Sparse grid-based polynomial chaos expansion for aerodynamics of an airfoil with uncertainties

Xiaojing WU,Weiwei ZHANG,Shufang SONG,Zhengyin YE

School of Aeronautics,Northwestern Polytechnical University,Xi’an 710072,China

1.Introduction

A vast amount of uncertainties exist in the practical aircraft design and application,which can cause fluctuations of aircraft performance.Therefore,it is important to take these uncertainties into account at the beginning of aircraft design.1,2Recently,many researches have concerned the topics.3–8The Uncertainty quantification(UQ)and uncertainty sensitivity analysis of aerodynamics are concerned in the paper.

Computational Fluid Dynamics(CFD)technology has been widely used to solve problems in fluid mechanics with the development of computer technology.Traditional CFD simulation is deterministic.However,a variety of uncertainties are inevitably introduced into CFD simulation due to the increasing complexity of the problems.This leads to the mismatch between the results of CFD simulation and the actual results.The UQ in CFD simulation has gained extensive attention in Ref.9.Sources and classifications of uncertainty in CFD were described in Ref.10.Several UQ strategies have been used in CFD,including Monte Carlo Simulation(MCS)method,moment method and Polynomial Chaos(PC)in Ref.11.MCS is a statistical method,which needs many samples to accurately quantify uncertainty.Moment methods are suitable to solve the problem of small parameter uncertainty or linear model.Recently,PC which is based on the spectral representation of the uncertainty has been adopted in UQ for fluid problems in Refs.12,13.PC methods can be divided into intrusive and non-intrusive ones according to the coupling ways with CFD solvers.In general,an intrusive approach is adopted to obtain unknown polynomial coefficients by projecting resulting equations into basis functions for different modes,and it requires the modification of CFD codes,which may be difficult and time-consuming for complex problems such as Navier–Stokes equations.To overcome the shortcomings of intrusive polynomial chaos,Non-Intrusive Polynomial Chaos(NIPC)has been developed.The CFD is regarded as a black box model without changing the CFD program codes in non-intrusive methods.There are two different sampling approaches to build NIPC:random sampling and deterministic sampling.Random sampling methods use MCS to evaluate the unknown coefficients,but their convergence rate is low.Deterministic sampling methods use the quadrature to evaluate the unknown coefficients.The quadrature-based methods are more efficient than random sampling methods for low-dimensional problems.However,they are inefficient for relatively high-dimensional problems because of the exponential rising of quadrature points with the increasing dimensions.The UQ based on PC and its applications in fluid mechanics were comprehensively reviewed in Refs.14,15.

Recently,the NIPC is sufficiently used for stochastic aerodynamic analysis with operational uncertainties.A subsonic aerodynamic analysis was conducted on a NACA0012 airfoil with an uncertain free stream velocity using a commercial flow solver in Ref.16.They proved that an uncertain free stream velocity led to the highest variation in pressure on the upper surface near the leading edge.Transonic stochastic response of two-dimensional airfoil to parameter uncertainty(Mach number Ma and angle of attack α)is focused using generalized Polynomial Chaos(gPC)in Ref.8.Two kinds of non-linearities are critical to transonic aerodynamics in their study:the shock characteristics and boundary-layer separation.A stochastic investigation of flows about NACA0012 airfoil was conducted at transonic speeds in Ref.17.A point-collocation NIPC method was used to quantify uncertainty of aerodynamic characteristics with uncertain variables Ma and α in the transonic-wing case in Ref.18.A stochastic fluid analysis on 3D wind blades considering the wind speed as an uncertain parameter was conducted in Ref.19.In their studies,when the flow separation appears,the separation vortex region corresponds to the maximum variation area which extends to the trailing edge and even to the whole suction side.However,the stochastic aerodynamic analysis considering geometric uncertainty was rarely involved.The geometric uncertainty on an aerodynamic surface resulting from manufacturing errors has significant effect on the aerodynamic performance.It is impractical to remove the impact of these geometric variations by improving the manufacturing tolerance due to the high cost of the precise surface manufacturing technique.In other words,the geometric uncertainty due to manufacturing errors is unavoidable.Therefore,it is important and necessary to conduct a stochastic aerodynamic analysis considering geometric uncertainty.Nevertheless,the description of the geometric uncertainty is difficult in a computing environment and many random variables are needed to represent aerodynamic surface fluctuations.Therefore,the quadrature-based NIPC method is inefficient in solving this high-dimensional stochastic problem.

