Bayesian parameter estimation of SST model for shock wave-boundary layer interaction flows with different strengths

Denggao TANG, Jinping LI, Fanzhi ZENG, Yao LI, Chao YAN

School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China

KEYWORDSBayesian calibration;Boundary layers;Compression corner;Sensitivity analysis;Shear-stress transport turbulence model;Shock waves

AbstractThe Shock Wave-Boundary Layer Interaction (SWBLI) flow generated by compression corner widely occurs in engineering.As one of the primary methods in engineering, the Reynolds Averaged Navier-Stokes (RANS) methods usually cannot correctly predict strong SWBLI flows.In addition to the defects of the eddy viscosity assumption, the uncertainty of the closure coefficients in RANS models often significantly impacts the simulation results.This study performs parametric sensitivity analysis and Bayesian calibration on the closure coefficients of the Menter k - ω Shear-Stress Transport(SST)model based on the SWBLI with different strengths.Firstly,the parametric sensitivity on prediction results is analyzed using the Sobol index.The results indicate that the Sobol indices of wall pressure and skin friction exhibited opposite fluctuation trends with the increase of SWBLI strength.Then, the Bayesian uncertainty quantification method is adopted to obtain the posterior probability distributions and Maximum A Posteriori (MAP) estimates of the closure coefficients and the posterior uncertainty of the Quantities of Interests (QoIs).The results indicate that the prediction ability for strong SWBLI of the SST model is significantly improved by using the MAP estimates, and the relative errors of QoIs are reduced dramatically.

1.Introduction

Shock Wave-Boundary Layer Interaction(SWBLI) is a common phenomenon involving complex flow structures on supersonic and hypersonic aircraft, which is one of the most prevalent challenges in fluid mechanics.1Supersonic compression corner and oblique SWBLI have proved to be the most challenging in all two-dimensional separated flows.2Therefore,the following work mainly analyzes these two types of flows.As a typical compressible turbulent separation form, the SWBLI embodies all the difficulties of turbulence, compressibility, and viscous-inviscid interaction phenomena.Meanwhile, the impact of SWBLI on flow is multifaceted, which is often a critical factor in determining the performance of aircraft or propulsion systems.For instance, excessive SWBLI will result in the inlet unstart of aircraft.This phenomenon widely exists at the first and second stage junction of the missile front and on the wing or fuselage of aircraft.Although identifying, understanding, and controlling separated flow are essential in aerodynamics, it is still challenging to predict flow separation.3,4In terms of flow mechanism, the SWBLI is mainly studied based on such accurate and detailed means as the Direct Numerical Simulation(DNS),Large Eddy Simulation(LES),etc.Nevertheless,these ways are accompanied by the difficulty of generating high-quality grids and unaffordable computational costs.As a result, these limitations make these high-fidelity methods only calculate some simple flows with low Reynolds number but cannot be applied to the engineering field.Fortunately, it is unnecessary to obtain the delicate flow structures in engineering, and the time-averaged results are enough.Therefore, it is not necessary to use computationally expensive methods.The RANS models are widely adopted in practical engineering applications due to their strong robustness, calculation efficiency, and lower request for computer hardware.

The Menter k - ω Shear-Stress Transport(SST)turbulence model is a two-equation eddy viscosity model that combines the best characteristics of the k - ω and k - ε turbulence models.5Consequently, the SST model is one of the most widely used models in engineering applications.For weak SWBLI flows, the SST, Spalart-Allmaras (SA), and other turbulence models behave similarly.6Nevertheless, for strong SWBLI flows, the SST turbulence model does not obtain satisfactory results, whether for separation or reattachment flow.7This study aims to recalibrate the parameters of the SST model for improving the simulation performance of the SWBLI while quantifying the uncertainty of the Quantities of Interests(QoIs).

