Quantification of Model Uncertainty in RANS Simulations: A Review

• Ensemble averaging or spatial filtering denotes the averaging/filtering operation.Source: Nomenclature · p. 2
• ||A||P denotes the norm of A weighted by covariance.Source: Nomenclature · p. 2
• Hadamard multiplication denotes element-wise multiplication.Source: Nomenclature · p. 3
• D· Dt denotes the material derivative.Source: Nomenclature · p. 3

Geometric and tensor notation

• Geometric, tensor, and coordinate notation includes barycentric coordinates, identity and rotation tensors, eigenvectors, spatial coordinates, time, mean-flow quantities, and real-number space.c1, c2, c3 are barycentric coordinates; I is the second-order identity tensor; Qδ is a rotation matrix; V contains eigenvectors; xi, x are spatial coordinates; t is time; q denotes mean flow features; P mean pressure; Ui, U mean velocity; and R real-number space.Source: Roman Letters · p. 3

Statistical notation

• Statistical notation includes covariance matrices or random-variable covariance, expectation, variance, probability distributions or mass functions, and random variables with their realizations.Cov denotes covariance; E[Z] expectation; Var[Z] variance; p(z) probability distribution; P discrete probability mass function; and Z, z a random variable and its realization.Source: Roman Letters · p. 3

Uncertainty and computational notation

• Uncertainty, inference, and numerical-method notation covers discrepancies, inference data, Gaussian processes and kernels, Kalman gains, covariance scales, state variables, random-field and stochastic-process terms, and LES or BMSA quantities.d measures observation–truth discrepancy; D is inference data; GP and K denote Gaussian processes and kernels; K is also a Kalman gain matrix; l is covariance-kernel length scale; β is a multiplicative discrepancy field; δ denotes discrepancies; ∆g is LES grid spacing or filter width; ˜Si is a BMSA scenario; z is an augmented state; wα are random-field expansion coefficients; and W is a Wiener process.Source: Roman Letters · p. 3

Operators and models

• Differential-equation and optimization notation covers material, linear, and nonlinear operators, objective functions, order estimates, and model sets or individual models.D· is the material derivative; L and N are linear and nonlinear differential operators; J is an objective function; O(·) means order of; and M, Mi denote models or sets of models.Source: Roman Letters · p. 3

Flow and turbulence variables

• Flow and turbulence notation defines pressure, velocity, turbulent kinetic energy, production, strain, transport, source terms, viscosity, and wing–body-juncture locations.The symbols include p and p′ for pressure and pressure fluctuation, u and u′ for velocity and fluctuation, k for turbulent kinetic energy, P for production, S for strain rate, Tω for turbulent-frequency transport, S for source terms, µ for dynamic viscosity, and P1, P2 for two locations.Source: Roman Letters · pp. 3–4

1.1. Landscape of turbulence modeling

• Turbulence-modeling approaches trade computational cost against resolved physics: DNS resolves all scales, LES resolves larger scales, and RANS models the full turbulent spectrum.LES is substantially cheaper than DNS for free-shear flows but remains expensive for high-Reynolds-number wall-bounded flows, motivating hybrid RANS/LES and wall-modeled LES.Source: §1.1 · pp. 5–6, Figure 1 · p. 6
• RANS requires closure models for Reynolds stresses generated by nonlinear convection, making closure choice and parameter calibration principal sources of predictive uncertainty.Different closure structures and coefficients are commonly selected or adjusted using expert judgment and canonical-flow calibration, which may not transfer to new flows.Source: §1.1 · p. 7
• RANS remains the engineering workhorse because its low computational cost and robustness outweigh persistent accuracy limitations and dependence on flow-specific tuning.Wind-tunnel testing stagnated partly because turbulence models remain a predictive bottleneck, while most widely used models were developed decades ago and lack generality.Source: §1.1 · pp. 7–8