To improve the computational efficiency of traditional quadrature-based NIPC method for high-dimensional problems,the sparse grid numerical integration is introduced to solve the coefficients of PC.The sparse grid technique extends one-dimensional formulae to higher dimensions by tensor product and then selects sampling points according to Smolyak theory in Ref.20.It has been widely used in numerical integration and interpolation21,22as well as data mining.23Recently,a new sparse grid-based method has been developed for UQ in Ref.24.From their research,it can be known that when the dimension of the uncertain variables is larger than 5,the computational cost required by the sparse grid method is much smaller than that required by tensor product method.A sparse grid interpolant was developed to solve the high-dimensional stochastic partial different equations in Ref.25.Sparse grid collocation schemes were applied to stochastic natural convection problems in Ref.26.Sparse grids-based stochastic approximations with applications to subsonic steady flow about a NACA0015 airfoil in the presence of geometrical and operational uncertainties with both simplified aerodynamics model and Reynolds-Averaged Navier–Stokes(RANS)simulation was presented in Ref.27.

In this paper,a Sparse Grid-based Polynomial Chaos(SGPC)method is constructed to UQ and sensitivity analysis for transonic aerodynamics considering airfoil geometric and operational uncertainties in detail.The paper is structured as follows:Section 2 introduces the mathematical formulations of the sparse grid technique;in Section 3,the SGPC is built;in Section 4,a stochastic aerodynamic analysis considering geometric and operational uncertainties is conducted in detail;Section 5 outlines several useful conclusions.

2.Sparse grid numerical integration

Sparse grid technique selects sampling points under Smolyak theory,which uses a weighted linear combination of special tensor products to reduce the grid size.The sparse grid has been successfully used for numerical integration.Locations and weights of the univariate quadrature points with a range of accuracy are determined in each dimension,and then the univariate quadrature point sets are extended to form a multi-dimensional grid using the sparse grid theory.The introduction of Sparse Grid Numerical Integration(SGNI)is as follows:

where⊗denotes the tensor product rule.

Quadrature points of multivariate integrals are all possible combinations of one-dimensional quadrature nodes(Nfull=nm1nm2···nmn).Thus,it is time-consuming for a problem with relatively high dimensions by the tensor product algorithm.

Since the full tensor product is inefficient for high dimensions,Smolyak theory is adopted because it can reduce the grid size with a weight linear combination of special tensor products.For the n-dimensional sampling points with p-level accuracy generated by the sparse grid technique,the tensor product rule based on the sparse grid is shown as

where|m|=m1+m2+ ···+mn.The bounded sum makes sure that tensor products exclude from full grids points that contribute less to the improvement of the required integration accuracy.The integration of multivariate nonlinear functionin terms of variable ξ by sparse grid can be computed by

where Psis the sum of all possible combinations of the multiple indices that satisfy q=p+n,n is the number of random dimensions,and ξ is the selected collocation points by sparse grid method.The weight alcorresponding to the lth collocation point is computed by

Table 1 shows the change of sparse grid collocation points(Nsg)from 2 to 10 dimensions with the 2-order accuracy Gaussian quadrature(Ntp).In a low-dimensional problem,with the same level p,the sparse grid approach needs more collocation points than the full grid approach.When n increases(e.g.n≥5),the number of collocation points of the sparse grid method is much smaller than that of the full grid.It can be observed that with the same polynomial exactness p the number of collocation points produced by the full grid scheme is(p+1)nwhich increases exponentially with dimensionality,whereas the sparse grid technique remarkably reduces the number of collocation points.

3.Sparse grid-based polynomial chaos

PC has been widely used in UQ,but the method is inefficient for high-dimensional problems.Therefore,the SGPC method is constructed to alleviate the computational burden for relatively high-dimensional problems.The method is introduced in detail in this section.

