Bayesian inference of mean velocity fields and turbulence models from flow MRI

A. Kontogiannis; A. Marsden; D. B. Ennis; M. Loecher; M. P. Juniper; P. Nair

REVIEW 3 major objections 5 minor 15 references

Bayesian inference of mean velocity fields and turbulence models from flow MRI

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Bayesian inversion of the RANS equations reconstructs a turbulent jet's mean velocity field and learns its eddy-viscosity parameters directly from flow MRI data.

desk verdict A legitimate proof of concept for turbulent Bayesian RANS inversion from flow MRI, but the 'without overfitting' claim is in-sample only and should be softened. read the letter →

arxiv 2412.11266 v1 pith:5DPK3BBQ submitted 2024-12-15 physics.flu-dyn cs.LGmath.OC

A. Kontogiannis , P. Nair , M. Loecher , D. B. Ennis , A. Marsden , M. P. Juniper This is my paper · ORCID

classification physics.flu-dyncs.LGmath.OC

keywords BayesianinversionReynolds-averagedNavier-StokesequationsflowMRIeddyviscositymodelturbulencecalibrationconfinedturbulentjetFDAnozzleLaplaceapproximation

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 aims to establish a proof of concept: a turbulent mean flow can be reconstructed and its turbulence model calibrated in one Bayesian pass, using only the coarse, noisy 3D velocity data that flow MRI provides. The authors solve an inverse Reynolds-averaged Navier–Stokes (RANS) problem in which the unknowns are the inlet boundary condition and the coefficients of an algebraic eddy-viscosity model; adjoint-based optimization under a Laplace approximation (a Gaussian fit to the posterior) finds the most probable parameters and their uncertainties. They test the method on a confined turbulent jet at Reynolds number 6500 in an idealized medical-device (FDA) nozzle, assimilating 4D flow MRI velocity data with 1 mm isotropic voxels. The reconstructed mean velocity field fits the measured data in both the jet-breakdown and pipe-flow regions, and the learned mixing-length parameters carry quantified error bars, which the authors read as evidence that the model is learning rather than overfitting. If this holds, it becomes possible to build RANS models whose turbulence closure parameters are informed directly by imaging data, without resolving turbulent eddies.

What carries the argument

The load-bearing machinery is the Bayesian inverse RANS problem with adjoint-accelerated Laplace approximation. The forward map $Q$ sends unknown parameters $x = (g_i, p)$ — the inlet Dirichlet velocity and the turbulence-model parameters — to a RANS solution $u$; the projection $S$ maps $u$ into the data space, and the mismatch with the flow MRI data $u^\star$ is measured under the Gaussian noise covariance $C_{u^\star}$. The objective is the negative log-posterior of Eqs. (2.2)–(2.3), minimized with adjoint gradients, and the posterior is Laplace-approximated around the MAP point to give the covariance of Eq. (2.4). Inside $Q$, turbulence is closed with the Boussinesq ansatz $\mu_t = \ell_c^2 \dot\gamma$, where $\dot\gamma$ is the shear-rate magnitude and the compound mixing length is $\ell_c = \alpha H_\eta(d_s - d_{s0}) + \beta(1-H_\eta)d_w$, with $H_\eta$ a smooth activation that selects the jet-breakdown or pipe-flow regime. The parameters $p = (\mu_\ell, \alpha, \beta, x_c, c, d_{s0})$ are exactly what the inversion learns.

What would settle it

Measure the full turbulent stress tensor in the same FDA nozzle at Reynolds number 6500 with particle image velocimetry, and compare it with the stress predicted by the inferred eddy-viscosity parameters; if the two disagree beyond measurement error, the inferred parameters are artifacts of the closure rather than true properties of the turbulence.

