Quantification of parametric uncertainty in γ-Reθmodel for typical flat plate and airfoil transitional flows

Ziming SONG, Zaijie LIU, Jiachen LU, Chao YAN

National Key Laboratory of Computational Fluid Dynamics, Beihang University, Beijing 100191, China

KEYWORDSAirfoil transition;Closure parameters;Sensitivity analysis;Uncertainty analysis;γ-Reθtransition model

AbstractOwing to the lack of physical knowledge of boundary layer transition, the γ-Reθtransition model introduces closure parameters, which increase the uncertainty of transition prediction.The objective of this work is to quantify the uncertainties of closure parameters in the quantities of interests and identify the key parameters.The six closure parameters in the uncertainty intervals are used as input variables,and the uncertainties of the output results are propagated by a stochastic expansion based on the point-collocation nonintrusive polynomial chaos method.The relative contribution of each parameter to uncertainty is evaluated by the Sobol index.The computational cases include natural and bypass transitional flows on zero-pressure-gradient flat plates, and subsonic and transonic flows around airfoils.For most cases,ce2,ca2,and ca1dominate the uncertainty,and the influence of σθtis also significant when the history effects of flow are evident.The contribution of parameters in airfoils is more complex than that in flat plates.The transonic airfoil case shows that flow separation dramatically changes the distribution of Sobol indices, which poses a challenge to the accurate prediction of transition.Generally, ce2and ca2are the key parameters of the γ-Reθmodel.

1.Introduction

The laminar-turbulent transition in boundary layers is one of the most challenging problems that remain in classical physics.Despite a century of interest, the physical mechanisms of the transition processes are still not fully understood.Because of its significant impact on skin friction drag, flow separation behavior, and aeroheating of aircraft, accurate prediction of transition is of great value for aircraft design.Current numerical simulation methods of transition include Direct Numerical Simulation(DNS),1Large-Eddy Simulation(LES),2transition correlation,3stability theory,4–6and transition models,7–13built on the Reynolds-Averaged Navier-Stokes (RANS) equations.Among these, the transition models are at present considered to be the most practical from an engineering point of view, owing to their cost-effectiveness and robustness.Research on transition models has made great progress in recent years.The γ-Reθtransition model10,14is currently-one of the most widely used transition models in engineering because it is compatible with unstructured grids and largescale parallel calculations.At the heart of this model are two additional transport equations.One is for the intermittency factor, designed to locally initiate the transition.The other is for a transition onset criterion, used to capture the nonlocal influence of turbulence intensity and pressure gradient.

Owing to the incomplete understanding of the physical processes involved in transition, some closure parameters are inevitably introduced in the construction of transition models.These parameters are calibrated by simple flows before being extended to complex flows.Consequently,the closure parameters increase the uncertainty in computational results.The γ-Reθmodel is no exception,with a total of eight closure parameters.The CFD Vision 2030 Study15stated that the absence of Uncertainty Quantification(UQ)has been a primary barrier to current CFD and highlighted the urgency to integrate UQ techniques into modern CFD software platforms.Therefore,the influence of model parameters should be considered in numerical simulation of transitions.Uncertainty quantification and sensitivity analysis of the model parameters are critical in improving the credibility of simulation results, as well as providing useful guidance for engineering applications and further research.

To the best of the authors’knowledge,the first study of the parametric uncertainty of RANS models was carried out by Godfrey and Cliff.16They ranked the sensitivity of the parameters in the Baldwin-Lomax, Spalart-Allmaras, and Wilcox’s k-ω turbulence model using a continuous sensitivity equation.Based on their work, Turgeon, et al.17–18concentrated on the parameters in the k-ε turbulence model.However, only bands of uncertainty intervals are yielded with continuous sensitivity equation.To obtain richer information, further studies on uncertainty were carried out.Platteeuw, et al.19determined the prior distribution of closure parameters for the k-ε turbulence model based on physical constraints combined with experimental data and DNS results,and explored the influence of the closure parameters using the Probabilistic Collocation(PC) method.Emory, et al.20conducted research on uncertainty in the k-ω Shear-Stress Transport (SST) model.Pecnik,et al.21–22evaluated the uncertainty of the γ-Reθtransition model in transonic flows of turbine blades.Hosder, et al.23demonstrated that Nonintrusive Polynomial Chaos (NIPC)method is able to obtain uncertainty and sensitivity information consistent with the traditional Monte Carlo(MC)method at a significantly reduced computational cost.Schaefer,et al.24evaluated the sensitivity of parameters in popularly employed turbulence models based on the NIPC method.An uncertainty study of aeroheating in Mars entry was undertaken by Wang25, Zhao26, et al.carried out a UQ analysis of transition models in hypersonic transitional aeroheating environments.Cheung, et al.27proposed a Bayesian uncertainty quantification method.Edeling, et al.28and Parish and Duraisamy29used the Bayesian method to study uncertainty in turbulence models.Zhang and Fu30proposed an efficient Bayesian method, applied it to UQ of the k-ω-γ transition model, and subsequently obtained the posterior distribution of the SST model parameters using a Bayesian inference procedure31.