The RANS model is derived based on some assumptions,resulting in many uncertainties of the predicted outcome.Xiao and Cinnella8classified the uncertainties of the turbulence model into parametric uncertainty and structural uncertainty.For the parametric uncertainty, it is a truism that the parameters of most turbulence models are determined partly by some elementary experiments and partly by analyses under some assumptions.9The recommended values of model parameters are called nominal values, but the nominal values may not be appropriate for all types of flows.8Therefore, the different kinds of flows should have different recommended values of the parameters to best match the turbulence model.In terms of structural uncertainty, the Boussinesq assumption has caused some defects.Some improvement methods are proposed to overcome these defects, such as compressibility correction,10,11length-scale correction,12,13and shockunsteadiness correction.14This study mainly focuses on parametric uncertainty analysis.In Computational Fluid Dynamics(CFD), the Bayesian inference technique and the Non-Intrusive Polynomial Chaos (NIPC) method are employed to assimilate experimental data or other high-fidelity data (e.g.,the DNS data or LES data) to infer turbulence model parameters while quantifying and reducing the uncertainty of the turbulence model.8Kato, et al.15adopted a data assimilation technique for estimating the parameters in the SST model.Cheung, et al.16adopted the Bayesian method to calibrate the turbulence model coefficients in the incompressible plate boundary layer for the first time.Li, et al.17,18evaluated the distinctions of different likelihood functions in the Bayesian calibration method based on supersonic jet interaction and introduced the Bayesian evidence to assess the k - ω models with different compressible correction types in SWBLI.In addition, Subbian,4Ray,19Zeng,20,21and Zhang,22et al.also performed similar analyses in different aspects of the CFD field using the Bayesian calibration method.There are more explanations for the research in the review by Xiao and Cinnella8.

The research contents of this paper are summarized as follows.First,the sensitivity analysis of parameters is carried out for the SWBLI with different strengths, and the key parameters that affect the flow are analyzed.Then, the parameters of the SST model are corrected by the Bayesian calibration method combined with the experimental pressure data of different cases, and the Maximum A Posteriori (MAP) estimates are used to predict all cases.The remainder of this paper is organized as follows.Section 2 describes the numerical model.In Section 3,the Bayesian framework is introduced.The calculation details, including experimental description, grid independence verification, and calibration details, are given in Section 4.In Section 5, the analysis results are discussed.The conclusions are drawn in Section 6.

2.Numerical method

All calculations in this study are based on an in-house finite volume solver,23–25which is a structured-grid cell-centered RANS code.The accuracy of the solver has been verified in many applications26,27.

Menter28improved the standard k - ω model and first proposed the SST turbulence model in 1994.The SST turbulence model combines the k - ω turbulence model and the k - ε turbulence model via blending function F1.Thus,the k - ω turbulence model with good robustness and accuracy is retained in the region near the wall, while the freestream independence of the k - ε turbulence model is used in the region outside the boundary layer.The SST turbulence model expression is

where k and ω are the turbulent kinetic energy and specific dissipation rate, respectively.

The production term P1is approximated by vorticity:

The function F1is defined as

The nominal values and prior ranges of the SST model parameters are shown in Table 1.On the premise that the CFD calculation converges, the prior ranges of parameters have been selected repeatedly to ensure that the prior distributions of QoIs can cover most of the experimental data.For κ,the boundary of the prior range is ±10% of the nominal value.For a1,the upper and lower boundary of the prior range are-10%–25%of the nominal value,respectively.In the previous prior calculations, it was found that selecting a larger value of a1often makes the prior distributions better cover the experimental data,so the upper boundary is set slightly farther than the lower boundary.The prior boundaries of the other parameters are ±15% of their nominal values.

3.Bayesian framework

This section mainly introduces the principle of the Bayesian calibration,the construction of surrogate model,and the parametric sensitivity analysis based on the Sobol index.More details can be found in the study of Debusschere, et al.29.

3.1.Surrogate model construction

The Markov Chain Monte Carlo (MCMC) method is employed for sampling in the process of Bayesian calibration.It is unaffordable that all samples are accurately solved by CFD.Therefore, it is compulsory to use high-efficiency ways

Table 1 Nominal values and prior ranges for parameters of SST model.

to replace the exact solution process of CFD with cost savings.As one of the ways, a surrogate model is constructed by the NIPC method to represent the relation between input variables(i.e., the parameters of the SST model) and output variables(i.e.,flow variables,such as the wall pressure).A random variable α*（x;ξ ）can be expressed as the following polynomial expansion:

Theoretically, the more the terms of the polynomial expansion are,the more accurate the result is.However,it is generally truncated with a limited number of the terms considering the difficulty of solving the polynomial coefficients.The number of expansion terms P+1 is related to the dimension n of the variables and the order O of the expansion terms, as shown in.

