{"id":"bf0924c0-c0fc-4c75-8270-9a17e6cba4c8","arxiv_id":"1908.04637","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new benchmark for fluid-rigid body interaction with a freely rotating obstacle is defined in 2D and 3D, along with reference intervals for key output quantities.","lead":"This paper proposes a new benchmark for fluid-rigid body interaction: a circle or sphere with a fixed center that can rotate freely in a channel flow, and gives reference intervals for drag, lift, torque, and angular velocity computed by several independent numerical methods. The benchmark gives the CFD community a standardized test case for validating solvers that couple fluid flow with rotating rigid bodies.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reference intervals assume a stable, unique attractor that is never verified; a Floquet/eigenvalue check would settle whether the benchmark states are well-posed.","rationale":"The proposed check is decisive because well-posedness of the benchmark is a necessary condition for the intervals to be meaningful as correctness targets. If Rot2d-2 has a single exponentially stable limit cycle, then the observed agreement of the finitely resolved methods and their convergence studies make the intervals credible, and I would not object further. If the Floquet spectrum reveals an additional neutral or unstable mode, or if the initial-condition runs land on different orbits, then the reference intervals in Table 4 are specific to a particular branch or initialization, and the benchmark would need to be restricted or redefined. The same logic applies to the steady cases. The reader's conditional verdict is therefore the right level: the paper is useful, but its central benchmark claim is contingent on a stability and uniqueness check that has not yet been performed.","tokens_in":21691,"tokens_out":8801,"duration_ms":97551,"concrete_test":"Use the HDG H(div) method (L=3,k=5) to integrate Rot2d-2 from at least three distinct initial states: quiescent flow, the Re=80 steady solution used in Section 3.5.2, and the same state with omega perturbed by +/-0.1. After discarding transients, compute CD,max, CL,max, omega*_max and St over a common window of several periods; if the values differ by more than the Table 4 interval widths, the benchmark is not a single attractor. Independently, time-step the linearized coupled fluid-omega system around the reference orbit for one period and compute the Floquet multipliers; the spectrum should contain one multiplier at 1 (phase invariance) and all others below 1-1e-2. For Rot2d-1 and Rot3d-1, compute the dominant eigenvalues of the linearization about the reported steady states; any eigenvalue with positive real part invalidates the stationary-limit interpretation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is only as strong as the reference intervals, and those intervals inherit an unverified attractor assumption. For Rot2d-1 and Rot3d-1 the reference is defined as the stationary limit, i.e. a steady solution with zero net torque; for Rot2d-2 the reference is a stable limit cycle with a well-defined period 1/f(CD). Existence, uniqueness, and stability of these states are not established. Section 2.3.1 admits the authors were unable to find a suitable stable configuration at higher Reynolds numbers in 3D, showing that well-posedness is not guaranteed by the geometry and parameters alone. The numerical evidence is mutual agreement among five in-house methods, which rules out many implementation errors but not a common modeling bias or a coexisting attractor. The interval construction also lacks an explicit inclusion rule: for example Table 3 HDG (L,k)=(0,5) gives CL,max=0.9555, while Table 4's interval is [0.9674,0.9686], so some runs are discarded by judgment. If the limit cycle had a second stable branch or a neutral direction, a correct solver could legitimately produce values outside the published intervals.