Closure and solution behaviour of Chandrasekhar's coupled scalar evolution equations for homogeneous axisymmetric turbulence

In 1950 S. Chandrasekhar reduced the dynamics of the two-point velocity correlation tensor of homogeneous axisymmetric turbulence — turbulence whose statistics are invariant under rotations about a single preferred axis — to a pair of coupled scalar evolution equations for two gauge-invariant defining scalars $Q_1(r,r\mu,t)$ and $Q_2(r,r\mu,t)$ , replacing the single von Kármán–Howarth equation of isotropic turbulence. The viscous terms are explicit, but the nonlinear source terms $S_1,S_2$ depend on third-order correlation scalar potentials and on the pressure–velocity correlation scalar, constrained by a differential compatibility condition; the paper derived the system but provided no closure and no solutions beyond the linear final period of decay, which Chandrasekhar treated in a companion paper. The open problem is to obtain a closure (or exact solution) of the coupled nonlinear scalar system that respects the compatibility condition derived in the paper and the realizability of the correlation tensor, and determine from initial and decay conditions whether the turbulence isotropizes or retains (and re-exhibits) anisotropy during decay. The linear final-period decay is solved, and substantial reformulated progress exists (EDQNM-type spectral closures, LES/DNS decay laws, dedicated experiments), but no known work closes or solves the nonlinear coupled system in the source formulation.

Background

Homogeneous turbulence is turbulent motion whose statistical properties are invariant under spatial translations, so that it can be characterized by correlation functions between velocity components $u_i$ and $u_j'$ at two points separated by a vector $\boldsymbol{\xi}$ . The simplest closure-free description is at the two-point level: the second-order correlation tensor $Q_{ij}=\overline{u_i u_j'}$ (the overline denoting a statistical average), whose evolution follows from the Navier–Stokes equations and involves, as an unclosed source, third-order velocity correlations and pressure–velocity correlations. For isotropic turbulence — statistically invariant under all rotations and reflections — Robertson's (1940) invariant theory reduces $Q_{ij}$ to one scalar function, and von Kármán & Howarth (1938) derived the single scalar evolution equation governing it; its nonlinear term (the triple correlation) is the classical turbulence closure problem, which remains unsolved in general.

Most turbulent flows possess a preferred direction (for example the mean-flow direction or an imposed strain axis), so full isotropy fails. Batchelor (1946) introduced axisymmetric turbulence — homogeneous turbulence whose statistics are invariant only under rotations about, and reflections in planes containing or perpendicular to, a fixed axis with unit vector $\boldsymbol{\lambda}$ — as the next case in order of symmetry, and showed that energy exchange between velocity components parallel and perpendicular to the axis is governed by the pressure–velocity correlation. Chandrasekhar (1950) completed the needed tensor algebra: any axisymmetric second-order tensor that is symmetric and solenoidal (divergence-free) in each index is determined, in a gauge-invariant way (i.e. uniquely, independent of the non-uniqueness — the gauge freedom — of the underlying scalar-potential representation), by two defining scalars $Q_1(r,r\mu)$ and $Q_2(r,r\mu)$ , where $r=|\boldsymbol{\xi}|$ is the separation distance and $\mu=(\boldsymbol{\xi}\cdot\boldsymbol{\lambda})/r$ is the cosine of the angle between the separation and the axis. Writing $D_r=\frac{1}{r}\frac{\partial}{\partial r}-\frac{\mu}{r^2}\frac{\partial}{\partial \mu}$ and $D_\mu=\frac{1}{r}\frac{\partial}{\partial\mu}$ , with doubled subscripts denoting iterated operators (so that $D_{\mu\mu}=D_\mu D_\mu=\frac{1}{r^2}\frac{\partial^2}{\partial\mu^2}$ , $D_{rr}=D_rD_r$ and $D_{r\mu}=D_rD_\mu$ ), the Cartesian Laplacian acting on the corresponding solenoidal fields induces on the defining scalars the mixed radial–angular operator $\Delta=r^2D_{rr}+2r\mu D_{r\mu}+D_{\mu\mu}+5D_r$ , which in plain partial derivatives reads $\Delta=\frac{\partial^2}{\partial r^2}+\frac{4}{r}\frac{\partial}{\partial r}+\frac{1-\mu^2}{r^2}\frac{\partial^2}{\partial\mu^2}-\frac{4\mu}{r^2}\frac{\partial}{\partial\mu}$ and which Chandrasekhar identified with the radial–angular operator of the ultraspherical harmonics (the higher-dimensional analogue of the spherical harmonics) in five dimensions. The tensor evolution equation then reduces to the coupled pair

