{'fraction_non_alphanumeric': 0.025085273816238284, 'fraction_numerical': 0.0067950474041844604, 'mean_word_length': 5.034683954619124, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A Bayesian Calibration-Prediction Method for Reducing Model-Form Uncertainties with Application in RANS Simulations

Model-form uncertainties in complex mechanics systems are a major obstacle for predictive simulations. Reducing these uncertainties is critical for stake-holders to make risk-informed decisions based on numerical simulations. For example, Reynolds-Averaged Navier-Stokes (RANS) simulations are increasingly used in mission-critical systems involving turbulent flows. However, for many practical flows the RANS predictions have large model-form uncertainties originating from the uncertainty in the modeled Reynolds stresses. Recently, a physics-informed Bayesian framework has been proposed to quantify and reduce model-form uncertainties in RANS simulations by utilizing sparse observation data. However, in the design stage of engineering systems, measurement data are usually not available. In the present work we extend the original framework to scenarios where there are no available data on the flow to be predicted. In the proposed method, we first calibrate the model discrepancy on a related flow with available data, leading to a statistical model for the uncertainty distribution of the Reynolds stress discrepancy. The obtained distribution is then sampled to correct the RANS-modeled Reynolds stresses for the flow to be predicted. The extended framework is a Bayesian calibration-prediction method. The merits of the proposed method are demonstrated on two flows that are challenging to standard RANS models. By not requiring observation data on the flow to be predicted, the present calibration-prediction method will gain wider acceptance in practical engineering application. While RANS modeling is chosen to demonstrate the merits of the proposed framework, the methodology is generally applicable to other complex mechanics models involving solids, fluids flows, or the coupling between the two, where model-form uncertainties are present in the constitutive relations.


