A Bi-Fidelity Approach for Uncertainty Quantification of Heat Transfer in a Rectangular Ribbed Channel

Abstract

This paper presents a bi-fidelity simulation approach to quantify the effect of uncertainty in the thermal boundary condition on the heat transfer in a ribbed channel. A numerical test case is designed where a random heat flux at the wall of a rectangular channel is applied to mimic the unknown temperature distribution in a realistic application. To predict the temperature distribution and the associated uncertainty over the channel wall, the fluid flow is simulated using 2D periodic steady Reynolds-Averaged Navier-Stokes (RANS) equations. The goal of this study is then to illustrate that the cost of propagating the heat flux uncertainty may be significantly reduced when two RANS models with different levels of fidelity, one low (cheap to simulate) and one high (expensive to evaluate), are used. The low-fidelity model is employed to learn a reduced basis and an interpolation rule that can be used, along with a small number of high-fidelity model evaluations, to approximate the high-fidelity solution at arbitrary samples of heat flux. Here, the low- and high-fidelity models are, respectively, the one-equation Spalart-Allmaras and the two-equation shear stress transport k–ω models. To further reduce the computational cost, the Spalart-Allmaras model is simulated on a coarser spatial grid and the non-linear solver is terminated prior to the solution convergence. It is illustrated that the proposed bi-fidelity strategy accurately approximates the target high-fidelity solution at randomly selected samples of the uncertain heat flux.

Bi-fidelity approximation for uncertainty quantification and sensitivity analysis of irradiated particle-laden turbulence.

“…This error bound may be used with small cost to assess the suitability of a given pair of low-and high-fidelity models to produce an accurate BF approximation. Instances of successful BF approximation viaÛ H in (20) have been reported in a number of recent studies [41,42,43,44,45].…”

“…The recent work in [41,42,43,44,45] builds a reduced basis (or low-rank) approximation of the HF QoI using LF model evaluations and a small set of selected HF samples. The HF reduced basis -consisting of realizations of the HF (vector-valued) solution at selected input samples -as well as an interpolation rule in this basis are determined from LF realizations.…”

“…The BF approximation of this work follows the approach of [43,45,44], which seeks to identify a low-rank representation of the HF QoI assisted by the realizations of its LF counterpart. While the QoIs in this work are scalar-valued, this BF approximation relies on vector-valued quantities u that describe the scalar-valued QoI.…”

“…In particular, the seminal work in [38] builds a Gaussian process approximation of the QoI based on an autoregressive model trained from nested observations of multiple, less expensive models. Each model of the sequence is related to the lower-fidelity model via a multiplicative constant and an additive Gaussian process correction term that are estimated from the lower-fidelity model evaluations as well as fewer realizations of the model itself.The recent work in [41,42,43,44,45] builds a reduced basis (or low-rank) approximation of the HF QoI using LF model evaluations and a small set of selected HF samples. The HF reduced basis -consisting of realizations of the HF (vector-valued) solution at selected input samples -as well as an interpolation rule in this basis are determined from LF realizations.…”

A low-rank control variate for multilevel Monte Carlo simulation of high-dimensional uncertain systems

“…Cost of pilot run. Recall that the construction of Z is based on the identification of the reduced basis U c −1 (using which U c is generated) and the associated least squares coefficients c −1 computed from (17). Following Algorithm 1, the former is based on the ID of the coarse data matrix U −1 consisting of N p pilot samples of q −1 ∈ R m −1 .…”