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Extended reading notes

Core claim

The central claim is that a compact algebraic eddy-viscosity closure, built on a compound mixing length that switches between a streamwise-distance law in the jet-breakdown region and a wall-distance law in the pipe-flow region, is descriptive enough for this confined jet that its parameters can be learned from mean-velocity data alone. Solving the Bayesian inverse RANS problem yields a maximum-a-posteriori parameter set whose modeled velocity field matches the flow MRI data in both regions, with posterior covariances that place error bars on every inferred parameter. Because the data are noisy and the priors are broad, the authors interpret the match as successful learning of turbulence-model parameters rather than overfitting of a flexible model. The paper further claims the method is general: any differentiable turbulence model, algebraic or multi-equation, can replace the closure used here, and the same machinery extends to unsteady turbulent flows.

Load-bearing premise

The whole inference rests on the assumption that a simple algebraic formula for the turbulence viscosity, built from a hand-chosen compound mixing length, can faithfully represent the turbulence of this jet in both the breakdown and pipe-flow regions, and this assumption is never checked against independent turbulence data.

Editorial extensions

If this is right

If the closure and inference are sound, the same algorithm can be run with one- or two-equation turbulence models such as $k$–$\varepsilon$; velocity data would constrain the RANS mean flow while turbulent-kinetic-energy data would constrain the turbulence model itself.
The assimilation produces a denoised, physically constrained mean velocity field at the model's finer resolution (0.75 mm) from 1 mm flow MRI data, effectively sharpening low-SNR scans without resolving turbulent eddies.
Quantified parameter uncertainties enable model comparison: different turbulence closures can be ranked by their marginal likelihood under the Laplace approximation rather than by ad hoc fit metrics.
Because any differentiable turbulence model can be inserted, the methodology extends to unsteady RANS, allowing time-resolved 4D flow MRI to be assimilated in the same framework.
For medical-device flows such as the FDA nozzle, this offers a path to device- or patient-specific closure parameters derived from routine MRI, reducing reliance on generic turbulence-model constants.

Reading between the lines

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

A sharp test the authors did not run: repeat the inference at a second flow rate in the same phantom and check whether the learned mixing-length parameters transfer; the paper itself warns that algebraic models are not expected to extrapolate well, so non-transfer would show the parameters are fitted constants rather than physical closure coefficients.
The local turbulent kinetic energy available from MR signal decay is an independent constraint the authors chose not to use; assimilating it would reduce the degeneracy between mixing-length amplitude and shear-rate distribution that a velocity-only fit may leave.
Because the posterior is only Laplace-approximated, a few MCMC samples around the MAP point would test whether the reported Gaussian error bars are reliable; if the posterior is skewed, the uncertainties on $\beta$ and $x_c$ would need revision.
The phantom's two pressure ports are an untapped validation channel: the inferred RANS solution predicts a pressure drop between the ports that can be compared with the catheter measurements without any additional imaging.

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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 legitimate proof of concept for inferring RANS model parameters and reconstructing a mean velocity field from flow MRI data in a turbulent confined jet. The novelty is the extension of the authors' earlier laminar and non-Newtonian Bayesian inversion work to a turbulent RANS setting with an algebraic eddy viscosity, and the FDA nozzle flow MRI dataset is a nice new test case. The experiment is carefully described, and the optimization convergence is shown. That part holds up.

The soft spot is the 'without overfitting' claim. The agreement shown in Figure 6(f) is in-sample: the discrepancy is measured on the very data used for optimization. The model has substantial compensatory freedom—the inlet profile gi is inferred as a full function, and the compound mixing length adds six parameters. With posterior uncertainties like beta = 19.2 ± 12.6 and xc = 3.25 ± 2.87 cm, the data barely constrain the turbulence parameters. The residual axial discrepancy of 2.08σ also suggests structured model error or misspecified noise, not the white Gaussian noise assumed in the likelihood. So the most likely parameters are a statement about a weakly identified Laplace mode, not about the physics. The stress-test note is on target.

That said, the authors are honest about the limitations: they explicitly say algebraic models are not expected to extrapolate well and flag the k-ε model as a next step. For a pilot study that is a reasonable posture. The central concept—that the Bayesian machinery can assimilate flow MRI data into a RANS problem and produce a plausible mean field—is still demonstrated, even if the specific parameter values are not reliable.