Introduction
Model-form uncertainties in complex mechanics systems are a major obstacle for predictive simulations. Reducing these uncertainties is critical for stake-holders to make riskinformed decisions based on numerical simulations. For example, the Reynolds-Averaged Navier-Stokes (RANS) simulations have been increasingly used in the design and analysis of mission-critical systems involving turbulent flows, thanks to the sustained growth of computational resources in the past decades. Examples include gas turbines, hydraulic turbines, aircrafts, and more recently the thermohydraulic system of nuclear power plants [1,2]. Although RANS solvers are more accurate than the state-of-the-art empirical methods and low-fidelity models [e.g., 3], their predictions still have significant uncertainties. For many practical flows in engineering, the uncertainty in the turbulence closure model embedded in the RANS equations is the dominant source for the uncertainties in the predicted Quantities of Interest (QoIs). This is referred to as model-form uncertainty, which is arguably the most challenging uncertainty to quantify in RANS models and in other complex mechanics models.
Several approaches that are generally applicable to different applications have been used to quantify the model-form uncertainties in RANS simulations. A widely adopted method in the engineering communities is the parametric perturbation or model ensemble approach, in which the baseline simulations are repeated by perturbing the coefficients in the turbulence model or by using several turbulence models. The scattering in the obtained ensemble is used as an empirical indication of the prediction uncertainty. However, this ad hoc approach tends to underestimate the true uncertainties, since different models usually share similar assumptions (e.g., the Boussinesq assumption) and thus have similar biases. The Bayesian calibration method of Kennedy and O'Hagan [4] is a widely used approach for quantifying model-form uncertainties. In their framework the discrepancy in the predicted QoI is modeled as a Gaussian process, whose hyperparameters are calibrated with data. However, this physics-neutral framework treats the numerical model as a black-box, which leads to inefficient use of calibration data and makes it difficult for the users to specify prior knowledge based on their insights of the model and the problem. In the past few years, some researchers have proposed open-box approaches to quantify the model-form uncertainties in RANS simulations, which are in contrast to the general-purpose, physics-neutral approaches described above. Dow and Wang [5] modeled the true eddy viscosity field as a Gaussian process, and used Direct Numerical Simulation (DNS) data to infer its hyperparameters and to find the optimal mean field that minimizes the misfit. Iaccarino and co-workers [6][7][8] introduced smooth perturbations on the Reynolds stress towards several limiting states in the physically realizable range [9,10]. Oliver et al. [11] introduced discrepancy to the Reynolds stress tensor, and modeled the discrepancy term with a stochastic differential equation. The essential differences among the three open-box approaches lie in the treatment of physical prior knowledge and calibration data. Dow and Wang [5] used DNS data to reduce uncertainties, resulting in a calibration-prediction procedure. In contrast, Iaccarino et al. [6][7][8] and Oliver et al. [11] focused on uncertainty quantification and propagation by exploiting the physical insight on the Reynolds stresses, and they did not directly use data to reduce uncertainties.
More recently, Xiao et al. [12] proposed an open-box, physics-informed, Bayesian framework for quantifying and reducing model-form uncertainties in RANS simulations. Uncertainties are introduced to the Reynolds stresses and are parameterized compactly with physical realizability and spatial smoothness guaranteed [6,11]. A Bayesian inference procedure based on an iterative ensemble Kalman method [13] is used to quantify and reduce the uncertainties by incorporating observation data. This method combines all sources of available information in a Bayesian framework including physical constraints, empirical knowledge, and observation data. In scenarios where observation data (e.g., sparse velocity measurements) are available, this framework provides a powerful method of predicting the whole-field velocity and other QoIs with quantified uncertainties.
However, in the design stage of engineering systems or devices, the target configuration (e.g., the new design of an aircraft or a gas turbine) has not been built yet, and thus measurement data are not available [14,15]. The framework of Xiao et al. [12] is not directly applicable to these scenarios. Without observation data the framework would degenerate to an uncertainty estimation procedure, and the potential of the Bayesian framework cannot be fully exploited. Therefore, in the present work we extend the original framework to predict flows with no observation data by utilizing data from a different yet related flow. Specifically, we first build a statistical model for the Reynolds stress discrepancy on a related flow by incorporating available data (i.e., infer its uncertainty distribution), and then we sample the inferred uncertainty distribution to predict the flow of interest. The extended framework is essentially a Bayesian calibration-prediction procedure. We consider two typical scenarios where observation data could be available on a related flow. (1) When mission-critical systems are developed, laboratory experiments are often performed on a down-scaled, geometrically similar model. Unfortunately, the experiments often have to be performed at a reduced Reynolds number than in the prototype, possibly due to limitations of the facilities or to avoid violating other similitudes (e.g., Mach number similarity). Although the Reynolds number similitude is not achieved, the experimental data obtained in the laboratory scale model can be valuable for quantifying uncertainties in the RANS-predicted Reynolds stresses. (2) Industrial product developments often consist of small incremental modifications of existing, well tested products, e.g., when a new gas turbine model is added to a family of existing products. In this case it is possible that measurement data, obtained from either experiments or field operations, are available on flows in a slightly different geometry from the flow to be predicted.
In the calibration-prediction framework outlined above, a pivotal assumption is that the Reynolds stress discrepancies in the two flows, the one used for calibration and the one to be predicted, can be described by the same statistical model. This assumption allows the extrapolation of Reynolds stress discrepancies from the calibration case to the prediction case, where "extrapolation" should be interpreted in a statistical sense. The extrapolation of the Reynolds stress discrepancy is justified if the two cases share overall physical characteristics despite the differences in their specific flow conditions (e.g., Reynolds number and geometry).
The same assumption is the basis of many wind tunnel experiments in which it is infeasible to achieve the prototype Reynolds number with the laboratory setup.
The objective of the present work is to explore the feasibility of such a calibrationprediction procedure. The performance of the proposed extension is evaluated on two canonical flows, the flow over periodic hills and the flow in a square duct, at several Reynolds numbers. In both cases the Reynolds stress discrepancies are calibrated in a lower Reynolds number flow with sparse observation of velocities and then are used to predict flows at higher Reynolds numbers. Moreover, we also explore the feasibility of extrapolating the Reynolds stress discrepancy calibrated in a square duct flow to the flow in a rectangular duct. The remaining of the paper is organized as follows. Section 2 outlines the model-form uncertainty quantification framework proposed in Xiao et al. [12] and presents the extension proposed in the current work. Section 3 presents and discusses numerical simulation results for two canonical flows that are challenging for standard RANS models. Finally, Section 4 concludes the paper.