The number of samples Ntrequired to solve the coefficients αi（x ）is given by Eq.(15).The samples are sampled by the optimized Latin hypercube method.According to the study of Hosder, et al.30, the oversampling ratio nP=2 and the order of the expansion term O=2 are generally selected.In this study, the sampling ratio nPand the order O of the expansion term are adjusted according to the accuracy of the surrogate model.

The specific form of the surrogate model can be obtained from Eq.(11), and the mean and variance of the Polynomial Chaos Expansion (PCE) can be obtained according to

3.2.Sensitivity analysis

Sensitivity analysis is mainly based on the Sobol index,31indicating the impact of the variables on the model output.

According to the work of Sudret32and Crestaux, et al.,33the variance of PCE can be decomposed as follows:

3.3.Bayesian calibration

where δ（x ）is the model error.Embedding model errors into the model itself is an innovative method, proposed by Sargsyan, et al.34The usual method is to use the first-order Gaussian-Hermite PCE.In other words, λ is augmented by a multivariate normal random variable as

where K is the number of expansion terms and O is the order of PC.Now the problem about the parameter λ is transformed into the problem about a parameter ^λ= （λ;α ）, which can be solved by the following Bayesian principle:

Fig.1 Sketch of supersonic channel35.

The exact high-dimensional posterior probability is difficult to calculate.Therefore, the MCMC method is usually employed to obtain samples from the posterior probability distributions of the closure coefficients.

4.Computational details

4.1.Geometry and freestream conditions

The two-dimensional SWBLI experiment conducted by Settles,et al.35,36is selected as the experimental data source for evaluating the turbulence models in this study.The ramps of 8?,16?,20?, and 24?were adopted in the experiment.The Princeton University 20 cm × 20 cm high Reynolds number channel is illustrated in Fig.1.The ramp angles considered in this study include the range from unseparated to significantly separated.The separated zone for the 16?ramp is tiny,and the flow is more accurately described as incipient separation.Cases 3 and 4 are used for the turbulence model evaluation.All compression corners are adopted for parametric sensitivity analysis of QoIs.The freestream conditions are illustrated in Table 237.The subscript ∞denotes the freestream conditions, the subscript 0 represents the stagnation conditions, the symbol δ represents the boundary-layer thickness, and the symbol xupstreamrepresents the distance from the inflection point of the ramp to the upstream.

An additional oblique SWBLI is selected as a verification case to verify the applicability of the results,which is recorded as Case 5.The experiment conducted by Reda, et al.38in the NASA Ames Unitary system is adopted as the data source in this study.The sketch of the shock generator and boundary conditions for Case 5 is shown in Fig.2.The freestream conditions are listed in Table 3.

Fig.2 Sketch of shock generator and boundary conditions for Case 5.

Table 3 Freestream conditions for Case 5.

Fig.3 Boundary conditions of Case 4.

4.2.Grid independence analysis

The influence of the grid is relatively significant for large separated flows.Therefore, the grid independence analysis is carried out for Cases 3 and 4 to eliminate the impact of grid resolution on the calculation results.The grids with different resolutions are generated, respectively.The number of grids in the flow direction is more than that in the normal direction,and the first cell of the grid near the wall satisfies y+<1.Fig.3 indicates the boundary conditions of Case 4.Fig.4 shows the results of the grid independence analysis.For Cases 3 and 4,the middle grids are enough to calculate the flow variables without the impact of the grid and adopted in the subsequent calculations.

Table 2 Freestream conditions for compression corner cases37.

Fig.4 Wall pressure and skin friction distributions using different grids.

