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Quantification of Model Uncertainty in RANS Simulations: A Review

Heng Xiao, Paola Cinnella

arXiv:1806.10434v2physics.flu-dynphysics.comp-ph

TL;DR

Model uncertainty remains a major obstacle to predictive RANS simulations, despite their expected continuing role in industrial CFD. This review synthesizes parametric and structural uncertainties and data-free and data-driven quantification methods. It highlights that nonparametric Reynolds-stress perturbations can span true solutions beyond parametric approaches, while calibration and model averaging retain important limitations.

  • Problem

    RANS simulations face structural and parametric model uncertainties that limit their predictive capability in industrial computational fluid dynamics.

  • Method

    The review examines uncertainty propagation, Bayesian inference, statistical calibration, model averaging, and data-driven corrections for RANS and scale-resolving simulations.

  • Results

    Nonparametric perturbations of Reynolds stresses, including eigen-directions, can span a range covering the true solution, unlike the described parametric approach.

  • Takeaways & Limitations

    RANS uncertainty quantification is strategically important for progressing toward certified numerical simulations of fluid flows.

  • Takeaways & Limitations

    Multi-model approaches can retain bias because they depend on a subjective finite model set, while coefficient calibration alone does not capture overall observational uncertainty.

Abstract

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In computational fluid dynamics simulations of industrial flows, models based on the Reynolds-averaged Navier--Stokes (RANS) equations are expected to play an important role in decades to come. However, model uncertainties are still a major obstacle for the predictive capability of RANS simulations. This review examines both the parametric and structural uncertainties in turbulence models. We review recent literature on data-free (uncertainty propagation) and data-driven (statistical inference) approaches for quantifying and reducing model uncertainties in RANS simulations. Moreover, the fundamentals of uncertainty propagation and Bayesian inference are introduced in the context of RANS model uncertainty quantification. Finally, the literature on uncertainties in scale-resolving simulations is briefly reviewed with particular emphasis on large eddy simulations.

Nomenclature

This nomenclature section defines notation for averaging, norms, multiplication, and material differentiation.

  • Ensemble averaging or spatial filtering denotes the averaging or filtering operation used in turbulence modeling.
  • The ||A||P notation denotes the P norm of A weighted by covariance.
  • The Hadamard symbol denotes element-wise multiplication.
  • D· Dt denotes the material derivative.

Roman Letters

This section lists Roman-letter notation for coordinates, operators, probabilities, states, models, and physical quantities.

  • c1, c2, and c3 are barycentric coordinates, while x_i and x denote spatial coordinates.
  • Cov, Var[Z], E[Z], P, and p(z) denote covariance, variance, expectation, probability mass, and probability distribution notation.
  • K(·,·), GP(·,·), and K denote a Gaussian-process kernel, Gaussian process, and Kalman gain matrix in EnKF.
  • L and N denote linear and nonlinear differential operators, while O(·) denotes order of magnitude.
  • M and M_i denote a set of models and an individual model; S̃_i denotes a scenario in BMSA.
  • p, P, q, R, S, and T_ω denote pressure, mean pressure, mean-flow features, observation-error covariance, strain-rate tensor, and turbulent-frequency transport.

1. Introduction

RANS reduces computational cost by modeling turbulent fluctuations, but its closure choices introduce structural and parametric uncertainties that limit predictive generality.

  • 1. Introduction: DNS resolves all turbulence scales at prohibitive cost, whereas RANS models the entire turbulent range; LES occupies an intermediate position by resolving larger scales.
  • 1. Introduction: The RANS modeling landscape has stagnated: wind-tunnel tests fell from 75 in the 1970s to 10 in the 1990s and then remained stagnant.
  • 1. Introduction: No turbulence model has achieved predictive generality, so flow-specific tuning and fudge functions remain indispensable in RANS simulations.
  • 1.3. Approaches for quantifying uncertainties in turbulence models: The review focuses on functional-form and parameter uncertainties and classifies quantification methods as parametric or nonparametric and data-free or data-driven.
  • 1.2. Origin of uncertainties in RANS models: Structural uncertainty arises from limitations of the Boussinesq assumption, especially in separated flows or flows with curvature or strong pressure gradients.
  • 1.2. Origin of uncertainties in RANS models: Parametric uncertainty reflects calibrated closure coefficients whose simple-flow data may be distant from practical applications and affected by measurement errors.
  • 1.2. Origin of uncertainties in RANS models: Parametric and structural uncertainties are epistemic, but reducing them can require more knowledge or data while risking excessive cost, poor robustness, or over-tuning.

