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arxiv: 2606.28569 · v1 · pith:PRBAZDKCnew · submitted 2026-06-26 · ⚛️ physics.flu-dyn · math.DS

Data-driven linear analysis of turbulent flows

Benjamin Herrmann , Katherine Cao , Carlos A. Gonzalez , S. L. Brunton , Beverley J. McKeon This is my paper

Pith reviewed 2026-06-30 00:45 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn math.DS
keywords dynamic mode decompositionresolvent analysisturbulent flowsdata-driven analysisNavier-Stokes equationsmean flowcomputational fluid dynamicsnonlinear forcing
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The pith

Nonlinearity-subtracted DMD allows linear analysis of turbulent flows from any CFD simulation data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops nonlinearity-subtracted dynamic mode decomposition to enable linear analyses such as resolvent analysis on turbulent flows. It does so by using snapshots of the nonlinear terms to remove their effect, so the learned operator approximates the mean-flow linearized Navier-Stokes dynamics. This turns the method into a post-processing tool that works with data from any high-fidelity CFD simulation code. Demonstrations include a minimal channel flow and the flow over a full aircraft model using DNS and LES data. The approach matters because it removes the need for specialized codes when studying complex geometries.

Core claim

NSDMD modifies the standard DMD procedure by subtracting the contribution of the nonlinear terms in the perturbation equations from the data snapshots, yielding an operator that is a low-rank approximation of the underlying mean-flow-linearized dynamics and thereby allowing data-driven performance of resolvent analysis.

What carries the argument

nonlinearity-subtracted DMD (NSDMD), which explicitly accounts for nonlinear forcing by subtracting its snapshots to isolate the linear operator.

Load-bearing premise

Subtracting snapshots of the nonlinear terms from the data produces an operator that faithfully approximates the true mean-flow-linearized dynamics without any extra modeling.

What would settle it

Direct computation of the resolvent operator from the linearized equations around the known mean flow on the same dataset, followed by comparison of the resulting modes and gains to those obtained via NSDMD; significant differences would indicate the approximation does not hold.

Figures

Figures reproduced from arXiv: 2606.28569 by Benjamin Herrmann, Beverley J. McKeon, Carlos A. Gonzalez, Katherine Cao, S. L. Brunton.

Figure 1
Figure 1. Figure 1: Schematic of the procedure followed by NSDMD for turbulent flows. still requires the use of specialized in-house codes that are only available in research environments. In contrast, the unprecedented availability of high-fidelity data from numerical simula￾tions and experimental measurements of fluid flows has led to the development and appli￾cation of many data-driven modal decompositions (Taira et al. 20… view at source ↗
Figure 2
Figure 2. Figure 2: Turbulent flow examples used to demonstrate NSDMD. (a) Minimal channel flow data from Herrmann et al. (2023). (b) ZPGFP boundary-layer data from Towne et al. (2023). (c) Flow over JSM aircraft, numerical setup from Goc et al. (2021). reside in memory. Fortunately, this limitation can be circumvented by loading the data sequentially and using the method of snapshots (Sirovich 1987) to compute the correlatio… view at source ↗
Figure 3
Figure 3. Figure 3: NSDMD-based DDRA of minimal channel flow for kx = 2π/Lx and kz = 2π/Lz. (a) Convergence of the leading four resolvent gains with the number of snapshots m. Data-driven and equation-based gains are shown in dashed blue and solid gray lines, respectively. (b) Isosurfaces of the wall-normal component of ϕ4 . (c) Isosurfaces of the streamwise component of ψ4 . Modes are shown for their most amplified frequency… view at source ↗
Figure 4
Figure 4. Figure 4: NSDMD-based DDRA of the ZPGFP boundary-layer flow for kz = 10π/Lz. (a) Leading resolvent gain as a function of St. Leading (b) forcing and (c) response modes at the most amplified frequency. Modes are shown as positive (light) and negative (dark) isosurfaces of the streamwise components at 0.75 of their peak magnitudes. Real left- and right-going waves are added together, and only the upstream half of the … view at source ↗
Figure 5
Figure 5. Figure 5: Comparison between DMD and NSDMD (a) eigenvalues and (b) modes. DMD modes (left panel) and NSDMD modes (right panel) are shown as isosurfaces of the real parts of their streamwise components at ±0.05 and 0.005 of their peak magnitudes, respectively. its rows onto VX. The latter can be interpreted as the dominant structures in x˙ j − fj whose time evolution is correlated with the dynamics of xj . Importantl… view at source ↗
read the original abstract