1.2. Origin of uncertainties in RANS models

• RANS model uncertainty includes structural uncertainty from chosen functional forms and parametric uncertainty from closure coefficients, while the review focuses on these two sources.Both are epistemic and can theoretically be reduced with better knowledge or data, but over-tuning parameters can mask structural inadequacy and cause over-fitting on new flows.Source: §1.2 · pp. 8, 11
• The Boussinesq assumption can fail in separated, curved, or strongly pressure-gradient flows, producing structural uncertainty in RANS predictions.Nonlinear eddy-viscosity and explicit algebraic Reynolds-stress models are examples of attempts to address linear eddy-viscosity limitations.Source: §1.2 · p. 9
• Parametric uncertainty arises because closure coefficients are calibrated on simple flows, affected by measurement error, selected for numerical reasons, or heuristically retuned for target flows.In the Baldwin–Lomax example, varying one of seven parameters changes the shock location and post-shock pressure on a transonic NACA0012 airfoil.Source: Figure 4 · p. 10, §1.2 · p. 11

1.3. Approaches for quantifying uncertainties in turbulence models

• The review primarily adopts a Bayesian framework, representing uncertainty as subjective probability while acknowledging alternative non-probabilistic treatments of epistemic uncertainty.Forward methods propagate prescribed distributions through RANS; backward methods assimilate data to infer coefficient distributions or model errors before subsequent prediction.Source: §1.3 · pp. 11–12
• RANS uncertainty-quantification methods divide into parametric versus non-parametric and forward data-free versus backward data-driven approaches, each balancing practicality, physical insight, and computational cost.Parametric methods are non-intrusive but constrained by baseline models, whereas non-parametric methods target modeled fields directly but are intrusive and grow in dimensionality with mesh size.Source: §1.3 · p. 12, Table 1 · p. 13

2. Fundamentals of probability and statistics for uncertainty quantification

• RANS uncertainty quantification uses forward propagation to map prior distributions on inputs through simulations and backward Bayesian inference to update inputs from noisy, biased, or incomplete data.The inferred posterior can subsequently be propagated to predict quantities of interest such as lift and drag coefficients.Source: Figure 6 · p. 15, §2.1 · p. 17

2.1. Representation, sampling, and propagation of model uncertainties

• Uncertain RANS quantities are represented as random variables, vectors, or fields, with distributions, moments, covariance structures, and spatial correlation describing model coefficients and discrepancies.The RANS model maps latent variables and uncertain parameters—vectors of coefficients or fields such as Reynolds stresses and eddy viscosity—to observable outputs.Source: §2.1 · pp. 16–17

2.2. Uncertainty propagation (Forward analysis)

• Monte Carlo propagation avoids dimensionality-dependent convergence degradation but converges at O(N −1/2), while spectral and surrogate methods can be faster yet face smoothness or dimensionality limits.Variance reduction, multilevel or multifidelity models, and surrogate mappings address expensive CFD evaluations, although multifidelity methods have mainly targeted input rather than model uncertainties.Source: §2.2 · pp. 17–18

2.3. Statistical inference (Backward analysis)

• Bayesian inference combines prior knowledge and the likelihood of observed data to compute a posterior, but each likelihood evaluation requires a costly RANS simulation.The posterior p(θ|D) is proportional to p(θ)p(D|θ), normalized by the total data probability p(D).Source: §2.3 · p. 18

2.3.1. Bayesian inference based on Markov chain Monte Carlo sampling

• MCMC targets high-posterior regions more effectively than plain Monte Carlo, but CFD-scale inference is impractical because sampling may require O(10^6) simulations.Each CFD likelihood evaluation can take hours or weeks; RANS jet-in-crossflow simulations required O(10^7) grid points and O(10^4) CPU hours. Surrogates and adaptive approximations reduce costs, while physical constraints can exclude regions whose fraction increases exponentially with dimension.Source: §2.3.1 · pp. 19–20

