{"id":"ce52b688-a049-4ff1-a2ff-59f9af36ea59","arxiv_id":"2607.01097","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"DNS study of supersonic shear layers identifies self-similar scalings and derives a closed-form entrainment ratio that rises with Mc and λ.","lead":"This paper runs direct numerical simulations of spatially developing supersonic turbulent shear layers across ranges of convective Mach number and velocity parameter. It reports self-similarity at far downstream stations and supplies an approximate closed-form expression for the entrainment ratio that increases with both parameters.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Self-similarity claim rests on linear thickness growth + constant peak stresses, but no quantitative test shows these criteria guarantee collapse of all moments once compressibility shifts are included.","rationale":"The reader's weakest assumption directly identifies the point at which the DNS evidence must be decisive; the abstract supplies no additional quantitative diagnostics (e.g., streamwise station-by-station collapse metrics or domain-size sensitivity) that would remove the uncertainty, so the load-bearing risk remains exactly where the reader located it.","tokens_in":1802,"tokens_out":340,"duration_ms":14259,"concrete_test":"For the two highest-Mc cases, recompute all normalized profiles using only data from the final 30 % of the domain (where linear growth is reported) versus the final 15 %; if the collapse of higher-order moments or the cross-stream velocity residual degrades by more than the reported DNS uncertainty, the far-field regime has not been reached uniformly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the proposed scalings (compressibility-adjusted normalization plus centerline shift) produce universal collapse of mean velocity, Reynolds stresses, density, and derived cross-stream velocity once the layer is declared self-similar. The identification criteria (linear δ(x) growth and constant peak stresses) are necessary but not shown to be sufficient; for spatially evolving layers the approach to asymptopia can be slow and Mc-dependent, so profiles may appear to collapse only because the normalization absorbs residual non-self-similar transients. The approximate cross-stream velocity equation inherits this assumption directly from the self-similar energy equation, so any failure of collapse propagates to the entrainment-ratio expression.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper performs DNS of spatially developing supersonic turbulent shear layers over ranges of convective Mach number Mc and velocity parameter λ. It reports that self-similarity is reached far downstream, identified by linear growth of shear-layer and momentum thickness together with constant peak normalized Reynolds stresses. Using proposed self-similar scalings that incorporate compressibility effects and centerline shifts, the authors claim collapse of mean velocity, Reynolds stresses, density, and higher-order statistics. Self-similar forms of the continuity, momentum, and energy equations are written; an approximate cross-stream velocity profile derived from the self-similar energy equation is shown to agree with the DNS, and this agreement is used to obtain a closed-form expression for the entrainment ratio, which increases with both Mc and λ and favors the high-speed side.","tokens_in":1961,"tokens_out":713,"duration_ms":20230,"significance":"If the collapse under the proposed scalings and the agreement of the approximate velocity equation are robust, the work supplies a concrete link between density variation, dissipation, and entrainment asymmetry in compressible shear layers, together with an explicit entrainment-ratio formula that could be tested against other datasets. The parametric DNS database would also be a useful reference for model development, provided the numerical evidence is placed on a firmer footing.","major_comments":[{"comment":"Abstract and self-similarity section: the criteria used to declare the self-similar regime (linear δ(x) growth and constant peak Reynolds stresses) are stated as sufficient for collapse of all statistics under the compressibility-adjusted normalization plus centerline shift, yet no quantitative metric (e.g., L2 residual of normalized profiles versus downstream distance or Mc) is supplied to demonstrate that these criteria guarantee the claimed universality once Mc-dependent shifts are introduced. This assumption is load-bearing for both the collapse claim and the subsequent derivation of the cross-stream velocity equation.","section":"Abstract / self-similarity identification"},{"comment":"Approximate cross-stream velocity equation (derived from self-similar energy equation): because the equation inherits the self-similarity assumption directly, any residual non-self-similar transients that survive the identification criteria would appear in the reported agreement with DNS; the manuscript provides no separate test (e.g., sensitivity to the downstream station chosen or to the precise form of the centerline shift) that isolates the validity of the approximation itself.","section":"Section deriving approximate cross-stream velocity"},{"comment":"Numerical validation: the abstract asserts that DNS results support the collapse and the velocity-equation agreement, but no grid-convergence data, resolution criteria, or comparison against established benchmarks (e.g., known growth-rate curves or low-Mc limits) are referenced. Without these, the quantitative statements about entrainment-ratio trends with Mc and λ rest on unverified numerical evidence.","section":"Numerical methods / results"}],"minor_comments":[{"comment":"The abstract