{"id":"0bd5bafd-2eb8-4bf8-b99f-ffa489c78818","arxiv_id":"2607.21736","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Stripping under-relaxation and time-step terms from the momentum interpolation mobility makes primal forces and adjoint sensitivities independent of solver parameters.","lead":"The paper proposes a small modification to the Momentum-Weighted Interpolation formula used in finite-volume CFD, removing the influence of the solver's relaxation factor and time-step size from computed forces and adjoint sensitivities. It matters because shape-optimization results in ship and flow engineering can currently shift by 20% or more and even change sign depending on solver settings, and the fix is straightforward to implement.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's weakest assumption is the equality of the interpolated and face-evaluated H/A term in the cancellation step. In the manuscript, the predictor flux (Eq. 12) and the face-based form (Eq. 13) are constructed from the same interpolated cell-centered H/A; the paper explicitly says these equations contain the same non-pressure-driven contribution. Therefore the cancellation is exact at the discrete level, not merely in a continuous limit. The fixed-point behavior of Eq. 16 reinforces this: the recursive terms cancel exactly when the solution converges, recovering the classical MWI with a purely spatial mobility, independent of relaxation and timestep. This is a strong analytical result. The genuine weaknesses are the contradictory Data Availability statement (\"No new data were generated\" despite extensive simulations) and the absence of any independent benchmark or open code, which the reader correctly flagged. These justify a CONDITIONAL verdict on completeness and reproducibility, but they do not constitute a technical concern about the central claim's correctness. Hence I disagree with the reader's identification of the H/A cancellation as the weakest assumption, while agreeing with the overall CONDITIONAL verdict.","tokens_in":14798,"tokens_out":19483,"duration_ms":186763,"concrete_test":"Implement Eq. 16 in an independent open-source collocated finite-volume solver (e.g., OpenFOAM) and run the 2D cylinder test case with momentum relaxation ω varying from 0.06 to 0.8 at fixed Δt, and Δt varying from 0.01 to 1 at fixed ω; then compute the drag coefficient and adjoint sensitivity. If the results remain constant within numerical precision across all parameter values, the central claim is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central derivation's cancellation of the non-pressure term H/A between Eq. 12 and Eq. 13 is exact by construction: both equations use the same interpolated cell-centered value, and the paper explicitly states that Eqs. 4 and 5 contain the same non-pressure-driven contribution. A fixed-point analysis of the recursive Eq. 16 shows that the history-dependent terms cancel at convergence, yielding a face flux that is independent of relaxation and timestep. No internal inconsistency or hidden assumption was found. The manuscript's weaknesses (contradictory Data Availability statement, no external code/data, reliance on self-cited solver validation) affect reproducibility and evidence quality but do not undermine the analytical result.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a correction to Momentum-Weighted Interpolation (MWI) that removes contributions of under-relaxation and time discretization from the diagonal momentum coefficient appearing in the pressure-driven face-flux correction. The derivation is presented in a recipe-like way: classical MWI (Eq. 6), relaxation-only (Appendix A), time-only (Appendix B), and the combined case (Eq. 16). The combined formulation introduces stored MWI corrections from the previous time level and outer iteration, so that the converged fixed point gives a face flux whose pressure-correction term scales with the spatial mobility only. The numerical experiments on a 2D laminar cylinder and a 3D turbulent ship hull (JBC) compare the classical and corrected formulations under variations of relaxation factor and pseudo-time-step size. The corrected formulation yields nearly flat integral and field-based quantities, whereas the classical formulation exhibits deviations of order 1% in forces, up to 25% in wake-related quantities, and sign changes in adjoint wake sensitivities.","tokens_in":14875,"tokens_out":19018,"duration_ms":168926,"significance":"If the central claim holds, the paper removes a real practical artifact: solver-parameter dependence of converged MWI fluxes in both primal and continuous-adjoint solvers. The derivation is analytic and parameter-free, with no fitted coefficients; the fixed-point cancellation of the recursive terms is a structural property, not a calibration result. The numerical study is carefully designed, especially