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REVIEW 3 major objections 6 minor 33 references

Large-eddy simulation and modeling of Taylor-Couette flow with an outer stationary cylinder

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that in turbulent Taylor-Couette flow with a stationary outer cylinder, the torque ratio grows as the square root of the Taylor number divided by the square of a Lambert-W function, a logarithmic-corrected non-power law.

desk verdict A genuinely new analytic torque law for TC flow, but the central asymptotic claim hangs on a single fitted closure constant; worth refereeing. read the letter →

arxiv 1908.06577 v1 pith:OITVKOUY submitted 2019-08-19 physics.flu-dyn

Wan Cheng , Dale I. Pullin , Ravi Samtaney This is my paper · ORCID
classification physics.flu-dyn PACS 47.27.Cn
keywords Taylor-Couetteflowturbulenttorquescalinglarge-eddysimulationconstantangularmomentumLambertWfunctionlogarithmicwalllayersrough-wallturbulenceNusseltnumber
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies the torque needed to sustain turbulent flow between a rotating inner cylinder and a stationary outer cylinder. It builds a minimalist three-zone picture: thin logarithmic wall layers at each cylinder, separated by a bulk region of constant angular momentum. The central result is a closed-form formula for the Nusselt number $Nu$ (the torque ratio): at fixed radius ratio $\eta<1$, $Nu$ grows like $\mathrm{Ta}^{1/2}$ divided by the square of the Lambert-W function of a quantity proportional to $\mathrm{Ta}^{1/4}$. That is a logarithmic correction to a half-power law, not a power law, so reported exponents below $0.5$ are only finite-range fits. If the model is right, the ultimate high-$\mathrm{Ta}$ state is nearly uniform angular momentum with vanishingly thin wall layers.

What carries the argument

The load-bearing construction is a one-dimensional, three-region radial model of the mean azimuthal velocity: region I is an inner-cylinder log layer with profile $U_\theta=\Omega_i R_i-u_{\tau i}(\kappa^{-1}\ln((r-R_i)u_{\tau i}/\nu)+A)$, region II has constant angular momentum $U_\theta=\frac{1}{2}\Omega_i R_i^2/r$, and region III is an outer log layer. The model is closed by the scaling relation $\delta_i=K u_{\tau i}/\Omega_i$, written as $\delta_i/d=\alpha Re_{\tau i}\eta/(Re_i(1-\eta))$ with $\alpha=2K=0.5$; this turns velocity matching at the inner-layer edge into a single algebraic equation for $Re_{\tau i}$. Solving that equation in the large-$Re_i$ limit introduces the Lambert-W function, whose sub-logarithmic growth produces the log-corrected torque law.

What would settle it

Measure $\delta_i$ and $u_{\tau i}$ separately in DNS or experiment at a second radius ratio, say $\eta=0.5$, over a range of $Re_i$: if $\delta_i\Omega_i/u_{\tau i}$ deviates from approximately 0.5 by more than the uncertainty, the closure—and with it the Lambert-W torque law—is falsified. Alternatively, bin existing $Nu(\mathrm{Ta})$ data at fixed $\eta$ by decade and check that the local slope drifts downward toward $1/2$ as $1/(\ln\mathrm{Ta})^2$ rather than settling on a constant exponent.