PC is a stochastic method based on the spectral representation of uncertainty.According to the spectral representation,the random function can be decomposed into deterministic and stochastic components.For example,a random variable(X)can be represented by

where αj(x)is the deterministic component and Ψj(ξ)is the random basis function corresponding to the jth mode.From Eq.(6),the random variable X is the function of the deterministic independent variable vector x and the n-dimensional standard random variable vector ξ =(ξ1，ξ2，···，ξn).PC expansion given by Eq.(6)should contain an in finite number of terms.In a practical computational context,PC is truncated in both order p and dimension n.The number of truncated terms is finite as follows:

The random basis function Ψjis chosen according to the type of the input random variable.For example,if the input uncertainty obeys Gauss distribution,the basis function is the multidimensional Hermite polynomial.If ξ is chosen to be uniform with the random variable,the basis function must be the Legendre polynomial.A complete description of gPC scheme is introduced in Ref.13.

The purpose of the PC method is to obtain unknown polynomial coefficients.Eq.(6)can be transformed to Eq.(8)by the inner product:

Table 1 Number of collocation points with different dimensions by 2-order accuracy of Gaussian quadrature(tensor product and sparse grid).

〈·〉represents the inner product which can be expressed by

Because of orthogonality,Eq.(8)can be transformed to

And then it can be derived:

There are two ways to solve the coefficients:intrusive and non-intrusive methods.The intrusive method computes unknown polynomial coefficients by projecting the resulting equations into basis functions for different modes.It requires the modification of the deterministic codes,which may be difficult,expensive,and time-consuming for complex computational problems.The non-intrusive method treats the CFD as a black box without changing the program code when propagating uncertainty.Steps of the non-intrusive method are as follows:

Step 1.Adopting relevant sampling methods to produce sample

Step 2.For each sample ξj,evaluate the uncertainty parameter λj.

Step 3.Use the selected N samples to determine the expansion coefficients by Galerkin projection.

Step 4.Given the computed coefficient αk,a polynomial approximation model can be built∑

Sampling approaches can be divided into random and deterministic sampling methods in Step 1.Random sampling methods use MCS to compute projection integrals,where the convergence rate is low.Deterministic sampling methods use the numerical quadrature to evaluate unknown coefficients.Using n-dimensional Gauss quadrature with q points in each dimension,the unknown coefficients can be computed by

One-dimensional integral is expanded into a highdimensional integral by tensor product in Eq.(12),the calculation times of which are qn(requiring(p+1)npoints for pth order chaos).For low-dimensional problems,the efficiency of the deterministic sampling method has been greatly improved compared with that of the random sampling method.However,the computational cost grows exponentially with the increasing dimensions.Thus,the NIPC method is inefficient in high-dimensional problems.Such shortcomings motivate us to improve NIPC method.If we use much less integral points than the conventional tensor product to solve the multi-dimensional integral Eq.(8),the computational cost of UQ can be reduced.It is well known that the SGNI can improve the computational efficiency to solve the multidimensional integral Eq.(8).From the description in Section 2,the sparse grid method is more efficient than the tensor product for high-dimensional integration(e.g.n≥5).Hence,we use the SGNI to replace the tensor product to solve Eq.(8)as follows:

4.Stochastic aerodynamic analysis

A stochastic aerodynamic analysis is conducted considering airfoil geometric and operational uncertainties.The operational uncertainty generally contains two parameters:Ma and α.However,the relatively high-dimensional random parameters are needed to represent aerodynamic surface fluctuations.In this section,SGPC and MCS methods are applied to the stochastic transonic aerodynamics analysis around a RAE2822 airfoil.Firstly,the deterministic flow computations are briefly introduced.