4.3.Baseline results

The SWBLI with four different strengths is calculated by the SST and SA models with nominal values before the Bayesian calibration of the turbulence model.The baseline results of all ramps are illustrated in Fig.5 and Fig.6,including the wall pressure distributions and density contours.After the supersonic flow passes through the ramp, a series of compression waves are generated due to compression and converge into a shock wave.The boundary layer behind the shock develops thicker, and the adverse pressure gradient increases.Meanwhile, the intensity of shock rises with the increase of ramp angle.The subsonic part of the boundary layer will propagate information upstream.Consequently, the separated flow occurs after the shock when the adverse pressure gradient is large enough to overcome its inertia.Finally, the shock wave moves forward to a new equilibrium position.The supersonic airflow bypasses the separation region and forms a new compression region and reattachment shock after reattachment.The main shock appears due to the merging of the separation and reattachment shock.Fig.6 shows that a slight separation(described as‘incipient separation’)occurs from the 16?ramp.The reattachment shock emerges at the 20?ramp, which leads to the apparent deviation of the separation point calculated by the turbulence models from the experimental measurements,so does the wall pressure.Fig.5 shows the wall pressure increases twice, such as Case 4, and it only increases once without separation flow,such as Case 1.Fig.6 shows that the airflow density increases sharply after passing through the shock.The separation shock at the new equilibrium position causes the first rise of the wall pressure, and the incoming flow impinges on the ramp and causes the second rise.For the 8?ramp, the prediction results between the SST and SA models are not noticeable.The apparent difference begins to appear at the 16?ramp.The prediction of separation point by the SA model is significantly more downstream than that by the SST model,and the SA model also predicts the pressure recovery rate more rapidly than the SST model on the ramp.Fig.5 indicates that(or the enlarged figure in Fig.4 shows more clearly) there is a slight kink near the inflection point in the strong SWBLI cases,such as Cases 3 and 4.The kink indicates that a secondary separation occurs within the separation bubble.39Fig.7 shows more details of the flow structures.

The density contour and flow structures of the baseline result of Case 5 are illustrated in Fig.8, including separation bubble, incident shock, reattachment shock, etc.The study of Xie, et al.40explained the detailed flow mechanism, which will not be described here.

4.4.Calibration details

The Bayesian parameter calibration is carried out using the experimental data of Cases 3 and 4 to assess the simulation ability of the SST model for the SWBLI cases with different strengths.Here,the length from the position of the separation shock wave to the inflection point of the ramp is used to represent the separation information, which is marked as LU.Meanwhile, the shock position measured in the experiment can also be obtained from the experimental wall pressure data.LUcan also be used as calibration data.The optimal Latin hypercube sampling method is used to sample from the prior distributions to establish the surrogate model.The training sets of Cases 3 and 4 contain 110 prior samples,and the verification sets contain 33 prior samples.Each prior sample needs a full CFD simulation.The accuracy of the surrogate model is evaluated by calculating the relative error of the surrogate model on the validation set.

Fig.9 shows the relative errors of the surrogate models of Cases 3 and 4.The relative error is defined by.

where Vvaldenotes the wall pressure of the corresponding sampling point output by CFD in the validation set,and Vsurrrepresents the value directly output by the surrogate model.

As will be mentioned below,among the nine parameters of the SST model, the first three parameters provide the primary source of the uncertainty of wall pressure (i.e.,κ,a1and β*).It is sufficient to analyze the posterior uncertainty of the wall pressure only by the three parameters.Therefore, within the allowable range of computing capacity, the PCE order of the surrogate model is increased by removing the parameter with a small Sobol index.Thus, the error of the surrogate model is reduced to ensure the accuracy of the evaluation results.The PC order O increases from 2 to 4, and the oversampling ratio nPincreases from 2 to 4.According to Eq.(15), when the PC order O and oversampling ratio nPchange,the number of the corresponding prior samples in the training set decreases from 110 to 105.In contrast, the number of samples included in the validation set remains unchanged to ensure the reliability of correlation analysis.Fig.9 shows that after increasing the order of the PCE,the relative error of the surrogate model decreases significantly.The separation point predicted by the SST model is mainly located in the region with strong nonlinearity.The separation shock in this region causes the wall pressure to change drastically, resulting in significant relative errors of the surrogate model.Although the relative errors for pressure in this region are significant for Cases 3 and 4,the data in this region are not used for calibration, but LUis used as the calibration data to replace the information reflected in this region.The relative errors of the surrogate model adopted to calculate the posterior uncertainty of wall pressure are less than 10 % at most measuring points in this region.The relative errors of the surrogate model for pressure outside this region and LUare both less than 6 %, which is enough for the MCMC sampling and related calculations.

