Wall-to-wall optimal transport in two dimensions

Andre N. Souza; Charles R. Doering; Ian Tobasco

REVIEW 3 major objections 5 minor 27 references

Wall-to-wall optimal transport in two dimensions

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Two-dimensional wall-to-wall heat-transport optimizers switch from Nu - 1 ~ Pe^2 to Nu ~ Pe^0.54, with a separable-ansatz upper bound of Pe^{6/11}.

desk verdict The paper's headline bound is honestly conditional, and the numerics don't quite verify the separability assumption the bound uses—still a worthwhile contribution for the convection-bounds crowd. read the letter →

arxiv 1908.02896 v2 pith:2DLPXHVA submitted 2019-08-08 physics.flu-dyn math.FAmath.OC

Andre N. Souza , Ian Tobasco , Charles R. Doering This is my paper · ORCID

classification physics.flu-dynmath.FAmath.OC

keywords wall-to-walloptimaltransportNusseltnumberPécletgradientascentseparableansatzRayleigh-Bénardconvectionvariationalboundsno-slipboundaryconditions

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 asks which incompressible flow between two parallel no-slip walls, at a fixed root-mean-square vorticity (the Péclet number $Pe$), transports the most heat. Using gradient ascent on a variational formulation, it computes two-dimensional steady optimizers up to $Pe\approx10^5$ and finds that the Nusselt number $Nu$, the total heat transport divided by the conductive value, grows as $Nu-1\sim Pe^2$ for small $Pe$ and then as $Nu\sim Pe^{0.54}$ for $Pe>10^3$. The computed optimizing fields are nearly separable, products of a wall-normal and a wall-parallel profile, and the dominant mode carries about 99% of the transport. Restricting the variational problem to exactly separable fields yields a conditional upper bound $Nu\lesssim Pe^{6/11}=Pe^{0.\overline{54}}$, matching the computed scaling. This matters because upper bounds on wall-to-wall transport convert directly into upper bounds on heat transport in Rayleigh-Bénard convection through $Pe^2=Ra(Nu-1)$.

What carries the argument

The engine is the symmetrized variational functional $$S = \langle |\nabla\eta|^2 + 2\xi u\cdot(\hat z-\nabla\eta) - |\nabla\xi|^2 + \mu($Pe^{2}$-|\nabla u|^2)\rangle,$$ obtained from the Nusselt functional by writing the temperature deviation as $\theta=\xi+\eta$ and the adjoint variable as $\phi=\xi-\eta$; its saddle points are the wall-to-wall optimizers. Gradient ascent on $S$—ascent in $\xi$ and $u$, descent in $\eta$—produces the computed flows. For the analytic bound, the key object is the Howard-type functional $$M[\$\sigma$,v;Pe] = \frac{(\langle v_3\$\sigma$\rangle)^2}{$Pe^{{-2}}$\langle|\nabla\$\sigma$|^2\rangle\langle|\nabla v|^2\rangle + \langle(v_3\$\sigma$-\langle v_3\$\sigma$\rangle)^2\rangle},$$ which bounds $Nu-1$ from above. Under the separable ansatz, elementary inequalities—Cauchy-Schwarz, the interpolation bound $[(\varphi'')^2][\varphi^2]\ge1$, and Howard's lemma—give a denominator bounded below by $\delta + Pe^{-2}\delta^{-8/3}$, whose minimum is $\sim Pe^{-6/11}$. That minimization is what produces the conditional upper bound $Nu\lesssim Pe^{6/11}$.

What would settle it

Run gradient ascent from many random initial conditions at $Pe\approx10^5$ and compare the best Nusselt number with the value on the published $Nu$–$Pe$ curve; if any flow exceeds that curve by more than numerical discretization error, the claimed optimal scaling fails. Alternatively, project the computed optimizer onto the separable subspace at that $Pe$ and measure $(N_1-N_2)/N_1$: the paper reports this transport error is at most 1%, so an independent computation that finds a non-separable optimizer with $(N_1-N_2)/N_1$ substantially larger than 1% would show that the separability assumption is the active restriction.

