Pith. sign in

REVIEW 4 major objections 5 minor 18 references

Linear instability of plane Couette and Poiseuille flows

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Plane Couette and Poiseuille flows can become linearly unstable at finite Reynolds numbers once disturbances are quasi-periodic rather than strictly periodic.

desk verdict The claimed finite-Reynolds linear instability is an artifact of finite-window averaging over modes that are not solutions of the equation; the paper's real question is worth a footnote, not a revision. read the letter →

arxiv 2506.04242 v1 pith:TQHFA3VC submitted 2025-05-28 physics.flu-dyn

Sergey G. Chefranov , Alexander G. Chefranov This is my paper
classification physics.flu-dyn PACS 47.20.Ft47.20.-k
keywords planeCouetteflowPoiseuillelinearhydrodynamicstabilitythresholdReynoldsnumberquasi-periodicdisturbancesenergymethoddissipativeinstabilitynegative-energy
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

Plane Couette flow has long been counted linearly stable at every Reynolds number, and plane Poiseuille flow unstable only above about 5772; experiments, however, put the transition near $\mathrm{Re}\approx 150$ and $\mathrm{Re}\approx 1080$. The paper claims both standard predictions are artifacts of the “normal” disturbance ansatz: a single common longitudinal period and separated $x$–$z$ variables. Relaxing that ansatz so that each transverse mode carries its own longitudinal period, and averaging the disturbance energy over a finite window, makes the linearized equations exponentially unstable above $\mathrm{Re}_{\mathrm{th}}\approx 139$ for Couette flow and $\mathrm{Re}_{\mathrm{th}}\approx 1035$ for Poiseuille flow, within a few percent of experiment. If the claim holds, the onset of shear-flow transition no longer requires finite-amplitude seeds: infinitesimal disturbances can grow through a dissipative instability.

What carries the argument

The argument is carried by the disturbance representation (2), $V_y = \sum_n A_n(x) \sin((2n+1)\pi z/2) e^{\lambda_1 \tau}$ (plus a similar cosine term for Couette flow), with mode-dependent longitudinal periods $T_n = 2\pi/\alpha_n$ where $\alpha_n = p_n \alpha_1$. Because the periods differ, the energy integral $I_2$ in Eqs. (6) and (10) contains cross-mode factors $\sin^2(\pi(p_m - p_n))$ that do not vanish; in the traditional single-period ansatz $p_m = p_n$ those terms vanish, leaving only stable eigenvalues. The energy method then converts the balance into the threshold condition $\mathrm{Re} > I_1/I_2$, and a minimization over $\alpha_1$, the period-ratio parameters, and the amplitude fall-off exponent $k$ produces the reported $\mathrm{Re}_{\mathrm{th}}$ values.

What would settle it

Numerically integrate the linearized equation (1) with the quasi-periodic ansatz (2) using the reported minimizing parameters for Couette flow ($k=1.7037$, $p=0.4859$, $N=100$) at $\mathrm{Re}=140$: if the finite-window energy does not grow exponentially, or if the growth disappears as the averaging window $T_{\max}$ is extended, the claimed instability is an artifact of the finite-window average rather than a property of the linearized dynamics.

Watch

Extended reading notes

Core claim

The paper’s central claim is that the classical linear-stability verdicts for plane Couette and plane Poiseuille flow are not intrinsic to the linearized equations but are forced by assuming disturbances of the “normal” form — periodic in the flow direction with a period common to all transverse modes, and with separated spatial variables. Keeping only the transverse velocity component $V_y$ and writing it as a sum of $\sin((2n+1)\pi z/2)$ modes whose longitudinal periods $T_n = 2\pi/\alpha_n$ differ from mode to mode, the authors derive from the energy balance a threshold condition $\mathrm{Re} > I_1/I_2$, where the cross-mode integral $I_2$ is nonzero precisely because $p_m \neq p_n$. Minimizing the threshold over wavenumber, period ratios, and the amplitude fall-off exponent yields $\mathrm{Re}_{\mathrm{th}} \approx 139.077$ for Couette flow (experiment: $150 \pm 5$) and $\mathrm{Re}_{\mathrm{th}} \approx 1035.3$ for Poiseuille flow (experiment: about 1080), whereas the traditional linear results are linear stability for Couette and $\mathrm{Re}_{\mathrm{th}} \approx 5772$ for Poiseuille. The instability mechanism is identified as dissipative instability of negative-energy disturbances, with the zero boundary conditions, not the viscous term itself, playing the key role.