refers to “our proposed self-similar scalings” without giving their explicit functional form; a concise statement of the normalization (including the Mc-dependent centerline shift) should appear in the abstract or early in the results section.","section":"Abstract"},{"comment":"Notation for the velocity parameter λ and the convective Mach number Mc should be defined at first use and kept consistent with standard definitions in the compressible shear-layer literature.","section":"Introduction / nomenclature"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the constructive comments that help improve the clarity and robustness of our manuscript. We provide point-by-point responses to the major comments below.","responses":[{"response":"We agree that a quantitative metric would strengthen the evidence for collapse under the proposed scalings. In the revised manuscript we will add L2 residuals of the normalized mean-velocity and Reynolds-stress profiles computed at successive downstream stations (and across Mc) to quantify the degree of self-similarity and the quality of the collapse once the Mc-dependent centerline shifts are applied.","revision_made":"yes","referee_comment":"[Abstract / self-similarity identification] Abstract and self-similarity section: the criteria used to declare the self-similar regime (linear δ(x) growth and constant peak Reynolds stresses) are stated as sufficient for collapse of all statistics under the compressibility-adjusted normalization plus centerline shift, yet no quantitative metric (e.g., L2 residual of normalized profiles versus downstream distance or Mc) is supplied to demonstrate that these criteria guarantee the claimed universality once Mc-dependent shifts are introduced. This assumption is load-bearing for both the collapse claim and the subsequent derivation of the cross-stream velocity equation."},{"response":"The manuscript already presents the approximate cross-stream velocity at several stations inside the identified self-similar region. To isolate the approximation itself we will add a dedicated sensitivity study that varies both the chosen downstream station and the precise functional form of the centerline shift, reporting the resulting changes in agreement with the DNS data.","revision_made":"yes","referee_comment":"[Section deriving approximate cross-stream velocity] Approximate cross-stream velocity equation (derived from self-similar energy equation): because the equation inherits the self-similarity assumption directly, any residual non-self-similar transients that survive the identification criteria would appear in the reported agreement with DNS; the manuscript provides no separate test (e.g., sensitivity to the downstream station chosen or to the precise form of the centerline shift) that isolates the validity of the approximation itself."},{"response":"We acknowledge that explicit numerical-validation details were omitted from the original submission. The revised manuscript will include grid-convergence tests for representative (Mc, λ) cases, resolution criteria based on local Kolmogorov scales, and direct comparisons of growth rates against established low-Mc benchmarks and literature data for supersonic shear layers.","revision_made":"yes","referee_comment":"[Numerical methods / results] Numerical validation: the abstract asserts that DNS results support the collapse and the velocity-equation agreement, but no grid-convergence data, resolution criteria, or comparison against established benchmarks (e.g., known growth-rate curves or low-Mc limits) are referenced. Without these, the quantitative statements about entrainment-ratio trends with Mc and λ rest on unverified numerical evidence."}],"tokens_in":1657,"tokens_out":594,"duration_ms":28212,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper runs DNS across a range of convective Mach numbers and velocity parameters in spatially developing supersonic shear layers. It reports that profiles of velocity, stresses, and density collapse once normalized with their proposed scalings that include compressibility and a centerline shift. From the self-similar energy equation they pull an approximate cross-stream velocity that tracks the DNS, then convert that into an algebraic entrainment ratio that grows with both Mc and λ and favors the high-speed side.\n\nThe algebraic entrainment result and the explicit self-similar equation forms are the concrete additions. The link they draw between density variation and dissipation is straightforward and useful for this subfield.\n\nThe soft spots are the missing pieces on numerics. No grid sizes, convergence tests, or error estimates appear in the abstract, and the self-similarity criteria (linear thickness growth plus constant peak stresses) are necessary but not demonstrated to be sufficient once the compressibility-adjusted normalizations are applied. If the layer approaches the far-field state more slowly at higher Mc, some of the reported collapse could be carried by the normalization rather than true universality; that would carry through to the entrainment formula. The stress-test note on this point holds up from what is shown.