2. Fundamentals of probability and statistics for uncertainty quantification

This section introduces probability and statistics for RANS model uncertainty quantification, then presents forward uncertainty propagation and backward Bayesian inference. It also compares propagation methods and explains why surrogate models and specialized sampling are needed for computationally expensive CFD simulations.

  • Probability and statistics: RANS uncertainty can be represented with random variables, random vectors, or random fields, with expectations, variances, and covariances characterizing their distributions.The uncertain variable θ may contain model coefficients or spatial fields such as Reynolds stress or eddy viscosity.
  • Uncertainty propagation and Bayesian inference: Forward uncertainty propagation samples a specified input distribution, evaluates the RANS model, and aggregates outputs to estimate uncertainty in quantities of interest.The inputs may include model coefficients or modeled fields, while outputs may include lift and drag coefficients.
  • Uncertainty propagation and Bayesian inference: Bayesian inference assimilates available, potentially noisy, biased, or incomplete output data to infer a posterior distribution for the uncertain inputs.The posterior updates the prior distribution after observing the data and can subsequently be propagated to predict outputs.
  • Uncertainty propagation methods: Spectral methods converge faster under smoothness conditions, whereas Monte Carlo methods avoid dimensionality penalties but converge uniformly slowly at O(N^-1/2).Spectral methods suffer from the curse of dimensionality because required evaluations can grow exponentially with parameter-space size.
  • Computational challenges: MCMC must exclude physically constrained regions that produce nonphysical or nonconvergent solutions, while the excluded fraction can increase exponentially with sample-space dimension.The surrogate approach used for this purpose is itself restricted to low-dimensional state spaces.

3. Parametric and multi-model approaches

RANS closure coefficients vary across models and calibration flows, motivating uncertainty quantification through forward propagation and Bayesian inference. These approaches show that parameter variability affects predictions and that single calibrated coefficients or models may not generalize reliably.

  • 3.1. Uncertainties in RANS model parameters: Experimental results suggest n = 1.3, corresponding to Cε2 = 1.77, while RNG k–ε and k–τ models use C̃ε2 = 1.68 and Cε2 = 1.83, respectively.The standard value gives Cε2 = 1.92, but the review notes that this is not a hard requirement.
  • 3.1. Uncertainties in RANS model parameters: DNS data support Cµ = 0.09 away from the wall, although the RNG k–ε model uses Cµ ≈0.085.The cited DNS result reports τ12 ≈−0.30k except close to the wall.
  • 3.1. Uncertainties in RANS model parameters: The nominal k–ε coefficients satisfy the von Karman constraint only approximately, yielding κ ≈0.43 instead of κ = 0.41; experiments place κ between 0.33 and 0.45.This range illustrates uncertainty even around commonly used standard values.
  • 3.2. Parametric uncertainty in RANS models: forward approaches: Forward sensitivity studies found uncertainty ranges large enough to encompass available flat-plate friction data, with outputs more sensitive to wall-function parameters than other model parameters.One study assigned κ ∼N(0.417, 0.0127) while keeping other quantities fixed.
  • 3.2. Parametric uncertainty in RANS models: forward approaches: Closure-coefficient variability significantly affected streamwise velocity near backward-step reattachment and turbulence intensity along the free shear layer.The reported effects were smaller or different for other output quantities, including reattachment location and pressure.
  • 3.3.1. Statistical inference of model parameters: Bayesian calibration estimates joint posterior distributions and parameter correlations, but gradient-based calibration returns deterministic coefficients and Hessian-based variances require intrusive second sensitivities.MCMC calibration of Spalart–Allmaras used velocity and skin-friction data from three boundary layers and 32,768 full Navier–Stokes calculations.