This paper is for people working on data assimilation in fluids, medical imaging, or turbulence model calibration. It doesn't validate the turbulence model, but it shows the inference pipeline working on a real turbulent flow MRI dataset. With a serious referee, the authors should be pushed to either add an independent validation (withheld data, a high-fidelity reference, or a predictive check) or substantially soften the overfitting claim. I'd send it to peer review rather than desk reject it; it's a useful data point, not a settled result.

Referee Report

3 major / 5 minor

Summary. The manuscript solves a Bayesian inverse RANS problem: it combines a finite-element RANS solver with an algebraic compound mixing-length eddy-viscosity closure (Eqs. 2.5–2.8) and uses adjoint-based gradient optimization plus Laplace's approximation (Eqs. 2.2–2.4) to infer the inlet boundary condition and six turbulence-model parameters from 3D flow-MRI data of a confined turbulent jet in the FDA nozzle at Re 6500. The authors report MAP estimates and uncertainties (Table 3), data-model discrepancies (Eq. 4.1, Figs. 5–6), and conclude that the algorithm reconstructs the mean flow and learns the most likely turbulence parameters 'without overfitting' (abstract; §4.1; §5).

Significance. If the central claims are fully supported, the paper would demonstrate a useful proof of concept: jointly reconstructing turbulent mean fields and calibrating a RANS closure from clinical-grade flow-MRI data, with quantified uncertainty, in a regime where laminar-assimilation tools fail. The experimental dataset (multi-VENC 4D-flow MRI, phantom construction, pressure ports) is a valuable contribution, and the algorithmic core—adjoint-accelerated Laplace-approximated Bayesian inference—is sound and clearly described. However, the validation presented is entirely in-sample and the parameter uncertainties are large; as it stands the paper establishes an interesting numerical pipeline but not the advertised 'without overfitting' generalization claim.

major comments (3)

[Abstract; §4.1, Eq. (4.1)] The claim of 'without overfitting' in the abstract and of a 'sufficiently descriptive' RANS model in §4.1 rests entirely on the data-model discrepancy (4.1), which is evaluated on the same flow-MRI data used to optimize gi and p. The MAP values (1.30, 1.31, 2.08)/σ are training residuals, not generalization errors. The model has ample capacity to fit those data even when the closure is wrong: gi is inferred over the full inlet and p contains six parameters (µℓ, α, β, xc, c, ds0). An independent check is required: hold-out voxels or velocity components, a synthetic-data twin, comparison with the measured pressure drop mentioned in §3.1, or another observable. The authors' own statement in §4.1 that the learned algebraic model 'is not expected to extrapolate well' further undercuts the blanket wording; the paper should be reframed as an in-sample assimilation proof of concept unless such a check is added.
[§4, Eqs. (2.2), (4.1)] At the MAP point the axial data-model discrepancy is 2.08σ (Sec. 4) with σ = 5 cm/s. Given the large number of voxels (40×45×169), a 2σ mean discrepancy is incompatible with the white-noise assumption ε ∼ N(0, σ²I) in Eq. (2.2), unless the residual is dominated by a small region. The paper neither plots the residual structure nor reports a spatial correlation test. This matters because the posterior covariance (2.4) is computed with that likelihood; if σ is misspecified the reported parameter uncertainties are unreliable. Please add residual diagnostics and either recalibrate σ or model correlated noise.
[§2, Table 3] The MAP uncertainties in Table 3 are extremely wide: β = 19.2 ± 12.6 and xc = 3.25 ± 2.87 cm, i.e., relative uncertainties of 66% and 88%. The claim that the algorithm 'learns the most likely turbulence model parameters' therefore overstates the information content. The posterior covariance in Eq. (2.4) is also a Laplace approximation around a possibly non-identified optimum; the paper should report the parameter correlation matrix or profile likelihoods, and dampen the conclusion to 'weakly constrained parameters with wide posterior.'

minor comments (5)