2.3.2. Approximate Bayesian inference based on MAP estimation

• Approximate Bayesian methods use MAP estimation when full posterior accuracy is unnecessary, with variational and ensemble approaches supporting parameter inference under RANS constraints.Iterative EnKF assimilates observations into prior ensembles to produce posterior-fitting states without derivatives, but its linear-model and Gaussian assumptions limit performance for nonlinear or non-Gaussian problems.Source: §2.3.2 · pp. 20–22

3.1. Uncertainties in RANS model parameters

• RANS closure coefficients lack universal values because their calibrated optima vary across canonical flows and models, revealing substantial parametric uncertainty.Examples include n = 1.3 corresponding to Cε2 = 1.77, κ values ranging from 0.33 to 0.45, and the physical constraint Cε2 > Cε1.Source: §3.1 · pp. 23–24

3.2. Parametric uncertainty in RANS models: forward approaches

• Forward uncertainty propagation shows that coefficient variability can affect recirculation, turbulence intensity, reattachment, pressure, and wall shear, but cannot explain all observed RANS discrepancies.For flat-plate flow, uncertain wall-function parameters produced intervals encompassing measured friction data, whereas channel-flow inaccuracies were attributed to structural uncertainty.Source: §3.2 · pp. 25–27
• Forward approaches require uncertain coefficient distributions that are often incomplete, subjective, or restricted to simple flows, motivating inverse inference from observable data.Sensitivity methods provide bands around nominal solutions, whereas probabilistic propagation estimates full output distributions from assigned joint coefficient distributions.Source: §3.2 · pp. 25, 27, §3.3.1 · p. 27

3.3.1. Statistical inference of model parameters

• Statistical inference finds that default literature coefficients may fall outside best-fit ranges, vary across datasets, and generally lack validity for predicting different flows.Bayesian calibration estimates joint posterior distributions and parameter correlations, while EnKF recovered known coefficients using ensembles of 100 CFD evaluations for synthetic data.Source: §3.3.1 · pp. 29, 31
• Bayesian calibration can represent model-form uncertainty through posterior variation across scenarios, but resulting p-box predictions may be overly conservative.Calibrating the Launder–Sharma model against 13 flat-plate datasets produced significantly varying likely coefficients and uncertainty bounds encompassing experimental intervals.Source: §3.3.1 · pp. 31–32

3.3.2. Accounting for structural uncertainties in RANS models

• Model inadequacy must be modeled separately from observational noise because even physically true parameters may produce inaccurate predictions under simplifying assumptions.Additive or multiplicative discrepancy terms can be spatially correlated; Cheung et al. used a multiplicative formulation that preserves the no-slip boundary condition.Source: §3.3.2 · pp. 32, 34
• The spatial correlation structure of model inadequacy affects Bayesian model selection, while calibrated corrections vary substantially across cases and may be too large for parameter adjustment alone to explain the data.Such inadequacy terms are specific to observed quantities, calibration geometries, and scenarios, limiting transfer to unobserved quantities, different geometries, or operating conditions.Source: §3.3.2 · pp. 34–35

3.3.3. Accounting for multiple models: Bayesian model selection and averaging

• Bayesian calibration found no single turbulence model consistently best across 13 boundary-layer datasets; model probabilities were highly flow-dependent, and algebraic models sometimes performed well despite Reynolds-stress models’ theoretical superiority.Oliver and Moser similarly found only slight preference for Chien’s k–ε model, with no dominant posterior winner.Source: §3.3.3 · p. 36, Figure 13 · p. 37
• Bayesian model averaging combines posterior predictive distributions from competing models, while BMSA additionally weights predictions by model posterior probabilities and calibration-scenario priors for a new scenario.BMSA averages I × K posterior predictive distributions and accounts for uncertainty in which calibration flow configuration is relevant.Source: §3.3.3 · pp. 36–37
• BMSA scenario priors encode presumed calibration-to-prediction similarity; uniform priors may overestimate predictive variance, whereas an empirical prior uses cross-model prediction scatter.The empirical scheme assigns larger prior weight to scenarios whose calibrated models predict the new quantity more similarly.Source: §3.3.3 · p. 38
• BMSA requires I × K stochastic CFD calculations, but propagating maximum-posterior parameter values reduces this to I × K deterministic simulations and enabled application to the ONERA M6 wing.This simplification replaces full posterior parameter propagation with one deterministic simulation per model–scenario combination.Source: §3.3.3 · p. 38
• Multi-model averaging remains biased by the subjective finite model set, but can reduce single-model bias, provide conservative predictions for unseen configurations, and estimate variance from model disagreement.The averaged result is expected to outperform the worst model without necessarily matching the unknown best model.Source: §3.3.3 · p. 38