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a set of benchmark problems for fluid-rigid body interaction, where a circular (2D) or spherical (3D) obstacle is fixed at its center of mass but is free to rotate. The geometry is chosen so that the fluid-solid interface does not move, allowing standard CFD methods to be applied. The paper defines four configurations (Rot2d-1 stationary, Rot2d-2 periodic, Rot2d-3 unsteady with a fixed time interval, and Rot3d-1 stationary) and provides reference intervals for drag, lift, torque, pressure difference, angular velocity, and Strouhal number. These intervals are computed with five different finite-element or hybrid discontinuous Galerkin methods, and the manuscript reports convergence studies, degrees of freedom, sparsity, and computation times. The main claimed contribution is a reproducible benchmark that quantitatively tests methods for fluid-particle interactions.","tokens_in":21842,"tokens_out":6913,"duration_ms":67937,"significance":"If the benchmark definitions are completed and the reference intervals are reproducible, this paper would be a valuable contribution to the CFD community, filling a gap for quantitative benchmarking of fluid-rigid body interaction. The use of several independent discretizations, systematic mesh and time-step refinement studies, and the public availability of code and data (DOI 10.5281/zenodo.3253455) are clear strengths. The reference values are computed outputs rather than fitted parameters, so there is no circularity. The paper also makes falsifiable predictions in the form of reference intervals, which is exactly what a benchmark should provide. The main weaknesses are that the well-posedness of the reference states is not established and that the interval-construction rule is not documented.","major_comments":[{"comment":"The initial condition for the angular velocity of the solid is not specified for any of the unsteady problems. Section 2.2.1 states for Rot2d-3 that 'the initial state is u(0)=0', but no omega(0) is given; for Rot2d-2 and Rot3d-1 no initial condition for the ODE (3) is stated. For Rot2d-3, which is integrated over the finite interval [0,8], the computed extrema depend directly on omega(0), so the benchmark is not reproducible without this datum. For Rot2d-2, the limit-cycle extrema may be independent of omega(0) after a sufficiently long transient, but the transient length and the precise definition of t0 should be made explicit. Please specify omega(0) for each unsteady case and, for Rot2d-2, state how the transient is discarded.","section":"Section 2.2.1 (Rot2d-2, Rot2d-3) and Section 2.3 (Rot3d-1)"},{"comment":"The reference intervals inherit an unverified assumption that each configuration has a unique, stable stationary or periodic solution (for Rot2d-1 and Rot3d-1 a steady state with zero net torque, for Rot2d-2 a stable limit cycle). This assumption is never tested. Section 2.3.1 explicitly states that 'we were unable to find a suitable stable configuration at higher Reynolds numbers' in 3D, demonstrating that well-posedness is not guaranteed by the geometry and parameters alone. The numerical evidence is mutual convergence of the five presented methods, which rules out many implementation errors but not a common modeling bias or a coexisting attractor. Please provide a stability analysis of the reference states (e.g., Floquet multipliers for the limit cycle in Rot2d-2, eigenvalue analysis of the linearized steady state for Rot2d-1 and Rot3d-1) or, failing that, a reasoned discussion with numerical diagnostics that support uniqueness and stability.","section":"Section 2.3.1 and Tables 2, 4, 6, 8"},{"comment":"The procedure for constructing the reference intervals from the raw results is not documented. For example, Table 3 reports HDG (L,k)=(0,5) with CL,max=0.9555, which lies far outside the Table 4 interval [0.9674,0.9686]; similarly, several coarse-mesh runs for Rot2d-1 (e.g., the first TH row in Table 1) are excluded from the Table 2 interval. The reader cannot tell whether the intervals are based on the converged finer runs only, whether some runs are regarded as outliers, or whether some other selection rule is used. This makes the reference intervals irreproducible. Please state explicitly which runs were included for each quantity and the criterion (e.g., a threshold on estimated discretization error) used to exclude others.","section":"Section 4, Tables 3 and 4"}],"minor_comments":[{"comment":"The EOLPS extrapolated row contains the entry 'change of sign' without specifying which quantity changes sign or how this affects the extrapolation; please clarify.","section":"Section 4, Table 5"},{"comment":"The text says 'reverence intervals'; this should be 'reference intervals'.","section":"Section 4, paragraph 'Rot-2d1'"},{"comment":"The phrase 'To eliviate the costs' should read 'To alleviate the costs'.","section":"Section 3.2"},{"comment":"There is a typo in 'rotating around it's centre of mass'; it should be 'its'.","section":"Section 2.3"},{"comment":"The problem names are