Watch

Extended reading notes

Core claim

The paper's claim is that the mean state of high-Reynolds-number Taylor-Couette flow with the outer cylinder at rest is captured quantitatively by matching two log-law layers against a known constant-angular-momentum core, $r u_\theta = \frac{1}{2}\Omega_i R_i^2$. The matching produces an algebraic equation for the inner friction Reynolds number $Re_{\tau i}$, and in the large-$Re_i$ limit the solution is $Re_{\tau i}\propto Re_i/W(Z_1)$ with $Z_1\propto \eta^{1/2}Re_i^{1/2}/(1-\eta)^{1/2}$. Converting to torque gives $Nu(\mathrm{Ta},\eta)=\kappa^2\eta^3\mathrm{Ta}^{1/2}/(4(1+\eta)^2 W(Z_2)^2)$ with $Z_2\propto \eta^{3/2}\mathrm{Ta}^{1/4}/((1-\eta)^{1/2}(1+\eta)^{3/2})$. At enormous $\mathrm{Ta}$ this becomes $Nu\sim 4\kappa^2\eta^3\mathrm{Ta}^{1/2}/((1+\eta)^2(\ln\mathrm{Ta})^2)$, a genuine non-power-law asymptote. The same model, extended with a Colebrook roughness function, predicts that a rough inner wall produces a fully rough plateau with $C_f$ independent of $Re_i$ and $Nu\sim\mathrm{Ta}^{1/2}$.

Load-bearing premise

The quantitative torque law rests on the assumption that the inner wall-layer thickness is proportional to the local friction velocity divided by the cylinder rotation speed, with the constant fixed at 0.5 from one DNS point; if that ratio depends on radius ratio or Reynolds number, the claimed asymptote fails.

Editorial extensions

If this is right

  • At fixed $\eta<1$, $Nu(\mathrm{Ta})$ has no power-law asymptote; the true large-$\mathrm{Ta}$ behavior is $Nu\sim \mathrm{Ta}^{1/2}/(\ln\mathrm{Ta})^2$, so local power-law fits with exponent below $0.5$ must drift downward as $\mathrm{Ta}$ increases.
  • The mean azimuthal flow approaches $u_\theta=\frac{1}{2}\Omega_i R_i^2/r$ across almost the whole gap, while the wall-layer thicknesses $\delta_i/d$ and $\delta_o/d$ shrink like the inverse of a Lambert-W function.
  • The model reproduces DNS, LES, and experimental $Nu(\mathrm{Ta},\eta)$ data for $\eta=0.5$, $0.72$, and $0.909$ across $\mathrm{Ta}\simeq10^{10}$ to $10^{13}$, including the decline of inner-wall boundary-layer measures with $Re_i$.
  • For a sand-grain-rough inner wall, the model predicts a fully rough plateau: $C_f$ independent of $Re_i$, $\delta_i/d$ independent of $Re_i$, and $Nu\sim\mathrm{Ta}^{1/2}$.
  • The model is intended for practical radius ratios roughly $0.6\le\eta<1$; it does not cover the plane-Couette limit $\eta\to1$, where the predicted $\delta_i/d$ would diverge.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the Lambert-W form is generic, then single-exponent fits to $Nu(\mathrm{Ta})$ are inherently temporary; the local slope should decrease logarithmically with $\mathrm{Ta}$, a drift that can be checked by decade-binning existing torque data.
  • The same three-region construction could be carried over to other rotating wall-bounded flows with roll-driven angular-momentum mixing, but the closure constant $K$ would need independent measurement rather than being assumed universal.
  • The rough-wall branch gives a concrete engineering target: torque measurements on roughened cylinders should plateau as $Re_i$ grows, a Moody-diagram-style saturation that existing Taylor-Couette facilities could test directly.
  • The deeper physical bet is that the constant-angular-momentum bulk persists to arbitrarily large $\mathrm{Ta}$; if Taylor rolls or the bulk itself change character at extreme driving, the asymptotic state would differ even if the wall-layer closure held.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper presents wall-resolved large-eddy simulations (LES) of Taylor–Couette flow with a rotating inner cylinder and a stationary outer cylinder at radius ratio η = 0.909 and inner Reynolds numbers up to Rei = 3×10^6, using the stretched-vortex subgrid model. The LES are verified against DNS at Rei = 10^5 and 3×10^5 and are then used to characterize the mean azimuthal velocity, turbulence intensities, and angular-momentum profiles. The authors propose a one-dimensional, three-region empirical model: log-law wall layers on each cylinder separated by a central region of constant angular momentum L = Ωi Ri^2/2. The model is closed by the scaling hypothesis δi = K uτ_i/Ωi, with the constant fixed to K = 0.25 (α = 2K = 0.5) using one DNS point. The model yields an analytic Lambert-W formula for Nu(Ta,η), predicts that Nu grows as Ta^{1/2} divided by the square of a Lambert-W function, and describes an asymptotic state of nearly constant angular momentum with vanishingly thin wall layers. An extension to sand-grain roughness predicts a fully rough regime with Nu ∼ Ta^{1/2} and Cf independent of Rei.