4.1. Motivation and overview

• Parametric approaches cannot represent secondary flows or misaligned Reynolds stresses when the Boussinesq assumption excludes the true solution, whereas non-parametric perturbations can span it [118].In wing–body juncture flow, perturbing full Reynolds stresses, including eigen-directions, covers the measured solution while perturbing turbulent kinetic energy does not.Source: §4.1 · pp. 39–40, Figure 17 · p. 42
• Non-parametric methods perturb either model forms, such as turbulent transport equations, or model outputs, such as turbulent viscosity and Reynolds stresses.Their algorithmic targets include PDE, intermediate-field, and parametric levels, whose uncertainty sources are coupled and difficult to rank across flows.Source: §4.1 · p. 40

4.2. Introducing uncertainties in turbulent transport equations

• Transport-equation approaches infer discrepancy fields in source terms, such as β(x) multiplying the k–ω production term, using DNS or experimental observations.Regularizing β toward 1 constrains the corrected model near the baseline and makes the otherwise high-dimensional inverse problem tractable.Source: §4.2 · pp. 42–43
• Correction profiles inferred across channel flows with Reτ = 395 to 4200 were qualitatively similar, supporting regression against nondimensional flow variables for transfer across related flows.A correction trained on S814 airfoil data at Re = 1 × 10^6 and Re = 2 × 10^6 was used to predict the S809 airfoil at Re = 2 × 10^6.Source: §4.2 · pp. 43–44
• Transport-equation corrections can generalize across model families, but a linear eddy-viscosity baseline remains constrained by the Boussinesq assumption.Thus, correcting transport equations does not remove structural limitations inherited from the selected baseline model.Source: §4.2 · p. 44

4.3. Introducing uncertainties in turbulent viscosity

• Given ground-truth mean velocities, an optimal eddy-viscosity field νopt_t is obtained by minimizing discrepancy between computed and ground-truth velocities, with positivity and smoothness constraints on νt.Source: §4.3 · p. 45
• Dow and Wang modeled logarithmic eddy-viscosity discrepancies with a Gaussian process, sampled fields for similar geometries, and propagated them through RANS to quantify velocity uncertainty.The method was extended to U-bend flow, but its ensemble predictions remained constrained by the baseline model’s Boussinesq assumption.Source: §4.3 · pp. 45–46

4.4. Introducing uncertainties in Reynolds stresses

• Reynolds-stress modeling error is the source of RANS model-form uncertainty for single-phase, fully turbulent flows without transition.Reynolds stress enters the RANS momentum equations through closure, and supplying the true stress theoretically yields the true velocity when numerical uncertainties are negligible and the equations are well-conditioned.Source: §4.4 · pp. 46–47

4.4.1. Stochastic differential equation of Reynolds stress discrepancy

• The stochastic-differential-equation approach models the Reynolds-stress discrepancy as a random field whose solution propagates uncertainty to velocities and other quantities of interest.Its production-diffusion structure resembles the Reynolds stress transport equation, while a stochastic residual replaces unclosed terms; cross-component and spatial correlations are represented, but extension beyond plane channels relies on heuristics.Source: §4.4.1 · pp. 47–48

