Analyzing natural convection in porous enclosure with polynomial chaos expansions: Effect of thermal dispersion, anisotropic permeability and heterogeneity

Fajraoui, Noura; Fahs, Marwan; Younès, Anis; Sudret, Bruno

doi:10.1016/j.ijheatmasstransfer.2017.07.003

Cited by 33 publications

(36 citation statements)

References 114 publications

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“…Thus in the recent years, there has been a growing interest in analysis of the effects of stochastic variations in the inputs on the outputs. There are multiple examples in the literature in which the uncertainty propagation techniques are combined with the deterministic numerical simulations [12][13][14][15][16][17][18][19][20][21][22][23]. It is popular to use the polynomial chaos expansion (PCE) for uncertainty prop-agation in which the output is approximated as a summation of polynomial basis which are functions of the stochastic inputs.…”

Section: Introductionmentioning

confidence: 99%

“…Uncertainty quantification (UQ) for various types of natural convection problems has been studied in the literature [15][16][17]20]. Maitre et al [15] used the zero-Mach-number model to simulate natural convection in a two dimensional differentially heated square cavity with uncertainty in the cold wall temperature.…”

Section: Introductionmentioning

confidence: 99%

“…It is found that the stochastic wall temperatures can extend the stability range of quasi-conduction states beyond the classical bifurcation point. Fajraoui et al [20] analyzed the natural convection of porous media in a two dimensional differentially heated square cavity with uncertainty in the Rayleigh number, permeability anisotropy ratio, dispersion coefficients and heterogeneity variation. PCE method is used to estimate the statistics and sensitivity of the output parameters such as temperature and Nusselt number distributions.…”

Section: Introductionmentioning

confidence: 99%

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Uncertainty quantification in three dimensional natural convection using polynomial chaos expansion and deep neural networks

Shahane

Aluru

Vanka

2019

International Journal of Heat and Mass Transfer

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This paper analyzes the effects of input uncertainties on the outputs of a three dimensional natural convection problem in a differentially heated cubical enclosure. Two different cases are considered for parameter uncertainty propagation and global sensitivity analysis. In case A, stochastic variation is introduced in the two non-dimensional parameters (Rayleigh and Prandtl numbers) with an assumption that the boundary temperature is uniform. Being a two dimensional stochastic problem, the polynomial chaos expansion (PCE) method is used as a surrogate model. Case B deals with non-uniform stochasticity in the boundary temperature. Instead of the traditional Gaussian process model with the Karhunen-Loève expansion, a novel approach is successfully implemented to model uncertainty in the boundary condition.The boundary is divided into multiple domains and the temperature imposed on each domain is assumed to be an independent and identically distributed (i.i.d) random variable. Deep neural networks are trained with the boundary 1 Corresponding Author temperatures as inputs and Nusselt number, internal temperature or velocities as outputs. The number of domains which is essentially the stochastic dimension is 4, 8, 16 or 32.Rigorous training and testing process shows that the neural network is able to approximate the outputs to a reasonable accuracy. For a high stochastic dimension such as 32, it is computationally expensive to fit the PCE. This paper demonstrates a novel way of using the deep neural network as a surrogate modeling method for uncertainty quantification with the number of simulations much fewer than that required for fitting the PCE, thus, saving the computational cost.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Uncertainty quantification in three dimensional natural convection using polynomial chaos expansion and deep neural networks

Shahane

Aluru

Vanka

2019

International Journal of Heat and Mass Transfer

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“…Use of deterministic simulations alone to analyze the engineering systems is incomplete due to the lack of precisely defined input data. Thus, there has been a growing interest [29][30][31][32][33][34][35][36][37][38] in coupling uncertainty propagation techniques with the deterministic numerical simulations to estimate the effects of stochastic variations in the input process parameters on the outputs. The polynomial chaos expansion is a popular method used to estimate the relation between input and output parameters.…”