inconsistent: 'Rot-2d1', 'Rot-2d2', and 'Rot-3d1' in Section 4 versus 'Rot2d-1' and 'Rot3d-1' in Section 2; please unify the notation.","section":"Section 4"},{"comment":"The plots show time histories but do not indicate which method's results are displayed; a legend or caption note would be helpful.","section":"Figure 3"}],"recommendation":"major_revision","confidential_remarks":"This is a useful benchmark paper with a solid methodological basis, but the missing initial condition for the angular velocity and the undocumented interval-construction rule are substantive reproducibility issues. The well-posedness concern is also legitimate and should be addressed with at least a stability check or a careful caveat. The manuscript is likely acceptable after these points are resolved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper does a useful, concrete thing: it defines a benchmark for fluid-rigid body interaction with a freely rotating obstacle, gives 2D and 3D configurations, and supplies reference intervals for drag, lift, torque, angular velocity, pressure drop, and Strouhal number, based on five independent finite-element implementations. The code and full data are on Zenodo. That is a real contribution. The FSI community has Schäfer–Turek for fixed obstacles and Turek–Hron for elastic structures, but no widely agreed set of reference values for a body that is free to rotate. This fills that gap.\n\nThe strongest part is the execution. The methods are genuinely different: high-order Taylor–Hood with grad-div stabilization, an H(div)-conforming hybrid DG method, equal-order with LPS, low-order Taylor–Hood, and a Taylor–Hood/scipy line. They report convergence studies, meshes, timesteps, dof counts, and runtimes. The converged values agree to tight intervals, e.g. CD to about 2e-5 in Rot2d-1. That is credible. The citation pattern is appropriate for a benchmark paper; the rotating-cylinder and rotating-sphere literature is there, including the free-rotation studies closest to this setup.\n\nThe paper is also honest about its limits: Section 2.3.1 says they could not find a suitable stable configuration at higher Reynolds numbers in 3D, so only one steady 3D case is offered.\n\nThe soft spots are real but not fatal. The reference values are self-consensus: five in-house codes, no independent group, no experiment. For laminar benchmarks this is normal practice, but it does mean the intervals inherit any common modeling bias. More importantly, the benchmark states are assumed to be unique stable attractors. No Floquet or eigenvalue analysis is given for the periodic case, and the steady cases are found by Newton rather than verified stable. The paper's own inability to find stable 3D states at higher Re shows this is not a trivial point. A coexisting attractor would make the published intervals misleading. I consider that a moderate concern, not a fatal one; the agreement across methods makes a hidden second attractor unlikely, but a Floquet check for Rot2d-2 would settle it.\n\nOne minor transparency issue: the intervals in Table 4 discard the coarsest HDG runs (e.g. CL,max=0.9555 at (L,k)=(0,5), and 0.96713 at (0,6)) without an explicit inclusion rule. That is defensible, but the authors should say how the interval endpoints are chosen.\n\nWho is this for: anyone developing or testing FSI codes for particles, rotating bodies, or monolithic/partitioned solvers. It deserves a serious referee. I would recommend accept after minor revision that addresses the interval-selection rule and either adds a stability check or clearly scopes the claim.","headline":"Useful, honestly executed benchmark for freely rotating bodies in flow; reference intervals are credible but rest on an unverified attractor assumption and self-consensus among five in-house codes.","tokens_in":22393,"tokens_out":3373,"would_cite":true,"duration_ms":32808,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M60","76M10","76D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proposes benchmark cases for a freely rotating circular or spherical obstacle in channel flow and gives reference intervals for drag, lift, torque and angular velocity.","keywords":["fluid-rigid body interaction","benchmark problem","reference intervals","freely rotating sphere","Navier-Stokes equations","finite element methods","fluid-structure interaction","code validation"],"falsifier":"Run one of the benchmark cases (for example Rot2d-2 at Reynolds number 100) with an independent discretization