Uncertainty quantification and global sensitivity analysis of double-diffusive natural convection in a porous enclosure

https://doi.org/10.1016/j.ijheatmasstransfer.2020.120291Get rights and content

Highlights

  • Double-diffusive convection in porous media is modeled under uncertainty.

  • An efficient strategy is developed to perform uncertainty and sensitivity analysis.

  • Stochastic and deterministic simulation outputs are compared.

  • Spatial maps of uncertainty and sensitivity indices are presented.

  • The parameters are ranked by order of influence on thermal and solute processes.

Abstract

In this paper, detailed uncertainty propagation analysis (UPA) and variance-based global sensitivity analysis (GSA) are performed on the widely adopted double-diffuse convection (DDC) benchmark problem of a square porous cavity with horizontal temperature and concentration gradients. The objective is to understand the impact of uncertainties related to model parameters on metrics characterizing flow, heat and mass transfer processes, and to derive spatial maps of uncertainty and sensitivity indices which can provide physical insights and a better understanding of DDC processes in porous media. DDC simulations are computationally expensive and UPA and GSA require large number of simulations, so an appropriate strategy is developed to reduce the computational burden. The approach is built on two pillars: (a) an efficient numerical simulator based on the Fourier series method that generates training data, and (b) polynomial chaos expansion (PCE) meta-models that are trained using the simulator data, and then replace the numerical model in UPA and GSA. Assuming that the Rayleigh number (Ra), the solutal to thermal buoyancy ratio (Nb) and the Lewis number (Le) are the uncertain input variables, the results of UPA show that the zones of high temperature and concentration variability are located in the regions where the flow is mainly driven by the buoyancy effects. GSA indicates that Nb is the most influential parameter affecting the temperature and concentration fields, followed respectively by Ra and Le. For the heat-driven flow case (Nb>1), the concentration field is more influenced by Le than Ra. For deeper understanding of uncertainty propagation, we estimate the bias introduced by replacing uncertain parameters by deterministic values. The resulting spatial maps of the difference between deterministic output and stochastic mean show that a deterministic approach leads to different zones where the temperature, concentration and velocity fields can be either overestimated or underestimated. The conclusions drawn in this work are likely to be helpful in different applications involving DDC in porous enclosures leading to convective circulation cells.

Introduction

Free convection or buoyancy driven flow caused by density variations in saturated porous media, has been studied extensively in the literature due to its wide applicability. These density variations may occur due to gradients in the fluid composition or temperature. Simultaneous occurrence of both gradients causes a type of flow known as double-diffuse convection (DDC), thermosolutal or thermohaline flow [36], [38]. Instances of DDC in porous media are widespread, ranging from large scale problems in geothermal engineering [29], CO2 sequestration [24], [25], nuclear waste disposal [21] and enhanced recovery of petroleum reservoirs [5], to small scale problems encountered in alloy solidification [8], fluidized beds and fuel cells [2]. DDC in porous media is often simulated by employing Darcy momentum conservation law with variable density, in conjunction with heat and mass transfer equations. Due to nonlinearity, DDC problems have no general analytical solutions and are often simulated using numerical models.

Use of numerical models has become widespread in the study and design of physical systems involving DDC in porous media. However, the structure and input parameters of these models are almost always prone to uncertainty resulting from simplifying assumptions, poor knowledge of underlying mechanisms, data insufficiency, natural stochasticity, etc. These sources of uncertainty propagate through the model and lead to uncertainty in model outputs. As stated by several previous studies (e.g. [35], [60]), the resulting model output uncertainties have negative effects on model reliability in practical applications, and this strengthens the need for a more in depth study of the critical issue of simulation under uncertainty. Addressing this issue involves at least two interrelated aspects: first, quantification of model output uncertainties resulting from the propagation of input uncertainties through the numerical model, and second, allocation of model output uncertainties to different sources of uncertainty in the model inputs [13], [40], [52]. The first aspect is known as uncertainty propagation analysis (UPA), and the second aspect requires global sensitivity analysis (GSA). GSA and UPA should be performed concurrently, as both are essential parts of the model development process in reliability analysis, robust design optimization, data-worth analysis, and risk assessment.