Load-bearing premise

The load-bearing assumption is that different transverse disturbance modes may carry different longitudinal periods and that the energy is averaged over a finite window; if one imposes the usual single common period for all modes, the cross-mode term that produces instability vanishes, and the flows return to their classical stable behavior.

Editorial extensions

If this is right

  • Linear theory alone would locate the onset of plane Couette transition at $\mathrm{Re}_{\mathrm{th}} \approx 139$, matching the experimentally observed $150 \pm 5$, without invoking finite-amplitude disturbances.
  • The plane Poiseuille threshold would move from the classical $\mathrm{Re}_{\mathrm{th}} \approx 5772$ down to $\approx 1035$, agreeing with the early experimental value of about 1080 to within 4%.
  • The stability threshold becomes a function of the disturbance’s period-ratio spectrum $p_n$ and amplitude fall-off exponent $k$, not just of amplitude, which explains the measured dependence of the Couette threshold on wire thickness in terms of wavelength ratios.
  • Because the mechanism is a dissipative instability of negative-energy disturbances, it works at infinitesimal amplitude and does not require a rigid finite-amplitude seed.
  • Direct numerical solution of the linearized equations with the non-separable ansatz should reproduce the predicted thresholds, which the paper identifies as a desirable check.

Reading between the lines

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

  • The same energy construction could be applied to other shear flows that classical linear theory declares stable, such as Hagen–Poiseuille pipe flow or boundary layers, potentially replacing finite-amplitude transition theories with a linear mechanism; the paper cites earlier work extending this idea to pipe flow.
  • A testable prediction of the paper’s logic is that intentionally seeding a Couette flow with two or more disturbances of different longitudinal wavelengths should trigger transition near $\mathrm{Re}\approx 139$, while a single-wavelength seed should remain stable — a direct experimental discriminator.
  • If the finite-window average is essential, then numerical simulations that impose periodic boundary conditions with a single fundamental period may suppress the instability, so fully resolved long-domain or quasi-periodic simulations could see linear growth where shorter periodic boxes do not.
  • The dependence of the threshold on the amplitude spectra $A_n$ suggests that experimental scatter in transition Reynolds numbers may encode the disturbance spectrum rather than noise, which could be tested by varying the spectral content of artificial disturbances.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The manuscript claims to overturn the classical linear-stability results for plane Couette and plane Poiseuille flow by replacing the normal-mode ansatz with a superposition of transverse modes having different longitudinal periods T_n and a single temporal growth rate λ1 (Eq. (2)), and by measuring disturbance energy over a finite longitudinal window T_max (Eq. (3)). From an energy balance the paper derives instability thresholds Reth≈139 for Couette flow and Reth≈1035 for Poiseuille flow (Eqs. (7)–(11)), and it compares these thresholds with experimental transition data. The central assertion is that abandoning the assumption of a common longitudinal periodicity produces a dissipative linear instability at finite Reynolds numbers.

Significance. If the result were correct, it would contradict a cornerstone of hydrodynamic stability theory: plane Couette flow is linearly stable at all Reynolds numbers within the standard normal-mode framework, and plane Poiseuille flow becomes linearly unstable only above Re≈5772. The paper's claimed agreement with experiments would therefore be remarkable. The manuscript makes its departure from standard theory explicit and provides closed-form expressions for the thresholds, which is a useful feature. However, the derivation is not a spectral analysis of the linearized operator, and the central ansatz is not shown to be an actual solution of the governing equation. Because the claimed instability rests on this ansatz, the significance is not established by the present manuscript.

