Uncertainty and Sensitivity Analysis for Reentry Flows with Inherent and Model-Form Uncertainties

Abstract

The objective of this paper is to introduce a computationally efficient methodology for the quantification of mixed (inherent and model-form) uncertainties and global sensitivity analysis (SA) in hypersonic reentry flow computations. The uncertainty-quantification (UQ) approach is based on the second-order UQ theory, using a stochastic response surface obtained with nonintrusive polynomial chaos. The global nonlinear SA is based on Sobol variance decomposition, which uses polynomial chaos expansions. The methodology was used to quantify the uncertainty and sensitivity information for surface heat flux to the spherical nonablating heat shield of a reentry vehicle at an angle of attack of 0 deg. Three uncertainty sources were treated in computational fluid dynamics simulations: inherent uncertainty in the freestream velocity, model-form uncertainty in the recombination efficiency used in partially catalytic wall-boundary condition, and model-form uncertainty in the binary-collision integrals. The SA showed that the velocity and recombination efficiency were the major contributors to the heat-flux uncertainty for the reentry case considered. The UQ and SA were performed with three different levels of input uncertainty in velocity, which revealed the importance of characterizing the velocity with well-defined uncertainty levels in the study of reentry flows because the variations in this quantity can drastically impact the accuracy of the heat-flux prediction. Nomenclature C = mass fraction CoV = coefficient of variation, D 1=2 = D = statistical variance H = enthalpy, J h = enthalpy, J=kg h D = dissociation enthalpy, J=kg k = multiplicative factor Le = Lewis number n = number of random variables p = pressure, N=m 2 _ q = heat transfer rate, W=m 2 R N = radius of curvature, m S = Sobol index S T = total Sobol indices T = temperature, K V = velocity, m=s x = lateral direction of the flow = spectral modes = stochastic output variable = recombination efficiency h f = heat of formation, J=kg = coefficient of viscosity, kg=m s = mean = standard random variable a = standard aleatory uncertain variable e = standard epistemic uncertain variable = density, kg=m 3 = random basis function 1;1 = diffusion collision integral 2;2 = viscosity collision integral Subscripts e = boundary layer edge o = total or stagnation condition sp = stagnation point w = wall 1 = freestream

Multifidelity Uncertainty Quantification of a Commercial Supersonic Transport

“…Upper and lower bounds of these intervals can be drawn from limited experimental data or from expert predictions and judgment. 18,19 An additional, special case of epistemic uncertainty is numerical error. This uncertainty is common in numerical modeling and is defined as a recognizable deficiency in any phase or activity of modeling and simulations that is not due to lack of knowledge of the physical system.…”

“…While an intrusive method may appear straightforward in theory, for complex problems this process may be time consuming, expensive, and difficult to implement as changing to the deterministic model are required. 18 In contrast, the non-intrusive approach can be easily implemented to construct a surrogate model that represents a complex computational simulation, because no modification to the deterministic model is required. The non-intrusive methods require only the response (or sensitivity) [23][24][25] values at selected sample points to approximate the stochastic response surface.…”

Robust Airfoil Optimization and the Importance of Appropriately Representing Uncertainty

“…Polynomial chaos (which has been used successfully many times in robust optimization [9,[24][25][26]) is an attractive propagation technique partly because it creates a response surface approximation to the actual system output, which can be used to propagate different types of uncertainties [9,11]. It obtains a spectral expansion of the output of a random process (which in the robust optimization case is the QOI, Y, at a given x x x: x x x design ) with respect to the uncertain input variable vector u u u = u 1 ,..., u n , which is truncated to M terms for practical implementation:…”

“…From the perspective of accurately quantifying the uncertainty on the QOI, it is important to represent these input uncertainties appropriately, as one might expect. Studies into uncertainty quantification in aerospace applications [9,10] illustrate that erroneous output uncertainty levels are predicted if all input uncertainties are assumed to be aleatory, leading to the development of methods for mixed interval and probabilistic uncertainty quantification techniques [11,12]. However, different quantifications of output uncertainty do not necessarily mean different outcomes of a robust optimization, and so in this paper we present an investigation into the effect of uncertainty representation on the results of robust airfoil shape optimizations.…”