in fixing the primal state for all adjoint evaluations, which isolates the MWI effect on sensitivities. The work is an incremental but useful extension of Kühl & Rung (2022), and the recipe-like presentation should make the correction easy to implement in other finite-volume codes. The main limitations are the absence of external code/data for reproducibility and a few inconsistencies in the printed equations, but these do not undermine the analytical core.","major_comments":[{"comment":"The first recursive term after the pressure-correction term is printed as \\omega d_F^n(\\dot{V}^{n,k-1}_F - \\tilde{V}^{n,k-1}_F). Subtracting Eq. (12) from Eq. (13) gives \\omega d_F^n(\\dot{V}^{n-1}_F - \\tilde{V}^{n-1}_F), not an n,k-1 term. Since Eq. (16) correctly uses the previous-time-level correction \\Delta\\dot{V}^{n-1}_F, this is likely a typographical error in a central equation, but it must be corrected.","section":"Section 3, Eq. (14)"},{"comment":"The under-relaxation term on the right-hand side is printed with v^{n,k}_{P,i}, while Eq. (9), obtained by dividing Eq. (8), has v^{n,k-1}_{P,i}. These two equations are inconsistent; the correct under-relaxation history term is v^{n,k-1}, as used in Eqs. (9) and (11). Please fix Eq. (8).","section":"Section 3, Eq. (8)"},{"comment":"The sentence 'the pressure-correction scaling becomes independent of both the chosen time-step size and the relaxation parameter' is not immediate from Eq. (16) alone, because the explicit pressure-correction coefficient is \\omega b_F/(1+d_F). The independence holds at the converged fixed point, where the recursive terms combine to produce \\dot{V} = \\tilde{V} - b_F[\\nabla p term]. This fixed-point reduction should be shown explicitly; as written, the claim appears too strong at the level of the transient equation.","section":"Section 3, after Eq. (16)"}],"minor_comments":[{"comment":"The statement 'No new data were generated or analyzed in this study' contradicts Section 4, which reports numerical results generated for this study. Please rephrase, e.g., 'No experimental data were generated; numerical data are available from the author upon request.'","section":"Section 9, Data Availability"},{"comment":"The H/A terms are written without the overbar/non-overbar distinction defined in the text. For clarity, explicitly mark the interpolated H/A in the predictor equations and state that the face-based form uses the same interpolated H/A by construction, so that the cancellation in Eqs. (6) and (16) is unambiguous.","section":"Section 2, Eqs. (3)-(5) and Section 3, Eqs. (12)-(13)"},{"comment":"The abstract and conclusions state wake deviations 'up to 20%', but Sec. 4.2 reports deviations up to approximately +25% for W and beyond -100% for S_W. Please harmonize these numbers.","section":"Abstract and Section 4.2"},{"comment":"The labels E1.1/E2.1 are reused with different meanings in the cylinder and ship sections. This is not fatal, but adding a prefix (e.g., C1.1, J1.1) would improve readability.","section":"Section 4.1 and 4.2, experiment labels"},{"comment":"The text says 'In complete analogy to Eqn. 22' before Eq. (13), but Eq. (22) appears only later in Appendix A. Please adjust the cross-reference or renumber.","section":"Section 3, reference to Eq. 22"}],"recommendation":"minor_revision","confidential_remarks":"The paper is analytically sound and the numerical evidence supports the claim, but the printed derivation contains two typos in core equations (Eqs. 8 and 14) that must be fixed. The fixed-point argument for parameter independence should be made explicit. The lack of external code/data and the self-citation-heavy validation limit reproducibility, but not the internal consistency. I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core result is real. Eq. 16 removes relaxation and time-step contributions from the MWI mobility while keeping the history terms as recursive corrections, and the same construction is applied consistently to the continuous-adjoint system. The derivation in Sec. 3 and the appendices is internally consistent; I spot-checked the algebra and it works. The numerical experiments are also convincing in what they claim: the classical formulation shows clear parameter dependence, the corrected one is flat across the tested ranges, and the field-sensitive quantities differ by up to roughly 20% with sign changes in adjoint sensitivities. There are no fitted parameters and no calibration, so the argument is not circular. What is new here is the unified recursive formula and its clean application to both primal and adjoint solvers. The paper also does a service by presenting the derivation recipe-style, which makes it transferable. Credit where due: the algebra is solid and the reported effect is practically relevant, especially for adjoint-based optimization where sign changes are dangerous. The soft spots are mostly about evidence and framing, not the math. The paper ships no code or data, and all demonstrations are inside the author's own FreSCo+ solver. That alone would be acceptable for a derivation-focused paper, but the Data Availability Statement says 'No new data were generated or analyzed,' which is plainly contradicted by the CFD simulations and figures. That needs to be fixed. Relatedly, the abstract and conclusions say 'physically consistent,' but no external validation against reference data is offered; the paper explicitly declines to validate against benchmarks. 