not represented among the paper's methods—say a spectral or lattice-Boltzmann code—and compare its extremal drag, lift and Strouhal number with the intervals in Tables 3 and 4; any value falling outside those intervals would show that the reference values depend on the participating methods rather than on the continuum solution.","tokens_in":21463,"feed_emoji":"🌀","tokens_out":12990,"duration_ms":104991,"temperature":0.7,"pith_summary":"The paper proposes a benchmark problem for fluid–rigid-body interaction: a circular (2D) or spherical (3D) obstacle whose center of mass is pinned but which rotates freely in a laminar channel flow. Because the fluid–solid partition never changes, the geometry stays fixed and a wide class of standard computational fluid dynamics codes can compute the coupled problem. Four configurations are defined—three in 2D (steady, periodic, and time-dependent inflow) and one stationary 3D case—together with a list of quantities of interest: drag, lift, torque, angular velocity, pressure difference and Strouhal number. Using several independent finite-element discretizations, the authors obtain reference intervals for each quantity, so that other codes can be validated against the benchmark.","feed_headline":"Freely rotating obstacle gets benchmark reference values","feed_subtitle":"Four 2D and 3D channel-flow cases give drag, lift, torque and spin intervals for validating fluid-structure codes.","key_machinery":"The load-bearing object is the rotational coupling law $J\\partial_t\\omega = T$, the balance between the body's moment of inertia and the fluid torque $T=\\int_I (x-c_S)\\times(\\sigma(u,p)\\,n)\\,ds$, with the interface velocity imposed as $u=\\omega\\times(x-c_S)$. Together with the choice of a fixed center of mass, this turns the fluid-structure interaction into a Navier-Stokes problem on a time-independent domain coupled to a single ODE for the angular velocity, so the geometric partition stays fixed and standard fluid solvers can be used. The reference quantities (dimensionless drag $C_D$, lift $C_L$, torque $C_T$, angular velocity $\\omega^*$, pressure difference $\\Delta p$, and Strouhal number $St$) are the functionals used to compare methods, and the convergence of the different discretizations to each other is what turns their ranges into reference intervals.","core_discovery":"The central claim is that these four configurations and the accompanying reference intervals constitute a usable quantitative benchmark for fluid-particle interaction methods, with the rotation coupling as the nontrivial ingredient. The paper defines the configurations by specifying the domain, obstacle size and position, inflow profiles, Reynolds numbers (20 and 100 in the 2D cases, 20 in 3D), and the rigid-body parameters, then reports the values computed by five different finite-element schemes. The reference intervals in Tables 2, 4, 6 and 8 are the ranges spanned by the converged computations; they are meant to serve as targets that an independent implementation should hit. The authors note that the 3D case is limited to a stationary configuration because no suitable stable periodic configuration was found at higher Reynolds numbers.","pith_inferences":["A natural extension would be to use the same fixed-center design for non-spherical bodies or multiple rotating obstacles, creating harder particle-flow benchmarks without changing the benchmark's core structure.","Because the geometry is fixed, the same cases could be used to isolate coupling errors from spatial discretization errors by comparing different fluid–solid coupling schemes on identical space discretizations.","The paper's remark that no stable 3D periodic configuration was found at higher Reynolds numbers suggests that an unsteady 3D benchmark would require a different geometry, forcing, or Reynolds-number regime.","The sign change in $\\omega_1^*$ indicates that near-zero reference quantities can be reported as intervals crossing zero; future benchmarks may want to add a secondary residual-based quantity to assess convergence in such cases."],"forward_implications":["A new fluid-particle solver can be validated by reproducing the intervals in Tables 2, 4, 6 and 8; matching them indicates that the torque coupling and force evaluation are implemented correctly.","Rot2d-1 is cheap to run, so it is suited as a routine regression test; the paper reports reference intervals as tight as about $10^{-5}$ for $C_D$ and $10^{-8}$ for $\\omega^*$.","Rot2d-2 gives a