4. Non-parametric approaches

Non-parametric approaches introduce uncertainty beyond calibrated coefficients or finite model ensembles, allowing RANS simulations to explore a broader solution space and represent structural model inadequacy. They perturb transport equations or modeled fields, and a wing–body junction example shows that perturbing full Reynolds stresses can encompass the true velocity solution when parametric perturbations cannot.

  • Motivation and overview: Parametric and multi-model approaches remain restricted to baseline models or selected model ensembles, potentially excluding parts of the true solution space.Linear eddy-viscosity models cannot represent features driven by Reynolds-stress anisotropy under the Boussinesq assumption, while finite ensembles span only their chosen models.
  • Motivation and overview: In wing–body junction flow, parametric velocity samples based on the Boussinesq assumption cannot encompass the truth, whereas Reynolds-stress perturbations can span a range covering it.The difference arises because the non-parametric approach perturbs Reynolds-stress eigen-directions as well as their magnitudes.
  • Motivation and overview: Non-parametric approaches introduce uncertainty into turbulent transport equations or modeled outputs such as turbulent viscosity and Reynolds stresses.The approaches are broadly divided between perturbing model forms and perturbing model-output fields.
  • Introducing uncertainties in turbulent transport equations: Transport-equation approaches infer a multiplicative discrepancy field β(x) in source terms using DNS or experimental observations of velocities or other quantities of interest.For the k–ω equation, β(x) modifies the production term; inferred discrepancies can correct the baseline model to agree with data.
  • Introducing uncertainties in turbulent transport equations: The optimization cost penalizes deviation of β from 1, constraining the corrected model near the baseline and reducing the search dimension in the high-dimensional discrepancy space.The inferred field can guide baseline-model improvement and data-driven correction schemes.
  • Introducing uncertainties in turbulent viscosity: A priori studies suggest that correction knowledge from one flow can extend to similar configurations, while viscosity estimates can support prediction improvement or discrepancy-based uncertainty quantification.The resulting correction function β(q) uses non-dimensionalized mean-flow variables and related quantities.

4.4. Introducing uncertainties in Reynolds stresses

The review introduces Reynolds-stress uncertainty through stochastic discrepancy fields or realizability-constrained perturbations. These approaches propagate physically constrained uncertainty into RANS predictions, with barycentric perturbations offering an efficient five-simulation scheme and data-driven methods improving predictions.

  • Reynolds stress is the principal modeled term in RANS momentum equations, so its modeling inaccuracy generates model-form uncertainty for fully turbulent single-phase flows.The Reynolds stress transport equation remains unclosed, requiring further modeling.
  • Two approaches characterize Reynolds-stress uncertainty: stochastic differential equations for discrepancy fields and realizability-constrained perturbations of single-point stresses.Both treat discrepancy as a random tensor field subject to physical constraints such as conservation laws or realizability.
  • SDE-based discrepancies provide full-field uncertainty with cross-component and spatial correlations, but their construction relies heavily on physical insight and modeling heuristics.The SDE mirrors the convection-diffusion-production structure of Reynolds stress transport equations while replacing unclosed terms with stochastic residual forcing.
  • Realizability maps constrain Reynolds-stress eigenvalue perturbations by representing limiting one-, two-, and three-component turbulence states within the barycentric triangle.Magnitude, shape, and orientation remain coupled characteristics of the same Reynolds stress tensor, so the realizability map alone does not directly bound every component.
  • The physics-based perturbation scheme estimates RANS model uncertainty using only five simulations and supports later statistical-inference and machine-learning methods.For large perturbations, samples may leave the barycentric triangle and require capping, slightly distorting the distribution.
  • In square-duct flow, incorporating data markedly improved velocity predictions at observed and unobserved cross-sections, and calibrated discrepancies transferred to higher Reynolds numbers and rectangular ducts.The review reports that empirical knowledge had effects similar to increasing the amount of observation data.

