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Spectral Methods in Fluid Dynamics

TLDR
Spectral methods have been widely used in simulation of stability, transition, and turbulence as discussed by the authors, and their applications to both compressible and incompressible flows, to viscous as well as inviscid flows, and also to chemically reacting flows are surveyed.
Abstract
Fundamental aspects of spectral methods are introduced. Recent developments in spectral methods are reviewed with an emphasis on collocation techniques. Their applications to both compressible and incompressible flows, to viscous as well as inviscid flows, and also to chemically reacting flows are surveyed. The key role that these methods play in the simulation of stability, transition, and turbulence is brought out. A perspective is provided on some of the obstacles that prohibit a wider use of these methods, and how these obstacles are being overcome.

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Citations
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A spectral/finite difference method for simulating large deformations of heterogeneous, viscoelastic materials

TL;DR: In this paper, a numerical algorithm is presented that simulates large deformations of heterogeneous, viscoelastic materials in two dimensions, based on a spectral/finite difference method and uses the Eulerian formulation including objective derivatives of the stress tensor in the rheological equations.
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A numerical resolution study of high order essentially non-oscillatory schemes applied to incompressible flow

TL;DR: In this article, high-order essentially non-oscillatory (ENO) schemes are applied to incompressible Euler and Navier-Stokes equations with periodic boundary conditions.
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Solitary-wave solutions of the Benjamin equation

TL;DR: A constructive proof of the existence of these waves together with a proof of their stability is developed and Continuation methods are used to generate a scheme capable of numerically approximating these solitary waves.
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Discontinuous Galerkin methods for general-relativistic hydrodynamics: formulation and application to spherically symmetric spacetimes

TL;DR: In this article, the authors have developed the formalism necessary to employ the discontinuous-Galerkin approach in general-relativistic hydrodynamics, which is first presented in a general four-dimensional setting and then specialized to the case of spherical symmetry within a $3+1$ splitting of spacetime.
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Direct numerical simulation of autoignition in a non-premixed, turbulent medium

TL;DR: In this article, a pressure-based method was proposed to simulate autoignition in an initially non-premixed medium under isotropic, homogeneous, and decaying turbulence.
References
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Book

Navier-Stokes Equations

Roger Temam
TL;DR: Schiff's base dichloroacetamides having the formula OR2 PARALLEL HCCl2-C-N ANGLE R1 in which R1 is selected from the group consisting of alkenyl, alkyl, alkynyl and alkoxyalkyl; and R2 is selected by selecting R2 from the groups consisting of lower alkylimino, cyclohexenyl-1 and lower alkynyl substituted cycloenenyl -1 as discussed by the authors.
Journal ArticleDOI

A spectral element method for fluid dynamics: Laminar flow in a channel expansion

TL;DR: In this article, a spectral element method was proposed for numerical solution of the Navier-Stokes equations, where the computational domain is broken into a series of elements, and the velocity in each element is represented as a highorder Lagrangian interpolant through Chebyshev collocation points.
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Numerical Simulation of Turbulent Flows

TL;DR: In this article, the Navier-Stokes equations are used to model the evolution of a turbulent mixing layer and turbulent channel flow in incompressible Newtonian fluids. And the results of simulations of homogeneous turbulence in uniform shear are presented graphically and discussed graphically.
Journal ArticleDOI

Spectral methods for problems in complex geometries

TL;DR: In this paper, a new iteration procedure is introduced to solve the full matrix equations resulting from spectral approximations to nonconstant coefficient boundary-value problems in complex geometries, and the work required to solve these spectral equations exceeds that of solving the lowest-order finite-difference approximation to the same problem by only O(N log N).

Improved turbulence models based on large eddy simulation of homogeneous, incompressible turbulent flows

TL;DR: In this paper, a subgrid scale similarity model is developed that can account for system rotation and the main effect of rotation is to increase the transverse length scales in the rotation direction, and thereby decrease the rates of dissipation.