'Solver-independent relative to the tested formulations' is what is actually shown. The one mathematical weak point is the cancellation of the H/A term between the predictor flux and the face-based flux. This is the standard MWI assumption, present in the classical formulation too, and the paper acknowledges the two reconstructions differ at the discrete level. So it is a limitation inherited from MWI, not a new flaw, but the paper could state more carefully that exactness on arbitrary skewed or polyhedral grids is not established. The citation pattern is fine: building on the author's own 2022 JCP paper is legitimate here, since the new formula is a direct extension. I would send this to peer review. The derivation deserves referee time, and the numerical evidence is strong enough to justify asking for the reproducibility fixes and the framing change. Recommend: accept after minor revision, with the data statement corrected, code or benchmark data made available, and the physical-consistency claim softened.","headline":"A clean, parameter-free fix for solver-induced MWI dependence; the derivation holds up and the numerics support it, but the paper needs a data-availability correction and a softer physical-consistency claim.","tokens_in":15374,"tokens_out":3399,"would_cite":true,"duration_ms":36028,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["76M12","76D05"],"pacs":["47.11.-j"],"model":"deepseek-v4-flash","headline":"Momentum-weighted interpolation can be made independent of solver relaxation and time-step parameters while preserving the original discretization.","keywords":["Momentum-Weighted Interpolation","Rhie–Chow interpolation","Pressure–velocity coupling","Continuous adjoint sensitivity","Under-relaxation dependence","Time-step dependence","Collocated finite-volume method","Solver parameter independence"],"falsifier":"Run the corrected MWI on a sequence of progressively more skewed or polyhedral meshes for the same cylinder or ship case at a fixed relaxation factor far from 1, and check whether the computed drag or wake quantity stays constant as mesh quality degrades; if the flux varies with mesh skewness while solver parameters are fixed, the cancellation underlying the corrected formulation is not exact at the discrete level.","tokens_in":14649,"feed_emoji":"🌊","tokens_out":6214,"duration_ms":57165,"temperature":0.7,"pith_summary":"Momentum-Weighted Interpolation (MWI) is the standard way collocated finite-volume solvers reconstruct face fluxes for pressure–velocity coupling. This paper shows that in typical implementations the diagonal momentum coefficient entering MWI includes under-relaxation and time-discretization contributions, so the computed face flux — and with it forces, wake quantities, and adjoint sensitivities — changes when the user changes relaxation or time-step, even though the spatial discretization is identical. It proposes a reformulation that removes those solver-induced terms from the mobility coefficient, replacing them with stored recursive correction terms from the previous iteration and time level. The result is a face flux whose pressure-correction part is unaffected by relaxation and time-step, applicable identically to primal and continuous-adjoint systems. Numerical tests on a 2D cylinder and a 3D turbulent ship hull show that the uncorrected form varies by about 1% for forces and up to about 20% for wake fields, with adjoint wake sensitivities even changing sign; the corrected form stays constant over wide parameter ranges.","feed_headline":"A simple fix removes solver-parameter bias from CFD face fluxes","feed_subtitle":"Relaxation and time-step changes move forces by ~1% and wake sensitivities by ~20%; the fix restores steady results.","key_machinery":"Momentum-Weighted Interpolation (MWI) itself — the face-flux reconstruction scheme that combines an interpolated cell-centered predictor with a face-based pressure-gradient correction weighted by a mobility coefficient b_F proportional to Omega/A_P. The key move is to divide the relaxed, time-discretized momentum equation by the spatial diagonal coefficient A_P only, so the mobility entering the pressure correction no longer contains 1/omega or 1/Delta-t; the previously absorbed terms become recursive multipliers acting on