High-Order Methods for Incompressible Fluid Flow

Deville, Michel; Fischer, Paul; Mund, E. H.

doi:10.1017/cbo9780511546792

Cited by 703 publications

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“…The side wall is thermally insulated. We apply a spectral element method and use the Nek5000 software package (23)(24)(25) to reproduce the gradients of the turbulent fields with high accuracy. On each of the 875,520 elements in runs RB1 and RB2 that cover the cylindrical convection cell, a spectral expansion of polynomial order N = 11 is applied in all three space directions.…”

Section: Numerical Modelmentioning

confidence: 99%

Enhanced enstrophy generation for turbulent convection in low-Prandtl-number fluids

Schumacher

Götzfried

Scheel

2015

Proc. Natl. Acad. Sci. U.S.A.

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Turbulent convection is often present in liquids with a kinematic viscosity much smaller than the diffusivity of the temperature. Here we reveal why these convection flows obey a much stronger level of fluid turbulence than those in which kinematic viscosity and thermal diffusivity are the same; i.e., the Prandtl number Pr is unity. We compare turbulent convection in air at Pr = 0.7 and in liquid mercury at Pr = 0.021. In this comparison the Prandtl number at constant Grashof number Gr is varied, rather than at constant Rayleigh number Ra as usually done. Our simulations demonstrate that the turbulent Kolmogorov-like cascade is extended both at the large-and small-scale ends with decreasing Pr. The kinetic energy injection into the flow takes place over the whole cascade range. In contrast to convection in air, the kinetic energy injection rate is particularly enhanced for liquid mercury for all scales larger than the characteristic width of thermal plumes. As a consequence, mean values and fluctuations of the local strain rates are increased, which in turn results in significantly enhanced enstrophy production by vortex stretching. The normalized distributions of enstrophy production in the bulk and the ratio of the principal strain rates are found to agree for both Prs. Despite the different energy injection mechanisms, the principal strain rates also agree with those in homogeneous isotropic turbulence conducted at the same Reynolds numbers as for the convection flows. Our results have thus interesting implications for small-scale turbulence modeling of liquid metal convection in astrophysical and technological applications. T urbulent convection depends strongly on the material properties of the working fluid that are quantified by the Prandtl number, the ratio of kinematic viscosity of the fluid to thermal diffusivity of the temperature, Pr = ν=κ. Compared with the vast number of investigations at Pr ≥ 1 (1, 2), the very-low-Pr regime appears almost as a "terra incognita" despite many applications. Turbulent convection in the Sun is present at Prandtl numbers Pr < 10 −3 (3-5). The Prandtl number in the liquid metal core of the Earth is Pr ∼ 10 −2 (6). Convection in material processing (7), nuclear engineering (8), or liquid metal batteries (9) has Prandtl numbers between 3 × 10 −2 and 10 −3 . Rayleigh-Bénard convection (RBC), a fluid flow in a layer that is cooled from above and heated from below, is a paradigm for all of these examples. One reason for significantly fewer low-Pr RBC studies is that laboratory measurements have to be conducted in opaque liquid metals such as mercury or gallium at Pr = 0.021 (10-12). The lowest value for a Prandtl number that can be obtained in optically transparent fluids is Pr = 0.2 for binary gas mixtures (13), i.e., an order of magnitude larger than in liquid metals. Direct numerical simulations (DNS) are currently the only way to gain access to the full 3D convective turbulent fields in low-Pr convection (14-18). These simulations turn out to become very demanding if the...

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Section: Numerical Modelmentioning

confidence: 99%

Enhanced enstrophy generation for turbulent convection in low-Prandtl-number fluids

Schumacher

Götzfried

Scheel

2015

Proc. Natl. Acad. Sci. U.S.A.

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“…This report compares the results for three LES codes: Nek5000, developed at Argonne National Laboratory (USA), and Cabaret and Conv3D, developed at the Moscow Institute of Nuclear Energy Safety at (IBRAE) in Russia. Nek5000 is based on the spectral element method (SEM), which is a high-order weighted residual technique that combines the geometric flexibility of the finite element method (FEM) with the tensor-product efficiencies of spectral methods [20,6]. Cabaret is a "compact accurately boundary-adjusting high-resolution technique" for fluid dynamics simulation [15].…”

Section: Introductionmentioning

confidence: 99%

CFD validation in OECD/NEA t-junction benchmark.

Obabko¹,

Fischer²,

Tautges³

et al. 2011

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“…Although the spectral element method is widely used in computational sciences and engineerings (cf. [13,19]), its application to the computational finance is still limited. Its main advantage is that, for a given accuracy, much less grid points, and in many case much less computing times, are needed comparing to low order methods.…”

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confidence: 99%

A New Spectral Element Method for Pricing European Options Under the Black–Scholes and Merton Jump Diffusion Models

Feng

Shen

2011

J Sci Comput

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We present a new spectral element method for solving partial integro-differential equations for pricing European options under the Black-Scholes and Merton jump diffusion models. Our main contributions are: (i) using an optimal set of orthogonal polynomial bases to yield banded linear systems and to achieve spectral accuracy; (ii) using Laguerre functions for the approximations on the semi-infinite domain, to avoid the domain truncation; and (iii) deriving a rigorous proof of stability for the time discretizations of European put options under both the Black-Scholes model and the Merton jump diffusion model. The new method is flexible for handling different boundary conditions and non-smooth initial conditions for various contingent claims. Numerical examples are presented to demonstrate the efficiency and accuracy of the new method.

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High-Order Methods for Incompressible Fluid Flow

Cited by 703 publications

References 0 publications

Enhanced enstrophy generation for turbulent convection in low-Prandtl-number fluids

Enhanced enstrophy generation for turbulent convection in low-Prandtl-number fluids

CFD validation in OECD/NEA t-junction benchmark.

A New Spectral Element Method for Pricing European Options Under the Black–Scholes and Merton Jump Diffusion Models

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