major comments (4)
  1. [Section 2, Eq. (2)] The ansatz in Eq. (2) is not a valid solution of Eq. (1) for N>1. Since the coefficients of Eq. (1) are independent of x, streamwise Fourier modes are uncoupled invariant subspaces of the linear dynamics. Substituting the superposition in Eq. (2) into Eq. (1) and using linear independence of the functions e^{iα_n x} shows that each mode must separately satisfy L_{α_n} φ_n = λ1 φ_n with the same λ1. The manuscript never verifies that the chosen transverse functions are eigenfunctions of L_{α_n} or that distinct α_n share a common eigenvalue; for U(z)=z and U(z)=1−z² they are not eigenfunctions. Therefore the exponent λ1 computed in Eqs. (4)–(6) is not an eigenvalue of the linearized operator, and the reported instability is not an instability of Eq. (1).
  2. [Section 3, Eqs. (3) and (6)] The energy in Eq. (3) is averaged over a finite longitudinal window T_max=1/α1 rather than over a common period or the infinite line. This finite-window average makes modes with different longitudinal periods non-orthogonal and produces the nonzero cross-term I2 in Eq. (6), which is proportional to sin(π(p_n−p_m)). For an actual solution of Eq. (1), the streamwise Fourier components evolve independently, and their energies decouple: for each Fourier mode the advective contribution has zero real part because ∫ U(z) v* ∂_x v dx dz is purely imaginary after integration by parts. The nonzero I2 is therefore an artifact of the averaging window, not a property of the linear dynamics.
  3. [Section 3.1, Eqs. (7)–(11)] The numerical thresholds Reth≈1035 and Reth≈139 are obtained by minimizing Eq. (7) with respect to α1 and then scanning the free parameters k and p while imposing the specific choices p_n=n^{k/8} and A_n^0=(n+1)^{-k/3}. The resulting threshold depends directly on this parametric family and on the truncation N. Since I2 would vanish identically in any Fourier-orthogonal representation, the nonzero numerator is created entirely by the ansatz and window choices. The agreement with experimental transition thresholds is therefore a consequence of parameter selection within the proposed ansatz rather than an independent prediction of the linearized equations.
  4. [Abstract and Section 2] The abstract states that the result is obtained by abandoning the assumption of longitudinal periodicity of disturbances, but Section 2, immediately after Eq. (2), explicitly introduces 'new periodic boundary conditions along the x axis, which are set individually for each n-th mode' with periods T_n=q_n/(k_n α_n). The theory therefore retains periodicity; what is abandoned is only the common period and the separation of variables. This discrepancy affects the framing of the claimed novelty and should be corrected.
minor comments (5)
  1. [Abstract and References] The Abstract cites [16,17] for the experimental value Reth≈150±5, but the Bottin et al. experimental papers are [7,8] in the reference list; the citation numbering is inconsistent.
  2. [Equation (1)] Equation (1) is typeset with garbled symbols (for example, 'ΔΔ' and 'zV'), making the displayed PDE difficult to verify; a clean, standard display of the equation is needed.
  3. [Section 3.1, Eq. (8)] The minimization of Eq. (7) with respect to α1 is asserted to be 'readily checked,' but no derivation is provided; since this step yields the compact formula (8), the calculation should be shown.
  4. [Figure 2 caption] The notation '1/2p' in the figure caption is ambiguous; it is not clear whether it means 1/(2p), p/2, or p raised to a power, and the convention should be stated explicitly.
  5. [Section 3.2] The relation between the sine and cosine branches in Eq. (2) and the wire-generated disturbances in the cited experiments is not explained, and the passage from A_n and B_n to the special case A_n=B_n is stated without physical or mathematical justification.

Circularity Check

3 steps flagged · score 8.0 of 10

The claimed instability threshold is manufactured by the non-solution ansatz (2) and the finite-window norm (3): the destabilizing term I2 is nonzero only because modes with different periods are averaged over a finite window, and the quoted thresholds are minima over freely tuned parameters k, p, N.

  1. self definitional [Section 2, Eq. (2) and the paragraph following it]
    "Let us seek a solution of linear equation (1) in the following form that satisfies zero boundary condition in the direction of the z axis on solid boundaries: ... (2) ... These relations in (2) determines new periodic boundary conditions along the x axis, which are set individually for each n-th mode (n=0,1,2,…,N)."

    Equation (1) is linear with x-independent coefficients, so each streamwise Fourier mode with wavenumber α_n is an invariant subspace. A sum of terms e^{λ1τ}φ_n(z)e^{iα_n x} can be a solution only if each α_n component separately satisfies the same eigenvalue problem with the same λ1; the paper never establishes this. Substituting the whole non-solution superposition into the energy balance makes the cross-term integral I2 a property of the arbitrary representation, not of the linearized dynamics.

  2. self definitional [Section 3, Eqs. (3) and (6)]
    "Let us consider on the basis of Eqs. (1), (2) the evoluition in time of the average (on the unit of mass) energy of disturbances: ... (3) ... Where in (3) T_max=1/α_0 may be suggested. ... I2 = ... (6), where p_n=α_n/α_1."

    The nonzero coupling integral I2 exists only because the energy is averaged over the finite window T_max=1/α_0 while each mode is assigned its own longitudinal period. Under the standard Fourier norm with a common period, or in the limit T_max→∞, the cross terms vanish and I2=0, so the instability condition Re>I1/I2 disappears. The instability is therefore created by the chosen averaging window and period assignment, i.e., by the definition of the norm rather than by Eq. (1).