[Eq. (4.1) and surrounding text] The list of discrepancies contains typos: E(¯uy) appears twice and E(¯u◦y) uses inconsistent bar/circle notation; please list E(ux), E(uy), E(uz) with consistent notation.
[§2.1, Table 3] The paper calls the model 'five-parameter' in several places, but p = (µℓ, α, β, xc, c, ds0) contains six parameters plus the inferred inlet profile gi; please correct the count.
[Table 3] The table header labels two columns as 'c [cm]'; one should be 'xc [cm]' and the other 'c [cm]'.
[§3.2, Eq. (2.2)] The Gaussian noise assumption should be justified for the magnitude-weighted multi-VENC averaging described in §3.2, since MRI phase noise is commonly modeled as Rician or Rayleigh; either cite supporting reasoning or note that the Gaussian likelihood is an approximation.
[Figure 6(e,f)] The discrepancy panels share the velocity colormap [0,100 cm/s], which compresses the residual structure; a symmetric or diverging colormap would better reveal where the model error actually remains.

Circularity Check

1 steps flagged · score 2.0 of 10

The Bayesian inference chain is not circular; the only self-referential element is that the 'sufficiently descriptive' claim is supported by agreement with the same data that were assimilated.

fitted input called prediction [Section 4.1, Discussion; Eq. (4.1) and objective (2.3)]
"Figure 6(f) shows that the algorithm has managed to find model parameters that cause the inferred velocity field to closely match the measured velocity field. ... This shows that this RANS model is sufficiently descriptive for this confined jet flow."

The 'closely match' is measured by the data-model discrepancy E(ui) in Eq. (4.1), evaluated on the same flow-MRI data u* that define the Bayesian objective J in Eq. (2.3). Reducing this discrepancy is exactly what the MAP optimization is constructed to do, so the reported agreement is the training error, not an independent test. The conclusion that the RANS model is 'sufficiently descriptive' therefore rests on the same data used to fit its parameters, which is a form of in-sample self-confirmation rather than a derivation-level equivalence.

full rationale

The formal derivation chain—Bayes' theorem (2.2), the MAP optimization (2.3), and the Laplace-approximated posterior covariance (2.4)—is internally consistent and does not reduce to its inputs by construction. The compound mixing-length model (2.7)-(2.8) is an explicit modeling ansatz, not a quantity derived from the flow-MRI data, so no self-definitional circularity is present. Self-citations to Kontogiannis et al. (2022, 2024a, 2024b) refer to independently published methodology and prior experimental validations; they are not used here to forbid alternative turbulence models or to smuggle in an unverified ansatz. The only genuinely self-referential aspect is the validation logic: the algorithm's success is demonstrated on the same data that were assimilated, and the phrase 'without overfitting' is asserted without a holdout or cross-validation. That weakens the evidential strength of the central claim but does not make the Bayesian inverse problem itself circular. Accordingly, the paper merits a low circularity score of 2 rather than a higher score reserved for cases where a prediction is forced by construction or by a self-citation chain.

Assumptions & free parameters 8 free parameters · 5 assumptions · 1 invented entities

The posterior inference rests on a Gaussian noise model, a Boussinesq algebraic turbulence closure, hand-chosen priors, and Laplace's approximation. The six viscosity parameters and the inlet boundary condition are fitted to the same MRI data, and no independent evidence is given for the compound mixing length model, so the fitted parameters are the main load-bearing ingredients.

free parameters (8)

mu_l (laminar dynamic viscosity) = 4.2 ± 0.15 mPa.s
Included in the parameter vector p in Sec 2.1 and reported in Table 3; prior and posterior coincide, so it is effectively pinned by the known fluid.
alpha (streamwise mixing length coefficient) = 0.722 ± 0.193
Inferred from flow MRI data via Eq 2.7; posterior uncertainty is about 27 percent of the estimate.
beta (wall-distance mixing length coefficient) = 19.2 ± 12.6
Inferred from flow MRI data via Eq 2.7; uncertainty is 66 percent of the estimate, so it is poorly constrained.
xc (activation center) = 3.25 ± 2.87 cm
Inferred from flow MRI data; uncertainty is comparable to the value.
c (activation width half-scale) = 7.32 ± 2.44 cm
Inferred from flow MRI data; controls the smooth transition region in Eq 2.8.
ds0 (streamwise offset) = 0.159 ± 0.0881 cm
Inferred from flow MRI data; offset in streamline distance.
inlet Dirichlet boundary condition gi (function) = not tabulated
Inferred as part of x=(gi,p) in Sec 4; its values and uncertainty are not reported, though it is part of the learned solution.
prior standard deviations of p = 0.15 mPa.s, 6, 15, 30 cm, 30 cm, 15 cm
Chosen by the authors in Table 3; these hyperparameters directly influence the posterior covariance and the MAP estimate.