From the above review, it can be seen that there are few parametric uncertainty studies of the γ-Reθtransition model,especially for flows around airfoils.Therefore, in this paper,an uncertainty and sensitivity analysis of γ-Reθmodel is carried out for boundary layer transition prediction for flat-plate and airfoil flows.First,the uncertainties of the Quantities of Interests (QoIs), such as the skin friction coefficient, the transition onset and length,and the separation bubble size are quantified.Then,the contributions of the closure parameters to the uncertainties are ranked by Sobol indices to identify the key parameters.To reduce the sample size, the point-collocation NIPC approach is utilized to represent and propagate the uncertainty.

The remainder of this paper is laid out as follows.Section 2 briefly introduces the governing equations of numerical method and the γ-Reθtransition model.The NIPC method and the Sobol index are described in Section 3.Section 4 gives the computational details, including freestream conditions,grid-independent validation, and uncertainty quantification details.Section 5 presents the results and discussion.Section 6 sums up the main conclusions.

2.Numerical methods

In this work, all the cases are simulated by an in-house solver called MI-CFD, developed by the authors.Its robustness and accuracy have been proven with a large number of complex flows.13,32–34The three-dimensional RANS equations are solved by the finite volume method on structured grids.The main algorithms of the code are shown below.

2.1.Governing equation

The RANS equations are obtained by relating the Reynolds stress and turbulent fluxes with the mean-flow variables using the Boussinesq eddy-viscosity hypothesis.It is worth mentioning that in the compressible case, Favre averaging is applied rather than Reynolds averaging.The conservation of mass equation is given as

2.2.γ-Reθtransition model

The γ-Reθtransition model gained widespread application and refinement10,14,35–39since it was first proposed by Menter,et al.40in 2002.The detailed construction and correlations were published in 2009,14which are consistent with the model description below.The method is based on the SST model41with additional transport equations for the intermittency factor γ and the local momentum thickness Reynolds number Reθt.The transport equations for γ and Reθtare given as

where S is the strain-rate magnitude,Flengthis an empirical correlation that controls the transition length, and Fonsetcontrols the transition onset location by turning on the intermittency factor production.At positions where the local vorticity Reynolds number Revexceeds the local transition onset criterion,Fonsetswitches from zero in a laminar boundary layer to one rapidly.Fonsetis constructed as a function of Revin the form described in Ref.14The destruction term Eγof γ is defined as

where Ω is the vorticity magnitude and Fturbis required to invalidate Eγoutside the laminar boundary layer or in the viscous sublayer.The transport equation for ～Reθtis essential as it relates the empirical correlation to the transition onset function, which hinges on the source termPθt, expressed as

3.Uncertainty method

Briefly, UQ is used to evaluate the effects of input parameters(e.g., geometric parameters, incoming flow parameters, and model parameters) on the output QoIs in a given system.The PC method is able to reduce the sample size significantly compared to conventional methods.The point-collocation NIPC method employed in this work is based on the PC method and improves the computational efficiency of the multidimensional uncertainty analysis without modifying the internal CFD code.The central idea of NIPC is to decompose the random response function α*into separable deterministic and stochastic parts42as

where x is the deterministic independent variable vector, ζ is the n-dimensional standard random variable vector, αiis the deterministic component,and ψiis the ith mode basis function.It should be noted that theoretically, the series in Eq.(15) is infinite, but in practice, it is cut off and discretely summed.43The number of the truncated terms is computed by26

where n is the number of random variables, p is the order of the Polynomial Chaos Expansion (PCE), and npis the oversampling ratio.The key to the NIPC method is the solution of the deterministic component αi.The point-collocation method firstly replaces the stochastic response function with the PCE,followed by a deterministic CFD evaluation at collocation points to obtain the left-hand term of Eq.(15),and eventually builds a linear system of equations and solves it for αi.The linear system of equations is shown as