Fig.6 Density contours near inflection point for all ramps.

Fig.7 Flow structure of supersonic compression corner.

Fig.8 Density contour and flow structure for Case 5.

After establishing the surrogate model, 3×105posterior samples will be obtained for each MCMC sampling.However, not all the posterior samples are valid.The first onefifth of the samples will be discarded to ensure convergence.One out of every-five consecutive posterior samples is selected to ensure the relative independence among the selected sample points.Therefore, the effective samples finally obtained are 48000.The kernel density estimation method is employed to obtain the posterior probability density distributions of the parameters of the SST model from the posterior samples.The posterior information is propagated forward by the surrogate model to obtain the posterior uncertainty of wall pressure.

5.Results and discussion

In this research,the parametric sensitivity analysis of the QoIs with the increase of the strength of SWBLI is first performed.Then the Bayesian parameter calibration is carried out for Cases 3 and 4.After that, the MAP estimates are applied to other cases to check whether the optimal parameters are applicable.Meanwhile, the calibration dataset only includes wall pressure,and it is analyzed whether the model parameters calibrated based on wall pressure can similarly improve the skin friction.

5.1.Sensitivity analysis of model parameters

Fig.9 Relative errors of surrogate models for Cases 3 and 4.

Fig.10 Wall pressure distributions of 110 prior samples.

Firstly, the global sensitivity analysis is carried out, and the Sobol index is adopted to characterize the sensitivity of QoIs to the closure coefficients.Sensitivity analysis is based on variance.The parameters with a large Sobol index will be used as the object of embedding model error.Fig.10 shows the distributions of 110 prior samples of Cases 3 and 4.The prior distributions indicate that the closure coefficients have a significant influence on the prediction ability of the turbulence model.The uncertainty of wall pressure and skin friction is mainly concentrated in the separation bubble region.The uncertainty in the separation zone is more significant than that in the reattachment zone.Fig.11 and Fig.12 show the Sobol index of the QoIs in the cases of no separation to significant separation to analyze the variation trend of parametric sensitivity when the SWBLI strength increases.The density contours of the baseline results are also provided in the pictures to correspond with the position of the flow structures.Fig.11 shows whether flow separation occurs or not, the parametersa1, κ and β*contribute the most to the uncertainty of wall pressure.For the significantly separated cases,the fluctuation degrees of the Sobol indices of pressure are considerably smaller than that without separation and incipient separation.And the total variance of wall pressure provided by a1also increases with the increase of separation bubble size,relatively.Fig.12 shows that for sensitivity analysis of skin friction, the fluctuation degrees of the Sobol indices increase with the increase of SWBLI strength,which is opposite to that of wall pressure.The Sobol indices of skin friction are straight lines nearly for Case 1.The three parameters a1,κ and β*provide most of the contribution to the uncertainty of skin friction, and in the cases of strong SWBLI, the variance provided by β1and σk1also accounts for a certain proportion.For the QoIs, the fluctuation of the Sobol indices in the separation zone and the flat plate zone is more significant than that in the reattachment zone and on the ramp.And the uncertainty provided by a1decreases while that provided by κ increases in the separation bubble.Meanwhile, the three parameters a1, κ and β*provide more than 96 % variance for the uncertainty of LU,which is indicated in Fig.13.Therefore,the three parameters a1,κ,and β*are used as the object of embedding model error.The results of the parametric sensitivity analysis of the QoIs are summarized in Table 4.

5.2.Calibration based on pressure and LUdata

In the previous work, it is assumed that the parameters of the SST model are uniformly distributed.This part of the research mainly obtains information from the experimental data to update the distribution of the model parameters.If less information is obtained from the experimental data, there is little difference between the posterior and prior probability distributions.The posterior uncertainty of the QoIs can be obtained from propagating the posterior samples forward by the surrogate model.The MAP estimates obtained by the Bayesian inference technique are applied to all other cases, and then we check whether the optimal values of parameters obtained are applicable.When the wall pressure and LUare adopted as the calibration dataset, we check whether the other QoIs can also be improved.