Watch

Extended reading notes

Core claim

The central claim is that in two dimensions with no-slip boundaries, the local maximizers of the wall-to-wall transport problem—flows maximizing convective heat transfer at prescribed enstrophy—switch scaling at $Pe\sim10$, entering a regime where $Nu\sim Pe^{0.54}$ that persists to at least $Pe\approx2.5\times10^5$. Numerically, the optimizers are nearly separable in their singular-value decomposition, with the first mode carrying more than 99% of $Nu-1$, and the horizontal profile resembles Jacobi elliptic functions rather than sinusoids. Under the explicit separable ansatz $u_1=-\Psi'(z)\varphi(x)$, $u_3=\Psi(z)\varphi'(x)$, $\xi=\Xi(z)\varphi'(x)$, the Howard-type functional is bounded above by $Pe^{6/11}=Pe^{0.\overline{54}}$, so the computation and the conditional analysis agree. The paper also notes that this cannot be the global asymptotic scaling because existing lower bounds give $\max Nu > C\,Pe^{2/3}/(\log Pe)^{4/3}$ as $Pe\to\infty$; the $Pe^{0.54}$ behavior is therefore presented as the optimal scaling in the computed moderate-$Pe$ range.

Load-bearing premise

The load-bearing premise is that the optimizing fields are exactly, or negligibly close to, separable products of wall-normal and wall-parallel profiles, $u_1=-\Psi'(z)\varphi(x)$, $u_3=\Psi(z)\varphi'(x)$, $\xi=\Xi(z)\varphi'(x)$; the paper provides no proof that non-separable parts contribute negligibly to transport over the computed $Pe$ range.

Editorial extensions

If this is right

For two-dimensional no-slip wall-to-wall flows, optimal transport follows $Nu\sim Pe^{0.54}$ over $Pe\in[10^3,10^5]$, which translates in Rayleigh-Bénard variables to the single-wavenumber upper-bound scaling $Nu\lesssim Ra^{3/8}$.
The optimizing fields are nearly separable, with the leading singular mode accounting for at least 99% of $Nu-1$; their horizontal profiles are Jacobi-elliptic-like rather than single sinusoids.
The conditional $Pe^{6/11}$ bound cannot describe the global maximum at arbitrarily large $Pe$, since rigorous lower bounds grow like $Pe^{2/3}/(\log Pe)^{4/3}$; the $Pe^{0.54}$ regime is a moderate-$Pe$ plateau that branching structures should eventually replace.
The optimal aspect ratio shrinks with enstrophy budget, with $\Gamma\sim Pe^{-0.37}$ in the nonlinear regime, so optimal cells narrow as $Pe$ increases.
The gradient-ascent framework transfers to other geometries and boundary conditions wherever Poisson and Stokes solvers are available.

Reading between the lines

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

A natural testable extension is to prove an energy estimate on the second singular mode's contribution to $Nu-1$; if that contribution can be bounded uniformly small up to $Pe\sim10^5$, the conditional $Pe^{6/11}$ bound would become unconditional over that range.
The near-separability of the optimizers suggests a practical control conclusion the paper does not draw: in two-dimensional settings, actuating a single large-scale roll (the dominant mode) should recover roughly 99% of the optimal heat transport, so low-order controllers may suffice.
The oscillatory relaxation of the local scaling exponent hints at a sequence of modal transitions; one could search for additional 'branched' optimizers slightly beyond $Pe=10^5$ and predict the onset where the exponent moves from $0.54$ toward $2/3$.
The same separable-ansatz machinery could be applied to three-dimensional wall-to-wall problems; a non-separable ansatz that produced a bound below $Pe^{6/11}$ in two dimensions would overturn the conclusion that the separable class captures the extremal scaling.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a solid, honest paper whose main result is explicitly conditional, and the gap the reader flagged is real: the numerics do not actually verify the exact separability ansatz used for the bound. Still worth sending out.

What is new: the conditional upper bound Nu ≲ Pe^{6/11} within a single-profile separable ansatz, and the systematic 2D no-slip computations up to Pe ~ 2.5e5 showing local optima with Nu ~ Pe^0.54. The numerical scaling by itself was already reported by Souza and confirmed by Motoki et al. and Ding & Kerswell; the paper says so. The theory section connecting the wall-to-wall functional to the background method and the Howard-Busse-Malkus functional, plus the conjectured duality gap, is a nice framing and the polynomial example is helpful.

The main soft spot is exactly what the stress-test note says. Section 4.1's dominant-mode measure uses different horizontal profiles φ1 and φ2 for the streamfunction and ξ. The ansatz (5.3)-(5.5) requires one φ for both u and ξ. The 99% transport-capture number does not constrain the difference between φ1 and φ2, so the Pe^{6/11} bound has no demonstrated bearing on the computed optimizers. The authors are explicit that the bound is conditional, but that condition is stronger than the numerical evidence supports. This is the load-bearing gap between the two headline claims.