When the oversampling ratio npis greater than 1,Eq.(17)is overdetermined and solved by the least-squares method.Once αiis available, the uncertain information of QoIs can be analyzed,including the mean μα*,the total variance δ,and the sensitivity parameters.The mean μα*,the total variance δ,and the standard deviation σ can be expressed as the terms of the PCE:

To describe the total contribution of the closure parameters, the Sobol index in the current work,STi, is the sum of the Sobol indices of a specific closure parameter:

In other words,STicontains the cross contribution between the parameters.For instance, when n=3, the total contribution of the first parameter (i=1) is expressed as

Therefore,the sum of the Sobol indices of all parameters in this paper exceeds 1.For comparison purposes, the Sobol index of each parameter is normalized in the following.

4.Computational details

4.1.Freestream conditions and grid validation

In this work,an uncertainty analysis of the γ-Reθmodel is carried out on natural and bypass transitional flows over zeropressure-gradient flat plates, and on subsonic and transonic flows around airfoils.The flat plate cases are the Schubauer and Klebanoff (S&K) flat plate,46and the T3A flat plate.The airfoil cases are the NLF0416 incompressible airfoil and the NLR7301 transonic airfoil.Their inlet flow conditions are displayed in Table 1 and Table 2.

4.1.1.S&K flat plate

The S&Kflat plate46has a low freestream turbulence intensity,so it is usually applied to verify the capability of transition models to simulate natural transition.Table 3 exhibits the three grids generated to analyze the grid convergence.The skin friction coefficients Cfof these three grids and the experimental results are presented in Fig.1.The results for the three grids are already very close, with the results of the medium and fine grids almost identical.This reveals that the medium grid has obtained convergence, which is selected for computation tosave computational effort, as shown in Fig.2.It is also found that the predicted transition onset of the γ-Reθmodel is generally consistent with experimental data, but the transition length is shorter than that in experiment.The reason may lie in the fact that the default value of ca1is too large for the S&K flat plate, and when transition onset is determined by Fonsetin Eq.(7), the intermittency factor γ reaches 1 rapidly,resulting in the short transition length.

Table 1 Freestream conditions for flat plates.

Table 2 Freestream conditions for airfoils.

4.1.2.T3A flat plate

The T3A flat plate is one of the European Research Community On Flow Turbulence And Combustion(ERCOFTAC)T3 series of plate experiments47,48.It has a zero streamwise pressure gradient with a freestream turbulence intensity of3.3 %, which is a standard test case for bypass transition.From the comparison between numerical and experimental Cfin Fig.3, it can be observed that the numerical result from the γ-Reθmodel is in good agreement with the experimental data.The grid of the T3A flat plate used in this paper is similar to that of the S&K, so the grid convergence test will not be repeated.The number of normal-wise nodes in the grid is 151, and the number of streamwise nodes is 701.

Table 3 Computational grids for S&K flat plate.

Fig.1 Grid convergence analysis of S&K flat plate.

Fig.2 Grid of S&K flat plate.

Table 4 Computational grids for NLF0416 airfoil.

4.1.3.NLF0416 airfoil

The NLF0416 airfoil is a high-lift and low-drag airfoil,characterized by a large range of laminar flow of greater than 30 %chord merely through favorable pressure gradients.It was originally designed for wings in general aviation, but it was later successfully applied to wind turbine blades49,50.Experiments were performed in the NASA Langley Tunnel50.The three grids used for grid convergence test are summarized in Table 4.The dimensionless heights of the first-layer grids y+are less than 1, and the transition region is additionally encrypted to better capture the transition position.Fig.4 presents Cfcalculated by these grids and experimental transition onset.It can be seen that the results for the medium and fine grids are nearly coincident, while a significant difference can be observed for the coarse grid.Therefore, the medium grid has achieved grid convergence and has been chosen as the computational grid,which is presented in Fig.5.The transition onset predicted by the γ-Reθmodel is further downstream in comparison with experimental results.