assumptions (5)

domain assumption White Gaussian noise model with zero mean and covariance sigma*I, sigma=5 cm/s
Invoked in Sec 4; flow MRI noise is actually Rician, and the Gaussian assumption can bias low-SNR regions.
domain assumption RANS equations with Boussinesq eddy viscosity closure describe the mean flow of the confined turbulent jet
Invoked in Sec 2.1 Eq 2.5-2.6; no validation against Reynolds-stress data is provided.
domain assumption Laplace approximation gives an adequate Gaussian posterior
Invoked in Sec 2 Eq 2.4; large posterior uncertainties suggest the posterior may be non-Gaussian.
domain assumption Geometry Omega and no-slip boundary from MRI magnitude segmentation are accurate
Invoked in Sec 4; segmentation errors propagate into all inferred quantities.
domain assumption Zero traction outlet BC and fixed inlet/outlet conditions are appropriate
Set in Sec 4; not validated against measured pressure data despite pressure ports.

invented entities (1)

Compound mixing length model with smooth activation H_eta
purpose: Defines algebraic eddy viscosity mu_t = l^2 dot_gamma to blend jet-breakdown scaling alpha(ds - ds0) with wall scaling beta*dw over two flow regions.
Introduced in Eq (2.7)-(2.8) for this confined jet; no independent dataset supports its form, and the authors note that algebraic models are not expected to extrapolate.

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Cite this review

Pith. "Pith review of Bayesian inference of mean velocity fields and turbulence models from flow MRI." pith.science (2026). https://pith.science/paper/5DPK3BBQ

@misc{pith2026241211266,
  author       = {Pith},
  title        = {Pith review of: Bayesian inference of mean velocity fields and turbulence models from flow MRI},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5DPK3BBQ}},
  note         = {Machine review of arXiv:2412.11266}
}

read the original abstract

We solve a Bayesian inverse Reynolds-averaged Navier-Stokes (RANS) problem that assimilates mean flow data by jointly reconstructing the mean flow field and learning its unknown RANS parameters. We devise an algorithm that learns the most likely parameters of an algebraic effective viscosity model, and estimates their uncertainties, from mean flow data of a turbulent flow. We conduct a flow MRI experiment to obtain mean flow data of a confined turbulent jet in an idealized medical device known as the FDA (Food and Drug Administration) nozzle. The algorithm successfully reconstructs the mean flow field and learns the most likely turbulence model parameters without overfitting. The methodology accepts any turbulence model, be it algebraic (explicit) or multi-equation (implicit), as long as the model is differentiable, and naturally extends to unsteady turbulent flows.

Discussion (0). Continue with ORCID to comment.

Reference graph

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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 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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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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, " * 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 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] [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]

& Kaiser, T

Casel, M., Oberleithner, K., Zhang, F., Zirwes, T., Bockhorn, H., Trimis, D. & Kaiser, T. L. 2022 Resolvent-based modelling of coherent structures in a turbulent jet flame using a passive flame approach. Combust. and Flame\/ 236 , 111695

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& Patz, S

Gudbjartsson, H. & Patz, S. 1995 The Rician distribution of noisy MRI data . Magn. Reson. Med.\/ 34 , 910--914

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2002 Approche bay\'esienne pour les probl\`emes inverses\/

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Jaynes, E. T. 2003 Probability Theory: The Logic of Science\/ . Cambridge University Press

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