Fig.11 Sobol indices of wall pressure for all ramps.

The uniform prior distributions are compared with the marginal posterior distributions of the model parameters obtained from Cases 3 and 4,which are shown in Fig.14.Consistent with the previous sensitivity analysis,more information obtained from the experimental data is the first three parameters a1, κ and β*, the posterior distributions are significantly different from the prior distributions.And the posterior distributions of other parameters are similar to the prior distributions.Therefore, only the first three parameters are analyzed here, i.e.,a1, κ and β*.Fig.11 shows that a value larger than the nominal value should be recommended for parameter a1,and this selection tendency of Case 4 is stronger than that of Case 3.For calibration with Case 3, the values of κ and β*are similar to the nominal values.Regarding Case 4,the value of β*is preferred less than the nominal value,while the recommended value of κ is larger than the nominal value.

The fourth-order accuracy surrogate model is used for the posterior uncertainty analysis to obtain more reliable results.The posterior samples obtained from the MCMC sampling are input into the surrogate model to calculate the posterior means and credible interval of wall pressure.In Fig.15, μpand σpare posterior means and posterior standard deviation,respectively.The total error includes the model error,posterior uncertainty,and surrogate error.Fig.15(a)and(b)show that the posterior means of wall pressure for Cases 3 and 4 are closer to the experimental values than the baseline results.Most of the experimental values are included in the 2σ credible interval.The posterior uncertainty of wall pressure in the reattachment zone and its subsequent ramp zone is smaller than that in the separation zone and the flat plate zone near the separation shock wave.Fig.15 (c) indicates that the posterior means of LUare much closer to the experimental data than the baseline results, which are nearly equal to the experimental values.

The decomposition of the total error is shown in Fig.16,which is consistent with the previous analysis.The posterior uncertainties of Cases 3 and 4 are mainly distributed in the region where the calculated separation points are located.The model error accounts for the main part of the posterior variance of Case 3, and the peaks appear near the separation point and the reattachment point.In the posterior variance of Case 4,apart from the model error,the surrogate error also accounts for a large part.In addition to the peak near the separation point and the reattachment point,there is a small peak before the separation point of the model error.

Fig.12 Sobol indices of skin friction for all ramps.

Fig.13 Sobol indices of LUfor Cases 3 and 4.

Table 4 Results of parametric sensitivity analysis of SST model.

5.3.Applicability analysis for other cases

Fig.14 Marginal posterior distributions of model parameters for Cases 3 and 4.

Fig.15 Posterior means of pressure and LUwith estimated errors for Cases 3 and 4.

To prove whether the MAP estimates of the model parameters obtained are applicable, the MAP estimates obtained from Case 3 are tagged as S1, and those obtained from Case 4 are marked as S2.It can be seen from Fig.17 that, regardless of which set of MAP estimates is used,the wall pressure is better than that of the baseline results.However,when S1 is selected for Cases 4 and 5,the predicted shock and separation positions are more upstream than the experimentally measured values.The predicted values and experimental values of other cases match well.When S2 is selected, the predicted shock position of Case 3 is to be slightly downstream from the measured value.The predicted shock positions of other cases are close to the experimental values.According to the previous analysis,the model parameter that has the most significant impact on the calculation results isa1, followed by κ andβ*.S1 and S2 have different recommendations for the three parameters, as shown in Fig.14.To prove the main reasons for the above difference, a1is selected from S1, and the other parameters are the nominal values.This set of parameters is marked as S3.κ and β*are selected from S1, and the other parameters are the nominal values.This set of parameters is denoted as S4.For S2,the same selection rules are adopted.The two new sets of parameters from S2 are recorded as S5 and S6,respectively.Because Cases 3 and 4 with the significant separation are more typical and challenging, these sets of parameters are used for Cases 3 and 4.The calculation results of wall pressure are shown in Fig.17.The results by CFD calculation using S3 and S5 showed significant differences, and the difference between the two sets of parameters only lies in the selection ofa1.And the results of S5 are better than those of S3.The difference between the calculation results of S4 and S6 is not noticeable, indicating that the effects on the predicted performance of the SST model of κ and β*respectively obtained from Cases 3 and 4 are similar.Meanwhile, it can be also found that the calculation results of S1 and S3 have significant differences,while S2 and S5 are similar.It is consistent with the previous parametric sensitivity analysis that the importance of a1increases with the increase of the SWBLI strength.