Other weaknesses are minor and standard: no code, no data, no error bars; the optima are local, so the computed scaling is a lower bound on the true max, not an upper bound. The conditional proof itself looks correct within its class, and the numerics are plausible given the resolution and the multiple starting points used.

Who should read it: anyone working on variational bounds for convection or the wall-to-wall problem. It is a useful contribution despite the gap, because it sharpens where the difficulty lies. I would accept it for peer review and let referees ask for data, error bars, and ideally a quantification of how different φ1 and φ2 actually are.

Referee Report

3 major / 5 minor

Summary. This paper addresses the wall-to-wall optimal transport problem in two dimensions with no-slip isothermal boundaries. The authors develop pseudo-spectral gradient-ascent schemes based on the variational formulation of the Nusselt number, and compute steady locally maximizing flows for Peclet numbers up to approximately 10^5, reporting a transition from Nu-1 ~ Pe^2 to Nu ~ Pe^{0.54} and an optimal aspect-ratio scaling Gamma ~ Pe^{-0.37}. They observe that the computed fields are nearly rank-one via singular value decomposition, which motivates a separable ansatz. Under that ansatz, they derive a conditional upper bound Nu <= C Pe^{6/11} and discuss connections to the background method, Howard-Busse-Malkus theory, and a conjectured duality gap. The paper is explicitly careful to label the analytic bound as conditional, and it notes that the computed finite-Pe scaling cannot persist for global optimizers as Pe tends to infinity in light of known lower bounds.

Significance. The paper is significant because it provides the most complete numerical map to date of locally optimal two-dimensional wall-to-wall transport in the no-slip case, identifies a clear finite-Pe scaling regime, and supplies a conditional analytic bound at the same exponent. The variational machinery connecting the wall-to-wall problem to the background method and Howard's functional is elegantly presented, and the polynomial example illustrating a possible duality gap is instructive. The authors are explicit that the analytic bound is conditional on the separable ansatz, and they correctly note that the computed Pe^{0.54} branch cannot represent global optimizers at asymptotically large Pe in light of the Tobasco-Doering lower bound. The main weaknesses are that the numerical evidence does not directly validate the common horizontal profile required by the ansatz, that the derivation of the bound contains an incorrect identity that must be repaired, and that the numerical scaling lacks rigorous error or resolution estimates.

major comments (3)

[Section 4.1 and Section 5.1, Eqs. (4.6)-(4.9) and (5.3)-(5.5)] The diagnostic (N1-N2)/N1 <= 0.01 reported in Section 4.1 uses two different horizontal profiles phi1 and phi2 in the leading-rank approximation (4.6)-(4.7), so it measures the fraction of transport carried by the leading singular triad rather than the error incurred by imposing the common-profile ansatz phi1 = phi2 = phi that is used in the bound of Section 5.1. Consequently the computed Nu ~ Pe^{0.54} branch is not demonstrated to lie within the separable class analyzed, and the claimed accord between the numerics and the Pe^{6/11} bound is not established. I ask the authors to add a direct test of the common-phi ansatz, for example by computing the transport of the projection of the optimal fields onto a single horizontal profile, or to soften the connection between the two results.
[Section 5.1, Eq. (5.11)] The identity <(u3 xi - <u3 xi>)^2> = integral (Psi Xi - 1)^2 dz is not correct when phi(x) is nonconstant: with the normalization <(phi')^2> = 1, horizontal averaging yields <(u3 xi - <u3 xi>)^2> = integral (Psi Xi)^2 <(phi')^4> dz - (integral Psi Xi dz)^2, which generally exceeds integral (Psi Xi - 1)^2 dz. The lower-bound argument still goes through if the equality is replaced by the inequality '>=' (justified by Jensen's inequality since <(phi')^4> >= <(phi')^2>^2 = 1), but the derivation as written is invalid and must be corrected.
[Section 4, Figure 2] The central numerical scaling Nu ~ Pe^{0.54} is inferred from local logarithmic slopes at the largest computed Pe, yet the paper reports no error bars, resolution or convergence studies, or statement of data availability. Because the local slope oscillates (bottom-left panel of Figure 2), the single value 0.544 at the largest Pe is not by itself a robust estimate of the exponent over the claimed range, and the manuscript should provide a quantitative uncertainty estimate or additional resolution checks.

minor comments (5)