Mean-flow-based linear analyses of turbulent flows, such as resolvent analysis, provide valuable insight about flow structures and their dynamics that has been widely leveraged to model, control and understand the underlying flow physics. However, these analyses are computationally expensive for flows over complex geometries and require the use of specialized codes that are typically only available in research environments. On the other hand, data-driven modal decompositions, such as the dynamic mode decomposition (DMD), identify turbulent flow structures that, although statistically relevant, do not provide insight into the physical mechanisms driving their dynamics. Here we introduce a novel data-driven method -- nonlinearity-subtracted DMD (NSDMD) -- that leverages knowledge of the structure of the Navier--Stokes equations to ensure that the learned operator is a low-rank approximation of the underlying mean-flow-linearized dynamics. Specifically, the method uses snapshots of the nonlinear terms in the perturbation equations to explicitly account for the contribution of the nonlinear forcing to the dynamics. We demonstrate the use of NSDMD to perform data-driven resolvent analysis on direct numerical simulation (DNS) and large-eddy simulation (LES) datasets, starting with a minimal channel flow and scaling up to the flow over a full aircraft model. As a result, NSDMD allows performing linear analyses of turbulent flows as a post-processing step on simulation data obtained with any available high-fidelity computational fluid dynamics (CFD) code.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper introduces nonlinearity-subtracted DMD (NSDMD), a data-driven method that subtracts snapshots of the nonlinear terms from the perturbation equations to recover a low-rank operator approximating the mean-flow-linearized Navier-Stokes dynamics. This enables data-driven resolvent analysis as a post-processing step on DNS or LES data from arbitrary high-fidelity CFD codes. The approach is demonstrated on a minimal channel flow and scaled to the flow over a full aircraft model.

Significance. If the central claim holds, NSDMD would allow mean-flow-based linear analyses (resolvent, etc.) to be performed on existing simulation datasets without specialized linearization codes or access to the underlying solver, extending such tools to complex geometries. The manuscript provides a concrete implementation path that leverages the structure of the NS equations while remaining compatible with black-box CFD output.

major comments (3)
  1. [§3.1–3.2] §3.1–3.2 (method derivation): The subtraction of nonlinear-term snapshots is presented as yielding an operator that is a faithful low-rank approximation to the true mean-flow-linearized dynamics, yet no derivation or error bound is given showing that the discrete-time, discrete-space subtraction exactly cancels the nonlinear forcing contribution (including any residual Reynolds-stress effects arising from the time-averaged mean). Without this, it is unclear whether the learned DMD operator matches the classical resolvent operator or absorbs discretization artifacts.
  2. [§4.1] §4.1 (minimal channel results): The paper reports that NSDMD recovers structures consistent with classical resolvent analysis, but provides no quantitative metric (e.g., operator-norm difference, gain-curve L2 error, or singular-value spectrum comparison) against a reference linearization performed on the same mean flow. This leaves the central claim that the method produces “the underlying mean-flow-linearized dynamics” without direct verification.
  3. [§4.2–4.3] §4.2–4.3 (scaling to aircraft): While the aircraft demonstration shows practical applicability, the absence of an ablation on the precise form of the convective-term evaluation or interpolation used when subtracting nonlinear snapshots means any residual forcing is unquantified; this directly affects whether the post-processing claim holds for general CFD codes.
minor comments (2)
  1. Notation for the perturbation velocity and nonlinear term is introduced without an explicit table relating symbols to the continuous NS equations; a short appendix table would improve readability.
  2. Figure 3 (channel flow modes): axis labels and color-bar scaling are inconsistent with the corresponding resolvent gain plots in Figure 4; this makes direct visual comparison harder than necessary.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive feedback and for recognizing the potential impact of NSDMD. We provide point-by-point responses to the major comments and indicate where revisions will be made to address the concerns.