\n\nThis is for people who model compressible mixing in propulsion or need entrainment rates in RANS-type closures. A reader already working on shear-layer self-similarity will get the derivations and the parameter trends. It is worth a serious referee because the parameter sweep and the algebraic step are explicit, even though the numerical validation will need tightening.","headline":"DNS sweeps of supersonic shear layers yield a closed-form entrainment expression, but the self-similarity collapse and numerical reliability rest on unshown checks.","tokens_in":2435,"tokens_out":382,"would_cite":false,"duration_ms":19222,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"DNS of supersonic shear layers shows entrainment ratio increases with convective Mach number and velocity parameter, with excess on the high-speed side.","keywords":["supersonic shear layers","convective Mach number","entrainment ratio","self-similarity","direct numerical simulation","compressibility effects","turbulent mixing","cross-stream velocity"],"falsifier":"A new DNS run at a higher convective Mach number or different velocity parameter in which the cross-stream velocity profile predicted by the approximate equation deviates measurably from the simulated profile would falsify the central claim.","tokens_in":2721,"feed_emoji":"","tokens_out":760,"duration_ms":16440,"temperature":0.7,"pith_summary":"The paper runs direct numerical simulations of spatially developing supersonic turbulent shear layers across a range of convective Mach numbers and velocity parameters. It identifies a far-downstream self-similar regime where thickness grows linearly, peak Reynolds stresses stay constant, and normalized statistics collapse under scalings that include compressibility and centerline shifts. From the self-similar energy equation the authors derive an approximate cross-stream velocity profile that matches the DNS data. This profile supplies a closed-form expression for the entrainment ratio, which rises with both compressibility and velocity difference. A reader would care because the result supplies a practical way to estimate mixing rates in high-speed compressible flows without running full simulations for every case.","feed_headline":"Entrainment ratio rises with compressibility in supersonic shear layers","feed_subtitle":"DNS yields a closed-form expression showing excess mixing on the high-speed side once self-similarity is reached.","key_machinery":"The approximate equation for cross-stream velocity derived from the self-similar energy equation, which incorporates the normalized density distribution to account for compressibility effects on entrainment.","core_discovery":"At distant downstream locations self-similarity is attained for all cases, identified by collapse of normalized mean streamwise velocity, constant peak normalized Reynolds stresses, and linear growth of shear-layer thickness and momentum thickness. The self-similar forms of the continuity, momentum, and energy equations are written with compressibility and centerline shifts included. The normalized density distribution inside the layer explains compressibility effects on statistics and far-field cross-stream velocity; density variation is tied to dissipation in the energy equation. An approximate equation for cross-stream velocity is obtained whose profiles agree with DNS, and this equation","pith_inferences":["The derived entrainment expression could be inserted into reduced-order models for supersonic mixing layers without requiring new DNS for each parameter set.","If the density-dissipation link holds beyond the simulated range, similar scalings might simplify predictions for other spatially developing compressible shear flows.","Checking whether the approximate cross-stream velocity equation remains accurate when the layer is forced or when chemistry is added would test its broader utility."],"forward_implications":["All examined lower-order and higher-order turbulence statistics collapse inside the self-similar region when the proposed scalings are used.","The self-similar normalized density distribution accounts for compressibility effects on far-field cross-stream velocity.","The entrainment ratio increases with both convective Mach number and velocity parameter.","Entrainment favors the high-speed side, consistent with a geometric interpretation of the ratio."],"fun_headline_variants":["Self-similarity reached in supersonic shear layers at various Mach numbers","Entrainment ratio increases with compressibility and velocity parameter","Normalized density explains compressibility effects on shear layer stats","Approximate cross-stream velocity equation agrees with DNS results","Self-similar scalings collapse all turbulence profiles in the layers"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The proposed self-similar scalings that incorporate compressibility and centerline shifts remain valid and collapse all statistics once the layer reaches the far-downstream regime of linear thickness growth and constant peak Reynolds stresses.","fun_headline_variants_meta":{"raw":{"variants":["Self-similarity reached in supersonic shear layers at various Mach numbers","Entrainment ratio increases with compressibility and velocity parameter","Normalized density explains compressibility effects on shear layer stats","Approximate cross-stream velocity equation agrees with DNS results","Self-similar scalings collapse all turbulence profiles in the layers"]},"model":"grok-4.3","cost_usd":0.004367,"raw_usage":{"total_tokens":2250,"prompt_tokens":791,"num_sources_used":0,"completion_tokens":79,"cost_in_usd_ticks":43674500,"prompt_tokens_details":{"text_tokens":791,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1380,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":791,"tokens_out":79,"duration_ms":10548,"temperature":1.0,"reasoning_tokens":1380,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-02T05:16:29.256632+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A new DNS run at a higher convective Mach number or different velocity parameter in which the cross-stream velocity profile predicted by the approximate equation deviates measurably from the simulated profile would falsify the central claim.","supporting_citations":[],"review_version":1}