1 more flagged steps
  1. fitted input called prediction [Section 3.1, Eqs. (8)–(9) and Fig. 2 caption]
    "For the sake of simplicity, let us restrict consideration to the case where the minimization of expression (8) is performed only for parameters k and p, while the other parameters are assumed to be fixed and sufficiently slowly increasing functions of the corresponding number n=3,4,…,N (for the convergence of series (9) for b, let p_n=n^{k/8} for n≥3). ... The minimum threshold Reynolds number in this case, Reth min=1035.3, is achieved at N=100 for k=0.675 и p=0.506."

    The headline predictions Reth≈1035 and Reth≈139 are the minima of Re_th=I1/I2 with respect to free parameters k, p, N, with convergence functions p_n=n^{k/8} and initial amplitudes A_n^0≤(n+1)^{-k/3} chosen by the authors. These functions and parameters are not determined by Eq. (1); they are adjustable degrees of freedom in the ansatz. Reporting the optimized minimum as a predicted transition threshold is presenting the tuned ansatz output as a derived result.

full rationale

The derivation is not a self-contained consequence of the linearized PDE. The ansatz in Eq. (2) is called a solution, but it is a superposition of modes with different longitudinal periods under a common growth rate λ1; for a linear equation with x-independent coefficients this is not a solution unless every mode separately shares the same eigenvalue, which is never shown. The only destabilizing term, I2 in Eqs. (4) and (6), arises from the finite-window average in Eq. (3), which makes modes with different periods non-orthogonal. The paper itself concedes in Section 5 that the instability mechanism is not the dissipative term in Eq. (1) but the zero boundary condition 'taken into account in representation of the disturbance field in the form of Eq. (2).' The numerical thresholds 1035 and 139 are then obtained by minimizing Re_th over k, p, N and over ad hoc convergence functions, so the experimental agreement is a selected extremum of a tunable expression rather than an independent prediction. The citations to the authors' own prior work on the same 'non-separable' approach are consistent with this picture but are not the main circular step; the main circularity is internal to Eqs. (2), (3), and (6).

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on four ad hoc modeling choices: a non-eigenfunction ansatz with mode-specific periods, a finite-window energy average, the restriction to V_y, and the convergence-enforcing forms for p_n and A_n^0. Together with minimization over α1, k, p, and N, these choices generate the critical Reynolds numbers. No new physical entity is introduced. The external experimental comparison does not remove the dependence of the result on these choices.

free parameters (4)
  • k = 0.675 (PP); 1.7037 (PC)
    Free exponent controlling mode-spectrum decay p_n=n^{k/8} and amplitude scale A_n^0; chosen by minimizing Re_th, and the threshold depends on k.
  • p = 0.506 (PP); 0.4859 (PC)
    Ratio of the longitudinal periods of the first two modes. It is minimized over, and Figs. 2-3 show the threshold varies from infinite to 139 as p changes, so the result is highly sensitive to this free choice.
  • N = 100
    Number of terms in the mode sum. The critical Re changes with N (e.g., PC 124.27 for N=2, 139.08 for N=100), so the published value is a truncation-dependent numerical result.
  • base wavenumber α1 = minimized (α1_min=sqrt(b/a))
    Normalized to 1 and then minimized; the final threshold is the minimum over α1 as well as k and p, so the reported number is a lower envelope, not a parameter-free prediction.
assumptions (4)
  • ad hoc to paper Disturbance ansatz with mode-dependent longitudinal periods and a common temporal growth rate λ1 (Section 2, Eq. (2)).
    This is not an eigenfunction expansion of Eq. (1). Since U(z) does not depend on x, streamwise Fourier modes are independent, so modes with different periods cannot share a single λ1. The non-orthogonality of the modes in the finite-window energy integral is exactly what makes the advective term I2 nonzero.
  • ad hoc to paper Energy is averaged over a finite longitudinal window T_max rather than over a common period or the infinite line (Section 3, Eqs. (3) and (6)).
    Boundary contributions at x=0 and x=T_max are retained and produce the apparent growth. On an infinite domain or with a common period these cross terms vanish and the standard energy balance gives damping.
  • domain assumption Only the spanwise velocity component V_y is nonzero; V_x and V_z are set to zero (Section 2, after Eq. (1)).
    The full linearized problem needs V_z and the continuity equation; this restriction excludes the Orr-Sommerfeld modes used in classical stability analyses and is not justified as the worst case.
  • ad hoc to paper Convergence is forced by setting p_n=n^{k/8} and A_n^0=(n+1)^{-k/3} with a single exponent k (Section 3.1, Eq. (9)).
    These are arbitrary functional forms chosen so the series converge and to leave one tunable parameter k; they are not derived from the physics, and the threshold depends on their choice.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Linear instability of plane Couette and Poiseuille flows." pith.science (2026). https://pith.science/paper/TQHFA3VC