\frac{\partial Q_1}{\partial t}=2\nu\Delta Q_1+S_1,\qquad \frac{\partial Q_2}{\partial t}=2\nu\big(\Delta Q_2+2D_{\mu\mu}Q_1\big)+S_2,

where $\nu$ is the kinematic viscosity. These are the axisymmetric replacement of the von Kármán–Howarth equation. The paper also derived (i) small-separation ( $r\to0$ ) Taylor expansions of $Q_1,Q_2$ whose coefficients $\alpha_{mn},\beta_{mn}$ determine the velocity variances, the curvature of longitudinal and transverse correlation coefficients, and (via the evolution equations) a generalized dissipation relation for $d\overline{u^2}/dt$ reducing to Taylor's isotropic formula; and (ii) the reduction of Batchelor's four axisymmetric integral scales to two one-dimensional functions $q_1(\mu),q_2(\mu)$ given by radial integrals of $Q_1,Q_2$ — the quantities an experiment or simulation should measure.

The nonlinear sources $S_1,S_2$ are the obstacle. They are gauge-invariant differential combinations of third-order correlation scalar potentials $T_k(r,r\mu)$ representing the triple velocity correlation and of the pressure–velocity scalar $\varpi(r,r\mu)$ defining the solenoidal vector $\rho^{-1}\overline{p u_i'}$ ; Chandrasekhar derived the differential compatibility condition that the intermediate gauge-invariant scalar functions built from the $T_k$ and $\varpi$ must satisfy in order for the nonlinear source tensor to be symmetric and solenoidal. The paper derived the coupled system but did not close it, solve it, or compare its measurable consequences with data. In a companion paper published the same month, Chandrasekhar solved only the final period of decay — the late, viscosity-dominated regime in which the nonlinear sources are neglected — obtaining explicit linear solutions and showing that a superposition of non-interacting isotropic and axisymmetric fields is dynamically compatible. Whether and when freely decaying axisymmetric turbulence isotropizes, or instead exhibits the empirically discussed persistence or reappearance of anisotropy during decay, is known to depend (in later reformulated treatments) on the large-scale structure of the initial data: Saffman (1967) showed that initial conditions with finite velocity impulse produce a low-wavenumber energy spectrum $E(k)\sim k^2$ (versus $\sim k^4$ for the 'Batchelor'-type data of Batchelor & Proudman (1956)), and this distinction controls the decay exponents and the fate of anisotropy at the large scales. Modern approximate treatments exist — notably eddy-damped quasi-normal Markovian (EDQNM) closures, a renormalized perturbative closure of the spectral two-point correlation equations, and direct or large-eddy simulations (DNS/LES) — but none closes or solves Chandrasekhar's physical-space coupled system under the compatibility constraint.

Problem Statement

Obtain a quantitative two-point theory of freely decaying homogeneous axisymmetric turbulence from Chandrasekhar's (1950) coupled scalar evolution equations

\frac{\partial Q_1}{\partial t}=2\nu\,\Delta Q_1+S_1,\qquad \frac{\partial Q_2}{\partial t}=2\nu\big(\Delta Q_2+2D_{\mu\mu}Q_1\big)+S_2,