4.4.2. Estimating uncertainty bounds guided by realizability maps

• Realizability-guided perturbations form an efficient physics-based scheme that explores limiting Reynolds-stress states using only five simulations and supports later inference and machine learning.The approach perturbs turbulent kinetic energy, eigenvalues, and eigenvectors while respecting their coupling and realizability constraints.Source: §4.4.2 · pp. 49–50

4.4.3. Systematic sampling of Reynolds stress uncertainty within realizability constraints

• Systematic sampling parameterizes turbulent-kinetic-energy and eigenvalue discrepancies as constrained random fields, while eigenvector perturbations use orthonormal rotations represented by Euler angles or quaternions.The parameterization preserves symmetric positive semidefinite stresses; the quaternion representation was found superior to Euler angles [141].Source: §4.4.3 · pp. 50–51

4.4.4. Random matrix approach for quantifying Reynolds stress uncertainty

• Random-matrix and physics-based perturbations are qualitatively similar for small perturbations, but random matrices avoid out-of-triangle samples and jointly constrain magnitude, shape, and orientation.The random-matrix method samples positive semidefinite Reynolds stresses from a maximum-entropy distribution, guaranteeing realizability without barycentric maps.Source: §4.4.4 · pp. 51–52

4.4.5. Quantifying and reducing Reynolds stress uncertainties with data

• Data assimilation markedly improves velocity predictions throughout square-duct cross-sections, including unobserved locations, and calibrated discrepancies transfer to higher Reynolds numbers and rectangular ducts.The ensemble Kalman method infers Reynolds stresses and full-field velocities from sparse velocity data while enforcing realizability, smoothness, and empirical discrepancy information.Source: §4.4.5 · pp. 52–53
• Data-driven Reynolds-stress correction improves predictions in periodic-hill separation and square-duct secondary flows, but ill-conditioning means small stress errors can cause large velocity errors [146].Learning the linear and nonlinear stress components separately, with the linear part treated implicitly, was proposed to improve conditioning.Source: §4.4.5 · pp. 53–54

Motivation

• Reynolds stress uncertainty quantification must model spatial discrepancy because the Reynolds stress divergence enters the RANS momentum equations, not just pointwise uncertainty bounds.Emory et al. specified spatial eigenvalue-perturbation fields using empirical knowledge of RANS-model performance for the target problem.Source: §4.5 · p. 54

Statistical spatial models

• Non-stationary Gaussian processes with Karhunen–Loeve expansions represent Reynolds stress discrepancy fields, but specifying their spatial distribution remains a major weakness.Xiao et al. used leading KL modes to approximate the perturbation field.Source: §4.5 · p. 55

Transport-equation models

• Edeling et al. modeled discrepancy using transport equations whose source terms describe deviations from the equilibrium assumed by linear eddy-viscosity models.The PDE coefficients can be calibrated with data and Bayesian inference, and the calibrated equations then support prediction.Source: §4.5 · p. 55

PDE-informed covariance

• Wu et al. derived a physically anchored covariance by transforming source-term covariance through a linearized discrepancy PDE, improving on purely statistical kernels.For L(δ)=S, the discrepancy covariance is induced through the inverse differential operator rather than specified directly for δ.Source: §4.5 · p. 55
• Figure 22 shows PDE-informed modes with streamline-aligned correlations, whereas squared-exponential modes exhibit nonphysical spatially isotropic structures in periodic-hill flow.The comparison uses the first three eigen-function modes from the squared-exponential and PDE-informed kernels.Source: Figure 22 · p. 56, §4.5 · p. 56

5. Uncertainties in large eddy simulations

• LES uncertainty arises from comparable SGS-model, boundary/initial-condition, and numerical-discretization sources, unlike RANS where model uncertainty dominates.For adequate meshes, most important turbulence scales are resolved except near walls, reducing SGS-model dominance.Source: §5 · p. 56
• Current LES UQ studies mainly perform uncertainty propagation and sensitivity analysis, often using polynomial-chaos, probabilistic-collocation, or sparse-grid surrogate models to reduce computational cost.LES studies generally have less computational capacity for extensive sampling than RANS studies because LES is much more expensive.Source: §5 · p. 57