The majority of studies on DDC in porous media use a scenario-based approach to analyze the effect of inputs variations on model outputs (e.g. [16], [28], [44], [47], [48], [59]), and despite the importance of the subject, few studies have addressed formal UPA and GSA for various types of DDC problems in porous media, or even done so in the broader context of DDC in bulk fluids. Key studies in this regard are reviewed in Table 1. Le Maitre et al. [30] discussed different techniques of UPA for compressible flows. Ganapathysubramanian and Zabaras [17] analyzed the effects of random parameters on natural convection in bulk-fluids. The same problem has been addressed in Venturi et al. [55] by assuming perturbed boundary conditions. Based on Monte Carlo method. Shome et al. [51] performed UPA for mixed convection in a circular tube. Fajraoui et al. [13] conducted UPA for natural convection in porous media. A parameter sensitivity analysis is presented in Shirvan et al. [50] for heat transfer in a porous solar cavity receiver. Shahane et al. [46] used deep neural networks to perform UPA for natural convection in a 3D box cavity. This review shows that existing studies are limited to purely thermal natural convection. Most of these studies in both porous media and bulk fluids, investigated scalar variables characterizing the overall heat transfer flux and velocity predictions, such as the average Nusselt number and maximum velocities. Hence, information on the spatial variability of model output uncertainties and sensitivity indices are not available, though similar studies have been done in Fajraoui et al. [13]. So, there are gaps in the investigation of UPA and GSA for problems of DDC in porous media. This study tends to address these gaps by performing detailed UPA and variance-based GSA on the widely adopted DDC benchmark problem of a square porous cavity with horizontal temperature and concentration gradients. The objective is to understand the impact of uncertainties related to model parameters on metrics characterizing flow, heat and mass transfer processes, and to derive spatial maps of uncertainty and sensitivity indices which can provide physical insights and a better understanding of DDC processes in porous media.

Perhaps the most important hindrance in the way of UPA and GSA of DDC problems is the computational cost. Due to nonlinearity and high dimensionality of DDC models, the unit cost of a forward simulation could be very high, depending on the time and space scale. Even for small time and space scales, simulation of DDC at high Rayleigh numbers requires dense computational grid and small time steps size, which contribute to a high CPU time. GSA and UPA both involve repetitive simulations by the numerical model, and hence the large number of DDC simulations required for obtaining accurate solutions may become computationally infeasible. To overcome the computational challenge, we (a) employ a highly efficient simulator based on the Fourier series solution [47] which allows for accurate solution with reduced degree of freedom, and (b) use the numerical model to train and validate polynomial chaos expansions (PCEs). The PCEs then replace the numerical model in UPA and GSA computations [41].

The structure of the present study is as follows. In Section 2, we describe our DDC problem and the governing equations of the mathematical model used in its simulations. Section 3 describes the UPA and GSA procedure and how PCEs are used in this context. In Section 4 the results of the UPA and GSA are presented with regards to each of the output quantities of interest (QoIs) and physical insight are provided. Finally, the summary and conclusions are provided in Section 5.

Section snippets

Problem statement and assumptions

Numerical models of DDC in porous media have been used in many industrial and environmental problems under realistic configurations. However in most theoretical studies, DDC in porous media is studied using the hypothetical problem of a porous enclosure. It is a very popular benchmark for DDC numerical codes (see the review by Corcione et al. [9]). Part of the popularity of this benchmark problem stems from its simplicity in terms of geometry and boundary conditions, and the fact that there are

Uncertainty propagation analysis (UPA)

UPA involves quantification of mode output uncertainty resulting from the propagation of input uncertainties through the model [6]. Monte Carlo simulation (MCS) is the most popular method for UPA in computational fluid dynamics [40]. MCS error is in the order of (1/nMC)) where nMC is the number of Monte Carlo samples [7]. Hence achieving an acceptably small level of error often requires a large number of deterministic model simulations, which may be computationally difficult. A surrogate

Results and discussions

We assume that Ra, Le and Nb are the uncertain input variables subject to UPA and GSA. As common in literature (e.g. [27], [47], [48], [54]), we assume that the range of variability for Ra is from 10 to 500, and Le is from 1 to 5, using uniform probability distribution in both cases. Note that these values are physically plausible. For Nb<1, the convective flow is mostly solute-driven (termed as case ‘SD’) while Nb>1 typifies opposing flows that are largely heat-driven (called case ‘HD’).

Summary and conclusions

We consider DDC in a porous square cavity and we investigate the effects of uncertainties related to physical parameters on model outputs characterizing flow and heat and mass transfer processes. Darcy's law is used to describe fluid flow in the porous media. The hydrodynamics and thermo-physical parameters can be regrouped in three dimensionless parameters which are the thermal Rayleigh number (Ra), the buoyancy ratio (Nb) and the Lewis number (Le). These parameters are assumed to be

CRediT authorship contribution statement

Mohammad Mahdi Rajabi: Conceptualization, Methodology, Investigation, Writing - original draft. Marwan Fahs: Conceptualization, Methodology, Writing - original draft, Funding acquisition. Aref Panjehfouladgaran: Investigation. Behzad Ataie-Ashtiani: Writing - review & editing, Supervision. Craig T. Simmons: Writing - review & editing, Supervision. Benjamin Belfort: Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

Mohammad Mahdi Rajabi and Marwan Fahs would acknowledge the support from the National School of Water and Environmental Engineering of Strasbourg (PORO6100 Grant). Behzad Ataie-Ashtiani and Craig T. Simmons acknowledge support from the National Centre for Groundwater Research and Training, Australia. Behzad Ataie-Ashtiani also appreciates the support of the Research Office of the Sharif University of Technology, Iran. The data used in this work are available on the GitHub repository: //github.com/fahs-LHYGES

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