Abstract
This paper presents an effective approach for uncertain aerodynamic analysis of airfoils via the polynomial chaos expansion (PCE). To achieve this, the multivariate polynomial is first setup to represent random factors within the aerodynamic model, whereas the expansion coefficient is expressed as the multivariate stochastic integral of the input random vector. In this regard, the statistical regression in conjunction with a small number of representative samples is employed to determine the expansion coefficient. Then, a combination of the PCE surrogate model with brutal-force Monte Carlo simulation allows to determine numerical results for the uncertain aerodynamic analysis. Potential applications of this approach are first illustrated by the uncertainty analysis of the Helmholtz equation with spatially varied wave-number random field, and its effectiveness is further examined by the uncertain aerodynamic analysis of the NACA 63-215 airfoil. Results for the small regression error and a close agreement between simulated and benchmark results have confirmed numerical accuracy and efficiency of this approach. It, therefore, has a potential to deal with computationally demanding aerodynamical models for the uncertainty analysis.
Highlights
With the fast development of computer and simulation techniques, numerical methods for the aerodynamic analysis of airfoils have been received considerable attentions during past decades [1, 2]
With numerical simulation results for random field of the wavenumber function, the procedure summarized in Section 2.4 is used for uncertainty analysis of the Helmholtz equation. e multivariate Hermit polynomials are first determined to represent the Gaussian random variables produced by the K-L expansion, and the polynomial chaos expansion (PCE) model for stochastic response of the Helmholtz equation is determined as n− 1 η(X; u, v) ak(u, v)φk(X), (24)
To implement the uncertain aerodynamic analysis, the probabilistic characteristics of input random factors are listed in Table 2. is includes the inflow velocity V∞, the air density ρ, and the viscosity parameter ν. e PCE algorithm is realized for the uncertain analysis, and the brutal-force Monte Carlo simulation with 104 samples is assumed to provide the benchmark result for the numerical verification
Summary
With the fast development of computer and simulation techniques, numerical methods for the aerodynamic analysis of airfoils have been received considerable attentions during past decades [1, 2]. 2. Uncertainty Analysis via the Polynomial Expansion Method e aerodynamic response of an airfoil (e.g., the pressure and the velocity field and the lift or the drag coefficient) would become stochastic, if input random variables are considered in the model function η(X; u, v, w). Examples of the random variable include the average wind speed, the air density, and the viscosity parameters, as shown in numerical examples To account for this input uncertainty, a regression-based polynomial chaos expansion method is presented as follows. Note that numerical realization of such large number of high-dimensional integrals is a computationally intensive task, given that the deterministic aerodynamic response η(X; u, v, w) is possibly spatially modelled based on a finite element scheme In this regard, the brutal-force MCS method is replaced with the statistical regression method as follows. Corresponding numerical realizations of the chaos polynomial set φi(x)ni −01 would be Polynomial Hermit Legendre Jacobi Laguerre
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