues in the stable region. The vortex surface cross-section has the form of four symmetric hyperbolic sheets with a simple equation $\left|y\right||x{|}^{\mu }=\text{const}$ in each quadrant of the tube cross-section ($xy$ plane).

We use the dilute gas approximation for the vorticity structures in a turbulent flow, assuming their size is much smaller than the mean distance between them. We vindicate this assumption by the scaling laws for the surface shrinking to zero in the extreme turbulent limit. We introduce the Gaussian random background strain for each vortex surface as an accumulation of a large number of small random contributions coming from other surfaces far away. We compute this self-consistent background strain, relating the variance of the strain to the energy dissipation rate.

We find a universal asymmetric distribution for energy dissipation. A new phenomenon is a probability distribution of the shape of the profile of the vortex tube in the $xy$ plane. This phenomenon naturally leads to the “multifractal” scaling of the moments of velocity difference $v(r_1)-v(r_2)$. More precisely, these moments have a nontrivial dependence of $n$, $log\mathrm{\Delta }r$, approximating power laws with effective index $\zeta (n,log\mathrm{\Delta }r)$. We derive some general formulas for the moments containing multidimensional integrals. The rough estimate of resulting moments shows the log–log derivative $\zeta (n,log\mathrm{\Delta }r)$ which is approximately linear in $n$ and slowly depends on $log\mathrm{\Delta }r$. However, the value of effective index is wrong, which leads us to conclude that some other solution of the CVS equations must be found. We argue that the approximate phenomenological relations for these moments suggested in a recent paper by Sreenivasan and Yakhot are consistent with the CVS theory. We reinterpret their renormalization parameter $\alpha \approx 0.95$ in the Bernoulli law $p=-\frac{1}{2}\alpha v^2$ as a probability to find no vortex surface at a random point in space.

We study the linear stability of contact discontinuities for the nonisentropic compressible Euler equations in two space dimensions. Assuming the jump of the tangential velocity across the discontinuity surface is sufficiently large, we derive a suitable energy estimate for the linearized boundary value problem. The found estimate extends to nonisentropic compressible flows the main result of Coulombel–Secchi for the isentropic Euler equations. Compared with this latter case, when the jump of the tangential velocity of the unperturbed flow attains a certain critical value in the region of weak stability, here an additional loss of regularity appears; this is related to the presence of a double root of the Lopatinskii determinant associated to the problem.

We consider supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients, Morando et al. recently derived a pseudo-differential equation that describes the time evolution of the discontinuity front of the vortex sheet. In agreement with the classical stability analysis, the problem is weakly stable if $|[v\cdot \tau ]|>2\sqrt{2}c$, and the well-posedness holds in standard weighted Sobolev spaces. Our aim in this paper is to improve this result, by showing the existence in functional spaces with additional weighted anisotropic regularity in the frequency space.

In this paper, we study Majda’s example for stability of mixed problems introduced in [A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Vol. 53 (Springer Verlag, 1984)]. After some transformation we analyze the stability of the problem by computing the roots of the Kreiss–Lopatinskiĭ determinant and recover the different cases as in Majda [Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Vol. 53 (Springer Verlag, 1984)]. Then, we focus on the weakly stable case and prove the a priori estimate of the solution. The proof follows by adapting the approach of Coulombel and Secchi [The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J.53 (2004) 941–1012] for the linear stability of 2D compressible vortex sheets with the simplification introduced by Chen et al. [Linear stability of compressible vortex sheets in two-dimensional elastodynamics, Adv. Math.311 (2017) 18–60]. Compared to [J.-F. Coulombel and P. Secchi, The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J.53 (2004) 941–1012], we improve the a priori energy estimate in that the solution is estimated in suitable weighted Sobolev spaces, anisotropic in the frequency space, whose definition reflects the properties of the associated Lopatinskiĭ determinant. Moreover, we show that this a priori energy estimate is optimal. This very simple example appears useful for the comprehension of the method of proof of the a priori estimate, beyond the technicalities of [R. M. Chen, J. Hu and D. Wang, Linear stability of compressible vortex sheets in two-dimensional elastodynamics, Adv. Math.311 (2017) 18–60; J.-F. Coulombel and P. Secchi, The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J.53 (2004) 941–1012].