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.
About: This article is published in Journal of Computational Physics.The article was published on 1984-06-01. It has received 2133 citations till now. The article focuses on the topics: Spectral element method & Spectral method.
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TL;DR: In this article, the authors present finite-difference schemes for the evaluation of first-order, second-order and higher-order derivatives yield improved representation of a range of scales and may be used on nonuniform meshes.
5,832 citations
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01 Jan 1987TL;DR: 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.
4,632 citations
TL;DR: Improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods, and a new family of stiffly stable schemes is employed in mixed explicit/implicit time-intgration rules.
1,341 citations
TL;DR: In this article, the spectral element method is used for the calculation of synthetic seismograms in 3D earth models using a weak formulation of the equations of motion, which are solved on a mesh of hexahedral elements.
Abstract: SUMMARY We present an introduction to the spectral element method, which provides an innovative numerical approach to the calculation of synthetic seismograms in 3-D earth models. The method combines the £exibility of a ¢nite element method with the accuracy of a spectral method. One uses a weak formulation of the equations of motion, which are solved on a mesh of hexahedral elements that is adapted to the free surface and to the main internal discontinuities of the model. The wave¢eld on the elements is discretized using high-degree Lagrange interpolants, and integration over an element is accomplished based upon the Gauss^Lobatto^Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix, which greatly simpli¢es the algorithm. We illustrate the great potential of the method by comparing it to a discrete wavenumber/re£ectivity method for layer-cake models. Both body and surface waves are accurately represented, and the method can handle point force as well as moment tensor sources. For a model with very steep surface topography we successfully benchmark the method against an approximate boundary technique. For a homogeneous medium with strong attenuation we obtain excellent agreement with the analytical solution for a point force.
1,184 citations
Cites methods from "A spectral element method for fluid..."
...The spectral element method discussed in this article has been used for more than 15 years in computational £uid dynamics (Patera 1984)....
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TL;DR: The spectral element method as discussed by the authors is a high-order variational method for the spatial approximation of elastic-wave equations, which can be used to simulate elastic wave propagation in realistic geological structures involving complieated free surface topography and material interfaces for two- and three-dimensional geometries.
Abstract: We present the spectral element method to simulate elastic-wave propagation in realistic geological structures involving complieated free-surface topography and material interfaces for two- and three-dimensional geometries. The spectral element method introduced here is a high-order variational method for the spatial approximation of elastic-wave equations. The mass matrix is diagonal by construction in this method, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy-momentum conserving scheme that can be put into a classical explicit-implicit predictor/multi-corrector format. Long-term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The associated Courant condition behaves as Δ tC < O ( nel−1/nd N −2), with nel the number of elements, nd the spatial dimension, and N the polynomial order. In practice, a spatial sampling of approximately 5 points per wavelength is found to be very accurate when working with a polynomial degree of N = 8. The accuracy of the method is shown by comparing the spectral element solution to analytical solutions of the classical two-dimensional (2D) problems of Lamb and Garvin. The flexibility of the method is then illustrated by studying more realistic 2D models involving realistic geometries and complex free-boundary conditions. Very accurate modeling of Rayleigh-wave propagation, surface diffraction, and Rayleigh-to-body-wave mode conversion associated with the free-surface curvature are obtained at low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves by three-dimensional (3D) surface topographies and the associated local effects on strong ground motion. Complex amplification patterns, both in space and time, are shown to occur even for a gentle hill topography. Extension to a heterogeneous hill structure is considered. The efficient implementation on parallel distributed memory architectures will allow to perform real-time visualization and interactive physical investigations of 3D amplification phenomena for seismic risk assessment.
1,183 citations
References
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01 Jan 1977
TL;DR: Spectral Methods Survey of Approximation Theory Review of Convergence Theory Algebraic Stability Spectral Methods Using Fourier Series Applications of algebraic stability analysis Constant Coefficient Hyperbolic Equations Time Differencing Efficient Implementation of Spectral Method as discussed by the authors.
Abstract: Spectral Methods Survey of Approximation Theory Review of Convergence Theory Algebraic Stability Spectral Methods Using Fourier Series Applications of Algebraic Stability Analysis Constant Coefficient Hyperbolic Equations Time Differencing Efficient Implementation of Spectral Methods Numerical Results for Hyperbolic Problems Advection-Diffusion Equation Models of Incompressible Fluid Dynamics Miscellaneous Applications of Spectral Methods Survey of Spectral Methods and Applications Properties of Chebyshev and Legendre Polynomial Expansions.
3,386 citations
TL;DR: In this paper, the velocity distribution and reattachment length of a single backward-facing step mounted in a two-dimensional channel were measured using laser-Doppler measurements.
Abstract: Laser-Doppler measurements of velocity distribution and reattachment length are reported downstream of a single backward-facing step mounted in a two-dimensional channel. Results are presented for laminar, transitional and turbulent flow of air in a Reynolds-number range of 70 < Re < 8000. The experimental results show that the various flow regimes are characterized by typical variations of the separation length with Reynolds number. The reported laser-Doppler measurements do not only yield the expected primary zone of recirculating flow attached to the backward-facing step but also show additional regions of flow separation downstream of the step and on both sides of the channel test section. These additional separation regions have not been previously reported in the literature.Although the high aspect ratio of the test section (1:36) ensured that the oncoming flow was fully developed and two-dimensional, the experiments showed that the flow downstream of the step only remained two-dimensional at low and high Reynolds numbers.The present study also included numerical predictions of backward-facing step flow. The two-dimensional steady differential equations for conservation of mass and momentum were solved. Results are reported and are compared with experiments for those Reynolds numbers for which the flow maintained its two-dimensionality in the experiments. Under these circumstances, good agreement between experimental and numerical results is obtained.
1,637 citations
TL;DR: In this article, the main flow and the captive eddy between it and the walls are analyzed, and it is concluded that the main role of the eddy is to shape the flow with a rather small energy exchange.
Abstract: Results of calculations and experiments on the flow of a viscous liquid through an axisymmetric conduit expansion are reported. The streamlines and vorticity contours are presented as functions of the Reynolds number of the flow. The dynamic interaction between the main flow and the captive eddy between it and the walls is analysed, and it is concluded that, for laminar flow, the main role of the eddy is that of shaping the flow with a rather small energy exchange.
272 citations
Book•
01 Jun 1983TL;DR: In this paper, a finite element solution methodology is derived, developed, and applied directly to the differential equation systems governing classes of problems in fluid mechanics, including turbulence closure and the solution of turbulent flows.
Abstract: Finite element analysis as applied to the broad spectrum of computational fluid mechanics is analyzed. The finite element solution methodology is derived, developed, and applied directly to the differential equation systems governing classes of problems in fluid mechanics. The heat conduction equation is used to reveal the essence and elegance of finite element theory, including higher order accuracy and convergence. The algorithm is extended to the pervasive nonlinearity of the Navier-Stokes equations. A specific fluid mechanics problem class is analyzed with an even mix of theory and applications, including turbulence closure and the solution of turbulent flows.
265 citations
TL;DR: In this article, the authors examined the performance of three steady-state finite difference formulations, namely: (i) the hybrid central/upwind differencing scheme, 2.
133 citations