5.1. Uncertainties in SGS models

• The Smagorinsky constant Cs should be treated as uncertain because it depends on the flow and filter, with an optimal value varying by grid refinement.Lucor et al. reconstructed solution statistics with typically 22 samples, while uncertainty propagation confirmed interactions between SGS modeling and discretization.Source: §5.1 · pp. 57–58

5.2. Uncertainties in the boundary conditions for LES

• LES boundary-condition uncertainty affects flow development, and LES is more sensitive to inlet Reynolds number than RANS, although modeling uncertainty can still dominate.Congedo et al. examined inlet bulk velocity, swirl ratio, and turbulent intensity; Carnevale et al. studied Reynolds-number sensitivity in pin-fin channel heat transfer.Source: §5.2 · pp. 58–59

5.3. Uncertainties due to the numerical discretization

• Implicit filtering creates strong interactions between SGS-model and numerical errors because the mesh and discretization jointly determine the effective filtering bandwidth.Studies propagated uncertainty in grid stretching, spanwise resolution, and explicit low-pass-filter weight.Source: §5.3 · p. 59
• LES UQ is expected to progress from data-free parametric studies toward data-driven, non-parametric methods and broader industrial deployment.The review anticipates equal emphasis on SGS, boundary-condition, and numerical uncertainties as computational resources and industrial LES use increase.Source: §5.3 · p. 59

• RANS uncertainty quantification has rapidly evolved, especially through statistical prediction-uncertainty estimation and data-driven uncertainty reduction enabled by computing, high-fidelity data, and statistical inference methods.These methods increasingly incorporate physical constraints and prior turbulence-model knowledge.Source: §6 · pp. 59–60
• Parametric and multi-model methods are robust, non-intrusive, relatively mature, and applicable to complex three-dimensional engineering flows when combined with surrogate models.They propagate or update uncertain closure coefficients and can average competing models using posterior probabilities and scenario similarity.Source: §6 · p. 60
• Non-parametric approaches represent model discrepancy with random fields, accounting for local differences between well-predicted equilibrium regions and difficult non-equilibrium regions.Their high-dimensional sampling and inference remain active research challenges.Source: §6 · p. 60
• Extrapolating discrepancy fields beyond training configurations remains delicate and potentially dangerous, despite recent data-assimilation and machine-learning applications.Sound physical constraints and physics-based transport equations for spatial correlations are proposed as promising safeguards.Source: §6 · p. 60
• Physics-informed integration of uncertainty quantification and data-driven modeling could produce RANS models that predict turbulent flows with quantified uncertainties, advancing certified CFD simulations.RANS remains central because its lower computational cost and greater robustness support industrial use over coming decades.Source: §6 · pp. 59, 61

Appendix A.1. Plain Monte Carlo sampling

• The method uses Monte Carlo simulations to obtain output distributions from uncertainty in model parameters.The input uncertainty is represented by p(θ), while the outputs are generated through repeated model evaluations.Source: Appendix A.1 · p. 76
• Plain Monte Carlo sampling propagates uncertainty by drawing model-parameter samples from the specified probability distribution p(θ).The algorithm is described as straightforward and begins with sampling from the prior distribution.Source: Appendix A.1 · p. 76
• The complete procedure consists of sampling, propagation, and aggregation.These three stages respectively draw parameter samples, compute outputs, and estimate the QoI distribution.Source: Appendix A.1 · p. 76
• For each sampled parameter set, the model is evaluated to obtain the corresponding outputs.The propagated outputs are denoted {y1, y2, · · · , yN}.Source: Appendix A.1 · p. 76
• The quantity-of-interest distribution is estimated by aggregating the propagated samples.Aggregation follows sampling and model evaluation in the stated procedure.Source: Appendix A.1 · p. 76
• The plain Monte Carlo uncertainty-propagation procedure is illustrated in Fig. 6a.The figure provides a pictorial representation of the algorithm.Source: Appendix A.1 · p. 76