[Abstract and Section 1] The abstract and several places describe the computed flows as 'maximizing' or 'optimal' without the qualifier 'locally'; since the optimization problem is non-convex and the method finds local maxima, please add 'locally' to these statements to avoid overstatement.
[Keywords] There is a typo in the keywords: 'variational methds' should be 'variational methods'.
[Introduction] In the first paragraph, 'complimentary role' should be 'complementary role'.
[Section 5.1] The inequalities in the derivation of the conditional bound omit all constants; since the result is an upper bound with an unspecified prefactor, a sentence stating that all constants are finite and depend only on the boundary conditions would clarify the logical status of the '>' signs.
[Section 2.6 and Section 5.1] The overline notation for horizontal averages is introduced in Section 2.6 but is reused in Section 5.1 in expressions such as phi^2 and (phi'')^2 without an explicit reminder of the convention; please make the notation self-contained in Section 5.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the numerical scaling is independently computed, and the separable-ansatz bound is explicitly conditional and not used to produce the numerics.

full rationale

The paper's central numerical result, Nu ~ Pe^0.54 for Pe > 10^3, comes from time-stepping gradient ascent solutions of the wall-to-wall Euler-Lagrange equations, not from the analytic bound. The Pe^{6/11} bound is derived only within an explicitly stated separable ansatz and is never claimed to be an unconditional proof. The authors say plainly: 'It should be emphasized that this is not a rigorous proof for the observed numerical scaling from the previous section since it must still be proven that all solutions in the range Pe ≲ 10^5 are indeed separable, or what is more likely the case, that their non-separable part contributes negligibly to transport.' They also state: 'What we lack is a proof that (5.3)-(5.5) are indeed valid assumptions over the given range of Pe.' The skeptical concern that the SVD evidence uses different horizontal profiles phi1 and phi2 while the ansatz uses a common phi is a genuine rigor gap, but it is an admitted limitation, not a circular reduction: the bound's conclusion is not assumed among its hypotheses. The variational identity (2.19) is attributed to prior same-group work but is rederived in Section 2.3 from the functional S, so the citation is not load-bearing. The lower bound cited from Tobasco & Doering (2017) and Doering & Tobasco (2019) is used only to argue that the computed Pe^0.54 scaling cannot persist as Pe -> infinity, which undermines rather than supports the paper's headline claim and is independent external support. No fitted parameter is renamed as a prediction, and no self-citation supplies the derivation of the main numerical result. The paper is self-contained in its numerical computations and explicitly transparent about the conditional status of the analytic part; there is no exhibited reduction of any claimed result to its own inputs.

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

The central numerical claim does not rely on fitted constants: Pe is the control parameter and the aspect ratio Gamma is optimized as part of the computation. The analytic bound uses no fitted exponents, but it rests on prior variational results from the same group and on the unproven separability assumption.

assumptions (4)

domain assumption The variational formula Nu-1 = max_xi min_eta S (Eq. 2.19) is a valid exact characterization of the Nusselt number for fixed velocity fields.
Invoked in Section 2.3, Eq. (2.19), citing Tobasco and Doering (2017) and Doering and Tobasco (2019); not proved in this paper.
domain assumption Time-independent flows suffice; time-dependent flows do not enhance transport beyond steady flows.
Authors restrict to steady flows in Section 2, citing heuristic evidence from the Lorenz model and preliminary computations, not a proof.
standard math Howard's lemma: integral of (Psi Xi - 1)^2 is bounded below by (integral (Psi'')^2 integral (Xi')^2)^{-1/4}.
Used in Section 5.1, Eq. (5.19), cited to Doering and Constantin (1996); stated without proof in this paper.
ad hoc to paper The separable ansatz (5.3)-(5.5) describes optimal fields to leading order in transport.
Motivated by singular value decompositions of numerical solutions; the authors state in Section 5 that they lack a proof of validity over the computed range of Pe.

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Cite this review

Pith. "Pith review of Wall-to-wall optimal transport in two dimensions." pith.science (2026). https://pith.science/paper/2DLPXHVA