Methodology of the Bayesian Calibration-Prediction Framework
Consider two flows that are closely related, e.g., flows in the same geometry but at a different Reynolds number, or flows in slightly different geometries. One flow (to be used for calibration) has some velocity observation data available, while the other flow (to be predicted) has no observation data. The overall idea of the proposed calibration-prediction framework can be summarized as follows. Baseline RANS simulations are first performed for both the calibration flow and the flow to be predicted, both having discrepancies in the RANS-predicted Reynolds stresses that are a priori unknown. In the present framework, we assume that the discrepancies in both cases have the same statistical distributions and model them as a random field. We first utilized prior knowledge and observation data to build a statistical model of the discrepancy. Subsequently, the obtained uncertainty distribution of the discrepancy is sampled and used to correct the baseline Reynolds stress in the prediction case, which is then propagated to uncertainties in velocities and in other QoIs by solving the RANS equations with the corrected Reynolds stresses.
Admittedly, since flows at different Reynolds numbers can have significantly different Reynolds stress magnitudes, direct extrapolation of the Reynolds stress discrepancy tensor field from one flow to another is likely to be problematic. Therefore, when extrapolating the Reynolds stress discrepancies to the flow to be predicted, we extrapolate the projections (magnitude, shape, orientation) of the discrepancies and not the componentwise discrepancies. Specifically, the log-discrepancy of the magnitude and discrepancies of the shapes parameters of the Reynolds stress tensors are extrapolated. These projections can be considered normalized quantities independent of the magnitude of the Reynolds stresses, and thus they can be extrapolated from one flow to another. Note that in the calibration of Reynolds stress discrepancies, the uncertainty space is also parameterized on the physical projections to guarantee the physical realizability (i.e., Reynolds stress tensor must correspond to a physically possible state [10]).
In Fig. 1 we summarize the probabilistic calibration-prediction framework for RANS modeling with uncertainty quantifications . The two components of the framework, calibration and prediction, are introduced in Sections 2.1 and 2.2, respectively.

Calibration by Building a Statistical Model of the Reynolds Stress Discrepancy
The calibration procedure uses the method of Xiao et al. [12] for quantifying and reducing model-form uncertainties. The procedure is briefly summarized here for completeness. In the framework, the true Reynolds stress τ (x) is modeled as a random tensorial field with  the spatial coordinate x as index and the RANS-predicted Reynolds stressτ rans (x) as prior mean. To ensure physical realizability, uncertainties are injected to the physically meaningful projections of the Reynolds stress tensor, which is transformed as follows [6]: where k is the turbulent kinetic energy which indicate the magnitude of τ ; I is the second order identity tensor; a is the anisotropy tensor; with λ 1 + λ 2 + λ 3 = 0 are the orthonormal eigenvectors and eigenvalues of a, respectively, indicating the shape and orientation of τ . After the decomposition, the eigenvalues λ 1 , λ 2 , and λ 3 are mapped to a Barycentric coordinates (C 1 , Consequently, all physically realizable states are enclosed in the Barycentric triangle shown in After the mapping, uncertainties are introduced to the mapped quantities k, ξ, and η by adding discrepancy terms to the corresponding RANS predictions, i.e., where δ k (x) is the log-discrepancy of the turbulent kinetic energy; δ ξ (x) and δ η (x) are discrepancies of the Reynolds stress shape parameters ξ and η, respectively. To avoid instability due to potential reverse diffusions, uncertainties are not introduced to the orientation (v 1 , v 2 , v 3 ) of the Reynolds stresses.
Another piece of prior information is the smooth spatial distribution of the Reynolds stresses. The smoothness is guaranteed by representing the Reynolds stress discrepancy fields δ k , δ ξ , and δ η (generically denoted as δ below) with smooth basis functions. Specifically, the prior distributions of the discrepancies are chosen as nonstationary zero-mean Gaussian random fields GP(0, K) (also know as Gaussian processes), and the basis set is chosen as the eigenfunctions of the kernel K [16]. This choice of basis function leads to the Karhunen-Loeve (KL) expansions of the random field. That is, the discrepancies can be represented as follows: Figure 2: Mapping between the Barycentric coordinate to the natural coordinate, transforming the Barycentric triangle enclosing all physically realizable states [7,9] to a square through standard finite element shape functions. Details of the mapping can be found in the appendix of ref. [12]. Corresponding edges in the two coordinates are indicated with matching colors.
where θ is the realized outcome, and the coefficients ω i (denoting ω k i , ω ξ i , and ω η i for discrepancy fields δ k , δ ξ , and δ η , respectively) are independent standard Gaussian random variables.
In practice the infinite series is truncated to m terms with m depending on the smoothness of the kernel K.
With the projections above, the discrepancies are parameterized by the coefficients ω k i , ω ξ i , ω η i with i = 1, 2, · · · , m. The uncertainty distributions of the coefficients are then inferred by using an iterative ensemble Kalman method [13,17]. In this method the prior distribution of Reynolds stresses (as parameterized by the coefficients) is first represented with samples drawn from the distribution. The collection of the samples, referred to as prior ensemble, is propagated to velocities by using a forward RANS solver. The Kalman filtering procedure is used to incorporate velocity observation data to the prediction, yielding a corrected ensemble. The procedure is repeated until statistical convergence is achieved. The converged posterior ensemble is a sample based representation of the uncertainty distributions of the Reynolds stresses and other QoIs.
The forward RANS solver tauFoam used in the Bayesian inference above basically computes velocities from a given Reynolds stress field. It is developed based on a conventional steady-state RANS solver in OpenFOAM [18] by replacing the turbulence modeling component (i.e., solution of the transport equations for turbulent quantities) with a supplied Reynolds stress field.