4.1.Deterministic flow computations

The deterministic steady-state flow solutions are computed by the RANS equations or method combined with the Spalart–Allmaras turbulence model.The cell-centered finite volume method is used for spatial discretization and AUSM-up scheme is used to evaluate the numerical flux.The implicit dual-time stepping method is used for temporal discretization.The symmetric Gauss–Seidel iterative time-marching scheme is applied in the pseudo-time step,and the second order accurate full implicit scheme is used to solve the equation in the physical time step.The developed flow solver is also used for unsteady flows.28,29

The computational mesh surrounding a RAE2822 airfoil is based on a structured C-grid.The 2D mesh is composed of two blocks,the sizes of which are 300×150(C block surrounding the airfoil)and 200×150.The chord of the airfoil is c=1 m and the far- field boundary is placed at a distance d=20c from the airfoil.Fig.1 shows the computing grids of RAE2822 airfoil and the reliability verification of the CFD program,in which Cpis the pressure coefficient.The geometry of airfoil keeps changing in the process of UQ,and Radial Basis Function(RBF)is used to mesh deformation.30

4.2.Geometric uncertainty

In aerospace engineering,despite advances of manufacturing engineering techniques,airfoilsvery often exhibitsome deviation from their intended shape due to the noisy manufacturing process.Moreover,it is often impractical to remove the impact of these geometric variations by improving the manufacturing tolerance due to the high cost of precise surface manufacturing techniques.In other words,the geometric uncertainty resulting from manufacturing errors is unavoidable.How to describe this geometric uncertainty in computing environment should be concerned.

Main geometric variation modes are obtained by Principal Component Analysis(PCA)with a large amount of geometric statistical measurement data of an airfoil in Refs.31,32.The description of the airfoil geometric uncertainty by PCA is shown as

where gnis the nominal geometry;¯g is the average geometric variation;viis the geometric mode shape;nsis the number of mode shapes used to represent the variation in geometry.The geometric mode can be computed by PCA based on measurement samples.σiis the ith singular value of the measurement snapshot matrix,which represents the geometric variability attributable to the ith mode.ziis a random parameter which obeys the standard normal distribution,and thus the product σiziis the stochastic contribution of the ith mode.

It is difficult to describe this geometric variation in the computing environment.A Gaussian random process simulation is used to obtain the geometric data in Ref.33.Then PCA is used to obtain main geometric modes.In addition,a variety of parametric methods have been employed to describe the geometric variation in aerodynamic design so far,such as PARSEC-11 geometry parameterization,34Class-Shape function Transformation(CST)method,35Free-Form Deformation(FFD)method,36and Hicks-Henne bump functions.37,38By changing the parameter of these parametric methods,the geometric variation is realized.Generally,these parametric methods need a lot of parameters to represent the airfoil shape.To reduce the dimensions of the variables,the PCA technology combined with airfoil parameterization is used to describe the geometric variation39,40used in the paper.Firstly,a parametric method is used to generate a set of sample data,and then the PCA is used to obtain the main deformation modes based on the generated sample data.In this paper,we use a parameterized representation of the airfoil ARE2822 by CST method with 12 parameters.The data of the measurement points on the airfoil surface are obtained by random perturbation of CST parameters.In this way,the PCA based sample data are conducted.Fig.2 shows the eigenvalue with the number of modes, which represents the geometric variation attributable to each mode.The smaller the eigenvalue of one mode is,the smaller the proportion of the mode to geometric variation is.The first 6 modes obtained by PCA are showed in Fig.3.It can be observed that these modes are global geometric deformation modes and present some typical geometric deformation.specifically,Mode 1 and Mode 2 are the scale modes in the thickness direction;Mode 3 and Mode 4 are translation modes of the maximum thickness in the axial direction;Mode 5 and Mode 6 are the extrusion modes of the upper surface.

4.3.Stochastic transonic aerodynamics analysis

In the paper,the uncertainties are specified by means of PDFs.The input uncertainties,which PDFs should obey,accord to the corresponding statistical information,which are found in the methods of statistical inference.In the current study,the uncertain geometric variables are zi(i=1,2,...,6)in Eq.(14)which obey the standard normal distribution according to Ref.26.The operational uncertainties(Ma and α)obey truncated Gaussian distribution.The Mach number has a 0.73 mean value and a±0.01 variability,and the angle of attack has a 2.5°mean value and a± 0.3°variability.The steady flow state is selected in a transonic region(Ma=0.73,α=2.5°,Re=3.0× 106),and the range of geometric uncertainty is showed in Fig.4.