Significance. If the central closure is accepted, the paper provides a simple, closed-form description of the torque scaling in turbulent Taylor–Couette flow with a stationary outer cylinder, including a concrete non-power-law asymptote that could organize existing experimental and numerical data. The new wall-resolved LES data at Rei up to 3×10^6 for η = 0.909 are a useful resource, and the rough-wall extension gives a testable Moody-diagram analogue for this geometry. The algebraic derivations are transparent and the final formulas are easy to use. The main caveat is that the quantitative predictions and the asymptotic state are contingent on the empirically calibrated closure (4.7), and the paper would be strengthened by a clear statement of that conditionality and by a sensitivity analysis of the fitted constant.

major comments (3)
  1. [4.1] The closure δi = K uτ_i/Ωi is the sole load-bearing assumption of the model, yet it is introduced without derivation, and the constant is fixed by matching a single DNS point (Rei = 10^5, Reτi = 1410). That same DNS point reappears in the validation (Table 2 and Fig. 7), so the agreement there is a consistency check rather than an independent confirmation. The comparisons at other η (experiments) and at higher Rei (present LES) do provide indirect support for K being independent of η and Ta over the tested range, but the asymptotic claims in §4.6 and the non-power-law form in Eq. (4.15) rely on (4.7) holding at arbitrarily large Ta, far beyond the validated range. Please add a sensitivity study with respect to α (e.g., how Nu(Ta,η) and the inferred asymptotic state vary for α = 0.25, 0.5, 0.75, 1.0), and either present direct evidence for the scaling δi ∝ uτ_i/Ωi from DNS/LES at several η and Rei or explicitly frame the asymptotic state and the Lambert-W law as model-based conjectures in the abstract and conclusions.
  2. [4.2] The approximate analytical solution leading to Eq. (4.15) is obtained after substituting α = 1/2 into Eq. (4.9) and then dropping subdominant terms, and the final formula contains no dependence on α. This may mislead readers into thinking the Lambert-W expression is parameter-free. In reality, for α ≠ 1/2 the argument of the Lambert function acquires a factor √α (through the 2αη term in the logarithmic argument), and the quantitative prediction changes. Please state explicitly that the functional form—the logarithmic correction to Ta^{1/2}—is insensitive to the calibrated value, but that the specific formula (4.15) and the numerical comparisons are tied to α = 0.5.
  3. [3.1] The LES validation of the model at high Rei rests on a computational domain of Δθ = π/10 and Ly = 2πd/3. As the authors themselves note in §3.4, Ostilla-Monico et al. (2015b) found that finite-domain effects on the near-wall log region and on turbulence statistics are non-negligible at moderate Reynolds numbers. Because the high-Rei LES (up to Rei = 3×10^6) are used to support the model’s Ta dependence, the paper should quantify or at least discuss the possible influence of this narrow domain on the mean azimuthal velocity and torque, or clearly state the resulting uncertainty in the high-Rei validation.
minor comments (6)
  1. [§2.1–2.2] The heading 'Numerical method' appears twice; the second occurrence should be renumbered or renamed to distinguish the curvilinear formulation.
  2. [§2.2] The citation 'Cheng et al. 2017, 2018, ?' contains a bare question mark; the reference list is incomplete and should be fixed.
  3. [§4.5, Eq. (4.19)] The definition of δ99 uses the inequality U(r) - U_target < 0.01, which is not sign-definite; it should likely be |U(r) - U_target|/U_target < 0.01.
  4. [throughout] The spelling of the author name 'Ostilla-Monico' is inconsistent (e.g., Ostilla-M´onico vs. Ostilla-Mnico) in the text and in the reference list.
  5. [§4.5] The sentence 'Once the parameters Reτi and ηi have been determined' should refer to Reτi and δi rather than ηi.
  6. [Abstract] The abstract contains the sentence 'With Ri,Ro the inner and outer radii respectively, the radius ratio is η = 0.909', which reads as if the paper only treats one radius ratio, whereas the model is applied for η = 0.5, 0.72, and 0.909. Please rephrase to avoid confusion.