4.1.4.NLR7301 airfoil

Fig.3 Comparison of skin friction coefficients Cffor T3A flat plate.

Fig.4 Grid convergence analysis of NLF0416 airfoil.

Fig.5 Grid of NLF0416 airfoil.

The NLR7301 airfoil is a classical supercritical airfoil with a blunt leading edge.Extensive wind tunnel experiments51performed on it provide reliable data for numerical verification.Fig.6 displays Cfcomputed by the γ-Reθmodel and the experimental transition onset.From the Cfdistribution, both the upper and lower surfaces of the NLR7301 airfoil have separation zones,and the transitions occur in these separation zones.Compared with the experimental data,the transition onset predicted by the γ-Reθmodel is matched for the lower surface and slightly advanced for the upper surface.The grid of the NLR7301 airfoil is similar to that of NLF0416, and the grid convergence verification is not repeated.The grid is shown in Fig.7, and the dimensionless heights of the first-layer grids y+are less than 1.

4.2.Uncertainty quantification details

Fig.6 Comparison of calculated skin friction coefficient Cfand experimental transition onset for NLR7301 airfoil.

Fig.7 Grid of NLR7301 airfoil.

The closure parameters of the γ-Reθtransition model have been specifically analyzed in Section 3.As pointed out above,the role of ce1is to control the highest value of the intermittency factor γ, which is equal to 1, and thus ce1is not considered.Although there are separation flows in the NLR7301 airfoil and s1is to control the separation bubble size, it is verified that s1has little effect on transition location,as shown in Fig.8.Hence, to reduce the computational cost,s1is not chosen.A total of six parameters are selected for uncertainty analysis in this work,namely,ca1,ca2,ce2,σf,cθt,and σθt.It is assumed that all the parameters have the same uncertainty interval of±30%,and the standard values and variation intervals are tabulated in Table 5.

Hosder,et al.52proved that satisfactory calculation results can be obtained when the oversampling ratio np=2, which is taken for sampling in the subsequent work.Multidimensional Legendre polynomials of order two (p=2) serve as the basis functions ψiowing to the boundedness of the closure parameters.Six closure parameters are analyzed (n=6), so the number of samples Ntis equal to 56 according to Eq.(16).Latin Hypercube Sampling (LHS)24is employed to obtain the optimal collocation points.

Fig.8 Comparison of Cfobtained by s1upper and lower limits.

5.Results and discussion

In this section, the four computational cases (S&K, T3A,NLF0416, and NLR7301) demonstrated above are used to quantify the uncertainty caused by the closure parameters of the γ-Reθtransition model, focusing on QoIs such as the skin friction coefficientCf,the transition onset,the length of transition region, and the separation bubble size.

5.1.S&K flat plate

Fig.9 shows the distribution of Cfversus x for the S&K flat plate, including the results of 56 samples (training cases) and the mean μα*with a 95 % C.I.It is worth noting that the dispersion of the training cases near the transition region is significant.To display the impact of the closure parameters on uncertainty more directly, Fig.10 shows the uncertainty of Cfalong the x-axis, with the benchmark result as a reference.From Fig.10, the uncertainty in the transition zone is great,indicating that transition is highly sensitive to closure parameters.By contrast, the uncertainty in the fully laminar or turbulent flow zone is extremely small or even close to zero.This reflects the validity of the γ-Reθmodel construction.Strikingly, the maximum uncertainty of Cfis as high as 200 %,which fully illustrates the importance of UQ analysis.The largest UQ is at x=0.726 m, where the training cases are in a complex state, including not starting transition, undergoing transition, and ending transition.The second peak of UQ is 169 % at x=0.664 m, where transition has just started.

Table 5 γ-Reθclosure parameters and corresponding variation intervals.

Fig.9 Cfdistribution of S&K flat plate.

At present,there are different criteria for the determination of transition onset.In the current work, the point where Cfdeviates from the laminar flow value to a certain threshold is considered as transition onset, and the point where Cfpeaks is considered as transition termination.The transition length is defined as the distance between transition onset and termination.Fig.11 shows that the UQ of transition length is slightly larger than that of transition onset, with values of 28.099%and 20.419%, respectively.