Prediction for Flows without Observation Data with Corrected Reynolds Stresses
The calibration procedure outlined in Section 2.1 builds a statistical model by inferring the uncertainty distribution of the Reynolds stress discrepancy, which is represented by the posterior ensemble obtained from the iterative Kalman method. Next, we use the obtained statistical model of the Reynolds stresses discrepancy to make predictions on flows with no observation data. To this end, we extrapolate the Reynolds stress discrepancy from the calibration case to the prediction case, and then we use the forward RANS solver to propagate the samples of corrected Reynolds stresses to the corresponding velocities and other QoIs.

Extrapolation to flow at a different Reynolds number
After calibrating the uncertainty distribution of the Reynolds stress discrepancy, it is straightforward to extrapolate to a flow with geometric similarity but not dynamic similarity (i.e., at a different Reynolds number). Here we merely emphasize two potential pitfalls as mentioned above. First, the analyst must ensure that the two flows indeed have the same overall characteristics. If the calibration case is a flow over an airfoil without separation while the flow to be predicted has a massive separation, the extrapolation can lead to incorrect results even though the two cases have the same geometry. Second, it is the discrepancies of the physical projections of the Reynolds stress that are extrapolated. Since these are also the discrepancies inferred in the calibration procedure, it does not pose any practical difficulties.

Extrapolation to flow in a different geometry
When the calibration case and the prediction case differ not only in Reynolds number but also in geometry, extrapolation of the Reynolds stress discrepancies poses additional challenges. That is, how to map a field from one geometry (e.g., a square) to another (e.g., a rectangle). The mapping scheme depends on the physical characteristics of the two flows.
The choice is inevitably case-specific and relies on the judgment of the analyst.
In Fig. 3 we use an example to illustrate the mapping scheme, where the Reynolds stress discrepancy calibrated on the flow in a square duct is used to predict the flow in a rectangular duct. Results for this case will be presented in Section 3.2. In the calibration ( Fig. 3a), the two contours symmetric along the diagonal indicate spatial distribution of Reynolds stress discrepancy due to particular features of the flow, e.g., vortices. The shaded region in Fig. 3b indicates that the Reynolds stress discrepancy in the prediction case, the flow in the rectangular geometry, is unknown before the knowledge from the square duct case is incorporated. There are at least three possbile schemes of extrapolating discrepancies calibrated in the square geometry to the rectangle: 1. Direct mapping. As shown in Fig. 3c, the obtained discrepancy field in the square is directly mapped to the lower half of the rectangle, while the Reynolds stress discrepancies in the upper half are still unknown. Prior knowledge can still be specified on the discrepancy in the unmapped region, but the uncertainty distribution in this region does not reflect information from the calibration case.
The mapping between the two geometries can be better understood by comparing the The analyst must rely on the physical understanding of the flow to be predicted (i.e., the flow in the rectangular geometry in Fig. 3b) to decide which mapping schemes described above are reasonable. Even though no data is directly available from the flow to be predicted, we still utilizes the two ingredients of a Bayesian framework, prior knowledge and data, to make predictions. The prior knowledge is formulated based on the relations between the flows in the calibration and prediction cases, while the data come from the calibration case.
Therefore, as with the original framework of Xiao et al. [12], the extended calibrationprediction procedure is a Bayesian framework.