stronger dynamic test through a periodic limit cycle, with vortex shedding and an additional Strouhal-number target.","Rot2d-3 tests time-dependent inflow and is particularly sensitive to the temporal discretization, making it a useful case for comparing time-stepping schemes.","Rot3d-1 extends the benchmark to three dimensions, where the near-zero component $\\omega_1^*$ changes sign between methods; the published interval tells users how much accuracy to expect for such a quantity."],"supporting_citations":[{"why":"It supplies the channel-with-cylinder geometry, parabolic inflow, and benchmarking approach that the 2D configurations are built on.","marker":"[1]"},{"why":"It provides the earlier FSI benchmark against which the present rigid-rotation benchmark is positioned.","marker":"[2]"},{"why":"It is the precedent for quantitative benchmark computations that report reference values for a two-phase flow problem.","marker":"[3]"},{"why":"It is a prior numerical study of a freely rotating sphere in a Newtonian fluid, giving context for the 3D case.","marker":"[5]"},{"why":"It reports earlier direct simulations of freely rotating cylinders, one strand of the literature the benchmark extends.","marker":"[7]"},{"why":"It studies shear-induced autorotation of a freely rotatable cylinder, another related configuration the benchmark draws on.","marker":"[12]"},{"why":"It justifies the volume-integral evaluation of drag and lift, which the authors say is more stable and accurate than boundary integrals.","marker":"[23]"}],"fun_headline_variants":["Rotating-sphere benchmark: drag, lift, torque, spin intervals","Fixed-center sphere benchmark for fluid-structure solvers","Numerical benchmark: freely rotating sphere in channel flow","Five schemes, one benchmark: fluid-rigid body reference data","Benchmark for rotating sphere: 2D and 3D reference intervals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The benchmark assumes that each configuration has a well-defined stable solution—a steady state, a limit cycle, or a prescribed time window—and that the reported reference intervals converge to that solution; the paper supports this only by observing that the participating methods agree with each other, not by proving existence, uniqueness, or convergence.","fun_headline_variants_meta":{"raw":{"variants":["Rotating-sphere benchmark: drag, lift, torque, spin intervals","Fixed-center sphere benchmark for fluid-structure solvers","Numerical benchmark: freely rotating sphere in channel flow","Five schemes, one benchmark: fluid-rigid body reference data","Benchmark for rotating sphere: 2D and 3D reference intervals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000845,"raw_usage":{"total_tokens":3605,"prompt_tokens":797,"completion_tokens":2808,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":413,"completion_tokens_details":{"reasoning_tokens":2730}},"tokens_in":413,"tokens_out":2808,"duration_ms":20946,"temperature":1.0,"reasoning_tokens":2730,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:36:19.111717+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run one of the benchmark cases (for example Rot2d-2 at Reynolds number 100) with an independent discretization not represented among the paper's methods—say a spectral or lattice-Boltzmann code—and compare its extremal drag, lift and Strouhal number with the intervals in Tables 3 and 4; any value falling outside those intervals would show that the reference values depend on the participating methods rather than on the continuum solution.","supporting_citations":[{"cited_title":"Sch¨ afer, S","cited_arxiv_id":null,"evidence_quote":"It supplies the channel-with-cylinder geometry, parabolic inflow, and benchmarking approach that the 2D configurations are built on."},{"cited_title":"Turek, J","cited_arxiv_id":null,"evidence_quote":"It provides the earlier FSI benchmark against which the present rigid-rotation benchmark is positioned."},{"cited_title":"Fabre, J","cited_arxiv_id":null,"evidence_quote":"It is a prior numerical study of a freely rotating sphere in a Newtonian fluid, giving context for the 3D case."},{"cited_title":"Ju´ arez, R","cited_arxiv_id":null,"evidence_quote":"It reports earlier direct simulations of freely rotating cylinders, one strand of the literature the benchmark extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It studies shear-induced autorotation of a freely rotatable cylinder, another related configuration the benchmark draws on."}],"review_version":1}