stored flux-correction differences, as in Eq. (16).","core_discovery":"The paper establishes that the classical MWI face flux inherits a spurious dependence on solver parameters because the mobility coefficient b_F is proportional to the inverse of the interpolated diagonal momentum coefficient A_P, which in practical solvers contains 1/omega and 1/Delta-t contributions. By writing the relaxed, time-integrated momentum balance in a divided form and applying the same elimination of the non-pressure term as in the classical derivation, the author derives a modified flux expression in which the pressure-correction mobility is scaled only by the spatial coefficient, while the temporal and relaxation effects reappear as coefficients multiplying stored flux-correctio","pith_inferences":["If the cancellation of the non-pressure term is exact only in the continuous limit, the residual dependence on relaxation might not vanish on highly skewed or otherwise low-quality meshes; a dedicated grid-quality sensitivity study would test this.","The recursive form of the corrected flux resembles a time filter or discrete integrator for the flux correction, suggesting a possible connection to deferred-correction and dual-time-stepping strategies where stored corrections are re-weighted.","A direct extension to higher-order time schemes (e.g., BDF2 or Crank–Nicolson) would require consistent recursive weights; the practical stability of the recursive term under strongly varying time steps has not been analyzed and is a natural next test.","The observed sign change in adjoint wake sensitivity implies that optimization loops using the classical formulation may converge to the wrong descent direction, making the corrected formulation necessary for reliable gradient-based design."],"forward_implications":["CFD results obtained with the corrected MWI are reproducible across different relaxation factors and time steps, removing a hidden source of variance between supposedly identical simulations.","Adjoint shape sensitivities become trustworthy when the adjoint solver requires small relaxation or pseudo-time steps, since the corrected form eliminates the parameter-induced sign changes observed in wake sensitivities.","The correction is a minimal change to existing MWI implementations: the pressure-correction term is scaled differently and previously stored flux corrections are reused, so it can be dropped into current segregated solvers without altering the spatial discretization.","Force-based quantities stabilize at the one-percent level, while field-based quantities such as wake homogeneity stabilize at the twenty-percent level, indicating that sensitivity studies benefit most from the correction.","Because the formulation is derived on unstructured polyhedral collocated grids and applies equally to primal and adjoint systems, it covers the class of industrial finite-volume solvers."],"fun_headline_variants":["Fix removes solver-induced bias from CFD face fluxes","CFD fluxes now independent of relaxation and time-step","Solver parameters no longer skew primal or adjoint results","New correction yields solver-independent forces and sensitivities","Momentum interpolation fix removes solver-parameter dependence"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The derivation requires that the non-pressure part of the momentum balance, when interpolated from cell centers to a face, equals the same term evaluated directly at the face; on a real collocated mesh these are different discrete reconstructions, so the cancellation is exact only in a formal continuous sense.","fun_headline_variants_meta":{"raw":{"variants":["Fix removes solver-induced bias from CFD face fluxes","CFD fluxes now independent of relaxation and time-step","Solver parameters no longer skew primal or adjoint results","New correction yields solver-independent forces and sensitivities","Momentum interpolation fix removes solver-parameter dependence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001198,"raw_usage":{"total_tokens":4805,"prompt_tokens":800,"completion_tokens":4005,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":544,"completion_tokens_details":{"reasoning_tokens":3931}},"tokens_in":544,"tokens_out":4005,"duration_ms":30464,"temperature":1.0,"reasoning_tokens":3931,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T06:51:50.089622+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the corrected MWI on a sequence of progressively more skewed or polyhedral meshes for the same cylinder or ship case at a fixed relaxation factor far from 1, and check whether the computed drag or wake quantity stays constant as mesh quality degrades; if the flux varies with mesh skewness while solver parameters are fixed, the cancellation underlying the corrected formulation is not exact at the discrete level.","supporting_citations":[],"review_version":1}