Resampling the posterior ensemble
Once we identify a scheme for mapping the discrepancies, we can sample the posterior distribution of the discrepancy obtained in the calibration and use them to make predic- The resampling procedure allows for the flexibility of using a larger number of samples for propagating the posterior uncertainties in the Reynolds stresses than that used in the inference procedure. More importantly, when simultaneously propagating uncertainties in the input along with model-form uncertainties, nested sampling or Sobol sampling [20] are needed for the uncertainty propagation. In this case it is important to be able to generate new samples for the Reynolds stress discrepancy from the obtained distribution. In the simulations presented in this work, however, the process can be simplified by using the same posterior ensemble obtained from the calibration for the uncertainty propagation in the prediction. The resampling algorithm is presented above for the completeness of the proposed framework.

Numerical Results and Discussions
In this section we demonstrate the capability of the proposed calibration-prediction method in two scenarios as described in Section 1. In the first scenario, the prediction flow and the calibration flow are geometrically similar but have different Reynolds numbers.
This scenario is relatively straightforward and is examined on two cases, the flow over periodic hills and the flow in a square duct. In the second scenario, the prediction case differs from the calibration case not only in Reynolds number, but also in geometry. Note that in all cases examined here, we assume that observation data are only available in the calibration case and not in the prediction case. Benchmark data in the prediction case are used only to assess the performance of the proposed method.

Prediction of Flow at Different Reynolds Numbers
Two canonical flows of engineering relevance, the flow over periodic hills and the flow in a square duct, are used to evaluate the performance of the proposed calibration-prediction method. Both flows are challenging for the standard RANS models.

Flow over periodic hills
The flow over periodic hills is widely utilized to evaluate the performance of turbulence models due to the comprehensive experimental and numerical benchmark data at a wide range of Reynolds numbers [21]. The geometry of the computational domain is shown in    region. This is related to a potential deficiency of extrapolating the log-discrepancy of the turbulent kinetic energy.
It is not clear if the log-discrepancy of the TKE, its discrepancy, or a combination of both is the best choice of quantity to extrapolate from the calibration to prediction case.
As the log-discrepancy is dimensionless while the discrepancy has a dimension of the TKE, the former seems to be preferred. However, the hypothetical case shown in Table 1 suggests otherwise. Suppose in a particular region the true turbulence intensity, defined as the ratio of the velocity fluctuations u to the reference velocity U b , is 10% in both the calibration and prediction cases, while the RANS simulations give 0.1% and 1%, respectively. The errors in the RANS predictions are in fact rather consistent for the two cases in that the flow has a high intensity turbulence while the predictions give low turbulence intensities.
In other words, the difference between 0.1% and 1% is not as significant as that between 1% and 10%. The true log-discrepancy would be 2 and 1 for the calibration and prediction cases, respectively. Hence, extrapolating the log-discrepancy to the prediction case would lead to a corrected turbulent intensity of 100% in the prediction case, which is drastically different from the truth, and such a correction can destabilize the simulation. In contrast, the absolute discrepancies are approximately the same for both cases and is thus a more appropriate quantity to extrapolate. On the other hand, one can easily devise a scenario where extrapolating the log-discrepancy is more reasonable. A more sophisticated scheme of extrapolation is needed and is the subject of future work. In view of the considerations above, the discrepancy of the turbulent kinetic energy is not extrapolated from the calibration cases to the prediction cases in the simulations presented below. In other words, the baseline RANS-modeled turbulent kinetic energy in the latter case is not perturbed in the prediction.