To further quantify the relative effects, Sobol indices are used to identify the sensitivity of the closure parameters.A parameter with a larger Sobol index makes a larger contribution to the total uncertainty.The Sobol indices for Cfare shown in Fig.12,and the values of benchmark Cfare also displayed to identify the significant closure parameters at particular points or regions.The contributions of parameters in the transition zone to uncertainty are ranked from the highest to the lowest as ce2,ca2,ca1,σθt,cθt,σf.The Sobol indices fluctuate in the transitional zone, showing a reciprocal pattern.It is obvious that ce2,ca2, and ca1are dominant contributors, all of which are parameters in the γ transport equation.Consistent conclusions can be drawn from the Sobol indices of the transition onset and length in Fig.13:ce2,ca2, and ca1have a great influence, while the contributions of σθt,cθt, σfare almost negligible.Therefore,from the present results,it is the parameters of the source terms Pγand Eγin the γ transport equation that contribute most to the uncertainty of natural transition: these are ce2,ca2, and ca1.

Fig.10 UQ of Cffor S&K plate.

Fig.11 UQs of transition onset and length for S&K plate.

5.2.T3A flat plate

Plots of the distribution of Cffor the T3A flat plate and their mean value μα*with a 95%C.I.are presented in Fig.14.It can be seen that the dispersion at transition termination is higher than that at transition onset.The distribution of UQ in Cfalong the x-axis is shown in Fig.15.Compared with the results for the S&K plate, the maximum UQ of the T3A flat plate is much smaller, about 24.4 %, at x=0.64 m.Again, this is a location that contains a variety of states,including undergoing and ending transitions.As can be observed in Fig.16, the uncertainty of transition length is about twice as great as that of transition onset (22.796 % and 10.191 %, respectively),which is consistent with the phenomenon in Fig.14.Compared with the results for the S&K plate,the uncertainty of transition length is not much different, but the uncertainty of transition onset for T3A flat plate is reduced by half.This illustrates that the γ-Reθmodel parameters are less sensitive to bypass transition onset than to natural transition.

Fig.12 Sobol indices of Cffor S&K flat plate.

Fig.13 Sobol indices for transition onset and length of S&K flat plate.

Fig.14 Cfdistribution for T3A flat plate.

Fig.15 UQ of Cffor T3A flat plate.

The Sobol indices for Cfof the T3A flat plate are presented in Fig.17.In most areas, ce2and ca2account for the largest proportion of uncertainty.What is striking is that the Sobol indices of ce2and ca2decrease rapidly near transition onset,while those of σθt, cθtincrease.The original ce2- and ca2-dominated state is then restored.To explain the results,we first recall the view of Abu-Ghannam and Shaw53,who suggest that transition onset is mainly influenced by the history of the pressure gradient and turbulence intensity rather than by the local values at transition.For the T3A plate, the turbulence intensity changes rapidly, and history effects are remarkable.In the γ-Reθmodel, σθtcontrols the lag between the local ～Reθtin the boundary layer and that in the freestream,thus controlling history effects10, which deciphers the sudden increase in the effect of σθtat transition onset.Another interesting finding is thatca1, which plays a key role in the Sobol indices of S&K,has little impact here.

Fig.16 UQs of transition onset and length for T3A plate.

From the histogram of the Sobol indices of the transition onset length in Fig.18,it can be concluded that the larger contributor to the uncertainty of transition onset is σθt, which is the parameter in the ～Reθttransport equation,and to transition length is ce2and ca2, which are parameters in the dissipation term Eγof the γ transport equation.This is in agreement with the distribution of Sobol indices of Cf.

5.3.NLF0416 airfoil

First, the baseline results for the NLF0416 airfoil are presented.The dimensionless pressure contour of NLF0416 airfoil is shown in Fig.19.For the upper surface, between 0.05c and 0.5c in the streamwise direction,a large area of low pressure is generated as a means of providing high lift.As can be seen from the comparison in Fig.20,the computed surface pressure coefficient Cpmatches well with the experimental result.and σθtincrease.The situation for the upper surface is slightly different,with ce2,ca2,and ca1remaining dominant,but with σθtalso becoming an important parameter in the first half of the transition region.

Fig.17 Sobol indices of Cffor T3A flat plate.

Fig.18 Sobol indices of transition onset and length for T3A flat plate.

Fig.19 Dimensionless pressure contour of NLF0416 airfoil.

Fig.20 Computed pressure distribution of NLF0416 airfoil compared with experimental data.