Appendix A.2. Exact Bayesian inference with Markov chain Monte Carlo sampling

• Metropolis–Hastings MCMC requires ergodicity, meaning any state-space set can reach any other with nonzero probability in finitely many steps.Source: Appendix A.2 · p. 76
• The procedure initializes θ(0), proposes z⋆ from q(z⋆|z(i)), evaluates p(z⋆)/p(z(i)), and accepts automatically when the ratio χ is at least one.Source: Appendix A.2 · p. 76
• Accepting some less-likely proposals lets the sampler explore posterior tails and improves mixing across different state-space regions.Source: Appendix A.2 · p. 77

Approximate Bayesian inference with iterative Ensemble Kalman method

• The augmented system state stacks the inferred parameter vector θ=[θ1, θ2, · · ·, θr] with the physical state, here represented by velocities.The velocity field is treated as the observed physical state for notational simplicity.Source: Appendix A.3 · p. 77
• The method samples prior parameter values, propagates each augmented state through the model, estimates ensemble statistics, and repeatedly applies analysis corrections until convergence.The augmented state combines unknown parameters θ with physical states; each correction updates both velocities and parameters using ensemble and observation-error covariances.Source: Appendix A.3 · p. 77
• The Kalman gain weights observations when forming corrected states from model predictions and measurements.The observation matrix H maps the system state space to the observation space, while covariance matrices determine the gain used for correction.Source: Appendix A.3 · pp. 77–78
• The observation operator maps simulated states to measured quantities, supporting point or volumetric velocities, line-integrated velocities, lift, drag, pressure coefficients, and surface coefficients.For the illustrated six-cell domain, three observations comprise a volumetric measurement, a point measurement averaged over cells, and a weighted line-integrated measurement.Source: Appendix A.3 · p. 78, Figure A.23 · p. 78

Composite model theory

• Composite model theory separates RANS simulators into reliable conservation laws and approximate Reynolds-stress closure models, placing uncertainty where it physically originates.This contrasts with the earlier blackbox Kennedy and O’Hagan framework, which introduced model inadequacy directly at quantities of interest or observations.Source: Appendix B · pp. 78–79

Open-box treatment of model inadequacy

• Open-box approaches emerged from composite model theory to enable physics-informed uncertainty quantification and statistical inference, unlike traditional physics-neutral blackbox approaches.The distinction is illustrated schematically in Fig. B.24.Source: Appendix B · p. 79, Figure B.24 · p. 79

Broader composite-model applications

• Composite models also occur in multiphase flow, climate, and weather simulations, where well-grounded conservation-law PDEs contain unclosed terms for unresolved physics [175, 176, 177–179].Examples include interphase mass and momentum exchanges and parameterized radiation, cloud, and boundary-layer processes.Source: Appendix B · p. 79

• DNS model evaluations compare RANS Reynolds stresses or solved fields against DNS statistics, but sampling and discretization errors can affect both a priori and a posteriori comparisons.Source: Appendix C · p. 80
• Statistical DNS quantities require averaging temporally uncorrelated instantaneous samples, while correlated or insufficient samples cause sampling errors and inadequate statistical convergence.Source: Appendix C · p. 80
• Sampling error is generally dominant in properly resolved DNS, but empirical mesh choices can couple sampling and discretization errors; Bayesian treatment has been proposed.Source: Appendix C · p. 80
• Propagated Reynolds stresses yielded significant velocity discrepancies in several flows but satisfactory agreement in square-duct flows, explained by flow-dependent sensitivity of mean velocities to stresses [186] [146].The contrasting outcomes reflect different model conditioning across flows.Source: Appendix C · p. 80
• DNS also inherits boundary-condition uncertainties, including inflow sensitivity in two-dimensional plane mixing-layer simulations.Source: Appendix C · p. 80