Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distinct transverse-response signatures of retained-spin, eliminated-spin, and polynomial Burnett-type surrogate closures

Satori Tsuzuki

Abstract

High-curvature observables in incompressible flows, including $k^4$-weighted spectra, can arise from explicit internal rotation, elimination of a fast spin variable, or polynomial higher-gradient closure. Building on a retained-spin micropolar closure derived separately from the Boltzmann--Curtiss equation, we show that these mechanisms are dynamically distinguishable in transverse linear response. In a fast-spin regime the retained-spin theory reduces to a one-field model with a rational $k$-dependent kernel whose low-$k$ expansion generates $k^4$ and $k^6$ terms, while preserving the large-$k$ roll-off of the eliminated degree of freedom. We compare four closures: incompressible Navier--Stokes, a polynomial Burnett-type surrogate, the explicit-spin micropolar theory, and the eliminated-spin rational-kernel theory. The explicit-spin theory has two poles, the eliminated-spin theory retains only the slow pole, and finite polynomial truncations fail qualitatively: a strict $k^4$ truncation becomes over-damped, while a matched $k^6$ truncation develops near-critical amplification and finite-$k$ instability. Many-particle event-driven simulations of perfectly rough spheres show that these observables are measurable and, in targeted campaigns, discriminating at the microscopic level: fixed-$k$ and multi-$k$ harmonic forcing resolve a finite spin-to-vorticity phase lag that strongly favors retained-spin dynamics over instantaneous adiabatic elimination, while the stronger-drive multi-$k$ vorticity response rejects a pure $k^2$ closure and favors the rational eliminated-spin kernel over a polynomial surrogate. Transverse response thus provides a practical diagnostic for separating retained rotational microphysics, eliminated-spin effective dynamics, and ordinary polynomial higher-gradient closures.

Distinct transverse-response signatures of retained-spin, eliminated-spin, and polynomial Burnett-type surrogate closures

Abstract

High-curvature observables in incompressible flows, including -weighted spectra, can arise from explicit internal rotation, elimination of a fast spin variable, or polynomial higher-gradient closure. Building on a retained-spin micropolar closure derived separately from the Boltzmann--Curtiss equation, we show that these mechanisms are dynamically distinguishable in transverse linear response. In a fast-spin regime the retained-spin theory reduces to a one-field model with a rational -dependent kernel whose low- expansion generates and terms, while preserving the large- roll-off of the eliminated degree of freedom. We compare four closures: incompressible Navier--Stokes, a polynomial Burnett-type surrogate, the explicit-spin micropolar theory, and the eliminated-spin rational-kernel theory. The explicit-spin theory has two poles, the eliminated-spin theory retains only the slow pole, and finite polynomial truncations fail qualitatively: a strict truncation becomes over-damped, while a matched truncation develops near-critical amplification and finite- instability. Many-particle event-driven simulations of perfectly rough spheres show that these observables are measurable and, in targeted campaigns, discriminating at the microscopic level: fixed- and multi- harmonic forcing resolve a finite spin-to-vorticity phase lag that strongly favors retained-spin dynamics over instantaneous adiabatic elimination, while the stronger-drive multi- vorticity response rejects a pure closure and favors the rational eliminated-spin kernel over a polynomial surrogate. Transverse response thus provides a practical diagnostic for separating retained rotational microphysics, eliminated-spin effective dynamics, and ordinary polynomial higher-gradient closures.

Paper Structure

This paper contains 21 sections, 71 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Retained-spin closure and fast-spin reduction
  3. Starting retained-spin equations
  4. Fast-spin elimination and effective one-field closure
  5. Transverse linear response and diagnostic models
  6. Linear response in the transverse sector
  7. Three diagnostic relations
  8. Diagnostic summary
  9. Reduced-model single-mode benchmarks
  10. Dispersion relations and Burnett stability
  11. Setting 1: free decay and instantaneous decay rate
  12. Setting 2: harmonic forcing and frequency response
  13. Diagnostic maps in the $(k,\Omega)$ plane
  14. Additional integrated response diagnostics
  15. Microscopic EDMD observability benchmarks
  16. ...and 6 more sections

Figures (18)

  • Figure 1: Transverse dispersion relations $s(k)$ (decay/growth rates) for two polynomial surrogate choices. Left: strict $k^4$ polynomial truncation $B^{(4)}$ (without the $k^6$ term), which remains stable but becomes increasingly over-damped at large $k$. Right: matched polynomial truncation $B^{(6)}$ (with the $k^6$ term), for which the damping coefficient changes sign and Model B becomes unstable for $k>k_{\mathrm{crit}}$. In both panels the left axis shows $\mathrm{Re}\,s(k)$ for Models A, C (slow/fast), and D, while Model B is plotted on a separate right axis.
  • Figure 2: Stability maps for Model B in the $(k,\Omega)$ plane. Left: strict $k^4$ polynomial truncation $B^{(4)}$, which is stable everywhere in the scanned domain. Right: matched truncation $B^{(6)}$, for which stable (decaying) parameter values occur for $k<k_{\mathrm{crit}}$ and unstable growth occurs for $k>k_{\mathrm{crit}}$. The boundary is independent of $\Omega$ because stability is controlled by the sign of the static damping coefficient.
  • Figure 3: Setting 1 (free decay): time evolution of the transverse vorticity mode amplitude $|\zeta(t;k)|$. Columns correspond to $k=1$, $k=2.3\simeq 0.96\,k_{\mathrm{crit}}$, and $k=4>k_{\mathrm{crit}}$ (for $B^{(6)}$). Top row: strict $k^4$ polynomial truncation $B^{(4)}$ (without the $k^6$ term). Bottom row: matched truncation $B^{(6)}$ (with the $k^6$ term). The top row stays stable but becomes increasingly over-damped at large $k$, whereas the bottom row becomes weakly damped near $k_{\mathrm{crit}}$ and unstable beyond it. Model C exhibits a short fast-spin transient and then follows Model D closely in both cases.
  • Figure 4: Setting 1: instantaneous decay rate $\lambda(t;k)$ defined in Eq. (\ref{['eq:inst_decay_rate']}). Columns correspond to the same $k$ values as in Fig. \ref{['fig:setting1_decay']}. Top row: $B^{(4)}$. Bottom row: $B^{(6)}$. A and D show the expected one-pole behavior ($\lambda$ essentially constant), while Model C shows a short-time transient associated with the fast pole and then converges to the slow rate. The contrast between the two polynomial surrogate variants is again clear: stable but increasingly rapid damping for $B^{(4)}$ versus near-critical slowing and loss of stability for $B^{(6)}$.
  • Figure 5: Setting 2: Bode amplitude $|\chi_{\zeta\zeta}(-i\Omega,k)|$. Columns correspond to $k=1$, $k=2.3\simeq 0.96\,k_{\mathrm{crit}}$, and $k=4>k_{\mathrm{crit}}$ (for $B^{(6)}$). Top row: $B^{(4)}$. Bottom row: $B^{(6)}$. The strict truncation remains stable but becomes increasingly suppressed at large $k$, while the matched truncation shows the largest diagnostic separation near $k_{\mathrm{crit}}$ at low $\Omega$ and ceases to admit a physical steady response for $k>k_{\mathrm{crit}}$.
  • ...and 13 more figures