Circularity Check

1 steps flagged · score 4.0 of 10

The model's closure constant α is calibrated to a single DNS point that is then reused as a validation entry; the Lambert-W and asymptotic claims otherwise follow from the stated closure, so the circularity is partial, not total.

  1. fitted input called prediction [§4.1, Eqs. (4.7)–(4.10); Table 2]
    "In their DNS of TC flow with η = 0.909, Ostilla-Mónico et al. (2016) report Reτi = 1410 at Rei = 10^5. Solving (4.9) with these parameters and with α = 0.25, 0.5, 0.75, 1.0 gives Reτi = 1529, 1418, 1360, 1323 respectively. For all subsequent calculations with the present model, we will use α = 0.5 which gives satisfactory agreement with DNS for this case."

    The free parameter α=2K in the closure δ_i = K u_τi/Ω_i is chosen so that the model reproduces the DNS value Reτi=1410 at Rei=10^5, η=0.909. Table 2 then lists that same DNS value (1410) alongside the model outputs (1418 from Eq. (4.10), 1426 from Eq. (4.12)) as an apparent validation, and the same point enters the Nu(Ta) comparison of Fig. 7. Consequently, the agreement at this anchor point is enforced by the calibration rather than predicted independently. The remaining rows, other radius ratios, and the Lambert-W asymptotic structure are not fixed by this one point, so the circularity is local and does not reduce the central claim.

full rationale

The core derivation is self-contained: Eqs. (4.1)–(4.6) define a three-region log-layer model; Eq. (4.7) is an explicitly stated closure, not a hidden one; substituting it produces Eq. (4.10), whose approximate solution (4.12) has Lambert-W form, and combining with Eq. (4.14) yields the Nu(Ta,η) law (4.15). This is a genuine derivation conditional on the closure, not a renaming of data. The only clear circular step is the reuse of the DNS anchor point used to set α=0.5: that point appears again in Table 2 and in the Nu validation, so that particular agreement is by construction. However, the asymptotic logarithmic correction in Eq. (4.17) is independent of α, and the model is compared with independent experiments and DNS/LES at other Rei and η. The uniform-angular-momentum bulk (4.20) is assumed in region II of the model, so that element of the abstract's asymptotic state is a premise; the derived content is the wall-layer thinning and turbulence-intensity decline. Thus the central claim has substantive independent content, and the overall circularity score is 4 rather than higher.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The model rests on one calibrated constant alpha, on the standard log-law description of wall turbulence, and on the assumed persistence of a constant-angular-momentum core. No new physical entities, forces, or conserved quantities are introduced.

