Mixed uncertainty quantification and its application in upwind scheme for computational fluid dynamics (CFD)
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摘要: 针对流体力学数值求解间断问题时，初始状态含有偶然和认知混合型的不确定性，将认知不确定度作为外层，偶然不确定度作为内层，分别使用非嵌入多项式混沌方法(non-intrusive polynomial chaos, NIPC)和概率盒(P-box)理论处理偶然不确定度和认知不确定度，发展了流体力学数值求解过程中，初始状态含有混合不确定度传播量化的一种方法。以迎风格式和黎曼解法器求解Sod问题为例，评估了左状态密度(偶然不确定度)和理想气体多方指数(认知不确定度)对模型输出结果的影响，有效验证了该方法的可行性。Abstract: Both aleatory uncertainty and epistemic uncertainty exist in the initial and boundary conditions when we numerically solve the CFD with sharp discontinuity. In this paper, mixed uncertainty quantification approaches are developed to deal with this situation. Specifically, the outer level uncertainty is linked to epistemic uncertainties, and the inner uncertainty is linked to aleatory uncertainty. The non-intrusive polynomial chaos method is utilized to cope with the aleatory uncertainties, while the P-box theory is used to deal with the epistemic uncertainties, and the upwind scheme and Riemann solver are used to solve the deterministic system. We apply this method to the Sod problem in the CFD, and acquire preferable effect. This method evaluates the influence of input uncertainty such as density (aleatory uncertainty) and polytrophic exponent (epistemic uncertainty) on the output uncertainty, the efficiency of this method is also proved. This method is also helpful in evaluating the degree of confidence and validation of the result from modeling and simulation by other models.
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图 4 t=0.22时Sod问题的期望和方差
Figure 4. Expectation and variance for the Sod problem at t=0.22
图 5 2种方法计算Sod可信区间的比较
Figure 5. Comparison of confidential intervals from two computational methods in solving the Sod problem
图 6 密度函数的累积分布函数族
Figure 6. Cumulative distribution function ensemble of density function
图 7 利用迎风格式构造的密度函数的P-box
Figure 7. P-box constructed by the upwind scheme for density function
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