Towards Uncertainty Quantification of LES and URANS for the Buoyancy-Driven Mixing Process between Two Miscible Fluids—Differentially Heated Cavity of Aspect Ratio 4

Abstract

Numerical simulations are subject to uncertainties due to the imprecise knowledge of physical properties, model parameters, as well as initial and boundary conditions. The assessment of these uncertainties is required for some applications. In the field of Computational Fluid Dynamics (CFD), the reliable prediction of hydrogen distribution and pressure build-up in nuclear reactor containment after a severe reactor accident is a representative application where the assessment of these uncertainties is of essential importance. The inital and boundary conditions that significantly influence the present buoyancy-driven flow are subject to uncertainties. Therefore, the aim is to investigate the propagation of uncertainties in input parameters to the results variables. As a basis for the examination of a representative reactor test containment, the investigations are initially carried out using the Differentially Heated Cavity (DHC) of aspect ratio 4 with Ra=2×109 as a test case from the literature. This allows for gradual method development for guidelines to quantify the uncertainty of natural convection flows in large-scale industrial applications. A dual approach is applied, in which Large Eddy Simulation (LES) is used as reference for the Unsteady Reynolds-Averaged Navier–Stokes (URANS) computations. A methodology for the uncertainty quantification in engineering applications with a preceding mesh convergence study and sensitivity analysis is presented. By taking the LES as a reference, the results indicate that URANS is able to predict the underlying mixing process at Ra=2×109 and the variability of the result variables due to parameter uncertainties.

Uncertainty Quantification of a Industrial Scale CFD Application Using Stochastic Spectral Methods

“…For solving the integral within the Fredholm equation, the discrete Karhunen-Loève method [20] was employed along with a uniform discretization of the field domain with B points. The expectation of the random field R (x, ω) was determined by utilizing OLS to first approximate the random field using PCE and taking the respective coefficients of the first PCE term αβ (1) = {α 1 β (1) , . .…”

“…The expectation of the squared error or the mean square error (MSE) was estimated by initially approximating it through PCE with the available computational runs. The extraction of the respective coefficients of the first PCE term αβ (1) provide an estimate of its expectation, as already described analogously in Section 2.2.2: The square root of the estimated MSE yields an approximation of the root-mean-square error RMSE = √ MSE. As global measure, the temporal mean RMSE, denoted by RMSE , is determined through the square root of the integral mean of the MSE over the whole time span of the model ∆t = t end − t start .…”

“…The impact of these uncertainties on responses needs to be quantified. Therefore, by using a generic test case, different methods were initially developed and qualified as suitable for the application to engineering applications [1,2,3,4]. Stochastic spectral methods, such as PCE and KLE, were proven a promising approach and form the basis for the approximation of stochastic results in the present work.…”

CFD Uncertainty Quantification using PCE-HDMR – Exemplary Application to a Buoyancy-Driven Mixing Process

“…To maintain consistency in the temperature field, a parabolic profile towards the corners and edges is applied, see [16]. The initial helium stratification is changed by variation of the initial mole fraction difference with the coordinate υ = y − 2H/3:…”

“…In the present work, the aim of the UQ method development for the description of stochastic processes is to produce accurate and efficient HDMR models that can represent stochastic results effectively while minimizing the number of required computations. Therefore, the PCE-HDMR approach, which combines Cut-HDMR with PCE, is applied to an intricate dynamic CFD system for the first time and continues the work of Wenig et al [16][17][18][19] in the field of uncertainty quantification for buoyancy-driven mixing processes between two miscible fluids. The PCE-HDMR approach basically involves the construction of high-dimensional stochastic models through a number of low-dimensional submodels, which are built by PCE.…”