From what is mentioned above,we know that the influence of parameters on the prediction ability of the turbulence model is very complicated.The results show that the MAP estimates obtained are applicable for the SWBLI flow.However, the MAP estimates corresponding to the given flow conditions are not necessarily the same as those under other conditions,so only the selection trends can be given here.In the SWBLI flow, the selection of a1is undoubtedly the most important.Fora1, it should preferably be larger than the nominal value and increase with the SWBLI strength.Since more information is obtained from Case 4 during calibration, selecting the parameter κ from the right side close to the nominal value and the parameter β*from the left side close to the nominal value is recommended.However, a problem is also shown here.For the cases with significant separation, limited by the turbulence model itself, the calculated results cannot be entirely consistent with the experimental values by adjusting the model parameters.Even though the calculated shock position is consistent with the experimental measurements,the wall pressure calculated by the SST model after the shock shows a steep upward trend.One possible reason is that the separation shock position is unsteady in the actual experimental measurement.14This unsteady phenomenon causes the wall pressure measured by the instrument to be a time-average result,which will ‘round off’the wall pressure rise caused by the shock.In essence, the RANS models cannot simulate this instability,and only a onefold shock position can be obtained so that the wall pressure is steeper than the measured.

The Bayesian calibration method is used to obtain the information from the pressure data to acquire the optimal values of the model coefficients matching with the corresponding flow.Both the wall pressure and the skin friction can reflect the size and location of the separation zone.Therefore, when the wall pressure is selected as the calibration dataset, the prediction performance of the turbulence model for both the wall pressure and skin friction can be improved.Fig.18 shows the skin friction calculated by the MAP estimates obtained from the wall pressure data and compares it with the baseline results.For the baseline results,the separation point predicted by the SA model is better than that predicted by the SST model.Still, the reattachment point predicted by these two models lags seriously.For the cases with significant separation,the skin friction in the boundary layer recovery region after reattachment is seriously underestimated.The skin friction calculated using the MAP estimates is better than the baseline results for predicting the separation point and the reattachment point more correctly and predicting the skin friction better in the recovery region on the ramp.However, for all compression corners, including the SA model, the prediction of skin friction by the turbulence models is larger than the experimental measurements in the flat plate region.The predicted results using the calibrated parameters are larger than the baseline results.From the above analysis, it can be seen that increasing a1conduces to the increase of the turbulence to delay the separation and increase the friction in the flat plate area and on the ramp.The effect of MAP estimates of different sets on skin friction is similar to that of wall pressure, which will not be repeated here.

Fig.16 Posterior variance decomposition of pressure for Cases 3 and 4.

To quantify the wall pressure and skin friction errors obtained by different models, the relative error is defined as.

where Qcfdrepresents the QoIs calculated and output by CFD,and QExprepresents the QoIs measured experimentally.

The errors of QoIs are tabulated in Table 5.It can be obtained that no matter which set of MAP estimates is used,the relative errors of wall pressure predicted by the SST turbulence model are smaller than those of the baseline results.The most apparent effect is that the error of Case 3 is reduced from 19.69% to 6.23% using the set of parameters S1, and the error of Case 4 is reduced from 18.87% to 7.70% using the set of parameters S2.The reason is that S1 and S2 are obtained from the wall pressure datasets of Cases 3 and 4,respectively,so the results improve the best.When S1 and S2 are used to predict the skin friction of case 1,the relative error is bigger than that of the baseline results, because the SST model with S1 and S2 improves the overall prediction of the skin friction of Case 1 compared with the baseline results.It enhances the recovery rate of the skin friction on the ramp, making the skin friction downstream of the ramp better match the experimental measurements, but the overall relative error increases.For other cases, the relative errors of skin friction predicted by S2 are smaller.