Flow in Square Duct
The fully developed turbulent flow in a square duct is a widely known case for which RANS models fail to predict the secondary flow induced by Reynolds stress imbalances [22].
A schematic is presented in Fig. 9   As in the periodic hill cases presented above, we only consider uncertainties in parameters ξ and η, which represent the anisotropy (i.e., shape) of the Reynolds stress tensor. This is because the secondary flow is primarily induced by the anisotropy of Reynolds stresses. Figure 10 shows the secondary velocities U z along four lines. It can be seen from Fig. 10a that the prior ensemble has a large range of scattering, which is attributed to the sensitivity of the secondary flow to the turbulence anisotropy. Compared to the prior ensemble mean, the posterior ensemble mean is much closer to the benchmark data, which suggests that the discrepancies δ ξ and δ η are successfully calibrated. With the calibrated discrepancies, predictions are made for the two flows at higher Reynolds numbers.    [23]. The baseline RANS result deviates from the experimental data because RANS models with isotropic eddy viscosity are not able to capture the stress anisotropy induced secondary flows. In contrast, the present method predicts an in-plane velocity magnitude comparable to the experimental data and captures its trend of spatial variation. Similarly, the prediction for the flow at Re = 2.5 × 10 5 along the same line is presented in Fig. 12a, which also shows a good agreement with the benchmark data, even though the Reynolds number in this case is more than an order of magnitude larger than that in the calibration case. Figure 11b shows the secondary velocity at Re = 8.3 × 10 4 along the diagonal of the channel with comparison to experimental data [24]. It can be seen that the prediction is significantly improved compared to the baseline RANS result. The prediction for the flow at Re = 2.5 × 10 5 as presented in Fig. 12b also demonstrates a satisfactory agreement with  the benchmark data, although the location of the velocity peak is slightly different from the benchmark results. A possible explanation for this difference is that the assumption of similar Reynolds stress discrepancies becomes less accurate with the departure of Reynolds number from that of the calibration flow. Despite this discrepancy, the prediction presented in Fig. 12a and 12b are still comparable to or better than those obtained from RANS solvers utilizing advanced turbulence models, including an explicit algebraic Reynolds stress model (EASM) [25] and a Reynolds stress transport model (RSTM) [26]. It can be seen from Fig. 12a that the secondary velocity U z of experimental data becomes negative between z/h = 0 and z/h = 0.1, a feature that is not captured by the explicit algebraic Reynolds stress model [25] or the Reynolds stress transport model [26]. Figure 12b shows that the velocity drop between z/h = 0.1 and z/h = 0.2 as in the experimental data is only captured by the present method, although the both Reynolds stress models capture the magnitude and general trend of the experimental data well. It is worth noting that by combining a RANS solver with a simple eddy viscosity model and indirect measurement data from a related flow (at a much lower Reynolds number), the present framework is able to achieve better predictions of in-plane velocities than advanced RANS models such as algebraic Reynolds stress model [25] and Reynolds stress transport models [26]. More assessment of advanced RANS models and comparison of their simulation results to experimental data can be found in ref. [25]. This comparison here highlights the merits of data in reducing model-form uncertainties in turbulence modeling.   and Re = 2.5 × 10 5 , are successfully captured as shown in Figs. 13a and 13b, respectively. In addition, two general trends can be observed from Fig. 13 as the Reynolds number increases.
First, the secondary flow penetrates further to the corner of the duct as the Reynolds number increases. That is, the corner region where secondary flow is absent becomes smaller with increasing Reynolds number. Second, the thickness of the secondary flow boundary layers decreases as the Reynolds number increases, i.e., the velocity gradient ∂U y /∂z near the bottom wall increases as the flow Reynolds number increases. Both trends are consistent with the findings from previous studies [23,27]. Again, it is emphasized that these trends are predicted without utilizing observation data from the high Reynolds number cases (Re = 8.3 × 10 4 or Re = 2.5 × 10 5 ).
In addition to the favorable quantitative agreement with experimental data presented above, we also found that the qualitative features of the flows at Reynolds numbers Re =