free parameters (1)
  • alpha (closure coefficient) = 0.5
    Introduced in Eq. (4.8) as alpha = 2K, where delta_i = K u_tau_i / Omega_i. The text reports that alpha = 0.5 gives the best match to DNS Re_tau_i = 1410 at Rei = 1e5, eta = 0.909. All quantitative model predictions inherit this calibration.
assumptions (3)
  • domain assumption The mean azimuthal velocity in each wall layer follows the classical log law with kappa = 0.4 and A = 4.5.
    Used in Eqs. (4.1) and (4.3) to construct the wall-layer profiles. These constants are imported from prior empirical knowledge and are not derived in this paper.
  • domain assumption A central region of constant angular momentum L = 0.5 exists between the wall layers and persists for arbitrarily large Taylor number.
    Stated in Section 4.1 and extrapolated in Section 4.6. The finite-Rei DNS and LES support a nearly constant L = 0.5 core, but the extension to asymptotically large Ta is an assumption.
  • domain assumption The wall layers are thin enough that matching the log-law profiles at r = Ri + delta_i and r = Ro - delta_o with the constant-angular-momentum profile is a valid approximation.
    The model's three-region construction requires this matching. The paper itself notes the approximation breaks as eta approaches 1, where delta_i/d diverges (Section 4.2).

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Cite this review

Pith. "Pith review of Large-eddy simulation and modeling of Taylor-Couette flow with an outer stationary cylinder." pith.science (2026). https://pith.science/paper/OITVKOUY

@misc{pith2026190806577,
  author       = {Pith},
  title        = {Pith review of: Large-eddy simulation and modeling of Taylor-Couette flow with an outer stationary cylinder},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OITVKOUY}},
  note         = {Machine review of arXiv:1908.06577}
}
abstract

We present wall-resolved large-eddy simulations (LES) of the incompressible Navier-Stokes equations together with empirical modeling for {turbulent} Taylor-Couette {(TC)} flow where the inner cylinder is rotating with angular velocity $\Omega_i$ and the outer cylinder is stationary. A simple empirical model of the turbulent, TC flow is developed consisting of near-wall, log-like turbulent wall layers separated by an annulus of constant angular momentum. The model is closed by a proposed scaling relation concerning the thickness of the wall layer on the inner cylinder. Model results include the Nusselt number $Nu$ (torque required to maintain the flow) and various measures of the wall-layer thickness as a function of both the Taylor {number} $Ta$ and $\eta$. These agree reasonably with experimental measurements, direct numerical simulation (DNS) and the present LES over a range of both $Ta$ and $\eta$. In particular, the model shows that, at fixed $\eta<1$, $Nu$ grows like $Ta^{1/2}$ divided by the square of the Lambert, (or Product-Log) function of a variable proportional to $Ta^{1/4}$. This cannot be represented by a power law dependence on $Ta$. At the same time the wall-layer thicknesses reduce slowly in relation to the cylinder gap. This suggests an asymptotic, very large $Ta$ state consisting of constant angular momentum in the cylinder gap with $u_\theta = 0.5\,\Omega_i\,R_i^2/r$, where $r$ is the radius, with vanishingly thin turbulent wall layers at the cylinder surfaces. An extension of the model to rough-wall turbulent wall flow at the inner cylinder surface is described. This shows an asymptotic, fully rough-wall state where the torque is independent of $Re_i/Ta$, and where $Nu\sim Ta^{1/2}$.

Figures

Figures reproduced from arXiv: 1908.06577 by the authors.