Fig.21 NLF0416 airfoil intermittency factor contours of baseline result with transition onset locations marked.

The Sobol indices with respect to transition onset and length are presented in Fig.25.For transition onset (see Fig.25(a)),the three most influential parameters for both surfaces are ce2,ca2,andca1.σθtis also an important parameter for the upper surface,which is consistent with the fact that σθthas a greater influence on Cfin the first half of the transition zone.For the transition length (see Fig.25(b)), the most influential parameter for the upper surface is ce2, followed by σθt.The Sobol indices of the remaining parameters are close,at around 0.12.For the lower surface,the three parameters that have the greatest influence on transition length are ca2,σθt, and ce2(in descending order).

5.4.NLR7301 airfoil

The baseline results for the NLR7301 airfoil are described below.Fig.26 shows the dimensionless pressure contour of the NLR7301 airfoil.It can be seen that the pressure rises steeply from 0.6 to 1.1 around x=0.61c on the upper surface and x=0.52c on the lower surface, indicating the presence of shock waves.Owing to the strong adverse pressure gradient induced by the shock, flow separation occurs in the boundary layer.Fig.27 shows the local enlargement and streamtraces of the separation zone.Cppredicted by the γ-Reθmodel is underpredicted at shock positions compared with the experimental result (see Fig.28).This discrepancy may arise from the approximations in the eddy-viscosity turbulence modeling,and consequently, the model fails to correctly predict separation bubble size behind the shock54.

From the intermittency factor contour of the NLR7301 baseline result (see Fig.29), it can be seen that the transition onset of the upper surface is at x=0.57c and that of the lower surface is at x=0.49c.Combined with the range of the separation region in Fig.27, it can be concluded that both upper and lower surface transitions occur in the separation zone.

Fig.22 Cfdistribution of NLF0416 airfoil.

Fig.23 UQs of transition onset and length for NLF0416 airfoil.

The Cfdistribution and the mean value with 95 % C.I.for the NLR7301 airfoil are plotted in Fig.30.The uncertainty of Cfin most areas of the upper surface is tiny.Only at the initial drop of Cf(x=0.43 m)and around the vortex core of the separation zone (x=0.6 m) does the variation of the closure parameters have any effect on Cf.On the lower surface,the larger uncertainties are still at the position of the initial descent of Cf(x=0.4 m) and around the vortex core in the separation zone (x=0.52 m).However, the magnitude of the overall uncertainty on the lower surface is greater than that on the upper surface, and the uncertainty on the lower surface remains evident in the fully turbulent zone.

The UQs of the transition onset and length are plotted in Fig.31.The UQ of transition onset on the upper surface is zero, indicating that the transition on the upper surface of the NLR7301 airfoil starts at the same position for all training cases and is not affected by changes in the closure parameters.The UQ of transition length on the upper surface is 1.532%.The respective UQs of 2.881% and 10.825% for transition onset and length on the lower surface are much greater than those on the upper surface.

As can be seen in Fig.32,the Sobol indices concerning Cfof the NLR7301 fluctuate sharply when Cfis less than 0.This fluctuation precedes transition onset and starts once Cfis below 0.Therefore, this fluctuation is related to flow separation rather than transition,which suggests that flow separation has a great influence on the sensitivity of the γ-Reθmodel parameters.In the other regions,the distributions of the Sobol indices are very similar to those of the S&K flat plate, with ce2,ca2, and ca1contributing the most.

Since the UQ of transition onset is zero for the upper surface, only the Sobol indices of transition length are analyzed for the upper surface.For the lower surface, where the transition ends at almost the same position, the majority of the UQ of transition length comes from the difference in transition onset.Therefore, only the Sobol indices for transition length on the lower surface are given, and the Sobol indices for transition onset are similar.The Sobol indices with respect to transition length are presented in Fig.33.For the upper surface,ca1and ca2dominate, followed by ce2.For the lower surface, the three parameters with the greatest contributions are, from the largest to the smallest,ca1,σθt, and ce2.

Fig.24 Sobol indices of Cffor NLF0416 airfoil.

Fig.25 Sobol indices of transition onset and length for NLF0416 airfoil.

Fig.27 Local enlargement and streamtraces of separation zone for NLR7301 airfoil.