Prediction of Flow in a Different Geometry
In Section 3.1 above, it has been demonstrated that the calibrated Reynolds stress discrepancy can be utilized to predict flows on a geometrically similar domain but at higher The obtained flow patterns with the three mapping schemes above largely reflect their respective physical assumptions as detailed above. The in-plane streamlines from experiments [28] are presented in Fig. 14a, which shows two vortices. The two vortices in the rectangular duct have different sizes, which is in contrast to the two symmetric vortices in the square duct. The major vortex is located at the upper right, while the minor structure is located at the lower left. The relative locations of the two vortices are similar to those in the square duct flow. The predicted streamlines based on the mapping schemes 1-3 are presented in Fig. 14b-14d, respectively. Figure 14b shows two vortices in the lower half of the duct similar to those in the square duct flow. It is noteworthy that a third vortex develops on the upper half of the duct, which is driven by the vortex below it. This predicted flow pattern is qualitatively different from that shown in Fig. 14a, suggesting that the assumption associated with scheme 1 is unphysical. Figure 14c shows that the prediction based on scheme 2 does capture the qualitative features of the major and the minor vortices, although the predicted center locations and the vortices sizes do not completely agree with the experimental results. Finally, the predicted streamlines based on scheme 3 are presented in Figure 14d, which has an even better agreement with the experimental data compared to that obtained with scheme 2. In particular, the center and the size of the minor vortex are very well predicted. The general pattern of the major vortex is also captured successfully, although the prediction quality is not as good as for the minor vortex. The remaining difference can be explained by the fact that the mapping scheme is linear, which may not be true due to the different influences of the boundary layer interactions.
Comparison of the three predictions above suggest that the assumed physical knowledge has a critical impact on the obtained predictions. An unphysical assumption as that behind scheme 1 leads to a qualitatively incorrect prediction of overall flow patterns. An assumption based on sound physical reasoning as in scheme 3 leads to the more favorable predictions. Figure 15 shows the comparison of secondary flow velocity magnitude contours between experimental data [28] and the prediction based on scheme 3. It can be seen that the predicted contour patterns are similar to those of the experimental data, and the predicted magnitude is also comparable to the experimental data. Admittedly, the predicted secondary flow field is not exactly the same as the experimental data, e.g., the velocity magnitude is greater than the experimental data at the near wall region and smaller than the experimental data away from the wall. A possible reason is that the prior physical knowledge incorporated in scheme 3 is still far from enough to accurately account for the additional physics introduced by the geometry change. However, it can be seen from Fig. 14 that the prediction qualitatively captures the flow pattern, which is totally absent in baseline RANS results, with much lower computational cost compared to high fidelity simulations. Therefore, we argue that the proposed method provides a practical approach to quickly and approximately (a) Experiment [28] (b) Scheme 1 (c) Scheme 2 (d) Scheme 3 Figure 14: Comparison of experimentally measured and the predicted secondary flow patterns (posterior mean) based on three schemes of mapping Reynolds stress discrepancy from the calibration case to prediction case. Panel (a) shows streamlines obtained from experiments [28]. Panels (b)-(d) show predicted streamlines based on schemes 1-3, respectively.
The baseline RANS prediction is omitted here as it predicts no secondary flows.
search the design space with low computational costs in the preliminary design stage of engineering systems. It will help identify the promising design candidates for the further investigations with high fidelity simulations and/or model experiments. The baseline RANS prediction is omitted here as it predicts no secondary flows.

Conclusion
Recently, Xiao et al. [12] proposed a framework for quantifying and reducing uncertainty in RANS simulations based on sparse observations of velocities. In the present work we extend the original framework to flows with no observation data, leading to a Bayesian calibration-prediction method. As in the original framework, the model-form uncertainty in RANS simulations are localized to the Reynolds stresses, which are modeled as a random field. The uncertainty distribution of the discrepancy is first calibrated with available data on a related flow (e.g., the flow in a geometrically similar domain or in a slightly different geometry). Subsequently, the obtained distribution is extrapolated to the prediction case and is sampled to correct the RANS-modeled Reynolds stresses. The proposed method is an algorithmically straightforward yet practically profound extension of the original framework. Since direct observation data are not required on the flow to be predicted, the range of applications is greatly expanded compared to the original framework. This is particularly relevant for turbulent flow simulations in support of engineering design, where the prototype system has not been build yet in the design stage.
An important assumption in the proposed method is that the flow used for calibration and the flow to be predicted are closely related. Specifically, they share the same overall characteristics despite the fact that the two flows are not dynamically similar (i.e., they have different Reynolds numbers) or not even geometrically similar (e.g., flow in a square duct versus flow in a rectangular duct). This assumption is valid in many scenarios in engineering design. On the other hand, the assumption also implies that the quality of the prediction inevitably depends on the judgment of the analyst. We argue that this is an intrinsic feature of Bayesian methods, and it is often an advantage for a framework to be able to incorporate insights and prior knowledge from the users, who are often experts of the model and the problem of concern.
4. The probability distribution functions and the corresponding cumulative density functions (CDF) for each component of the random vector a is estimated from the samples, i.e., {α i } N i=1 , by using kernel density estimation techniques.
5. Generate N samples for each of the component from the estimated CFD by using standard sampling techniques [19]. 6. Reconstruct sample {ω i } N i=1 of size N from the resampled coefficients {α i } N i=1 and the basis {φ j } 3m j=1 .