4.3.1.Verification of SGPC

Table 2 Results of uncertainty quantification of aerodynamic coefficients.

Table 3 Comparison of CFD calls among selected methods.

A convergence study of SGPC has been performed by the stochastic transonic aerodynamic analysis.The polynomial order p is increased to enhance the accuracy of SGPC.Moreover,MCS is also used in this analysis,which aims to validate the accuracy and efficiency of SGPC.The number of CFD calls of MCS is 5000.

Figs.5 and 6 show the standard deviations of the pressure coefficient Cpdistribution and skin-friction coefficient Cfdistribution along the chord and they also re flect the convergence of SGPC method.It can be seen that the standard deviation distributions of Cpand Cfcoincide well with the results of MCS when SGPC reaches the 3-order accuracy.Table 2 also gives the convergence of SGPC for lift coefficient CLand drag coefficient CD.It can be observed that when p reaches 2-order accuracy,the Standard Deviation(StD)of aerodynamic coefficients is convergent and coincides well with the results of MCS.The convergence study of SGPC verifies the accuracy and correctness of SGPC to the stochastic aerodynamic analysis.

The computational cost of SGPC should also be concerned for the stochastic transonic aerodynamic analysis.From Section 3,it can be learned that the computational cost of SGPC is related with parameters n and p.Table 3 shows computational costs of SGPC,NIPC and MCS for the 8-dimensional problem.It can be observed that SGPC is more efficient than the other two.It must be emphasized here that NIPC method is not used to the stochastic aerodynamic analysis because of the prohibitive computational cost.The CFD calls of NIPC is obtained according to the formula(p+1)n.

4.3.2.Uncertainty quantification of transonic aerodynamics

Now,the in fluence of input uncertainties on transonic flows is studied.Fig.7 shows the mean and fluctuations of Cpon the airfoil surface.It is well known that the shock wave exists in the transonic flow.The flow characteristics before and after the shock wave change dramatically with the input uncertainties.Therefore,it can be observed that a strong spurious effect appears near the shock wave foot.Fig.8 shows the mean and fluctuation of Cfon the airfoil surface.The skin-friction behavior displays discrepancies from the pressure coefficient distribution.The variation of Cfis similar to that of Cpbefore and in the shock disturbance region.However,the difference occurs in the downstream region of shock wave;the standard deviation has larger magnitudes below the shock disturbance region,and then it gradually decreases.This indicates that the flow characteristics not only near the shock wave but also in the boundary-layer separation are sensitive to geometric and operational uncertainties.

Moreover,Figs.7 and 8 show the corresponding local PDF pro files at the selected five chord locations.It is well known that once the mean position of the shock wave is constrained by the separated shear layer,the probability density function of the solution may exhibit a bifurcation corresponding to a jump in the solution.Therefore,the stochastic response shows a bimodal nature at x/c=0.56.

4.3.3.Global sensitivity analysis of transonic aerodynamics

Global Sensitivity Analysis(GSA)should be applied to study how uncertainty of the model output can be apportioned to different sources of uncertainty of the model input.A sensitivity analysis is performed by using the Sobol’s analysis,which analyzes the individual and coupled effects between the random parameters(a deep introduction of the Sobol’s analysis can be referred to in Ref.41).When a PC model is built,the Sobol’s analysis based on the PC can be directly calculated from the expansion coefficients.41In the paper,the aerodynamic load distribution and aerodynamic coefficients are included for Sobol’s analysis.

Figs.9 and 10 show the results of Sobol’s analysis of Cpand Cf.The partial standard deviation and standard deviation caused by the coupling effect for each uncertainty variable are given.It can be observed that Ma is important to the flow characteristics both in the vicinity of the shock wave and in the boundary-layer separation.Now,we concern the contribution of the geometric deformation modes.It can be observed that deformation modes of the upper surface are much important than those of the lower surface to aerodynamic load distribution.Specifically,the scale mode(Mode 1)and extrusion mode(Mode 5)are important to the flow characteristic of shock wave and boundary-layer separation,but the translation mode(Mode 3)is only important to the flow characteristic of shock wave.Moreover,it can also be observed that the coupling effect among geometric deformation modes is not negligible near the shock wave.

