Source
Parish and Duraisamy (2017), Physical Review Fluids 2: 014604, LIT-1259, read in full in NOTE-973 from arXiv:1611.03311v2. §§III–V and Appendix B, Figs. 2–11.
What was actually shown
Memory length (Burgers, §III.B.1, Fig. 2). Sixty runs over ν ∈ {0.05, …, 0.0005}, cutoff k_c ∈ {8, 16, 32} and amplitude U₀* ∈ {1, 2, 5, 10}, against 4096-mode DNS. For each, τ₀ of the first-order model (FM1) was fitted by least squares on the dissipation rate over t ∈ [0, 2]. Plotted against 1/ρ(∂R/∂u), the time-averaged inverse spectral radius of the resolved Jacobian, the fits fall on one line for k_c = 16 and 32 and on a steeper one for k_c = 8. The paper gives τ₀ ≈ 0.2/ρ. Its reading: higher Reynolds number shortens the memory, coarser resolution lengthens it, and “the inferred memory length should not be viewed as a physical parameter.” Appendix B samples the slope as a Gaussian (the fits are bimodal) and finds the predicted energy and dissipation “well concentrated”, the subgrid term less so.
Burgers (Figs. 3–5). With 32 modes against 2048-mode DNS (ν = 0.01, U₀* = 1, τ₀ = 0.135), FM1–FM3 follow the energy, the dissipation rate with both its peaks, and the mean subgrid term. FM2 and FM3 also follow the spectrum at t = 2 at all wavenumbers. The t-model overpredicts the subgrid term after t ≈ 0.5, and Smagorinsky gets its time course wrong. In the coarsest extra case (k_c = 8, U₀* = 5), FM1 misses the second dissipation peak.
Navier–Stokes. All runs at 32³ or 64³ with τ₀ = 0.1.
- Isotropic turbulence (64³, from 512³ DNS, Re_λ ≈ 200, t = 4, Fig. 6). Spectra are comparable to dynamic Smagorinsky. In the subgrid transfer spectrum, FM1 alone puts the drain near the cutoff (k ≈ 33) as DNS does (k ≈ 31). Its peak is about −5 × 10⁻⁴ against DNS’s −3.5 × 10⁻⁴, and it shows slight backscatter below k ≈ 17. Both Smagorinsky models drain at low k. The paper: “it is perhaps not prudent to conclude that the M-Z-based models performed significantly better than the Smagorinsky models.”
- Taylor–Green vortex (32³, Re = 800 and 1600, Figs. 7–8). Smagorinsky dissipates from t = 0, though the flow starts resolved. The t-model over-dissipates after early times and needed a smaller time step. FM1 follows energy and dissipation, including the double peak near t = 8, and the spectra at t = 5 and 10.
- Channel flow (Re_τ = 180, 32 × 64 × 32, Fig. 9, pressure effects of coarse-graining neglected). FM1 and dynamic Smagorinsky give similar mean profiles and Reynolds stresses, both near the DNS of Moser, Kim and Mansour and better than no model.
What could have failed: FM1 has one constant, held at 0.1 across all four Navier–Stokes flows. It could have done as badly as the t-model on the Taylor–Green vortex, and the Burgers fits could have scattered with no relation to the resolved timescale.
What this does not say
- Not that the closure is more accurate than dynamic Smagorinsky in general. On isotropic turbulence and channel flow it is comparable; its advantage is shown on a flow that starts resolved, where, in the paper’s words, traditional subgrid models “do not perform well” because they are “designed for fully developed turbulent flows”.
- Not “excellent”, quantitatively. No error norm is reported anywhere. The comparisons are curves, and this account is read from them.
- Not that the memory length is a property of the flow. It moves with the cutoff. Part of Fig. 2’s collapse may be dimensional: at fixed k_c every timescale of Burgers scales with 1/(U₀k_c) and 1/(νk_c²), as ρ does. The paper does not test τ₀ against another timescale.
- Not that the τ₀ = 0.1 of the Navier–Stokes runs follows from the Jacobian rule. The paper names a second heuristic (the time step scaled by the ratio of grid size to Kolmogorov scale) and does not say which, if either, produced 0.1.
Connections
- THEORY-760 is what the closure is: a finite Markovian augmentation with an exponential kernel and no error bound. This account is how well it worked.
- CLAIM-331. Turbulence is the third source of slowness in the claim’s note of 2026-10-19: no gap between resolved and unresolved scales. Here the fitted memory is about a fifth of the fastest resolved timescale. So the eliminated scales’ influence decays faster than any resolved mode moves, a ratio set by the cutoff, not a spectral gap. That fits the claim’s “characteristically slower”, and shows the ordering is graded and moves with the chosen cut.
- QUESTION-219. What the closure carries is the energy flux across the cutoff (T^{SGS} = u*·w^{(0)}). It gets the flux’s location right, which Smagorinsky does not, by carrying it as a relaxing state variable. Eyink and Aluie (LIT-1239, unread) are the record’s pending source on how local that transfer is.