Section: Introductionmentioning

confidence: 99%

Finite volume simulation framework for die casting with uncertainty quantification

Shahane

Aluru

Ferreira

et al. 2019

Applied Mathematical Modelling

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The present paper describes the development of a novel and comprehensive computational framework to simulate solidification problems in materials processing, specifically casting processes. Heat transfer, solidification and fluid flow due to natural convection are modeled. Empirical relations are used to estimate the microstructure parameters and mechanical properties. The fractional step algorithm is modified to deal with the numerical aspects of solidification by suitably altering the coefficients in the discretized equation to simulate selectively only in the liquid and mushy zones. This brings significant computational speed up as the simulation proceeds. Complex domains are represented by unstructured hexahedral elements. The algebraic multigrid method, blended with a Krylov subspace solver is used to accelerate convergence. State of the art uncertainty quantification technique is included in the framework to incorporate the effects of stochastic variations in the input parameters. Rigorous validation is presented using published 1 Corresponding Author

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“…Heterogeneity introduces terms involving the space derivative of the permeability that require specific permeability-depth relationships to be addressed analytically. This was achieved by considering a stratified domain characterized via the Gardner exponential permeability function [63], which is widely used in layered aquifers, e.g., [64][65][66][67][68][69][70]. Solving the flow and transport equations in the spectral space is a challenging task, especially when large numbers of Fourier modes should be used to obtain oscillation-free solutions.…”

Section: Introductionmentioning

confidence: 99%

A Generalized Semi-Analytical Solution for the Dispersive Henry Problem: Effect of Stratification and Anisotropy on Seawater Intrusion

Fahs

Koohbor

Belfort

et al. 2018

Water

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The Henry problem (HP) continues to play a useful role in theoretical and practical studies related to seawater intrusion (SWI) into coastal aquifers. The popularity of this problem is attributed to its simplicity and precision to the existence of semi-analytical (SA) solutions. The first SA solution has been developed for a high uniform diffusion coefficient. Several further studies have contributed more realistic solutions with lower diffusion coefficients or velocity-dependent dispersion. All the existing SA solutions are limited to homogenous and isotropic domains. This work attempts to improve the realism of the SA solution of the dispersive HP by extending it to heterogeneous and anisotropic coastal aquifers. The solution is obtained using the Fourier series method. A special hydraulic conductivity-depth model describing stratified heterogeneity is used for mathematical convenience. An efficient technique is developed to solve the flow and transport equations in the spectral space. With this technique, we show that the HP can be solved in the spectral space with the salt concentration as primary unknown. Several examples are generated, and the SA solutions are compared against an in-house finite element code. The results provide high-quality data assessed by quantitative indicators that can be effectively used for code verification in realistic configurations of heterogeneity and anisotropy. The SA solution is used to explain contradictory results stated in the previous works about the effect of anisotropy on the saltwater wedge. It is also used to investigate the combined influence of stratification and anisotropy on relevant metrics characterizing SWI. At a constant gravity number, anisotropy leads to landward migration of the saltwater wedge, more intense saltwater flux, a wider mixing zone and shallower groundwater discharge zone to the sea. The influence of stratified heterogeneity is more pronounced in highly anisotropic aquifers. The stratification rate and anisotropy have complementary effects on all SWI metrics, except for the depth of the discharge zone.

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Analyzing natural convection in porous enclosure with polynomial chaos expansions: Effect of thermal dispersion, anisotropic permeability and heterogeneity

Cited by 33 publications

References 114 publications

Uncertainty quantification in three dimensional natural convection using polynomial chaos expansion and deep neural networks

Uncertainty quantification in three dimensional natural convection using polynomial chaos expansion and deep neural networks

Finite volume simulation framework for die casting with uncertainty quantification

A Generalized Semi-Analytical Solution for the Dispersive Henry Problem: Effect of Stratification and Anisotropy on Seawater Intrusion

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