For the NLR7301 airfoil,in addition to Cf,transition onset and length of transition region, the separation bubble size is also an important QoI.The separation bubble size is defined as the distance between the x-coordinates of two points where Cfis equal to 0.The UQ of the separation bubble is 1.913 %for the upper surface and 23.936 % for the lower surface, as shown in Fig.34.The Sobol indices of the closure parameters are shown in Fig.35.The parameters that contribute the most to the uncertainty in separation bubble size for both surfaces are ca2,ce2, and ca1, and the distribution of contributions among the parameters for the lower surface is more even.

Fig.28 Computed pressure distribution of NLR7301 airfoil compared with experimental data.

Fig.29 NLR7301 airfoil intermittency factor contours of baseline result with transition onset locations marked.

Fig.30 Cfdistribution of NLR7301 airfoil.

Fig.31 UQs of transition onset and length for NLR7301 airfoil.

5.5.Discussion

If the closure parameters with Sobol indices greater than 0.1 are assumed significant, all closure parameters significant to QoIs are included in Table 6.The parameters are listed from top to bottom in order of contribution.The parameters in bold type are significant for every QoI in the same case,and the last row is a summary of these.For clarity, overlapping images of the parameters significant to each case are presented in Fig.36,which shows the parameters significant to both the S&K and T3A flat plates, both the NLF0416 and NLR7301 airfoils,and the γ-Reθmodel.Table 7 ranks the significant parameters based on the number of appearances for the 12 QoIs in Table 6,and in order of contribution when the number is equal.

Fig.32 Sobol indices of Cffor NLR7301 airfoil.

Fig.33 Sobol indices of transition length for NLR7301 airfoil.

€As can be seen, ce2and ca2are the two most significant parameters, affecting all QoIs for all cases.The next isca1,which also has a greater influence for the cases except for the T3A flat plate.ce2,ca2, and ca1are all closure parameters of source terms in the γ transport equation, indicating that this equation is more critical than the ～Reθttransport equation.For the T3A flat plate and NLF0416 airfoil,σθtis a larger contributor.This is due to the modeling of the flow history effects,which is explained particularly in Section 5.2.Compared with the situation in which several parameters dominate in the flat plate cases, the Sobol indices of parameters for the airfoils are more complex, and the differences between contributions of parameters are less distinct.We suggest that priority should be given to ce2and ca2when using the γ-Reθmodel in future refinements.

Fig.34 UQs of separation bubble size for NLR7301 airfoil.

Fig.35 Sobol indices of separation bubble size for NLR7301 airfoil.

6.Conclusions

Fig.36 Closure parameters significant to all QoIs of computational cases.

In this paper, uncertainty quantification and sensitivity analysis of the closure parameters in the γ-Reθmodel have been carried out on four computational cases of typical flat plates and airfoils.A second-order stochastic expansion based on the point-collocation NIPC method has been employed to represent and propagate the uncertainties of Quantities of Interests(QoIs), including the skin friction coefficientCf, transitiononset, length of transition region, and separation bubble size.Meanwhile, the relative contributions of closure parameters to uncertainty have been evaluated using Sobol indices.For each computational model, calculations have been performed for 56 training cases to obtain the uncertainty results.The main conclusions are drawn as follows:

Table 6 Closure parameters significant to QoIs of all cases.

Table 7 Number of appearances of each significant closure parameter for QoIs.

(1) The UQ results indicate that variations of the γ-Reθmodel parameters have significant effects on boundary layer transition on flat plates and smaller effects on airfoils in this work.

(2) As revealed by the Sobol indices of Cffor the NLR7301 airfoil,flow separation can cause dramatic fluctuation in the distribution of the Sobol indices,which poses a challenge to the γ-Reθmodel with regard to predicting transition accurately.

(3) In most areas,ce2,ca2,and ca1have the dominant role for the uncertainty of Cf.But at times,the influence of σθtis significant, and the influence of ca1decreases correspondingly,especially when the history effects of transition are evident.

(4) For flat plates, the Sobol indices are usually dominated by a few parameters, with the remaining parameters contributing little.For airfoils, the sensitivity results are more complex,and the differences among the contributions of the parameters are not obvious.

(5) Generally, ce2and ca2make the largest contributions to the uncertainty of QoIs, followed by ca1.These are all parameters in the γ transport equation, indicating the importance of this equation.

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.

Acknowledgements

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