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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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Computation of electromagnetic scattering with a non‐conforming discontinuous spectral element method

TL;DR: In this paper, a high-order quadrilateral discontinuous spectral element method (DSEM) is proposed to solve electromagnetic scattering problems by approximating Maxwell's equations in the time-domain with a highorder Quadrilateral Discriminative Spectral Element Method.
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The analysis and simulation of compressible turbulence

TL;DR: In this article, it is shown that even if the divergence of the initial velocity field is negligibly small, it can grow rapidly on a non-dimensional time scale which is the inverse of the fluctuating Mach number.
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Efficient Spectral-Galerkin Methods III: Polar and Cylindrical Geometries

TL;DR: Several extremely efficient and accurate spectral-Galerkin methods for second- and fourth-order equations in polar and cylindrical geometries based on appropriate variational formulations which incorporate naturally the pole condition(s).
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Sensitive dependence on initial conditions in transition to turbulence in pipe flow

TL;DR: In this article, the authors show that the transition to turbulence is connected with the formation of a chaotic saddle in the phase space of the system and quantify a sensitive dependence on initial conditions and find in a statistical analysis that the distribution of turbulent lifetimes follows an exponential law.
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A new triangular and tetrahedral basis for high-order (hp) finite element methods

TL;DR: The foundations of a new hierarchical modal basis suitable for high-order (hp) finite element discretizations on unstructured meshes is described, based on a generalized tensor product of mixed-weight Jacobi polynomials.
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.
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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.