@misc{pith2026190802896,
  author       = {Pith},
  title        = {Pith review of: Wall-to-wall optimal transport in two dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2DLPXHVA}},
  note         = {Machine review of arXiv:1908.02896}
}

abstract

Gradient ascent methods are developed to compute incompressible flows that maximize heat transport between two isothermal no-slip parallel walls. Parameterizing the magnitude of velocity fields by a P\'eclet number $\text{Pe}$ proportional to their root-mean-square rate-of-strain, the schemes are applied to compute two-dimensional flows optimizing convective enhancement of diffusive heat transfer, i.e., the Nusselt number $\text{Nu}$ up to $\text{Pe} \approx 10^5$. The resulting transport exhibits a change of scaling from $\text{Nu}-1 \sim \text{Pe}^{2}$ for $\text{Pe} < 10$ in the linear regime to $\text{Nu} \sim \text{Pe}^{0.54}$ for $\text{Pe} > 10^3$. Optimal fields are observed to be approximately separable, i.e., products of functions of the wall-parallel and wall-normal coordinates. Analysis employing a separable ansatz yields a conditional upper bound $\lesssim \text{Pe}^{6/11} = \text{Pe}^{0.\overline{54}}$ as $\text{Pe} \rightarrow \infty$ similar to the computationally achieved scaling. Implications for heat transfer in buoyancy-driven Rayleigh-B\'enard convection are discussed.

Figures

Figures reproduced from arXiv: 1908.02896 by the authors.

**Figure 1.** Optimal no-slip solutions for different enstrophy budgets in a single cell. The black contour lines are the streamlines and the colours represent the temperature field. From left to right, top to bottom the P´eclet numbers are 4.0 × 10−1 , 4.0 × 100 , 4.0 × 101 , 4.0 × 102 , 4.0 × 103 , 4.0 × 104 . The domain size in the horizontal direction x shrinks as the enstrophy budget increases. (iv) Machine Accuracy: Utilisi… view at source ↗

**Figure 2.** Computed optimal Nusselt number (Nu) and aspect ratio (Γ) as a function of the enstrophy budget (Pe), for no-slip boundary conditions. Top Left: Log-Log plot of Pe vs Nu-1. Bottom Left: The instantaneous slope of the top left plot, Pe vs d log(Nu − 1)/(d log Pe). Top Right: Log-Log plot of Pe vs Γ. Bottom Right: The instantaneous slope of the top right plot, Pe vs d log(Γ)/(d log Pe). The last instantaneous slope fo… view at source ↗

**Figure 3.** The stream function ψ and its first three modes at Pe ≈ 2.4 × 104 . The modes are ordered left to right by the magnitude of their singular values starting with the largest. function ψ defined by (−∂zψ, ∂xψ) = (u1, u3) this would mean ψ(x, z) ≈ Ψ(z)φ(x). (4.1) We investigate this possibility by minimizing the functional A[Ψ, φ] = 1 Γ Z Γ 0 Z 1 0 (ψ(x, z) − Ψ(z)φ(x))2 dxdz (4.2) subject to appropriate constraints on Ψ… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: [Colour Online] The field ξ = θ + ϕ (top), the vertical velocity u3, and their first three modes at Pe ≈ 2.4 × 104 . The outer products are ordered with respect to their singular values [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]

**Figure 5.** Figure 5: The first singular vectors in the singular value decomposition of the ξ field (top) and the vertical velocity u3 (bottom). The respective products of these singular vectors yields the first approximations in [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

Discussion (0). Continue with ORCID to comment.

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ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence ...
[2]

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" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...
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Pith tools

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

[1] [1]

, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence ...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

[3] [3]

Alben, Silas 2017 Improved convection cooling in steady channel flows . Phys. Rev. Fluids 2 , 104501

work page 2017

[4] [4]

2001 Chebyshev and Fourier Spectral Methods\/ , 2nd edn

Boyd, J.P. 2001 Chebyshev and Fourier Spectral Methods\/ , 2nd edn. Dover

work page 2001

[5] [5]

1969 On H oward's bound for heat transport by turbulent convection

Busse, F.H. 1969 On H oward's bound for heat transport by turbulent convection . Journal of Fluid Mechanics 37 , 457--477

work page 1969

[6] [6]

2019 Exhausting the background approach for bounding the heat transport in rayleigh-b\'enard convection

Ding, Zijing & Kerswell, Richard R. 2019 Exhausting the background approach for bounding the heat transport in rayleigh-b\'enard convection . preprint

work page 2019

[7] [7]

Doering, C. R. & Constantin, P. 1996 Variational bounds on energy dissipation in incompressible flows. III . C onvection . Physical Review E 53 , 5957--5981

work page 1996

[8] [8]

& Tobasco, Ian 2019 On the optimal design of wall-to-wall heat transport

Doering, Charles R. & Tobasco, Ian 2019 On the optimal design of wall-to-wall heat transport . to appear in Communications on Pure and Applied Mathematics d oi:10.1002/cpa.21832

work page doi:10.1002/cpa.21832 2019