The coordinate systems involved in the compression corner are shown in Fig.19.The profile of velocity U at several stations of Case 4 is given based on the (s, n) coordinates.These two coordinates are the same upstream of the inflection point.Fig.20 (a)–Fig.20(c) compare the dimensionless profiles of velocity U, turbulent kinetic energy k, and turbluent eddy viscosity μtat different stations.The previous work shows that the separation region of the baseline results is much larger than the experimental measurement.Therefore, the U profile predicted by the baseline is significantly different from the experimental measurement, which is more prone to loss, especially at s/δ=-2.76,-1.33,-0.44,and 0.The U profile of the calibrated prediction at the s/δ=-2.76 station overlaps with the experimental measurement, and it has also been significantly improved in other stations.It can be seen from Fig.20(b)that before separation, the turbulent kinetic energy k predicted by the baseline in the buffer layer and the outside region(y/δ>0.0006; y+>6) is smaller than that by the calibrated prediction.Fig.20(c) shows that from the outer layer to the outer boundary of the boundary layer, the turbluent eddy viscosity μtpredicted by the baseline is also smaller than that predicted by the calibrated result, and the calibrated eddy viscosity increases more rapidly.The contour of the dimensionless turbluent eddy viscosity μtis shown in Fig.21.

6.Conclusions

In the research of this paper, the Bayesian calibration method based on the experimental wall pressure is used to analyze the SST model.Firstly, the surrogate model is constructed by the NIPC method.For the cases of SWBLI with different strengths, the parametric sensitivity analysis of the SST model is carried out based on the Sobol index.Then, the parameters of the SST model are calibrated using the wall pressure data from the compression corner experiment with different angles.The MAP estimates obtained from the calibration are applied to other cases.An additional oblique SWBLI case is introduced to verify the applicability of the MAP estimates.The main conclusions are drawn as follows:

Fig.17 Wall pressure calculated using different sets of parameters for all cases.

(1) The compression corner flow is complex, and the SST model cannot accurately calculate the flow variables of largeangle compression corners.The uncertainty of the QoIs is mainly concentrated on and near the separation bubble region,and the posterior uncertainty in the separation zone and the flat plate zone near the shock wave is more considerable than that in the reattachment zone and on the ramp.

(2) The three parameters a1, κ and β*provide the primary source of the uncertainty of the QoIs.With the increase of separation bubble size, the fluctuation degree of the overall Sobol indices decreases for wall pressure and increases for skin friction.The fluctuation of the Sobol indices in the separation zone and the flat plate zone is more complex than that in the reattachment zone and on the ramp.And the uncertainty provided by a1decreases while that provided by κ increases in the separation bubble.The increase in the SWBLI strength leads to a rise in the proportion of the uncertainty supplied bya1.

Fig.18 Skin friction calculated using different sets of parameters for all cases.

Table 5 Relative errors of QoIs obtained by different methods for all cases.

Fig.19 Various coordinates are involved in the compression corner.

(3) Compared with the baseline results, the prediction results of the calibrated SST model with MAP estimates match better to the experimental data, and the relative errors of the QoIs are reduced dramatically.The 2σ credible interval can cover most of the experimental values.For a1,it should preferably be larger than the nominal value and increase with the increase of SWBLI strength.The improvement still exists for different cases,which verifies the applicability of the calibrated values.

(4) The SST model is prone to underestimate the Reynolds stress.The turbulence near the wall is increased by adjusting the parameters to increase the eddy viscosity.After the Bayesian calibration, the prediction performance of the SST model is enhanced for the reverse pressure gradient flows.However,the SST model cannot simulate the inherent shock instability in SWBLI by adjusting parameters, and the predicted wall pressure in the separation zone is still too large.Therefore,the prediction performance of the SST model can be improved by adjusting parameters, but the defects of the RANS models still need to be solved from other facets.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This study was supported by the National Numerical WindTunnel Project (No.NNW2019ZT1-A03) and the National Natural Science Foundation of China (No.11721202).

Fig.20 Comparison of dimensionless profiles of velocity,turbulent kinetic energy,and turbulent eddy viscosity μtbetween baseline and calibrated prediction at different stations for Case 4.

Fig.21 Comparison of eddy viscosity between baseline and calibrated prediction for Case 4.