for the gauge-invariant defining scalars $Q_1(r,r\mu,t)$ and $Q_2(r,r\mu,t)$ of the two-point velocity correlation tensor, where $\Delta=r^2D_{rr}+2r\mu D_{r\mu}+D_{\mu\mu}+5D_r$ , equivalently $\Delta=\frac{\partial^2}{\partial r^2}+\frac{4}{r}\frac{\partial}{\partial r}+\frac{1-\mu^2}{r^2}\frac{\partial^2}{\partial\mu^2}-\frac{4\mu}{r^2}\frac{\partial}{\partial\mu}$ , with $D_r=\frac{1}{r}\frac{\partial}{\partial r}-\frac{\mu}{r^2}\frac{\partial}{\partial\mu}$ , $D_\mu=\frac{1}{r}\frac{\partial}{\partial\mu}$ and doubled subscripts denoting iterated operators ( $D_{rr}=D_rD_r$ , $D_{r\mu}=D_rD_\mu$ , $D_{\mu\mu}=D_\mu D_\mu$ ), and $S_1,S_2$ are nonlinear sources built from six third-order correlation scalar potentials $T_1,\dots,T_6$ — the defining scalar potentials of the third-order correlation tensor, which is symmetric in two indices and solenoidal in the third — and from the pressure–velocity scalar $\varpi$ , which defines the solenoidal vector $P_i=\rho^{-1}\overline{p\,u_i'}$ through $P_i=-(r\mu D_r+D_\mu)\varpi\,\xi_i+(r^2D_r+r\mu D_\mu+2)\varpi\,\lambda_i$ . The gauge-invariant combinations

\begin{aligned} s_1&=\big(r^2D_r+r\mu D_\mu+5\big)T_4+\big(r\mu D_r+D_\mu\big)T_5,\\ s_2&=\big(r\mu D_r+D_\mu\big)T_2+D_\mu\big(T_3+\varpi\big)-D_\mu T_5+\big(r^2D_r+r\mu D_\mu+5\big)T_6,\\ s_3&=\big(r\mu D_r+D_\mu\big)T_6+D_r\big(T_3+\varpi\big)-D_\mu T_4+\big(r^2D_r+r\mu D_\mu+5\big)T_1,\\ s_4&=\big(T_3+\varpi\big)+T_5 \end{aligned}

of these potentials must satisfy the differential compatibility condition

\big(r^2D_r+r\mu D_\mu+4\big)s_3+\big(r\mu D_r+D_\mu\big)s_2=D_\mu s_1-D_r s_4,

which is the condition derived in Chandrasekhar (1950) for the nonlinear source tensor to be symmetric and solenoidal. The intermediate scalars $\sigma_1=s_1-s_2-r\mu s_3$ , $\sigma_2=s_2+D_\mu(s_4+r\mu s_2+r^2s_3)$ and $\sigma_3=s_3+D_r(s_4+r\mu s_2+r^2s_3)$ are the defining scalars of the source tensor $X_{ij}$ , and the two scalar source terms appearing in the evolution equations above are their even combinations $S_1(r,r\mu)=\sigma_1(r,+\mu)+\sigma_1(r,-\mu)$ and $S_2(r,r\mu)=\sigma_2(r,+\mu)+\sigma_2(r,-\mu)$ . The overarching objective is a solution theory of decaying homogeneous axisymmetric turbulence at the two-point level, comprising (i) a closure (or exact solution) of the system — that is, a specification of $S_1,S_2$ , either expressed in terms of $Q_1,Q_2$ or solved jointly with the third-order and pressure–velocity scalars, such that the induced nonlinear source tensor is symmetric and solenoidal — equivalently, such that $S_1,S_2$ are derivable from third-order and pressure–velocity potentials whose gauge-invariant combinations $s_1,\dots,s_4$ satisfy the differential compatibility condition stated above — and that preserves the realizability of the correlation tensor — that it be a genuine covariance function, with non-negative variances and Cauchy–Schwarz-type bounds on its components — the closure being approximate (model-based) or exact, subject to the compatibility condition and the realizability of the correlation tensor; and (ii) a determination of the behaviour of solutions during decay — which initial conditions and decay regimes produce isotropization and which produce persistent or re-emergent anisotropy, including the decay laws for the energy and the directional integral scales. The admissible data are those of the source paper: $Q_1,Q_2$ and the third-order potentials are taken continuous and bounded in $r$ and $r\mu$ , and in particular nonsingular at $r=0$ and free of non-integrable singularities or distributional components, with $Q_1,Q_2$ analytic in $r$ near $r=0$ (admitting the small-separation Taylor expansions) and decaying sufficiently rapidly as $r\to\infty$ for the radial integrals defining $q_1(\mu),q_2(\mu)$ to converge. The linear final period of decay ( $S_1=S_2=0$ ) was solved in Chandrasekhar's companion paper; the open problem concerns closures that determine $S_1,S_2$ rather than leaving them unclosed.

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