4.5. Spatial correlations in Reynolds stress discrepancy

Spatial variation in Reynolds-stress discrepancy matters because its divergence enters the RANS momentum equations. The review describes transport-based and covariance-based approaches that impose physically informed spatial correlations, while noting that specifying the spatial distribution remains difficult.

  • RANS uncertainty studies often bound Reynolds stress at a single point, although prediction uncertainty also depends on spatial discrepancy variation.The divergence of the Reynolds-stress field appears directly in the RANS momentum equations.
  • Xiao et al. represented Reynolds-stress discrepancy with a non-stationary Gaussian process and Karhunen–Loève expansions, but spatial-distribution specification remains a major weakness.The expansion approximates the perturbation field using leading modes.
  • Return-to-eddy-viscosity models use transport equations with source terms to describe deviations from the equilibrium state assumed by linear eddy-viscosity models.Their coefficients can be calibrated with data and Bayesian inference before prediction.
  • PDE-informed covariance models derive discrepancy structure from a linearized differential operator and kernels for unclosed source terms, rather than specifying a purely statistical kernel directly.The generalized form is L(δ) = S, where δ is the discrepancy field and S represents unclosed source terms.
  • Physics-informed covariance better accounts for discrepancy spatial correlation than a squared exponential kernel, producing streamline-aligned rather than spatially isotropic modes over periodic hills.The streamline alignment reflects convection in the physics-informed covariance structure.

5. Uncertainties in large eddy simulations

LES uncertainty arises from SGS models, inputs, and numerical discretization, with strong interactions between modeling and numerical errors. The reviewed literature remains dominated by parametric propagation and sensitivity studies because LES is computationally expensive.

  • LES uncertainty includes SGS-model parameters, initial and boundary conditions, and numerical discretization, unlike RANS where model uncertainty clearly dominates.Input and numerical uncertainties are distinct from model uncertainties.
  • Because LES is much more expensive than RANS, most uncertainty studies propagate assumed input distributions or perform sensitivity analysis rather than high-dimensional inference.These studies investigate output sensitivity to uncertain inputs while limiting sample counts and computational cost.
  • 5.1. Uncertainties in SGS models: The Smagorinsky constant should be treated as uncertain because it depends on the specific flow and filter, despite commonly being chosen between 0.1 and 0.2.The SGS viscosity also depends on strain rate and grid size through the algebraic Smagorinsky model.
  • 5.1. Uncertainties in SGS models: Uncertainty propagation across grid resolutions found an optimal Smagorinsky constant for each refinement level, confirming interactions between SGS modeling and numerical discretization.The result was reported from a study using 22 samples.
  • 5.2. Uncertainties in the boundary conditions for LES: LES results were more sensitive than RANS results to inlet Reynolds number in turbulent pipe flow with an axisymmetric expansion.The associated study varied inlet bulk velocity, swirl ratio, and turbulent intensity.
  • Implicit-filtered LES couples modeling and numerical errors because the mesh determines local filtering bandwidth, making scale separation difficult in practical simulations.This coupling is especially relevant when the mesh is part of the model.

6. Conclusions and future research

The review organizes RANS model-uncertainty methods into parametric and non-parametric approaches and surveys a rapidly developing field shaped by computation, data, and statistical inference. It identifies locality and high-dimensional random-field inference as central challenges for future data-driven prediction.

  • RANS remains strategically important for industrial CFD because it offers lower computational cost and greater robustness than scale-resolving methods.The review emphasizes uncertainty quantification as important for the longer-term goal of certified CFD simulations.
  • Recent RANS research increasingly uses statistical approaches to estimate turbulence-model prediction uncertainty and data-driven methods to reduce it.This development has been fostered by increased computing resources, high-fidelity data, and physically guided statistical inference.
  • Parametric approaches: Parametric approaches randomize closure coefficients and propagate or update them with observations, offering non-intrusive workflows but remaining constrained by the chosen baseline model.Surrogate models can make sampling feasible for complex three-dimensional engineering flows.
  • Non-parametric approaches: Non-parametric approaches represent local RANS discrepancy with random fields, addressing flow regions where a single global calibration may be inadequate.Their high-dimensional sampling and inference remain active areas with substantial open challenges.
  • Future research: Data assimilation and machine learning have begun extrapolating estimated discrepancy fields to configurations relatively close to training flows, with physics-informed methods aimed at quantified-uncertainty prediction.The review presents this direction as a potential path toward certified CFD simulations.