Figure 1
Figure 1. Flow configuration for Taylor-Couette flow with rotating inner cylinder (Ωi 6= 0) and stationary outer cylinder (Ωo = 0). Ri is the radius of the inner cylinder, Ro is the radius of the outer cylinder, d = Ro − Ri. Rei Nθ Nr Ny T a Reτi ri∆θ + ∆r + min ∆y + 1 × 105 256 256 1024 1.108 × 1010 1.400 × 103 34.3 0.75 5.74 3 × 105 512 512 1536 9.969 × 1010 3.908 × 103 47.9 0.54 10.7 6 × 105 1024 512 2048 3.988 × 1011 7.28… view at source ↗
Figure 2
Figure 2. Visualization of an instantaneous flow field in a radial-spanwise plane: r 0 = (r − Ri)/d, θ. Rei = 105 , η = 0.909. (a), streamlines of the azimuthal-averaged flow field (ur, uy); (b), instantaneous azimuthal velocity field at mid-span plane; (c), instantaneous span-wise velocity field at mid-span plane. parameters defining the flow are then the radius ratio η and the inner-cylinder Reynolds number Rei . The inner-… view at source ↗
Figure 3
Figure 3. Comparison of LES with DNS by Ostilla-M´onico et al. (2016). (a), mean flow velocity profiles U + at Rei = 105 ; (b), turbulent intensities (u 0 θu 0 θ ) +,(u 0 yu 0 y) +,(u 0 ru 0 r) + at Rei = 105 . (c), U + at Rei = 3 × 105 ; (b), turbulent intensities at Rei = 3 × 105 . Square symbols: DNS by Ostilla-M´onico et al. (2016). Solid lines with filled squares: present LES. N = NθNrNz is 1/32 of that for the correspon… view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Mean velocity profile for all present LES. (a), the mean azimuthal velocity versus r +; (b), parameter Ξ versus 2r 0 with r 0 = (r−Ri)/d. Lines for different Rei: , 105 , ; 3 × 105 ; 6 × 105 ; , 106 ; , 3 × 106 . 100 101 102 103 104 0 2 4 6 8 10 (u’u’)+ r + (a) [PITH_…
Figure 5
Figure 5. Figure 5: Turbulent intensities for all present LES. (a), versus r +; (b), versus 2r 0 . Lines for different Rei: , 105 ; , 3 × 105 ; , 6 × 105 ; , 106 ; , 3 × 106 . all higher Rei LES extends to about 1% of the half gap, which is consistent with DNS at Rei up to 3 × 105 . 3.4. …
Figure 6
Figure 6. Figure 6: Nondimensional angular momentum profiles. L = r Uθ/(Ωi R 2 i ) lines for different Rei: , 105 ; , 3 × 105 ; , 6 × 105 ; , 106 ; , 3 × 106 . of the present study. The issue is interesting as indicated by the substantial vari￾ation in profile shapes indicated in figure 1…
Figure 7
Figure 7. Figure 7: Nu versus T a. Open symbols; experiment η = 0.909 ( ), 0.72(◦) Van Gils et al. (2011, 2012); η = 0.5 (4) Merbold et al. (2013). ; DNS of η = 0.909 by Ostilla-M´onico et al. (2016) . N; present LES of η = 0.909. Lines: from (4.15). ; η = 0.909. ; η = 0.72. ;η = 0.5 [PI…
Figure 8
Figure 8. Figure 8: Nu versus η. Dashed line T a = 1011. Solid line T a = 1012. Symbols key; see figure 7 4.4. Angular momentum profiles It is straightforward to calculate profiles of the angular momentum L from the model. When normalized such that L = rUθ/(Ωi R2 i ), this gives, with r 0…
Figure 9
Figure 9. Figure 9: Radial angular momentum profiles. Model compared with DNS and LES. (a); Rei = 105 . (b), Rei = 3 × 106 . , model prediction; , present LES; , DNS by Ostilla-M´onico et al. (2016). II: L = 0.5, δi/d 6 r 0 6 1 − δ0/d, III: L =  (1−η)r 0 η + 1 2 η Reτ1 Rei [PITH_FULL_I…
Figure 10
Figure 10. Figure 10 [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: Skin-friction coefficient Cf for different roughness level. Left : Cf versus Rei for η = 0.909. Solid lines: model prediction. Top to bottom  = ks/d = 4 × 10−3 , 2 × 10−3 , 10−3 , 4 × 10−4 , 10−4 , 10−5 , 10−6 , 0. , DNS by Ostilla-M´onico et al. (2016); N, present L…
Figure 12
Figure 12. Figure 12: Nu versus T a for rough, inner cylinder walls. Left η = 0.5, right η = 0.909. Solid lines: model prediction. Dashed line; slope 1/2. Symbol key; see figures 7 and 11. Hence for fully rough-wall, turbulent wall layer flow, the skin friction, and there￾fore the torque r…

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Reviewed August 14, 2026 · model on record in the stance chip above.