Figs.11 and 12 show the results of Sobol’s analysis of aerodynamic coefficients.It can be observed that α is the most important to lift coefficients.To drag characteristic,Ma is the most important;geometric modes of the upper surface and angle of attack also have obvious influences.

4.4.Stochastic aerodynamic investigation considering geometric uncertainty in different flow regimes

A further stochastic aerodynamic investigation is conducted in different flow regimes.Three flow states are selected:Case A,subsonic flow with a small angle of attack(Ma=0.5,α=2.5°);Case B,subsonic flow with a large angle of attack(Ma=0.5,α =20°);and Case C,transonic flow with a small angle of attack(Ma=0.73, α =2.5°).SGPC with 3-order accuracy is used in the stochastic aerodynamic analysis.At first,the three kinds of flow states have been employed in the steady flow computation,respectively.The streamline of the flow field is shown in Fig.13.It can be observed that the flow is smooth and there is no separation along the airfoil in Case A.There is a separation vortex above the upper surface of the airfoil in Case B.In Case C,the separation appears on the aft part of the airfoil because of the boundary layer separation.

Fig.14 shows error bars of pressure and friction coefficient distribution of RAE2822 airfoil in the three states.In case A,variations of Cpand Cfare the same,and the maximum fluctuations mainly lie in the upper surface near the leading edge.In Case B,we can also observe that the maximum fluctuations mainly display on the upper surface near the leading edge.Fluctuations in other regions are even smaller than those in Case A.This indicates that flow characteristics in the separation region are intensive to this geometric uncertainty.In Case C,variation regularities of Cpand Cfare different.For Cp,the variation mainly appears in the shock wave disturbance region,and for Cf,besides this region,it also exists in the boundary layer separation region.

Table 4 shows uncertainty analysis results of aerodynamic coefficients.The Cov of CDis relatively small for Case A and Case B while it is relatively large for Case C.The Cov of CLis small for all the three cases.This means that subsonic aerodynamic coefficients are not sensitive to the geometric uncertainty.The drag is sensitive to the geometric uncertainty,but the lift is not in the transonic region.

Table 4 Uncertainty quantification of aerodynamic coefficients in three different flow states.

5.Conclusions

In the paper,the sparse grid-based polynomial chaos expansion was used for UQ and GSA for transonic aerodynamics with geometric and operational uncertainties.The method is much more efficient than NIPC and MCS in dealing with relatively high-dimensional stochastic aerodynamic problems.The accuracy and efficiency of the SGPC are verified by the stochastic transonic aerodynamic analysis.It is proved that the method is more efficient than NIPC and MCS.It is observed that the flow characteristics in the vicinity of the shock wave and the boundary-layer separation region are sensitive to uncertainties because of the nonlinear flow characteristics of shock wave and boundary-layer separation.By uncertainty sensitivity analysis(Sobol’s analysis),it can be learnt that Ma is important to the flow characteristics both in the vicinity of the shock wave and in the boundary-layer separation;the geometric deformation modes of the upper surface are much important than those of the lower surface to aerodynamic load distribution.Specifically,the scale mode(Mode 1)and extrusion mode(Mode 5)are important to the flow characteristic of shock wave and boundary-layer separation,but the translation mode(Mode 3)is only important to the flow characteristic of shock wave.The coupling effect is not negligible in the vicinity of the shock wave foot.

Moreover,a stochastic aerodynamic investigation considering geometric uncertainty is conducted by the SGPC in different flow states.It can be concluded that fluctuations of subsonic aerodynamic characteristics are mainly concentrated in the airfoil head.When the flow is in the transonic condition,the affected region shifts from the upper surface near the leading edge to the shock wave and boundary-layer separation region because of the nonlinear characteristics of the shock wave and boundary-layer separation behind the shock wave.

Acknowledgements

This work was supported by the National Natural Science Foundation of China(No.11572252),the ‘111” Project of China(No.B17037),and the National Science Fund for Excellent Young Scholars(No.11622220).

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