@misc{pith2026250604242,
  author       = {Pith},
  title        = {Pith review of: Linear instability of plane Couette and Poiseuille flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TQHFA3VC}},
  note         = {Machine review of arXiv:2506.04242}
}
read the original abstract

It is shown that linear instability of plane Couette flow can take place even at finite Reynolds numbers which meets with known experimental data. This new result of the linear theory of hydrodynamic stability is obtained only due by abandoning traditional assumption of the longitudinal periodicity of disturbances in the flow direction.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

18 extracted references · 14 canonical work pages

  1. [1]

    A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (Gidrometeoizdat, St. Petersburg, 1992; MIT, Cambridge, MA, 1971), Vol. 1

    work page 1992

  2. [2]

    S. A. Orszag, J. Fluid Mech. 50, 689 (1971). Doi 10.1017/S0022112071002842

    doi:10.1017/s0022112071002842 1971

  3. [3]

    S. A. Orszag and L. C. Kells, J. Fluid Mech. 96, 159 (1980). doi 10.1017/S0022112080002066

    doi:10.1017/s0022112080002066 1980

  4. [4]

    S. G. Chefranov and A. G. Chefranov, arXiv: 1007.1097v1 [physics.flu-dyn]

    work page Pith review arXiv

  5. [5]

    S. G. Chefranov and A. G. Chefranov, J. Exp. Theor. Phys. 119, 331 (2014). doi 10.1134/S1063776114070127

    work page doi:10.1134/s1063776114070127 2014

  6. [6]

    S. G. Chefranov and A. G. Chefranov, Dokl. Phys. 60, 327 (2015). doi 10.1134/S1028335815070083

    work page doi:10.1134/s1028335815070083 2015

  7. [7]

    Bottin, O

    S. Bottin, O. Dauchot, and F. Daviaud, Phys. Rev. Lett. 79, 4377 (1997). doi 10.1103/PhysRevLett.79.4377

    work page doi:10.1103/physrevlett.79.4377 1997

  8. [8]

    Bottin, O

    S. Bottin, O. Dauchot, F. Daviaud, and P. Manneville, Phys. Fluids 10, 2597 (1998). doi 10.1063/1.869773

    work page doi:10.1063/1.869773 1998

Show all 18 references
  1. [9]

    Lamb, Hydrodynamics (Cambridge Univ

    H. Lamb, Hydrodynamics (Cambridge Univ. Press, Cambridge, 1993)

    1993

  2. [10]

    S. G. Chefranov, JETP Lett. 73, 274 (2001). doi 10.1134/1.1374259

    2001 doi

  3. [11]

    S. G. Chefranov, Phys. Rev. Lett. 93, 254801 (2004). doi 10.1103/PhysRevLett.93.254801

    2004 doi

  4. [12]

    S. J. Davies and C. M. White, Proc. R. Soc. London A 119, 92 (1928). doi 10.1098/rspa.1928.0086

    1928

  5. [13]

    Couliou and R

    M. Couliou and R. Monchaux, Phys. Fluids 27, 034101 (2015). doi 10.1063/1.4914082 12

    2015 doi

  6. [14]

    Onorato and D

    M. Onorato and D. Proment, Phys. Rev. Lett. 107, 184502 (2011). doi 10.1103/PhysRevLett.107.184502

    2011 doi

  7. [15]

    Zakharov, J

    V. Zakharov, J. Appl. Mech. Tech. Phys. 9, 190 (1968). doi 10.1007/BF00913182

    1968 doi

  8. [16]

    T. B. Benjamin and J. E. Feir, J. Fluid Mech. 27, 417 (1967). doi 10.1017/S002211206700045X

    1967 doi

  9. [17]

    Rolland, arXiv: 1401.3586v1 [physics.flu-dyn]

    J. Rolland, arXiv: 1401.3586v1 [physics.flu-dyn]

    arXiv

  10. [18]

    Zammert and B

    S. Zammert and B. Eckhardt, arXiv: 1312.6783v1 [physics.flu-dyn]. 19. L. D. Landau, Zh. Eksp. Teor. Fiz. 11, 592 (1941) 13

    1941 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.