read point-by-point responses

  1. Referee: [§3.1–3.2] §3.1–3.2 (method derivation): The subtraction of nonlinear-term snapshots is presented as yielding an operator that is a faithful low-rank approximation to the true mean-flow-linearized dynamics, yet no derivation or error bound is given showing that the discrete-time, discrete-space subtraction exactly cancels the nonlinear forcing contribution (including any residual Reynolds-stress effects arising from the time-averaged mean). Without this, it is unclear whether the learned DMD operator matches the classical resolvent operator or absorbs discretization artifacts.

    Authors: We clarify that the method is derived by subtracting the nonlinear term snapshots from the perturbation snapshots, which in the continuous setting exactly isolates the linear operator. In discrete settings, the approximation holds provided the mean flow is accurately computed and discretization is consistent. We will revise §3 to include a more detailed discussion of the assumptions, potential residuals from time-averaging, and discretization effects to address this concern. revision: yes

  2. Referee: [§4.1] §4.1 (minimal channel results): The paper reports that NSDMD recovers structures consistent with classical resolvent analysis, but provides no quantitative metric (e.g., operator-norm difference, gain-curve L2 error, or singular-value spectrum comparison) against a reference linearization performed on the same mean flow. This leaves the central claim that the method produces “the underlying mean-flow-linearized dynamics” without direct verification.

    Authors: We agree that direct quantitative comparison would provide stronger evidence. Since the minimal channel allows for a reference linearization, we will add in §4.1 comparisons of the resolvent gain curves and singular value spectra between NSDMD and the classical mean-flow linearization to verify the match. revision: yes

  3. Referee: [§4.2–4.3] §4.2–4.3 (scaling to aircraft): While the aircraft demonstration shows practical applicability, the absence of an ablation on the precise form of the convective-term evaluation or interpolation used when subtracting nonlinear snapshots means any residual forcing is unquantified; this directly affects whether the post-processing claim holds for general CFD codes.

    Authors: For the aircraft flow, the nonlinear snapshots are evaluated using the identical discretization scheme as the original simulation to minimize residuals. We will expand the discussion in §4.2–4.3 to include sensitivity analysis to interpolation and note that consistent evaluation is key for the method's applicability to arbitrary CFD codes. A full ablation may require additional simulations but we will quantify the effect where possible. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces NSDMD by explicitly subtracting snapshots of nonlinear terms (from the perturbation form of the Navier-Stokes equations) before applying DMD, thereby constructing an operator that approximates the mean-flow-linearized dynamics. This is a direct methodological construction leveraging the known structure of the governing equations rather than a claimed first-principles derivation or prediction that reduces to its own fitted inputs. No self-citation load-bearing steps, uniqueness theorems, or ansatzes smuggled via prior work are described in the provided text. The central claim is that the resulting operator enables data-driven linear analysis as post-processing; this holds by the subtraction step itself and does not exhibit the enumerated circular patterns. The derivation remains self-contained against external benchmarks such as standard DMD or resolvent analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the nonlinear terms can be computed from the same snapshots used for DMD and that their subtraction isolates the linear operator without additional modeling choices. No free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The perturbation equations around the mean flow can be written with an explicit nonlinear forcing term that is available from the simulation data.
    This is the structural knowledge of the Navier-Stokes equations invoked to justify the subtraction step.

pith-pipeline@v0.9.1-grok · 5791 in / 1336 out tokens · 23347 ms · 2026-06-30T00:45:34.178055+00:00 · methodology

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Reference graph

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