Appendix A.1. Plain Monte Carlo sampling

Plain Monte Carlo propagates uncertainty by sampling model parameters, evaluating the model for each sample, and aggregating the outputs into a quantity-of-interest distribution.

  • Monte Carlo samples model parameters from the specified prior probability distribution.
  • Each sampled parameter set is propagated through the model to obtain corresponding outputs.
  • The propagated samples are aggregated to estimate the distribution of the quantity of interest.
  • The procedure is illustrated in Fig. 6a.

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

Exact Bayesian inference with MCMC uses ergodic sampling and Metropolis–Hastings proposals to explore the posterior, including both high-probability regions and tails.

  • MCMC sampling requires ergodicity so that any state-space set can be reached from any other with nonzero probability in finite steps.
  • Metropolis–Hastings initializes a state, proposes a new state from q(z⋆|z(i)), evaluates posterior density, and computes the density ratio.
  • Accepting less likely states helps explore posterior tails and improves mixing across different regions.

Appendix A.3. Approximate Bayesian inference with iterative Ensemble Kalman method

The iterative Ensemble Kalman method infers parameters by repeatedly propagating an ensemble, comparing predicted fields with observations, correcting states, and iterating toward convergence.

  • The augmented system state stacks unknown parameters with physical states such as velocity fields.
  • EnKF inversion begins by drawing M parameter samples from their prior distributions.
  • Each ensemble member is propagated, then the ensemble mean and covariance are estimated from the propagated states.
  • The analysis step compares predicted physical fields with observations and uses covariances to compute the Kalman gain.
  • Each sample is corrected using the Kalman gain, updating velocities and parameters in the analyzed state.
  • Propagation and analysis repeat until convergence is achieved.
  • The observation matrix maps the state space to the observation space and can represent point or derived measurements.

Appendix B. Composite model theory and openbox treatment of model inadequacy

Composite model theory locates RANS model uncertainty in approximate closure terms, motivating open-box, physics-informed approaches instead of treating the simulator as a black box.

  • Reynolds stress is identified as the source of uncertainty in the RANS equations.
  • RANS combines rigorous conservation equations with approximate closure models for unresolved or unknown physics.
  • Composite model theory separates the simulator into numerical equations and closure models, placing uncertainty where it originates physically.
  • Open-box approaches contrast with earlier black-box treatments that introduced model inadequacy directly into quantities of interest or observations.

Appendix C. Uncertainties in DNS and their impact on RANS modeling

DNS is a key reference for evaluating turbulence models, but sampling, discretization, and boundary-condition uncertainties complicate comparisons with RANS. Specified-stress propagation can also produce flow discrepancies that depend on how sensitively mean velocities respond to Reynolds stresses.

  • DNS turbulence-model evaluations compare RANS Reynolds stresses or solved fields with DNS quantities, either a priori or a posteriori.DNS data are widely treated as a reference standard, although the comparisons inherit DNS uncertainties.
  • Statistical moments from DNS require averaging sufficiently many temporally uncorrelated instantaneous samples, so correlated or insufficient samples produce sampling errors.
  • Even properly resolved DNS is usually dominated by sampling error, while empirically chosen meshes can couple sampling and discretization errors.A Bayesian approach has been proposed to account for sampling errors when estimating discretization errors.
  • Specified-Reynolds-stress propagation can disagree with DNS velocities in some flows but agree satisfactorily in fully developed square-duct flows because flow conditioning changes sensitivity to Reynolds stresses.
  • DNS comparisons also contain uncertainty from boundary-condition specification, including probabilistic sensitivity studies of inflow conditions.
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